The Unknotting Number Is Not Additive

The Unknotting Number Is Not Additive
divisbyzero.com

Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”?

For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary?

You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts. But does it make them any less real?

Sorry for rambling off topic like a meth addict, just hoping someone can enlighten me.

> https://www.youtube.com/watch?v=tCUK2zRTcOc

Translated transcript:

  Physics is a "Real Science". It deals with reality. Math is a structural science. It deals with the structure of thinking. These structures do not have to exist. They can exist, but they don't have to. That's a fundamental difference. The translation of mathematical concepts to reality is highly critical, I would say. You cannot just translate it directly, because this leads to such strange questions like "what would happen if we take the law of gravitation by old Newton and let r^2 go to zero?". Well, you can't! Because Heisenberg is standing down there.

The prime property of numbers is also very real, a number N is prime if and only if arranging N items on a rectangular, regular grid can only be done if one of the sides of the rectangle is 1. Multiplication and addition are even more simply realized.

The infinity of natural numbers is not as real, if what we mean by that is that we can directly experience it. It's a useful abstraction but there is, according to that abstraction, an infinity of "natural" numbers that mankind will not be able to ever write down, either as a number or as a formula. So infinity will always escape our immediate perception and remain fundamentally an abstraction.

Real numbers are some of the least real of the numbers we deal with in math. They turn out to be a very useful abstraction but we can only observe things that approximate them. A physical circle isn't exactly pi times its diameter up to infinity decimals, if only because there is a limit to the precision of our measurements.

To me the relationship between pi and numbers is not so unnatural but I have to look at a broader set of abstractions to make more sense of it, adding exponentials and complex numbers - in my opinion the fact that e^i.pi = 1 is a profound relationship between pi and natural numbers.

But abstractions get changed all the time. Math as an academic discipline hasn't been around for more than 10,000 years and in that course of time abstractions have changed. It's very likely that the concept of infinity wouldn't have made sense to anyone 5,000 years ago when numbers were primarily used for accounting. Even 500 years ago the concept of a number that is a square root of -1 wouldn't have made sense. Forget aliens from another planet, I'm pretty sure we wouldn't be able to comprehend 100th century math if somehow a textbook time-traveled to us.