CTK Insights

21 Feb

Mathematical Legends

The bible takes six words to describe the evolution of the Hebrew tribe in Egypt from 70 souls to a people (Exodus 1.7). The closest translation I found among several is this: "... and the sons of Israel have been fruitful, and they teem, and multiply, and are very very mighty ..." In Hebrew, the four verbs are practically synonymous and the conveyed meaning is strengthened by the double "very very". What is the reason the bible found it necessary to use six words? Would not two or three suffice. Since everything in the bible has a meaning and a purpose, Rabbi Shlomo Yitzhaki (1040 - 1105) whose commentary became a standard fixture of the Hebrew Bible interprets the sentence to say that Hebrew women were producing six babies on every delivery. Bible commentaries and the rest of the vast Hebrew literature contain (if not overflow with) views and stories like that, legends that became an article of faith.

Rabbi Berel Wein quotes [Patterns in Jewish History, p. 167] the famous Rabbi Menahem Mendel Morgenstern of Kotzk who said

He who believes all of these tales is a fool and yet he who states that they could not have occurred is a non-believer.

and then proceeds to ponder that indeed the very existence of such legends reflects on historical events or persons. He corroborates that point of view with an anecdote:

Two men once exchanged stories about a great sage for whom fantastic claims of spirituality and piety were being made. One asked, "Do you really believe that story?" The other replied, "No, I do not. But no one tells such stories about the two of us.

This set me to thinking about the many legends - unconfirmed but widely told stories - which are abundant in math folklore. The story of Archimedes' death by a sword of a Roman soldier, irate for being ignored by Archimedes who was drawing math figures in the sand, is most likely untrue but sheds light on Archimedes' degree of concentration. Newton's conceiving of gravity due to a fall of an apple is a likely metaphor for Newton having insights that to other people might have appeared "out of the blue".

A quick search on the web brought to my attention an entertaining paper Life on the Mathematical Frontier: Legendary Figures and Their Adventures by Roger Cooke - a math professor at the University of Vermont - (Notices of the AMS, Volume 57, Number 4 (April 2010), 464-475). The paper was rather congenial to the idea I had in mind.

Cooke observes that legends arise also in modern times:

Who has not heard the "explanation" of the absence of a Nobel Prize in mathematics?

One of the modern legends is the story of George Bernard Dantzig (1914–2005) I have mentioned in a previous post. Dantzig confirms the story in an interview [More Mathematical People, p. 67], so what makes it a legend? In time, the story acquired a life of its own. Cooke recollects that the very same story was told to him by his roommate, but the central figure of it was John Milnor.

Some legends, Cooke says, have undoubtedly been lost. For example, L. A. Lyusternik reminisces that in Luzin’s Moscow school it was customary to invent exotic proofs of the infinitude of primes. Only a few have been documented. Among the surviving proofs is one ascribed to Khinchin. The proof is based on Euler’s formula

\prod_{p}(1-\frac{1}{p^{2}})^{-1}=\sum_{n=1}\frac{1}{n^{2}}=\frac{\pi^{2}}{6},

where the product on the left is over all the primes. If the set of primes were finite, the left-hand side of this formula would be a rational number, and hence \pi^{2} would be rational. (The reciprocal of this expression gives the probability that two random integers are mutually prime.)

Actually, there are quite a few collections of mathematical anecdotes bordering on being legends. As H. Eves wrote of N. Wiener in his [Mathematical Circles Revisited, pp 174-175]

In time his brilliance and his eccentricity became woven into M.I.T. campus mythology, and a host of stories and legends sprang up about him.

This is the one I like best (355°):

It happened that the Wieners moved to a new house, in the same neighborhood as the former residence. Knowing her husband absentmindedness, Mrs. Wiener gave him careful and written instructions for reaching the new house. However at the close of the day, Professor Wiener could not find the written instructions and of course did not remember them. Hence, seeking something familiar, he set off in the direction of his former residence. Presently he spied a young child and asked her" "Little girl, can you tell me where the Wieners have moved to?" "Yes, Daddy," came the reply, "Mommy said you'd probably be here so she sent me over to show you the way home."

Rabbi Wein writes (p. 168)

The Jewish people as a whole possess a strong collective memory. ... This memory bank has been fed by stories about great people, significant events and terrible tragedies that have occurred over the millennia of Jewish life. These legends, whether completely accurate or not, help us recall the core event and/or person about which they revolve and, in so doing, keep our memory of the past alive enabling us to deal so much better with our present situations and challenges.

According to Philip Davis (the first chapter of the collection Essays in Humanistic Mathematics),

Mathematics, like literature, has metaphor. Mathematics, like poetry, has ambiguity. Mathematics possess aesthetic component ... Mathematics has paradox. Mathematics has mystery and can convey awe. Mathematics has: a sense of outcome, a feeling of rightness, a sense of catharsis. ... Mathematics has history. Like anthropology and literature, mathematics embodies mythologies.

Mathematics has collective memory that enlivens its subject and makes it a human endeavor. Sometimes I think that mathematics education, though, has no memory, see, for example, a blog by Julie Mack (with thanks to Gary Davis.) But this is a separate story.

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