Quantslut: A laywoman’s musings on the “hard” sciences and math

I’ve been learning about Fermat’s Last Theorem (FLT) lately, and it’s fascinating. Almost everyone remembers learning the Pythagorean Theorem at some point in school: a² + b² =c² describes the way two sides of a right triangle relate to its hypotenuse. In 1670, the French mathematician Fermat guessed that if you switch those exponents to anything higher than 2, you’ll never be able to solve the equation with nice, clean, integers for a, b, and c. He wrote this conjecture in the margin of a book he was reading, adding that he had a proof, but no room to write it there. So for centuries, mathematicians hacked away at the pieces of the puzzle until finally, in 1995, Andrew Wiles did the last, very difficult chunk of it, and won the Field’s Medal (math’s Nobel). But that’s not the end of the story—mathematicians tend to prove things over and over again in different ways, uncovering new aspects of an idea each time.  So the search is on for simpler proofs of the FLT, ones that Fermat and his peers—if we traveled back in time to meet them—would understand.

Mathematician  Colin McLarty, writing for specialists, has shown that the implications of Wiles’ proof are profound and will take a long time to work out. As a mathematical outsider (who learned much of what I know about the FLT from an article written by the great math popularizer Ian Stewart, whose books I recommend), it strikes me as odd that a simple-sounding statement should prove so fiendishly hard to prove. It made me wonder whether the FLT is in some deep way about exponentiation itself—meaning about what multiplication and addition, and duplication or identity, actually are. Those are topics number theorists do indeed ponder—and there are wild and woolly areas of math where the answers are not what you’d expect.  (Addition, for example, is no longer commutative in the ordinal transfinite.)

The fact that FLT is “trivially false” (as one says in the math trade) for infinite cardinals, and other oddities of infinite arithmetic, are one of the topics taken up by the late Bruno Augenstein in an excellent article on the Links Between Physics and Set Theory. That’s been another of my favorite reads this month, since I’ve wondered for a while whether the apparently-super-abstract math of set theory has practical implications. Pure math often does; but pressing too hard on this point can get one accused of Platonism, the now-unfashionable belief that math exists outside our minds. Citing a number of great mathematicians who leaned in that direction, however, Augenstein suggests we take the possibility seriously. Among Augenstein’s points are that the empty set can be used to generate nearly the whole structure of mathematics, and the vacuum may be similarly productive for physics; that seminal set theorist Paul Cohen introduced to that field notions of partial information and information content that are also relevant for physics; and that the ideas of foundational thinkers including Georg Cantor, David Hilbert and Kurt Goedel contributed to both disciplines. Most provocatively of all, he argues that a truly robust theorization of quantum mechanics would require the Axiom of Choice—one of the most productively ambiguous and controversial items in the set theorists’ toolkit—to be true.

Two other books have also intrigued me lately. One is Jan von Plato’s The Great Formal Machinery Works, which focuses on aforementioned über-logician Kurt Gödel.  He’s best known for proving that any mathematical language complex enough to be interesting will be able to express more than it can prove (because there will always be sentences equivalent to “This sentence is false,” which is true if false, and false if true). Von Plato shows that along the way to his Incompleteness Theorems (which simultaneously broke mathematics and showed it new ways to thrive), Gödel made contributions to the infant science of coding that made modern computing possible.  It’s a great correction to the popular origin-narrative of computing that cites only Babbage, Lovelace and Turing, and like Augenstein’s piece, it illuminates links between “pure” math and the technological world in which we live.

Finally, I’ve been riveted by what I can grasp of The History of Continua, an essay collection edited by Stuart Shapiro and Geoffrey Hellman which surveys the history of human thinking about continuity from ancient Greece to the present day. Continuity has played a big role in the development of modern math—perhaps most famously through the Continuum Hypothesis, which is like the Voynich manuscript of set theory—a puzzle whose solution may be irrelevant for practical purposes, but which never ceases to fascinate. The most mathematically influential version of the continuum is the late-nineteenth century one associated with the aforementioned Cantor, which imagines a continuum as a smooth and infinitely divisible collection of points.  More recent approaches to the topic, however, think of continua as potentially “gunky” instead—full of clumps and bumps, stops and starts, and clogs. In addition to being thrilled that the slangy “gunk” is now a mathematical term of art, I was reminded of Felix Hausdorff’s 1898 book, Chaos in Cosmic Selection, where he makes similar suggestions about the nature of time.  Maybe these “new” mathematical conceptions are revisiting the byways explored by the greats of generations past—and perhaps applying them to all the problems for which continuity matters will generate new insights.

~ Kimberly Gladman ~