**book excerptise: a
book unexamined is wasting trees **

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Lakatos, Imre;**

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Proofs and Refutations**

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Thomas Nelson and Sons, Edinburgh. 1964 (Reprint: British J. Philosophy of Science v.XIV:53-56, 1963-64)**

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ISBN 0521290384
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topics: | philosophy | science | logic | math

When I first encountered this book at a reading circle run by Mohini Mullick in the 90s, I found it to be a powerful deconstruction of the myth of logical progression in science. For me this text is the bible of post-modernism - with its destructive critique of formalism as a process, as a de-humanization of ourselves, and with the ultimate realization that concepts like "truth" itself are subjective. It mainly argues against the formalist dogma that mathematics proceeds via logic from proof to proof The book operates at many levels; framed in a story of a teacher unfolding Euler's theorem in geometry with a group of students, the discourse follows a trajectory through the tortuous history of this proof, outlining the creative aspects of mathematical proof - e.g. the role of observation (p.15) or even taste (p.103-4); the human power-play in mathematics - how a proof is not a matter of pure logic but of a majority decision, and how the acceptability of a proof rises with its elegance (p.9) - AM dec 08

'Proofs and Refutations' essay was a much amended and improved version of chapter 1 of Lakatos's 1961 Cambridge Ph.D. thesis. When he died in 1974, he was planning a revised edition. This was ultimately brought out in 1976, edited by his students John Worrall and Elie Zahar. It includes parts of Chapter 2 from his thesis, relating to Poincare's vector-algebraic proof of the Descartes-Euler conjecture, and more details on Cauchy's proof of the theorem on the limit of convergent series.

Carnap demands that (a) 'philosophy is to be replaced by the logic of science ...', (b) 'the logic of science is nothing other than the logical syntax of the language of science ...', (c) 'metamathematics is the syntax of mathematical language'. - p.2-3

[A]ccording to the formalist concept of mathematics, there is no history of mathematics proper. Any formalist would basically agree with Russell's 'romantically' put but seriously meant remark, according to which Boole's Laws of Thought (1854) was the 'first book ever written on mathematics'.... None of the creative periods... would be admitted into the formalist heaven, where mathematical theories dwell like the seraphim, purged of all the impurities of earthly uncertainty... for some 'mixtures of mathematics and something else' we can find formal systems 'which include them in a certain sense' then they too may be admitted. On these terms Newton had to wait four centuries until Peano, Russell, and Quine helped him onto heaven by formalizing the Calculus. Dirac is more fortunate: Schwartz saved his soul during his lifetime. - p.3 Tarski uses the term 'deductive sciences'explicitlyas a shorthand for 'formalised deductive sciences'. ... the subject matter of metamathematics is confined to formalized deductive disciplines because non-formalised deductive sciences are not suitable objects for scientific investigation at all. ... Nobody will doubt that some problems about a mathematical theory can only be approached after it has been formalised, just as some problems about human beings (say concerning their anatomy) can only be approached after their death. But few will infer from this that human beings are 'suitable for scientific investigation; only when they are 'presented in "dead" form', and that biological investigations are confined in consequence to the discussion of dead human beings. - p.4- 5, footnote But formalist philosophy of mathematics has very deep roots. It is the latest link in the long chain ofdogmatistphilosophies of mathematics. For more than two thousand years there has been an argument between {\it dogmatists} andsceptics. The dogmatists hold that ... we can attain truth and know that we have attained it. The sceptics on the other hand either hold that we cannot attain truth at all ... or that we cannot know if we attain it or that we have attained it. - p.6 Nothing is more characteristic of a dogmatist epistemology than its theory of error. For if some truths are manifest, one must explain how anyone can be mistaken about them, in other words, why the truths are not manifest to everybody. According to its particular theory of error, each dogmatist epistemology offers its particular therapeutics to purge minds from error. Cf Popper [1963], Introduction. [IDEA: Useful in debates against fanatics.] - p.34 foot "a sick mind, twisting in pain" - part of the Stoic theory of error p.35 [The modest aim of this case-study] is to elaborate the point that informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. - p.6 --- [ELEGANCE: IDEA/EXTENSION: If the informal process of mathematical discovery can be studied, why not the question of mathematical elegance or beauty: "After Cauchy's proof it became absolutely indubitable that the elegant relation V + F = E + 2 applies to all sorts of polyhedra," [Jonquieres 1890a] - p.9 footnote. What constitutes elegance? And its opposite: "I turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives." [Hermite, in letter to Stieltjes 1893] - p.21 footnote. Also p.60: Weierstrassian rigour triumphed over its reactionary monster-barring and lemma-hiding opponents who used slogans like 'the dullness of rigour','artificiality versus beauty', etc. Also, p.103: Why are steps 6 and 7, i.e. incorporation of included surfaces and ring-shapes - not an increase in depth? "RHO: not every increase in content is also an increase in depth: think of (6) and (7)!" followed by the footnote that "Quite a few mathematicians cannot distinguish the trivial from the non- trivial." (p.103). There is also the quote from von Neumann about the 'danger of degeneration' from such trivialities, who thought that it would not be so bad 'if the discipline is under the influence of men with an exceptionally well-developed taste' [1947]. This quote is passed on by Lakatos passim (p.104), but is this 'taste' not an extremely mysterious yet IMPORTANT aspect of all scientific advance? --- TEACHER: I do not think that ['proof'] establishes the truth of the conjecture... I propose to retain the time-honoured technical term 'proof' for athought-experiment - or 'quasi-experiment' - which suggests a decomposition of the original conjecture into subconjectures or lemmas, thusembeddingit in a possibly quite distant body of knowledge. - p.10 "Just send me the theorems, then I shall find the proofs" - Chrysippus to Cleanthes. Riemann: "If only I had the theorems! Then I should find the proofs easily enough." - p.11 TEACHER: You are interested only in proofs which 'prove' what they set out to prove. I am interested in proofs even if they do not accomplish their intended task. Columbus did not reach India but he discovered something quite interesting. - p.15 "As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons, The properties of the numbers known today have been mostly discovered by observation"... editors intro to Euler [1753]. - p.11. [The cube-in-cube counterexample was discovered through observation.] Both Lhulier and Hessel were led to their discovery by mineralogical collections in which they noticed some double crystals... Lhulier acknowledges the stimulus of the crystal collection of his friend Professor Picret [1812]. Hessel refers to lead sulphide cubes enclosed in transluscent calcium fluoride crystals [1832]. - p.15 DELTA: So really you showed us two polyhedra - two surfaces, one completely inside the other. A woman with a child in her womb is not a counterexample to the thesis that human beings have one head. - p.16 "Researches dealing with ... functions violating laws which one hoped were universal, were regarded almost as the propagation of anarchy and chaos where past genberations had sought order and harmony" [Saks 1933]. The similarly fierce battle that raged later between opponents and protagonists of modern mathematical logic and set theory was a direct continuation of this. - p.21 foot ALPHA: It is strange to think that once upon a time [V-E+F=2] was a wonderful guess, full of challenge and excitement. Now, because of your weird shifts of meaning, it has turned into a poor convention, a despicable piece of dogma. (He leaves the classroom. - p.23 GAMMA: I think that if we want to learn about anything really deep, we have to study it not in its 'normal', regular, usual form, but in its critical state, in fever, in passion.... If you want to know ordinary polyhedra, study their lunatic fringe. This is how one can carry mathematical analysis into the heart of the subject. - p.25 [IDEA: Qualitative reasoning: the importance of tangencies/alignments.] Poniard [1908]: 'Logic sometimes makes monsters. Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives, etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular cases. There remains for them only a very small corner.' - p.24 Newton[1717]: If no exception occur from phenomena, the conclusion may be pronounced generally. But if at any time afterwards any exception should occur, it may then begin to be pronounced with such exceptions as occur. - p.30 We first guessed that for all polyhedra V-E+F=2, because we found it to be true for cubes, octahedra, pyramids, and prisms. We certainly cannot accept 'this miserable way of inferring from the special to the general'. - p.30. Quote from [Able 1826]: "In Higher Analysis very few propositions are proved with definitive rigor. One finds everywhere the miserable way of inferring from the special to the general, and it is a marvel that such procedure leads only rarely to what are called paradoxes. It is really very interesting to look for the reason. In my opinion the reason is to be found in the fact that analysts have been mostly occupied with functions that can be expressed as power series. As soon as other functions enter - which certainly is rarely the case - one does not get on any more and as soon as one starts drawing false conclusions, an infinite multitude of mistakes will follow, all supporting each other ... " - p.30 Many working mathematicians are puzzled about what proofs are for if they do not prove.... Applied mathematicians usually try to solve this dilemma by a shamefaced but firm belief that the proofs of the pure mathematicians are 'complete'. and soreallyprove. Pure mathematicians, however, know better - they have such respect only for the complete proofs of logicians. [e.g. Hardy 1928]: 'There is strictly speaking no such thing as mathematical proof; we can, in the last analysis, do nothing but point; ... proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils.' ... G.Polya points out that proofs, even if incomplete, establish connections between mathematical facts and this helps us keep them in our memory: proofs yield a mnemotechnic system [1945]. - p.31- 32 BETA: Not 'guesswork' this time, but insight! TEACHER: I abhor your pretensions 'insight'. I respect conscious guessing, because it comes from the best human qualities: courage and modesty. - p.32 Poinsot was certainly brainwashed some time between 1809 and 1858 ... now he sees examples where he previously saw counterexamples. The self- criticism had to be surreptitious, cryptic, because in scientific tradition there are no patterns available for articulating such volte- faces. [EXCLUSION BY RE-DEFINITION = Monsterbarring; DOMAIN EXCLUSION = EXCEPTION BARRING. Monsterbarring: Using this method one can eliminate any counterexample to the original conjecture by a sometimes deft bug always ad hoc redefinition of the polyhedron, of its defining terms, of the defining terms of its defining terms. - p.25 Our naive conjecture was 'All polyhedra are Eulerian'. The monster-barring method defends this by reinterpreting its terms in such a way that at the end we have amonster-barring theorem: 'All polyhedra are Eulerian'. But the identity of the linguistic expressions of the naive conjecture and the monster-barring theorem hides, behind surreptitious changes in the meaning of the terms, an essential improvement. The exception-barring method introduced an element which is really extraneous to the argument: convexity. Theexception-barring theoremwas: 'All convex polyhedra are Eulerian'. The lemma-incorporating method relied on the argument - i.e. on the proof - and on nothing else. It virtuallysummed up the proof in the lemma-incorporating theorem: 'All simple polyhedra with simply- connected faces are Eulerian'. This shows that ...one does not prove what one has set out to prove.Therefore no proof should conclude with the words: 'Quod erat demonstrandum'. - Alpha, p.44 There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it. -p.43 We already agreed to omit, that is, 'hide', trivially true lemmas. - p.49 The standard expression for this is 'we assume familiarity with lemmas of typex.' Cauchy, e.g. did not even notice that his celebrated [1821] presupposed 'familiarity' with thetheory of real numbers.... Not so Weierstrass and his school: textbooks of formal mathematics now contain a new chapter on the theory of real numbers where these lemmas are collected. ... More rigorous textbooks narrow down background knowledge even further: Landau, in the introduction to his famous [1930], assumes familiarity only with'logical reasoning and German language'.One wonders when 'the author confesses ignorance about the field x' will replace the authoritarian euphemism 'the author assumes familiarity with the field x': surely when it is recognized that knowledge has no foundations. - p.49 foot Russell, Principles of Mathematics, 1903: It is one of the chief merits of proofs that they instill a certain scepticism as to the result proved.' H.G. Forder [1927]: 'The virtue of a logical proof is not that it compels belief, but that it suggests doubts'.- p.52 foot The analogy between political ideologies and scientific theories is then more far-reaching than is commonly realised: political ideologies which first may be debated (and perhaps accepted only under pressure) may turn into unquestioned background knowledge even in a single generation: the critics are forgotten (and perhaps executed) until a revolution vindicates their objections. [Provides case studies with Euclid and Newton. "The peak of Euclid's authority was reached in the Age of Enlightenment. Clairaut urges his colleagues not to 'obscure proofs and disgust readers' by stating evident truths: Euclid did so only in order to convince 'obstinate readers' [1741]." - p.53 foot I have the right to put forward any example that satisfies the conditions of your argument and I strongly suspect that what you call bizarre, preposterous examples are in fact embarrassing examples, prejudicial to your theorem. (G. Darboux [1874]). - p.54 I am terrified by the hoard of implicit lemmas. It will take a lot of work to get rid of them. (G. Darboux [1883] - p.54 ALPHA: There is still the irrefutable master-theorem:'All polyhedra on which one can perform the thought-experiment,or briefly,all Cauchy-polyhedra, are Eulerian.My approximate proof analysis drew the borderline of the class of Cauchy-polyhedra with a pencil that - I must admit - was not particularly sharp. Now eccentric counterexamples teach us to sharpen our pencil. But first:no pencil is absolutely sharp( and if we overdo sharpening it may break); secondly,pencil-sharpening is not creative mathematics.- p.55 But surely 'at each stage of the evolution our fathers also thought they had reached it [absolute rigor]? If they deceived themselves, do we not likewise cheat ourselves? - p.56 Changes in the criterion of 'rigor of the proof' engender major revolutions in mathematics. Pythagoreans held that rigorous proofs can only be arithmetical. They however discovered a rigorous proof that root(2) was 'irrational'. When the scandal eventually leaked out, the Criterion was changed: arithmetical 'intuition' was discredited and geometrical intuition took its place. This meant a major and complicated reorganization of mathematical knowledge (e.g. the theory of proportions). In the eighteenth century 'misleading' figures brought geometrical proofs into disrepute, and the nineteenth century saw arithmetical intuition re-enthroned with the help of the cumbersome theory of real numbers. Today the main dispute is about what is rigorous and what not in set-theoretical and meta-mathematical proofs... - p.56 ALPHA: Ever more eccentric counterexamples will be countered by ever more trivial lemmas - yielding a vicious infinity of ever longer and clumsier theorems. ... GAMMA: At a certain point we may reach truth and then the flow of refutations will stop. But of course we shall not know when. Only refutations are conclusive - proofs are a matter of psychology. LAMBDA: I still trust that the light of absolute certainty will flash up when refutations peter out! KAPPA: But will they? What if God created polyhedra so that all true universal statements about them - formulated in human language - are infinitely long? Is it not blasphemous anthropomorphism to assume that (divine) true theorems are of finite length? ... Truth is only for God. - p.58 Different levels of rigor differ only about where they draw the line between the rigor of proof-analysis and the rigor of proof, i.e. about where criticism should stop and justification should start. 'Certainty is never achieved'; 'foundations' are never found - but the 'cunning of reason' turns each increase in rigor into an increase in content, in the scope of mathematics. - p.60 [IDEA: Gaussian Sphere: Legendre's proof (p.64) entails mapping the polyhedron onto a sphere containing the polyhedron. What is the class of objects for which this can be done? Is this identical to the Gaussian Sphere models, recently revived by Ziv/Malik et al?] OMEGA: My quest is not only for certainty but also for finality. The theorem has to be certain - there must not be any counterexamples {\it within} its domain; but it has also to be final; there must not be any examplesoutsideits domain. I want to draw a dividing line between examples and counterexamples, and not just between a safe domain of a few examples on the one hand and a mixed bag of examples and counterexamples on the other. LAMBDA: Or, you want the conditions of the theorem to be not only sufficient, but also necessary! - p.67 [A proof must explain the Eulerian-ness of the great stellated duodecahedron] 'More questions may be easier to answer than just one question. A new more ambitious problem may be easier to handle than the original problem.' - p.72, Polya[1945]. "Inventor's Paradox" ZETA: You have fallen in love with the problem of finding out where God drew the firmament dividing Eulerian from non-Eulerian polyhedra. But there is no reason to believe that the term 'Eulerian' occured in God's blueprint of the universe at all. - p.73 ZETA: Like most mathematicians I cannot count. I just tried to count the edges and vertices of a heptagon. I found first 7 edges and 8 vertices, and then again 8 edges and 7 vertices... - p.77 BETA: Then what suggested V-E+F=2 to me, if not the facts listed in my table? TEACHER: I shall tell you. .. You had three or four conjectures which in turn were quickly refuted. Your table was built up in the process of testing and refuting these conjectures. Naive conjectures are not inductive conjectures: we arrive at them by trial and error, through conjectures and refutations. - p.78 [Polya: Mathematics and Plausible Reasoning, 2 vols, 1954; v.1 contains a detailed analysis of the Euler polyhedron problem. p.79 foot discusses how Lakatos improves on Polya; the central point is that polya 'never questioned that science is inductive, and because of his correct vision of deep analogy between scientific and mathematical heuristic he was led to think that mathematics is also inductive.'] [Section IV, p.75-100+, rushes too quickly I felt, through some momentous changes. Also, would have done better to introduce the manifold/Holes topological version of the formula, which simplifies the Sigma(ek) business. The end itself, with the teacher leaving the room, leaves one unsatisfied. The drama of the earlier sections is also lost in section IV] LAMBDA: Do you really think that (1) is the single axiom from which all the rest follows? That deduction increases content? [one vertex is one vertex: axiom 1] - p.86 ALPHA: Of course! Isn't this the miracle of the deductive thought- experiment? If once you have got hold of a little truth, deduction expands it infallibly into a tree of knowledge. If a deduction does not increase the content I would not call it deduction, but 'verification'; 'verification differs from true demonstration precisely because it is purely analytic and because it is sterile'. - p.86-87, quote from [Poniard 1902].

concepts get reformulated as proofs progress.] PI: By the time the Descartes-Euler conjecture was put forward, the concept of polyhedron included all sorts of convex polyhedra and even some concave polyhedra. But it certainly did not include polyhedra which were not simple, or polyhedra with ringshaped faces. For the polyhedra that they [the monsterbarrers] had in mind, the conjecture was true as it stood and the proof was flawless. ... The refutationists ... stretched the concept of polyhedron, to cover objects that were alien to the intended representation. Their refutation revealed no error in the original conjecture, no mistake in the original proof: it revealed the likelihood of a new conjecture which nobody had stated or thought of before. ... Imagine a different situation, where the definition fixed the intended interpretation of 'polyhedron' correctly. Then it would have been up to the refutationists to devise ever longermonster-including definitionsfor say, 'complex polyhedra': 'A complex polyhedron is an aggregate of (real) polyhedra such that each two of them are soldered by congruent faces'. 'The faces of complex polyhedra can be complex polygons that are aggregates of (real) polygons such that each two of them are soldered by congruent edges'... SIGMA: I never dreamt that concept-formation might lag behind an unintendedly wide definition! -p.89-91 Often, as soon as concept-stretching refutes a proposition, the refuted proposition seems such an elementary mistake that one cannot imagine that great mathematicians could have made it. This important characteristic of concept-stretching refutation explains why respectful historians, because they do not understand that concepts grow, create for themselves a maze of problems. After saving Cauchy by claiming that he 'could not possibly miss' polyhedra which are not simple and that therefore he 'categorically' (!) restricted the theorem to the domain of convex polyhedra, the respectful historian now has to explain why Cauchy's borderline was 'unnecessarily' narrow.... So [they] explain away a mistake Cauchy never made. Other historians proceed in a different way. They say that before the point where the correct conceptual framework (i.e. the one they know) was reached there was only a 'dark age' with 'seldom, if ever, sound' results. This point in the theory of polyhedra is Jordan's proof (1866) according to Lebesgue [1923]; it is Poincare's (1895) according to [Bell 1945, p.460]. - p.93

'Mathematics is the art of giving the same name to different things. ... If one chooses the right language, one is surprised to learn that the proofs made for a known object apply immediately to many new objects, without the slightest change - one can even retain the names' Poniard [1908], but "Darboux, in his [1874] came close to this idea." 94 When the physicists started to talk about "electricity," or the physicians about "contagion," these terms were vague, obscure, muddled. The terms that the scientists use today, such as "electric charge," "electric current," "fungus infection," "virus infection," are incomparably clearer and more definite. Yet what a tremendous amount of observation, how many ingenious experiments lie between the two terminologies, and some great discoveries too. - Polya [1954] v.1p.55 - p.95 The problem of universals should be reconsidered in view of the fact that, as knowledge grows, languages change. - p.98

on the need for "mathematical taste" and "mathematical critics" to stem the tide of pretentious trivialities in mathematical literature: quotes Polya [1954, v.1,p.30]: Shallow, cheap generalization is 'more fashionable nowadays than it was formerly. It dilutes a little idea with a big terminology. It would be very easy to quote examples, but I don't want to antagonize people.' p.104. For any proposition there is always some sufficiently narrow interpretation of its terms, such that it turns out true, and some sufficiently wide interpretation such that it turns out false. The first interpretation may be called thedogmatist, verificationist or justificationist interpretation,and the second thesceptical, critical or refutationists interpretation.- p.105

There is a simple pattern of mathematical discovery - or of the growth of informal mathematical theories. It consists of the following stages: (1) Primitive conjecture. (2) Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures or lemmas). (3) 'Global* counterexamples {counterexamples to the primitive conjecture) emerge. (4) Proof re-examined: the 'guilty lemma' to which the global counterexample is a 'local' counterexample is spotted. This guilty lemma may have previously remained 'hidden' or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - the improved conjecture - supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature. [first noted by Seidel in 1847, in connection with the Cauchy conjecture] Starting from the certainty just achieved, that the theorem is not universallyvalid, and hence that its proof must rest on some extra hidden assumption, one then subjects the proof to a more detailed analysis. It is not very difficult to discover the hidden hypothesis. One can then infer backwards that this condition expressed by the hypothesis is not satisfied by series which represent discontinuous functions, since only thus can the agreement between the otherwise correct proof sequence, and what has been on the other hand established, be restored. [Lakatos remarks that these steps are not infallible - sometimes, step 4 may precede step 3 - an ingenious proof analysis may suggest the counterexample. ] Subsequent to these four steps, the theorem is further examined, along with other related theorems. The counterexamples are eventually turned into new examples for new theories - new fields of inquiry open up.

PRIMITIVE CONJECTURE: the limit of any convergent series of continuous functions is itself continuous. It was Cauchy [1821] who gave the first proof of this conjecture, whose truth had been taken for granted and assumed therefore not to be in need of any proof throughout the eighteenth century. It was regarded as the special case of the ' axiom' according to which 'what is true up to the limit is true at the limit'. We find the conjecture and its proof in Cauchy's celebrated (p. 131). Cauchy's view of Fourier: that talented but woolly and unrigorous dilettante... 130

Abel 1826: 'It seems to me that there are some exceptions to Cauchy's theorem', and immediately gives the example of the series sin(x) - sin(2x)/2 + sin(3x)/3+... [this sequence of continuous functions is discontinuous owing to the oscillating terms] Abel adds that 'as it is known, there are many more examples like this*. His response to these counterexamples is to start guessing: 'What is the safe domain of Cauchy's theorem ?' His answer to this question is this: the domain of validity of the theorems of analysis in general, and that of the theorems about the continuity of the limit function in particular, is restricted to power series. All the known exceptions to this basic continuity principle were trigonometrical series, and so he proposed to withdraw analysis to within the safe boundaries of power series, thus leaving behind Fourier's cherished trigonometrical series as an uncontrollable jungle - where exceptions are the norm and successes miracles. 133 [It was not until 1847 that Seidel looked at the assumptions implicit in Cauchy's "proof" and came up with the need to distinguish two kinds of convergence, now called: * pointwise convergence: given a sequence of functions f1... fn ..., then if for all x, fn(x) convergest to some function f(x), then f is the limit of fn (it is independent of x). * uniform convergence: that for every epsilon there is an N s.t. the sum of the first N terms differs from the pointwise limit by less than epsilon the latter is a stronger notion - every uniform convergent sequence is pointwise convergent but not vice versa - e.g. for x IN [0,1), x^n is pointwise convergent but is not independent of x, and hence is not uniformly convergent. Similarly, Abel's counter-example sin(x) - sin(2x)/2 + sin(3x)/3+... is a sequence of continuous functions that is discontinuous because it is not uniformly convergent.]

Lakatos's contribution to the philosophy of mathematics was, to put it simply, definitive: the subject will never be the same again. (...) Lakatos made us think instead about what most research mathematicians do. He wrote an amazing philosophical dialogue around the proof of a seemingly elementary but astonishingly deep geometrical idea pioneered by Euler. It is a work of art - I rank it right up there with the dialogues composed by Hume or Berkeley or Plato. He made us see a theorem, a mathematical fact, coming into being before our eyes. - Ian Hacking, in a review of For and Against Method by Lakatos/Feyerabend Mathematics is the art of giving the same name to different things... When the language is well chosen, we are astonished to learn that all te proofs made for a certain object apply immediately to many new objects; there is nothing to change, not even the words, since the names have become the same. - Jules Henri PoincarĂ© [1908, p.375] p. 91 ---blurb Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. ---bio Imre Lakatos was born in Hungary as Imre Lipsitz in 1922. Active in the Communist Party in Hungary after World War II he worked in the Ministry of Education. He earned his Ph.D from Debrecen University in 1947. Expelled from the Communist Party in 1950, he was interned for three years. He fled Hungary in 1956, and was awarded a Rockefeller Fellowship to study at Cambridge, where he completed another Ph.D. He became a lecturer at the London School of Economics where Karl Popper was a great influence on him. Lakatos died in 1974. - http://www.complete-review.com/reviews/lakatosi/pandr.htm

from http://www.rbjones.com/rbjpub/philos/bibliog/lakato76.htm The purpose of this book is: to approach some problems of the methodology of mathematics. where methodology qua logic of discovery is intended. The work, though primarily philosophical in intent, appears also to be in part: historical giving logically reconstructed case studies in the development of mathematical theories, with "real history" in the footnotes. sociological studying the behaviour of mathematicians and partially classifying the recurrent features in their practice of mathematics educational analysis and polemic considering the impact of the presentation of mathematical developments on the comprehension by students of the processes involved, arguing the merits of presentations which retain more of the original structure of the discoveries. mathematical methodology arguing that mathematicians practice heuristic rather than deductive methods mathematical philosophy arguing against formalism and "dogmatism". In relation to this objective the material is illuminating, consisting primarily of case studies showing how a putative mathematical discovery can evolve through a series of conjectures, proofs, refutations and reformulations. In this aspect the work has more the character of sociology than philosophy. The case studies are interpreted through the introduction of special terminology describing recurrent features in the examples cited: * local and global counterexamples * monster-barring * exception-barring * piecemeal exclusions * strategic withdrawal * lemma-incorporation * proof-generated theorem * concept-stretching Alongside and interweaved in this perspective on mathematical discovery we also discover doctrines of a more philosophical nature. In particular: the core of this case-study will challenge mathematical formalism Its modest aim is to elaborate the point that informal, quasi-empirical mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism, by the logic of proof and refutation. It is clear also that Lakatos is attacking: * Formalism * dogmatist philosophies of mathematics * meta-mathematics * the deductivist approach --- Ernest Gellner described the philosopher's lectures as 'intelligible, fascinating, dramatic and above all conspicuously amusing'. - http://www.lse.ac.uk/resources/LSEHistory/lakatos.htm ---blurb Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

amitabha mukerjee