WEBVTT 00:00:01.909 --> 00:00:06.209 Hi. It seems natural that tying and untying a 00:00:06.209 --> 00:00:08.470 shoe would have been discovered at the same time. 00:00:08.650 --> 00:00:12.169 One action creates the knot, the other reverses 00:00:12.169 --> 00:00:16.070 it. Mathematics is filled with similar pairs 00:00:16.070 --> 00:00:20.609 of doing and undoing. We add 3 plus 4 equals 00:00:20.609 --> 00:00:24.969 7. Then subtraction undoes the addition. 7 minus 00:00:24.969 --> 00:00:29.230 4 returns to 3. Multiplication is repeated addition. 00:00:29.410 --> 00:00:34.229 4 plus 4 plus 4. equals 12. And division undoes 00:00:34.229 --> 00:00:40.170 multiplication. 12 divided by 3 equals 4. Mathematics 00:00:40.170 --> 00:00:42.509 keeps building these layers. Repeated multiplication 00:00:42.509 --> 00:00:47.530 becomes exponentiation. 4 times 4 times 4 equals 00:00:47.530 --> 00:00:52.670 4 cubed, which equals 64. And then logarithms 00:00:52.670 --> 00:00:57.929 undo exponentiation. At least, that's how we 00:00:57.929 --> 00:01:00.859 think about them today. But here's our paradox. 00:01:02.000 --> 00:01:05.000 Exponentials and logarithms were not originally 00:01:05.000 --> 00:01:08.760 discovered together. In fact, they emerged from 00:01:08.760 --> 00:01:13.400 completely different worlds. In the early 1600s, 00:01:13.420 --> 00:01:17.239 John Napier invented logarithms as a practical 00:01:17.239 --> 00:01:21.500 tool to simplify massive calculations used in 00:01:21.500 --> 00:01:25.900 astronomy and navigation. His goal was efficiency. 00:01:27.209 --> 00:01:31.390 Logarithms transformed multiplication into addition, 00:01:31.670 --> 00:01:37.150 making difficult calculations far easier. Meanwhile, 00:01:37.310 --> 00:01:40.090 a completely different story was unfolding with 00:01:40.090 --> 00:01:43.769 future actuaries in mind. Mathematicians studying 00:01:43.769 --> 00:01:47.650 compound interest and continuous growth kept 00:01:47.650 --> 00:01:52.609 encountering a strange number that was near 2 00:01:52.609 --> 00:01:59.709 .71828... Later, we gave this number the letter 00:01:59.709 --> 00:02:04.689 E. And this number emerged naturally from growth 00:02:04.689 --> 00:02:09.689 itself. One discovery came from simplifying calculations. 00:02:10.050 --> 00:02:13.590 The other came from studying growth and change. 00:02:14.129 --> 00:02:17.969 Different people, different problems, different 00:02:17.969 --> 00:02:22.270 centuries. At first, these ideas seemed unrelated. 00:02:23.050 --> 00:02:28.979 And then... Euler entered the story. Euler discovered 00:02:28.979 --> 00:02:33.479 something truly amazing, especially at the time. 00:02:34.039 --> 00:02:37.280 Log rhythms and exponential growth were deeply 00:02:37.280 --> 00:02:40.599 connected. They were inverses of each other. 00:02:40.759 --> 00:02:44.919 The same hidden structure had been sitting underneath 00:02:44.919 --> 00:02:49.900 both ideas all along. I just wonder about the 00:02:49.900 --> 00:02:54.590 aha moment Euler had in this discovery. suddenly 00:02:54.590 --> 00:02:58.789 mathematics became more unified than anyone had 00:02:58.789 --> 00:03:03.030 imagined. And that is one of the reoccurring 00:03:03.030 --> 00:03:07.349 paradoxes of math. Truth often appears in scattered 00:03:07.349 --> 00:03:10.629 pieces before we recognize the deeper pattern 00:03:10.629 --> 00:03:15.030 connecting them. Life works a bit this way too. 00:03:15.189 --> 00:03:18.250 Sometimes we may see how things are connected 00:03:18.250 --> 00:03:21.849 quickly, but many times the connection requires 00:03:21.849 --> 00:03:26.189 time. But if we take time to notice, we can discover 00:03:26.189 --> 00:03:29.530 that some experiences that seem completely unrelated 00:03:29.530 --> 00:03:34.610 later reveal hidden connections. A difficult 00:03:34.610 --> 00:03:39.830 season later produces compassion. Failure creates 00:03:39.830 --> 00:03:45.810 wisdom. Loss deepens gratitude. At the time, 00:03:45.810 --> 00:03:50.110 these pieces may feel disconnected. Only later 00:03:50.110 --> 00:03:54.969 do we begin to see the larger story. Adam Grant 00:03:54.969 --> 00:03:58.169 talks about the importance of rethinking, being 00:03:58.169 --> 00:04:02.150 willing to let new understanding reshape old 00:04:02.150 --> 00:04:06.909 assumptions. Mathematics trains us to do exactly 00:04:06.909 --> 00:04:11.770 that. It repeatedly reminds us that reality may 00:04:11.770 --> 00:04:14.530 be more connected than our first impressions 00:04:14.530 --> 00:04:18.670 suggest, that beneath the scattered pieces, there 00:04:18.670 --> 00:04:22.089 may be hidden coherence waiting to be discovered. 00:04:23.470 --> 00:04:27.829 And perhaps hope grows when we learn not to cling 00:04:27.829 --> 00:04:31.089 too tightly to our current understanding. Because 00:04:31.089 --> 00:04:35.350 reality may still be revealing deeper connections 00:04:35.350 --> 00:04:38.610 that our eyes cannot yet see.