The Marginal Note That Sparked One of Math’s Greatest Mysteries

First things first, I want to apologize for the significant gap in time between the last post and this one.  Things have been a bit hectic, but now that everything is back on track the blog will be updated with regularity.  With the introductory posts out of the way, content will be more frequent and less lengthy.

This week I want to focus on a particular type of provenance evidence, marginalia.  People write in books for a number of different reasons, but these marginal notes are most often textual commentary or study aides.  Occasionally, when reading through the notes someone left in an old text, it feels almost as if that person is speaking to you, personally, through the veil of ages.  Marginal note-takers often word their annotations in a particular voice that addresses either the author of the text, or the reader directly.   Notes in the latter case are likely intended to serve as reminders to the annotator himself, but when the notes outlive the note-taker they can seem to speak directly to any future reader.

This particular quality of voice lends some marginal notes a tantalizing air of mystery, when they seem to be speaking directly to us, but (since they were usually written for the writer himself) are cryptic and reference persons or events unknown to us.  Let’s look at a somewhat well-known example that shows just how important and frustrating marginalia can be: the story of Fermat’s Last Theorem.

Pierre de Fermat was a seventeenth century French lawyer and an amateur mathematician.  He made a number of contributions to several fields of mathematics, and like many scholars of his time, would frequently scrawl copious amounts of notes in the margins of the various books and manuscripts in his personal mathematical library.

Fermat would often annotate books in his collection claiming to have worked out proofs to mathematical problems posed therein.  After his death, his son published special editions of many of these books, editions which included the text of Fermat’s marginal annotations.  The notes outlived their creator, and allowed the mathematician to speak to future scholars long after he was gone.

Readers of these annotations were often curious about the bold claims Fermat made, that he had worked out proofs of a number of mathematical conjectures.  As people read through his work, however, or attempted to solve the problems themselves, they realized that Fermat’s claims were seemingly based in reality.  Eventually, every problem that Fermat claimed was solvable was proven by future mathematicians – every problem but one.

In his copy of Diophantus’s Arithmetica, Fermat wrote a particularly vague and vexing annotation next to the explanation of one problem.  Addressing the problem posed by Diophantus of how to split a given square number into two other squares (as in Pythagoras’s famous equation a2+b2=c2), Fermat made the bold claim that “it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers.  I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

No proof of this conjecture was ever found in Fermat’s other notes or surviving writings, and unlike the mathematician’s other striking claims, this supposed lost proof of Fermat’s theorem would elude mathematicians for centuries.  A solution was not found for this conjecture until Andrew Wiles finally cracked the case in 1995.

Science and math journalist Simon Singh wrote a book on the history of Fermat’s problem, and in the video below he weaves the tale of the 358-year hunt for a proof sparked by this infamous marginal note.

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