Papers that debunk common myths in the history of mathematics

Question

What are some good papers that debunk common myths in the history of mathematics?

To give you an idea of what I'm looking for, here are some examples.

Tony Rothman, "Genius and biographers: The fictionalization of Evariste Galois," Amer. Math. Monthly 89 (1982), 84–106. Debunks various myths about Galois, in particular the idea that he furiously wrote down all the details of Galois theory for the first time the night before he died.

Jeremy Gray, "Did Poincaré say 'set theory is a disease'?", Math. Intelligencer 13 (1991), 19-22. Debunks the myth that Poincaré said, "Later generations will regard Mengenlehre as a disease from which one has recovered."

Colin McLarty, Theology and its discontents: The origin myth of modern mathematics. Debunks the myth that Gordan denounced Hilbert's proof of the basis theorem with the dismissive sentence, "This is not mathematics; this is theology!"

Some might say that my question belongs on the Historia Matematica mailing list; however, besides the fact that I don't subscribe to Historia Matematica, I think that the superior infrastructure of MathOverflow actually makes it a better home for the list I hope to create. Still, maybe someone should let the Historia Matematica mailing list know that I'm asking the question here.

Answer 1

For over a century, books published a picture of Legendre that was not in fact a picture of Legendre. There's an AMS notices article. Does a picture count as a myth?

Answer 2

This entry is inspired by sigfpe's example of "picture as a myth".

There is an even more outrageous picture scam: for centuries, mathematics books featured

this picture of Euclid of Megara, whereas the author of the Elements was Euclid of Alexandria!

P.S. I offered a prize to any student who can find a glaring error in our linear algebra text. No one succeeded, although in fairness to them, sleazy publisher cropped the picture.

Answer 3

Misconceptions about the Golden Ratio
George Markowsky
College Math Journal: Volume 23, Number 1, Pages: 2-19 1992

first paragraph...

The golden ratio, also called by different authors the golden section [Cox], golden number [Fi4], golden mean [Lin], divine proportion [Hun], and division in extreme and mean ratios [Smi], has captured the popular imagination and is discussed in many books and articles. Generally, its mathematical properties are correctly stated, but much of what is presented about it in art, architecture, literature, and esthetics is false or seriously misleading. Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high school geometry textbooks such as [Ser] make many incorrect statements about the golden ratio.

Answer 4

There is a nice article by Brian Hayes in the American Scientist (May-June 2006), "Gauss' Day of Reckoning," in which he looks at the history of the so-called "baby Gauss" story (that Gauss amazes his teacher by summing the first 100 positive integers).

There is a famous anecdote about Euler embarrassing Diderot in Catherine the Great's court. He claimed to have mathematical proof of the existence of God, when in fact he just stated mathematical nonsense (which Diderot did not understand): "Monseur, $(a+b^n)/n=x$ donc Dieu existe; répondez!" B. H. Brown tracked down the source of this myth in "The Euler-Diderot Anecdote" (Amer. Math. Monthly, Vol 49, 1942, reprinted in William Dunham's The genius of Euler: reflections on his life and work (2007))

Finally, there is a mathematical urban legend that I thought was surely was false, but is apparently true (according to Snopes). This is the story about the student who comes late to class and sees the homework written on the board. After a lot of effort he solves the problems. Only later did he discover that they were not homework, but open problems. It turns out that the student in the story was George Dantzig. Snopes cites a 1986 interview with Dantzig from the College Mathematics Journal.

Answer 5

Historians like nothing better than to debunk other historians, so there are plenty of papers shooting down one myth or another. I enjoy these papers as much as the next person, but sometimes the debunking becomes the new myth. So, if I may turn the question round a bit, here is a case where I think the original "myth" has merit, and I'm not convinced by the "debunking".

"Myth". The ancient Mesopotamians who compiled Plimpton 322 were looking for Pythagorean triples and had a powerful method for finding them. For example, they found the triple (13500,12709,18541).

"Debunking". By Eleanor Robson in Historia Mathematica 28 (2001) 167-206, which may be viewed here.

Answer 6

One of the biggest myths in number theory is that work on Fermat's last theorem played a large role in the development of ideal theory and algebraic number theory. In fact it was much loftier goals such as the quest for higher reciprocity laws that were the true sources of inspiration. For references see e.g. Lemmermeyer's book "Reciprocity Laws" p. 15 (notes on Lagrange).

Speaking of FLT, recall the legend that Kummer submitted a false proof based on the erroneous assumption that cyclotomic number rings were UFDs. Edwards argued that this may be a myth and put forth an argument that, instead, Kummer's mistake was based on a simple error not related to any UFD assumption. However R. Bolling recently discovered new evidence that seems to lend strong support to the veracity to the claim that Kummer did in fact mistakenly assume facts equivalent to unique factorization in various rings of cylotomic integers in one of his early papers (which was not an attempted proof of FLT). So only part of this legend is actually true.

The above two paragraphs don't really do justice to the complex history. For a more faithful rendition I highly recommend that the interested reader also consult Franz Lemmermeyer's recent paper [1] which, imho, is one of the most interesting historical works on number theory in quite some time.

By the way, far less known than the constructivity of Euclid's proof that there are infinitely many primes (cf. M. Hardy below) is the striking fact that Euclid's constructive proof generalizes quite widely - namely to any infinite ring having fewer units than elements. For this little-known proof see my post here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1209616#p1209616 http://google.com/groups?selm=y8zk5f3rn4e.fsf%40nestle.csail.mit.edu

[1] Franz Lemmermeyer. Jacobi and Kummer's ideal numbers.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg v. 79, 2, 2009, 165-187. http://dx.doi.org/10.1007/s12188-009-0020-5

Answer 7

"Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, pages 44‒52, by me and Catherine Woodgold, debunks the widespread belief that Euclid's proof of the infinitude of primes is a proof by contradiction. The proof that Euclid actually wrote is simpler and better than the proof by contradiction often attributed to him.

Answer 8

I like this question very much - many popular history of mathematics books seem filled with legends more than history.

I would expand the question "What are some good papers that debunk common myths...?" to include books as well as papers. Maybe the best books to debunk myths are books that directly discuss and include the primary source material (in translation, perhaps). A fantastic recent book in this spirit is "The Mathematics of Egypt, China, India, and Islam: A Sourcebook," edited by Victor J. Katz. This is a great value as well! For example, an scholarly translation of the Chinese "Nine Chapters on the Mathematical Art" costs around 350 dollars on Amazon -- but one can instead find it in this sourcebook, together with a multitude of other translated texts, for around 50 or 60 dollars.

To demonstrate that this book addresses your specific question, on pages 467-477 you can find a translation of the Bijaganita of Bhaskara II. At the end is Verse 129, the end of which is translated, "Hence, for the sake of brevity, the square-root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated. And otherwise, when one has set down those parts of the figure there, [merely] seeing [it is sufficient].

The author of this section then mentions "These verses are presumably the ultimate source of the widespread legend that Bhaskara gave a proof of the Pythagorean theorem containing only the square figure shown in figure 4.19 and the word 'Behold!' "

The figure (4.19 in the book) is not among the verses in an old text, as far as I know. I think that Indian texts of the period were traditionally written on palm leaves (which degrade somewhat quickly), and copied every generation or two, so we don't have very old texts. In any case, the "Behold!" legend for the Pythagorean theorem seems to be a myth or at least a vigorous embellishment of history.

Answer 9

There is a rumor (a myth?) in holomorphic dynamics about a rivalry between Fatou and Julia competing for the 1918 Grand Prize of the french science academy. In fact Fatou never submitted a memoir for the Prize.

What we now call the Julia set was first defined in a note in Comptes Rendus by Montel, and is the starting point for the work of Fatou and Julia. Fatou published a few notes on the subject in the Comptes Rendus before Julia, who started a quarrel, claiming priority for the results. Fatou didn't try to fight back, apparently because he didn't bother, and the Academia fullfilled the wish of Julia. This may be the source of the rumor.

You may also know the Parseval formula about the relationship between the square of the L2 norm of a function and its Fourier coefficients. This formula is due to Fatou. H. Lebesgue did the case of a bounded function, and Fatou extended it to any square integrable functions. This made a big impression on Lebesgue.

This is exposed in a recent book by Michele Audin entitled "Fatou, Julia, Montel (...)"

Answer 10

Check out "Life on the Mathematical Frontier: Legendary Figures and Their Adventures" by Roger Cooke in the April 2010 issue of the Notices (Volume 57, Number 4, pages 464–475), available online at http://www.ams.org/notices/201004/rtx100400464p.pdf .

I hope someone will follow up on Cooke's "Modest Proposal" (see page 473): "I would like to invite some ambitious mathematician with time on his/her hands to write a monograph on the role played by legends in the mathematical community. Or perhaps someone would be willing to set up the mathematical equivalent of the snopes.com website, where mathematical “urban legends” can be checked out."

Answer 11

Gauss made a famous statement:

I protest first of all against the use of an infinite quantity as a completed one, which is never permissible in mathematics. The infinite is only a façon de parler, where one is really speaking of limits to which certain ratios come as close as one likes while others are allowed to grow without restriction.

This statement is often used to claim that Gauss was opposed to the use of infinite sets. In the paper "Gauss on Infinity", (Historia Mathematica 6 (1979), 430-436), W. C. Waterhouse shows that Gauss was actually talking about the sloppy use of "circles of infinite radius" in a fallacious geometric proof by H. Schumacher of the parallel postulate. Infinite sets do not even come up, but rather it is the imprecise use of what we would now formalize as "unlimited non-standard real numbers" that Gauss is objecting to.

The article goes on to show that, curiously, it seems that it was Lipschitz, when writing to Cantor, who first interpreted the above statement by Gauss as criticizing the use of infinite sets.

I will now add my own statement going beyond Waterhouse's paper. It seems that Gauss's full criticism of Schumacher's proof, not just the excerpt commonly used, is saying that non-Euclidean planes have non-zero curvature, and it seems likely to me that if done with non-standard analysis, that is what Schumacher's proof would show, once the fallacious step was removed.

Answer 12

Here is an paper on this subject: Mathematical Myths G. A. Miller National Mathematics Magazine, Vol. 12, No. 8 (May, 1938), pp. 388-392

This is available on jstor

Answer 13

Was Cantor Surprised? published in Monthly is debunking (or trying to do so) that Cantor was so surprised when he discovered $I=[0,1]$ and $I^2$ have the same cardinality that he said “I see it, but I don’t believe it!”. The abstract of the paper reads as follows:

Abstract. We look at the circumstances and context of Cantor’s famous remark, “I see it, but I don’t believe it.” We argue that, rather than denoting astonishment at his result, the remark pointed to Cantor’s worry about the correctness of his proof.

Answer 14

There have been recent articles in The Mathematical Intelligencer about the well-known rivalry/animosity between Erdős and Selberg. Didier adds this link, pointing to an article of Graham and Spencer that appeared in The Mathematical Intelligencer. Emerton adds this link, pointing to an article by Goldfeld on the controversy.

Answer 15

David Fowler's book "The Mathematics Of Plato's Academy: A New Reconstruction" sets out to deconstruct the myth that "the early Pythagoreans based their mathematics on commensurable magnitudes, but their discovery of the phenomenon of incommensurability (the irrationality of the square root of 2) showed that this was inadequate; this provoked problems in the foundation of mathematics that were not resolved before the discovery of the proportion theory that we find in Book V of Euclid's Elements". The arguments and conclusions are complex and interesting; here is a review with some of the main points: http://www.maa.org/publications/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction.

Answer 16

I think the myth that Kolmogorov was the first to come up with the idea of basing probability theory on measure theory is very common. This myth is debunked in

The Sources of Kolmogorov’s Grundbegriffe

G.Shafer and V.Vovk, The Sources of Kolmogorov’s Grundbegriffe, Statist. Sci. Volume 21, Number 1 (2006), 70-98. Online (free).

Answer 17

Frank Nelson Cole's own 1903 paper: On The Factoring of Large Numbers, discussed in another MO question, debunks the myth that he factored $M_{67}$ with "3 years of Sundays" using only trivial trial division.

What is yet to be debunked, but many have found suspicious, is whether he presented his results to the AMS by "silently multiplying the factors together."

Added Later

Izzy Grosof, in what appears to be the good-natured SIGBOVIK proceedings of April 1, 2019 (link to PDF), estimates the time complexity of various methods of multiplication - by performing the multiplication of the factors of $M_{67}$ with pencil-and-paper. Grosof concludes that if Cole did multiply at the chalkboard, he might have used lattice multiplication, which Grosof was able to do error-free in about 10 minutes.

(H/T to @Arcorann for pointing this out on hsm.stackexchange.)

Answer 18

Blaszczyk, Katz, and Sherry, "Ten Misconceptions from the History of Analysis and Their Debunking," http://arxiv.org/abs/1202.4153

The questions they address are:

1. Were the founders of calculus working in a numerical vacuum?

2. Was Berkeley’s criticism coherent?

3. Were d'Alembert’s anticipations ahead of his time?

4. Did Cauchy replace infinitesimals by rigor?

5. Was Cauchy’s 1821 "sum theorem" false?

6. Did Weierstrass succeed in eliminating infinitesimals?

7. Did Dedekind discover the essence of continuity?

8. Who invented Dirac’s delta function?

9. Is there continuity between Leibniz and Robinson?

10. Is Lakatos’ take on Cauchy tainted by Kuhnian relativism?

The main thread of their revisionist picture is that 17th- and 18th-century analysis had foundations that were far more rigorous and well-defined than most modern mathematicians have been led to believe.

Answer 19

There is the widespread myth that Cauchy gave an epsilon-delta definition of continuity. For Cauchy's definition, see http://u.cs.biu.ac.il/~katzmik/bradleypage26.pdf

To respond to Timothy Chow's comment, there is a famous essay by historian Judith Grabiner entitled "who gave you the epsilon?" with the implied answer being "Cauchy". This can be viewed at http://www.ams.org/mathscinet-getitem?mr=691368

It should be noted that not all historians agree with Grabiner. Thus, Schubring in a recent text comments: "I am criticizing historiographical approaches like that of Judith Grabiner where one sees epsilon-delta already realized in Cauchy". This can be viewed at http://link.springer.com/article/10.1007/s10699-015-9424-0

Answer 20

I would expand the question "What are some good papers that debunk common myths...?" to include web pages as well as papers. Naturally, there is a couple of candidate web pages in the reference section of the note mentioned by sigfpe. Another candidate might be http://www.snopes.com/science/nobel.asp where the myth of the rivalry between A. Nobel and Mittag-Leffler is debunked (to some extent).

Answer 21

Donald E. Knuth's "Johann Faulhaber and sums of powers" debunks common belief (mentioned at at Mathworld, for example) that Faulhaber was first to discover Bernoulli formula and even (to some extent) Bernoulli numbers.

Answer 22

Historians of mathematics get very cross if you dare to say "Pythagoras's theorem." You have to say, "The Pythagorean theorem," to emphasize that it wasn't the brainwave of a man called Pythagoras. Indeed, that fact, as well as the irrationality of the square root of 2, were (I read) probably known long before Pythagoras, and the only Greek proofs we know of came much later. As for the story that the penalty for revealing the irrationality of the square root of 2 to outsiders was death ...

Answer 23

B. H. Brown debunks in [1] the myth that "Algebra was Hebrew to Diderot" (cf. E. T. Bell, Men of Mathematics, NY, 1937, pp. 146-147).

Gillings mentions in his note that it is D. Thiébault's account the only authority in the Euler-Diderot anecdote (even though "Thiébault himself was not convinced of the truth of it...").

It is important to add that in Thiébault's version of the story there is no explicit reference to the name of L. Euler.

References

[1] B. H. Brown. The Euler-Diderot Anecdote. Amer. Math. Monthly. Vol. 49, Issue 5, 1942, pp. 302-303.

[2] R. J. Gillings. The So-called Euler-Diderot Incident. Amer. Math. Monthly. Vol. 61, Issue 2, 1954, pp. 77-80.

Answer 24

Theodor Nenu and I have a paper addressing the question of whether Alan Turing proved the undecidability of the halting problem in his seminal 1936 paper on computable numbers, in which he introduces Turing machines.

Joel David Hamkins and Theodor Nenu, “Did Turing prove the undecidability of the halting problem?”, 18 pages, 2024, Mathematics arXiv:2407.00680, JDH blog post.

Abstract. We discuss the accuracy of the attribution commonly given to Turing (1936) for the computable undecidability of the halting problem, eventually coming to a nuanced conclusion.

From the introduction:

The halting problem is the decision problem of determining whether a given computer program halts on a given input, a problem famously known to be computably undecidable. In the computability theory literature, one quite commonly finds attribution for this result given to Alan Turing (1936), and we should like to consider the extent to which these attributions are accurate. After all, the term halting problem, the modern formulation of the problem, as well as the common self-referential proof of its undecidability, are all—strictly speaking—absent from Turing’s work. However, Turing does introduce the concept of an undecidable decision problem, proving that what he calls the circle-free problem is undecidable and subsequently also that what we call the symbol-printing problem, to decide if a given program will ever print a given symbol, is undecidable. This latter problem is easily seen to be computably equivalent to the halting problem and can arguably serve in diverse contexts and applications in place of the halting problem—they are easily translated to one another. Furthermore, Turing laid down an extensive framework of ideas sufficient for the contemporary analysis of the halting problem, including: the definition of Turing machines; the labeling of programs by numbers in a way that enables programs to be enumerated and also for them to be given as input to other programs; the existence of a universal computer; the undecidability of several problems that, like the halting problem, take other programs as input, including the circle-free problem, the symbol-printing problem, and the infinite-symbol-printing problem, as well as the Hilbert-Ackermann Entscheidungsproblem. In light of these facts, and considering some general cultural observations, by which mathematical attributions are often made not strictly for the exact content of original work, but also generously in many cases for the further aggregative insights to which those ideas directly gave rise, ultimately we do not find it unreasonable to offer qualified attribution to Turing for the undecidability of the halting problem. That said, we also find it incorrect to suggest that one will find a discussion of the halting problem or a proof of its undecidability in Turing (1936).

Answer 25

Myths can be such nice inventions, one should invent more of them!

Answer 26

This is a very old question, but a favourite paper of mine along the debunking lines is Peter Neumann’s “A lemma that is not Burnside’s”.

http://www.appliedprobability.org/data/files/TMS%20articles/4_2_11.pdf

In which he observes that the orbit counting lemma (the number of orbits of a permutation group is equal to the average number of fixed points of its elements), which was frequently referred to as “Burnside’s Lemma”, is not actually due to Burnside.

His goal - to correct the misattribution before it was too late - seems to have been realised, because nowadays it is normally called the Cauchy-Frobenius Lemma or just the “orbit counting theorem”.

Answer 27

There are stories about work of Banach being written up by people other than Banach. Details and debunking links available on another MO thread, Who wrote up Banach's Thesis?

Answer 28

A recent one could be Peter Milne's "On Gödel Sentences and What They Say" Philosophia Mathematica (III) 15 (2007), 193–226. doi:10.1093/philmat/nkm015 debunking the myth that Gödel sentences are true because they say of themselves that they are unprovable.

Sorry, I realize this is not exactly what was being asked. However, there is still some analogy with the pattern "someone was believed to do or say something while in fact he/she didn't".

Answer 29

One persistent (and quite annoying) myth is the one about the incident between Leonard Euler and Denis Diderot visiting the court of Catherine II at St.Petersburg in 1773. This article documents its origin and how the anecdote was inflated by various authors, till E.T.Bell’s usual tabloid style.

The so-called Euler-Diderot incident, R.J.Gillings

AMM, Vol. 61, No. 2 (1954), pp. 77-80

Answer 30

This myth is in living memory and I have not seen it debunked in print. The myth is that the Appel-Haken proof of the four colour theorem was controversial when it was published because it was a computer proof. This makes a good story but is it true?