Talk:Mathematical folklore

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Apology in advance - a detailed and polite response to each and every one of the points in this Talk file was eaten by a crash - so this will be less polite

This article really suffers from a fundamental confusion over three different entities: mathematics in its current form, calculation, and physics.

No, YOU suffer from a fundamental confusion that mathematics, calculation and physics are distinct, one shared by others as a symptom of scientism. Older cultures based their conceptions of prediction of events not on absolute trust in human cognition and 'proof' but on more 'mystic' or 'ethical' conceptions. That bias is extant throughout what you have written, as is your arrogance:

When you state: ...their conceptions of prediction of events... you are not talking about mathematics. Mathematics doesn't predict that there are an infinite number of primes; it proves it.

The concept of prediction vs. proof is a modern one, that of science vs. math. It should be covered in the article. No, I would not confuse the concepts in an article about physics or mathematics. It is however necessary to treat them together in an article about FOLK mathematics, where this "confusion" (your word, a perjorative one) was common, and probably necessary, given their technologies.

modern... didn't the Greeks prove statements?

Physics predicts events; so your comments in this regard indicate a confusion - you are attempting to define mathematics as something to do with the real world; but that is not the the domain of mathematics. Chas zzz brown 01:18 Jan 17, 2003 (UTC)

That is your own personal philosophy of mathematics. Not mine. And not that of every practitioner creating proofs either.

It is quite interesting, from a philosophical and historical viewpoint, to observe "naive" mathematical ideas and how they inform the current state of mathematics; but this article is littered with confusion and bias.

Nonsense. It is your statements that are littered with, at best, unthinking groupthink probably inherited from your math teachers. While modern physics can be compared to other ideas and called "naive" due to provably better results, it's not at all clear that what distincts folk from modern mathematics is anything but social. More on that below:

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. 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.

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

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I removed:

These could be said to have been carried into modern times by a single surviving copy of manuscripts in an Arab library, reprinted in Latin by Diophantus and delivered to Pierre de Fermat.

It's quite a stretch to imply that the axiomatic approach, as begun by Euclid, would not have proceeded without some single document read by Fermat in the 1600s. This is during a time when many were already attempting to prove the independence Euclid's 5th postulate - a far more relevant activity to the idea of a formalized axiomatization than the document discussed. Chas zzz brown 05:36 Jan 16, 2003 (UTC)

That is not implied. If you think it is, change the wording don't remove the point - this is the historical link between the whole community of modern mathematicians and the ancient proofs, at least according to Simon Singh who wrote roughly this in "Fermat's Enigma".

Why are the practices of Pierre de Fermat less relevant than Euclid's 5th postulate? You are looking at the body of proofs, which is only one way to look at mathematics. If you look at the body of mathematicians, you would be more interested in Diophantus than the 5th postulate.

The premise of this article appears to be that there is something called "folk mathematics" which is different than something else called "mathematical practice". The relevant differences between the two appear to rotate around two features: the axiomatic approach, and the process of peer review. This paragraph attempted to state that an important element of modern "mathematical practice" is derived from a single document read by Fermat. Since Fermat contributed very little to the theory of axiomatics, and since Fermat's impact on peer review would not appear to be affected one way or the other by the Diophantus document, I fail to see what relevance it has to the argument. On the other hand, to demonstrate that there was indeed already at the time an interest in the formalization of understanding what is meant by "axiomatic system", I note that there was at the time of Fermat many other mathematicians whose work on axiomatics is relevant to the development of "mathematical practice"; mathematicians whose goal was to publish for the review of others, in an approximation of the modern manner. Chas zzz brown 01:18 Jan 17, 2003 (UTC)

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It is hard to imagine how mathematics might be different without these events, or the development of early algebra and the abacus in China, primitive accounting in Sumeria, the discovery of zero in India, and the modern number system from Arabia. See History of mathematics for more.

1. Early algebra, in terms of its influence on mathematical thought in China as well as the rest of the world today, is more due to the work of Arabic scholars (see Al-Khwarizmi) than from Chinese.

Ask anyone in China. That's a Western bias, tothe degree Islam is Western. An abacus is evidence of understanding of algebra.

Please see the mathematical definition of algebra. An abacus is a machine used for calculation.—The preceding unsigned comment was added by [[User:{{{1}}}|{{{1}}}]] ([[User talk:{{{1}}}|talk]] • [[Special:Contributions/{{{1}}}|contribs]]) .

Ancient China had discovered Gaussian Elimination , Chinese Remainder Theorem (it is called Han Xin Counting Soldiers, as I recall), calculated pi to find it lies between 3.1415926 and 3.1415927 (Zu Chong Zhi) in the fifth century, discovered (but did not prove) Shang Gao(Pythagoras) Theorem before Pythagoras and invented various algorithms.Hillgentleman 19:38, 29 August 2006 (UTC)

And there is no implication that the Chinese practices spread to the West, it's simply one thing on alist of things.

1. I find it quite easy to imagine mathematics in its current form without the development of the abacus, or for that matter, primitive accounting in Sumeria; as the whole point of this article appears to be that such mathematical ideas as "counting" and "calculation", and the economic pressures that supported this activity, arose independently in many parts of the world.

Yes, that is a main point, and should be stated more clearly, as you have.

1. It seems like this article should be retitled "folk calculation" rather than "folk mathematics", since most of what is discussed here relates to the application of calculations to real world problems. For example, the use of the digit "0", while vastly simplifying the notation of numbers and the basic calculations of addition and multiplication, sheds absolutely no light on questions of geometry, topology, calculus, analysis, discrete mathematics, or number theory - in short, any of the major elements of modern mathematics. While the naive reader may assume otherwise, calculation is not the primary activity of mathematics.

The field is called 'folk mathematics' and that term originates elsewhere, so don't complain to me. Calculation, physics, and mathematical proofs are three different practices TODAY, but they were not so in most cultures of any interest. And none of them existed apart from beliefs about mysticism:

For this reason, I don't see how it would be hard to imagine how mathematics might be without some particular feature of this earlier conglomeration - these earlier formulations were largely discarded as inessential to what is now considered mathematics. 13 is a prime number, whether it is written base 10, or as XIII, or any other form. Chas zzz brown 01:18 Jan 17, 2003 (UTC)

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Even in the cultures where modern mathematics was developed, such as the ancient Greek, folk mathematics existed alongside that of the elites, even next door, since many mathematicians, e.g. Pythagoras, kept their discoveries secret, and shared them only with their inner circle.

"Folk mathematicians", in the form of computer scientists who use a heuristic approach to math rather than an axiomatic one, still exist today (and one could argue that they form a more elite group than are mathematicians in general - they certainly get paid more). It is not the "secrecy" of "elites" that created the folk mathematician as described here; rather, it was the focus on applications and calculation, as opposed to proof from fundamental principles, that appears to be the ultimate distinction.

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Even Pierre de Fermat kept secret most of his proofs until others had discovered them.

This is the reason that many were "irked" with Fermat - he disliked proving his statements rigorously, and enjoyed tweaking professional mathematicians by stating he had a proof and challenging them to come up with one as well. This was at a time when the "culture of mathematics" already required publishable proof of assertions; his reticence to provide these proofs is supposed to be evidence of... what? That Fermat was secretely a folk mathematician? That some people are annoying? Chas zzz brown 05:36 Jan 16, 2003 (UTC)

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Nonetheless, commerce, engineering, calendar creation and the prediction of eclipse and stellar progression were very well understood by almost all of these ancient cultures, and until the arrival of rival civilizations with knowledge of modern mathematics and physics, they survived without most of our ideas about math or any mathematical practice (proof, peer review, publication, citation).

What is your agenda here? The dinosaurs thrived for millions of years until they were utlimately wiped out by a meteor, and yet they too had no formalized, axiomatic approach to mathematics. So what? Mathematics is not about survival; it is a particular human activity, like music. It is instructive to note which aspects of mathematical understanding arise independently of a particular cultural history; but to seek for the reasons of a society's survival or ultimate destruction in terms of their mathematical "sophistication" seems far off the point here.

In addition, to state that stellar progression was "well understood" is more a statement of physics than mathematics - given the instruments at hand, they were capable of astoundingly accurate predictions, but accuracy does not imply understanding. Chas zzz brown

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Mathematically, such structures rely on some basic geometry such as pi, concepts of angles (especially of repose), and simple figures like arches, which require master (sic) of both ideas.

It has not been demonstrated that a "mastery" of pi (whatever that means) or the angle of repose is neccessary to construct an arch. This can just as easily be seen as a cut-and-try approach from engineering; and since most engineers of the day were trained as part of an apprenticeship, this seems a far more likely source for the use of these ideas.

This is not to say that, after the fact, these structures are not great achievements; but they do no therefore imply a great knowledge of the mathematical abstractions involved. Chas zzz brown 05:36 Jan 16, 2003 (UTC)

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A cynical assumption about ancient practices is that they worked only because human life was cheap - there was labor in abundance and there was little or no consequence if people died in collapses because some structure was poorly understood. This seems to have been debunked in recent years, especially by the unearthing of workers' quarters at the Great Pyramids, and workers' graves, all of which seems to indicate that the Pyramid builders were not luckless slaves, but a well paid professional work force who was better off than the main population.

Do you have a reference indicating that anyone has ever seriously suggested that "because human life was cheap", structures like the pyramids were rebuilt again and again until they remained standing? Even if this were true, it disregards the great cost in building such structures above and beyond any waste of human life, as well as the obvious loss in prestige that would accompany the failure of any project so large and so fraught with religious meaning. Chas zzz brown 05:36 Jan 16, 2003 (UTC)

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So one could say that American measurements are a leftover of older systems that were themselves a folk practice, and reflected only a limited understanding of what a measurement was, e.g. definitions of a 'foot' varied until it was finally well-defined - in terms of the meter - in the 19th century.

First, this has nothing to do with mathematics or counting systems - it has to do with measurement systems (perhaps considered as part of naive physics), and their ultimate standardization. The implication that "folk mathematicians" didn't understand the importance of measurements is absurd as well - how would a carpenter make a door if that carpenter didn't first measure the opening for which the door was intended?

Secondly, from a mathematical viewpoint, there is nothing in the abstract to distinguish between base 12 systems or base 10 systems, with one system being more "folk mathematical" and the other more "modern". If we calculated in base 12, then it would make more "sense" to measure in base 12 - but this again comes to ease of calculation, and has no bearing on mathematics itself. Chas zzz brown 05:36 Jan 16, 2003 (UTC)

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Another is to determine if an ancient (or any older) proof would stand the test of today's rigorous methods, or if it must be considered a piece of folk mathematics, or an error of process.

What is older than "ancient"? Pre-historic? This paragraph appears to posit that either a piece of mathematics is rigorously proven in the modern sense, or else it is "folk mathematics", or else it is an error. This seems to me to be a false trichotomy - a mathematical result can be guessed at through inuition (folk mathematics), incorrectly proven through an erroneous process, and then rigourously proven. The interesting thing about folk mathematics is not so much their results, as the process by which they are attained, and what that tells us about human cognition - which should be the point of this article. Chas zzz brown 05:36 Jan 16, 2003 (UTC)

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This example illustrates a problem with the idea of folk mathematics: to a future or alien being, today's mathematics might well appear as crude and inexact, overly certain of itself, naive, even ignorant. A key tenet of anthropology is to avoid assumptions of superiority, and to understand the real gains that may be possible when adopting some simpler but less powerful system of thinking, or of choosing actions.

The implication is here is that there is something called "folk mathematics" that is currently derided as crude, inexact, overly certain of itself, naive, even ignorant, etc.; and that this arises from a general impression of cultural superiority. This is a straw man argument - what is apparent is that the goals of modern mathematics are different than the goals of "folk mathematics". Estimating the ratio of a circle's circumference to its diameter as 22/7 is crude and inexact - but when your goal is to ensure you've got sufficient material to cover a cylinder with a known radius, it's perfectly acceptable as an approximation. Contrariwise, while most mathematicians today consider Galois theory to be a great mathematical triumph, it would probably not be considered of particular interest to your average Sumerian accountant - or even your average IT professional.

It is simply not true that pi can expressed as the ratio of two integers; but this doesn't mean that people were "stupid" for not proving this in ancient times; they simply had a different agenda. At the same time, that does not then imply we should adopt a simpler, less powerful system of thinking about mathematics - and the question of how we choose our actions is in a completely different category.

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Another key study in folk mathematics is counting systems?. The most common base for mathematics was often 12 in such systems not 10, which was the standard for the Greeks and Romans - and the modern Metric system. The Babylonians used a base 60 counting system for most of their calculations. English measures were based on 12ths until they were finally supplanted by metric in the 20th century - which in the United States has not yet happened.

What do different number systems have to do with this? (and don't link plurals, please!) English measures were based on EVERYTHING, 12, 16, 3, etc. -- Tarquin 00:10 Jan 17, 2003 (UTC)

Different number systems are clearly evidence of different thinking.

Again, the modern notions of proof, calculation, physics, measurements, number systems or bases, are of course very different concerns of moderns. All of these were fused into less distinct practices in ancient cultures, and even in some modern cultures (correctly noted by Chas Brown above). It's entirely inappropriate to make confused objections about the fact that folk mathematical concepts don't divide up the world of thought into neat packages the way we did in the 20th century (but may not always do in future, see cognitive science of mathematics). It's just more arrogance to state that things like number bases etc. 'in the abstract' can't possibly make a difference to what we call 'mathematics', or that abstract algebra is somehow more a part of mathematics than standard systems of weights/measures. That's clearly wrong historically and probably in theory too.

It is not "arrogance", it's by definition that abstract algebra is part of mathematics, and standard systems of weights and measures are not. Standards system of weights and measures are features of the real world - they are not mathematical objects in the modern sense; the "kilogram" is not somehow derived from set theory, it's an agreed upon physical onject which is actually sitting somewhere. You are again confusing what we now call mathematics with some combination of engineering, physics, and calculation. Chas zzz brown 01:18 Jan 17, 2003 (UTC)

Choice of number system does NOT affect mathematics. You could do algebra in roman numerals if you really felt like it. -- Tarquin 10:16 Jan 17, 2003 (UTC)

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Mathematicians speak of something called a folk theorem and of mathematical folklore. That does not mean "non-professional mathematical practices, especially of aboriginal or ancient peoples outside the Mediterranean basin". (Why "the Mediterranean basin", by the way?) Rather, a folk theorem is a result that is well-known among people familiar with the relevant area of mathematical research even though it is not found in any published source. A folk theorem circulates only by word-of-mouth without benefit of publication. That doesn't mean it non-professional; the people among whom it circulates are usually professionals. Michael Hardy 02:18 Jan 17, 2003 (UTC)

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Let's face it -- we have another crank on our hands. :( -- Tarquin 10:16 Jan 17, 2003 (UTC)

At least it makes a break from FWappner. Chas zzz brown 20:29 Jan 17, 2003 (UTC)

He's not new though. It's user:24 who is currently being banned from the site. Whenever you see someone making links to sacred geometry or foundation ontology, you know it's him. Another favorite of his is the crank theory that Euler's identity somehow has a deep relevance to the philosophy of mathematics, never quite explained. AxelBoldt 02:02 Jan 24, 2003 (UTC)

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I was startled to read that Kurt Goedel, in his first paper on undecidability, wrote about a hypothetical "Martian mathematics". I know only secondary sources, and since Goedel's original argument is of course considered needlessly complicated, I am not encouraged to look at the original. Maybe this claim will change that. Has anyone heard this claim before? Michael Hardy 20:21 Jan 17, 2003 (UTC)

I haven't heard of it before. Chas zzz brown 20:29 Jan 17, 2003 (UTC)

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I would like quick confirmation of the first sentence

especially of aboriginal or ancient peoples outside the Mediterranean basin.

AxelBoldt 02:02 Jan 24, 2003 (UTC)

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I have now divided this up, sending half to ethno-cultural studies of mathematics. I'd naturally like to redirect the back-links properly; but the whole area seems to be a snarl of assertions about this and that. I'm not really sure where to start. Of course there should be some way to interface what the mathematics community says it is doing, with some critical views from outside. Also it is not uninteresting to identify what 'traditional' mathematics was, and what 'modern' mathematics is; many of the users of mathematics adhere to attitudes which are better described as traditional rather than modern.

Charles Matthews 12:41, 24 Nov 2004 (UTC)

[edit] Folk theorem

Hey folks! Folk theorem currently redirects here. The term "Folk theorem" is used for a particular result regarding first the iterated Prisoner's dilemma and now a more general series of results. I really don't know to much about the area, but I'm sure of the term. Anyway, I wanted to know if you would prefer a disambiguation page at Folk theorem or would you rather I just take the name over and put maybe a note at the top? -Kzollman 23:53, Jun 2, 2005 (UTC)

I created a disambig page linas 20:31, 27 October 2005 (UTC)

I think the most famous example of a folk theorem, as defined on this page, is folk theorem (game theory). Shouldn't this page make mention of it, at least as an example of how a folk theorem can become a major result?Smmurphy 21:06, 27 October 2005 (UTC)

I added another disambig notice. Note that I changed the redirects to be disambig pages Click on folk theorem to see. As to fame ... never heard of the game theory version, whereas math undergrads encounter the definition given here ... linas 21:21, 27 October 2005 (UTC)

I'm just trying to bring this in line with commonly accepted wikipedia article structure guidelines, that's all. linas 21:23, 27 October 2005 (UTC)

As for fame, compare [1] to [2], the word folk theorem refers very often to that specific class of theorems having to do with feasible outcomes of repeated games. I think the article should give the particular example of the most common folk theorem.Smmurphy 21:59, 27 October 2005 (UTC)

I guess I feel a disambiguation page is a bit much, when a simple link will suffice with one less click, and can put into context why folk theorem sends the user here and not there.Smmurphy 22:03, 27 October 2005 (UTC)

[edit] RENAMING SUGGESTION: Mathematical Folklore

• I have never heard anybody using the term "folk mathematics". Usually I hear people say, "it is in the folklore" or "in the literature".
• The word "folk" confused a lot of people.
• The word "mathematical" gives the correct tone.

                          'Suggestion:  1.  Mathematical Folklore
                                                  2.  Folklore Theorems  in MathematicsHillgentleman 19:32, 29 August 2006 (UTC)

[edit] Examples?

Can we have some examples of "theorems, definitions, proofs, or mathematical facts or techniques" which come under mathematical folklore? --Kprateek88(Talk | Contribs) 17:05, 2 November 2006 (UTC)