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Math prof: Be careful what we do with infinity. Weird things can happen.


From Eugenia Cheng at ScienceFriday:

What has gone wrong? The problem is that we have manipulated equations as if infinity were an ordinary number, without knowing if it is or not. One of the first things we’re going to see in this book is what infinity isn’t, and it definitely isn’t an ordinary number. We are gradually going to work our way toward finding what type of “thing” it makes sense for infinity to be. This is a journey that took mathematicians thousands of years, involving some of the most important developments of mathematics: set theory and calculus, just for starters.

The moral of that story is that although the idea of infinity is quite easy to come up with, we have to be rather careful what we do with it, because weird things start happening. And that was just the beginning of the weird things that can happen. We’re going to look at all sorts of weird things that happen with infinity, with infinite collections of things, hotels with infinite rooms, infinite pairs of socks, infinite paths, infinite cookies. Some weird things are like 1 = 0, not just weird, but undesirable. So we try to build our mathematical ideas to avoid those. But other weird things don’t contradict logic, they just contradict normal life. Those weird things don’t cause problems to our logic, they just cause problems to our imagination. More.

A photograph of the Greek letter pi, created as a large stone mosaic embedded in the ground.
pi in mosaic, Berlin/Holger Motzkau

Math has driven men mad. Some people get really pumped about pi too.

Note: The article is an excerpt from Cheng’s Beyond Infinity: An Expedition to the Outer Limits of Mathematics

See also: Is celeb number pi “normal”? No. If it were, it wouldnt be a celeb. It would be down there with 318.

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Math academicians are guilty of using math in situations where it can't work, and guilty of misteaching math to kids, but misusing infinity is NOT one of their sins. In general mathematicians are cautious around infinity, treating it as "a very large number" or a limit that can't be approached. polistra
News, this article about a year past may be helpful: https://uncommondescent.com/mathematics/fyi-ftr-on-ehrlichs-unified-overview-of-numbers-great-and-small-ht-ds/ KF PS: I have found it helpful to think of Mathematics as in essence the logic of structure and quantity. (Where, space is a sort of structure.) kairosfocus

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