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  • Prof. Arthur Mattuck

Departments

  • Mathematics

As Taught In

  • Differential Equations
  • Mathematical Analysis
  • Mathematical Logic

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Introduction to analysis, course description.

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space.

MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology. Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication.

Graph of the Intermediate Value Theorem.

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Supplement to Analysis

Definitions and descriptions of analysis.

The older a word, the deeper it reaches. (Wittgenstein NB , 40) { §6.5 }

This supplement collects together various definitions and descriptions of analysis that have been offered in the history of philosophy (including all the classic ones), to indicate the range of different conceptions and the issues that arise. (There are also some remarks on related topics such as analyticity, definition, and methodology more generally.) In most cases, abbreviated references are given; full details can be found in the Annotated Bibliography on Analysis, in the section mentioned in curly brackets after the relevant definition or description. Where there is more than one passage quoted from a particular author, passages are numbered in chronological order of composition (as far as that can be determined).

  • Cambridge Dictionary of Philosophy , 1999, ed. Robert Audi

Concise Oxford Dictionary , 1976, ed. J. B. Sykes

  • Dictionary of Philosophy and Psychology , 1925, ed. James Mark Baldwin

A Kant Dictionary , 1995, by Howard Caygill

Oxford dictionary of philosophy , 1996, by simon blackburn, philosophielexikon , 1997, ed. a. hügli and p. lübcke, routledge encyclopedia of philosophy , 1998, entry under ‘analytical philosophy’ by thomas baldwin, routledge encyclopedia of philosophy , 1998, entry under ‘conceptual analysis’ by robert hanna, alexander of aphrodisias, arnauld, antoine and nicole, pierre, ayer, a. j., bentham, jeremy, bergson, henri, bos, henk j. m., bradley, f. h., brandom, robert b., carnap, rudolf, cassirer, ernst, cohen, l. jonathan, collingwood, r. g., davidson, donald, de chardin, teilhard, derrida, jacques, descartes, rené, frege, gottlob, geertz, clifford, hegel, georg w.f., heidegger, martin, hobbes, thomas, hodges, wilfrid, holton, gerald, husserl, edmund, kant, immanuel, lakatos, imre, leibniz, gottfried wilhelm, lichtenberg, georg christoph, locke, john, lodge, david, mendelssohn, moses, moore, g. e., newton, isaac, nietzsche, friedrich, poincaré, jules henri, polya, george, quine, w.v.o., rorty, richard, rosen, stanley, russell, bertrand, ryle, gilbert, schiller, friedrich, sellars, wilfrid, soames, scott, stebbing, l. susan.

  • Strawson, F. Peter

Urmson, J. O.

Whitehead, alfred north, wilson, john cook, wittgenstein, ludwig, 1. definitions of analysis, cambridge dictionary of philosophy , 2nd ed., 1999, ed. robert audi.

the process of breaking up a concept, proposition, linguistic complex, or fact into its simple or ultimate constituents. { §1.1 }
1. Resolution into simpler elements by analysing (opp. synthesis ); statement of result of this; … 2. (Math.) Use of algebra and calculus in problem-solving. { §1.1 }

Dictionary of Philosophy and Psychology , 1925, ed. James Mark Baldwin, Vol. I

The isolation of what is more elementary from what is more complex by whatever method. { §1.1 }
Kant combines two senses of analysis in his work, one derived from Greek geometry, the other from modern physics and chemistry. Both remain close to the original Greek sense of analysis as a ‘loosening up’ or ‘releasing’, but each proceed in different ways. The former proceeds ‘lemmatically’ by assuming a proposition to be true and searching for another known truth from which the proposition may be deduced. The latter proceeds by resolving complex wholes into their elements. { §4.5 }
The process of breaking a concept down into more simple parts, so that its logical structure is displayed. { §1.1 }
Auflösung, Zerlegung in Bestandteile, im Gegensatz zu Synthese. { §1.1 }
Philosophical analysis is a method of inquiry in which one seeks to assess complex systems of thought by ‘analysing’ them into simpler elements whose relationships are thereby brought into focus. { §1.1 }
The theory of conceptual analysis holds that concepts – general meanings of linguistic predicates – are the fundamental objects of philosophical inquiry, and that insights into conceptual contents are expressed in necessary ‘conceptual truths’ (analytic propositions). { §1.1 }

Annotated Bibliography, §1.1

2. Descriptions of Analysis

And he [Aristotle] called them Analytics because the resolution of every compound into those things out of which the synthesis [is made] is called analysis . For analysis is the converse of synthesis. Synthesis is the road from the principles to those things that derive from the principles, and analysis is the return from the end to the principles. For geometers are said to analyze when, beginning from the conclusion they go up to the principles and the problem, following the order of those things which were assumed for the demonstration of the conclusion {1}. But he also uses analysis who reduces composite bodies into simple bodies {2}, and he analyzes who divides the word into the parts of the word {3}; also he who divides the parts of the word into the syllables {4}; and he who divides these into their components {5}. And they are severally said to analyse who reduce compound syllogisms into simple ones {6}, and simple ones into the premisses out of which they get their being {7}. And further, resolving imperfect syllogisms into perfect ones is called analyzing {8}. And they call analysis the reducing of the given syllogism into the proper schemata {9}. And it is especially in this meaning of analysis that these are entitled Analytics , for he describes for us a method at the end of the first book with which we shall be able to do this. ( Commentary on Aristotle’s Prior Analytics , §1.2.1 (7, lines 11-33); tr. in Gilbert 1960, 32; the square brackets are in the original translation, the curly brackets have been added here to highlight the nine senses that Alexander distinguishes) { §2.4 , §3.2 }

it is not the same thing to take an argument in one’s hand and then to see and solve its faults, as it is to be able to meet it quickly while being subjected to questions; for what we know, we often do not know in a different context. Moreover, just as in other things speed or slowness is enhanced by training, so it is with arguments too, so that supposing we are unpractised, even though a point is clear to us, we are often too late for the right moment. Sometimes too it happens as with diagrams; for there we can sometimes analyse the figure, but not construct it again: so too in refutations, though we know on what the connexion of the argument depends, we still are at a loss to split the argument apart. ( SR , 16, 175a20-30) { §2.4 }

We must next explain how to reduce syllogisms to the figures previously described; this part of our inquiry still remains. For if we examine the means by which syllogisms are produced, and possess the ability to discover them, and can also analyse [ analuoimen ] the syllogisms when constructed into the figures previously described, our original undertaking will be completed. (( PrA , I, 32, 46b40-47a6; Tredennick tr. slightly modified) { §2.4 }

Thus it is evident (1) that the types of syllogism which cannot be analysed in these figures [viz., second figure syllogisms into the third figure, and third figure syllogisms into the second figure] are the same as those which we saw could not be analysed into the first figure; and (2) that when syllogisms are reduced to the first figure these alone are established per impossibile .

It is evident, then, from the foregoing account [taken as including the discussion prior to chapter 45] how syllogisms should be reduced; and also that the figures can be analysed into one another. ( PrA , I, 45, 51a40-b5; Tredennick tr., substituting ‘analysed’ for ‘resolved’) { §2.4 }

If it were impossible to prove truth from falsehood, it would be easy to make analyses [ analuein ]; for then the propositions would convert from necessity. Let A be something that is the case; and if A is the case, then these things are the case (things which I know to be the case—call them B ). From the latter, then, I shall prove that the former is the case. (In mathematics conversion is more common because mathematicians assume nothing incidental—and in this too they differ from those who argue dialectically—but only definitions.) ( PoA , I, 12, 78a6-13) { §2.4 }

We deliberate not about ends but about means. For a doctor does not deliberate whether he shall heal, nor an orator whether he shall convince, nor a statesman whether he shall produce law and order, nor does any one else deliberate about his end. Having set the end, they consider how and by what means it is to be attained; and if it seems to be produced by several means they consider by which it is most easily and best produced, while if it is achieved by one only they consider how it will be achieved by this and by what means this will be achieved, till they come to the first cause, which in the order of discovery is last. For the person who deliberates seems to inquire and analyse in the way described as though he were analysing a geometrical construction (not all inquiry appears to be deliberation—for instance mathematical inquiries—but all deliberation is inquiry), and what is last in the order of analysis seems to be first in the order of becoming. And if we come on an impossibility, we give up the search, e.g. if we need money and this cannot be got; but if a thing appears possible we try to do it. ( NE , III, 3, 1112b8-27) { §2.4 }

The art of arranging a series of thoughts properly, either for discovering the truth when we do not know it, or for proving to others what we already know, can generally be called method.

Hence there are two kinds of method, one for discovering the truth, which is known as analysis , or the method of resolution , and which can also be called the method of discovery . The other is for making the truth understood by others once it is found. This is known as synthesis , or the method of composition , and can also be called the method of instruction .

Analysis does not usually deal with the entire body of a science, but is used only for resolving some issue. ( LAT , 233-4) { §4.1 }

Now analysis consists primarily in paying attention to what is known in the issue we want to resolve. The entire art is to derive from this examination many truths that can lead us to the knowledge we are seeking.

Suppose we wondered whether the human soul is immortal, and to investigate it we set out to consider the nature of the soul. First we would notice that it is distinctive of the soul to think, and that it could doubt everything without being able to doubt whether it is thinking, since doubting is itself a thought. Next we would ask what thinking is. Since we would see nothing contained in the idea of thought that is contained in the idea of the extended substance called body, and since we could even deny of thought everything belonging to body - such as having length, width, and depth, having different parts, having a certain shape, being divisible, etc. - without thereby destroying the idea we have of thought, from this we would conclude that thought is not at all a mode of extended substance, because it is the nature of a mode not to be able to be conceived while the thing of which it is a mode is denied. From this we infer, in addition, that since thought is not a mode of extended substance, it must be the attribute of another substance. Hence thinking substance and extended substance are two really distinct substances. It follows from this that the destruction of one in no way brings about the destruction of the other, since even extended substance is not properly speaking destroyed, but all that happens in what we call destruction is nothing more than the change or dissolution of several parts of matter which exist forever in nature. Likewise it is quite easy to judge that in breaking all the gears of a clock no substance is destroyed, although we say that the clock is destroyed. This shows that since the soul is in no way divisible or composed of parts, it cannot perish, and consequently is immortal.

This is what we call analysis or resolution . We should notice, first, that in this method - as in the one called composition - we should practice proceeding from what is better known to what is less known. For there is no true method which could dispense with this rule.

Second, it nevertheless differs from the method of composition in that these known truths are taken from a particular examination of the thing we are investigating, and not from more general things as is done in the method of instruction. Thus in the example we presented, we did not begin by establishing these general maxims: that no substance perishes, properly speaking; that what is called destruction is only a dissolution of parts; that therefore what has no parts cannot be destroyed, etc. Instead we rose by stages to these general notions.

Third, in analysis we introduce clear and evident maxims only to the extent that we need them, whereas in the other method we establish them first, as we will explain below.

Fourth and finally, these two methods differ only as the route one takes in climbing a mountain from a valley differs from the route taken in descending from the mountain into the valley, or as the two ways differ that are used to prove that a person is descended from St. Louis. One way is to show that this person had a certain man for a father who was the son of a certain man, and that man was the son of another, and so on up to St. Louis. The other way is to begin with St. Louis and show that he had a certain child, and this child had others, thereby descending to the person in question. This example is all the more appropriate in this case, since it is certain that to trace an unknown genealogy, it is necessary to go from the son to the father, whereas to explain it after finding it, the most common method is to begin with the trunk to show the descendants. This is also what is usually done in the sciences where, after analysis is used to find some truth, the other method is employed to explain what has been found.

This is the way to understand the nature of analysis as used by geometers. Here is what it consists in. Suppose a question is presented to them, such as whether it is true or false that something is a theorem, or whether a problem is possible or impossible; they assume what is at issue and examine what follows from that assumption. If in this examination they arrive at some clear truth from which the assumption follows necessarily, they conclude that the assumption is true. Then starting over from the end point, they demonstrate it by the other method which is called composition . But if they fall into some absurdity or impossibility as a necessary consequence of their assumption, they conclude from this that the assumption is false and impossible.

This is what may be said in a general way about analysis, which consists more in judgment and mental skill than in particular rules. ( LAT , 236-8) { §4.1 }

It is advisable to stress the point that philosophy, as we understand it, is wholly independent of metaphysics, inasmuch as the analytic method is commonly supposed by its critics to have a metaphysical basis. Being misled by the associations of the word ‘analysis’, they assume that philosophical analysis is an activity of dissection; that it consists in ‘breaking up’ objects into their constituent parts, until the whole universe is ultimately exhibited as an aggregate of ‘bare particulars’, united by external relations. If this were really so, the most effective way of attacking the method would be to show that its basic presupposition was nonsensical. For to say that the universe was an aggregate of bare particulars would be as senseless as to say that it was Fire or Water or Experience. It is plain that no such possible observation would enable to veify such an assertion. But, so far as I know, this line of criticism is in fact never adopted. The critics content themselves with pointing out that few, if any, of the complex objects in the world are simply the sum of their parts. They have a structure, an organic unity, which distinguishes them, as genuine wholes, from mere aggregates. But the analyst, so it is said, is obliged by his atomistic metaphysics to regard an object consisting of parts a , b , c , and d , in a distinctive configuration as being simply a + b + c + d , and thus gives an entirely false account of its nature.

If we follow the Gestalt psychologists, who of all men talk most constantly about genuine wholes, in defining such a whole as one in which the properties of every part depend to some extent on its position in the whole, then we may accept it as an empirical fact that there exist genuine, or organic, wholes. And if the analytic method involved a denial of this fact, it would indeed be a faulty method. But, actually, the validity of the analytic method is not dependent on any empirical, much less any metaphysical, presupposition about the nature of things. For the philosopher, as an analyst, is not directly concerned with the physical properties of things. He is concerned only with the way in which we speak about them.

In other words, the propositions of philosophy are not factual, but linguistic in character – that is, they do not describe the behaviour of physical, or even mental, objects; they express definitions, or the formal consequences of definitions. Accordingly, we may say that philosophy is a department of logic. For we shall see that the characteristic mark of a purely logical inquiry is that it is concerned with the formal consequences of our definitions and not with questions of empirical fact.

It follows that philosophy does not in any way compete with science. The difference in type between philosophical and scientific propositions is such that they cannot conceivably contradict one another. And this makes it clear that the possibility of philosophical analysis is independent of any empirical assumptions. That it is independent of any metaphysical assumptions should be even more obvious still. For it is absurd to suppose that the provision of definitions, and the study of their formal consequences, involves the nonsensical assertion that the world is composed of bare particulars, or any other metaphysical dogma.

What has contributed as much as anything to the prevalent misunderstanding of the nature of philosophical analysis is the fact that propositions and questions which are really linguistic are often expressed in such a way that they appear to be factual. A striking instance of this is provided by the proposition that a material thing cannot be in two places at once. This looks like an empirical proposition, and is constantly invoked by those who desire to prove that it is possible for an empirical proposition to be logically certain. But a more critical inspection shows that it is not empirical at all, but linguistic. It simply records the fact that, as the result of certain verbal conventions, the proposition that two sense-contents occur in the same visual or tactual sense-field is incompatible with the proposition that they belong to the same material thing. And this is indeed a necessary fact. But it has not the least tendency to show that we have certain knowledge about the empirical properties of objects. For it is necessary only because we happen to use the relevant words in a particular way. There is no logical reason why we should not so alter our definitions that the sentence ‘A thing cannot be in two places at once’ comes to express a self-contradiction instead of a necessary truth. (1936, 75-7) { §6.7 }

From our assertion that philosophy provides definitions, it must not be inferred that it is the function of the philosopher to compile a dictionary, in the ordinary sense. For the definitions which philosophy is required to provide are of a different kind from those which we expect to find in dictionaries. In a dictionary we look mainly for what may be called explicit definitions; in philosophy, for definitions in use . ...

We define a symbol in use , not by saying that it is synonymous with some other symbol, but by showing how the sentences in which it significantly occurs can be translated into equivalent sentences, which contain neither the definiendum itself, nor any of its synonyms. A good illustration of this process is provided by Bertrand Russell’s so-called theory of descriptions, which is not a theory at all in the ordinary sense, but an indication of the way in which all phrases of the form ‘the so-and-so’ are to be defined. ( Ibid ., 80-1) { §6.7 }

[A serious mistake in my account in Language, Truth and Logic ] was my assumption that philosophical analysis consisted mainly in the provision of ‘definitions in use’. It is, indeed, true that what I describe as philosophical analysis is very largely a matter of exhibiting the inter-relationship of different types of propositions; but the cases in which this process actually yields a set of definitions are the exception rather than the rule. ...

... Thus, when Professor Moore suggests that to say that ‘existence is not a predicate’ may be a way of saying that ‘there is some very important difference between the way in which “exist” is used in such a sentence as “Tame tigers exist” and the way in which “growl” is used in “Tame tigers growl”’, he does not develop his point by giving rules for the translation of one set of sentences into another. What he does is to remark that whereas it makes good sense to say ‘All tame tigers growl’ or ‘Most tame tigers growl’ it would be nonsense to say ‘All tame tigers exist’ or ‘Most tame tigers exist’. Now this may seem a rather trivial point for him to make, but in fact it is philosophically illuminating. For it is precisely the assumption that existence is a predicate that gives plausibility to ‘the ontological argument’; and the ontological argument is supposed to demonstrate the existence of a God. Consequently Moore by pointing out a peculiarity in the use of the word ‘exist’ helps to protect us from a serious fallacy; so that his procedure, though different from that which Russell follows in his theory of descriptions, tends to achieve the same philosophical end. (1946, 31-3) { §6.7 }

By the word paraphrasis may be designated that sort of exposition which may be afforded by transmuting into a proposition, having for its subject some real entity, a proposition which has not for its subject any other than a fictitious entity. ( EL , 246) { §5.6 }

By intuition is meant the kind of intellectual sympathy by which one places oneself within an object in order to coincide with what is unique in it and consequently inexpressible. Analysis, on the contrary, is the operation which reduces the object to elements already known, that is, to elements common both to it and other objects. To analyse, therefore, is to express a thing as a function of something other than itself. All analysis is thus a translation, a development into symbols, a representation taken from successive points of view from which we note as many resemblances as possible between the new object which we are studying and others which we believe we know already. In its eternally unsatisfied desire to embrace the object around which it is compelled to turn, analysis multiplies without end the number of its points of view in order to complete its always incomplete representation, and ceaselessly varies its symbols that it may perfect the always imperfect translation. It goes on, therefore, to infinity. But intuition, if intuition is possible, is a simple act. (1903, 6-7) { §5.1 }

[Analysis] operates always on the immobile, whilst intuition places itself in mobility, or, what comes to the same thing, in duration. There lies the very distinct line of demarcation between intuition and analysis. The real, the experienced and the concrete are recognised by the fact that they are variability itself, the element by the fact that it is invariable. And the element is invariable by definition, being a diagram, a simplified reconstruction, often a mere symbol, in any case a motionless view of the moving reality. (1903, 40-1) { §5.1 }

Modern science is neither one nor simple. It rests, I freely admit, on ideas which in the end we find clear; but these ideas have gradually become clear through the use made of them; they owe most of their clearness to the light which the facts, and the applications to which they led, have by reflection shed on them - the clearness of a concept being scarcely anything more at bottom than the certainty, at last obtained, of manipulating the concept profitably. At its origin, more than one of these concepts must have appeared obscure, not easily reconcilable with the concepts already admitted into science, and indeed very near the borderline of absurdity. This means that science does not proceed by an orderly dovetailing together of concepts predestined to fit each other exactly. True and fruitful ideas are so many close contacts with currents of reality, which do not necessarily converge on the same point. However the concepts in which they lodge themselves manage somehow, by rubbing off each other's corners, to settle down well enough together. (1903, 74) { §5.1 }

It may help to be reminded that many philosophers who might allow themselves to be described as “analysts” have been strongly influenced by the work of Russell, Moore, and Wittgenstein. For while all three have been engaged in “clarification of meaning” they have done so in different and distinctive ways; and the resulting divergences in conceptions of philosophical method have not yet been reconciled. This makes it hard to give any simple account of what is meant today by “philosophical analysis”. (1950a, 2) { §6.1 }

A man who had to describe “philosophical analysis” might resort to talking about a climate of opinion. The weather, he might say, is congenial to empiricists, naturalists, agnostics; the well acclimatized have admired the two Principia’s and the Tractatus and have read a hundred pages of Hume for one of Kant. Here rhetoric is viewed with suspicion and enthusiasm barely tolerated; this is a land of “prose writers, hoping to be understood” [J.M. Keynes, A Treatise on Probability , 1921, preface].

... If a formula or a slogan is wanted, it is easy enough to say that these writers (like Russell, Moore, and Wittgenstein before them) are engaged in clarification of meaning . ... And if those who are best at the work of clarification might feel embarrassed to provide a satisfactory analysis of “analysis”, that is perhaps no cause for apology or alarm. For it is a mark of life to resist arbitrary confinement, and “philosohical analysis” is still much alive. (1950a, 12-13) { §6.1 }

Analysis comprises mathematical methods for finding the solutions (in geometry: the constructions) of problems or the proofs of theorems, doing so by introducing unknowns. (2001, 129) { §4.2 }

It is a very common and most ruinous superstition to suppose that analysis is no alteration, and that, whenever we distinguish, we have at once to do with divisible existence. It is an immense assumption to conclude, when a fact comes to us as a whole, that some parts of it may exist without any sort of regard for the rest. Such naive assurance of the outward reality of all mental distinctions, such touching confidence in the crudest identity of thought and existence, is worthy of the school which so loudly appeals to the name of Experience. ... If it is true in any sense (and I will not deny it) that thought in the end is the measure of things, yet at least this is false, that the divisions we make within a whole all answer to elements whose existence does not depend on the rest. It is wholly unjustifiable to take up a complex, to do any work we please upon it by analysis, and then simply predicate as an adjective of the given these results of our abstraction. These products were never there as such, and in saying, as we do, that as such they are there, we falsify the fact. You can not always apply in actual experience that coarse notion of the whole as the sum of its parts into which the school of ‘experience’ so delights to torture phenomena. If it is wrong in physiology to predicate the results, that are reached by dissection, simply and as such of the living body, it is here infinitely more wrong. The whole that is given to us is a continuous mass of perception and feeling; and to say of this whole, that any one element would be what it is there, when apart from the rest, is a very grave assertion. We might have supposed it not quite self-evident, and that it was possible to deny it without open absurdity. ( PL , §64/ WLM , 77-8) { §5.6 }

judgement is the differentiation of a complex whole, and hence always is analysis and synthesis in one. ( AR , 149/ WLM , 158) { §5.6 }

At any moment my actual experience, however relational its contents, is in the end non-relational. No analysis into relations and terms can ever exhaust its nature or fail in the end to belie its essence. What analysis leaves for ever outstanding is no mere residue, but is a vital condition of the analysis itself. Everything which is got out into the form of an object implies still the felt background against which the object comes, and, further, the whole experience of both feeling and object is a non-relational immediate felt unity. The entire relational consciousness, in short, is experienced as falling within a direct awareness. This direct awareness is itself non-relational. It escapes from all attempts to exhibit it by analysis as one or more elements in a relational scheme, or as that scheme itself, or as a relation or relations, or as the sum or collection of any of these abstractions. And immediate experience not only escapes, but it serves as the basis on which the analysis is made. Itself is the vital element within which every analysis still moves, while, and so far as, and however much, that analysis transcends immediacy. ( ETR , 176/ WLM , 280-1) { §5.6 }

I would rather now lay more stress on the logical vice of all Analysis and Abstraction – so far as that means taking any feature in the Whole of Things as ultimately real except in its union with the Whole. ( Collected Works of F.H. Bradley: Selected Correspondence 1905-1924 , Bristol, Thoemmes Press, 1999, 275)

Analysis and synthesis I take in the end to be two aspects of one principle … Every analysis proceeds from and on the basis of a unity ... The point before us is the question as to how, without separation in its existence, we can discriminate ideally in analysis. ( ETR , 300)

Socratic method is a way of bringing our practices under rational control by expressing them explicitly in a form in which they can be confronted with objections and alternatives, a form in which they can be exhibited as the conclusions of inferences seeking to justify them on the basis of premises advanced as reasons, and as premises in further inferences exploring the consequences of accepting them. (2000, 56) { §6.9 }

I think of analytic philosophy as having at its center a concern with semantic relations between what I will call ‘vocabularies’. … Its characteristic form of question is whether and in what way one can make sense of the meanings expressed by one kind of locution interms of the meanings expressed by another kind of locution. So, for instance, two early paradigmatic projects were to show that everything expressible in the vocabulary of number-theory, and again, everything expressible using definite descriptions, is expressible already in the vocabulary of first-order quantificational logic with identity.

The nature of the key kind of semantic relation between vocabularies has been variously characterized during the history of analytic philosophy: as analysis, definition, paraphrase, translation, reduction of different sorts, truth-making, and various kinds of supervenience—to name just a few contenders. In each case, however, it is characteristic of classical analytic philosophy that logical vocabulary is accorded a privileged role in specifying these semantic relations. It has always been taken at least to be licit to appeal to logical vocabulary in elaborating the relation between analysandum and analysans —target vocabulary and base vocabulary—and, according to stronger versions of this thesis, that may be the only vocabulary it is licit to employ in that capacity. I will refer to this aspect of the analytic project as its commitment to ‘ semantic logicism ’. (2006, Lecture One, §1) { §6.9 }

What I want to call the “classical project of analysis”, then, aims to exhibit the meanings expressed by various target vocabularies as intelligible by means of the logical elaboration of the meanings expressed by base vocabularies thought to be privileged in some important respects—epistemological, ontological, or semantic—relative to those others. This enterprise is visible in its purest form in what I have called the “core programs” of empiricism and naturalism, in their various forms. In my view the most significant conceptual development in this tradition—the biggest thing that ever happened to it—is the pragmatist challenge to it that was mounted during the middle years of the twentieth century. Generically, this movement of thought amounts to a displacement from the center of philosophical attention of the notion of meaning in favor of that of use : in suitably broad senses of those terms, replacing concern with semantics by concern with pragmatics . ( Ibid ., Lecture One, §2) { §6.9 }

the analysis or, more precisely, quasi-analysis of an entity that is essentially an indivisible unit into several quasi-constituents means placing the entity in several kinship contexts on the basis of a kinship relation, where the unit remains undivided. (1928a, §71; English tr. by Rolf A. George slightly altered) { §6.7 }

The logical analysis of a particular expression consists in the setting-up of a linguistic system and the placing of that expression in this system. (1936, 143) { §6.7 }

That part of the work of philosophers which may be held to be scientific in its nature—excluding the empirical questions which can be referred to empirical science—consists of logical analysis. The aim of logical syntax is to provide a system of concepts, a language, by the help of which the results of logical analysis will be exactly formulable. Philosophy is to be replaced by the logic of science —that is to say, by the logical analysis of the concepts and sentences of the sciences, for the logic of science is nothing other than the logical syntax of the language of science . (1937, xiii) { §6.7 }

The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction. We call this the task of explicating, or of giving an explication for, the earlier concept … (1947, 8-9) { §6.7 }

By the procedure of explication we mean the transformation of an inexact, prescientific concept, the explicandum , into a new exact concept, the explicatum . Although the explicandum cannot be given in exact terms, it should be made as clear as possible by informal explanations and examples. ...

The term ‘explicatum’ has been suggested by the following two usages. Kant calls a judgement explicative if the predicate is obtained by analysis of the subject. Husserl, in speaking about the synthesis of identification between a confused, nonarticulated sense and a subsequently intended distinct, articulated sense, calls the latter the ‘Explikat’ of the former. (For both uses see Dictionary of philosophy [1942], ed. D. Runes, p. 105). What I mean by ‘explicandum’ and ‘explicatum’ is to some extent similar to what C.H. Langford calls ‘analysandum’ and ‘analysans’: “the analysis then states an appropriate relation of equivalence between the analysandum and the analysans” [Langford 1942, 323 { §6.4 }]; he says that the motive of an analysis “is usually that of supplanting a relatively vague idea by a more precise one” ( ibid ., p. 329).

(Perhaps the form ‘explicans’ might be considered instead of ‘explicatum’; however, I think that the analogy with the terms ‘definiendum’ and ‘definiens’ would not be useful because, if the explication consists in giving an explicit definition, then both the definiens and the definiendum in this definition express the explicatum, while the explicandum does not occur.) The procedure of explication is here understood in a wider sense than the procedures of analysis and clarification which Kant, Husserl, and Langford have in mind. The explicatum (in my sense) is in many cases the result of analysis of the explicandum (and this has motivated my choice of the terms); in other cases, however, it deviates deliberately from the explicandum but still takes its place in some way; this will become clear by the subsequent examples. (1950, 3) { §6.7 }

[T]he sense of all objective judgments reduces to a final original relation, which can be expressed in different formulations as the relation of “form” to “content”, as the relation of “universal” to “particular”, as the relation of “validity [ Geltung ]” to “being [ Sein ]”. Whatever designation one may finally choose here, what is alone decisive is that the basic relation itself is to be retained as a strictly unitary relation, which can only be designated through the two opposed moments that enter into it – but never constructed out of them, as if they were independent constituents present in themselves. The original relation is not to be defined in such a way that the “universal” somehow “subsists” next to or above the “particular” – the form somehow separate from the content – so that the two are then melded with one another by means of some or another fundamental synthesis of knowledge. Rather, the unity of mutual determination constitutes the absolutely first datum, behind which one can go back no further, and which can only be analyzed via the duality of two “viewpoints” in an artificially isolating process of abstraction. It is the basic flaw of all metaphysical epistemologies that they always attempt to reinterpret this duality of “moments” as a duality of “elements”. (1913, 13-14; cited and tr. by Friedman 2000, 34) { §5.4 }

conceptual analysis typically relates one kind of reason for using a certain word to another. (1986, 51) { §6.9 }

When philosophical analysis proceeds from intuitively sanctioned premisses to a reasoned conclusion, it may be described as moving from analysandum to analysans. It seeks to ensure that any muddles or inconsistencies in our unreasoned inclinations and passive prejudices are replaced by an explicitly formulated, consciously co-ordinated, adequately reasoned, and freely adopted system of acceptable principles. (1986, 96) { §6.9 }

Socrates was essentially the inventor of a method. ... His revolt against the study of nature was essentially a revolt against observation in favour of thought; and whereas mathematical method, as an example of thought, had already been discovered by his predecessors, his own discovery was that a similar method, for which he invented an appropriate technique, could be applied to ethical questions. This technique, as he himself recognized, depended on a principle which is of great importance to any theory of philosophical method: the principle that in a philosophical inquiry what we are trying to do is not to discover something of which until now we have been ignorant, but to know better something which in some sense we knew already; not to know it better in the sense of coming to know more about it, but to know it better in the sense of coming to know it in a different and better way—actually instead of potentially, or explicitly instead of implicitly, or in whatever terms the theory of knowledge chooses to express the difference: the difference itself has been a familiar fact ever since Socrates pointed it out. (1933, 10-11) { §5.6 }

[The] work of disentangling and arranging questions, which ... I [call] analysis, may be alternatively described as the work of detecting presuppositions. ... The analysis which detects absolute presuppositions I call metaphysical analysis; but as regards procedure and the qualifications necessary to carry it out there is no difference whatever between metaphysical analysis and analysis pure and simple ... (1940, 39-40) { §5.6 }

It is only by analysis that any one can ever come to know either that he is making any absolute presuppositions at all or what absolute presuppositions he is making.

Such analysis may in certain cases proceed in the following manner. If the inquirer can find a person to experiment upon who is well trained in a certain type of scientific work, intelligent and earnest in his devotion to it, and unaccustomed to metaphysics, let him probe into various presuppositions that his ‘subject’ has been taught to make in the course of his scientific education, and invite him to justify each or alternatively to abandon it. If the ‘inquirer’ is skilful and the ‘subject’ the right kind of man, these invitations will be contemplated with equanimity, and even with interest, so long as relative presuppositions are concerned. But when an absolute presupposition is touched, the invitation wil be rejected, even with a certain degree of violence.

The rejection is a symptom that the ‘subject’, co-operating with the work of analysis, has come to see that the presupposition he is being asked to justify or abandon is an absolute presupposition; and the violence with which it is expressed is a symptom that he feels the importance of this absolute presupposition for the kind of work to which he is devoted. This is what ... I called being ‘ticklish in one’s absolute presuppositions’; and the reader will see that this ticklishness is a sign of intellectual health combined with a low degree of analytical skill. A man who is ticklish in that way is a man who knows, ‘instinctively’ as they say, that absolute presuppositions do not need justification. ( Ibid. , 43-4) { §5.6 }

metaphysical analysis, the discovery that certain presuppositions actually made are absolute presuppositions, is an integral part or an indispensable condition, you can put it whichever way you like, of all scientific work.( Ibid. , 84) { §5.6 }

In philosophy we are used to definitions, analyses, reductions. Typically these are intended to carry us from concepts better understood, or clear, or more basic epistemologically or ontologically, to others we want to understand. The method I have suggested fits none of these categories. I have proposed a looser relation between concepts to be illuminated and the relatively more basic. (‘Radical Interpretation’, 1972, Inquiries into Truth and Interpretation , Oxford: Oxford University Press, 2001, 137)

Unlike the primitives who gave a face to every moving thing, or the early Greeks who defined all the aspects and forces of nature, modern man is obsessed by the need to depersonalise (or impersonalise) all that he most admires. There are two reasons for this tendency. The first is analysis , that marvellous instrument of scientific research to which we owe all our advances but which, breaking down synthesis after synthesis, allows one soul after another to escape, leaving us confronted with a pile of dismantled machinery, and evanescent particles. The second reason lies in the discovery of the sidereal world, so vast that it seems to do away with all proportion between our own being and the dimensions of the cosmos around us. ( The Phenomenon of Man , 1955, 282; tr. Bernard Wall, Fontana, 1965; tr. first publ. 1959)

Up until now the idea of philosophy remained defined in a formal way as an idea of an infinite task theoria . Could a history of this infinite theoretical life, which merges itself in its efforts and failures with a simple realization of the self , take on the value of a genetic description? Will the history of the “transcendental motive” through all the stages of European philosophy, enlighten us at last on the genesis of transcendental subjectivity? But such a history presupposes the possibility of such a going backward, the possibility of finding again the originary sense of the former presents as such. It implies the possibility of a transcendental “regression” ( Ruckfrage ) through a history that is intelligible and transparent to consciousness, a history whose sedimentations can be unmade and remade without alteration. ( The Problem of Genesis in Husserl's Philosophy , The University of Chicago Press, 2003, 161; tr. Marian Hobson)

[discussing his ‘Rule Four’: “ We need a method if we are to investigate the truth of things ”] … the human mind has within it a sort of spark of the divine, in which the first seeds of useful ways of thinking are sown, seeds which, however neglected and stifled by studies which impede them, often bear fruit of their own accord. This is our experience in the simplest of sciences, arithmetic and geometry: we are well aware that the geometers of antiquity employed a sort of analysis which they went on to apply to the solution of every problem, though they begrudged revealing it to posterity. At the present time a sort of arithmetic called ‘algebra’ is flourishing, and this is achieving for numbers what the ancients did for figures. ( Rules for the Direction of the Mind , in PW , I, 16-17) { §4.2 }

As for the method of demonstration, this divides into two varieties: the first proceeds by analysis and the second by synthesis.

Analysis shows the true way by means of which the thing in question was discovered methodically and as it were a priori , so that if the reader is willing to follow it and give sufficient attention to all points, he will make the thing his own and understand it just as perfectly as if he had discovered it for himself. But this method contains nothing to compel belief in an argumentative or inattentive reader; for if he fails to attend even to the smallest point, he will not see the necessity of the conclusion. Moreover there are many truths which - although it is vital to be aware of them - this method often scarcely mentions, since they are transparently clear to anyone who gives them his attention.

Synthesis, by contrast, employs a directly opposite method where the search is, as it were, a posteriori (though the proof itself is often more a priori than it is in the analytic method). It demonstrates the conclusion clearly and employs a long series of definitions, postulates, axioms, theorems and problems, so that if anyone denies one of the conclusions it can be shown at once that it is contained in what has gone before, and hence the reader, however argumentative or stubborn he may be, is compelled to give his assent. However, this method is not as satisfying as the method of analysis, nor does it engage the minds of those who are eager to learn, since it does not show how the thing in question was discovered.

It was synthesis alone that the ancient geometers usually employed in their writings. But in my view this was not because they were utterly ignorant of analysis, but because they had such a high regard for it that they kept it to themselves like a sacred mystery.

Now it is analysis which is the best and truest method of instruction, and it was this method alone which I employed in my Meditations . As for synthesis, which is undoubtedly what you are asking me to use here, it is a method which it may be very suitable to deploy in geometry as a follow-up to analysis, but it cannot so conveniently be applied to these metaphysical subjects.

The difference is that the primary notions which are presupposed for the demonstration of geometrical truths are readily accepted by anyone, since they accord with the use of our senses. Hence there is no difficulty there, except in the proper deduction of the consequences, which can be done even by the less attentive, provided they remember what has gone before. Moreover, the breaking down of propositions to their smallest elements is specifically designed to enable them to be recited with ease so that the student recalls them whether he wants to or not.

In metaphysics by contrast there is nothing which causes so much effort as making our perception of the primary notions clear and distinct. Admittedly, they are by their nature as evident as, or even more evident than, the primary notions which the geometers study; but they conflict with many preconceived opinions derived from the senses which we have got into the habit of holding from our earliest years, and so only those who really concentrate and meditate and withdraw their minds from corporeal things, so far as is possible, will achieve perfect knowledge of them. Indeed, if they were put forward in isolation, they could easily be denied by those who like to contradict just for the sake of it. (‘Second Set of Replies’, in PW , II, 110-11) { §4.2 }

[interpolated into the text of the Elements ] What is analysis and what is synthesis. Analysis is the assumption of that which is sought as if it were admitted [and the arrival] by means of its consequences at something admitted to be true. Synthesis is an assumption of that which is admitted [and the arrival] by means of its consequences at something admitted to be true. ( E , Book XIII, Prop. 1; Vol. III, 442, where Heath comments on the interpolation) { §2.2 }

[In replying to the objections that Husserl had raised in his Philosophie der Arithmetik (1891) to Frege’s Grundlagen definitions] If words and combinations of words refer to [ bedeuten ] ideas, then for any two of them there are only two possibilities: either they designate the same idea or they designate different ideas. In the former case it is pointless to equate them by means of a definition: this is ‘an obvious circle’; in the latter case it is wrong. These are also the objections the author raises, one of them regularly. A definition is also incapable of analysing the sense, for the analysed sense just is not the original one. In using the word to be explained, I either think clearly everything I think when I use the defining expression: we then have the ‘obvious circle’; or the defining expression has a more richly articulated sense, in which case I do not think the same thing in using it as I do in using the word to be explained: the definition is then wrong. One would think that a definition was unobjectionable in the case where the word to be explained had as yet no sense at all, or where we were asked explicitly to regard its sense as non-existent so that it was first given a sense by the definition. But in the last case too, the author refutes the definition by reminding us of the difference between the ideas (p. 107). To evade all objections, one would accordingly have to create a new verbal root and form a word out of it. This reveals a split between psychological logicians and mathematicians. What matters to the former is the sense of the words, as well as the ideas which they fail to distinguish from the sense; whereas what matters to the latter is the thing itself: the Bedeutung of the words. The reproach that what is defined is not the concept but its extension actually affects all mathematical definitions. For the mathematician, it is no more right and no more wrong to define a conic as the line of intersection of a plane with the surface of a circular cone than to define it as a plane curve with an equation of the second degree in parallel coordinates. His choice of one or the other of these expressions or of some other one is guided solely by reasons of convenience and is made irrespective of the fact that the expressions have neither the same sense nor evoke the same ideas. I do not intend by this that a concept and its extension are one and the same, but that coincidence in extension is a necessary and sufficient criterion for the occurrence between concepts of the relation that corresponds to identity [ Gleichheit ] between objects. ( RH , 319-20/ FR , 225-6) { §6.2 }

We come to definitions . Definitions proper must be distinguished from elucidations [ Erläuterungen ]. In the first stages of any discipline we cannot avoid the use of ordinary words. But these words are, for the most part, not really appropriate for scientific purposes, because they are not precise enough and fluctuate in their use. Science needs technical terms that have precise and fixed Bedeutungen , and in order to come to an understanding about these Bedeutungen and exclude possible misunderstandings, we provide elucidations. Of course in so doing we have again to use ordinary words, and these may display defects similar to those which the elucidations are intended to remove. So it seems that we shall then have to provide further elucidations. Theoretically one will never really achieve one’s goal in this way. In practice, however, we do manage to come to an understanding about the Bedeutungen of words. Of course we have to be able to count on a meeting of minds, on others’ guessing what we have in mind. But all this precedes the construction of a system and does not belong within a system. In constructing a system it must be assumed that the words have precise Bedeutungen and that we know what they are. ( LM , 224/ FR , 313) { §6.2 }

We have ... to distinguish two quite different cases :

1. We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a ‘constructive definition’ [‘ aufbauende Definition ’], but we prefer to call it a ‘definition’ tout court .

2. We have a simple sign with a long-established use. We believe that we can give a logical analysis [ Zerlegung ] of its sense, obtaining a complex expression which in our opinion has the same sense. We can only allow something as a constituent of a complex expression if it has a sense we recognize. The sense of the complex expression must be yielded by the way in which it is put together. That it agrees with the sense of the long established simple sign is not a matter for arbitrary stipluation, but can only be recognized by an immediate insight. No doubt we speak of a definition in this case too. It might be called an ‘analytic definition’ [‘ zerlegende Definition ’] to distinguish it from the first case. But it is better to eschew the word ‘definition’ altogether in this case, because what we should here like to call a definition is really to be regarded as an axiom. In this second case there remains no room for an arbitrary stipulation, because the simple sign already has a sense. Only a sign which as yet has no sense can have a sense arbitrarily assigned to it. So we shall stick to our original way of speaking and call only a constructive definition a definition. According to that a definition is an arbitrary stipulation which confers a sense on a simple sign which previously had none. This sense has, of course, to be expressed by a complex sign whose sense results from the way it is put together.

Now we still have to consider the difficulty we come up against in giving a logical analysis when it is problematic whether this analysis is correct.

Let us assume that A is the long-established sign (expression) whose sense we have attempted to analyse logically by constructing a complex expression that gives the analysis. Since we are not certain whether the analysis is successful, we are not prepared to present the complex expression as one which can be replaced by the simple sign A . If it is our intention to put forward a definition proper, we are not entitled to choose the sign A , which already has a sense, but we must choose a fresh sign B , say, which has the sense of the complex expression only in virtue of the definition. The question now is whether A and B have the same sense. But we can bypass this question altogether if we are constructing a new system from the bottom up; in that case we shall make no further use of the sign A – we shall only use B . We have introduced the sign B to take the place of the complex expression in question by arbitrary fiat and in this way we have conferred a sense on it. This is a definition in the proper sense, namely a constructive definition.

If we have managed in this way to construct a system for mathematics without any need for the sign A , we can leave the matter there; there is no need at all to answer the question concerning the sense in which – whatever it may be – this sign had been used earlier. In this way we court no objections. However, it may be felt expedient to use sign A instead of sign B . But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition. We must therefore explain that the sense in which this sign was used before the new system was constructed is no longer of any concern to us, that its sense is to be understood purely from the constructive definition that we have given. In constructing the new system we can take no account, logically speaking, of anything in mathematics that existed prior to the new system. Everything has to be made anew from the ground up. Even anything that we may have accomplished by our analytical activities is to be regarded only as preparatory work which does not itself make any appearance in the new system itself.

Perhaps there still remains a certain unclarity. How is it possible, one may ask, that it should be doubtful whether a simple sign has the same sense as a complex expression if we know not only the sense of the simple sign, but can recognize the sense of the complex one from the way it is put together? The fact is that if we really do have a clear grasp of the sense of the simple sign, then it cannot be doubtful whether it agrees with the sense of the complex expression. If this is open to question although we can clearly recognize the sense of the complex expression from the way it is put together, then the reason must lie in the fact that we do not have a clear grasp of the sense of the simple sign, but that its outlines are confused as if we saw it through a mist. The effect of the logical analysis of which we spoke will then be precisely this – to articulate the sense clearly. Work of this kind is very useful; it does not, however, form part of the construction of the system, but must take place beforehand. Before the work of construction is begun, the building stones have to be carefully prepared so as to be usable; i.e. the words, signs, expressions, which are to be used, must have a clear sense, so far as a sense is not to be conferred on them in the system itself by means of a constructive definition.

We stick then to our original conception: a definition is an arbitrary stipulation by which a new sign is introduced to take the place of a complex expression whose sense we know from the way it is put together. A sign which hitherto had no sense acquires the sense of a complex expression by definition. ( LM , 227-9/ FR , 317-8) { §6.2 }

Analysis … is sorting out the structures of signification … and determining their social ground and import. ( The Interpretation of Cultures , New York: Basic Books, 1973, 9)

Cultural analysis is (or should be) guessing at meanings, assessing the guesses, and drawing explanatory conclusions from the better guesses, not discovering the Continent of Meaning and mapping out its bodiless landscape. ( Ibid ., 20)

The analysis of an idea, as it used to be carried out, was, in fact, nothing else than ridding it of the form in which it had become familiar. To break an idea up into its original elements is to return to its moments, which at least do not have the form of the given idea, but rather constitute the immediate property of the self. This analysis, to be sure, only arrives at thoughts which are themselves familiar, fixed, and inert determinations. But what is thus separated and non-actual is an essential moment; for it is only because the concrete does divide itself, and make itself into something non-actual, that it is self-moving. The activity of dissolution is the power and work of the Understanding , the most astonishing and mightiest of powers, or rather the absolute power. The circle that remains self-enclosed and, like substance, holds its moments together, is an immediate relationship, one therefore which has nothing astonishing about it. But that an accident as such, detached from what circumscribes it, what is bound and is actual only in its context with others, should attain an existence of its own and a separate freedom—this is the tremendous power of the negative; it is the energy of thought, of the pure ‘I’. Death, if that is what we want to call this non-actuality, is of all things the most dreadful, and to hold fast what is dead requires the greatest strength. Lacking strength, Beauty hates the Understanding for asking of her what it cannot do. But the life of Spirit is not the life that shrinks from death and keeps itself untouched by devastation, but rather the life that endures it and maintains itself in it. It wins its truth only when, in utter dismemberment, it finds itself. It is this power, not as something positive, which closes its eyes to the negative, as when we say of something that it is nothing or is false, and then, having done with it, turn away and pass on to something else; on the contrary, Spirit is this power only by looking the negative in the face, and tarrying with it. This tarrying with the negative is the magical power that converts it into being. This power is identical with what we earlier called the Subject, which by giving determinateness an existence in its own element supersedes abstract immediacy, i.e. the immediacy which barely is, and thus is authentic substance: that being or immediacy whose mediation is not outside of it but which is this mediation itself. ( PS , ‘Preface’, §32, 18-19)

[Summary of above passage offered by J.N. Findlay] The analysis of an idea is the removal of its familiarity, its reduction to elements that are the true possessions of the thinking self. In such reduction the idea itself changes and renders itself unreal. The force which effects analysis is that of the Understanding, the most remarkable and absolute of powers, the power of the thinking self and also of death. It is above all marvellous that this thinking self should be able to isolate, and to look at apart, what can only exist as an aspect or ‘moment’ in a living whole. Thinking Spirit can, however, only grasp such a whole by first tearing it into parts, each of which it must look at separately for a while, before putting them back in the whole. The thinking self must destroy an immediate, existent unity in order to arrive at a unity which includes mediation, and is in fact mediation itself. (‘Analysis of the Text’, §32, in PS , 499) { §5.2 }

What we are trying to bring to light here by means of phenomenological analysis in regard to the intentional structure of production is not contrived and fabricated but already present in the everyday, pre-philosophical productive behaviour of the Dasein. In producing, the Dasein lives in such an understanding of being without conceiving it or grasping it as such. (1927, §12, 114-15) { §5.8 }

every method by which we investigate the causes of things is either compositive, or resolutive, or partly compositive, partly resolutive. And the resolutive is usually called analytic, while the compositive is usually called synthetic. ( Logica , ‘On Method’, §1, 289) { §4.1 }

What philosophers seek to know. Philosophers seek scientific knowledge either simply or indefinitely, that is, they seek to knkow as much as they can when no definite question is proposed or the cause of some definite phenomenon or at least to discover something definite, such as what the cause of light is, or of heat, or gravity, of a figure which has been proposed, and similar things; or in what subject some proposed accident inheres; or which of many accidents is above all conducive to the production of some proposed effect; or in what way particular proposed causes ought to be conjoined in order to produce a definite effect. Because of the variety of the things sought for, sometimes the analytic method, sometimes the synthetic method, and sometimes both ought to be applied.

The first part, by which principles are found, is purely analytic. Seeing that the causes of all singulars are composed from the causes of universals or simples, it is necessary for those who are looking simply for scientific knowledge, which consists of the knowledge of the causes of all things insofar as this can be achieved, to know the causes of universals or those accidents which are common to all bodies, that is, to every material thing, before they know the causes of singular things, that is, of the accidents by which one thing is distinguished from another. Again, before the causes of those things can be known, it is necessary to know which things are universals. But since universals are contained in the nature of singular things, they must be unearthed by reason, that is, by resolution. For example, let any conception or idea of a singular thing be proposed, say a square. The square is resolved into: plane, bounded by a certain number of lines equal to one another, and right angles . Therefore we have these universals or components of every material thing: line, plane (in which a surface is contained), being bounded, angle, rectitude , and equality . If anyone finds the causes or origin of these, he will put them together as the cause of the square. Again, if he proposes to himself the conception of gold, the ideas of being solid, visible, and heavy (that is, of tending to the center of the earth or of motion downwards) and many others more universal than gold itself, which can be resolved further until one arrives at the most universal, will come from this by resolution. And by this same method of resolving things into other things one will know what those things are, of which, when their causes are known what those things are, of which, when their causes are known and composed one by one, the causes of all singular things are known. We thus conclude that the method of investigating the universal notions of things is purely analytic. ( Ibid ., §§ 3-4, 291-5) { §4.1 }

The method of scientific knowledge, civil as well as natural, [starting] from sense-experience and [going] to principles is analytic; while [starting] from principles is synthetic. ( Ibid ., §7, 301) { §4.1 }

it is obvious that in the investigation of causes there is a need partly for the analytic method, partly for the synthetic method. The analytic method is needed for understanding the circumstances of the effect one by one; the synthetic method for putting together those things which, single in themselves, act as one. ( Ibid ., §10, 311) { §4.1 }

that art of geometers which they call logistic is ... the method according to which by supposing that the thing asked about is true they come upon in reasoning either things known [to be true], from which they can prove the truth of the thing sought, or [they come upon] impossibilities, from which it can be understood that what was supposed [to be true] was false. ( Ibid ., §19, 329) { §4.1 }

[Logical analysis] stands somewhere between translating and paraphrasing. ( Logic , Harmondsworth: Penguin, 1977, 86)

The terms “analysis” and “synthesis” bring to mind, on the one hand, certain methodological practices in the works of Plato, Descartes, Newton, Kant, Hegel, and others and, on the other hand, techniques in fields as disparate as chemistry and logic, mathematics and psychology. The width of this spectrum of associations alerts us to the realization that at the base of these two related terms there lies a specific methodological thema-antithema ... pair. Indeed, it is one of the most pervasive and fundamental ones, in science and outside. This chapter attempts to uncover and identify this thematic content, to clarify the meanings and uses of the terms “analysis” and “synthesis”, and especially to distinguish among four general meanings: (1) Analysis and Synthesis, and particularly synthesis, used in the grand, cultural sense, (2) Analysis and Synthesis used in the reconstitutional sense (e.g., where an analysis, followed by a synthesis, re-establishes the original condition), (3) Analysis and Synthesis used in the transformational sense (e.g., where the application of Analysis and Synthesis advances one to a qualitatively new level), and (4) Analysis and Synthesis used in the judgmental sense (as in the Kantian categories and their modern critiques). (1998, 111) { §5.5 }

The point of view of function is the central one for phenomenology; the investigations radiating from it comprise almost the whole phenomenological sphere, and in the end all phenomenological analyses somehow enter into its service as component parts or preliminary stages. In place of analysis and comparison, description and classification restricted to particular experiences [ Erlebnisse ], the particulars are considered from the “teleological” point of view of their function, to make possible “synthetic unity”. ( IPP , I, §86; Kersten’s tr. modified) { §5.8 }

Explication is penetration of the internal horizon of the object by the direction of perceptual interest. In the case of the unobstructed realization of this interest, the protentional expectations fulfill themselves in the same way; the object reveals itself in its properties as that which it was anticipated to be, except that what was anticipated now attains original givenness. A more precise determination results, eventually perhaps partial corrections, or—in the case of obstruction—disappointment of the expectations, and partial modalization. ( EJ , §22, 105) { §5.8 }

The process of explication in its originality is that in which an object given at first hand is brought to explicit intuition. The analysis of its structure must bring to light how a twofold constitution of sense [ Sinngebung ] is realized in it: “object as substrate” and “determination α ...”; it must show how this constitution of sense is realized in the form of a process which goes forward in separate steps, through which, however, extends continuously a unity of coincidence —a unity of coincidence of a special kind, belonging exclusively to these sense-forms. ( EJ , §24a, 114) { §5.8 }

§1. MATHEMATICS ARRIVES AT ALL ITS DEFINITIONS SYNTHETICALLY, WHEREAS PHILOSOPHY ARRIVES AT ITS DEFINITIONS ANALYTICALLY

There are two ways in which one can arrive at a general concept: either by the arbitrary combination of concepts, or by separating out that cognition which has been rendered distinct by means of analysis. Mathematics only ever draws up its definitions in the first way. For example, think arbitrarily of four straight lines bounding a plane surface so that the opposite sides are not parallel to each other. Let this figure be called a trapezium . The concept which I am defining is not given prior to the definition itself; on the contrary, it only comes into existence as a result of that definition. Whatever the concept of a cone may ordinarily signify, in mathematics, the concept is the product of the arbitrary representation of a right-angled triangle which is rotated on one of its sides. In this and in all other cases the definition obviously comes into being as a result of synthesis .

The situation is entirely different in the case of philosophical definitions. In philosophy, the concept of a thing is always given, albeit confusedly or in an insufficiently determinate fashion. The concept has to be analysed; the characteristic marks which have been separated out and the concept which has been given have to be compared with each other in all kinds of contexts; and this abstract thought must be rendered complete and determinate. For example, everyone has a concept of time. But suppose that that concept has to be defined. The idea of time has to be examined in all kinds of relation if its characteristic marks which have been abstracted have to be combined together to see whether they yield an adequate concept; they have to be collated with each other to see whether one characteristic mark does not partly include another within itself. If, in this case, I had tried to arrive at a definition of time synthetically, it would have had to have been a happy coincidence indeed if the concept, thus reached synthetically, had been exactly the same as that which completely expresses the idea of time which is given to us. ( IDP , 2:276-7/ TP , 248-9) { §4.5 }

The true method of metaphysics is basically the same as that introduced by Newton into natural science and which has been of such benefit to it. Newton’s method maintains that one ought, on the basis of certain experience and, if need be, with the help of geometry, to seek out the rules in accordance with which certain phenomena of nature occur. ( IDP , 2:286/ TP , 259) { §4.5 }

What I am chiefly concerned to establish is this: in metaphysics one must proceed analytically throughout, for the business of metaphysics is actually the analysis of confused cognitions. If this procedure is compared with the procedure which is adopted by philosophers and which is currently in vogue in all schools of philosophy, one will be struck by how mistaken the practice of philosophers is. With them, the most abstracted concepts, at which the understanding naturally arrives last of all, constitute their starting point, and the reason is that the method of the mathematicians, which they wish to imitate throughout, is firmly fixed in their minds. This is why there is a strange difference to be found between metaphysics and all other sciences. In geometry and in the other branches of mathematics, one starts with what is easier and then one slowly advances to the more difficult operations. In metaphysics, one starts with what is the most difficult: one starts with possibility, with existence in general, with necessity and contingency, and so on – all of them concepts which demand great abstraction and close attention. And the reason for this is to be sought chiefly in the fact that the signs for these concepts undergo numerous and imperceptible modifications in use; and the differences between them must not be overlooked. One is told that one ought to proceed synthetically throughout. Definitions are thus set up right at the beginning, and conclusions are confidently drawn from them. Those who practise philosophy in this vein congratulate each other for having learnt the secret of thorough thought from the geometers. What they do not notice at all is the fact that geometers acquire their concepts by means of synthesis , whereas philosophers can only acquire their concepts by means of analysis – and that completely changes the method of thought. ...

Metaphysics has a long way to go yet before it can proceed synthetically. It will only be when analysis has helped us towards concepts which are understood distinctly and in detail that it will be possible for synthesis to subsume compound cognitions under the simplest cognition, as happens in mathematics. ( IDP , 2:289-90/ TP , 262-3) { §4.5 }

Such a system of pure (speculative) reason I hope myself to deliver under the title Metaphysics of Nature , which will be not half so extensive but will be incomparably richer in content than this critique, which had first to display the sources and conditions of its possibility, and needed to clear and level a ground that was completely overgrown. Here I expect from my reader the patience and impartiality of a judge , but there I will expect the cooperative spirit and assistance of a fellow worker ; for however completely the principles of the system may be expounded in the critique, the comprehensiveness of the system itself requires also that no derivative concepts should be lacking, which, however, cannot be estimated a priori in one leap, but must be gradually sought out; likewise, just as in the former the whole synthesis of concepts has been exhausted, so in the latter it would be additionally demanded that the same thing should take place in respect of their analysis , which would be easy and more entertainment than labor. ( CPR , Axxi) { §4.5 }

I understand by an analytic of concepts not their analysis, or the usual procedure of philosophical investigations, that of analyzing [ zergliedern ] the content of concepts that present themselves and bringing them to distinctness, but rather the much less frequently attempted analysis [ Zergliederung ] of the faculty of understanding itself, in order to research the possibility of a priori concepts by seeking them only in the understanding as their birthplace and analyzing its pure use in general; for this is the proper business of a transcendental philosophy; the rest is the logical treatment of concepts in philosophy in general. We will therefore pursue the pure concepts into their first seeds and predispositions in the human understanding, where they lie ready, until with the opportunity of experience they are finally developed and exhibited in their clarity by the very same understanding, liberated from the empirical conditions attaching to them. ( CPR , A65-6/B90-1) { §4.5 }

[in offering a refutation of Mendelssohn’s proof of the persistence of the soul] If we take the above propositions in a synthetic connection, as valid for all thinking beings, as they must be taken in rational psychology as a system, and if from the category of relation, starting with the proposition “All thinking beings are, as such, substances” we go backward through the series of propositions until the circle closes, then we finally come up against the existence of thinking beings, which in this system are conscious of themselves not only as independent of external things but also as being able to determine themselves from themselves (in regard to the persistence belonging necessarily to the character of a substance). But from this it follows that idealism , at least problematic idealism, is unavoidable in that same rationalistic system, and if the existence of external things is not at all required for the determination of one’s own existence in time, then such things are only assumed, entirely gratuitously, without a proof of them being able to be given.

If, on the contrary, we follow the analytic procedure, grounded on the “I think” given as a proposition that already includes existence in itself, and hence grounded on modality, and then we take it apart so as to cognize its content, whether and how this I determines its existence in space or time merely through it, then the propositions of the rational doctrine of the soul begin not from the concept of a thinking being in general but from an actuality; and from the way this is thought, after everything empirical has been detached from it, it is concluded what pertains to a thinking being in general ... ( CPR , B416-19) { §4.5 }

Give a philosopher the concept of a triangle, and let him try to find out in his way how the sum of its angles might be related to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and in it the concept of equally many angles. Now he may reflect on this concept as long as he wants, yet he will never produce anything new. He can analyze [ zergliedern ] and make distinct the concept of a straight line, or of an angle, or of the number three, but he will not come upon any other properties that do not already lie in these concepts. But now let the geometer take up this question. He begins at once to construct a triangle. Since he knows that two right angles together are exactly equal to all of the adjacent angles that can be drawn at one point on a straight line, he extends one side of his triangle, and obtains two adjacent angles that together are equal to two right ones. Now he divides the external one of these angles by drawing a line parallel to the opposite side of the triangle, and sees that here there arises an external adjacent angle which is equal to an internal one, etc. In such a way, through a chain of inferences that is always guided by intuition, he arrives at a fully illuminating and at the same time general solution of the question. ( CPR , A716-7/B744-5) { §4.5 }

although a mere plan that might precede the Critique of Pure Reason would be unintelligible, undependable, and useless, it is by contrast all the more useful if it comes after. For one will thereby be put in the position to survey the whole, to test one by one the main points at issue in this science, and to arrange many things in the exposition better than could be done in the first execution of the work.

Here then is such a plan subsequent to the completed work, which now can be laid out according to the analytic method , whereas the work itself absolutely had to be composed according to the synthetic method , so that the science might present all of its articulations, as the structural organization of a quite peculiar faculty of cognition, in their natural connection. ( PFM , 4:263/ 13) { §4.5 }

In the Critique of Pure Reason I worked on this question [Is metaphysics possible at all?] synthetically , namely by inquiring within pure reason itself, and seeking to determine within this source both the elements and the laws of its pure use, according to principles. This work is difficult and requires a resolute reader to think himself little by little into a system that takes no foundation as given except reason itself, and that therefore tries to develop cognition out of its original seeds without relying on any fact whatever. Prolegomena should by contrast be preparatory exercises; they ought more to indicate what needs to be done in order to bring a science into existence if possible, than to present the science itself. They must therefore rely on something already known to be dependable, from which we can go forward with confidence and ascend to the sources, which are not yet known, and whose discovery not only will explain what is known already, but will also exhibit an area with many cognitions that all arise from these same sources. The methodological procedure of prolegomena, and especially of those that are to prepare for a future metaphysics, will therefore be analytic . ( PFM , 4:274-5/ 25-6) { §4.5 }

[interpreting the method of analysis in ancient Greek geometry] Rule of analysis and synthesis: Draw conclusions from your conjecture, one after the other, assuming that it is true. If you reach a false conclusion, then your conjecture was false. If you reach an indubitably true conclusion, your conjecture may have been true. In this case reverse the process, work backwards, and try to deduce your original conjecture via the inverse route from the indubitable truth to the dubitable conjecture. If you succeed, you have proved your conjecture. (1978a, 72-3) { §2.2 }

Synthesis is when, beginning from principles and running through truths in order, we discover certain progressions and form tables, as it were, or sometimes even general formulae, in which the answers to what arises later can be discovered. Analysis, however, goes back to principles solely for the sake of a given problem, just as if nothing had been discovered previously, by ourselves or by others. It is better to produce a synthesis, since that work is of permanent value, whereas when we begin an analysis on account of particular problems we often do what has been done before. However, to use a synthesis which has been established by others, and theorems which have already been discovered, is less of an art than to do everything by oneself by carrying out an analysis; especially as what has been discovered by others, or even by ourselves, does not always occur to us or come to hand. There are two kinds of analysis: one is the common type proceeding by leaps, which is used in algebra, and the other is a special kind which I call ‘reductive’. This is much more elegant, but is less well-known. In practice, analysis is more necessary, so that we may solve the problems which are presented to us; but the man who can indulge in theorising will be content to practice analysis just far enough to master the art. For the rest, he will rather practise synthesis, and will apply himself readily only to those questions to which order itself leads him. For in this way he will always progress pleasantly and easily, and will never feel any difficulties, nor be disappointed of success, and in a short time he will achieve much more than he would ever have hoped for at the outset. ( USA , 16-17) { §4.4 }

Primary truths are those which either state a term of itself, or deny an opposite of its opposite. For example, ‘A is A’, or ‘A is not not-A’ ...

All other truths are reduced to primary truths by the aid of definitions—i.e. by the analysis of notions; and this constitutes a priori proof , independent of experience. ...

The predicate or consequent, therefore, is always in the subject or antecedent, and this constitutes the nature of truth in general, or, the connexion between the terms of a proposition, as Aristotle also has observed. In identities this connexion and inclusion of the predicate in the subject is express, whereas in all other truths it is implicit and must be shown through the analysis of notions, in which a priori demonstration consists. ( PT , 87-8) { §4.4 }

There are two kinds of truths , those of reason and those of fact . Truths of reason are necessary and their opposite is impossible; truths of fact are contingent and their opposite is possible. When a truth is necessary, its reason can be found by analysis, resolving it into simpler ideas and truths, until we come to those that are primitive. ( M , §33; tr. R. Latta) { §4.4 }

Our whole philosophy is rectification of colloquial linguistic usage. ( Aphorisms , 115) { §4.5 }

Writing is an excellent means of awakening in every man the system slumbering within him; and everyone who has ever written will have discovered that writing always awakens something which, though it lay within us, we failed clearly to recognize before. ( Ibid ., 119) { §4.5 }

Whichever way you look at it, philosophy is always analytical chemistry. The peasant employs all the propositions of the most abstract philosophy, only he employs them enveloped, concealed, compounded, latent, as the chemist and physicist says; the philosopher gives us the propositions pure. ( Ibid ., 162) { §4.5 }

There are therefore three ways whereby we get the complex Ideas of mixed Modes . 1. By Experience and Observation of things themselves. Thus by seeing two Men wrestle, or fence, we get the Idea of wrestling or fencing. 2. By Invention , or voluntary putting together of several simple Ideas in our own Minds: So he that first invented Printing, or Etching, had an Idea of it in his Mind, before it ever existed. 3. Which is the most usual way, by explaining the names of Actions we never saw, or Notions we cannot see; and by enumerating, and thereby, as it were, setting before our Imaginations all those Ideas which go to the making them up, and are the constituent parts of them. For having by Sensation and Reflection stored our Minds with simple Ideas , and by use got the Names, that stand for them, we can by those Names represent to another any complex Idea , we would have him conceive; so that it has in it no simple Idea , but what he knows, and has, with us, the same name for. For all our complex Ideas are ultimately resolvable into simple Ideas , of which they are compounded, and originally made up, though perhaps their immediate Ingredients, as I may so say, are also complex Ideas . Thus the mixed Mode , which the word Lye stands for, is made of these simple Ideas : 1. Articulate Sounds. 2. Certain Ideas in the Mind of the Speaker. 3. Those words the signs of those Ideas . 4. Those signs put together by affirmation or negation, otherwise than the Ideas they stand for, are in the mind of the Speaker. I think I need not go any farther in the Analysis of that complex Idea , we call a Lye : What I have said is enough to shew, that it is made up of simple Ideas : And it could not be an offensive tediousness to my Reader, to trouble him with a more minute enumeration of every particular simple Idea , that goes to this complex one; which, from what has been said, he cannot but be able to make out to himself. The same may be done in all our complex Ideas whatsoever; which however compounded, and decompounded, may at last be resolved into simple Ideas , which are all the Materials of Knowledge or Thought we have or can have. ( Essay , II, xxii, 9) { §4.3 }

Analysis has a way of unravelling the self: the longer you pull on the thread, the more flaws you find. ( Therapy , London, 31)

The certainty of mathematics is based upon the general axiom that nothing can be and not be at the same time. In this science each proposition such as, for example, “A is B”, is proven in one of two ways. Either one unpacks the concepts of A and shows “A is B”, or one unpacks the concepts of B and infers from this that not-B must also be not-A. Both types of proof are thus based upon the principle of contradiction, and since the object of mathematics in general is magnitude and that of geometry in particular extension , one can say that in mathematics in general our concepts of magnitude are unpacked and analyzed, while in geometry in particular our concepts of extension are unpacked and analyzed. In fact, since geometry lays nothing else as its basis than the abstract concept of extension and derives all its conclusions from this single source – deriving them, to be sure, in such a way that one recognizes distinctly that everything maintained in it is necessarily connected by the principle of contradiction with the abstracted concept of extension, there is no doubt that all geometric truths that geometry teaches us to unpack or untangle from the concept of extension must be encountered all tangled up in it. For what else can the profoundest inferences do but analyze a concept and make distinct what was obscure? Such inferences cannot bring in what is not to be found in the concept, and it is easy to see that it is also not possible, by means of the principle of contradiction, to derive from the concept what is not to be found in it. In the concept of extension, for example, there lies the inner possibility that a space is limited by three straight lines in such a way that two of them include a right angle. For it follows from the essence of extension that it is capable of many sorts of limitations and that the assumed sort of limitation of one of its level planes contains no contradiction. If one subsequently shows that the concept of this assumed limitation or of a right-angled triangle necessarily entails that the square of the hypotenuse is such-and-such, then it must have also been possible to find this truth originally and implicitly in the initial concept of extension. Otherwise it could never have been derived from it by means of the principle of contradiction. The idea of extension is inseparable from the idea of the possibility of such a limitation, as was previously assumed, and the limitation is in turn necessarily connected to the concept of the equality of the aforesaid square. Thus, this truth also lay tangled up, as one might say, in the original concept of extension, but it escaped our attention and could not be distinctly known and distinguished until, through analysis, we unpacked all the parts of this concept and separated them from one another. The analysis of concepts is for the understanding nothing more than what the maginfying glass is for sight. It does not produce anything that was not to be found in the object. But it spreads out the parts of the object and makes it possible for our senses to distinguish much that they would otherwise not have noticed. The analysis of concepts does nothing different from this; it makes the parts and members of these concepts, which were previously obscure and unnoticed, distinct and recognizable, but it does not introduce anything into the concepts that was not already to be found in them. (1763, §1/ PW , 257-8) { §4.5 }

It seems necessary, then, to regard the world as formed of concepts. These are the only objects of knowledge. They cannot be regarded fundamentally as abstractions either from things or from ideas; since both alike can, if anything is to be true of them, be composed of nothing but concepts. A thing becomes intelligible first when it is analysed into its constituent concepts. ( NJ , 8) { §6.4 }

It appears to me that in Ethics, as in all other philosophical studies, the difficulties and disagreements, of which its history is full, are mainly due to a very simple cause: namely to the attempt to answer questions, without first discovering precisely what question it is which you desire to answer. I do not know how far this source of error would be done away, if philosophers would try to discover what question they were asking, before they set about to answer it; for the work of analysis and distinction is often very difficult: we may often fail to make the necessary discovery, even though we make a definite attempt to do so. But I am inclined to think that in many cases a resolute attempt would be sufficient to ensure success; so that, if only this attempt were made, many of the most glaring difficulties and disagreements in philosophy would disappear. ( PE , vii) { §6.4 }

My point is that ‘good’ is a simple notion, just as ‘yellow’ is a simple notion; that, just as you cannot, by any manner of means, explain to any one who does not already know it, what yellow is, so you cannot explain what good is. Definitions of the kind that I was asking for, definitions which describe the real nature of the object or notion denoted by a word, and which do not merely tell us what the word is used to mean, are only possible when the object or notion in question is something complex. You can give a definition of a horse, because a horse has many different properties and qualities, all of which you can enumerate. But when you have enumerated them all, when you have reduced a horse to his simplest terms, then you no longer define those terms. They are simply something which you think of or perceive, and to any one who cannot think of or perceive them, you can never, by any definition, make their nature known. ( PE , 7) { §6.4 }

As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy. And although the arguing from Experiments and Observations by Induction be no Demonstration of general Conclusions; yet it is the best way of arguing which the Nature of Things admits of, and may be looked upon as so much the stronger, by how much the Induction is more general. And if no Exception occur from Phænomena, the Conclusion may be pronounced generally. But if at any time afterwards any Exception shall occur from Experiments, it may then begin to be pronounced with such Exceptions as occur. By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis: and the Synthesis consists in assuming the Causes discover’d, and establish’d as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations. ( Opticks , Book Three, Part I, 404-5) { §4.1 }

All concepts in which an entire process is semiotically telescoped elude definition. ( On the Genealogy of Morals , 1887, tr. Walter Kaufmann, New York: Random House, 1968, 80)

the most valuable insights are methods . ( The Antichrist , 1895, §13)

The so-called Treasury of Analysis [ analuomenos ] .. is, in short, a special body of doctrines furnished for the use of those who, after going through the usual elements, wish to obtain the power of solving theoretical problems, which are set to them, and for this purpose only is it useful. It is the work of three men, Euclid the author of the Elements , Apollonius of Perga, and Aristaeus the Elder, and proceeds by the method of analysis and synthesis.

Now analysis is the way from what is sought—as if it were admitted—through its concomitants [ akolouthôn ] in order to something admitted in synthesis. For in analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent [ proêgoumenon ] of the latter, until we on our backward way light upon something already known and being first in order. And we call such a method analysis, as being a solution backwards [ anapalin lysin ].

In synthesis, on the other hand, we suppose that which was reached last in analysis to be already done, and arranging in their natural order as consequents [ epomena ] the former antecedents [ proêgoumena ] and linking them one with another, we in the end arrive at the construction of the thing sought. And this we call synthesis.

Now analysis is of two kinds. One seeks the truth, being called theoretical. The other serves to carry out what was desired to do, and this is called problematical. In the theoretical kind we suppose the thing sought as being and as being true, and then we pass through its concomitants [ akolouthôn ] in order, as though they were true and existent by hypothesis, to something admitted; then, if that which is admitted be true, the thing sought is true, too, and the proof will be the reverse of analysis. But if we come upon something false to admit, the thing sought will be false, too. In the problematic kind we suppose the desired thing to be known, and then we pass through its concomitants [ akolouthôn ] in order, as though they were true, up to something admitted. If the thing admitted is possible or can be done, that is, if it is what the mathematicians call given, the desired thing will also be possible. The proof will again be the reverse of analysis. But if we come upon something impossible to admit, the problem will also be impossible. ( PAC , tr. in Hintikka and Remes 1974, 8-10) { §2.2 }

For we should remember that if a person goes on analyzing names into words, and inquiring also into the elements out of which the words are formed, and keeps on always repeating this process, he who has to answer him must at last give up the inquiry in despair … But if we take a word which is incapable of further resolution, then we shall be right in saying that we at last reached a primary element, which need not be resolved any further. (‘Cratylus’, Benjamin Jowett (trans.), in Hamilton and Cairns (ed.), Collected Dialogues , New York: Pantheon Books, 421e)

Then, said I, is not dialectic the only process of inquiry that advances in this manner, doing away with hypotheses, up to the first principle itself in order to find confirmation there? And it is literally true that when the eye of the soul is sunk in the barbaric slough of the Orphic Myth, dialectic gently draws it forth and leads it up, employing as helpers and cooperators in this conversation the studies and sciences which we enumerated, which we called sciences often from habit, though they really need some other designation, connoting more clearness than opinion and more obscurity than science. ‘Understanding’ I believe was the term we employed. But, I presume we shall not dispute about the name when things of such moment lie before us for consideration. (‘Republic VII’, Paul Shorey (trans.), Ibid. , 533d)

Understand then, said I, that by the other section of the intelligible I mean that which the reason lays hold of by the power of dialectic, treating its assumptions not as absolute beginnings but literally as hypotheses, underpinnings, footings and springboards so to speak, to enable it to rise to that which requires no assumption and is the starting point of all, and after attaining to that again taking hold of the first dependencies from it, so to proceed downward to the conclusion, making no use whatever of any object of sense but only of pure ideas moving on through ideas to ideas and ending with ideas. (‘Republic VI’, Paul Shorey (trans.), Ibid ., 511b)

In mathematics logic is called analysis , and analysis means division , dissection . It can have, therefore, no tool other than the scalpel and the microscope. (‘Intuition and Logic in Mathematics’, 1900, in William Ewald, ed., From Kant to Hilbert , Oxford: Oxford University Press, 1996, 1018)

Nonmathematical illustration [of the method of analysis described by Pappus] . A primitive man wishes to cross a creek; but he cannot do so in the usual way because the water has risen overnight. Thus, the crossing becomes the object of a problem; “crossing the creek’ is the x of this primitive problem. The man may recall that he has crossed some other creek by walking along a fallen tree. He looks around for a suitable fallen tree which becomes his new unknown, his y . He cannot find any suitable tree but there are plenty of trees standing along he creek; he wishes that one of them would fall. Could he make a tree fall across the creek? There is a great idea and there is a new unknown; by what means could he tilt the tree over the creek?

This train of ideas ought to be called analysis if we accept the terminology of Pappus. If the primitive man succeeds in finishing his analysis he may become the inventor of the bridge and of the axe. What will be the synthesis? Translation of ideas into actions. The finishing act of the synthesis is walking along a tree across the creek.

The same objects fill the analysis and the synthesis; they exercise the mind of the man in the analysis and his muscles in the synthesis; the analysis consists in thoughts, the synthesis in acts. There is another difference; the order is reversed. Walking across the creek is the first desire from which the analysis starts and it is the last act with which the synthesis ends. (1957, 145) { §2.2 }

beauty and order are common to all branches of mathematics, as are the method of proceeding from things better known to things we seek to know and the reverse path from the latter to the former, the methods called analysis and synthesis. ( CEE , 8/6-7) { §2.2 }

as Nous is set over understanding and dispenses principles to it from above, perfecting it out of its own riches, so in the same way dialectic, the purest part of philosophy, hovers attentively over mathematics, encompasses its whole development, and of itself contributes to the special sciences their various perfecting, critical, and intellective powers—the procedures, I mean, of analysis, division, definition, and demonstration. Being thus endowed and led towards perfection, mathematics reaches some of its results by analysis, others by synthesis, expounds some matters by division, others by definition, and some of its discoveries binds fast by demonstration, adapting these methods to its subjects and employing each of them for gaining insight into mediating ideas. Thus its analyses are under the control of dialectic, and its definitions, divisions, and demonstrations are of the same family and unfold in conformity with the way of mathematical understanding. It is reasonable, then, to say that dialectic is the capstone of the mathematical sciences. It brings to perfection all the intellectual insight they contain, making what is exact in them more irrefutable, confirming the stability of what they have established and referring what is pure and incorporeal in them to the simplicity and immateriality of Nous, making precise their primary starting-points through definitions and explicating the distinctions of genera and species within their subject-matters, teaching the use of synthesis to bring out the consequences that follow from principles and of analysis to lead up to the first principles and starting-points. ( CEE , 42-3/35-6) { §2.2 }

Magnitudes, figures and their boundaries, and the ratios that are found in them, as well as their properties, their various positions and motions—these are what geometry studies, proceeding from the partless point down to solid bodies, whose many species and differences it explores, then following the reverse path from the more complex objects to the simpler ones and their principles. It makes use of synthesis and analysis, always starting from hypotheses and first principles that it obtains from the science above it and employing all the procedures of dialectic—definition and division for establishing first principles and articulating species and genera, and demonstrations and analyses in dealing with the consequences that follow from first principles, in order to show the more complex matters both as proceeding from the simpler and also conversely as leading back to them. ( CEE , 57/46) { §2.2 }

[Euclid’s Elements ] contains all the dialectical methods: the method of division for finding kinds, definitions for making statements of essential properties, demonstrations for proceeding from premises to conclusions, and analysis for passing in the reverse direction from conclusions to principles. ( CEE , 69/57) { §2.2 }

there are certain methods that have been handed down, the best being the method of analysis, which traces the desired result back to an acknowledged principle. Plato, it is said, taught this method to Leodamas, who also is reported to have made many discoveries in geometry by means of it. A second is the method of diaeresis , which divides into its natural parts the genus proposed for examination and which affords a starting-point for demonstration by eliminating the parts irrelevant for the establishment of what is proposed. This method also Plato praised as an aid in all the sciences. A third is the reduction to impossibility, which does not directly show the thing itself that is wanted but by refuting its contradictory indirectly establishes its truth. ( CEE , 211-12/165-6) { §2.2 }

for problems one common procedure, the method of analysis, has been discovered, and by following it we can reach a solution; for thus it is that even the most obscure problems are pursued. ( CEE , 242/189) { §2.2 }

In general we must understand that all mathematical arguments proceed either from or to the starting-points, as Porphyry somewhere says. Those that proceed from the starting-points are themselves of two kinds, as it happens, for they proceed either from common notions, that is, from self-evident clarity alone, or from things previously demonstrated. Those that proceed to the starting-points are either affirmative of them or destructive. But those that affirm first principles are called “analyses”, and their reverse procedures “syntheses” (for it is possible from those principles to proceed in orderly fashion to the thing sought, and this is called “synthesis”); when they are destructive, they are called “reductions to impossibility”, for it is the function of this procedure to show that something generally accepted and self-evident is overthrown. There is a kind of syllogism in it, though not the same as in analysis ... ( CEE , 255/198-9) { §2.2 }

A maxim of shallow analysis prevails: expose no more logical structure than seems useful for the deduction or other inquiry at hand. In the immortal words of Adolf Meyer, where it doesn’t itch don't scratch.

On occasion the useful degree of analysis may, conversely, be such as to cut into a simple word of ordinary language, requiring its paraphrase into a composite term in which other terms are compounded with the help of canonical notation. When this happens, the line of analysis adopted will itself commonly depend on what is sought in the inquiry at hand; again there need be no question of the uniquely right analysis, nor of synonymy. (1960, §33, 160-1) { §6.9 }

This construction [of the ordered pair as a class, such as Wiener’s identification of the ordered pair x , y > with the class {{ x }, { y , Λ}}] is paradigmatic of what we are most typically up to when in a philosophical spirit we offer an “analysis” or “explication” of some hitherto inadequately formulated “idea” or expression. We do not claim synonymy. We do not claim to make clear and explicit what the users of the unclear expression had unconsciously in mind all along. We do not expose hidden meanings, as the words ‘analysis’ or ‘explication’ would suggest; we supply lacks. We fix on the particular functions of the unclear expression that make it worth troubling about, and then devise a substitute, clear and couched in terms to our liking, that fills those functions. Beyond those conditions of partial agreement, dictated by our interests and purposes, any traits of the explicans come under the head of “don’t-cares” … Under this head we are free to allow the explicans all manner of novel connotations never associated with the explicandum. …

Philosophical analysis, explication, has not always been seen in this way. Only the reading of a synonymy claim into analysis could engender the so-called paradox of analysis, which runs thus: how can a correct analysis be informative, since to understand it we must already know the meanings of its terms, and hence already know that the terms which it equates are synonymous? The notion that analysis must consist somehow in the uncovering of hidden meanings underlies also the recent tendency of some of the Oxford philosophers to take as their business an examination of the subtle irregularities of ordinary language. And there is no mistaking the obliviousness of various writers to the point about the don’t-cares. …

... explication is elimination . We have, to begin with, an expression or form of expression that is somehow troublesome. It behaves partly like a term but not enough so, or it is vague in ways that bother us, or it puts kinks in a theory or encourages one or another confusion. But also it serves certain purposes that are not to be abandoned. Then we find a way of accomplishing those same purposes through other channels, using other and less troublesome forms of expression. The old perplexities are resolved.

According to an influential doctrine of Wittgenstein’s, the task of philosophy is not to solve problems but to dissolve them by showing that there were really none there. This doctrine has its limitations, but it aptly fits explication. For when explication banishes a problem it does so by showing it to be in an important sense unreal; viz., in the sense of proceeding only from needless usages. (1960, §53, 258-60) { §6.9 }

This brings us to the second of the five turning points, the shift from terms to sentences. The medievals had the notion of syncategorematic words, but it was a contemporary of John Horne Tooke who developed it into an explicit theory of contextual definition; namely, Jeremy Bentham. He applied contextual definition not just to grammatical particles and the like, but even to some genuine terms, categorematic ones. If he found some term convenient but ontologically embarrassing, contextual definition enabled him in some cases to continue to enjoy the services of the term while disclaiming its denotation. He could declare the term syncategorematic, despite grammatical appearances, and then could justify his continued use of it if he could show systematically how to paraphrase as wholes all sentences in which he chose to imbed it. Such was his theory of fictions: what he called paraphrasis, and what we now call contextual definition. The term, like the grammatical particles, is meaningful as a part of meaningful wholes. If every sentence in which we use a term can be paraphrased into a sentence that makes good sense, no more can be asked. (1975, 68-9) { §5.6 }

The issue is: is there such an activity as “conceptual analysis” or can philosophers do no more than describe usage and, perhaps, make recommendations for change in usage? One’s answer to this question will determine whether one thinks that Wittgenstein was wrong to give up on the idea of a systematic theory of meaning, and Quine right to suggest that the very notion of “meaning” was a hangover of Aristotelean essentialism. If they were right, it is hard to hang on to the idea that “conceptual clarity” is a goal of philosophical inquiry … Metaphilosophical issues hover in the wings of the debates over whether the content of an assertion varies from utterer to utterer and from audience to audience. If it does not, if something remains invariable – the concepts expressed by the words that make up the sentence – then perhaps there really are entities with intrinsic properties which philosophical analysis can hope to pin down. But, if content does vary in this way, then concepts are like persons - never quite the same twice, always developing, always maturing. You can change a concept by changing usage, but you cannot get a concept right, once and for all. (‘Analytic and Conversational Philosophy’, Philosophy as Cultural Politics , Cambridge: Cambridge University Press, 2007, 122-3)

Analysis, to be sure, is articulation rather than dissolution. (1980, 8) { §1.2 , §5.8 }

we must see where we are going , or what will “count” as the successful resolution to the given exercise of analysis. … Analysis is the admittedly indispensable road to our destination, but it is no more the destination than it is the intention to begin the voyage. One could perhaps say that the destination is an articulated structure. But we know that we have reached the destination only when we recognize a given articulation as the explanation of that structure. We cannot see that an analysis explains a structure by performing an additional step in the analysis. At some point we must see that we are finished. And to see an analysis is not to analyze. It is rather to see an articulated structure as a unity, whole, or synthesis. ( Ibid ., 9) { §1.2 , §5.8 }

If to understand is to possess an explanation, and if an explanation is an analysis, it remains the case that an analysis is intelligible because it is also a synthesis. Explanation may be called “recollection” in the Platonic sense because it is the process of retracing, by the method of counting and measuring, the joints of an internally articulated unity, one prefigured within the initial formulation of the entire analytical exercise. In slightly more prosaic terms, analysis is never merely the application of rules. It is also at once a seeing of which rules to apply and how to apply them. This is what it means to say that analysis is also synthesis. And this is why it is false to say, as is at least implied by so much contemporary analytical philosophy, that we begin with intuitions and then replace them with ever more sophisticated analyses. Not only is it false to say this, but strictly speaking, it is meaningless. If “to mean” is “to provide an analysis”, there is no analysis of analysis without ingredient intuition. Without intuition, there is at each stage nothing to analyze. Intuition (of syntheses or unities) without analysis is mute, but analysis without intuition is inarticulate as well as blind: the sounds it utters cannot be distinguished from noise. ( Ibid ., 9-10) { §1.2 , §5.8 }

analysis is a cognitive activity and it cannot be coherently understood except by recourse to intuition. There is a non-discursive context of analysis . ( Ibid ., 27) { §1.2 , §5.8 }

conceptual analysis is rooted in intuitions which cannot be replaced by the process of analysis but which regulate that process. ( Ibid ., 48) { §1.2 , §5.8 }

That all sound philosophy should begin with an analysis of propositions, is a truth too evident, perhaps, to demand a proof. That Leibniz’s philosophy began with such an analysis, is less evident, but seems to be no less true. ( PL , 8) { §6.3 }

It is necessary to realize that definition, in mathematics, does not mean, as in philosophy, an analysis of the idea to be defined into constituent ideas. This notion, in any case, is only applicable to concepts, whereas in mathematics it is possible to define terms which are not concepts. Thus also many notions are defined by symbolic logic which are not capable of philosophical definition, since they are simple and unanalyzable. ( POM , ch. 2, §31, 27) { §6.3 }

For the comprehension of analysis, it is necessary to investigate the notion of whole and part, a notion which has been wrapped in obscurity—though not without certain more or less valid logical reasons—by the writers who may be roughly called Hegelian. ( POM , ch. 16, §133, 137) { §6.3 }

I have already touched on a very important logical doctrine, which the theory of whole and part brings into prominence—I mean the doctrine that analysis is falsification. Whatever can be analyzed is a whole, and we have already seen that analysis of wholes is in some measure falsification. But it is important to realize the very narrow limits of this doctrine. We cannot conclude that the parts of a whole are not really its parts, nor that the parts are not presupposed in the whole in a sense in which the whole is not presupposed in the parts, nor yet that the logically prior is not usually simpler than the logically subsequent. In short, though analysis gives us the truth, and nothing but the truth, yet it can never give us the whole truth. This is the only sense in which the doctrine is to be accepted. In any wider sense, it becomes merely a cloak for laziness, by giving an excuse to those who dislike the labour of analysis. ( POM , ch. 16, §138, 141) { §6.3 }

We are sometimes told that things are organic unities, composed of many parts expressing the whole and expressed in the whole. This notion is apt to replace the older notion of substance, not, I think, to the advantage of precise thinking. The only kind of unity to which I can attach any precise sense—apart from the unity of the absolutely simple—is that of a whole composed of parts. But this form of unity cannot be what is called organic; for if the parts express the whole or the other parts, they must be complex, and therefore themselves contain parts; if the parts have been analyzed as far as possible, they must be simple terms, incapable of expressing anything except themselves. A distinction is made, in support of organic unities, between conceptual analysis and real division into parts. What is really indivisible, we are told, may be conceptually analyzable. This distinction, if the conceptual analysis be regarded as subjective, seems to me wholly inadmissible. All complexity is conceptual in the sense that it is due to a whole capable of logical analysis, but is real in the sense that it has no dependence upon the mind, but only upon the nature of the object. Where the mind can distinguish elements, there must be different elements to distinguish; though, alas! there are often different elements which the mind does not distinguish. The analysis of a finite space into points is no more objective than the analysis (say) of causality into time-sequence + ground and consequent, or of equality into sameness of relation to a given magnitude. In every case of analysis, there is a whole consisting of parts with relations; it is only the nature of the parts and the relations which distinguishes different cases. Thus the notion of an organic whole in the above sense must be attributed to defective analysis, and cannot be used to explain things.

It is also said that analysis is falsification, that the complex is not equivalent to the sum of its constituents and is changed when analyzed into these. In this doctrine, as we saw in Parts I and II, there is a measure of truth, when what is to be analyzed is a unity. A proposition has a certain indefinable unity, in virtue of which it is an assertion; and this is so completely lost by analysis that no enumeration of constituents will restore it, even though itself be mentioned as a constituent. There is, it must be confessed, a grave logical difficulty in this fact, for it is difficult not to believe that a whole must be constituted by its constituents. For us, however, it is sufficient to observe that all unities are propositions or propositional concepts, and that consequently nothing that exists is a unity. If, therefore, it is maintained that things are unities, we must reply that no things exist. ( POM , ch. 53, §439, 466-7) { §6.3 }

What we want to be clear about is the twofold method of analysis of a proposition, i.e. , first taking the proposition as it stands and analyzing it, second taking the proposition as a special case of a type of propositions. Whenever we use variables, we are already necessarily concerned with a type of propositions. E.g. “ p ⊃ q ” stands for any proposition of a certain type. When values are assigned to p and q , we reach a particular proposition by a different road from that which would have started with those values plus implication, and have so built up the particular proposition without reference to a type. This is how functions come in. (‘Fundamental Notions’, 1904, in 1994, 118) { §6.3 }

We ought to say, I think, that there are different ways of analysing complexes, and that one way of analysis is into function and argument, which is the same as type and instance. ( Ibid ., 256) { §6.3 }

The fundamental epistemological principle in the analysis of propositions containing descriptions is this: Every proposition which we can understand must be composed wholly of constituents with which we are acquainted. ( KAKD , 159) { §6.3 }

when we say ‘the author of Waverley was Scott’ we mean ‘one and only one man wrote Waverley, and he was Scott’. Here the identity is between a variable, i.e. an indeterminate subject (‘he’), and Scott; ‘the author of Waverley’ has been analysed away, and no longer appears as a constituent of the proposition. ( KAKD , 165) { §6.3 }

Analysis may be defined as the discovery of the constituents and the manner of combination of a given complex. The complex is to be one with which we are acquainted; the analysis is complete when we become acquainted with all the constituents and with their manner of combination, and know that there are no more constituents and that that is their manner of combination. We may distinguish formal analysis as the discovery of the manner of combination, and material analysis as the discovery of the constituents. Material analysis may be called descriptive when the constituents are only known by description, not by acquaintance. ( TK , 119) { §6.3 }

Philosophy, if what has been said is correct, becomes indistinguishable from logic as that word has now come to be used. The study of logic consists, broadly speaking, of two not very sharply distinguished portions. On the one hand it is concerned with those general statements which can be made concerning everything without mentioning any one thing or predicate or relation, such for example as ‘if x is a member of the class α and every member of α is a member of β , then x is a member of the class β , whatever x , α , and β may be.’. On the other hand, it is concerned with the analysis and enumeration of logical forms , i.e. with the kinds of propositions that may occur, with the various types of facts, and with the classification of the constituents of facts. In this way logic provides an inventory of possibilities, a repertory of abstractly tenable hypotheses. ( SMP , 84-5) { §6.3 }

The essence of philosophy as thus conceived is analysis, not synthesis. To build up systems of the world, like Heine’s German professor who knit together fragments of life and made an intelligible system out of them, is not, I believe, any more feasible than the discovery of the philosopher’s stone. What is feasible is the understanding of general forms, and the division of traditional problems into a number of separate and less baffling questions. ‘Divide and conquer’ is the maxim of success here as elsewhere. ( SMP , 86) { §6.3 }

Kant, under the influence of Newton, adopted, though with some vacillation, the hypothesis of absolute space, and this hypothesis, though logically unobjectionable, is removed by Occam’s razor, since absolute space is an unnecessary entity in the explanation of the physical world. Although, therefore, we cannot refute the Kantian theory of an a priori intuition, we can remove its grounds one by one through an analysis of the problem. Thus, here as in many other philosophical questions, the analytic method, while not capable of arriving at a demonstrative result, is nevertheless capable of showing that all the positive grounds in favour of a certain theory are fallacious and that a less unnatural theory is capable of accounting for the facts.

Another question by which the capacity of the analytic method can be shown is the question of realism. Both those who advocate and those who combat realism seem to me to be far from clear as to the nature of the problem which they are discussing. If we ask: ‘Are our objects of perception real and are they independent of the percipient?’ it must be supposed that we attach some meaning to the words ‘real’ and ‘independent’, and yet, if either side in the controversy of realism is asked to define these two words, their answer is pretty sure to embody confusions such as logical analysis will reveal. ( SMP , 90-1) { §6.3 }

The supreme maxim in scientific philosophizing is this:

Wherever possible, logical constructions are to be substituted for inferred entities.

Some examples of the substitution of construction for inference in the realm of mathematical philosophy may serve to elucidate the uses of this maxim. Take first the case of irrationals. In old days, irrationals were inferred as the supposed limits of series of rationals which had no rational limit; but the objection to this procedure was that it left the existence of irrationals merely optative, and for this reason the stricter methods of the present day no longer tolerate such a definition. We now define an irrational number as a certain class of ratios, thus constructing it logically by means of ratios, instead of arriving at it by a doubtful inference from them. Take again the case of cardinal numbers. Two equally numerous collections appear to have something in common: this something is supposed to be their cardinal number. But so long as the cardinal number is inferred from the collections, not constructed in terms of them, its existence must remain in doubt, unless in virtue of a metaphysical postulate ad hoc . By defining the cardinal number of a given collection as the class of all equally numerous collections, we avoid the necessity of this metaphysical postulate, and thereby remove a needless element of doubt from the philosophy of arithmetic. A similar method, as I have shown elsewhere, can be applied to classes themselves, which need not be supposed to have any metaphysical reality, but can be regarded as symbolically constructed fictions.

The method by which the construction proceeds is closely analogous in these and all similar cases. Given a set of propositions nominally dealing with the supposed inferred entities, we observe the properties which are required of the supposed entities in order to make these propositions true. By dint of a little logical ingenuity, we then construct some logical function of less hypothetical entities which has the requisite properties. The constructed function we substitute for the supposed inferred entities, and thereby obtain a new and less doubtful interpretation of the body of propositions in question. This method, so fruitful in the philosophy of mathematics, will be found equally applicable in the philosophy of physics, where, I do not doubt, it would have been applied long ago but for the fact that all who have studied this subject hitherto have been completely ignorant of mathematical logic. I myself cannot claim originality in the application of this method to physics, since I owe the suggestion and the stimulus for its application entirely to my friend and collaborator Dr Whitehead, who is engaged in applying it to the more mathematical portions of the region intermediate between sense-data and the points, instants and particles of physics.

A complete application of the method which substitutes constructions for inferences would exhibit matter wholly in terms of sense-data, and even, we may add, of the sense-data of a single person, since the sense-data of others cannot be known without some element of inference. This, however, must remain for the present an ideal, to be approached as nearly as possible, but to be reached, if at all, only after a long preliminary labour of which as yet we can only see the very beginning. ( RSDP , 115-6) { §6.3 }

In the special sciences, when they have become fully developed, the movement is forward and synthetic, from the simpler to the more complex. But in philosophy we follow the inverse direction: from the complex and relatively concrete we proceed towards the simple and abstract by means of analysis, seeking, in the process, to eliminate the particularity of the original subject-matter, and to confine our attention entirely to the logical form of the facts concerned. ( OKEW , 189-90) { §6.3 }

The nature of philosophic analysis … can now be stated in general terms. We start from a body of common knowledge, which constitutes our data. On examination, the data are found to be complex, rather vague, and largely interdependent logically. By analysis we reduce them to propositions which are as nearly as possible simple and precise, and we arrange them in deductive chains, in which a certain number of initial propositions form a logical guarantee for all the rest. ( OKEW , 214) { §6.3 }

the chief thesis that I have to maintain is the legitimacy of analysis. ( PLA , 189) { §6.3 }

it is very important to distinguish between a definition and an analysis. All analysis is only possible in regard to what is complex, and it always depends, in the last analysis, upon direct acquaintance with the objects which are the meanings of certain simple symbols. It is hardly necessary to observe that one does not define a thing but a symbol. ( PLA , 194) { §6.3 }

Analysis is not the same thing as definition. You can define a term by means of a correct description, but that does not constitute an analysis. ( PLA , 196) { §6.3 }

The business of philosophy, as I conceive it, is essentially that of logical analysis, followed by logical synthesis. ( LA , 341) { §6.3 }

Ever since I abandoned the philosophy of Kant and Hegel, I have sought solutions of philosophical problems by means of analysis; and I remain firmly persuaded, in spite of some modern tendencies to the contrary, that only by analysing is progress possible. ( MPD , 11) { §6.3 }

Philosophy must then involve the exercise of systematic restatement. But this does not mean that it is a department of philology or literary criticism.

Its restatement is not the substitution of one noun for another or one verb for another. That is what lexicographers and translators excel in. Its restatements are transmutations of syntax, and transmutations of syntax controlled not be desire for elegance or stylistic correctness but by desire to exhibit the forms of the facts into which philosophy is the enquiry.

I conclude, then, that there is, after all, a sense in which we can properly enquire and even say “what it really means to say so and so”. For we can ask what is the real form of the fact recorded when this is concealed or disguised and not duly exhibited by the expression in question. And we can often succeed in stating this fact in a new form of words which does exhibit what the other failed to exhibit. And I am for the present inclined to believe that this is what philosophical analysis is, and that this is the sole and whole function of philosophy. (1932, 100) { §6.8 }

I have no special objection to or any special liking for the fashion of describing as ‘analysis’ the sort or sorts of conceptual examination which constitute philosophizing. But the idea is totally false that this examination is a sort of garage inspection of one conceptual vehicle at a time. On the contrary, to put it dogmatically, it is always a traffic inspector’s examination of a conceptual traffic-block, involving at least two streams of vehicles hailing from the theories, or points of view or platitudes which are at cross-purposes with one another. (1953, 32) { §6.8 }

It is certain that when I wrote “Systematically Misleading Expressions” I was still under the direct influence of the notion of an “ideal language”—a doctrine according to which there were a certain number of logical forms which one could somehow dig up by scratching away at the earth which covered them. I no longer think, especially not today, that this is a good method. I do not regret having traveled that road, but I am happy to have left it behind me. (In Rorty 1967, 305) { §6.8 }

alas! intellect must first destroy the object of Inner Sense if it would make it its own. Like the analytical chemist, the philosopher can only discover how things are combined by analysing them, only lay bare the workings of spontaneous Nature by subjecting them to the torment of his own techniques. In order to lay hold of the fleeting phenomenon, he must first bind it in the fetters of rule, tear its fair body to pieces by reducing it to concepts, and preserve its living spirit in a sorry skeleton of words. Is it any wonder that natural feeling cannot find itself again in such an image, or that in the account of the analytical thinker truth should appear as paradox? ( AE , I, 4) { §5.2 }

analysis without synopsis must be blind. (‘Time and the World Order’, in Herbert Feigl and Grover Maxwell, (eds.), Minnesota Studies in the Philosophy of Science III , Minneapolis: University of Minnesota Press, 1962, 527)

[in discussing Ryle 1953 { Quotation }] Personally, I have no axe to grind about what it takes to analyze a concept. Very likely, there are different sorts of cases. It may well be that sometimes what we want from an analysis is the tracing of the sort of intricate web of conceptual relations in which Ryle delights. But there is little reason for thinking that this is always so—at least, if analysis is construed as whatever it is that philosophers do to solve their problems. What strikes me as worrisome is Ryle’ tendency to use the web metaphor as a rationale for rejecting the old, Russellian conception of analysis, with its emphasis on precisely formulated logical forms, and replacing it with methodology which, in some cases, may degenerate into a recipe for generating a conceptual fog. It is all well and good to recognize that sometimes the concepts philosophers deal with will be vague, imprecise, and open-ended, with close conceptual connections to other concepts of the same sort. We do have to be able to deal with such cases—perhaps along the lines Ryle suggests. What is not good is a prior ideological commitment to blurred edges, indirectness, and an unwillingness to separate tangential from central issues. Sometimes Ryle and other ordinary language philosophers seem to go too far in this direction; substituting one confining orthodoxy about analysis for another. When this happens, central philosophical points get missed ... (2003, II, 80-1) { §6.1 }

Philosophical analysis is a term of art. At different times in the twentieth century, different authors have used it to mean different things. What is to be analyzed (e.g., words and sentences versus concepts and propositions), what counts as a successful analysis, and what philosophical fruits come from analysis are questions that have been vigorously debated since the dawn of analysis as a self-conscious philosophical approach. Often, different views of analysis have been linked to different views of the nature of philosophy, the sources of philosophical knowledge, the role of language in thought, the relationship between language and the world, and the nature of meaning—as well to more focused questions about necessary and apriori truth. Indeed the variety of positions is so great as to make any attempt to extract a common denominator from the multiplicity of views sterile and not illuminating.

Nevertheless analytic philosophy—with its emphasis on what is called “philosophical analysis”—is a clear and recognizable tradition. Although the common core of doctrine uniting its practitioners scarcely exceeds the platitudinous, a pattern of historical influence is not hard to discern. The tradition begins with G.E. Moore, Bertrand Russell, and Ludwig Wittgenstein (as well as Gottlob Frege, whose initial influence was largely filtered through Russell and Wittgenstein). These philosophers set the agenda, first, for logical positivists such as Rudolf Carnap, Carl Hempel, and A.J. Ayer and then later for Wittgenstein, who in turn ushered in the ordinary language school led by Gilbert Ryle and J.L. Austin. More recently the second half of the twentieth century has seen a revival of Russellian and Carnapian themes in the work of W.V. Quine, Donald Davidson, and Saul Kripke. Analytic philosophy, with its changing views of philosophical analysis, is a trail of influence ... (2005, 144) { §6.1 }

In my opinion Logical Positivism fails in its treatment of analysis. Wittgenstein and the other Logical Positivists talk much about analysis, but they do not consider the various kinds of analysis, nor do they show in what sense philosophy is the analysis of facts. They make use of analytic definition of a symbolic expression, and of the analytic clarification of a concept, but they do not distinguish between them. They also employ postulational analysis. But they do not seem to understand directional analysis, and, accordingly, they fail to apprehend the need for it. In this way they depart, in my opinion, from the practice of Moore. Not only is their conception of analysis defective, but, further, their conception of the kinds of facts to be analysed is inadequate. They treat all facts as linguistic facts . Hence, they suppose that the first problem of philosophy is to determine the principles of symbolism, and from these principles to draw limits with regard to what we can think. This assumption has two important consequences. First, it leads to the view that philosophy is ‘the activity of finding meaning’, to quote Schlick’s statement. The second consequence is that they are apt to place too much reliance upon the construction of postulational systems. (1933b, 82-3) { §6.6 }

Strawson, Peter F.

An analysis, I suppose, may be thought of as a kind of breaking down or decomposing of something. So we have the picture of a kind of intellectual taking to pieces of ideas or concepts; the discovering of what elements a concept or idea is composed and how they are related. Is this the right picture or the wrong one—or is it partly right and partly wrong? That is a question which calls for a considered response … ( Analysis and Metaphysics , Oxford: Oxford University Press, 1992, 2)

If we took this notion [of analysis as decomposition] completely seriously for the case of conceptual analysis—analysis of ideas—we should conclude that our task was to find ideas that were completely simple, that were free from internal conceptual complexity; and then to demonstrate how the more or less complex ideas that are of interest to philosophers could be assembled by a kind of logical or conceptual construction out of these simple elements. The aim would be to get a clear grasp of complex meanings by reducing them, without remainder, to simple meanings. Thus baldly stated, this may seem a rather implausible project. And so it is. Nevertheless it, or some close relation of it, has been, and is, taken seriously. Even when not taken to the lengths I have just described, it continues to exercise a certain influence on the philosophical mind. ( Ibid. 18)

Among the philosophers who were most influential in England in the period between the two world wars were the analysts. Their analytic theories were sometimes associated with the metaphysical view which Russell called logical atomism, sometimes with the supposedly anti-metaphysical doctrines of logical positivism, and sometimes, as in the case of G. E. Moore, the analytic practice had no clearly defined dogmatic background at all. But they were united at least in the view that analysis was at least one of the most important tasks of the philosopher; and by analysis they meant something which, whatever precise description of it they chose, at least involved the attempt to rewrite in different and in some way more appropriate terms those statements which they found philosophically puzzling. (1956, vii) { §6.1 }

analysis is a familiar philosophical method. I shall not attempt to offer you a complete historical account of analytic philosophy. Even the minute examination of a particular analytic philosopher, or group of analytic philosophers, would not be of great interest. I propose rather to sketch, in broad strokes, four major forms of philosophical analysis which I think important to distinguish carefully from one another. I shall call the first of these: classical analysis. It corresponds, roughly, to the traditional method of analysis used by English philosophers, a method which Russell did so much to develop. I shall then examine three other, more recent forms of philosophical analysis: (1) the type of analysis which involves the construction of artificial languages; (2) the type of analysis practiced by Wittgenstein in his later period; (3) the type of analysis which characterizes present-day Oxford Philosophy.

The fundamental notion of classical analysis is that propositions couched in ordinary language are correct, in the sense that they are not objectionable in principle. They are neither logically nor metaphysically absurd. On the other hand, insofar as the form of these propositions of ordinary language hides their true meaning, they are neither metaphysically nor logically satisfactory. The task of the analyst is, therefore, to reformulate them so that this meaning will be clearly and explicitly presented, rather then to reject them. To analyze, is to reformulate,—to translate into a better wording. (1962, 294-5) { §6.1 }

The logical positivism of the Vienna Circle did not modify the methodology of classical analysis. However, because of the anti-metaphysical standpoint which was characteristic of positivism, it could not accept the notion of the goal of analysis as metaphysical discovery. For the positivists of this school, the goal of philosophical analysis is to clarify the language of science, a clarification which would result from, for example, elucidating the relationships between observation and theory, or between scientific concepts at different levels of abstraction. ( Ibid ., 296) { §6.1 }

A second school [or third school, after ‘classical analysis’ and logical positivism] was inspired (largely, but not entirely) by the thought of Wittgenstein in his later period. Wittgenstein had himself been led tothis new point of view in his criticism of his own Tractatus Logico-Philosophicus ( Logische-Philosophische Abhandlung ), a book which itself espoused implicitly a certain form of classical analysis. According to Wittgenstein, classical analysis rested upon a false conception of language and of thought. ...

... for an analyst of this sort, philosophical problems do not result from ignorance of the precise meaning of a concept, but from an entirely false conception of its function. ... Such a false conception is what Ryle calls a “category mistake”. To resolve a philosophical problem, one should exhibit the generic character of the concepts involved in it, rather than attempting to give a perfect definition or explication of these concepts. ...

This conception of philosophical analysis—of analysis as the resolution of conceptual enigmas—has sometimes been condescendingly called “therapeutic positivism”. ( Ibid ., 297-9) { §6.1 }

The fourth method of analysis ... is that of Oxford Philosophy. ...

The analytic philosophers of the Cambridge School—for example, Russell and Wittgenstein—came to philosophy after considerable work in the sciences and in mathematics. Philosophy of mathematics was the first topic to which Russell applied his classical method of analysis. But the Oxford philosophers came to their subject, almost without exception, after extensive study of classics. Thus they were naturally interested in words, in syntax, and in (idioms. They did not wish to use linguistic analysis simply to resolve philosophical problems; they were interested in the study of language for its own sake. Therefore these philosophers are, perhaps, both more given to making linguistic distinctions, and better at finding such distinctions, than most. Ibid ., 299) { §6.1 }

Many English philosophers (including many who owe allegiance to Oxford Philosophy) would place themselves at a position between that of Wittgenstein and the view I have just sketched. It may therefore be in point to indicate briefly the principal differences between the two schools:

(1) Wittgensteinian analysis has, for its sole end, the resolution of philosophical enigmas. If there were no such enigmas, there would be no need for analysis. For Oxford, on the other hand, analysis has an intrinsic value.

(2) According to Wittgenstein and his disciples, all that is necessary is to exhibit the generic character of the concepts which we analyze. For Oxford, a minute analysis is indispensable.

(3) For Wittgenstein, analysis is the only useful method in philosophy. For Oxford, it is only one among others, and no one claims that it is sufficient, by itself, to resolve all philosophical problems. ( Ibid ., 301) { §6.1 }

It is not sensible to ask for the method of making one‘s fortune (or of ruining oneself); there are many. It is no more sensible to ask “What is the analytical method?” There is not one “analytic philosophy”. There are several. ( Ibid ., 301 [closing sentences]) { §6.1 }

The primary weapon is analysis. And analysis is the evocation of insight by the hypothetical suggestions of thought, and the evocation of thought by the activities of direct insight. In this process the composite whole, the interrelations, and the things related, concurrently emerge into clarity. ( Essays in Science and Philosophy , New York: Philosophical Library, 1947, 157)

Analysis is often understood to imply a whole of which the parts are explicitly known before the analysis; but logical elements are for our ordinary consciousness only implicit: we use them without reflecting on them, just as we use grammatical distinctions long before we have any knowledge of grammar. Logic does not merely analyse: it makes explicit what was implicit. ( Statement and Inference , Oxford: Oxford University Press, 1926, 49)

The hypothetical process therefore combines in itself both the method of discovery and the proof, and is the proper scientific exposition. The non-hypothetical proof to which we are accustomed is a sort of scientific pedantry, and it is consequently a great mistake first to give what is called analysis, which corresponds to the hypothetical process, and then to follow it by a synthesis, which is the non-hypothetical part, thus putting aside analysis as if it were a sort of accident. It is an error because it conceals the true process of thinking. ( Ibid. , 560)

I have changed my views on “atomic” complexes: I now think that qualities, relations (like love) etc. are all copulae! That means I for instance analyse a subject-predicate proposition, say, “Socrates is human” into “Socrates” and “something is human”, (which I think is not complex). The reason for this is a very fundamental one. I think that there cannot be different Types of things! In other words whatever can be symbolized by a simple proper name must belong to one type. And further: every theory of types must be rendered superfluous by a proper theory of symbolism: For instance if I analyse the proposition Socrates is mortal into Socrates, mortality and (∃x,y) ∈ 1 (x,y) I want a theory of types to tell me that “mortality is Socrates” is nonsensical, because if I treat “mortality” as a proper name (as I did) there is nothing to prevent me to make the substitution the wrong way round. But if I analyse (as I do now) into Socrates and (∃x).x is mortal or generally into x and (∃x) φx it becomes impossible to substitute the wrong way round because the two symbols are now of a different kind themselves. What I am most certain of is not however the correctness of my present way of analysis, but of the fact that all theory of types must be done away with by a theory of symbolism showing that what seem to be different kinds of things are symbolized by different kinds of symbols which cannot possibly be substituted in one another’s places. I hope I have made this fairly clear!

Propositions which I formerly wrote ∈ 2 (a,R,b) I now write R(a,b) and analyse them into a,b and (∃x,y)R(x,y) [with (∃x,y)R(x,y) marked in the text as “not complex”] ( NB , 121-2) { §6.5 }

How is it reconcilable with the task of philosophy, that logic should take care of itself? If, for example, we ask: Is such and such a fact of the subject-predicate form?, we must surely know what we mean by “subject-predicate form”. We must know whether there is such a form at all. How can we know this? “From the signs”. But how? For we haven’t got any signs of this form. We may indeed say: We have signs that behave like signs of the subject-predicate form, but does that mean that there really must be facts of this form? That is, when those signs are completely analysed? And here the question arises again: Does such a complete analysis exist? And if not : then what is the task of philosophy?!!? ( NB , 2) { §6.5 }

Our difficulty now lies in the fact that to all appearances analysability, or its opposite, is not reflected in language. That is to say: We can not , as it seems, gather from language alone whether for example there are real subject-predicate facts or not. But how COULD we express this fact or its opposite? This must be shewn . ( NB , 10) { §6.5 }

The trivial fact that a completely analysed proposition contains just as many names as there are things contained in its reference [ Bedeutung ]; this fact is an example of the all-embracing representation of the world through language. ( NB , 11) { §6.5 }

The completely analysed proposition must image its reference [ Bedeutung ]. ( NB , 18) { §6.5 }

A question: can we manage without simple objects in LOGIC?

Obviously propositions are possible which contain no simple signs, i.e. no signs which have an immediate reference [ Bedeutung ]. And these are really propositions making sense, nor do the definitions of their component parts have to be attached to them.

But it is clear that components of our propositions can be analysed by means of a definition, and must be, if we want to approximate to the real structure of the proposition. At any rate, then, there is a process of analysis . And can it not now be asked whether this process comes to an end? And if so: What will the end be?

If it is true that every defined sign signifies via its definitions then presumably the chain of definitions must some time have an end. [Cf. TLP 3.261.]

The analysed proposition mentions more than the unanalysed.

Analysis makes the proposition more complicated than it was, but it cannot and must not make it more complicated than its meaning [ Bedeutung ] was from the first.

When the proposition is just as complex as its reference [ Bedeutung ], then it is completely analysed.

But the reference [ Bedeutung ] of our propositions is not infinitely complicated. ( NB , 46) { §6.5 }

But it also seems certain that we do not infer the existence of simple objects from the existence of particular simple objects, but rather know them—by description, as it were—as the end-product of analysis, by means of a process that leads to them. ( NB , 50) { §6.5 }

Let us assume that every spatial object consists of infintely many points, then it is clear that I cannot mention all these by name when I speak of that object. Here then would be a case in which I cannot arrive at the complete analysis in the old sense at all; and perhaps just this is the usual case.

But this is surely clear: the propositions which are the only ones that humanity uses will have a sense just as they are and do not wait upon a future analysis in order to acquire a sense.

Now, however, it seems to be a legitimate question: Are–e.g.–spatial objects composed of simple parts; in analysing them, does one arrive at parts that cannot be further analysed, or is this not the case?

—But what kind of question is this?—

Is it , A PRIORI, clear that in analysing we must arrive at simple components—is this, e.g., involved in the concept of analysis— , or is analysis ad infinitum possible?—Or is there in the end even a third possibility? ( NB , 62) { §6.5 }

In a proposition a thought can be expressed in such a way that elements of the propositional sign correspond to the objects of the thought.

I call such elements ‘simple signs’, and such a proposition ‘completely analysed’. ( TLP , 3.2, 3.201) { §6.5 }

A proposition has one and only one complete analysis. ( TLP , 3.25) { §6.5 }

It is obvious that the analysis of propositions must bring us to elementary propositions which consist of names in immediate combination.

This raises the question how such combination into propositions comes about. ( TLP , 4.221) { §6.5 }

If we know on purely logical grounds that there must be elementary propositions, then everyone who understands propositions in their unanalysed form must know it. ( TLP , 5.5562) { §6.5 }

A proposition is completely logically analysed if its grammar is made completely clear: no matter what idiom it may be written or expressed in. ( PR , 51; cf. BT , 308) { §6.5 }

Logical analysis is the analysis of something we have, not of something we don’t have. Therefore it is the analysis of propositions as they stand . ( PR , 52) { §6.5 }

a mathematical proof is an analysis of the mathematical proposition. ( PR , 179) { §6.5 }

Complex is not like fact. For I can, e.g., say of a complex that it moves from one place to another, but not of a fact.

But that this complex is now situated here is a fact. ...

A complex is composed of its parts, the things of a kind which go to make it up. (This is of course a grammatical proposition concerning the words ‘complex’, ‘part’ and ‘compose’.)

To say that a red circle is composed of redness and circularity, or is a complex with these component parts, is a misuse of these words and is misleading. (Frege was aware of this and told me.) It is just as misleading to say the fact that this circle is red (that I am tired) is a complex whose component parts are a circle and redness (myself and tiredness).

Neither is a house a complex of bricks and their spatial relations; i.e. that too goes against the correct use of the word. ( PR , 301-2) { §6.5 }

When I say: “My broom is in the corner”,—is this really a statement about the broomstick and the brush? Well, it could at any rate be replaced by a statement giving the position of the stick and the position of the brush. And this statement is surely a further analysed form of the first one.—But why do I call it “further analysed”?—Well, if the broom is there, that surely means that the stick and brush must be there, and in a particular relation to one another; and this was as it were hidden in the sense of the first sentence, and is expressed in the analysed sentence. Then does someone who says that the broom is in the corner really mean: the broomstick is there, and so is the brush, and the broomstick is fixed in the brush?—If we were to ask anyone if he meant this he would probably say that he had not thought specially of the broomstick or specially of the brush at all. And that would be the right answer, for he meant to speak neither of the stick nor of the brush in particular. Suppose that, instead of saying “Bring me the broom”, you said “Bring me the broomstick and the brush which is fitted on to it.”!—Isn’t the answer: “DO you want the broom? Why do you put it so oddly?”——Is he going to understand the further analysed sentence better?—This sentence, one might say, achieves the same as the ordinary one, but in a more roundabout way.— Imagine a language-game in which someone is ordered to bring certain objects which are composed of several parts, to move them about, or something else of that kind. And two ways of playing it: in one (a) the composite objects (brooms, chairs, tables, etc.) have names, as in (15); in the other (b) only the parts are given names and the wholes are described by means of them.—In what sense is an order in the second game an analysed form of an order in the first? Does the former lie concealed in the latter, and is it now brought out by analysis?—True, the broom is taken to pieces when one separates broomstick and brush; but does it follow that the order to bring the broom also consists of corresponding parts? ...

To say, however, that a sentence in (b) is an ‘analysed’ form of one in (a) readily seduces us into thinking that the former is the more fundamental form; that it alone shews what is meant by the other, and so on. For example, we think: If you have only the unanalysed form you miss the analysis; but if you know the analysed form that gives you everything.—But can I not say that an aspect of the matter is lost on you in the latter case as well as the former? ( PI , §§ 60, 63) { §6.5 }

Our investigation is therefore a grammatical one. Such an investigation sheds light on our problem by clearing misunderstandings away. Misunderstandings concerning the use of words, caused, among other things, by certain analogies between the forms of expression in different regions of language.—Some of them can be removed by substituting one form of expression for another; this may be called an “analysis” of our forms of expression, for the process is sometimes like one of taking a thing apart.

But now it may come to look as if there were something like a final analysis of our forms of language, and so a single completely resolved form of every expression. That is, as if our usual forms of expression were, essentially, unanalysed; as if there were something hidden in them that had to be brought to light. When this is done the expression is completely clarified and our problem solved.

It can also be put like this: we eliminate misunderstandings by making our expressions more exact; but now it may look as if we were moving towards a particular state, a state of complete exactness; and as if this were the real goal of our investigation. ( PI , §§ 90-1) { §6.5 }

We are not analysing a phenomenon (e.g. thought) but a concept (e.g. that of thinking), and therefore the use of a word. ( PI , §383) { §6.5 }

A list of key works on analysis (monographs and collections) can be found in the

Annotated Bibliography, §1.2 .

Copyright © 2014 by Michael Beaney < michael . beaney @ hu-berlin . de >

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  • Terence Tao 0

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Presents a foundation text on real analysis for undergraduate and graduate students of mathematics

Emphasizes on number systems and set theory than in comparable analysis

Is the first book of a two-volume textbook on real analysis

Part of the book series: Texts and Readings in Mathematics (TRIM, volume 37)

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Table of contents (11 chapters)

Front matter, introduction.

Terence Tao

Starting at the Beginning: The Natural Numbers

Integers and rationals, the real numbers, limits of sequences, infinite sets, continuous functions on  \({{\textbf{r}}}\), differentiation of functions, the riemann integral, back matter.

This is the first book of a two-volume textbook on real analysis. Both the volumes—Analysis I and Analysis II—are intended for honors undergraduates who have already been exposed to calculus. The emphasis is on rigor and foundations. The material starts at the very beginning—the construction of number systems and set theory (Analysis I, Chaps. 1–5), then on to the basics of analysis such as limits, series, continuity, differentiation, and Riemann integration (Analysis I, Chaps. 6–11 on Euclidean spaces, and Analysis II, Chaps. 1–3 on metric spaces), through power series, several variable calculus, and Fourier analysis (Analysis II, Chaps. 4–6), and finally to the Lebesgue integral (Analysis II, Chaps. 7–8). There are appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) is in two quarters of twenty-five to thirty lectures each.

  • natural numbers
  • integers and rationals
  • real numbers
  • limits of sequences
  • continuous functions
  • differentiation of functions
  • Riemann integral

Terence Tao has been a professor of Mathematics at the University of California Los Angeles (UCLA), USA, since 1999, having completed his Ph.D. under Prof. Elias Stein at Princeton University, USA, in 1996. Tao's areas of research include harmonic analysis, partial differential equations, combinatorics, and number theory. He has received a number of awards, including the Salem Prize in 2000, the Bochner Prize in 2002, the Fields Medal in 2006, the MacArthur Fellowship in 2007, the Waterman Award in 2008, the Nemmers Prize in 2010, the Crafoord Prize in 2012, and the Breakthrough Prize in Mathematics in 2015.  Terence Tao also currently holds the James and Carol Collins chair in Mathematics at UCLA and is a fellow of the Royal Society, the Australian Academy of Sciences (the corresponding member), the National Academy of Sciences (a foreign member), and the American Academy of Arts and Sciences. He was born in Adelaide, Australia, in 1975.

Book Title : Analysis I

Book Subtitle : Fourth Edition

Authors : Terence Tao

Series Title : Texts and Readings in Mathematics

DOI : https://doi.org/10.1007/978-981-19-7261-4

Publisher : Springer Singapore

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Hindustan Book Agency 2022

eBook ISBN : 978-981-19-7261-4 Published: 22 February 2023

Series ISSN : 2366-8717

Series E-ISSN : 2366-8725

Edition Number : 2

Number of Pages : X, 340

Topics : Analysis

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Mathematical Analysis I

what is analysis 1

Elias Zakon

Copyright Year: 2004

Publisher: The Trillia Group

Language: English

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Reviewed by Kent Neuerburg, Professor of Mathematics, Southeastern on 12/9/19

This first volume offers an excellent introduction to analysis. There is easily sufficient material for a two-semester undergraduate course beginning with ideas from set theory and the real numbers, the author moves immediately to vector spaces... read more

Comprehensiveness rating: 4 see less

This first volume offers an excellent introduction to analysis. There is easily sufficient material for a two-semester undergraduate course beginning with ideas from set theory and the real numbers, the author moves immediately to vector spaces and metric spaces; in particular, covering both E^n and C^n. Once the properties of these spaces have been covered the text advances to functions, limits, and continuity while also covering basic topological ideas such as compactness and connectedness. Infinite sequences and series, including power series, are also discussed. Finally, differentiation and antidifferentiation are tackled. Topics here include L'Hopital's Rule, Taylor's series, rectifiable curves, and integral definitions of certain functions (e.g., ln(x) and the inverse trigonometric functions). More advanced topics, such as the theory of metric spaces, Riemann-Stieltjes integration, and Lebesgue theory are not covered in this volume, but are reserved for the author's Mathematical Analysis II. The index is very detailed and the entries are hyperlinked to the text.

Content Accuracy rating: 5

This book appears to be quite well-written and error-free.

Relevance/Longevity rating: 5

Mathematical analysis is a cornerstone of mathematics. As such, the content of this book is highly relevant to any mathematical scientist. The text provides a solid foundation for students of mathematics, physics, chemistry, or engineering. The level of depth and rigor is appropriate for an undergraduate audience and would form a solid basis for future study at the graduate level.

Clarity rating: 3

Originally published in 1975, this is the one area in which the text shows its age. The book leans heavily on the notation of symbolic logic. Modern students may require some extra assistance in parsing some the notation in the statements of definitions, theorems, and problems. To his credit, the author often explains in words a definition or theorem before giving the formal statement using the symbolism.

Consistency rating: 5

The vocabulary and notation used in this text is standard. The author maintains consistent notation and vocabulary throughout the text making it easy to transition from one topic to the next. The author does extensively utilize the notation of symbolic logic; however, this notation is introduced in the very first chapter during a review of the basics of set theory.

Modularity rating: 5

The book is broken into sections of reasonable length each covering a particular topic. Problem sets follow every few sections and are focused on the content of those sections. As is true for most mathematics texts, this book should be treated in a sequential manner; however, the author does note those sections that can be omitted without a loss of continuity. Also, with appropriate development of notation, individual chapters of this book could be used in conjunction with or to supplement another text.

Organization/Structure/Flow rating: 5

The text is organized in five chapters each having several chapters (from nine to seventeen). The book is designed to be covered in the order written, though the author does indicate those sections that can (or even should) be omitted from an introductory course. The topics are developed in a logical order as the text builds later concepts from earlier ideas in a traditional manner.

Interface rating: 5

The presentation and typesetting of formulas, equations, etc. is very well-done. The figures, graphs, and images are clear and integrated into the text. References to earlier results as well as entries in the index are hyperlinked to allow the reader to easily navigate the text.

Grammatical Errors rating: 5

I found no grammatical errors in the text.

Cultural Relevance rating: 5

Since this is a book on abstract mathematics, there are no cultural references in the text.

Table of Contents

  • Chapter 1. Set Theory
  • Chapter 2. Real Numbers. Fields
  • Chapter 3. Vector Spaces. Metric Spaces
  • Chapter 4. Function Limits and Continuity
  • Chapter 5. Differentiation and Antidifferentiation

Ancillary Material

About the book.

This award-winning text carefully leads the student through the basic topics of Real Analysis. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, Taylor's theorem, total variation, rectifiable arcs, and sufficient conditions of integrability. Well over 500 exercises (many with extensive hints) assist students through the material.

For students who need a review of basic mathematical concepts before beginning "epsilon-delta"-style proofs, the text begins with material on set theory (sets, quantifiers, relations and mappings, countable sets), the real numbers (axioms, natural numbers, induction, consequences of the completeness axiom), and Euclidean and vector spaces; this material is condensed from the author's Basic Concepts of Mathematics, the complete version of which can be used as supplementary background material for the present text.

About the Contributors

Elias Zakon, As a research fellow at the University of Toronto, he worked with Abraham Robinson. In 1957, he joined the mathematics faculty at the University of Windsor, where the first degrees in the newly established Honours program in Mathematics were awarded in1960. While at Windsor, he continued publishing his research results in logic and analysis. In this post-McCarthy era, he often had as his house-guest the prolific and eccentric mathematician Paul Erdos, who was then banned from the United States for his political views. Erdos would speak at the University of Windsor, where mathematicians from the University of Michigan and other American universities would gather to hear him and to discuss mathematics. While at Windsor, Zakon developed three volumes on mathematical analysis,which were bound and distributed to students. His goal was to introduce rigorous material as early as possible; later courses could then rely on this material. We are publishing here the latest complete version of the second of these volumes, which was used in a two-semester class required of all second-year Honours Mathematics students at Windsor.

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Module 8: Analysis and Synthesis

What is analysis, learning objective.

  • Explain the basics of analysis

Diagram of three parts in a whole.

Figure 1 . Analysis consists of breaking something down and taking a close look at each of its parts while looking for themes, patterns, and assumptions.

Critical thinking skill analysis is the process of methodically breaking something down to gain a better understanding of it. Analysis also includes the ability to connect pieces of information as the basis for generalization or explanation. Analytical assignments in college often couple analysis with the critical thinking skills of interpretation and evaluation.

Analysis can be applied to content but can also cover form, function, and context. For example, an analysis assignment in an art appreciation class might ask you to analyze the subject and iconography of a painting, but also expect you to analyze the use of shape, space, color, and texture (form), as well as the artist’s intended purpose (function) and the culture or time period in which the work was created (context).

While each academic discipline characterizes the analytic process to suit its needs, the essential skills of analysis are the following:

  • Breaking down information or artifacts into component parts
  • Uncovering relationships among those parts
  • Determining motives, causes, and underlying assumptions
  • Making inferences and finding evidence to support generalizations

The Language of Analytical Assignments

Although analysis is ubiquitous in college, students sometimes fail to recognize when they are being asked to apply analysis. Often that confusion stems from differences in vocabulary across different disciplines.

For example, each of the verbs in the following list may indicate or denote some type of analysis:

Although this list is a good start, these aren’t the only verbs that signal something is happening with analysis. Another way to tell whether an assignment is asking for analysis is this: If the assignment asks you to determine how the parts of something relate to the whole, how something works, what something means, or why it’s important , the assignment is asking you to analyze. Below is a list of sample analytic assignments that meet these criteria.

How the parts relate to the whole:

  • Classify problems to identify the appropriate algorithms.
  • Determine how well a feminist interpretation is supported by evidence contained in a work.

How something works:

  • Recognize flaws, inconsistencies, and logical fallacies in an opinion editorial.
  • Distinguish between facts and assumptions in a scientific report.

What something means:

  • Interpret quantitative relationships in a graph.
  • Analyze data/situations to identify root problems.

Why something is important:

  • Assess alternative solutions to the health care crisis.
  • Separate relevant from irrelevant information in testimony.

analysis : the process of methodically breaking something down to gain a better understanding of it

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  • What Is Analysis? and The Language of Analytic Assignments. Authored by : Karen Forgette. Provided by : University of Mississippi. License : CC BY-SA: Attribution-ShareAlike

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Definition of analysis noun from the Oxford Advanced American Dictionary

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Find the answers with Practical English Usage online, your indispensable guide to problems in English.

  • 3 [ uncountable ] = psychoanalysis In analysis, the individual resolves difficult emotional conflicts.

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Definition of analysis

  • anatomizing
  • deconstruction

Examples of analysis in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'analysis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

borrowed from Medieval Latin, borrowed from Greek análysis "loosing, releasing, breaking something down into its elements, solution of a problem," from analýein "to loosen, undo, dissolve, resolve into constituent elements," from ana- ana- + lýein "to loosen, undo" — more at lose

1581, in the meaning defined at sense 2

Phrases Containing analysis

analysis of variance

  • activation analysis
  • content analysis
  • analysis situs
  • in the final / last analysis
  • defy analysis
  • systems analysis
  • self - analysis
  • quantitative analysis
  • Fourier analysis
  • factor analysis
  • discourse analysis
  • numerical analysis
  • neutron activation analysis
  • meta - analysis
  • cluster analysis
  • transactional analysis
  • qualitative analysis
  • high analysis
  • harmonic analysis
  • philosophical analysis

Dictionary Entries Near analysis

Cite this entry.

“Analysis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/analysis. Accessed 22 Feb. 2024.

Kids Definition

Kids definition of analysis.

derived from Greek, from analyein "to break up," from ana - "up" and lyein "to loosen"

Medical Definition

Medical definition of analysis, more from merriam-webster on analysis.

Nglish: Translation of analysis for Spanish Speakers

Britannica English: Translation of analysis for Arabic Speakers

Britannica.com: Encyclopedia article about analysis

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Your Modern Business Guide To Data Analysis Methods And Techniques

Data analysis methods and techniques blog post by datapine

Table of Contents

1) What Is Data Analysis?

2) Why Is Data Analysis Important?

3) What Is The Data Analysis Process?

4) Types Of Data Analysis Methods

5) Top Data Analysis Techniques To Apply

6) Quality Criteria For Data Analysis

7) Data Analysis Limitations & Barriers

8) Data Analysis Skills

9) Data Analysis In The Big Data Environment

In our data-rich age, understanding how to analyze and extract true meaning from our business’s digital insights is one of the primary drivers of success.

Despite the colossal volume of data we create every day, a mere 0.5% is actually analyzed and used for data discovery , improvement, and intelligence. While that may not seem like much, considering the amount of digital information we have at our fingertips, half a percent still accounts for a vast amount of data.

With so much data and so little time, knowing how to collect, curate, organize, and make sense of all of this potentially business-boosting information can be a minefield – but online data analysis is the solution.

In science, data analysis uses a more complex approach with advanced techniques to explore and experiment with data. On the other hand, in a business context, data is used to make data-driven decisions that will enable the company to improve its overall performance. In this post, we will cover the analysis of data from an organizational point of view while still going through the scientific and statistical foundations that are fundamental to understanding the basics of data analysis. 

To put all of that into perspective, we will answer a host of important analytical questions, explore analytical methods and techniques, while demonstrating how to perform analysis in the real world with a 17-step blueprint for success.

What Is Data Analysis?

Data analysis is the process of collecting, modeling, and analyzing data using various statistical and logical methods and techniques. Businesses rely on analytics processes and tools to extract insights that support strategic and operational decision-making.

All these various methods are largely based on two core areas: quantitative and qualitative research.

To explain the key differences between qualitative and quantitative research, here’s a video for your viewing pleasure:

Gaining a better understanding of different techniques and methods in quantitative research as well as qualitative insights will give your analyzing efforts a more clearly defined direction, so it’s worth taking the time to allow this particular knowledge to sink in. Additionally, you will be able to create a comprehensive analytical report that will skyrocket your analysis.

Apart from qualitative and quantitative categories, there are also other types of data that you should be aware of before dividing into complex data analysis processes. These categories include: 

  • Big data: Refers to massive data sets that need to be analyzed using advanced software to reveal patterns and trends. It is considered to be one of the best analytical assets as it provides larger volumes of data at a faster rate. 
  • Metadata: Putting it simply, metadata is data that provides insights about other data. It summarizes key information about specific data that makes it easier to find and reuse for later purposes. 
  • Real time data: As its name suggests, real time data is presented as soon as it is acquired. From an organizational perspective, this is the most valuable data as it can help you make important decisions based on the latest developments. Our guide on real time analytics will tell you more about the topic. 
  • Machine data: This is more complex data that is generated solely by a machine such as phones, computers, or even websites and embedded systems, without previous human interaction.

Why Is Data Analysis Important?

Before we go into detail about the categories of analysis along with its methods and techniques, you must understand the potential that analyzing data can bring to your organization.

  • Informed decision-making : From a management perspective, you can benefit from analyzing your data as it helps you make decisions based on facts and not simple intuition. For instance, you can understand where to invest your capital, detect growth opportunities, predict your income, or tackle uncommon situations before they become problems. Through this, you can extract relevant insights from all areas in your organization, and with the help of dashboard software , present the data in a professional and interactive way to different stakeholders.
  • Reduce costs : Another great benefit is to reduce costs. With the help of advanced technologies such as predictive analytics, businesses can spot improvement opportunities, trends, and patterns in their data and plan their strategies accordingly. In time, this will help you save money and resources on implementing the wrong strategies. And not just that, by predicting different scenarios such as sales and demand you can also anticipate production and supply. 
  • Target customers better : Customers are arguably the most crucial element in any business. By using analytics to get a 360° vision of all aspects related to your customers, you can understand which channels they use to communicate with you, their demographics, interests, habits, purchasing behaviors, and more. In the long run, it will drive success to your marketing strategies, allow you to identify new potential customers, and avoid wasting resources on targeting the wrong people or sending the wrong message. You can also track customer satisfaction by analyzing your client’s reviews or your customer service department’s performance.

What Is The Data Analysis Process?

Data analysis process graphic

When we talk about analyzing data there is an order to follow in order to extract the needed conclusions. The analysis process consists of 5 key stages. We will cover each of them more in detail later in the post, but to start providing the needed context to understand what is coming next, here is a rundown of the 5 essential steps of data analysis. 

  • Identify: Before you get your hands dirty with data, you first need to identify why you need it in the first place. The identification is the stage in which you establish the questions you will need to answer. For example, what is the customer's perception of our brand? Or what type of packaging is more engaging to our potential customers? Once the questions are outlined you are ready for the next step. 
  • Collect: As its name suggests, this is the stage where you start collecting the needed data. Here, you define which sources of data you will use and how you will use them. The collection of data can come in different forms such as internal or external sources, surveys, interviews, questionnaires, and focus groups, among others.  An important note here is that the way you collect the data will be different in a quantitative and qualitative scenario. 
  • Clean: Once you have the necessary data it is time to clean it and leave it ready for analysis. Not all the data you collect will be useful, when collecting big amounts of data in different formats it is very likely that you will find yourself with duplicate or badly formatted data. To avoid this, before you start working with your data you need to make sure to erase any white spaces, duplicate records, or formatting errors. This way you avoid hurting your analysis with bad-quality data. 
  • Analyze : With the help of various techniques such as statistical analysis, regressions, neural networks, text analysis, and more, you can start analyzing and manipulating your data to extract relevant conclusions. At this stage, you find trends, correlations, variations, and patterns that can help you answer the questions you first thought of in the identify stage. Various technologies in the market assist researchers and average users with the management of their data. Some of them include business intelligence and visualization software, predictive analytics, and data mining, among others. 
  • Interpret: Last but not least you have one of the most important steps: it is time to interpret your results. This stage is where the researcher comes up with courses of action based on the findings. For example, here you would understand if your clients prefer packaging that is red or green, plastic or paper, etc. Additionally, at this stage, you can also find some limitations and work on them. 

Now that you have a basic understanding of the key data analysis steps, let’s look at the top 17 essential methods.

17 Essential Types Of Data Analysis Methods

Before diving into the 17 essential types of methods, it is important that we go over really fast through the main analysis categories. Starting with the category of descriptive up to prescriptive analysis, the complexity and effort of data evaluation increases, but also the added value for the company.

a) Descriptive analysis - What happened.

The descriptive analysis method is the starting point for any analytic reflection, and it aims to answer the question of what happened? It does this by ordering, manipulating, and interpreting raw data from various sources to turn it into valuable insights for your organization.

Performing descriptive analysis is essential, as it enables us to present our insights in a meaningful way. Although it is relevant to mention that this analysis on its own will not allow you to predict future outcomes or tell you the answer to questions like why something happened, it will leave your data organized and ready to conduct further investigations.

b) Exploratory analysis - How to explore data relationships.

As its name suggests, the main aim of the exploratory analysis is to explore. Prior to it, there is still no notion of the relationship between the data and the variables. Once the data is investigated, exploratory analysis helps you to find connections and generate hypotheses and solutions for specific problems. A typical area of ​​application for it is data mining.

c) Diagnostic analysis - Why it happened.

Diagnostic data analytics empowers analysts and executives by helping them gain a firm contextual understanding of why something happened. If you know why something happened as well as how it happened, you will be able to pinpoint the exact ways of tackling the issue or challenge.

Designed to provide direct and actionable answers to specific questions, this is one of the world’s most important methods in research, among its other key organizational functions such as retail analytics , e.g.

c) Predictive analysis - What will happen.

The predictive method allows you to look into the future to answer the question: what will happen? In order to do this, it uses the results of the previously mentioned descriptive, exploratory, and diagnostic analysis, in addition to machine learning (ML) and artificial intelligence (AI). Through this, you can uncover future trends, potential problems or inefficiencies, connections, and casualties in your data.

With predictive analysis, you can unfold and develop initiatives that will not only enhance your various operational processes but also help you gain an all-important edge over the competition. If you understand why a trend, pattern, or event happened through data, you will be able to develop an informed projection of how things may unfold in particular areas of the business.

e) Prescriptive analysis - How will it happen.

Another of the most effective types of analysis methods in research. Prescriptive data techniques cross over from predictive analysis in the way that it revolves around using patterns or trends to develop responsive, practical business strategies.

By drilling down into prescriptive analysis, you will play an active role in the data consumption process by taking well-arranged sets of visual data and using it as a powerful fix to emerging issues in a number of key areas, including marketing, sales, customer experience, HR, fulfillment, finance, logistics analytics , and others.

Top 17 data analysis methods

As mentioned at the beginning of the post, data analysis methods can be divided into two big categories: quantitative and qualitative. Each of these categories holds a powerful analytical value that changes depending on the scenario and type of data you are working with. Below, we will discuss 17 methods that are divided into qualitative and quantitative approaches. 

Without further ado, here are the 17 essential types of data analysis methods with some use cases in the business world: 

A. Quantitative Methods 

To put it simply, quantitative analysis refers to all methods that use numerical data or data that can be turned into numbers (e.g. category variables like gender, age, etc.) to extract valuable insights. It is used to extract valuable conclusions about relationships, differences, and test hypotheses. Below we discuss some of the key quantitative methods. 

1. Cluster analysis

The action of grouping a set of data elements in a way that said elements are more similar (in a particular sense) to each other than to those in other groups – hence the term ‘cluster.’ Since there is no target variable when clustering, the method is often used to find hidden patterns in the data. The approach is also used to provide additional context to a trend or dataset.

Let's look at it from an organizational perspective. In a perfect world, marketers would be able to analyze each customer separately and give them the best-personalized service, but let's face it, with a large customer base, it is timely impossible to do that. That's where clustering comes in. By grouping customers into clusters based on demographics, purchasing behaviors, monetary value, or any other factor that might be relevant for your company, you will be able to immediately optimize your efforts and give your customers the best experience based on their needs.

2. Cohort analysis

This type of data analysis approach uses historical data to examine and compare a determined segment of users' behavior, which can then be grouped with others with similar characteristics. By using this methodology, it's possible to gain a wealth of insight into consumer needs or a firm understanding of a broader target group.

Cohort analysis can be really useful for performing analysis in marketing as it will allow you to understand the impact of your campaigns on specific groups of customers. To exemplify, imagine you send an email campaign encouraging customers to sign up for your site. For this, you create two versions of the campaign with different designs, CTAs, and ad content. Later on, you can use cohort analysis to track the performance of the campaign for a longer period of time and understand which type of content is driving your customers to sign up, repurchase, or engage in other ways.  

A useful tool to start performing cohort analysis method is Google Analytics. You can learn more about the benefits and limitations of using cohorts in GA in this useful guide . In the bottom image, you see an example of how you visualize a cohort in this tool. The segments (devices traffic) are divided into date cohorts (usage of devices) and then analyzed week by week to extract insights into performance.

Cohort analysis chart example from google analytics

3. Regression analysis

Regression uses historical data to understand how a dependent variable's value is affected when one (linear regression) or more independent variables (multiple regression) change or stay the same. By understanding each variable's relationship and how it developed in the past, you can anticipate possible outcomes and make better decisions in the future.

Let's bring it down with an example. Imagine you did a regression analysis of your sales in 2019 and discovered that variables like product quality, store design, customer service, marketing campaigns, and sales channels affected the overall result. Now you want to use regression to analyze which of these variables changed or if any new ones appeared during 2020. For example, you couldn’t sell as much in your physical store due to COVID lockdowns. Therefore, your sales could’ve either dropped in general or increased in your online channels. Through this, you can understand which independent variables affected the overall performance of your dependent variable, annual sales.

If you want to go deeper into this type of analysis, check out this article and learn more about how you can benefit from regression.

4. Neural networks

The neural network forms the basis for the intelligent algorithms of machine learning. It is a form of analytics that attempts, with minimal intervention, to understand how the human brain would generate insights and predict values. Neural networks learn from each and every data transaction, meaning that they evolve and advance over time.

A typical area of application for neural networks is predictive analytics. There are BI reporting tools that have this feature implemented within them, such as the Predictive Analytics Tool from datapine. This tool enables users to quickly and easily generate all kinds of predictions. All you have to do is select the data to be processed based on your KPIs, and the software automatically calculates forecasts based on historical and current data. Thanks to its user-friendly interface, anyone in your organization can manage it; there’s no need to be an advanced scientist. 

Here is an example of how you can use the predictive analysis tool from datapine:

Example on how to use predictive analytics tool from datapine

**click to enlarge**

5. Factor analysis

The factor analysis also called “dimension reduction” is a type of data analysis used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. The aim here is to uncover independent latent variables, an ideal method for streamlining specific segments.

A good way to understand this data analysis method is a customer evaluation of a product. The initial assessment is based on different variables like color, shape, wearability, current trends, materials, comfort, the place where they bought the product, and frequency of usage. Like this, the list can be endless, depending on what you want to track. In this case, factor analysis comes into the picture by summarizing all of these variables into homogenous groups, for example, by grouping the variables color, materials, quality, and trends into a brother latent variable of design.

If you want to start analyzing data using factor analysis we recommend you take a look at this practical guide from UCLA.

6. Data mining

A method of data analysis that is the umbrella term for engineering metrics and insights for additional value, direction, and context. By using exploratory statistical evaluation, data mining aims to identify dependencies, relations, patterns, and trends to generate advanced knowledge.  When considering how to analyze data, adopting a data mining mindset is essential to success - as such, it’s an area that is worth exploring in greater detail.

An excellent use case of data mining is datapine intelligent data alerts . With the help of artificial intelligence and machine learning, they provide automated signals based on particular commands or occurrences within a dataset. For example, if you’re monitoring supply chain KPIs , you could set an intelligent alarm to trigger when invalid or low-quality data appears. By doing so, you will be able to drill down deep into the issue and fix it swiftly and effectively.

In the following picture, you can see how the intelligent alarms from datapine work. By setting up ranges on daily orders, sessions, and revenues, the alarms will notify you if the goal was not completed or if it exceeded expectations.

Example on how to use intelligent alerts from datapine

7. Time series analysis

As its name suggests, time series analysis is used to analyze a set of data points collected over a specified period of time. Although analysts use this method to monitor the data points in a specific interval of time rather than just monitoring them intermittently, the time series analysis is not uniquely used for the purpose of collecting data over time. Instead, it allows researchers to understand if variables changed during the duration of the study, how the different variables are dependent, and how did it reach the end result. 

In a business context, this method is used to understand the causes of different trends and patterns to extract valuable insights. Another way of using this method is with the help of time series forecasting. Powered by predictive technologies, businesses can analyze various data sets over a period of time and forecast different future events. 

A great use case to put time series analysis into perspective is seasonality effects on sales. By using time series forecasting to analyze sales data of a specific product over time, you can understand if sales rise over a specific period of time (e.g. swimwear during summertime, or candy during Halloween). These insights allow you to predict demand and prepare production accordingly.  

8. Decision Trees 

The decision tree analysis aims to act as a support tool to make smart and strategic decisions. By visually displaying potential outcomes, consequences, and costs in a tree-like model, researchers and company users can easily evaluate all factors involved and choose the best course of action. Decision trees are helpful to analyze quantitative data and they allow for an improved decision-making process by helping you spot improvement opportunities, reduce costs, and enhance operational efficiency and production.

But how does a decision tree actually works? This method works like a flowchart that starts with the main decision that you need to make and branches out based on the different outcomes and consequences of each decision. Each outcome will outline its own consequences, costs, and gains and, at the end of the analysis, you can compare each of them and make the smartest decision. 

Businesses can use them to understand which project is more cost-effective and will bring more earnings in the long run. For example, imagine you need to decide if you want to update your software app or build a new app entirely.  Here you would compare the total costs, the time needed to be invested, potential revenue, and any other factor that might affect your decision.  In the end, you would be able to see which of these two options is more realistic and attainable for your company or research.

9. Conjoint analysis 

Last but not least, we have the conjoint analysis. This approach is usually used in surveys to understand how individuals value different attributes of a product or service and it is one of the most effective methods to extract consumer preferences. When it comes to purchasing, some clients might be more price-focused, others more features-focused, and others might have a sustainable focus. Whatever your customer's preferences are, you can find them with conjoint analysis. Through this, companies can define pricing strategies, packaging options, subscription packages, and more. 

A great example of conjoint analysis is in marketing and sales. For instance, a cupcake brand might use conjoint analysis and find that its clients prefer gluten-free options and cupcakes with healthier toppings over super sugary ones. Thus, the cupcake brand can turn these insights into advertisements and promotions to increase sales of this particular type of product. And not just that, conjoint analysis can also help businesses segment their customers based on their interests. This allows them to send different messaging that will bring value to each of the segments. 

10. Correspondence Analysis

Also known as reciprocal averaging, correspondence analysis is a method used to analyze the relationship between categorical variables presented within a contingency table. A contingency table is a table that displays two (simple correspondence analysis) or more (multiple correspondence analysis) categorical variables across rows and columns that show the distribution of the data, which is usually answers to a survey or questionnaire on a specific topic. 

This method starts by calculating an “expected value” which is done by multiplying row and column averages and dividing it by the overall original value of the specific table cell. The “expected value” is then subtracted from the original value resulting in a “residual number” which is what allows you to extract conclusions about relationships and distribution. The results of this analysis are later displayed using a map that represents the relationship between the different values. The closest two values are in the map, the bigger the relationship. Let’s put it into perspective with an example. 

Imagine you are carrying out a market research analysis about outdoor clothing brands and how they are perceived by the public. For this analysis, you ask a group of people to match each brand with a certain attribute which can be durability, innovation, quality materials, etc. When calculating the residual numbers, you can see that brand A has a positive residual for innovation but a negative one for durability. This means that brand A is not positioned as a durable brand in the market, something that competitors could take advantage of. 

11. Multidimensional Scaling (MDS)

MDS is a method used to observe the similarities or disparities between objects which can be colors, brands, people, geographical coordinates, and more. The objects are plotted using an “MDS map” that positions similar objects together and disparate ones far apart. The (dis) similarities between objects are represented using one or more dimensions that can be observed using a numerical scale. For example, if you want to know how people feel about the COVID-19 vaccine, you can use 1 for “don’t believe in the vaccine at all”  and 10 for “firmly believe in the vaccine” and a scale of 2 to 9 for in between responses.  When analyzing an MDS map the only thing that matters is the distance between the objects, the orientation of the dimensions is arbitrary and has no meaning at all. 

Multidimensional scaling is a valuable technique for market research, especially when it comes to evaluating product or brand positioning. For instance, if a cupcake brand wants to know how they are positioned compared to competitors, it can define 2-3 dimensions such as taste, ingredients, shopping experience, or more, and do a multidimensional scaling analysis to find improvement opportunities as well as areas in which competitors are currently leading. 

Another business example is in procurement when deciding on different suppliers. Decision makers can generate an MDS map to see how the different prices, delivery times, technical services, and more of the different suppliers differ and pick the one that suits their needs the best. 

A final example proposed by a research paper on "An Improved Study of Multilevel Semantic Network Visualization for Analyzing Sentiment Word of Movie Review Data". Researchers picked a two-dimensional MDS map to display the distances and relationships between different sentiments in movie reviews. They used 36 sentiment words and distributed them based on their emotional distance as we can see in the image below where the words "outraged" and "sweet" are on opposite sides of the map, marking the distance between the two emotions very clearly.

Example of multidimensional scaling analysis

Aside from being a valuable technique to analyze dissimilarities, MDS also serves as a dimension-reduction technique for large dimensional data. 

B. Qualitative Methods

Qualitative data analysis methods are defined as the observation of non-numerical data that is gathered and produced using methods of observation such as interviews, focus groups, questionnaires, and more. As opposed to quantitative methods, qualitative data is more subjective and highly valuable in analyzing customer retention and product development.

12. Text analysis

Text analysis, also known in the industry as text mining, works by taking large sets of textual data and arranging them in a way that makes it easier to manage. By working through this cleansing process in stringent detail, you will be able to extract the data that is truly relevant to your organization and use it to develop actionable insights that will propel you forward.

Modern software accelerate the application of text analytics. Thanks to the combination of machine learning and intelligent algorithms, you can perform advanced analytical processes such as sentiment analysis. This technique allows you to understand the intentions and emotions of a text, for example, if it's positive, negative, or neutral, and then give it a score depending on certain factors and categories that are relevant to your brand. Sentiment analysis is often used to monitor brand and product reputation and to understand how successful your customer experience is. To learn more about the topic check out this insightful article .

By analyzing data from various word-based sources, including product reviews, articles, social media communications, and survey responses, you will gain invaluable insights into your audience, as well as their needs, preferences, and pain points. This will allow you to create campaigns, services, and communications that meet your prospects’ needs on a personal level, growing your audience while boosting customer retention. There are various other “sub-methods” that are an extension of text analysis. Each of them serves a more specific purpose and we will look at them in detail next. 

13. Content Analysis

This is a straightforward and very popular method that examines the presence and frequency of certain words, concepts, and subjects in different content formats such as text, image, audio, or video. For example, the number of times the name of a celebrity is mentioned on social media or online tabloids. It does this by coding text data that is later categorized and tabulated in a way that can provide valuable insights, making it the perfect mix of quantitative and qualitative analysis.

There are two types of content analysis. The first one is the conceptual analysis which focuses on explicit data, for instance, the number of times a concept or word is mentioned in a piece of content. The second one is relational analysis, which focuses on the relationship between different concepts or words and how they are connected within a specific context. 

Content analysis is often used by marketers to measure brand reputation and customer behavior. For example, by analyzing customer reviews. It can also be used to analyze customer interviews and find directions for new product development. It is also important to note, that in order to extract the maximum potential out of this analysis method, it is necessary to have a clearly defined research question. 

14. Thematic Analysis

Very similar to content analysis, thematic analysis also helps in identifying and interpreting patterns in qualitative data with the main difference being that the first one can also be applied to quantitative analysis. The thematic method analyzes large pieces of text data such as focus group transcripts or interviews and groups them into themes or categories that come up frequently within the text. It is a great method when trying to figure out peoples view’s and opinions about a certain topic. For example, if you are a brand that cares about sustainability, you can do a survey of your customers to analyze their views and opinions about sustainability and how they apply it to their lives. You can also analyze customer service calls transcripts to find common issues and improve your service. 

Thematic analysis is a very subjective technique that relies on the researcher’s judgment. Therefore,  to avoid biases, it has 6 steps that include familiarization, coding, generating themes, reviewing themes, defining and naming themes, and writing up. It is also important to note that, because it is a flexible approach, the data can be interpreted in multiple ways and it can be hard to select what data is more important to emphasize. 

15. Narrative Analysis 

A bit more complex in nature than the two previous ones, narrative analysis is used to explore the meaning behind the stories that people tell and most importantly, how they tell them. By looking into the words that people use to describe a situation you can extract valuable conclusions about their perspective on a specific topic. Common sources for narrative data include autobiographies, family stories, opinion pieces, and testimonials, among others. 

From a business perspective, narrative analysis can be useful to analyze customer behaviors and feelings towards a specific product, service, feature, or others. It provides unique and deep insights that can be extremely valuable. However, it has some drawbacks.  

The biggest weakness of this method is that the sample sizes are usually very small due to the complexity and time-consuming nature of the collection of narrative data. Plus, the way a subject tells a story will be significantly influenced by his or her specific experiences, making it very hard to replicate in a subsequent study. 

16. Discourse Analysis

Discourse analysis is used to understand the meaning behind any type of written, verbal, or symbolic discourse based on its political, social, or cultural context. It mixes the analysis of languages and situations together. This means that the way the content is constructed and the meaning behind it is significantly influenced by the culture and society it takes place in. For example, if you are analyzing political speeches you need to consider different context elements such as the politician's background, the current political context of the country, the audience to which the speech is directed, and so on. 

From a business point of view, discourse analysis is a great market research tool. It allows marketers to understand how the norms and ideas of the specific market work and how their customers relate to those ideas. It can be very useful to build a brand mission or develop a unique tone of voice. 

17. Grounded Theory Analysis

Traditionally, researchers decide on a method and hypothesis and start to collect the data to prove that hypothesis. The grounded theory is the only method that doesn’t require an initial research question or hypothesis as its value lies in the generation of new theories. With the grounded theory method, you can go into the analysis process with an open mind and explore the data to generate new theories through tests and revisions. In fact, it is not necessary to collect the data and then start to analyze it. Researchers usually start to find valuable insights as they are gathering the data. 

All of these elements make grounded theory a very valuable method as theories are fully backed by data instead of initial assumptions. It is a great technique to analyze poorly researched topics or find the causes behind specific company outcomes. For example, product managers and marketers might use the grounded theory to find the causes of high levels of customer churn and look into customer surveys and reviews to develop new theories about the causes. 

How To Analyze Data? Top 17 Data Analysis Techniques To Apply

17 top data analysis techniques by datapine

Now that we’ve answered the questions “what is data analysis’”, why is it important, and covered the different data analysis types, it’s time to dig deeper into how to perform your analysis by working through these 17 essential techniques.

1. Collaborate your needs

Before you begin analyzing or drilling down into any techniques, it’s crucial to sit down collaboratively with all key stakeholders within your organization, decide on your primary campaign or strategic goals, and gain a fundamental understanding of the types of insights that will best benefit your progress or provide you with the level of vision you need to evolve your organization.

2. Establish your questions

Once you’ve outlined your core objectives, you should consider which questions will need answering to help you achieve your mission. This is one of the most important techniques as it will shape the very foundations of your success.

To help you ask the right things and ensure your data works for you, you have to ask the right data analysis questions .

3. Data democratization

After giving your data analytics methodology some real direction, and knowing which questions need answering to extract optimum value from the information available to your organization, you should continue with democratization.

Data democratization is an action that aims to connect data from various sources efficiently and quickly so that anyone in your organization can access it at any given moment. You can extract data in text, images, videos, numbers, or any other format. And then perform cross-database analysis to achieve more advanced insights to share with the rest of the company interactively.  

Once you have decided on your most valuable sources, you need to take all of this into a structured format to start collecting your insights. For this purpose, datapine offers an easy all-in-one data connectors feature to integrate all your internal and external sources and manage them at your will. Additionally, datapine’s end-to-end solution automatically updates your data, allowing you to save time and focus on performing the right analysis to grow your company.

data connectors from datapine

4. Think of governance 

When collecting data in a business or research context you always need to think about security and privacy. With data breaches becoming a topic of concern for businesses, the need to protect your client's or subject’s sensitive information becomes critical. 

To ensure that all this is taken care of, you need to think of a data governance strategy. According to Gartner , this concept refers to “ the specification of decision rights and an accountability framework to ensure the appropriate behavior in the valuation, creation, consumption, and control of data and analytics .” In simpler words, data governance is a collection of processes, roles, and policies, that ensure the efficient use of data while still achieving the main company goals. It ensures that clear roles are in place for who can access the information and how they can access it. In time, this not only ensures that sensitive information is protected but also allows for an efficient analysis as a whole. 

5. Clean your data

After harvesting from so many sources you will be left with a vast amount of information that can be overwhelming to deal with. At the same time, you can be faced with incorrect data that can be misleading to your analysis. The smartest thing you can do to avoid dealing with this in the future is to clean the data. This is fundamental before visualizing it, as it will ensure that the insights you extract from it are correct.

There are many things that you need to look for in the cleaning process. The most important one is to eliminate any duplicate observations; this usually appears when using multiple internal and external sources of information. You can also add any missing codes, fix empty fields, and eliminate incorrectly formatted data.

Another usual form of cleaning is done with text data. As we mentioned earlier, most companies today analyze customer reviews, social media comments, questionnaires, and several other text inputs. In order for algorithms to detect patterns, text data needs to be revised to avoid invalid characters or any syntax or spelling errors. 

Most importantly, the aim of cleaning is to prevent you from arriving at false conclusions that can damage your company in the long run. By using clean data, you will also help BI solutions to interact better with your information and create better reports for your organization.

6. Set your KPIs

Once you’ve set your sources, cleaned your data, and established clear-cut questions you want your insights to answer, you need to set a host of key performance indicators (KPIs) that will help you track, measure, and shape your progress in a number of key areas.

KPIs are critical to both qualitative and quantitative analysis research. This is one of the primary methods of data analysis you certainly shouldn’t overlook.

To help you set the best possible KPIs for your initiatives and activities, here is an example of a relevant logistics KPI : transportation-related costs. If you want to see more go explore our collection of key performance indicator examples .

Transportation costs logistics KPIs

7. Omit useless data

Having bestowed your data analysis tools and techniques with true purpose and defined your mission, you should explore the raw data you’ve collected from all sources and use your KPIs as a reference for chopping out any information you deem to be useless.

Trimming the informational fat is one of the most crucial methods of analysis as it will allow you to focus your analytical efforts and squeeze every drop of value from the remaining ‘lean’ information.

Any stats, facts, figures, or metrics that don’t align with your business goals or fit with your KPI management strategies should be eliminated from the equation.

8. Build a data management roadmap

While, at this point, this particular step is optional (you will have already gained a wealth of insight and formed a fairly sound strategy by now), creating a data governance roadmap will help your data analysis methods and techniques become successful on a more sustainable basis. These roadmaps, if developed properly, are also built so they can be tweaked and scaled over time.

Invest ample time in developing a roadmap that will help you store, manage, and handle your data internally, and you will make your analysis techniques all the more fluid and functional – one of the most powerful types of data analysis methods available today.

9. Integrate technology

There are many ways to analyze data, but one of the most vital aspects of analytical success in a business context is integrating the right decision support software and technology.

Robust analysis platforms will not only allow you to pull critical data from your most valuable sources while working with dynamic KPIs that will offer you actionable insights; it will also present them in a digestible, visual, interactive format from one central, live dashboard . A data methodology you can count on.

By integrating the right technology within your data analysis methodology, you’ll avoid fragmenting your insights, saving you time and effort while allowing you to enjoy the maximum value from your business’s most valuable insights.

For a look at the power of software for the purpose of analysis and to enhance your methods of analyzing, glance over our selection of dashboard examples .

10. Answer your questions

By considering each of the above efforts, working with the right technology, and fostering a cohesive internal culture where everyone buys into the different ways to analyze data as well as the power of digital intelligence, you will swiftly start to answer your most burning business questions. Arguably, the best way to make your data concepts accessible across the organization is through data visualization.

11. Visualize your data

Online data visualization is a powerful tool as it lets you tell a story with your metrics, allowing users across the organization to extract meaningful insights that aid business evolution – and it covers all the different ways to analyze data.

The purpose of analyzing is to make your entire organization more informed and intelligent, and with the right platform or dashboard, this is simpler than you think, as demonstrated by our marketing dashboard .

An executive dashboard example showcasing high-level marketing KPIs such as cost per lead, MQL, SQL, and cost per customer.

This visual, dynamic, and interactive online dashboard is a data analysis example designed to give Chief Marketing Officers (CMO) an overview of relevant metrics to help them understand if they achieved their monthly goals.

In detail, this example generated with a modern dashboard creator displays interactive charts for monthly revenues, costs, net income, and net income per customer; all of them are compared with the previous month so that you can understand how the data fluctuated. In addition, it shows a detailed summary of the number of users, customers, SQLs, and MQLs per month to visualize the whole picture and extract relevant insights or trends for your marketing reports .

The CMO dashboard is perfect for c-level management as it can help them monitor the strategic outcome of their marketing efforts and make data-driven decisions that can benefit the company exponentially.

12. Be careful with the interpretation

We already dedicated an entire post to data interpretation as it is a fundamental part of the process of data analysis. It gives meaning to the analytical information and aims to drive a concise conclusion from the analysis results. Since most of the time companies are dealing with data from many different sources, the interpretation stage needs to be done carefully and properly in order to avoid misinterpretations. 

To help you through the process, here we list three common practices that you need to avoid at all costs when looking at your data:

  • Correlation vs. causation: The human brain is formatted to find patterns. This behavior leads to one of the most common mistakes when performing interpretation: confusing correlation with causation. Although these two aspects can exist simultaneously, it is not correct to assume that because two things happened together, one provoked the other. A piece of advice to avoid falling into this mistake is never to trust just intuition, trust the data. If there is no objective evidence of causation, then always stick to correlation. 
  • Confirmation bias: This phenomenon describes the tendency to select and interpret only the data necessary to prove one hypothesis, often ignoring the elements that might disprove it. Even if it's not done on purpose, confirmation bias can represent a real problem, as excluding relevant information can lead to false conclusions and, therefore, bad business decisions. To avoid it, always try to disprove your hypothesis instead of proving it, share your analysis with other team members, and avoid drawing any conclusions before the entire analytical project is finalized.
  • Statistical significance: To put it in short words, statistical significance helps analysts understand if a result is actually accurate or if it happened because of a sampling error or pure chance. The level of statistical significance needed might depend on the sample size and the industry being analyzed. In any case, ignoring the significance of a result when it might influence decision-making can be a huge mistake.

13. Build a narrative

Now, we’re going to look at how you can bring all of these elements together in a way that will benefit your business - starting with a little something called data storytelling.

The human brain responds incredibly well to strong stories or narratives. Once you’ve cleansed, shaped, and visualized your most invaluable data using various BI dashboard tools , you should strive to tell a story - one with a clear-cut beginning, middle, and end.

By doing so, you will make your analytical efforts more accessible, digestible, and universal, empowering more people within your organization to use your discoveries to their actionable advantage.

14. Consider autonomous technology

Autonomous technologies, such as artificial intelligence (AI) and machine learning (ML), play a significant role in the advancement of understanding how to analyze data more effectively.

Gartner predicts that by the end of this year, 80% of emerging technologies will be developed with AI foundations. This is a testament to the ever-growing power and value of autonomous technologies.

At the moment, these technologies are revolutionizing the analysis industry. Some examples that we mentioned earlier are neural networks, intelligent alarms, and sentiment analysis.

15. Share the load

If you work with the right tools and dashboards, you will be able to present your metrics in a digestible, value-driven format, allowing almost everyone in the organization to connect with and use relevant data to their advantage.

Modern dashboards consolidate data from various sources, providing access to a wealth of insights in one centralized location, no matter if you need to monitor recruitment metrics or generate reports that need to be sent across numerous departments. Moreover, these cutting-edge tools offer access to dashboards from a multitude of devices, meaning that everyone within the business can connect with practical insights remotely - and share the load.

Once everyone is able to work with a data-driven mindset, you will catalyze the success of your business in ways you never thought possible. And when it comes to knowing how to analyze data, this kind of collaborative approach is essential.

16. Data analysis tools

In order to perform high-quality analysis of data, it is fundamental to use tools and software that will ensure the best results. Here we leave you a small summary of four fundamental categories of data analysis tools for your organization.

  • Business Intelligence: BI tools allow you to process significant amounts of data from several sources in any format. Through this, you can not only analyze and monitor your data to extract relevant insights but also create interactive reports and dashboards to visualize your KPIs and use them for your company's good. datapine is an amazing online BI software that is focused on delivering powerful online analysis features that are accessible to beginner and advanced users. Like this, it offers a full-service solution that includes cutting-edge analysis of data, KPIs visualization, live dashboards, reporting, and artificial intelligence technologies to predict trends and minimize risk.
  • Statistical analysis: These tools are usually designed for scientists, statisticians, market researchers, and mathematicians, as they allow them to perform complex statistical analyses with methods like regression analysis, predictive analysis, and statistical modeling. A good tool to perform this type of analysis is R-Studio as it offers a powerful data modeling and hypothesis testing feature that can cover both academic and general data analysis. This tool is one of the favorite ones in the industry, due to its capability for data cleaning, data reduction, and performing advanced analysis with several statistical methods. Another relevant tool to mention is SPSS from IBM. The software offers advanced statistical analysis for users of all skill levels. Thanks to a vast library of machine learning algorithms, text analysis, and a hypothesis testing approach it can help your company find relevant insights to drive better decisions. SPSS also works as a cloud service that enables you to run it anywhere.
  • SQL Consoles: SQL is a programming language often used to handle structured data in relational databases. Tools like these are popular among data scientists as they are extremely effective in unlocking these databases' value. Undoubtedly, one of the most used SQL software in the market is MySQL Workbench . This tool offers several features such as a visual tool for database modeling and monitoring, complete SQL optimization, administration tools, and visual performance dashboards to keep track of KPIs.
  • Data Visualization: These tools are used to represent your data through charts, graphs, and maps that allow you to find patterns and trends in the data. datapine's already mentioned BI platform also offers a wealth of powerful online data visualization tools with several benefits. Some of them include: delivering compelling data-driven presentations to share with your entire company, the ability to see your data online with any device wherever you are, an interactive dashboard design feature that enables you to showcase your results in an interactive and understandable way, and to perform online self-service reports that can be used simultaneously with several other people to enhance team productivity.

17. Refine your process constantly 

Last is a step that might seem obvious to some people, but it can be easily ignored if you think you are done. Once you have extracted the needed results, you should always take a retrospective look at your project and think about what you can improve. As you saw throughout this long list of techniques, data analysis is a complex process that requires constant refinement. For this reason, you should always go one step further and keep improving. 

Quality Criteria For Data Analysis

So far we’ve covered a list of methods and techniques that should help you perform efficient data analysis. But how do you measure the quality and validity of your results? This is done with the help of some science quality criteria. Here we will go into a more theoretical area that is critical to understanding the fundamentals of statistical analysis in science. However, you should also be aware of these steps in a business context, as they will allow you to assess the quality of your results in the correct way. Let’s dig in. 

  • Internal validity: The results of a survey are internally valid if they measure what they are supposed to measure and thus provide credible results. In other words , internal validity measures the trustworthiness of the results and how they can be affected by factors such as the research design, operational definitions, how the variables are measured, and more. For instance, imagine you are doing an interview to ask people if they brush their teeth two times a day. While most of them will answer yes, you can still notice that their answers correspond to what is socially acceptable, which is to brush your teeth at least twice a day. In this case, you can’t be 100% sure if respondents actually brush their teeth twice a day or if they just say that they do, therefore, the internal validity of this interview is very low. 
  • External validity: Essentially, external validity refers to the extent to which the results of your research can be applied to a broader context. It basically aims to prove that the findings of a study can be applied in the real world. If the research can be applied to other settings, individuals, and times, then the external validity is high. 
  • Reliability : If your research is reliable, it means that it can be reproduced. If your measurement were repeated under the same conditions, it would produce similar results. This means that your measuring instrument consistently produces reliable results. For example, imagine a doctor building a symptoms questionnaire to detect a specific disease in a patient. Then, various other doctors use this questionnaire but end up diagnosing the same patient with a different condition. This means the questionnaire is not reliable in detecting the initial disease. Another important note here is that in order for your research to be reliable, it also needs to be objective. If the results of a study are the same, independent of who assesses them or interprets them, the study can be considered reliable. Let’s see the objectivity criteria in more detail now. 
  • Objectivity: In data science, objectivity means that the researcher needs to stay fully objective when it comes to its analysis. The results of a study need to be affected by objective criteria and not by the beliefs, personality, or values of the researcher. Objectivity needs to be ensured when you are gathering the data, for example, when interviewing individuals, the questions need to be asked in a way that doesn't influence the results. Paired with this, objectivity also needs to be thought of when interpreting the data. If different researchers reach the same conclusions, then the study is objective. For this last point, you can set predefined criteria to interpret the results to ensure all researchers follow the same steps. 

The discussed quality criteria cover mostly potential influences in a quantitative context. Analysis in qualitative research has by default additional subjective influences that must be controlled in a different way. Therefore, there are other quality criteria for this kind of research such as credibility, transferability, dependability, and confirmability. You can see each of them more in detail on this resource . 

Data Analysis Limitations & Barriers

Analyzing data is not an easy task. As you’ve seen throughout this post, there are many steps and techniques that you need to apply in order to extract useful information from your research. While a well-performed analysis can bring various benefits to your organization it doesn't come without limitations. In this section, we will discuss some of the main barriers you might encounter when conducting an analysis. Let’s see them more in detail. 

  • Lack of clear goals: No matter how good your data or analysis might be if you don’t have clear goals or a hypothesis the process might be worthless. While we mentioned some methods that don’t require a predefined hypothesis, it is always better to enter the analytical process with some clear guidelines of what you are expecting to get out of it, especially in a business context in which data is utilized to support important strategic decisions. 
  • Objectivity: Arguably one of the biggest barriers when it comes to data analysis in research is to stay objective. When trying to prove a hypothesis, researchers might find themselves, intentionally or unintentionally, directing the results toward an outcome that they want. To avoid this, always question your assumptions and avoid confusing facts with opinions. You can also show your findings to a research partner or external person to confirm that your results are objective. 
  • Data representation: A fundamental part of the analytical procedure is the way you represent your data. You can use various graphs and charts to represent your findings, but not all of them will work for all purposes. Choosing the wrong visual can not only damage your analysis but can mislead your audience, therefore, it is important to understand when to use each type of data depending on your analytical goals. Our complete guide on the types of graphs and charts lists 20 different visuals with examples of when to use them. 
  • Flawed correlation : Misleading statistics can significantly damage your research. We’ve already pointed out a few interpretation issues previously in the post, but it is an important barrier that we can't avoid addressing here as well. Flawed correlations occur when two variables appear related to each other but they are not. Confusing correlations with causation can lead to a wrong interpretation of results which can lead to building wrong strategies and loss of resources, therefore, it is very important to identify the different interpretation mistakes and avoid them. 
  • Sample size: A very common barrier to a reliable and efficient analysis process is the sample size. In order for the results to be trustworthy, the sample size should be representative of what you are analyzing. For example, imagine you have a company of 1000 employees and you ask the question “do you like working here?” to 50 employees of which 49 say yes, which means 95%. Now, imagine you ask the same question to the 1000 employees and 950 say yes, which also means 95%. Saying that 95% of employees like working in the company when the sample size was only 50 is not a representative or trustworthy conclusion. The significance of the results is way more accurate when surveying a bigger sample size.   
  • Privacy concerns: In some cases, data collection can be subjected to privacy regulations. Businesses gather all kinds of information from their customers from purchasing behaviors to addresses and phone numbers. If this falls into the wrong hands due to a breach, it can affect the security and confidentiality of your clients. To avoid this issue, you need to collect only the data that is needed for your research and, if you are using sensitive facts, make it anonymous so customers are protected. The misuse of customer data can severely damage a business's reputation, so it is important to keep an eye on privacy. 
  • Lack of communication between teams : When it comes to performing data analysis on a business level, it is very likely that each department and team will have different goals and strategies. However, they are all working for the same common goal of helping the business run smoothly and keep growing. When teams are not connected and communicating with each other, it can directly affect the way general strategies are built. To avoid these issues, tools such as data dashboards enable teams to stay connected through data in a visually appealing way. 
  • Innumeracy : Businesses are working with data more and more every day. While there are many BI tools available to perform effective analysis, data literacy is still a constant barrier. Not all employees know how to apply analysis techniques or extract insights from them. To prevent this from happening, you can implement different training opportunities that will prepare every relevant user to deal with data. 

Key Data Analysis Skills

As you've learned throughout this lengthy guide, analyzing data is a complex task that requires a lot of knowledge and skills. That said, thanks to the rise of self-service tools the process is way more accessible and agile than it once was. Regardless, there are still some key skills that are valuable to have when working with data, we list the most important ones below.

  • Critical and statistical thinking: To successfully analyze data you need to be creative and think out of the box. Yes, that might sound like a weird statement considering that data is often tight to facts. However, a great level of critical thinking is required to uncover connections, come up with a valuable hypothesis, and extract conclusions that go a step further from the surface. This, of course, needs to be complemented by statistical thinking and an understanding of numbers. 
  • Data cleaning: Anyone who has ever worked with data before will tell you that the cleaning and preparation process accounts for 80% of a data analyst's work, therefore, the skill is fundamental. But not just that, not cleaning the data adequately can also significantly damage the analysis which can lead to poor decision-making in a business scenario. While there are multiple tools that automate the cleaning process and eliminate the possibility of human error, it is still a valuable skill to dominate. 
  • Data visualization: Visuals make the information easier to understand and analyze, not only for professional users but especially for non-technical ones. Having the necessary skills to not only choose the right chart type but know when to apply it correctly is key. This also means being able to design visually compelling charts that make the data exploration process more efficient. 
  • SQL: The Structured Query Language or SQL is a programming language used to communicate with databases. It is fundamental knowledge as it enables you to update, manipulate, and organize data from relational databases which are the most common databases used by companies. It is fairly easy to learn and one of the most valuable skills when it comes to data analysis. 
  • Communication skills: This is a skill that is especially valuable in a business environment. Being able to clearly communicate analytical outcomes to colleagues is incredibly important, especially when the information you are trying to convey is complex for non-technical people. This applies to in-person communication as well as written format, for example, when generating a dashboard or report. While this might be considered a “soft” skill compared to the other ones we mentioned, it should not be ignored as you most likely will need to share analytical findings with others no matter the context. 

Data Analysis In The Big Data Environment

Big data is invaluable to today’s businesses, and by using different methods for data analysis, it’s possible to view your data in a way that can help you turn insight into positive action.

To inspire your efforts and put the importance of big data into context, here are some insights that you should know:

  • By 2026 the industry of big data is expected to be worth approximately $273.4 billion.
  • 94% of enterprises say that analyzing data is important for their growth and digital transformation. 
  • Companies that exploit the full potential of their data can increase their operating margins by 60% .
  • We already told you the benefits of Artificial Intelligence through this article. This industry's financial impact is expected to grow up to $40 billion by 2025.

Data analysis concepts may come in many forms, but fundamentally, any solid methodology will help to make your business more streamlined, cohesive, insightful, and successful than ever before.

Key Takeaways From Data Analysis 

As we reach the end of our data analysis journey, we leave a small summary of the main methods and techniques to perform excellent analysis and grow your business.

17 Essential Types of Data Analysis Methods:

  • Cluster analysis
  • Cohort analysis
  • Regression analysis
  • Factor analysis
  • Neural Networks
  • Data Mining
  • Text analysis
  • Time series analysis
  • Decision trees
  • Conjoint analysis 
  • Correspondence Analysis
  • Multidimensional Scaling 
  • Content analysis 
  • Thematic analysis
  • Narrative analysis 
  • Grounded theory analysis
  • Discourse analysis 

Top 17 Data Analysis Techniques:

  • Collaborate your needs
  • Establish your questions
  • Data democratization
  • Think of data governance 
  • Clean your data
  • Set your KPIs
  • Omit useless data
  • Build a data management roadmap
  • Integrate technology
  • Answer your questions
  • Visualize your data
  • Interpretation of data
  • Consider autonomous technology
  • Build a narrative
  • Share the load
  • Data Analysis tools
  • Refine your process constantly 

We’ve pondered the data analysis definition and drilled down into the practical applications of data-centric analytics, and one thing is clear: by taking measures to arrange your data and making your metrics work for you, it’s possible to transform raw information into action - the kind of that will push your business to the next level.

Yes, good data analytics techniques result in enhanced business intelligence (BI). To help you understand this notion in more detail, read our exploration of business intelligence reporting .

And, if you’re ready to perform your own analysis, drill down into your facts and figures while interacting with your data on astonishing visuals, you can try our software for a free, 14-day trial .

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“Data is Everywhere” , in sheets, in social media platforms, in product reviews and feedback, everywhere. In this latest information age it’s created at blinding speeds and, when data is analyzed correctly, can be a company’s most valuable asset. “ To grow your business even to grow in your life, sometimes all you need to do is Analysis! ” If your business is not growing, then you have to look back recognize your mistakes, and make a plan again without repeating those mistakes. And even if your business is growing, then you have to look forward to making the business grow more.

All you need to do is analyze your business data and business processes. The process of studying the data to find out the answers to how and why things happened in the past. Usually, the result of data analysis is the final dataset, i.e. a pattern, or a detailed report that you can further use for Data Analytics . In this article, we will explore What is Data Analysis? , How it works , the types of data analysis , and the Tools required for data analysis .

Table of Content

Why data analysis is important, types of data analysis methods, what is the data analysis process, top data analysis tools, how to become data analyst.

Before jumping into the term “ Data Analysis”, let’s discuss the term “Analysis” . Analysis is a process of answering “How?” and “Why?” . For example, how was the growth of XYZ Company in the last quarter? Or why did the sales of XYZ Company drop last summer? So to answer those questions we take the data that we already have. Out of that, we filter out what we need. This filtered data is the final dataset of the larger chunk that we have already collected and that becomes the target of data analysis . Sometimes we take multiple data sets and analyze them to find a pattern. For example, take summer sales data for three consecutive years. I found out if that fall in sales last summer was because of any specific product that we were selling or if it was just a recurring problem. It’s all about looking for a pattern. We analyze things or events that have already happened in the past.

Let’s say you own a business and sell daily products. Your business model is pretty simple. You buy products from the supplier and sell them to the customer. Let’s assume the biggest challenge for your business is to find the right amount of stock at the given time. You can’t stock excess dairy products as they are perishable and if they go bad you can’t sell them, resulting in a direct loss for you. At the same time, you can not understock as it may result in the loss of potential customers. But data analytics can help you in predicting the strength of your customers at a given time. Using that result, you can sufficiently stock your supplies, in turn, minimizing the loss. In simple words, using data analysis, you can find out the time of the year when your store has the least or the most customers. Using this info, you can stock your supplies accordingly. So these are some reasons why analysis of data is important.

The major Data Analysis methods are:

  • Descriptive Analysis
  • Diagnostic Analysis
  • Predictive Analysis
  • Prescriptive Analysis
  • Statistical Analysis

1. Descriptive Analysis

A Descriptive Analysis looks at data and analyzes past events for insight as to how to approach future events. It looks at the past performance and understands the performance by mining historical data to understand the cause of success or failure in the past. Almost all management reporting such as sales, marketing, operations, and finance uses this type of analysis.

Example: Let’s take the example of DMart, we can look at the product’s history and find out which products have been sold more or which products have large demand by looking at the product sold trends, and based on their analysis we can further make the decision of putting a stock of that item in large quantity for the coming year.

2. Diagnostic Analysis

Diagnostic analysis works hand in hand with Descriptive Analysis . As descriptive Analysis finds out what happened in the past, diagnostic Analysis, on the other hand, finds out why did that happen or what measures were taken at that time, or how frequently it has happened. it basically gives a detailed explanation of a particular scenario by understanding behavior patterns.

Example: Let’s take the example of Dmart again. Now if we want to find out why a particular product has a lot of demand, is it because of their brand or is it because of quality. All this information can easily be identified using diagnostic Analysis.

3. Predictive Analysis

Information we have received from descriptive and diagnostic analysis, we can use that information to predict future data. it basically finds out what is likely to happen in the future. Now when future data doesn’t mean we have become fortune-tellers, by looking at the past trends and behavioral patterns we are forecasting that it might happen in the future.

Example: The best example would be Amazon and Netflix recommender systems. You might have noticed that whenever you buy any product from Amazon, on the payment side it shows you a recommendation saying the customer who purchased this has also purchased this product that recommendation is based on the customer purchase behavior in the past. By looking at customer past purchase behavior analyst creates an association between each product and that’s the reason it shows recommendation when you buy any product.   

4. Prescriptive Analysis

This is an advanced method of Predictive Analysis . Now when you predict something or when you start thinking out of the box you will definitely have a lot of options, and then we get confused as to which option will actually work. Prescriptive Analysis helps to find which is the best option to make it happen or work. As predictive Analysis forecast future data, Prescriptive Analysis on the other hand helps to make it happen whatever we have forecasted. Prescriptive Analysis is the highest level of Analysis that is used for choosing the best optimal solution by looking at descriptive, diagnostic, and predictive data.

Example: The best example would be Google’s self-driving ca r, by looking at the past trends and forecasted data it identifies when to turn or when to slow down, which works much like a human driver.

5. Statistical Analysis

Statistical Analysis is a statistical approach or technique for analyzing data sets in order to summarize their important and main characteristics generally by using some visual aids. This approach can be used to gather knowledge about the following aspects of data:

  • Main characteristics or features of the data.
  • The variables and their relationships.
  • Finding out the important variables that can be used in our problem.

A Data analysis has the ability to transform raw available data into meaningful insights for your business and your decision-making. While there are several different ways of collecting and interpreting this data, most data-analysis processes follow the same six general steps.

  • Specify Data Requirements
  • Collect Data
  • Clean and Process the Data
  • Analyse the Data
  • Interpretation

1. Specify Data Requirements

In step 1 of the data analysis process define what you want to answer through data. This typically stems from a business problem or questions, such as

  • How can we reduce production costs without sacrificing quality?
  • How do customers view our brand?
  • How can we increase sales opportunities using our current resources?

2. Collect Data

  • Find Your Source : Determine what information can be collected from existing sources, and what you need to find elsewhere.
  • Standardize Collection : Create file storage and naming system ahead of time.
  • Keep Track : Keep data organized in a log with dates and add any source notes as you go.

3. Clean and Process the Data

Ensure your data is correct and usable by identifying and removing any errors or corruption.

  • Monitor Errors : Keep a record and look at trends of where most errors are coming from.
  • Validate Accuracy : Research and invest in data tools that allow you to clean your data in real-time.
  • Scrub for Duplicate Data : Identify and remove duplicates so you save time during analysis.
  • Delete all Formatting : Standardise the look of your data by removing any formatting styles.

4. Analyse the Data

Different data analysis techniques allow you to understand, interpret, and derive conclusions based on your business question or problem. 

5. Interpretation

As you interpret the result of your data, ask yourself these key questions:

  • Does the data answer your question? How?
  • Does the data help you defend against any objections? How?
  • Are there any limitations or angles you haven’t considered?

Data Analysis can be used to report to different people:

  • A primary collaborator or client
  • Executive and business leaders
  • A technical supervisor  
  • Keep it Succinct: Organize data in a way that makes it easy for different audiences to skim through it to find the information most relevant to them.
  • Make it Visual: Use data visualizations techniques, such as tables and charts, to communicate the message clearly.
  • Include an Executive Summary : This allows someone to analyze your findings upfront and harness your most important points to influence their decisions.

Data analysis tools make it easier for users to process and manipulate data, analyze the relationships and correlations between data sets, and it also helps to identify patterns and trends for interpretation. Below is the list of some popular tools explain briefly:

  • SAS : SAS was a programming language developed by the SAS Institute for performed advanced analytics, multivariate analyses, business intelligence, data management, and predictive analytics. , SAS was developed for very specific uses and powerful tools are not added every day to the extensive already existing collection thus making it less scalable for certain applications.
  • Microsoft Excel : It is an important spreadsheet application that can be useful for recording expenses, charting data, and performing easy manipulation and lookup and or generating pivot tables to provide the desired summarized reports of large datasets that contain significant data findings. It is written in C# , C++ , and .NET Framework , and its stable version was released in 2016.
  • R  :It is one of the leading programming languages for performing complex statistical computations and graphics. It is a free and open-source language that can be run on various UNIX platforms, Windows, and macOS . It also has a command-line interface that is easy to use. However, it is tough to learn especially for people who do not have prior knowledge about programming.
  • Python :It is a powerful high-level programming language that is used for general-purpose programming . Python supports both structured and functional programming methods. Its extensive collection of libraries make it very useful in data analysis . Knowledge of Tensorflow , Theano , Keras , Matplotlib , Scikit-learn , and Keras can get you a lot closer to your dream of becoming a machine learning engineer.
  • Tableau Public :Tableau Public is free software developed by the public company “ Tableau Software ” that allows users to connect to any spreadsheet or file and create interactive data visualizations. It can also be used to create maps, dashboards along with real-time updation for easy presentation on the web. The results can be shared through social media sites or directly with the client making it very convenient to use.
  • RapidMiner : RapidMiner is an extremely versatile data science platform developed by “RapidMiner Inc”. The software emphasizes lightning-fast data science capabilities and provides an integrated environment for the preparation of data and application of machine learning, deep learning, text mining, and predictive analytical techniques. It can also work with many data source types including Access, SQL , Excel, Tera data, Sybase , Oracle, MySQL , and Dbase.
  • Knime  :Knime, the Konstanz Information Miner is a free and open-source data analytics software. It is also used as a reporting and integration platform. It involves the integration of various components for Machine Learning and data mining through the modular data-pipe lining. It is written in Java and developed by KNIME.com AG . It can be operated in various operating systems such as Linux, OS X, and Windows.

To become a data analyst you must require least a bachelor’s degree. To those who are at higher level , you may require a master’s degree. You also need to developed skills such as : Statistical Analysis , Data Visualization , Data Cleaning , Database Mnagement , and MS-Excel. Start with internships to gain experience and make projects that will demonstrate your skills. The files of Data Analytics is changing rapidly So, you need to keep yourself updated as according to the time by taking online sessions, attending workshops, or reading related books and articles published. As you grow in the field of data science you might find the specific industries to work with and you can explore more in-depth about Data Analysis.

FAQs on Data Analyst?

Q1. how does data analysis differ from data science.

Data Science includes a wider range of activities like create a new algorithm, building predictive models, and harnessing data whereas in Data Analysis includes a performing and processing of datasets that are already existing.

Q2. What tools do data analysts typically use?

The main tools data analysts typically uses that are Excel, SQL , Python , Power BI, R, and Tableau and so on .

Q3. What is the role of a data analyst in a company?

A Data Analyst manage data to help company to make informed decisions. They gather techniques and processes to analyse data . They also communicate their data to stack holders.

Q4. Do I need to Know programming to be a data analyst?

It’s not always compulsory to learn a programming language for this role but languages like Python and R can play a significant role to automate task, and handle large datasets.

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8.3: What Is Analysis?

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Learning Objective

  • Explain the basics of analysis

Diagram of three parts in a whole.

Critical thinking skill analysis is the process of methodically breaking something down to gain a better understanding of it. Analysis also includes the ability to connect pieces of information as the basis for generalization or explanation. Analytical assignments in college often couple analysis with the critical thinking skills of interpretation and evaluation.

Analysis can be applied to content but can also cover form, function, and context. For example, an analysis assignment in an art appreciation class might ask you to analyze the subject and iconography of a painting, but also expect you to analyze the use of shape, space, color, and texture (form), as well as the artist’s intended purpose (function) and the culture or time period in which the work was created (context).

While each academic discipline characterizes the analytic process to suit its needs, the essential skills of analysis are the following:

  • Breaking down information or artifacts into component parts
  • Uncovering relationships among those parts
  • Determining motives, causes, and underlying assumptions
  • Making inferences and finding evidence to support generalizations

The Language of Analytical Assignments

Although analysis is ubiquitous in college, students sometimes fail to recognize when they are being asked to apply analysis. Often that confusion stems from differences in vocabulary across different disciplines.

For example, each of the verbs in the following list may denote some type of analysis:

Although this list is a good start, these aren’t the only verbs that denote analysis. Another way to tell whether an assignment is asking for analysis is this: If the assignment asks you to determine how the parts of something relate to the whole, how something works, what something means, or why it’s important , the assignment is asking you to analyze. Below is a list of sample analytic assignments that meet these criteria.

How the parts relate to the whole:

  • Classify problems to identify the appropriate algorithms.
  • Determine how well a feminist interpretation is supported by evidence contained in a work.

How something works:

  • Recognize flaws, inconsistencies, and logical fallacies in an opinion editorial.
  • Distinguish between facts and assumptions in a scientific report.

What something means:

  • Interpret quantitative relationships in a graph.
  • Analyze data/situations to identify root problems.

Why something is important:

  • Assess alternative solutions to the health care crisis.
  • Separate relevant from irrelevant information in testimony.

https://assessments.lumenlearning.co...essments/20269

analysis : the process of methodically breaking something down to gain a better understanding of it

Contributors and Attributions

  • What Is Analysis? and The Language of Analytic Assignments. Authored by : Karen Forgette. Provided by : University of Mississippi. License : CC BY-SA: Attribution-ShareAlike
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United Airlines says after a ‘detailed safety analysis’ it will restart flights to Israel in March

FILE - Two United Airlines Boeing 737s are parked at the gate at the Fort Lauderdale-Hollywood International Airport in Fort Lauderdale, Fla., July 7, 2022. United Airlines said Wednesday, Feb. 21, 2024, that it plans to resume flights to Israel in March, reviving a route that was suspended in October 2023 at the start of the Israel-Hamas war. (AP Photo/Wilfredo Lee, File)

FILE - Two United Airlines Boeing 737s are parked at the gate at the Fort Lauderdale-Hollywood International Airport in Fort Lauderdale, Fla., July 7, 2022. United Airlines said Wednesday, Feb. 21, 2024, that it plans to resume flights to Israel in March, reviving a route that was suspended in October 2023 at the start of the Israel-Hamas war. (AP Photo/Wilfredo Lee, File)

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CHICAGO (AP) — United Airlines says it plans to resume flights to Israel next month, reviving a route that was suspended in October at the start of the Israel-Hamas war .

The airline said Wednesday that it will start flights from Newark, New Jersey, to Tel Aviv with a stop in Munich on March 2 and March 4. United said it hopes to begin daily service on March 6 and to add a second daily flight as soon as May.

American Airlines and Delta Air Lines also stopped flying to Tel Aviv after the war started and have not announced when service might resume. Germany’s Lufthansa and its affiliates Austrian Airlines and Swiss brought back flights to Tel Aviv in January, followed by Air France. Other European carriers have said they plan to restart flights to Israel this spring.

United said it conducted “a detailed safety analysis” and consulted security experts and government officials in both countries before deciding to resume the flights. The airline said it also worked with the two unions that represent its pilots and flight attendants.

The Chicago-based airline said it will evaluate whether to resume flights this fall to Israel from San Francisco, Chicago and Dulles airport outside Washington, D.C.

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