WEBVTT 00:00:01.929 --> 00:00:05.429 One of my favorite things about mathematics is 00:00:05.429 --> 00:00:10.710 that it is incredibly creative. When people think 00:00:10.710 --> 00:00:15.089 of math, they often imagine a rigid system where 00:00:15.089 --> 00:00:18.789 every rule has already been decided. But that's 00:00:18.789 --> 00:00:22.850 not actually how mathematics begins. Mathematicians 00:00:22.850 --> 00:00:26.750 invented notation. We've created the definitions 00:00:26.750 --> 00:00:30.269 and we've decided what assumptions or axioms 00:00:30.269 --> 00:00:33.219 we're going to start with. So in that sense, 00:00:33.320 --> 00:00:38.280 mathematics is an act of creativity. For example, 00:00:38.460 --> 00:00:42.140 someone, specifically think Leonard Euler, had 00:00:42.140 --> 00:00:45.479 to decide that we would use the symbol pi for 00:00:45.479 --> 00:00:47.820 the ratio of a circle's circumference to its 00:00:47.820 --> 00:00:51.899 diameter. Someone had to define what a vector 00:00:51.899 --> 00:00:55.500 is. Someone had to formalize the idea of a group, 00:00:55.600 --> 00:01:00.600 or a field, or even a matrix. These were not 00:01:00.600 --> 00:01:03.679 discovered lying around in nature. They were 00:01:03.679 --> 00:01:07.219 human inventions designed to help us describe 00:01:07.219 --> 00:01:11.799 patterns more clearly. But here is the part that 00:01:11.799 --> 00:01:16.200 absolutely fascinates me. Once we've chosen our 00:01:16.200 --> 00:01:20.799 definitions and assumptions, our freedom ends. 00:01:21.420 --> 00:01:25.540 And at that point, we do not get to decide what 00:01:25.540 --> 00:01:31.359 comes next. Logic takes over. We do not get to 00:01:31.359 --> 00:01:34.180 vote on the theorems we would like to be true. 00:01:34.760 --> 00:01:38.700 We do not get to ignore conclusions that make 00:01:38.700 --> 00:01:42.900 us uncomfortable. We simply follow the logical 00:01:42.900 --> 00:01:46.359 consequences of the choices that we have already 00:01:46.359 --> 00:01:49.959 made. And this is why in the history of mathematics, 00:01:50.359 --> 00:01:54.180 we have uncovered so many surprising discoveries. 00:01:54.900 --> 00:01:57.280 We've talked about them, irrational numbers, 00:01:57.560 --> 00:02:00.739 non -Euclidean geometry. and different sizes 00:02:00.739 --> 00:02:05.799 of infinity, all challenge our intuition. And 00:02:05.799 --> 00:02:09.439 many mathematicians initially resisted these 00:02:09.439 --> 00:02:15.919 ideas because they felt wrong. But logic did 00:02:15.919 --> 00:02:20.879 not care how anyone felt. If the reasoning was 00:02:20.879 --> 00:02:24.120 sound, then the conclusions had to be accepted. 00:02:24.680 --> 00:02:28.560 And I think that is one of the most humbling 00:02:28.560 --> 00:02:33.259 lessons that math teaches us. Our intuition is 00:02:33.259 --> 00:02:36.180 valuable. It often points us towards interesting 00:02:36.180 --> 00:02:40.400 questions, but intuition is not our final authority. 00:02:41.060 --> 00:02:45.939 Truth is. So as we work through problems in our 00:02:45.939 --> 00:02:49.120 work, and as you work through your problems, 00:02:49.280 --> 00:02:52.520 you will probably have moments where you're convinced 00:02:52.520 --> 00:02:56.539 an answer must be right, and then only to discover 00:02:56.539 --> 00:03:00.159 you have a mistake in your reasoning. I know 00:03:00.159 --> 00:03:04.060 I've certainly had many of those moments. They 00:03:04.060 --> 00:03:07.120 are frustrating, but they're also one of the 00:03:07.120 --> 00:03:10.620 ways mathematics shapes us. It teaches us to 00:03:10.620 --> 00:03:13.699 ask a different question. It's not what do I 00:03:13.699 --> 00:03:17.460 want to be true, but given these assumptions, 00:03:17.900 --> 00:03:22.080 what must be true? And that's a very different 00:03:22.080 --> 00:03:25.680 way of thinking. And it is the foundation for 00:03:25.680 --> 00:03:30.789 this week's theme on humility. To me, humility 00:03:30.789 --> 00:03:35.250 is not thinking less of myself, but it is recognizing 00:03:35.250 --> 00:03:38.969 that once the assumptions are set, all of us, 00:03:38.990 --> 00:03:42.270 the students, the professors, the professional 00:03:42.270 --> 00:03:46.270 mathematicians alike, we all stand under the 00:03:46.270 --> 00:03:52.689 same authority, the authority of logic. Wonderful. 00:03:52.810 --> 00:03:55.539 Well, tomorrow, we're going to look... at one 00:03:55.539 --> 00:03:58.560 of the biggest reasons that humility is necessary 00:03:58.560 --> 00:04:03.460 in mathematics. Our intuition is often wrong, 00:04:03.520 --> 00:04:06.560 and that is perfectly okay.