Mind and Matter by John Urschel & Louisa Thomas

Author:John Urschel & Louisa Thomas
Language: eng
Format: epub, pdf
Publisher: Penguin Publishing Group
Published: 2019-05-13T16:00:00+00:00

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OR AT LEAST, that was the idea.

Before practice, while most of the students—and even most of my teammates—were still sleeping, I’d wake up early and work on problems or read. I’d do the same before going to bed. Most of the books I was reading were textbooks, but occasionally I’d come across a book or essay that took a broader perspective. I was always more interested in mathematics than philosophy, more drawn to physics than metaphysics, but some of the deeper questions became inescapable when I started working more seriously with proofs. For instance, if proofs were to be founded on perfectly solid foundations, built only from axioms and statements that have been proven before, where does the chain of logic begin? At the end of the nineteenth century and early twentieth, a German mathematician named David Hilbert had led an effort to formalize proofs so that they existed only on axiomatic grounds. He wanted to build a system statement by statement, one that would make intuition unnecessary. The human element would disappear; there would be no reliance on common sense or innate understanding. We would be left with only well-defined facts. He had given a talk once in which he said, “In mathematics there is no ignorabimus”—Latin for “we will not know.” He went on, “We must know. We shall know.” But as it turned out, that was not true.

A young Austrian mathematician, Kurt Gödel, showed why. He discovered that any axiomatic system that relies on basic arithmetic will be incomplete. There will always be statements about the natural numbers that are true but unprovable within the formal system. To show this, he used a brilliant—and rigorous—proof. Soon after, he wrote a second paper extending the results of the first, showing that a system cannot demonstrate its own consistency. Gödel had been in part inspired by a famous paradox that I had come across as a kid doing puzzles: “This sentence is false.” (It’s known as the liar’s paradox, and it stretches back to ancient Greece.) If the sentence is true, then it is false, because that is the claim that it is making. But if it is false, then it is true, because that is also what it is saying. It can’t be both true and false, and yet . . . Gödel created a kind of formal analog to the paradox using self-referential logical propositions to prove that there are true propositions that cannot be proven within arithmetic.

Many people thought that Gödel’s results meant the end of objective truth. That horrified Gödel. Like his good friend Einstein, who rejected interpretations of relativity that elevated subjectivity, Gödel absolutely believed in the existence of a mathematical reality. But his work did leave open a role for intuition in mathematics.

I thought about the role of intuition in mathematics a lot. Reason and imagination can influence and reinforce each other. Intuition will not take you very far if you cannot support it with sound theory, but theory has its limits, if it is not constantly questioned and invigorated by intuition.

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