WEBVTT 00:00:02.089 --> 00:00:05.509 One of the ways mathematics has changed me is 00:00:05.509 --> 00:00:08.869 that I kind of now expect things to be connected. 00:00:09.789 --> 00:00:12.429 So when I first studied algebra and geometry, 00:00:12.710 --> 00:00:16.050 they seemed like different subjects. Algebra 00:00:16.050 --> 00:00:19.010 was about equations, variables, and symbolic 00:00:19.010 --> 00:00:23.309 rules. Geometry was about shapes, angles, and 00:00:23.309 --> 00:00:27.329 diagrams. One lived in a world of symbols, and 00:00:27.329 --> 00:00:32.179 the other lived in a world of pictures. came 00:00:32.179 --> 00:00:36.700 the coordinate plane. With an x -axis and a y 00:00:36.700 --> 00:00:40.899 -axis, an equation could suddenly become a shape. 00:00:41.320 --> 00:00:44.960 For example, the equation y equals 2x plus 1 00:00:44.960 --> 00:00:48.719 was not only a rule for manipulating symbols, 00:00:49.020 --> 00:00:53.500 it was also a line. The equation y equals x squared 00:00:53.500 --> 00:00:57.100 became a parabola. The equation x squared plus 00:00:57.100 --> 00:01:02.490 y squared equals 25 drew a circle. and the connection 00:01:02.490 --> 00:01:05.769 worked in both directions. An equation could 00:01:05.769 --> 00:01:08.890 produce a picture, and a picture could reveal 00:01:08.890 --> 00:01:13.810 information about an equation. This almost feels 00:01:13.810 --> 00:01:18.049 magical. Two subjects I had placed in separate 00:01:18.049 --> 00:01:21.150 compartments turned out to be different ways 00:01:21.150 --> 00:01:25.430 of describing the same thing. Halsberg gave me 00:01:25.430 --> 00:01:29.129 one language, and geometry gave another language. 00:01:29.849 --> 00:01:32.950 and then the coordinate system allowed them to 00:01:32.950 --> 00:01:37.590 speak to each other. The mathematics became richer 00:01:37.590 --> 00:01:40.810 because of this connection. An equation might 00:01:40.810 --> 00:01:44.269 be difficult to understand symbolically, but 00:01:44.269 --> 00:01:47.269 became crystal clear when I could see it in its 00:01:47.269 --> 00:01:50.909 graph. A geometric relationship might be easier 00:01:50.909 --> 00:01:53.790 to analyze after translating it into algebra. 00:01:54.530 --> 00:01:58.700 Neither representation replaced the other. Each 00:01:58.700 --> 00:02:01.640 allowed me to notice something that the other 00:02:01.640 --> 00:02:05.900 made less visible. And this has influenced the 00:02:05.900 --> 00:02:08.939 way I think beyond math. For example, when I 00:02:08.939 --> 00:02:12.879 encounter two ideas that seem unrelated, I'm 00:02:12.879 --> 00:02:16.060 just more inclined now to ask whether there might 00:02:16.060 --> 00:02:19.479 be a bridge between them. Perhaps they are not 00:02:19.479 --> 00:02:22.759 competing explanations, but different representation 00:02:22.759 --> 00:02:26.310 of the same underlying reality. Perhaps each 00:02:26.310 --> 00:02:29.009 perspective preserves something that the other 00:02:29.009 --> 00:02:32.969 leaves out. And of course, not everything connects 00:02:32.969 --> 00:02:36.590 nearly. And sometimes we force comparisons that 00:02:36.590 --> 00:02:39.810 do not belong. But mathematics has taught me 00:02:39.810 --> 00:02:43.610 not to assume that the boundaries that I first 00:02:43.610 --> 00:02:48.270 see are real or permanent. The categories may 00:02:48.270 --> 00:02:52.229 have been created for convenience. A change in 00:02:52.229 --> 00:02:56.469 language or viewpoint or representation may reveal 00:02:56.469 --> 00:03:00.270 what appeared separate was now connected all 00:03:00.270 --> 00:03:04.530 along. And this is one reason why math feels 00:03:04.530 --> 00:03:08.449 less like a collection of isolated facts and 00:03:08.449 --> 00:03:12.789 more like a landscape. The further I travel, 00:03:12.909 --> 00:03:17.610 the more paths begin to cross. Ideas that I learned 00:03:17.610 --> 00:03:22.990 years apart suddenly meet. A symbol becomes a 00:03:22.990 --> 00:03:26.090 shape, a shape becomes a relationship, and a 00:03:26.090 --> 00:03:29.490 relationship opens the way to another question. 00:03:30.509 --> 00:03:33.789 Math has taught me that understanding often arrives 00:03:33.789 --> 00:03:38.469 when I find the right connection. And once that 00:03:38.469 --> 00:03:43.389 connection appears, both sides look different.