How Do Mathematicians Know Their Proofs Are Correct?
The Joy of Why
Steven Strogatz, Janna Levin and Quanta Magazine
4.9 • 577 Ratings
🗓️ 13 July 2022
⏱️ 29 minutes
🧾️ Download transcript
Summary
Just as scientists test hypotheses, mathematicians prove or disprove conjectures. But what makes a proof stronger than a guess? What does evidence look like in the world of mathematics? Hear Melanie Matchett Wood, professor of mathematics at Harvard University, explain how probability helps to guide number theorists toward certainty.
Transcript
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| 0:00.0 | I'm Steve Strogatz and this is The Joy of Why. |
| 0:06.0 | A podcast from Quantum Magazine that takes you into some of the biggest unanswered questions in math and science today. |
| 0:13.0 | In this episode, we're going to be talking about evidence in mathematics. |
| 0:18.0 | What kinds of evidence do mathematicians use? What leads them to suspect that |
| 0:23.1 | something might be true before they have a watertight proof? It might sound like a paradox, |
| 0:28.8 | but it turns out that reasoning based on probability theory, the study of chance and randomness, |
| 0:35.5 | can sometimes lead to what mathematicians are really after, |
| 0:39.1 | which is certainty, not just probability. For example, in the branch of math known as number |
| 0:44.3 | theory, there's a long history of using randomness to help mathematicians guess what's true. |
| 0:50.1 | Now probability is being used to help them prove what's true. |
| 0:56.9 | We'll be focusing here on prime numbers. |
| 1:00.5 | You probably remember prime numbers, right? You learned about them at school. |
| 1:07.3 | A prime number is a whole number greater than one that can only be divided by one and itself. |
| 1:13.5 | For instance, seven or 11, those are prime numbers, but 15 is not because 15 can be divided evenly by three or by five. You could think of prime numbers as sort of like the elements |
| 1:19.8 | in the periodic table of chemistry in the sense that they are the indivisible atoms that make up |
| 1:25.8 | all the other numbers. |
| 1:32.8 | Prime numbers seem like they should be simple, but some of the biggest mysteries in math are questions about prime numbers. In some cases, questions that have been around for hundreds of |
| 1:37.1 | years. There's really something very subtle about primes. They seem to live in a borderland |
| 1:43.8 | between order and randomness. |
| 1:46.4 | My guest today will help us understand more about the nature of evidence in math, |
| 1:50.5 | and especially how and why randomness can tell us so much about prime numbers, |
| 1:55.7 | and why models based on probability can be so useful at the cutting edge of number theory. |
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