WEBVTT 00:00:02.160 --> 00:00:05.639 Hi, mathematics has taught me that when something 00:00:05.639 --> 00:00:10.679 feels impossible to imagine, the limitation may 00:00:10.679 --> 00:00:14.539 belong to my intuition rather than to reality. 00:00:15.380 --> 00:00:21.420 A wonderful example is infinity. For a long time, 00:00:21.440 --> 00:00:25.760 I used to think of infinity as simply endlessness, 00:00:25.899 --> 00:00:30.239 something unimaginably large. that keeps going 00:00:30.239 --> 00:00:35.000 and going and going forever. If two collections 00:00:35.000 --> 00:00:38.479 were both infinite, I assume that they must be 00:00:38.479 --> 00:00:41.740 the same size. After all, how could anything 00:00:41.740 --> 00:00:46.719 be larger than something that never ends? Well, 00:00:46.820 --> 00:00:50.200 mathematics offers a surprising answer. Some 00:00:50.200 --> 00:00:54.939 infinities are larger than others. Consider the 00:00:54.939 --> 00:00:58.820 counting numbers 1, 2, 3, 4, and so on. And now, 00:00:59.119 --> 00:01:02.219 Consider only the even numbers, 2, 4, 6, 8, and 00:01:02.219 --> 00:01:05.459 so on. Now, it seems that there should be fewer 00:01:05.459 --> 00:01:08.799 even numbers because they make up only part of 00:01:08.799 --> 00:01:12.200 the counting numbers. But every counting number 00:01:12.200 --> 00:01:17.040 can be paired with exactly one even number. 1 00:01:17.040 --> 00:01:23.200 with 2, 2 with 4, 3 with 6, and so on. Neither 00:01:23.200 --> 00:01:28.099 list ever runs out. So in the mathematical sense, 00:01:28.730 --> 00:01:32.489 These two infinite collections have the same 00:01:32.489 --> 00:01:37.829 size. Wow. Okay, the rational numbers. These 00:01:37.829 --> 00:01:39.730 are the numbers that can be written as fractions. 00:01:40.170 --> 00:01:44.049 They, of course, are also infinite. Between any 00:01:44.049 --> 00:01:46.769 two rational numbers, we can always find another. 00:01:47.069 --> 00:01:50.189 They seem to be everywhere along the number line. 00:01:51.750 --> 00:01:55.250 Remarkably, however, they can still be arranged 00:01:55.250 --> 00:02:00.260 into a list. Their infinity is the same size 00:02:00.260 --> 00:02:06.680 as our counting numbers. Okay, then we encounter 00:02:06.680 --> 00:02:10.099 the irrational numbers, numbers such as square 00:02:10.099 --> 00:02:14.080 root of 2 and pi. These cannot be expressed as 00:02:14.080 --> 00:02:18.280 ratios of whole numbers. Now together, the rational 00:02:18.280 --> 00:02:21.139 and irrational numbers, they form the real number 00:02:21.139 --> 00:02:25.710 line. But the real numbers cannot be placed into 00:02:25.710 --> 00:02:31.509 a complete list, even an infinite one. So their 00:02:31.509 --> 00:02:37.550 infinity is larger. This is one of those mathematical 00:02:37.550 --> 00:02:41.969 discoveries that makes me feel it's very strange, 00:02:42.050 --> 00:02:48.879 but it's also very spacious. The number line 00:02:48.879 --> 00:02:53.039 appears so simple when we draw it, and yet hidden 00:02:53.039 --> 00:02:56.939 within it are levels of infinity that ordinary 00:02:56.939 --> 00:03:02.319 intuition would never predict. Ideas like this 00:03:02.319 --> 00:03:06.020 have changed me because they have made me less 00:03:06.020 --> 00:03:11.000 confident that the limits of my imagination are 00:03:11.000 --> 00:03:18.300 the limits of what is possible. It is formed 00:03:18.300 --> 00:03:22.939 by experience with finite objects in an everyday 00:03:22.939 --> 00:03:27.319 world. It is not always prepared for infinity. 00:03:28.860 --> 00:03:32.919 Mathematics lets me move beyond that first intuition. 00:03:33.280 --> 00:03:37.340 It gives me tools for approaching ideas that 00:03:37.340 --> 00:03:41.099 I cannot picture and language for distinctions 00:03:41.099 --> 00:03:45.639 I would not otherwise know how to make. I may 00:03:45.639 --> 00:03:49.780 never be able to visualize infinity, but mathematics 00:03:49.780 --> 00:03:52.680 has taught me that understanding does not always 00:03:52.680 --> 00:03:55.620 begin with being able to imagine the answer. 00:03:56.340 --> 00:04:00.659 Sometimes understanding begins when I allow my 00:04:00.659 --> 00:04:03.199 imagination to be changed.