Science & Tech

# Onward

Terence Tao

Lesson time 04:41 min

We are all part of a larger math story. Terence is one of many mathematical figures who influences our understanding of the world, and there will be many after him. The human tendency to ask questions means math and science are ever-evolving.

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Topics include: Onward Becoming Part of the Story

Terence Tao

Teaches Mathematical Thinking

World-renowned mathematician Dr. Terence Tao teaches you his approach to everyday problem-solving—without complex equations or formulas.

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﻿[MUSIC PLAYING] - Mathematics is actually the most cumulative subject in human knowledge, really. We are using mathematics that was developed 2,000, 3,000 years ago. We have-- the theorems of Pythagoras or Euclid, we still use today, the basis of modern mathematics. And that's more true in mathematics than almost any other discipline. For example, in science, you know, Aristotle thought that all matter was made up of four elements, Earth, air, fire, and water, which is completely incorrect. And modern physics is not really based on Aristotle in physics. But modern mathematics is based on ancient Greek mathematics and even more ancient mathematics than that. We stand on the shoulders of giants. Mathematics is famous for studying problems which had no practical application in mind. But many years later, a scientist or engineer or someone working on a practical problem realized that some mathematician studied this problem decades earlier and could use that mathematics to solve the problem. I think Eugene Wigner once called this "the unreasonable effectiveness of mathematics in the physical sciences." One classical example is in, I think, the 18th century. People started studying what are called non-Euclidean geometries. So the usual geometry, which is called Euclidean geometry, are straight lines and points. When you have two lines pointing in the same direction, which are parallel, they go on forever. They never get closer, or they get further apart. They just stay at the same distance forever. But people just started asking, what if space was curved? Are there geometries where parallel lines eventually cross or where the parallel lines diverge? And this seemed like a purely theoretical pastime. Because, you know, clearly, the world was-- the actual world was flat. But then centuries later, Albert Einstein realized that in order to make his theory of gravity work, space had to be curved. So he asked some mathematician friends, is there an existing theory of curved space? And there was. People had developed it earlier. And it turned out to be exactly what was needed to develop what we now call Einstein's general theory of relativity. There are insights and breakthroughs by very smart people, but they don't come from nowhere. They come from a very patient process of working out what everyone else has done, talking to other people, and then eventually, naturally, you see the way forward. There was a great mathematician, Alexander Grothendieck, who made this analogy that a hard math problem is like a walnut. The natural tendency for some people is to take a-- to open the walnut you see, hit it with a sledgehammer. And some people actually are like sledgehammers. They can do that. But more often what happens, the process is more like you soak this walnut in water for a long time, and it gets softer and softer. And eventually, this shell becomes so soft, you can just peel it apart with your hands. And the soluti...