Strange Paradoxes in Maths │ The History of Mathematics with Luc de Brabandère


Can you cross a square? Is someone lying or
not? Discover the strange paradoxes revealed by maths. Subscribe to follow the series, the
history of maths, on our YouTube channel now. Sometimes, when you are a logician or a mathematician,
you face strange problems. Problems you really don’t know how to solve. Those problems
are called paradoxes. Let’s start with logic. If I tell you I am
a liar, and if this is true, I am not a liar. If this is not true, then I am a liar. You cannot
escape the loop. This is called a paradox. In mathematics, you also have paradoxes. A
famous was Zeno. He told his friends it’s impossible to cross a square. It’s impossible
to go from A to B. Yes, he said, because if you want to go from A to B, you first have
to go half way. And then, half way. And then, half way. You never can reach the other end
of the square, because you can go as far as you want, there is still a little distance,
you have to start with half way. That was Zeno’s paradox. And paradoxes in logic were “solved” by Bertrand Russel and the new kind of logic. Paradoxes in mathematics,
at least this one, were solved by, at the same time, Leibniz and Newton with the invention of calculus,
and you will see in the next video how they did it. Let’s have another example of a beautiful paradox in mathematics. Let’s take a right
angle triangle again, A, B, C. I will prove to you that AC=AB + BC. You say “No, it’s not
possible”. Yes, and I will prove it to you. Let’s start with A C. I propose to build this
like a stair, like this, and you replace the straight line by your stair. If you look at
the sum of the horizontal pieces, it’s exactly the distance here (BC). If you look at the
sum of the vertical pieces, it’s exactly the distance here (AB). Now, if you go to a very,
very thin stair, with very limited pieces of steps, you’ll finally have AC=AB + BC. It’s
another paradox and it’s important because mathematicians make progress when they embrace
paradoxes. Next time we will explain how that scourge
of school children, calculus, was invented.

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