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What you are struggling with here is the definition of existence. And Platonism has this problem in general, you rarely get a sharp definition of what it means

by danbruc 11y ago

What you are struggling with here is the definition of existence. And Platonism has this problem in general, you rarely get a sharp definition of what it means that something exists. Natural numbers exist. I know what natural numbers refers to but I have no idea what existence refers to. When we talk about things in the physical world I can come up with a definition of existence, say anything I can - at least in principle - interact with. This definition is far from perfect and has quite a few issues itself but it is at least a start. But what does the existence of mathematical objects even mean?

skybrian 11y ago
Yes, exactly. The essay is attempting to tie the moment of creation for a concept to some event in the real world. Certainly there is something physical going on which we call a "discovery" or an "invention", but it gets fuzzier the more you look at it. We can objectively say that 1492 was an important year of first contact and we should talk about what happened. But it wasn't "discovering the New World" from the Native Americans' point of view. Both in science and math, publishing a paper can be an important event and it's a physical process. But the first math paper published may not correspond to the first time someone had an idea or made an observation. If we're going to talk about a concept as a thing that exists in time and space, we need to take into account that ideas may be created, shared, and forgotten many times. Perhaps we can think of Platoism as a way to avoid getting bogged down in the history of an idea by declaring it irrelevant for our purposes.
DougMerritt 11y ago
> but I have no idea what existence refers to. A lot of brilliant famous mathematicians in history have been befuddled on that point as well. I think it's a linguistic issue, which perhaps you are hinting at. If I say "that rock exists", in the real world, that meaning of "exists" is parallel to, yet different than if I say "prime numbers exist". It's hardly the first time that one word meant more than one thing. The linguistic "exists" is a "deictic" -- I can point to the thing. The mathematical "exists" means "come, let us reason together, and I can demonstrate to you abstractly that this one thing is inescapably implied by this starting point". (Phrasing borrowed from Leibniz's commentary on Calculus Ratiocinator) They both have a similar final state, I suppose: that we are convinced of the "existence" of a thing, but nonetheless the existence in each case is in a separate realm.
- ttctciyf 11y ago
  I like to consider the mathematical use of 'exists' as something quite similar to its use in sentences like "a solution to such-and-such problem exists" - basically, an assertion that if you do X then Y will result - not so dissimilar from your "come let us reason together..." explanation, perhaps.

by danbruc 11y ago

skybrian 11y ago

Yes, exactly. The essay is attempting to tie the moment of creation for a concept to some event in the real world. Certainly there is something physical going on which we call a "discovery" or an "invention", but it gets fuzzier the more you look at it. We can objectively say that 1492 was an important year of first contact and we should talk about what happened. But it wasn't "discovering the New World" from the Native Americans' point of view. Both in science and math, publishing a paper can be an important event and it's a physical process. But the first math paper published may not correspond to the first time someone had an idea or made an observation. If we're going to talk about a concept as a thing that exists in time and space, we need to take into account that ideas may be created, shared, and forgotten many times. Perhaps we can think of Platoism as a way to avoid getting bogged down in the history of an idea by declaring it irrelevant for our purposes.

DougMerritt 11y ago

> but I have no idea what existence refers to. A lot of brilliant famous mathematicians in history have been befuddled on that point as well. I think it's a linguistic issue, which perhaps you are hinting at. If I say "that rock exists", in the real world, that meaning of "exists" is parallel to, yet different than if I say "prime numbers exist". It's hardly the first time that one word meant more than one thing. The linguistic "exists" is a "deictic" -- I can point to the thing. The mathematical "exists" means "come, let us reason together, and I can demonstrate to you abstractly that this one thing is inescapably implied by this starting point". (Phrasing borrowed from Leibniz's commentary on Calculus Ratiocinator) They both have a similar final state, I suppose: that we are convinced of the "existence" of a thing, but nonetheless the existence in each case is in a separate realm.

ttctciyf 11y ago
I like to consider the mathematical use of 'exists' as something quite similar to its use in sentences like "a solution to such-and-such problem exists" - basically, an assertion that if you do X then Y will result - not so dissimilar from your "come let us reason together..." explanation, perhaps.