-1- wrote: ↑Fri Aug 10, 2018 2:51 pm

WTF, you still haven't answered MY question.

@-1- You raised a number of actually good points. I do have responses which I'll post later.

By the way are you asking me for the sin of 2 radians? Or 2 pi radians? I confess I don't understand the point of the question either way but I just want to be sure. Once you clarify I'll respond. But I do admit I have no idea how that fits into our discussion. The sin is the sin. It's a function that inputs a real number and outputs a real number in the range [0,1]. Or it inputs a complex number and outputs a complex number.

Ok anyway I'll write more later. I thought your point about the square and the line segment was the most interesting thing you've said so far. I do have a response but it will take me a while to write up, it's kind of involved in my mind. But bottom line, is it the same 4? Yes.

There is only one 4 in the Platonic world of abstractions, namely the thing pointed to by all the expressions

- '4',

- '2+2',

- '3.999...",

- "the smallest n for which there are at least two distinct nonisomorphic groups of order n",

- etc.

All those various representations we humans could ever write down, all "point to" the abstract, Platonic ideal of the number 4.

And if you're not a Platonist but rather believe that the objects of mathematics have existence only insofar as the symbols talk about them, even then

there is a unique set of real numbers; and a unique number 4. In other worlds whatever your philosophy of math, there is only one real number 4, which can be used a lot of different ways.

It can be used as an age, like a 4 year old kid. It can be an area, Farmer Brown's 4 acres and 3 cows. It can be a length, the Euclidean distance between x = 3 and x = 7 on the real line. Or a fancier way to say the same thing, it's the Lebesgue measure of the interval [0,4]. And yes it is ALSO the Lebesgue measure of the square you described!

You gotta get this! The number 4 is the number 4. What people USE it for is something else. And yes very much more on point, the 4 that represents an area in the plane is the

same exactly 4 as the 4 that represents the length of a line segment.

This is how mathematicians regard your question. It's the same 4. You do not have to LIKE that fact, but I am simply describing how math works. The length of a line is defined as a real number; and the area of a square is defined as a real number. There is only one real number 4. There aren't two of them.

When a botanist sees the number 4, it's the petals on a flower. When a biologist sees 4, it's 4 bacteria or 4 elephants. A chemist has 4 electron shells. A classical physicist has 4 feet times 100 pounds = 400 foot-pounds of work done. A quantum physicist has 4 as some amplitude.

But a mathematician studies the number 4 itself. The pure Platonic ideal of the number 4.

Even if you don't have an affinity or an interest for mathematical abstraction; at least you can agree that

mathematicians have such an affinity!

Well anyway that's not a bad response considering I was only posting to say I'll respond better later! But yes, numbers in math are pure. If you want to say, "Oh, well, physics is all that matters and math is bullshit," ok, take that position and go in peace.

But if the reductio to absurdity of that position does gnaw on your conscience a bit, just remember that science is about the world, and math is about abstractions. The fact that many practical people have a use for the number 4 is of no interest to the mathematician. The mathematician studies the number 4.