Lex Fridman: commitment

Transcript context

…The following is a conversation with Joel David Hamkins, a mathematician and philosopher specializing in set theory, the foundation of mathematics, and the nature of infinity. He is the number one highest rated user on MathOverflow, which I think is a legendary accomplishment. MathOverflow, by the way, is like StackOverflow but for research mathematicians. He is also the author of several books, including Proof in The Art of Mathematics and Lectures on the Philosophy of Mathematics. And he has a great blog, infinitelymore.xyz. This is a super technical and super fun conversation about the foundation of modern mathematics and some mind-bending ideas about infinity, nature of reality, truth, and the mathematical paradoxes that challenged some of the greatest minds of the 20th century. I have been hiding from the world a bit, reading, thinking, writing, soul-searching, as we all do every once in a while. But mostly, just deeply focused on work and preparing mentally for some challenging travel I plan to take on in the new year. Through all of it, a recurring thought comes to me, how damn lucky I am to be alive and to get to experience so much love from folks across the world. I want to take this moment to say thank you from the bottom of my heart for everything, for your support, for the many amazing conversations I’ve had with people across the world. I got a little bit of hate and a whole lot of love, and I wouldn’t have it any other way. I’m grateful for all of it. This is the Lex Fridman Podcast. To support it, please check out our sponsors in the description, where you can also find ways to contact me, ask questions, give feedback, and so on. And now, dear friends, here’s Joel David Hamkins. Some infinities are bigger than others. This idea from Cantor at the end of the 19th century, I think it’s fair to say, broke mathematics before rebuilding it. I also read that this was a devastating and transformative discovery for several reasons. So one, it created a theological crisis, because infinity is associated with God, how could there be multiple infinities? Also, Cantor was deeply religious himself. Second, there’s a kind of mathematical civil war. The leading German mathematician, Kronecker, called Cantor a corrupter of youth and tried to block his career. Third, many fascinating paradoxes emerged from this, like Russell’s paradox, about the set of all sets that don’t contain themselves, and those threatened to make all of mathematics inconsistent. Finally, on the psychological and personal side, Cantor’s own breakdown. He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis. So laying that all out on the table, can you explain the idea of infinity, that some infinities are larger than others, and why was this so transformative to mathematics? Well, that’s a really great question. I would want to start talking about infinity and telling the story much earlier than Cantor actually, because, I mean, you can go all the way back to Ancient Greek times when Aristotle emphasized the potential aspect of infinity as opposed to the impossibility, according to him, of achieving an actual infinity. Archimedes’ method of exhaustion where he is trying to understand the area of a region by carving it into more and more triangles, say, and sort of exhausting the area and thereby understanding the total area in terms of the sum of the areas of the pieces that he put into it. And it proceeded on this kind of potential understanding of infinity for hundreds, thousands of years. Almost all mathematicians were potentialists only and thought that it was incoherent to speak of an actual infinity at all. Galileo is an extremely prominent exception to this, though he argued against this sort of potentialist orthodoxy in The Dialogue of Two New Sciences. Really lovely account there that he gave. In many ways, Galileo was anticipating Cantor’s developments, except he couldn’t quite push it all the way through and ended up throwing up his hands in confusion, in a sense. The Galileo paradox is the idea or the observation that if you think about the natural numbers, I would start with zero, but I think maybe he would start with one. The numbers one, two, three, four, and so on, and you think about which of those numbers are perfect squares. So zero squared is zero and one squared is one and two squared is four, three squared is nine, 16, 25, and so on. And Galileo observed that the perfect squares can be put into a one-to-one correspondence with all of the numbers. I mean, we just did it. I associated every number with its square. And so it seems like on the basis of this one-to-one correspondence that there should be exactly the same number of squares, perfect squares, as there are numbers, and yet there are all the gaps in between the perfect squares, right? And this suggests that there should be fewer perfect squares, more numbers than squares because the numbers include all the squares plus a lot more in between them, right? And Galileo was quite troubled by this observation because he took it to cause a kind of incoherence in the comparison of infinite quantities, right? Another example is, if you take two line segments of different lengths, and you can imagine drawing a kind of foliation, a fan of lines that connect them. So the endpoints are matched from the shorter to the longer segment, and the midpoints are matched and so on. So spreading out the lines as you go. And so every point on the shorter line would be associated with a unique, distinct point on the longer line in a one-to-one way.…

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