Discover how the concepts of zero and infinity revolutionized mathematics.It was stupid and boring.

A Black woman professor narrated. They always seems to find Blacks and women for these shows. Not sure why. Does PBS have a lot of Black viewers? Is it trying to get more?

I doubt it. My guess is that the typical WHite liberal PBS viewer gets a good feeling of social justice when a Black woman is lecturing.

Much of the show was about the invention of the Zero. It attributed it to India, and said that Persians and other middle easterners brought it to Europe.

But what was the invention? The use of 0 as a placeholder, or as a counting number, or as a number on the same footing as other numbers?

I looked for some statement from India or Persia saying something like: The counting numbers are { 0, 1, 2, ... }, and for any such numbers, A + B = B + A.

That would show that the author considered 0 to be a number just like 1 and 2.

But on the contrary, the show itself did not even do that. The moderator kept referring to the counting numbers as 1, 2, 3, ..., and not including 0.

Any who says that has still not grasped the invention of 0. 0 is a counting number. If I ask you how many apples you have, and you have none, then you answer 0. Maybe you answer -2, if you owe 2 apples. If you say there is no answer, then you have not accepted the 0.

Even the business pages of a typical newspaper rarely treat zero as a number. It will often avoid it with various euphemisms.

The show eventually moved on to infinity, but that was not any better. It gave a faulty version of Cantor's diagonal proof of the uncountability of the reals.

Suppose you list .4, .49999..., .009, .0009, ... . It said to add 1, mod 10, to each diagonal digit. That gives .50000. That is the same real number as the 2nd item on the list, with a different decimal representation.

A good proof must somehow take into account that real numbers can have two decimal representations.

Cantor's orginal proof did not use diagonalization.

The show went on to Zeno's paradoxes and Hilbert's hotel. It was all fairly trivial.

Since it tried to trace the origin of the zero, I thought that it might tell us who invented infinity?

It did talk about approximating π as a limit of an infinite sequence. I guess that idea goes back to the ancient Greeks. The invention of infinitesmal calculus required limits. Those ideas were made rigorous centuries later. Cantor introduced the concept of different infinite cardinals. I am not sure who really first had the modern concept.

These PBS shows appears to be expensively produced, but you can find lots of free YouTube videos that explain the math much better, and are more entertaining.

Both of Cantor's proofs were hoopy nonsense and were debunked by Feferman. Cantor is about the redundant metaphysics of metamathematics and not mathematics proper. Ultrafinitists just see ordinary algebra, like the Risch algorithm. Maple can do all the math humans can and more. It doesn't matter whether you include the axiom of choice because paradoxes occur from assuming a contradiction of completed infinities. See Stan Wagon on this point:

ReplyDeletehttps://www.cambridgeblog.org/2017/04/it-is-best-to-accept-the-banach-tarski-paradox/

https://www.amazon.com/Light-Logic-Computation-Philosophy/dp/0195080300 http://www.ccas.ru/alexzen/papers/CANTOR-2003/BSL-3ENG.pdf