WEBVTT 00:00:01.710 --> 00:00:04.889 Hi, yesterday we ended with a question. For almost 00:00:04.889 --> 00:00:08.130 200 years, mathematicians used calculus with 00:00:08.130 --> 00:00:13.250 extraordinary success. It just worked. But eventually 00:00:13.250 --> 00:00:18.010 they went back and asked, why does it work? Newton 00:00:18.010 --> 00:00:21.329 and Leibniz developed calculus in the late 1600s, 00:00:21.329 --> 00:00:25.789 and it was incredibly powerful. Using calculus, 00:00:26.190 --> 00:00:29.699 mathematicians could describe motion They could 00:00:29.699 --> 00:00:32.640 calculate areas and volumes, they could study 00:00:32.640 --> 00:00:35.899 changing quantities, and eventually they modeled 00:00:35.899 --> 00:00:39.079 everything from planetary orbits to electricity. 00:00:39.880 --> 00:00:43.859 There was only one problem. Some of its foundations 00:00:43.859 --> 00:00:50.240 were a bit shaky. Early on in calculus, we frequently 00:00:50.240 --> 00:00:54.640 talked about quantities becoming infinitely small. 00:00:55.399 --> 00:00:59.359 What exactly is an infinitely small quantity? 00:00:59.640 --> 00:01:04.840 Is it zero? If it's zero, how can we divide by 00:01:04.840 --> 00:01:09.379 it? And if it isn't zero, well, how small is 00:01:09.379 --> 00:01:14.200 it? Well, for generations, mathematicians used 00:01:14.200 --> 00:01:18.299 calculus successfully despite questions like 00:01:18.299 --> 00:01:21.959 these. The answers worked and the applications 00:01:21.959 --> 00:01:25.959 worked. But remember where we began this week. 00:01:26.159 --> 00:01:30.920 We fine -tuned things we care about, and mathematicians 00:01:30.920 --> 00:01:35.680 cared enough about calculus to ask what was really 00:01:35.680 --> 00:01:41.239 happening underneath it. Well, during the 1800s, 00:01:41.260 --> 00:01:45.659 mathematicians, including Cauchy and Weisterhaus, 00:01:45.939 --> 00:01:49.799 began putting these ideas on firmer foundations. 00:01:51.400 --> 00:01:56.120 One of the key ideas was the limit. Now remember 00:01:56.120 --> 00:01:59.579 a series from earlier this week. We had one half 00:01:59.579 --> 00:02:02.200 plus one fourth plus one eighth plus one sixteen 00:02:02.200 --> 00:02:06.379 plus dot dot dot. That equals one. But we've 00:02:06.379 --> 00:02:09.479 been saying that the sum gets closer and closer 00:02:09.479 --> 00:02:13.039 to one. And that sounds reasonable. But what 00:02:13.039 --> 00:02:18.800 exactly does closer and closer mean? So here 00:02:18.800 --> 00:02:22.599 is the fine tuning. Give me any distance from 00:02:22.599 --> 00:02:26.800 one that you want, say a tenth or a millionth 00:02:26.800 --> 00:02:30.960 or even a trillionth or something unimaginably 00:02:30.960 --> 00:02:35.520 smaller. So if I go far enough in the process, 00:02:35.580 --> 00:02:40.860 I can get within that distance and stay there. 00:02:41.680 --> 00:02:47.319 Now, getting closer forever has a precise mathematical 00:02:47.319 --> 00:02:51.919 meaning. Notice what happened. Mathematicians 00:02:51.919 --> 00:02:54.979 did not fine -tune calculus because calculus 00:02:54.979 --> 00:02:58.479 had failed. Quite the opposite. It had already 00:02:58.479 --> 00:03:01.960 helped transform mathematics and science. They 00:03:01.960 --> 00:03:05.659 fine -tuned it because working wasn't enough. 00:03:05.840 --> 00:03:10.960 They wanted to understand why it worked. And 00:03:10.960 --> 00:03:13.979 to me, that's one of the most endearing features 00:03:13.979 --> 00:03:18.889 of the history of mathematics. May come first, 00:03:19.090 --> 00:03:23.689 and then understanding deepens, and then rigor 00:03:23.689 --> 00:03:27.509 follows. And that takes us back to yesterday. 00:03:27.750 --> 00:03:31.629 The truth did not change, but our understanding 00:03:31.629 --> 00:03:36.430 did. So maybe that's what fine -tuning often 00:03:36.430 --> 00:03:39.509 looks like. We return to something that we think 00:03:39.509 --> 00:03:44.050 we know. We look more closely. We ask better 00:03:44.050 --> 00:03:47.680 questions. And sometimes we discover that an 00:03:47.680 --> 00:03:51.680 idea we thought we understood contains another 00:03:51.680 --> 00:03:57.259 layer that we had not seen before. Okay, tomorrow 00:03:57.259 --> 00:03:59.740 we will finish our week by returning to where 00:03:59.740 --> 00:04:03.159 we began. Why do we keep fine -tuning things 00:04:03.159 --> 00:04:05.539 that are already true?