Proofs and Refutations: The Logic of Mathematical Discovery

Cambridge Philosophy Classics

by Imre Lakatos, John Worrall (Editor), Elie Zahar (Editor)

Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.

Genres: Philosophy, Mathematics, Science, Nonfiction, Logic, Education, History

196 pages, Paperback

First published January 1, 1976

About the author

Philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.
More at Wikipedia.

Reviews

[Michael Nielsen · Rating 5 out of 5]

Radically changed my idea of what mathematical definitions and proofs are, and where they come from. In particular, Lakatos convincingly refutes the idea that definitions come before theorems and proofs (as often seems the case). Rather, they arise out of repeated back-and-forth interplay between conjectures and proof-ideas.

That's a pretty abstract- and weird-sounding review. The book itself is incredibly readable, incredibly fun, and by the end will (if you're anything like me) have caused an earthquake in your worldview. So ignore the weirdness of my last paragraph, and just go read the book. It's amazing.

[Ben Labe · Rating 5 out of 5]

Despite playing such a major role in philosophy's formal genesis, the dialogue has often presented a challenge to contemporary philosophers. Many are apt to shy away from it due to its apparent levity and lack of rigor. However, the dialogue possesses significant didactic and autotelic advantages. At its best, it can reveal without effort the dialectic manner in which knowledge and disciplines develop. This way, the reader has a chance to experience the process.

"Proofs and Refutations" is a paragon of dialogical philosophy. Using just a few historical case studies, the book presents a powerful rebuttal of the formalist characterization of mathematics as an additive process in which absolute truth is gradually arrived at through infallible deductions. The "logic of discovery," he claims, is a much messier affair. Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. It is only through a dialectical process, which Lakatos dubs the method of "proofs and refutations," that mathematicians finally arrive at the subtle definitions and absolute theorems that they later end up taking so much for granted.

Lakatos contrasts the formalist method of approaching mathematical history against his own, consciously "heuristic" approach. Instead of treating definitions as if they have been conjured up by divine insight to allow the mathematician to deduce theorems from the bottom up, the heuristic approach recognizes the very top down aspect of performing mathematics, by which definitions develop as a consequence of the refinement of proofs and their related concepts. Ultimately, the naive conjecture (the top) is where the mathematician begins, and it is only after the process of "proofs and refutations" has finalized that we are even prepared to present mathematics as beginning from first principles and flourishing therefrom.

[Gwern · Rating 4 out of 5]

Surprisingly interesting, like Wittgenstein if he wrote in a human fashion, and longer than one would think possible given how straightforward the problem initially appears.

[Devi · Rating 5 out of 5]

It is common for people starting out in Mathematics, by the time they've mastered Euclidean Geometry or some other first rigorous branch, to believe in its complete infallibility. If something is mathematically proven we know beyond any shadow of a doubt that it is true because it follows from elementary axioms. Lakatos argues that this view misses quite a lot of how mathematical ideas historically have emerged. His main argument takes the form of a dialogue between a number of students and a teacher. The dialogue itself is very witty and entertaining to read. The students put forward attempts of proofs that correspond to the historical development of Euler's conjecture about polyhedra. We see how new definitions emerge, like simply connected, from the nature of the naive, but incomplete, proofs of the conjecture. We also see how generally it is the refutations, the counterexamples, that help us in the development by forcing us to specify more conditions in the theorems, using more specific definitions and hint at further developments of the theorem.

This book is warmly recommended to anyone who does mathematics, is interested in philosophy of mathematics (or science) or simply enjoys a well-written dialogue about philosophical questions. The mathematics is generally (except in the appendices about analysis) quite elementary and doesn't require any prior knowledge, though it will feel more familiar if you have some experience with mathematical proofs.

[Vasil Kolev · Rating 5 out of 5]

I would have to reread this some day. This book describes a lot of what I found missing while studying mathematics in the university, mainly the reasoning for the way proofs were, and the overall reasoning for the definitions and terms used. The book looks into those from the purely mathematical standpoint, and shows that they can be a lot easier to grasp and understand.

(the other part, the actual usage of most of it, doesn't seem to exist in a single book, but in bits and pieces in the actual areas where the different mathematical methods/ideas are used)

[Andrew]

Many of you, I'm guessing, have some math problems. You didn't do so hot in higher-level math, are more comfortable with the subjectivity of the written word, and view the process of mathematical discovery from a position of respect and distance.

What Lakatos shows you is that math is not the rigid formalistic system you may conceive of, but something far more fluid, something prone to frequent revision, something that must always have its underpinnings challenged in order to reach mathematical truth. So in this dialogue, he exposes those challenges in order to arrive at a better understanding of Euler's theorem.

What's important here, for the non-mathematically inclined, is to understand how we apply those same formalisms to our day-to-day thought. How we "monster-bar" by claiming that an exception to the rule is irrelevant or (worse) "proves the rule." How our arguments contain hidden lemmas that underpin our thought even if we don't expect them to. And it teaches us how interesting things can get when you scratch beneath the surface.

Shit, I think I might get a tattoo of that ferocious "urchin" on the book cover.

[Douglas · Rating 5 out of 5]

This is an excellent, though very difficult, read. It reminds me of Ernest Mach's "Science of Mechanics"--the latter is not in the form of a dialogue.

Having heard Lakatos speak I can see how the book's dialogue format fits in with his style which is to the point and voluble. He makes you think about the nature of proof, kind of along the lines of the great Morris Kline--still an occasional presence during my graduate school days at New York University--and who's wonderful book, "Mathematics and the Loss of Certainty" reinvigorated my love for mathematics; because it showed mathematics didn't have to be presented in the dry theorem-lemma-proof style that has had it in a strangle hold since the 20th century predominance of the rigorists (called formalists by Lakatos).

But back to Lakatos. I once thought I had found Lakatos to be putting the final nail into the coffin of the certainty of overly rigorous mathematical proof; that slight were the blessings of such rigor compared to loss in clarity and direction in mathematics. This poverty of rewards is the explicit claim of Kline, whom I had read years before coming across Lakatos. Both men believed that claims by its proponents to the contrary, rigor was more obfuscation than clarification. Indeed the distinctive feature of Lakatos' work is to skewer the rigorists with their own tools including their tedious "microanalysis." Which is why I say the reader is in for a slow ascent.

Such a view fit in with my own frustration over rigorism which diverts the student from the rich meat of mathematical ideas towards the details of the implements by which it is to be served. As an enthusiastic but relatively feeble intellect--at least by the standards of today's ultra-competitive modern university wizards--I felt cheated. I know I can understand many great mathematical ideas but I am put off by the reliance on logical primness often leading to roundabout "proofs," merely for the sake of a certain notion of rigor. (Indeed, according to mathematical logicians almost all the proofs encountered in say, a good textbook on mathematical analysis like Walter Rudin's or Paul Halmos', aren't proofs at all but merely informal arguments. Unfortunately, with the spread of computer science, their influence on the whole body of mathematics is gaining sway!)

Hence when I put quotes around the word proof, as I just did, I was following Lakatos. He gave me the reassurance to go on reading and seeking mathematical presentations which preserved the spirit of the amateur and the enthusiast. Here is Lakatos talking about the formalists,

"Formalism denies the status of mathematics to most of what has been commonly understood to be mathematics, and can say nothing about its growth. None of the 'creative' periods and hardly any of the 'critical' periods of mathematical theories would be admitted into the formalist heaven, where mathematical theories dwell like the seraphim, purged of all the impurities of earthly uncertainty."

By creative periods, Lakatos has in mind no less than those which issued from the ranks of men of the caliber of Isaac Newton; whom he said had to wait nearly four centuries to be "helped into heaven" by the likes of Russell, Quine and Peano. What a rogue! And much to my liking. I might add listening to Lakatos--as can be done on the internet--infects the listener with this roguish enthusiasm and may make you want to read this book all the more. But I warn you, it's a slow go itself.

I believe Lakatos' basic diagnosis is essentially correct. Unfortunately, he choose Popper as his model. I am not a philosopher and so I make no pretense to speak authoritatively about this. Instead I follow--and point the reader towards--a wonderful essay by the little-known Australian philosopher, David Stove, entitled, "Cole Porter and Karl Popper: the Jazz Age in the Philosophy of Science". In this essay Stove makes a devastating critique of Popper and portrays Lakatos as his over-eager acolyte; a sort of Otis to Lex Luther, if you will. And like Otis, it appears that, by taking Popper's argument too far, Lakatos incurred the disapproval, if not emnity, of the former. But Stove also makes the point that Lakatos was, in fact, only carrying "Popperism" to its logical conclusion for Popper could not find a way to place a limit to his notions of falsifiability and bracketing.

According to Roger Kimball's review of Stove, "Who was David Stove", (New Criterion, March 1997), "In [Popper's] philosophy of science, we find the curious thought that falsifiability, not verifiability, is the distinguishing mark of scientific theories; this means that, for Popper, one theory is better than another if it is more dis-provable than the other. 'Irrefutability,' he proclaimed, 'is not a virtue of a theory . . . but a vice.' Popper denied that we can ever legitimately infer the unknown from the known; audacity, not caution, was for him of the essence of science; far from being certain, the conclusions of science, he said, were never more than guesswork ('we must regard all laws and theories … as guesses')"

It hardly needs to be said that scientists--almost to a man--line up with Popper's notion of falsifiability. Stove attempts to show how this has lead to what he calls irrationalism; by which he means the destruction of the intellect. It is this destruction, not irrefutability as Popper claims, that has lead to the ascendancy of bogus ideas such as Marxism, feminism and, lately, deconstructionism.

And this is why, even though I recommend Lakatos' book, ultimately I must back away from it. Though I find his critique of rigor appealing it comes at too high a price if I also have to accept the attendant irrationalism. I think we need to revert to an older point of view, echoed as well in the writings of the late Mortimer Adler, who also had some points to pick along these lines with modern philosophy and who would have us hearken back to the concreteness of Aristotle.

It does seem that the prevailing belief that we cannot really know anything--that there is uncertainty even in mathematical proof--has something to do with the loss of confidence in Western civilization itself; that the return to verifiability from falsifiability would herald a return to the old confidence in not only Western civilization but the idea of civilization itself. Today all we have is culture and that allows no judgment as to progress of mankind--except as an outworking of an all-encompassing statism. With culture in the place of civilization there can be no question of the transcendent that applies to all men. There can only be man-the-organism exhibiting behavior much as beavers or wasps build dams and nests. The difference between man and animals is thus a matter of degree and not of kind. Did Lakatos know he was doing all this? I don't think so but interesting as Proofs and Refutations is, it exhibits a view as blinded as 20th century thought itself.

[Nick · Rating 5 out of 5]

Math as evolving social construct. Truth itself evolves. And Lakatos knows the history of eulers theorem, presents it as a classroom discussion making us realize that nothing is ever static in mathematics.

[Flx]

A very compelling and ingeniously structured argument about math. Actual mathematical discovery is messy and iterative. Axioms and definitions are not static and assumed, but continually adapted to fit new results and counter examples.

The structure is a Socratic classroom dialogue about the Euler characteristic of polyhedrons, a seemingly simple result dating back to 1537 that could be taught to an elementary schooler. Yet the complexity of the dialogue that is revealed through footnotes to map onto a 500 year debate over the characteristic reveals writing a proof is not definitive.

Mathematics is not done by writing down your axioms and then like a computer deriving the intended result, a "formalist" account which I was sympathetic to. Instead, creative minds across hundreds of years come up with counter examples, reformulations and then further counter examples that slowly begin to approximate formalist mathematical truth. Yet these counter examples, at least from the case of the Euler characteristic, seem never entirely possible to rule out. They require mathematical creativity, being able to conceive of shapes that violate the result. They could not be discovered by pure logic.

A suprisingly fun book with a format that should be imported to other domains. Why not a dialogue with footnotes tracing theories of economic growth, or following the web of letters exchanged between early modern political philosophers?


Notes:
Calling all the students greek letters instead of real names perhaps bought me to the limits of my appreciation for mathematical abstraction. Alpha arguing with Beta was much harder for me to track then say Alex arguing with Bettie.

The students were passionate! Resorting to make calling and aggression when Rho had the wrong understanding of "monster" counter examples. If only my own university seminars were this heated...

[Conrad · Rating 5 out of 5]

By far one of the best philosophical texts I've read. It takes a theory about the sides of a polyhedron by Euler and uses dialogue form to show how the methods of inquiry of a handful of different theoreticians fall apart when attempting to prove or disprove the proposition. I've never gotten past Algebra II, and I still understood most of the book, though to be sure I missed out on the bits of calculus here and there, and didn't know enough about math to discern which dialogue participant stood for which philosopher. Definitely worthwhile.

[Erica · Rating 4 out of 5]

I rated this book 4 stars but it would be more accurate to call it 4 stars out of 5 for a mathematics book or for a school book or for a required reading book. Portions of Proofs and Refutations were required reading for one of my classes for my master's degree, but I liked it enough that I finished it after the course was completed. I really enjoyed wrestling with the idea that "proofs" can not be the perfect ideal that mathematics and mathematicians should strive for. Lakatos argues that proofs are either far too limited to be of any use, or else they invariable let in some "monsters". I can see my self re-reading this book in the future, but I would not recommend it to anyone in my social circle. I would recommend it to anyone with an interest in mathematics and philosophy.

[Jake · Rating 5 out of 5]

To quote Northrop Frye, we go see MacBeth to learn what it feels like for a man to gain a kingdom but lose his soul. We develop mathematical definitions, examples, theorems, and proofs to meet human needs through heuristics. We assume, incorrectly that mathematics are solid continents of rules and facts, but what we observe are loosely connected archipelagos of calibrated and stable forms where those islands are in constant risk of being retaken by the sea. Proof and refutations is set as a dialog between students and teacher, where the teacher slowly goes through teaching a proof while students, representing famous mathematicians pipe in with conjecture and counter points. Lakatos goes through great pains, to succinctly convey the broad perspective of the students(Euler, Cauchy, Poincare, etc).

[Quiver · Rating 4 out of 5]

'Trendsetting' —the beginning of mathematical practice.

[Jake · Rating 4 out of 5]

What does it mean to prove something? I don’t know. But now I know I don’t know

[Keith Peters · Rating 4 out of 5]

I got into an argument recently with a PhD about the I infallibility of math. My argument came down a bit on the human endeavor at the heart of math. This book is a bit awkward and suffers from the hubris I see in basically all philosophy texts but covers my concerns well.

[ltcomdata · Rating 3 out of 5]

A book about the meaning and philosophy of mathematical proofs.

The most important lesson from this book is the idea of proof-based theorems. That is, one should look at one's proof, and pin down exactly what properties are used, and then based on that thorough examination, state one's theorem accordingly. In this view of things, the theorem statement becomes secondary to the proof idea, which then takes precedence as the most important part of the mathematical work.

To create the most apt theorem statement, the proof is examined for 'hidden assumptions', 'domain of applicability', and even for sources of definitions. Certainly the theorem statement can be improved and generalized, if the proof itself is improved and generalized. That is, the proof always takes precedence.

The book goes through the Euler theorem that relates edges, vertices, and faces in polyhedra (V - E + F = 2). As it turns out, the proofs generated by earlier mathematicians (Euler and Cauchy, among others) did not entirely apply to all the polyhedra in their most general examples. Is the theorem wrong, then? This book answers with a resounding "no!" --- because the proof is not wrong, it is simply miss-applied to more general general definitions of polyhedra than were intended by the theorem creators.

And indeed, that is one of the main messages to take away from reading this book. Definitions stretch as the history of mathematics rolls on; quite often slowly, and imperceptibly, so that when old theorems are seen in the light of the new (stretched) definitions, suddenly the proof is seen to be false, or to assume a 'hidden lemma'. In fact, the definitions themselves have become more generally encompassing without this fact being consciously realized by the mathematicians working with the new definitions. Thus the old proofs are seen as 'obviously' assuming a 'hidden lemma'.

One particularly enlightening application of this 'proof-first' method comes via the proof of Cauchy that the limit of a sequence of continuous functions is continuous. His proof (still the standard proof in beginning analysis) contained a 'hidden lemma'. Indeed, when other mathematicians discovered that the theorem was not true in general, and his proof was checked for errors, this 'hidden lemma' was discovered. The discovery led to the definitional distinction between 'point-wise convergence' and 'uniform convergence'. And the exact condition necessary for Cauchy's proof to be correct became the definition of uniform convergence.

[0xd34df00d · Rating 4 out of 5]

First of all, a disclaimer: I'm not a working mathematician, I'm not discovering anything, I'm not building theories, concepts, or paradigms. I just read math books and work out some exercises, that's all. Despite that, I also happen to be a formalist: if it isn't mechanized, it isn't a proof, but merely an intuition graft.

That out of the way, the Platonic (Socratic? Socrato-Platonic?) dialogs themselves are a pleasure to read, if interpreted as a short history of mathematical rigor and an excursion into the attitudes towards this rigor. But, alas, it's not a mere book about the history of the Eulerian polyhedra.

Irrespective of Lakatos position, one (big, IMO) omission is the interpretability and applicability of theorems (which is ironic, given that the book is largely about the sociology of math). Sure, you can lemma-incorporate all free variables in your proof into the theorem statement, making it one big conjunction like `(p : Polyhedron) → Stretchable p ∧ Simply-Connected p ∧ … → Eulerian p`, but is it really more useful than asking for a mere "convex polyhedron without ring-faces"? Especially at the informal, "intended meaning"-allowed level, it's a lot easier to check that a given polyhedron is convex than checking all those proof-generated conditions.

Or forget polyhedra and take a simpler object of study, like natural numbers. There's a theorem about prime numbers distribution, infinitely precise and almost as infinitely useless, claiming `(p : ℕ) → Prime p ⟺ p ≡ 2 ∨ p ≡ 3 ∨ p ≡ 5 ∨ …`. A much more relaxed theorem, like the one saying that n-th prime number is asymptotically n ln n, is also much more useful.

A similar thread is the claimed symmetry between overapproximating theorems (requiring too little and proving too much) and underapproximating (requiring too much and proving too little). This looks downstream of a more general stance appearing to be winning in the dialogs: that is, that the theorems shall have the form A ⟺ B, not mere A → B. The dialogs seem to resolve towards the proper theorem (with a proper proof) establishing sufficient conditions also. For instance, Omega says a certain proof that a certain class of polyhedra is Eulerian doesn't work because there are Eulerian polyhedra outside of that class: "[I didn't tell you about Gergonne's proof that Gergonne polyhedra are Eulerian] because I immediately refuted it by non-Gergonnian polyhedra that are Eulerian." Or, later, Omega explicitly says:

My quest is not only for certainty but also for finality. The theorem has to be certain – there must not be any counterexamples within its domain; but it has also to be final: there must not be any examples outside its domain.

To me, being non-final is in no way a reason to discount a theorem. Sure, decidability is better than semidecidability, but semidecidability is infinitely better than nothing. Semidecidable things are useful per se (and not merely semi-useful).

Lakatos position itself, I think, clamps mathematics, history of mathematics, sociology of mathematics, and pedagogy of mathematics, into one big pile. Sure, one might argue they aren't separable. I argue they are, but arguing about that proper is outside the scope of this review, which is already long enough.

For instance, this clamped math/sociology leaks (albeit weakly, and you have to squint to notice) via phrases like the "intended meaning [of polyhedra]". I'm not sure it's reasonable to mix what amounts to basically "how do we know different people agree when they speak of the same thing" with "how do we ensure the derivation is sound and a proof actually proves the thing". These are adjacent problems, but they are not the same. The first one is field-invariant (that famous question of "what do we mean by red?"), the second one is what math is. Mixing them makes solving them harder, not easier. The only thing made easier by mixing them is showing that the problem exists.

Anyway, me being a formalist, I can't help but note most of Lakatos' objections are reduced as much as theoretically (pun intended) possible if math is treated rigorously (albeit they are not reduced to zero, sadly). Decidability? Checkable. "Intended meaning"? Here it is, in symbols, and what remains is the problem of translation (that remains anyway for anything non-trivial). Kappa's words about the infinite regress, claiming "Thus you will slide on to another infinite slope: you will be forced to admit of each ‘particular linguistic form’ of your true theorem that it was not precise enough […] you never get out of vagueness"? Formalism again. With formalism, you only ever have to trust the meta-axioms (and the translation, but more on that later).

And thus Kappa's words:

Why not accept that our ability to specify what we mean is nil, therefore our ability to prove is nil? If you want mathematics to be meaningful, you must resign of certainty. If you want certainty, get rid of meaning. You cannot have both.

is a false dichotomy, really — it's more like a spectrum, since "uncertainty" is not binary. You can trust the "intended meaning" and the informal, intuitive proof (which all are specific to a domain, so the more domains you do, the bigger is your trust surface), or you can trust merely the meta-axioms (which are all-encompassing by definition).

Curiously, the process of mechanization can be considered an ultimate proofs-and-refutations process, with sub-second latency of your favorite proof assistant versus months and years of the community of flesh-and-blood-mathematicians. You try to prove a lemma, the computer screams at you, you adjust the proof or the definitions or the premises, rinse, repeat.

But this requires formalization in the first place, by construction. In a way, the most Lakatosian thing is the most anti-Lakatosian thing.

Also, reframing the main thesis in a formal framework helps (or, at least, helped me). Exception-barring is just adding extra arguments to the theorem statement. Lemma-incorporation is finding free variables in the proof body and making it well-scoped by lifting them, again, to the theorem statement (which makes it equivalent in a sense to exception-barring, really, although coming from epistemically opposite starting points).

But besides the dialogs there are also the appendices. Curiously, they all point towards formalism as well. For instance, deconstructing the concept of uniform convergence via "making explicit the functional dependencies"? That's the job for the type checker!

The other point I haven't touched is mathematical pedagogy, and here I half-agree with Lakatos and the editors, but for different reasons. While they say that history of mathematical concepts, necessarily showing failed proofs, is important for education, I doubt it is. Doing several versions of a proof, refining the concepts in the meanwhile, and so on, is doubtfully fruitful. What is fruitful is building the intuition why a certain concept is useful, and it doesn't have to come from failed proofs (after all, a proof is always arbitrary in a certain sense).

I absolutely hate textbooks that are just "definition-theorem-proof" in a loop, they suck, that's bad pedagogy and good sleeping pills, they should not exist. What is good pedagogy, though, is arriving at concepts via the generalizations. At least, that's what the books I've most enjoyed do — like Leinster's one on CT, or Aluffi's one on algebra, or TAPL, or the likes. Good pedagogy, good motivation for a concept, that is, does not require retracing the steps of generations of mathematicians before.

And, me being a (hopefully) honest formalist, I can't help but emphasize again that the problem of translation still remains (but then it's also present in the "intuitive", "polyhedra-as-social-phenomenon" approach, just in a latent form). Moreover, that's precisely the worldview-level contribution of the book for me personally: I knew and kept in active memory that ultimately you have to trust your meta-axioms (your ZFC, or whatever the foundational logic you do), but turns out you really need to trust those funny symbols on paper (or in Unicode) really representing your applied problem, and this is something that's super easy to forget if you're tunnel-vision-fixed on foundations. So, sure, you still have to trust something even if you mechanized everything. The problem is, without formalism and mechanization you have to trust a lot more.

But then maybe applied math isn't real math anyway.

All in all, this book made me think hard to argue, and it also reminded me of the problem of translation, and the dialogs are genuinely fun to read, so I might disagree with Lakatos all I want, but it is a good book. And I'll have to re-read it in a couple of years to appreciate it fully.

And there's also a phrase buried somewhere that I should remind myself more often: "we can appreciate a poem without considering it to be perfect". I probably spend too much time with the machines, but I start to forget how does it feel to appreciate art for what it is, and not hold it to some abstract standards of rigor.

[Eryk Banatt · Rating 3 out of 5]

I picked this up seeing it on a list of Robb Seaton's favorite books". I think I can describe it as "Plato's The Republic meets Philosophy meets History of Mathematics" and that sentence can more or less describe the entirety of the book.

I will admit that the book was a bit challenging for me, and I suspect I will revisit this book when I get a bit better at math, but for what it was I think it was quite readable and I enjoyed it. It was a little dry at times but the dialogue was very interesting and posed some very interesting questions about the way people have approached solving problems throughout history.

A line I thought was pretty interesting is the following:

Of course I would. I certainly wouldn’t call a whale a fish, a radio a noisy box (as aborigines may do), and I am not upset when a physicist refers to glass as a liquid. Progress indeed replaces naive classification by theoretical classification, that is, by theory-generated (proof-generated, or if you like, explanation-generated) classification. Conjectures and concepts both have to pass through the purgatory of proofs and refutations. Naive conjectures and naive concepts are superseded by improved conjectures (theorems) and concepts (proof-generated or theoretical concepts) growing out of the method of proofs and refutations. And as theoretical ideas and concepts supersede naive ideas and concepts, theoretical language supersedes naive language.

This quote reminded me a lot of a great blogpost I read once, The Categories were Made for Man, Not Man for the Categories. The cool part of this part of this passage is the idea that statements have different consistency values depending on the language in which you talk about them - you have certain things that might be true in a naive language (i.e. finned creatures in the ocean are called fish and a whale is a fish) that may be untrue when you drill down into a different language (i.e. whales and tunas are not in the same taxonomic classification and therefore only one can be a fish).

Overall pretty readable for what it is - will revisit again someday.

[Nick Black]

Amazon third-party 2009-04-15. I'm excited about this one, riding in as it does on a ringing recommendation of Conrad's (although I'm a bit puzzled by his tagging of House of Leaves with "masterpieces"). Looks to contain echoes of Halmos's Automathography and Davis's The Mathematical Experience; we'll see.

[Aleks Veselovsky · Rating 3 out of 5]

Although I appreciates Lakatos' classroom discussion style as original I had a hard time keeping up with the development of the conversation and keeping the original question in mind. Nevertheless, I can name a few lessons learned. I think that the use of counterexamples is underutilized in the classroom and Lakatos shows how useful it can be. The idea that the definition creates the mathematical meaning is a another powerful one, and I think it would be interesting to do an activity where students could come up with initial definitions and then try to rewrite them to make them more broad or more narrow.

[Kelly John · Rating 5 out of 5]

Probably one of the most important books I've read in my mathematics career. This short, but inspiring read discusses not a particular theorem or proof in mathematics, but rather the process of how mathematics is developed from an initial idea, hypothesis, monster-barring, expansion of the theorem, etc.

It really shows and demonstrates how you can take a really simple relation and build it up to create an extensive and interesting theory (and possibly) field of mathematics one step at a time.

If you are going into mathematics at a University level, I would highly recommend this book.

[Lucille Nguyen · Rating 4 out of 5]

A fun little read (admittedly, I don't like the Socratic dialogue style of literature) about the genesis and function of mathematical discovery. A lot of it is spent trying to argue against formalists and dogmatists, which is interesting, because my intellectual journey leads me rather to to a formalist style of mathematical philosophy (from Haskell Curry, more appropriately called structuralism today). Gives a lot of food for thought about the development and growth of mathematics.

[Jake · Rating 3 out of 5]

This deserves a higher rating, but the math was beyond my meager understanding so I struggled a bit. The philosophy was good though. Science and math make progress by conjectures leading to proofs which are refuted with counterexamples. Then the conjectures can be modified and tightened up to make theories. Written in Socratic dialogue.

[Jonathan · Rating 5 out of 5]

The very idea of mathematical truth and the changing notions of rigour and proof are all discussed with stunning clarity. Imre Lakatos has written a highly readable book that ought to be read and re-read, to remind current and future generations of mathematicians that mathematics is not a quest for knowledge with an actual end, but shared cultural, even psychological, human activity.

[Robb Seaton · Rating 5 out of 5]

Begins strong with a deconstruction of the Euler characteristic, but soon gets bogged down in philosophy, along with a troubling amount of relativism, although I'm not entirely clear about what Lakatos intends when he writes about truth, certainty, and progress.

[Arron · Rating 5 out of 5]

This book is a wonderful blend of philosophy, history, pedagogy, and interesting mathematics. Worth multiple readings.

[John Lepp · Rating 4 out of 5]

Honestly, I prefer it to MSRP.

[Gage Hoefer · Rating 4 out of 5]

Not for everybody- this made me reconsider some of the fundamental ideas I held about mathematics and their foundations. The dialectic lives on even in math apparently.

[T · Rating 4 out of 5]

"The proof proves the theorem, but it leaves the question open of what is the theorem's domain of validity. We can determine this domain by stating carefully the 'exceptions' (this euphemism is characteristic of the period). These exceptions are then written into the formulation of the theorem. [...] This is why Euclid has been the evil genius particularly for the history of mathematics, and for the teaching of mathematics, both on the introductory and the creative levels." (140)

"Mathematical activity is human activity" (146)

Imre Lakatos argues that logicists and formalists misrepresent mathematical discovery as a purely rule-finding, dehistoricised mission. However, for Lakatos, mathematics develops through proofs and refutations, a dialectical process which processes through struggle and experimentation. It isn't that mathematicians or scientists follow a linear process of building knowledge, rather they stumble past problems developing arguments that encapsulate the problem, until a new formula is formed. "X+Y=Z", becomes "X+Y=Z in all cases bar...", until a new formula comes along and proves that "X+Y=Z+N".

I daren't guess how little of this book I truly understood, in fact I must admit that my knowledge of maths is embarrassingly poor, but this book did open up my mind to a much more humanistic understanding of maths and science. I have always wondered if there was a "philosophy of maths" in the way that there is of science, sociology, and other fields, and Lakatos has explained that there is. Mathematics is a human activity, and thus the search for certitude is part of the dialectical evolution of knowledge. This book will absolutely be one that I return to.

[Allan Olley · Rating 5 out of 5]

This is a fascinating and epochal contribution to the philosophy of mathematics. Most of the text is told as a dialogue between imagined students each with the name of a Greek letter (Alpha, Beta etc.) under the guidance of Teacher. They discuss the Eulerian conjecture that Vertices minus Edges Plus Faces equals two (V-E+F=2), attempting to prove it and discussing a host of counter examples and how they might be dealt with. In this discussion the nature and purpose of mathematical proof and the best methods and heuristics of mathematical investigation. In the dialogue accentuated in the footnotes an argument is made for a rejection of a dogmatic, fixed and certain notion of the product of the mathematical exercise.

This book is a triumph of the dialogue format. The discussion is both engaging and understandable. One begins o speculate oneself how one might respond to objections and counter examples. Some frustration comes with the fact that you can't get a further explanation of some technical point or a response to some idea that comes to you from the interlocuters.

The Appendices are written in a more standard discursive format. The history of mathematics offered as implicit in this book seems a bit potted and probably only hints at some of the messy back and forth at work in actual historical debates. Also the description of philosophically dogmatic positions (ones that focus on or claim absolute truth for some mathematical claims) as authoritarian seems either speculative or unwarranted in its implications.