WEBVTT 00:00:01.819 --> 00:00:05.660 Hi. Yesterday we ended with a question. The rational 00:00:05.660 --> 00:00:09.119 numbers, those fractions, are everywhere on the 00:00:09.119 --> 00:00:12.660 number line. And yet, in terms of measure, they 00:00:12.660 --> 00:00:16.920 take up no space at all. So what happens if we 00:00:16.920 --> 00:00:23.239 remove every rational number? What is left? Well, 00:00:23.359 --> 00:00:26.640 it's the irrational numbers. Numbers like square 00:00:26.640 --> 00:00:31.530 root of 2 and pi and e. But here is the surprising 00:00:31.530 --> 00:00:35.909 part. What's left is not just something. It is 00:00:35.909 --> 00:00:40.350 almost everything. In the language of measure, 00:00:40.630 --> 00:00:45.070 almost every number on the number line is irrational. 00:00:46.710 --> 00:00:50.170 Think about how strange that is. The fractions 00:00:50.170 --> 00:00:54.109 are everywhere. No matter how far we zoom in, 00:00:54.229 --> 00:00:58.070 we always find them. And yet, in another mathematical 00:00:58.070 --> 00:01:01.789 sense, Almost every number is not a fraction. 00:01:02.969 --> 00:01:06.769 Once again, our intuition needs a little fine 00:01:06.769 --> 00:01:11.890 -tuning. Try to sit on that for a bit. Okay, 00:01:11.930 --> 00:01:14.890 for now, let's shift our attention. For the first 00:01:14.890 --> 00:01:16.989 four days, we've been fine -tuning mathematical 00:01:16.989 --> 00:01:21.510 ideas. Today, I want to look at how mathematicians 00:01:21.510 --> 00:01:27.150 fine -tune mathematics itself. Think about the 00:01:27.150 --> 00:01:29.859 Pythagorean theorem. You know, a squared plus 00:01:29.859 --> 00:01:33.620 b squared equals c squared. You've probably learned 00:01:33.620 --> 00:01:36.760 it in school. We've known it for thousands of 00:01:36.760 --> 00:01:40.060 years. And once someone proves it, it's proven. 00:01:40.180 --> 00:01:45.780 We're done. Right? Well, apparently mathematicians 00:01:45.780 --> 00:01:50.000 did not get the memo. There are hundreds of different 00:01:50.000 --> 00:01:53.920 proofs of the Pythagorean theorem. And all that 00:01:53.920 --> 00:01:58.510 raises an interesting question. Why? I mean, 00:01:58.530 --> 00:02:01.909 if we already know something is true, why keep 00:02:01.909 --> 00:02:07.390 proving it? Well, sometimes a new proof is simpler. 00:02:07.849 --> 00:02:11.610 Sometimes it reveals a connection that an earlier 00:02:11.610 --> 00:02:16.330 proof hid. Sometimes a new proof is simply more 00:02:16.330 --> 00:02:20.090 beautiful. And that is a word we mathematicians 00:02:20.090 --> 00:02:24.430 actually use, beautiful. A proof can be completely 00:02:24.430 --> 00:02:30.080 correct and still feel. Clumsy. Maybe it uses 00:02:30.080 --> 00:02:33.879 more machinery than necessary. Or maybe it gets 00:02:33.879 --> 00:02:37.280 us to the destination but really hides why the 00:02:37.280 --> 00:02:43.300 result is true. And then someone discovers another 00:02:43.300 --> 00:02:48.960 proof. A bit shorter. A bit cleaner. A connection 00:02:48.960 --> 00:02:52.419 that was hidden suddenly becomes visible. And 00:02:52.419 --> 00:02:56.949 when you see it, you almost cannot imagine. proving 00:02:56.949 --> 00:03:02.090 it any other way. The mathematician G .H. Hardy 00:03:02.090 --> 00:03:07.009 has said, there is no permanent place in the 00:03:07.009 --> 00:03:12.849 world for ugly mathematics. That's kind of a 00:03:12.849 --> 00:03:14.990 remarkable thing for a mathematician to say. 00:03:15.550 --> 00:03:18.389 He's not talking about whether the answer is 00:03:18.389 --> 00:03:22.870 correct. He's talking about beauty. So perhaps 00:03:22.870 --> 00:03:26.139 that's another reason why we fine -tune things. 00:03:26.259 --> 00:03:30.520 We fine -tune because we're searching for beauty. 00:03:31.219 --> 00:03:35.379 And personally, I've spent a lot of time learning 00:03:35.379 --> 00:03:38.159 different proofs of the Pythagorean theorem, 00:03:38.280 --> 00:03:43.840 all because of its beauty. Think about the artist 00:03:43.840 --> 00:03:46.379 that we talked about earlier this week. At some 00:03:46.379 --> 00:03:49.300 point, the painting already looks like the subject, 00:03:49.500 --> 00:03:54.860 but the artist keeps working. A little more shadow. 00:03:54.960 --> 00:03:59.379 A slight change in perspective. A touch of light. 00:04:00.000 --> 00:04:03.759 The painting wasn't necessarily wrong before, 00:04:03.900 --> 00:04:09.080 but each adjustment can reveal depth that was 00:04:09.080 --> 00:04:14.300 not visible before. And mathematicians do something 00:04:14.300 --> 00:04:19.199 similar. We return to an old theorem. We turn 00:04:19.199 --> 00:04:23.649 it around. We look for another perspective. Not 00:04:23.649 --> 00:04:26.310 necessarily because the old mathematics was wrong, 00:04:26.490 --> 00:04:28.670 but because there might be something beautiful 00:04:28.670 --> 00:04:34.069 hiding inside that we have not seen yet. And 00:04:34.069 --> 00:04:36.810 that takes us back to where we started this week. 00:04:36.990 --> 00:04:42.829 We fine -tune things we care about. The Pythagorean 00:04:42.829 --> 00:04:45.629 theorem does not become more true when we discover 00:04:45.629 --> 00:04:49.550 a better truth. The truth has not changed, but 00:04:49.550 --> 00:04:55.439 our understanding And sometimes our ability to 00:04:55.439 --> 00:05:00.000 see its beauty has changed as well. And I live 00:05:00.000 --> 00:05:06.199 to see more beauty in math. Okay, tomorrow we 00:05:06.199 --> 00:05:09.319 will look at one of the greatest examples in 00:05:09.319 --> 00:05:12.839 the history of mathematics. For almost 200 years, 00:05:12.920 --> 00:05:18.180 mathematicians used calculus with extraordinary 00:05:18.180 --> 00:05:24.579 success. It worked. But eventually, they went 00:05:24.579 --> 00:05:29.139 back and asked, why does it work?