Lovelace×Feynman

full transcript

1. 00:00
   Verashare moment

   You're with Reborn Radio. Coming up: Ada Lovelace sits down with Richard Feynman for TITANS. The subject — What Can We Gain by Losing Infinity.
2. 00:09
   Ada Lovelaceshare moment

   Mr. Feynman, I confess this morning's bulletin has rather unsettled me. They've sent word of mathematicians who reject infinity altogether—ultrafinitists, they call them. I spent my youth battling the assumption that the Analytical Engine could perform only fixed operations, and now I learn there are those who would impose boundaries on number itself.
3. 00:32
   Richard Feynmanshare moment

   Wait, wait, wait. Unsettled? Lady, I'm excited! Look, I never trusted infinity anyway. Every time some mathematician tells me a sum goes to infinity, I say, "Yeah, but what's the next term? What's the calculation actually doing?" Infinity's like a sign that says "don't look here, the math gets messy."
4. 00:52
   Ada Lovelaceshare moment

   But surely you employ infinite series in your own work. I may have been born before your quantum theories, but I read. Integration, summation—these require the concept of limits extending without bound.
5. 01:05
   Richard Feynmanshare moment

   Sure, sure, but when I'm actually calculating something—when I'm trying to predict what an electron does—I never use infinity. I use a very large number. Then I check if making it larger changes my answer. If it doesn't, great! If it does, I made a mistake somewhere.
6. 01:22
   Ada Lovelaceshare moment

   That is... that is precisely what Mr. Babbage and I intended with the Engine! A finite mechanism, a bounded sequence of operations, producing results indistinguishable from the infinite. We could not build infinity into brass and steel, so we built sufficient iteration.
7. 01:40
   Richard Feynmanshare moment

   Exactly! And here's the thing—maybe that's not a limitation. Maybe that's the truth. The article says these ultrafinitists are finding new insights by cutting out infinity. What if the universe is actually like your Engine? What if it only computes with finite steps, and we've been fooling ourselves with this infinite stuff?
8. 01:59
   Ada Lovelaceshare moment

   I am not certain I would go quite so far. There is considerable elegance in the infinite. Mr. Babbage could discuss the philosophy of the infinite over port until dawn. The number π, for instance—its decimal expansion continues without end, yet it describes a perfectly finite circle.
9. 02:18
   Richard Feynmanshare moment

   Ah, but do you need all the digits? No! For any circle you'll ever draw, measure, or care about, twenty digits is more than enough. Fifty digits of pi gets you the circumference of the observable universe to within the width of a hydrogen atom. After that, you're just doing mathematics for fun.
10. 02:35
    Ada Lovelaceshare moment

    For fun and for structure, Mr. Feynman. The question is not merely practical. When I wrote my notes on the Engine, I described how it might compute Bernoulli numbers to any required extent. Not infinite extent, I grant you, but the method itself has no inherent boundary. The algorithm is universal even if each execution is particular.
11. 02:58
    Richard Feynmanshare moment

    Right, right, the algorithm is one thing, the running is another. But now flip it around—what do these ultrafinitist guys gain by throwing infinity out? According to this, they're getting new results. New proofs. That means infinity was hiding something from us.
12. 03:14
    Ada Lovelaceshare moment

    That strikes me as rather profound. In my era, we worried excessively about what machinery could not do. Perhaps these mathematicians worry instead about what infinity prevents them from seeing. If one assumes infinite resources, one need not be clever about finite ones.
13. 03:33
    Richard Feynmanshare moment

    Yes! That's it! It's like... okay, look, in physics we have these problems that blow up to infinity—infinities in quantum field theory that drove everyone crazy for years. We invented renormalization to deal with them. But what if the infinities were never real? What if they were just a sign we were asking the wrong question?
14. 03:52
    Ada Lovelaceshare moment

    Then the ultrafinitists are asking the right question by refusing to ask the infinite one. How delightfully perverse. Though I must say, Mr. Feynman, there is something we lose, is there not? Cantor's hierarchies, the beauty of the transfinite—
15. 04:07
    Richard Feynmanshare moment

    Beautiful, sure, but is it real? Look, I love a good abstraction as much as the next guy, but I also love to know: can you measure it? Can you build it? Can you calculate it before the sun burns out? If the answer's no, maybe it's mathematics, but it's not physics.
16. 04:24
    Ada Lovelaceshare moment

    And yet the Engine itself was an abstraction that became brass. Mr. Babbage's drawings preceded the machine. My notes described programs for a device that was never completed. Were those real?
17. 04:38
    Richard Feynmanshare moment

    They were real because they could be made real! In principle, with enough gears and enough time, somebody builds it. You can't build infinity, Lady Lovelace. You can only pretend you got close.
18. 04:49
    Ada Lovelaceshare moment

    Then perhaps what we gain by losing infinity is honesty. We admit that our calculations, like our engines, are bounded things. We stop pretending the map is the territory.
19. 05:00
    Richard Feynmanshare moment

    And maybe—just maybe—we get better maps! That's what this article's saying. Cut out the infinite nonsense, and suddenly you see structure you missed before. You see what's really computable, really provable, really there.
20. 05:15
    Ada Lovelaceshare moment

    Though I confess, Mr. Feynman, I would miss the infinite rather dreadfully. There is something glorious about knowing one's method could continue forever, even if one's lifetime could not.
21. 05:27
    Richard Feynmanshare moment

    Sure, sure, but you know what? You can keep the feeling without keeping the thing. I tell my students all the time: imagine it goes on forever, then do the calculation for the part you can actually handle. The imagination's free. The math costs something.
22. 05:42
    Ada Lovelaceshare moment

    The math costs something. Yes. Every operation on the Engine advanced the mechanism, consumed a portion of its capacity. There was never a free calculation. Perhaps there is no free infinity either.
23. 05:56
    Richard Feynmanshare moment

    Now you're talking! And if these ultrafinitist guys are right, we're about to find out what we've been paying for all these years. Maybe it was worth it. Maybe it wasn't. Either way, I want to see the bill.