Re:The existence of the square root of two (2001)
by antonvs on Monday December 25, 2017 @04:46PM
(#16005848)
> The underlying issue here seems to be a conflict between a thin notion of existence and a fat one.
The fundamental issue is described at the end of the article:
> "I didn't just define the square root of two. Rather, I defined an entire number system and showed, by which I mean actually proved, that the square root of 2 exists in that system.
> "Oh, well if that's what you mean by the existence of the square root of two, then I suppose I accept it."
This seems to be the key point of the article, and it's quite correct. Any concept is meaningless without a framework in which it is defined. We can, by the mechanism described in the above quote, define all sorts of mathematical concepts that demonstrably "exist" in such frameworks.
To then talk about existence in the physical world requires mapping these concepts from their abstract frameworks into the physical world. In the case of the square root of 2, we might point out that the length of the diagonal of a physical version of the unit square "is" sqrt(2), where "is" means something like, "as close to the mathematical definition as it is possible to get given physical limitations."
Separating our formalized abstractions of the world from the physical objects they sometimes are intended to describes helps achieve clarity and avoids confusing platonic abstractions with physical existence.
The fundamental issue is described at the end of the article:
> "I didn't just define the square root of two. Rather, I defined an entire number system and showed, by which I mean actually proved, that the square root of 2 exists in that system.
> "Oh, well if that's what you mean by the existence of the square root of two, then I suppose I accept it."
This seems to be the key point of the article, and it's quite correct. Any concept is meaningless without a framework in which it is defined. We can, by the mechanism described in the above quote, define all sorts of mathematical concepts that demonstrably "exist" in such frameworks.
To then talk about existence in the physical world requires mapping these concepts from their abstract frameworks into the physical world. In the case of the square root of 2, we might point out that the length of the diagonal of a physical version of the unit square "is" sqrt(2), where "is" means something like, "as close to the mathematical definition as it is possible to get given physical limitations."
Separating our formalized abstractions of the world from the physical objects they sometimes are intended to describes helps achieve clarity and avoids confusing platonic abstractions with physical existence.