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There are numerous popular folklore stories in mathematics, and it is an interesting question to understand the accuracy of these folk stories.

Folklore stories, and urban legends are subborn things, and especially because of their oft-repeated status become unquestioned and appear sensible.

I am looking for widely accepted folklore stories in mathematics that are either substantially more complicated, have contested components, are embellished, apocryphal, or known to be false. I am also interested in folklore notions that been widely accepted through reproduction, but are scantly sourced and are used variably or vaguely.

Example 1: Cole's "Talk without words"

Corry noted in the difficulty and trustworthiness of Bell's account of Cole's famous "talk without words leading to standing ovation" (see also Conifold's more detailed discussion), where the "no words" aspect at the minimum is apocryphal. He notes that:

His account, at any rate, became an accepted mathematical urban legend that has been repeated over and over again, often extending the three years of Bell to "twenty years of Sunday afternoons"

In fact the problem is well-captured by Miller (1938) who writes on Bell's writing:

In a review of E. T. Bell's Men of Mathematics, published in the Mathematical Gazette, volume 21, pages 311-312 (1937), the following statements appear: "There are numerous errors of detail, unimportant in a stimulant; there is a trace, it seems to me, of an error in principle; but the author succeeds in his main aim, to portray mathematicians three-dimensionally by setting them in or against their historical background, social, cultural and philosophical." The main part of this quotation for the present purpose is the remark that the errors of detail are unimportant in a stimulant. This is probably true as regards some of the readers concerned, but there are others who will be very much annoyed by errors of detail, and whose interest in the book will be greatly diminished when they become convinced that they cannot assume that the author took a reasonable amount of care to avoid misleading remarks even when they are striking.

  • Miller, G. A. (1938). Mathematical myths. National Mathematics Magazine, 12(8), 388-392.

Example 2: "Emmy Noether made topology algebraic"

As an example for a complicated story is the relationship of Emmy Noether and the algebraization of homology.

The folklore notion is "Emmy Noether made topology algebraic". Dieudonne wrote an article making that point (Dieudonné, J. (1984). Emmy Noether and algebraic topology. Journal of Pure and Applied Algebra, 31(1-3), 5-6.) To which Mac Lane responded that there is a problem, in that Vietoris had used the notion independently of Noether and Hopf (Mac Lane, S. (1986). Topology becomes algebraic with Vietoris and Noether. Journal of Pure and Applied Algebra, 39, 305-307.) However, even Mac Lane's correction is problematized by Weyl using group notions as early as 1923 in an admittedly obscure Spanish-language journal (H. Weyl: Analysis Situs Combinatorio, Rev. math. Hisp. amer. 5 (1923), 209-218, 241-248, 278-279; 6 (1924), 33-41.). An updated perspective of the issue can be found in McLarty (2006) (Emmy Noether’s ‘set theoretic’ topology: From Dedekind to the rise of functors. The architecture of modern mathematics: Essays in history and philosophy, 211-35.) Following McLarty, Noether's contribution is pointing out the simplicity of the group quotient vis-a-vis matrix-based methods. For this reason this seems to be an example of a substantially complicated story simplified in folklore.

Example 3: "Poincare is the last universal mathematician"

An example of a folklore notion is that of "Poincare is the last universal mathematician", where I am personally unaware of strong articulations that justify this attribution as historical, or even seriously contextualizes this folklore notion. The problem with tracing folklore notions is that they are usually exempt from citations and other forms of sourcing and propagate by unsourced repetition. Even variation (is it just Poincare or also Hilbert?) do not tend to disturb the propagation of the folklore notion.

For example we find in Dalmedico (2013). Mathematics in the twentieth century. In Science in the twentieth century (pp. 651-667). Routledge.:

It has been said that Poincare and Hilbert were the last 'universal mathematicians' in the sense that they worked in or at least followed the development of almost all fields of mathematics.

The very notion of the "last universal" is transportable. For example "last universal genius" or "last universal scholar" has been applied to Leibniz as well as Leonardo da Vinci, Goethe, Alexander von Humboldt, and Helmholtz. For a popular exposition on the transportability of the "last universal" see O'Neill (2017):

ARISTOTLE, of course, was the “last man to know everything” – everything useful to know about the world during his lifetime. No wait, it was Leonardo da Vinci. Or was it Goethe, or his equally brilliant Teutonic contemporary Alexander von Humboldt?

The trope of the last universal polymath is a common one – along with the idea that, as our compendium of knowledge grew, at some point it outstripped the capacity of one brain to house it.

It seems to capture a notion of historical recognition that managed to enter cultural reproduction. For example, Hermann Grassmann might be considered a German polymath of Helmholtz breath but is not broadly known as a historically recognized figure in the same sense as Helmholtz, hence does not serve as a popular folklore notion carrier.

What are other folklore notions or folklore stories in these senses?

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  • $\begingroup$ There are lots of theoretically unmotivated rules of thumb in science. One good example is from statistics and concerns p-value based significance levels. P-values of 0.05 or less are the 'once in a blue moon' cutoff for nonrandom, significance of results, a rule of thumb which originating with Fisher a century ago and which statisticians continue to employ. The problem is, as noted by the late Joseph Kadane (CMU) wrt forensic statistics in a paper published in the Journal of the American Statistical Association some years ago, this rule has no grounding in statistical theory. $\endgroup$
    – DJohnson
    Commented Jun 28 at 2:03
  • $\begingroup$ @djohnson It is intriguing, that an absolute value choice may be arbitrary but its interpretability and relative properties are not. p<=0.05 means it's a <=5% chance of being wrong. Knowing this is why we don't pick p<=0.95. We know how to interpret these numbers. The actual precise number may be arbitrary but that we know that the 0.01 standard (used in medicine) is tighter, is not arbitrary. If we wanted higher degrees of certainty we could go 0.005 or 0.001 and we'd have an expectation (pun intended) in which direction of certainty we are going and by how much. $\endgroup$
    – Georg Essl
    Commented Jun 28 at 7:22
  • $\begingroup$ @GeorgEssl "p<=0.05 means it's a <=5% chance of being wrong." seems to be an arbitrary interpretation. You can use these statistics to estimate things that in reality have probability 0 or 1, such as what is the probability there is a string of six 9's in the first 1000 digits of pi. And in reality most results in certain areas of medicine are not reproducible. $\endgroup$
    – David
    Commented Jun 28 at 7:52
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    $\begingroup$ See also mathoverflow.net/q/26613/454 $\endgroup$ Commented Jun 28 at 19:20
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    $\begingroup$ @GeorgEssl Maybe this is at the level of nitpicking, but even aside David's comment, that's not actually what a p-value means, though that's a common misconception. p-value means that if the null hypothesis is true, then the probability of seeing results at least as far from the mean as what you measured is p. It doesn't measure the likelihood of the null hypothesis, it measures the likelihood of getting your measured values given the null hypothesis; a subtle difference, but an important one. $\endgroup$
    – Idran
    Commented Jul 1 at 14:32

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The most famous such stories (which are demonstrably untrue) are:

  1. About Archimedes burning Roman ships with some mirrors, (this is technically impossible), and

  2. About Newton's apple. They even show THAT apple tree in Cambridge to the tourists, completely ignoring the fact that the discovery was made by Newton in the estate of his mother in Woolsthorpe, and that apple played no role in it. The story about apple was presumably made up by Newton's niece who told it to Valtaire, and Voltaire popularized it.

Remark. There is little doubt that all stories told about ancient mathematicians (Euclid, Pythagoras, Thales, Archimedes etc.) are made up.

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  • $\begingroup$ Pythagoras has its own question ... hsm.stackexchange.com/questions/391/… $\endgroup$ Commented Jun 28 at 19:08
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    $\begingroup$ What's the demonstration that shows that burning ships with mirrors is technically impossible? Just curious because there's been some news articles popping up about a 12-year-old creating a demonstration that it might work. I've cooked food with mirrors and burned paper with a magnifying glass. What's the evidence or proof that this is impossible? $\endgroup$
    – JimmyJames
    Commented Jun 28 at 20:58
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    $\begingroup$ Wikipedia has an article, Isaac Newton's apple tree, which says that there is a specific tree at Woolsthorpe Manor associated with the story, and that Newton himself often told the story. There are references for both claims that I didn't check. The article agrees with you that Voltaire heard it from Newton's niece and popularized it. The version where the apple hit him on the head is false of course. $\endgroup$
    – benrg
    Commented Jun 28 at 21:41
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    $\begingroup$ Burning ships is clearly technically possible. Do you mean impractical or unfeasible? $\endgroup$
    – pipe
    Commented Jun 28 at 22:27
  • $\begingroup$ @JimmyJames: What was the distance between the mirror and your food you cooked or paper you burned? $\endgroup$ Commented Jun 29 at 13:51
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The Pythagorean Hippasus getting thrown overboard is a favorite. Historians of math have had many good laughs about this one.

Another good one is about Cantor, Dedekind, and Weierstrass eliminating infinitesimals from mathematics.

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    $\begingroup$ Yeah I should definitely watch not going overboard with being irrational! $\endgroup$
    – Georg Essl
    Commented Jun 28 at 7:24
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    $\begingroup$ When I was a kid, my math teacher confidently told us that all numbers could be represented as ratios/fractions. Being a little nerd, I piped up and asked "what about irrational numbers?" to which she very angrily and loudly said: "numbers are not irrational!" Later, when I heard the Hippasus story, it rang true, and I felt a sense of deep comradery with him. Can't we just pretend it's true? $\endgroup$
    – JimmyJames
    Commented Jun 28 at 21:13
  • $\begingroup$ I'm not sure why the laughs. A number of sources--written long after the time of Pythagoras, but still ancient--mention similar, but somewhat conflicting accounts about someone being lost at sea, or being treated as dead by his comrades, for having revealed a mathematical secret of some sort. Modern authors have embellished the story to a ridiculous extent, but the story isn't made of whole cloth. $\endgroup$ Commented Jul 9 at 17:13
  • $\begingroup$ "Modern authors have embellished the story to a ridiculous extent" : That's why the laughs :-) @WillOrrick $\endgroup$ Commented Jul 10 at 15:59
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The story that Gauss at some age (as young as 5) found the sum $1+2+\dots+100$ in an instant is probably inaccurate.

The earliest version of this story is probably from an 1856 biography by Sartorius von Waltershausen, which states merely that at some unknown age of at least nine, Gauss was asked to sum some (unknown) arithmetic series:

Gauss besuchte zuerst 1784, nachdem er sein siebentes Lebensjahr zurückgelegt, die Catharinen-Volksschule ... In dieser Schule, ... blieb der junge Gauss zwei Jahre lang ohne durch etwas Ausserordentliches aufzufallen. Erst nach jener Zeit brachte es der Gang des Unterrichtes mit sich, dass auch er in die Rechenklasse eintrat ... Es ereignete sich hier ein Umstand ... und den er uns in seinem hohen Alter mit grosser Freude und Lebhaftigkeit öfter erzählt hat. ... Der junge Gauss war kaum in die Rechenclasse eingetreten, als Büttner die Summation eine arithmetischen Reihe aufgab. Die Aufgabe war indess kaum ausgesprochen als Gauss die Tafel mit den im niedern Braunschweiger Dialekt gesprochenen Worten auf den Tisch wirft: “Ligget se’.” (Da liegt sie.)

Google translated:

Gauss attended the Catharinen elementary school in 1784 after he had passed his seventh year ... At this school... the young Gauss stayed for two years without attracting attention to anything extraordinary. It was only after that time that the course of the lessons meant that he too entered the arithmetic class ... A circumstance happened here ... and which he often told us in his old age with great joy and animation. ... The young Gauss had hardly entered the arithmetic class when Büttner [the teacher] gave up the summation of an arithmetic series. The task had scarcely been said when Gauss threw the tablet on the table with the words spoken in the Lower Brunswick dialect: “Ligget se’”. (There she lies.)

In the above original version of the story, Gauss was at least nine. But subsequent versions of the story use different ages—perhaps most commonly seven, but one version has Gauss as young as five.

Brian Hayes has collected 145 “Versions of the Gauss Schoolroom Anecdote”.

It seems that Franz Mathé (1906) was the first to add the embellishment that the arithmetic series was $1 + 2 + \dots + 100$.

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Of course there must so many incorrect folklores and incorrect associations especially in pre-historical mathematics. One common thing that circulates in applied mathematics is the full name of sinc as cardinal sine or sine cardinal function. The British mathematician, Philip M. Woodward, who introduced the sinc function in 1952, never called it that or explained the full name in his paper, however, the myth propagates everywhere.

Woodward, P. M.; Davies, I. L. (March 1952). "Information theory and inverse probability in telecommunication" (PDF). Proceedings of the IEE – Part III: Radio and Communication Engineering. 99 (58): 37–44.

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    $\begingroup$ Nice one! As someone who uses sinc functions in signal interpolations I may well have committed this "cardinal sin". $\endgroup$
    – Georg Essl
    Commented Jun 27 at 17:59
  • $\begingroup$ There are many more, especially origins of sine, cosines etc. Earlier the terms were associated to Arabic words but now South Asians also claim ownership. $\endgroup$
    – ACR
    Commented Jun 27 at 18:06
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The usual story of the discovery by the Pythagoreans of the incommensurability and the crisis in mathematical thought that followed has been questioned, and it seems nowadays that there isn’t a solid evidence of this vulgate version.

There is no sound basis to state that the discovery should be attribute to Pythagoreans and, in particular, that the discovery of incommensurability caused a ‘foundation crisis’ of mathematics.

An example of this traditional version can be found in Wikipedia:

Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it . Wikipedia, Square root of 2

The historian of mathematics D.H. Fowler disputed the standard story of this mathematical discovery by Pythagoreans, and the fact that the discovery of the phenomenon of incommensurability came as a shock. In his article ‘The story of the discovery of incommensurability, revisited’, pp.221-235 of Trends in the Historiography of Science, ed K. Gavroglu, J. Christianidis, & E. Nicoliaidis, Boston: Kluwer,1994 Fowler writes:

This farrago of mutually inconsistent stories, which appear for the first time in a source of doubtful reliability and relevance dating from some nine centuries after the time of Pythagoras, is the main evidential base for much of what has been written about the discovery of incommensurability! I add the slightly perplexed comment that the most recent and authoritative study of the role of mathematics in neo-Pythagorism, D.J. O’Meara’s book Pythagoras Revived, Mathematics and Philosophy in Late Antiquity, does not even seem to mention incommensurability.

The foundation crisis. Again, I find Freudenthal, Knorr and other writers very convincing when they argue that, far from being a period of crisis and confusion, the early fourth century was an extraordinary period of creativity, especially in Plato’s circle; we have no historical evidence for any of the postulated difficulties of a ‘foundation crisis’[…]

The implications of the discovery of incommensurability. Just what precisely are the supposed problems raised by the discovery of incommensurability ? As far as I know, no Greek text, early or late, tells us clearly of the mathematical difficulties raised by the phenomenon. Aristotle wrote of the innocent's surprise […] (pp. 5-6)

More recently, in a similar vein, the historian of Greek science Lucio Russo, in his book The Forgotten Revolution, Springer, 2004 says that the attribution of the discovery of incommensurability to Pythagoreans has no serious basis, and that the ensuing crisis in mathematical thought is more or less a legend.

Russo distinguishes between the discovery by the Pythagoreans of an aporia concerning the measurement of the diagonal of a square and its side, and the notion of incommensurability (or the irrationality of $\sqrt {2}$)

He writes:

Another important example, traditionally attributed to the Pythagorean school, is the discovery of incommensurability. This is often quoted as the demonstration of the irrationality of root of $2$. But the original argument should not be blurred with its later development. One reconstruction of the early state of affairs is the following. We know that early Pythagoreans thought that every segment was made up of finitely many points. If we construct a square whose side is made up of an odd number of points, say $k$, we can ask whether the number of $n$ points of the diagonal is even or odd. Since the square of $n$, by the Pythagorean theorem, is $2k^2$. And therefore even, and only an even number can have an even square, it can be easily concluded that $n$ is even. But someone must have noticed that if $n$ were even its square would be a multiple of $4$, whereas $2k^2$ is not such a multiple if $k$ is odd; therefore $n$ must be odd. Since both reasoning appear convincing, but an odd number cannot be even, they did not know what to conclude. […]

But if the reconstruction above is correct, the Pythagoreans had not proved anything by contradiction; they had simply reached a contradiction (to their chagrin!), while finding the parity of the diagonal [ …]

Only by abandoning the Pythagorean notion that a line is made up of a succession of points- and therefore abandoning the Pythagorean program of basing explanation about real objects on the concept of an integer number – can one attain the idea that two segments not admit a common subdivision - may be incommensurable. There is no evidence for classifying this step as being Pythagorean. That neither Plato nor Aristotle, in any of the several passages they discuss the problems posed by incommensurability, ever attributed its discovery to Pythagoreans is a strong indication that neither should we (Russo, cit., pp. 35-36, emphasis mine)$^1$

As for the issue of ‘foundation crisis’ Russo writes:

The widespread idea that the discovery of incommensurability shook the foundations of mathematics is based on the assumption that in the fifth century B.C. mathematics already existed in our sense; it must be arisen by analogy with the shaking of foundations of mathematics at the turn of the twentieth century. What could have be shaken at that time was the original Pythagorean philosophical framework, which wished to be the foundation (in a sense very different from the one used by today’s scholars) of much more than our mathematics. The studies that we call mathematical today would have continued without much trouble, precisely because they did not have a monolithic foundation. (Russo, pp. 37).


$^1$ The pages of the book of Lucio Russo are referred to the Kindle edition.

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Here's one.
That the golden ratio $(1+\sqrt5)/2$ was used in architecture and art before the 20th century. See this MO post .

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    $\begingroup$ I don't understand why George Markowsky hates the golden ratio so much... Seriously: the fact that a numerical coincidence does not imply that a specific real number was used does not even mean that that real number was definitely not used. What can one want? Can one want an artist to explicitly state the real numbers used in a painting or construction? In the specific case, Luca Pacioli wrote a book about the golden ratio and the illustrations in the book are by... Leonardo da Vinci. More than Leonardo da Vinci, what is desired? Vincent van Gogh? Pablo Picasso? $\endgroup$
    – M. Lonardi
    Commented Jul 6 at 16:32
  • $\begingroup$ I don't think that paper alone shows that it is false to claim that the golden ratio was "used in architecture and art before the 20th century". That paper only lists (and possibly debunks) the "misconceptions" that the golden ratio was used in/by Great Pyramid, Parthenon, Leonardo (plus a few other painters), UN Building, Aeneid. $\endgroup$
    – user103496
    Commented Jul 10 at 4:01
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One may say that the following is more about physics than mathematics, but, for what it's worth, Littlewood wrote in his "Mathematician's Miscellany":

I heard an account of the battle of the Falkland Islands (early in the 1914 war) from an officer who was there. The German ships were destroyed at extreme range, but it took a long time and salvos were continually falling 100 yards to the left. The effect of the rotation of the earth is similar to ' drift ' and was similarly incorporated in the gun-sights. But this involved the tacit assumption that Naval battles take place round about latitude 50$^{\circ}$ N. The double difference for 50$^{\circ}$ S. and extreme range is of the order of 100 yards.

The story has been repeated frequently, but it looks like this is a myth https://doi.org/10.1119/5.0106008

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In physics we have many folklore stories that are most possibly not true:

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The Babylonian clay tablet known as Plimpton 322, dating to c. 1800 BCE, has been used to support the notion that the Pythagorean theorem preceded Pythagoras (fl. 500 BCE) by more than a millennium. But Robson deconstructs this conclusion, suggesting that interpretations of Plimpton 322 are more varied than merely anticipating the Pythagorean theorem.

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  • $\begingroup$ I find your interpretation of the linked article hard to square with the following statement of Robson in Mathematics in Ancient Iraq: A Social History. Princeton University Press.: p. 109 "Many Old Babylonian mathematical practitioners … knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste ‘algebra’ on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia." (from Wikipedia's Pyth The article) $\endgroup$ Commented Jul 9 at 12:27
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If is an oft repeated story that Descartes conceived of what we now call Cartesian coordinates while laying in bed and observing a fly. For example, from René Descartes and the Fly on the Ceiling:

The coordinate system we commonly use is called the Cartesian system, after the French mathematician René Descartes (1596-1650), who developed it in the 17th century. Legend has it that Descartes, who liked to stay in bed until late, was watching a fly on the ceiling from his bed. He wondered how to best describe the fly's location and decided that one of the corners of the ceiling could be used as a reference point.

More to the point, nowhere in Descartes' La Géométrie does Descartes employ the eponymous coordinate system. As pointed out in the HSM posting When do we see for the first time the use of Cartesian coordinates?, the use of coordinates was first introduced by Oresme and Descartes' inspiration was not a fly but most likely Descartes got the idea of coordinates from Apollonius of Perga's Conica.

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