Comment by yujzgzc
5 days ago
Yes these knots are real and can be experienced with a simple piece of rope.
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.
I see infinity all the time. Go look at a one point perspective drawing.
I think that is a little like pi. There is a limit to what we can measure. In a real life drawing on paper the "one point" is not dimensionless. There is a limit to what we can draw.
The "one point" in "one point perspective" isn't drawn at all, rather it is the point where all lines going into the page perpendicular to the viewing plane eventually converge to. Eg if you were to stand on a set of straight train tracks (don't do this) you would see both rails (and any roads or whatever else is parallel to them) converge to a point somewhere on the horizon line. The artists call it the "vanishing point", the mathematicians call it "the point at infinity".
Indeed with the point at infinity you can simplify geometry by dispensing with Euclid's 5th postulate. There are no parallel lines, any two lines intersect at a single point just the same way as any two points are intersected by a single line, and the intersection points of the lines we call "parallel" simply happen to be "at infinity" (outside the set of ordinary finite coordinates).
The vanishing point in a perspective drawing is a point with a value that is literally beyond the finite coordinates of any object. And you don't need to be looking at a drawing to see it.
In a certain regard its an accounting trick. Saying parallel lines meet at infinity is literally like saying "lets schedule this meeting for never", except the mathematicians added an actual box to the calendar for a date called "never" as an accounting hack, but the hack works so well you really have to wonder if it might actually be a real date or if its just an incredibly useful fiction.
Aren't all numbers just incredibly useful fictions?
Why is a date called never / a point at infinity any different?
https://i.pinimg.com/originals/20/7b/ae/207bae64d2488373fd4a...
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