Proofs and Refutations: The Logic of Mathematical Discovery

Cambridge Philosophy Classics

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

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 the logic of proofs and refutations.

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

188 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.

[Felix Brn]

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.

[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.

[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.

[Richard Thompson · Rating 4 out of 5]

I was drawn to math at the high school level because it seemed to be the one area of study that offered absolute truths. I was OK with the idea that they were truths about abstractions, and that they got muddy when applied in the real world. But I loved the idea that you could start with a set of basic axioms and rules of logic and use them to uncover wild universes of structure and connections that were consistent and true and that always built on one another to expand human knowledge.

Then I learned about Godel. Some truths cannot be proved. No axiomatic system is complete. My faith was further dashed by Russell's barber who shaves everyone who does not shave himself. Oh no! My beautiful world of mathematical perfection was melting away. Perhaps even in math there are unresolvable paradoxes. And now Lakatos - proofs are not to be trusted; there are often (perhaps always) counterexamples and refutations. What we thought was absolute is never so. But under the system of proofs and refutations described here, this is a strength, not a weakness. Every proof is just a test, an entryway for stretching and expanding our conjectures and for adjusting our proof and revisiting our definitions and assumptions so as to get better stronger proofs and find answers to questions that we were not initially even asking. It makes math feel like science. There is still great hope for advancing knowledge, but it's a different kind of hope. I'll have to get used to it.

[Philip Naw · Rating 5 out of 5]

I would like to give this book a 4.5/5 but I rounded up and gave it a 5/5.

The good:
I have a background in set theory or axiomatics, and so the material in this book initially appeared quite shocking to me. I have studied Hegel for quite some time now, but Lakatos' book introduced me to a new side of the dialectical method -- yes, this book will teach you the method of "Proofs and Refutations" which is, a dialectical method of mathematical discovery. The book is profoundly deep, in a philosophical way, and it was not too difficult, which is probably why I enjoyed it so much.

The bad:
The book is written almost like a story, but although the action and dialogue are meant to appear natural, the book is a little corny or hard to follow/relate sometimes. At some parts of the book, the amount of prerequisite mathematical knowledge is small, then suddenly takes a giant leap into undefined (but commonly known in advanced mathematics literature), so it can be a little difficult. Anyways, the bad is so negligible, I would say that if you like philosophy and if you like math, AND if you want a new perspective -- read this very readable book.

[An Te · Rating 4 out of 5]

In sum, mathematics form and definitions are not immutable. One must freely cast a system. Scary right, as most conceive as mathematics as fixed and unchanging in its findings. Not so. Mathematics is experimental in nature and responds to where formal mathematics take the discipline. It is a very human affair. Faith in one's proofs then lay the 'foundation' stones for future mathematics to be based upon.

Maths as an act of faith. Well sure, Godel and Tarski prove as much. Mathematics is not truth but the search for a consistent procedure to capture the order and creativity of our God. Let a mathematical system arise, let faith and trust arise. One and the other are co-terminous, in my humble opinion.

(* has sin marred our ability to comprehend the consistency of God and his creation... this was pondered by Medievalist thinkers. Such lovely questions!)