This Man Is About to Blow Up Mathematics

Harvey Friedman is about to bring incompleteness and infinity out of quarantine.

By JORDANA CEPELEWICZ

Feb 23 2017

http://nautil.us/issue/45/power/this-man-is-about-to-blow-up-mathematics

It is 7 o’clock in the morning and Harvey Friedman has just sent an email to an unspecified number of recipients with the subject line “stop what you are doing.” It features a YouTube link to a live 1951 broadcast of a concert by the famous Russian pianist Vladimir Horowitz. “There is a pattern on YouTube of priceless gems getting taken down by copyright claims,” Friedman writes, “so I demand (smile) that you stop everything you are doing, including breathing, eating, thinking, sleeping, and so forth, to listen to this before it disappears.”

His comment takes its place at the top of a chain of emails stretching back months, with roughly as many messages sent at 3 a.m. as at noon or 9 p.m. The haphazard correspondence covers a wide range of topics, from electronic music editing to an interdisciplinary field Friedman calls “ChessMath.” At one point, he proposes to record at home, by himself, a three-part “Emotion Concert.” Anonymous piano players on the email thread discuss their own thoughts on the lineup.

As diverse as the topics in the email history are, Friedman asks the same question of them all: What are their basic constituents and what laws govern them? He seems to be searching for the right vocabulary—“the right way,” he says, “of talking about what the fundamental ideas are, to black-box the ad hoc technicalities and get to the real meat of the thing.”

That is not to say all of these topics are equal. There is one that is nearest and dearest to Friedman’s heart: the foundations of mathematics, which concerns itself with the consistency, unity, and structure of mathematics itself. The field has occupied Friedman since his teenage years, when he first read Bertrand Russell’s Introduction to Mathematical Philosophy. (If you’re thinking it’s not an easy read, you’re right: “Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes…”) And it consumes him still as a 68-year-old retired math professor living on a leafy street in suburban Columbus, Ohio, sleeping for a few hours at a time, twice a day, so as to free up time to think.

The foundations of mathematics is also a field—in stark contrast to the casual and light tone of Friedman’s emails—that has been in crisis for nearly a century. In 1931, the Austrian mathematician and philosopher Kurt Gödel proved that any logical system adequate to develop basic arithmetic gives rise to statements that cannot be proven true or false within that system. One such statement: that the system itself is consistent. In other words, no system can ever prove itself to be free of contradiction. The result seemed to present an insurmountable problem for mathematicians, not so much because it prevented them from ever knowing whether the system their work is built on is consistent (so far there haven’t been inconsistencies), but because it meant their fundamental logic had significant limitations.

Any hope for a unified formal theory of mathematics, an endeavor championed by the mathematician David Hilbert in the 19th century and into the 20th (and taken up by many others), was dashed. The foundations of mathematics could never be as secure as Hilbert wanted: Gödel had effectively shown that every axiomatic system, no matter how comprehensive, is vulnerable to irreparable holes. Filling those holes by creating a stronger system would only yield new statements that cannot be proven—so that an even stronger system would be needed, and so on, ad infinitum.

And so something odd happened: Mathematicians chose to move on. Incompleteness, they decided, had no direct bearing on their own work. The axioms commonly known as ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice) that constitute today’s most commonly used foundation of mathematics provides a rigorous framework for proving theorems. In fact, ZFC turned out to be so comprehensive that most mathematicians today don’t use the entire extent of its machinery anyway. “You can carry out Hilbert’s program in a pretty sweeping way,” says Stephen Simpson, a mathematician at Vanderbilt University, “for something like 85 percent of mathematics.” Statements whose proofs do require something stronger than ZFC are long-winded and esoteric—contrived, artificial renderings of the self-referential sentence “I am not provable” and the like. Philosophically interesting, but safely ignored when doing “core” mathematics.

[snip]