Talk:Mathematical practice

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Can't work with long talk files for some reason. Points made here are also applicable to folk mathematics.

Examples? Chas zzz brown 01:18 Jan 17, 2003 (UTC)

You state yourself correctly that Fermat was defying an emerging concept of peer review and proof, in keeping his own proofs secret.

Defying? "Out of step with" is more like it; see my comments below.

You state also correctly that these are the two main differences between our modern ideas of science and mathematics. In fact one can go quite a bit further to say that mathematics has proof, and peer review, science has peer review, and empirical evidence, and the social sciences have empirical evidence, and correlation (not causal proof). Folk mathematics was not likley to distinguish these much, if at all.

This discussion regards the article "folk mathematics", as distinct from "mathematical practice". What exactly is the distinction? I know you feel that it is not mathematical practice; fine, then what precisely is it? All I've seen so far is that it is a very vague description of a group of things that today we would call calculation, engineering, and physics - none of which are what we mean now by mathematics. Thus, the argument that somehow modern mathematics has been affected by "folk mathematics" is, to me, spurious - it is rather of interest because it sheds some light on why certain ideas, like counting, appear to be innate.

The question is, where does 'proof' and 'peer review' lead us to different conclusions than less rigorous quasi-empirical and social methods?

What social methods and quasi-empirical methods common in "folk mathematics" are of application to the problems of modern mathematics? Could they be applied to problems of set theory, abstract algebra, topology, analysis, etc.? No; because as you note above, these are the concerns of modern mathematics, not "folk mathematics", which has an entirely different agenda.

Putnam wrote a paper on this called "What is Mathematical Truth" (1975, reprinted in 1998) that is worth reading on this. In his view, not often, mathematics has always been at least quasi-empirical, and has always accepted oddballs (like Fermat) based on their reputation, as opposed to a formal process of proof (remember that Fermat's Last conjecture was in fact known as Fermat's Last Theorem on the assumption he MUST HAVE BEEN RIGHT BECAUSE OF HIS REPUTATION) right into modern times.

The conclusion that people believed that Fermat must have been right because of his reputation is just not true. First, Fermat didn't publicly proclaim that he had a proof; this appears as a note, probably to himself, in the margin of a book. This is as opposed to, for example, Goldbach's conjecture, which Goldbach publically claimed to believe was true, but did not claim he had a proof of; or Heawood's (incorrect) assertion that he had a complete proof of the four color theorem. Secondly, there is plenty of reason to believe that Fermat thought he had a proof, but later changed his mind (his later published proofs of subsets of the problem would have been unnecessary if he indeed had a proof of the entire conjecture). After the marginal note was discovered after Fermat's death, people believed that Fermat might well have had a proof. However, despite the name, it was not considered an actual theorem (for example, other works would not accept it as the basis of their own proofs, as they would with other theorems) until Wile's (corrected) proof.

There is a distinction among the mathematicians I know between believing that something (for example Goldbach's conjecture) is true, and having it proven to be true. Most mathematicians (if it's in a field they care about) have opinions - sometimes strong ones - about the truth of the various open problems in their fields; but they always make a distinction between these beliefs, and the concept of a proof of these conjectures.

On the 'groupthink' question, because you see the word 'mathematics' in the title, you assume that the exact distinctions applicable to the modern mathematical practice (including most critically axiomatic proof and peer review) must be applicable in detail to the practices or names of the practices describeed. This is obvious nonsense, else we would have 'folk abstract algebra', 'folk group theory', 'folk statistics', 'folk calculation', 'folk calendars', and twenty other articles that could be picked to bits for not obeying some distinction associated in modern times with 'statistics' versus 'probability' for example. That is clearly not the right thing to do.

Well, why then are you using modern mathematical phrases such as "algebra" in an article about "folk mathematics"? If you attempt to define it by using these terms, then I think it's fair to require you to use the usual meaning of these terms; when you use those terms inconsistently, I don't feel it's improper for me to make corrections.

Your use of the term "groupthink" is usually as an ad hominem argument. If your arguments are valid, defend them. If my arguments are invalid, disprove them. When you apply the "groupthink" label, that doesn't make my arguments any less valid; it's simply an elitist way of calling me a name.

(abacus as evidence of calculation not of any knowledge of algebra)

Again, you apply a modern definition to an ancient practice/device. Do not tell me an abacus can't be used to solve simple problems with unkonwns, it can, although there is little evidence that the practice was very widespread. Admittedly this is like taking a Rubik's Cube as evidence of group theory.

Exactly - group theory is a method of solving the Rubik's cube, not the other way around. Lots of people, without knowing any group theory themselves, have come up with solutions for the Rubik's cube. Their solutions can also be derived using group theory. That doesn't mean that having a Rubik's cube means you know group theory.

An abacus is a tool that is useful for performing calculations - no more, no less. What you choose to do with that capability is a different matter. Likewise, the knowledge that there is a method of solving problems through the abstraction of unknown quantities into symbols, and then performing manipulations on the symbols, is a useful tool that we could call "algebra". Whether you perform those manipulations on an abacus or a slide rule or by moving stones around is independent of this knowledge.

(Fermat resisted the emergence of modern mathematical practice)

A very good point, worth making in the article. Fermat thought results mattered, and proof was a game, more of a cleverness or status competition. Fermat may have been the last working folk mathematician... his 'Last Theorem' not being called a Conjecture is evidence for that. Also there is the fact that when Peano wrote his Axioms of Arithmetic, and others attempted to add the Axiom of Choice, he demanded 'proof' for them in terms of his existing axioms, but had not provided it for his own. This raised surprisinly little controversy.

... and this relates to "folk mathematics" how? It's an interesting period in the history of the developments which have led to our current understanding of the axiomatic approach. Hoiwever, I didn't mean to imply that I thought that Fermat was "resisting the emergence of modern mathematical practice". In my opinion, Fermat was not trying to "champion" some older, more "traditional" approach to mathematics. His "Last Theorem" was not some kind of published challenge to the world; his announcement of his proof consisted of a note in the margin of a copy of Diophantus. There were many, many proofs that he did publish; and there is certainly no evidence that his actions were part of an attempt to "challenge" the mathematical "paradigm"; instead there is evidence that he found the publishing process tedious and not to his liking, and that, as an amateur mathematician, he preferred playing the maveric.

As far as Peano, etc.; again, I think it's interesting, but I just don't see any tie-in to anything that could be called "folk mathematics" Chas zzz brown 04:02 Jan 17, 2003 (UTC)

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Reading the above, I realize I was starting to fall into the trap of the deadly Flame, so let me start again, from an encyclopedic point of view: With both this article, and the folk mathematics article, I'm just not clear, after reading the article, what is intended by "mathematical practice". I can imagine a sociological study of mathematicians and their behaviour, but I honestly don't see how that is relevant to assertions about whether Cantor's diagonal proof, for example, really is a proof. Or is it the history of how it came to be that Cantor was the person who provided it at that particular point in history, but again, I just don't see the connection that then negates the validity of "truth" from a mathematical standpoint.

If you could clarify the answers to questions like these in the article, then at least I'd know what I was arguing about!! Cheers Chas zzz brown 08:54 Jan 17, 2003 (UTC)

For the second time, a crash wiped out detailed responses to all of the above. I am not going to attempt to discuss anything at length in any talk file ever again. My apologies, as this is an interesting topic well worth discussing.

Before pressing Save page or Show preview, you could copy the text in the edit window to an editor and save it on your own computer. - Patrick 12:39 Jan 17, 2003 (UTC)

If you can't see the relevance of the Peano example to practice, perhaps you can see the relevance of the Pythagoras example. Also the fact that Euclid certainly thought he was doing something more than proving a little game of Euclidean geometry, back when we thought the world was a lot more Euclidean. The mystic implications of this (see sacred geometry for a very small subset of the folk mathematics issues) were very important to Euclid, to Pythagoras, and even to Peano.

Even if *I* didn't see the relevance, I'd still have to acknowledge that the book "New Directions in the Philosophy of Mathematics", Thomas Tymoczko, 1998 edition, starts with five papers "Challenging Foundations" by Hersh, Lakatos, Putnam, Thom and Goodman, plus two by Polya that are collected there precisely because they believe that THERE ARE NO foundations of mathematics. These are not lightweights and my copy of this book is very heavily marked up after two reads this past summer. What is said in foundations of mathematics reflects the conclusions, and what is not said in philosophy of mathematics reflects the fact that AxelBoldt hasn't read this book. So, if there are no foundations, just what is math based on?

The second half of the book, titled "mathematical practice", is admittedly poorly summarized by my article here, but the papers are diverse, by Wang, Lakatos, Davis, Hersh. Then there's "the evolution of mathematical practice", three more papers by Wilder, Grabiner, Kitcher. Then some discussion of computers and what they mean to mathematical practice, by Tymoczko, De Millo, Lipton, and Perlis and Chaitlin.

Finally "Proof as a source of truth" by Resnik, "proof and progress in mathematics" by [[Thurston" and "Does V equal L" by Maddy.

Wittgenstein, Austin, Quine and Grice are mentioned in sections on "mathematics as a language", which is particularly fascinating.

If you care to read this book, perhaps my crash problem will have ended by then, and I can come back and dare to add text to this page, e.g. first names to those authors.

If not, find someone who has, and discuss these matters with them, not me, as I have done the best I have time for, in the article, and in these responses.

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Regarding "Mathematical Practice as the Practice of Mathematics": What does this mean? What's the point? That "Mathematical Practice" should mean mathematics as practiced by everyone in day to day life? In my non-expert mind Mathematical Practice refers to "how do professional mathematicians prove theorems, how they interact with one another, how do they share ideas and engage in debate and controversy, and (yes I'm loathe to mention it) what paradigms do they as a group maintain?" Should there be a fork b.w. "Mathematical Practice" more in-line with the characteristics and behaviours of professional mathemeticians, and another article more in-line with folk mathematics? Reuben Hersh and his (excellent) The Mathematical Experience. I might be Ok with keeping the current stuff here with some rewording and just changing the title from "Mathematical Practice as the Practice of Mathematics" to something a little less redundant. Thanks!--63.138.93.195 18:18, 23 April 2006 (UTC)