WEBVTT 00:00:01.009 --> 00:00:04.629 Hi. Don't you just love it when you struggle 00:00:04.629 --> 00:00:07.450 understanding something in math for a long time 00:00:07.450 --> 00:00:11.810 and finally the light bulb turns on? This happened 00:00:11.810 --> 00:00:14.810 to me this week. I've known for a couple of years 00:00:14.810 --> 00:00:18.329 that mathematicians state that the set of integers 00:00:18.329 --> 00:00:22.789 and set of even numbers are the same size. Now 00:00:22.789 --> 00:00:26.109 that does not seem possible since the even numbers 00:00:26.109 --> 00:00:30.190 are only half of the integers. But finally this 00:00:30.190 --> 00:00:34.159 week, I looked at it in a different way. I considered 00:00:34.159 --> 00:00:36.960 what happens if you take the positive integers 00:00:36.960 --> 00:00:41.159 and multiply them by 2. The result is the set 00:00:41.159 --> 00:00:44.679 of even integers. Multiplying numbers in a set 00:00:44.679 --> 00:00:47.799 by a constant 2 should not change the number 00:00:47.799 --> 00:00:51.820 of items in a set. Another slightly different 00:00:51.820 --> 00:00:55.140 perspective is to refer to the integers as yards. 00:00:55.340 --> 00:01:00.840 So 1 is 1 yard, 2 is 2 yards, etc. Then we take 00:01:00.840 --> 00:01:04.379 this pattern to infinity. But what if we take 00:01:04.379 --> 00:01:08.379 those same distances and measure feet? Then we 00:01:08.379 --> 00:01:13.480 have 3 feet, 6 feet, 9 feet, dot, dot, dot. We're 00:01:13.480 --> 00:01:16.799 not changing the size of the set, just scaling 00:01:16.799 --> 00:01:20.959 the measurement of the set. Thus, scaling the 00:01:20.959 --> 00:01:24.079 integers should not change the number of items 00:01:24.079 --> 00:01:28.560 in a set. So the light bulb came on and it all 00:01:28.560 --> 00:01:33.569 made sense. I like math not only because understanding 00:01:33.569 --> 00:01:36.530 enhances my knowledge and challenges my thinking 00:01:36.530 --> 00:01:40.370 as to what is true, but it also challenges me 00:01:40.370 --> 00:01:44.790 to be patient and give these paradox time to 00:01:44.790 --> 00:01:47.790 sink in. I have to think about them from different 00:01:47.790 --> 00:01:51.450 angles, consider different perspectives. But 00:01:51.450 --> 00:01:56.109 once the light bulb turns on, I feel like I joined 00:01:56.109 --> 00:01:59.709 this special club of math people. that have also 00:01:59.709 --> 00:02:02.650 wrestled with these paradoxes and have seen it 00:02:02.650 --> 00:02:06.290 through. Understanding a paradox is different 00:02:06.290 --> 00:02:09.949 than just solving a difficult problem. To understand 00:02:09.949 --> 00:02:13.110 paradoxes, we have to let go of one intuition, 00:02:13.349 --> 00:02:16.289 like the even set of numbers is the same size 00:02:16.289 --> 00:02:20.169 as the evens and the odds, and adopt a new intuition. 00:02:22.129 --> 00:02:25.370 Eugenia Cheng takes this a step further and says 00:02:25.370 --> 00:02:29.199 that math is not always identifying what is true 00:02:29.199 --> 00:02:34.379 or not true, but identifying in what sense something 00:02:34.379 --> 00:02:38.759 is true. This seems to be a good muscle to not 00:02:38.759 --> 00:02:41.979 only understand paradoxes, but to grow in healthier 00:02:41.979 --> 00:02:45.900 relationships. Sometimes other people's perspectives 00:02:45.900 --> 00:02:49.699 seem difficult to understand, almost like a paradox. 00:02:50.300 --> 00:02:54.639 I find it is easy to quickly dismiss their perspective 00:02:54.639 --> 00:02:59.219 as wrong, but my math training, encourages me 00:02:59.219 --> 00:03:02.939 to stay in the conversation longer and understand 00:03:02.939 --> 00:03:07.479 in what sense they are right and in what sense 00:03:07.479 --> 00:03:12.460 do I need to expand my thinking. Paradoxes are 00:03:12.460 --> 00:03:16.099 not only fun puzzles to solve, but they remind 00:03:16.099 --> 00:03:18.979 me to invite new perspectives.