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For the article to say that the "Existence" claim is "tolerably clear" is rather disingenuous. The basis for this statement in the article is that you can forma

by pdonis 5y ago

For the article to say that the "Existence" claim is "tolerably clear" is rather disingenuous. The basis for this statement in the article is that you can formalize the "Existence" claim with an existential quantifier; but the whole point of an existential quantifier in mathematics is that it has no meaning unless you already know what domain is being quantified over. And in the absence of some well-defined mathematical domain, the existential quantifier basically becomes a tautology: it means "object x exists in the domain of whatever objects exist". Given this, it's no wonder that such "existence" claims have always been problematic and have always been viewed with skepticism by many non-philosophers and even many philosophers.

joe_the_user 5y ago
Yeah, I think that the reality of mathematical "objects" is very relative. That you can apparently construct an approximation to a Turing machine to any degree says that things behaving like finite mathematical objects can be readily constructed. But I don't think that would satisfy a Platonist.
mjburgess 5y ago
Your observation here really, I think, solves your own problem. We shouldn't think of there being a universal quantifier (ie., a universal "Exists") -- but a variety of ones depending on the domain. We can construct these domains by enumerating objects which share (eg.,) a property. So there is the physical domain of spatio-temporal objects, fictional domain of fictional objects, etc. The question of whether something exists then becomes a "thin" one: does the relevant quantifier apply? In this sense most things exist -- and this to me feels kinda obviously correct. Debates are mostly confusions over the domain: are mathematical objects in the physical domain? NO. And no one thinks they are. '2' has no spatio-temporal properties. If we just drop the idea of a privileged quantifier, everything becomes much clearer. There is no "really real", or "really exists!". There are just enumerable domains of objects of which we can form claims.
- pdonis 5y ago
  > Your observation here really, I think, solves your own problem. It's not my problem; I am not a mathematical Platonist, so I have no issue at all with the idea that any claim about mathematical objects "existing" is simply in a different domain from claims about physical objects "existing". However, for someone who is a mathematical Platonist, there is a problem, because mathematical Platonism appears to be claiming that "exist" is not domain-specific. For example, the article under discussion formalizes the "existence" claim as "Ex Mx" (I don't have the backwards "E" on my keyboard), where "Mx" is the predicate "x is a mathematical object"--i.e., it separates the "existence" claim (the quantifier) from the "domain" claim (the predicate). This only makes sense if the quantifier is not domain-specific but general--i.e., if the quantifier is the same one for all possible domains, so that the existence claim is an absolute one. Which, as I noted in my previous comment, and as you appear to agree, is simply not how existential quantifiers in mathematics are supposed to work.
  - mjburgess 5y ago
    Maybe the issue isnt the platonists but the physicalists. I suspect platonists would be happy with their own domain. The issue is physicalists want a privileged quantifier E_physical, and they want a hierarchy of quantifiers where everything reduces to E_physical. Ie., E_mathematical is "really just" a subset of E_physical. The worry of the physicalist is that Ex means x has spatiotemporal properties; so '2' has to "be somewhere".

by pdonis 5y ago

joe_the_user 5y ago

Yeah, I think that the reality of mathematical "objects" is very relative. That you can apparently construct an approximation to a Turing machine to any degree says that things behaving like finite mathematical objects can be readily constructed. But I don't think that would satisfy a Platonist.

mjburgess 5y ago

Your observation here really, I think, solves your own problem. We shouldn't think of there being a universal quantifier (ie., a universal "Exists") -- but a variety of ones depending on the domain. We can construct these domains by enumerating objects which share (eg.,) a property. So there is the physical domain of spatio-temporal objects, fictional domain of fictional objects, etc. The question of whether something exists then becomes a "thin" one: does the relevant quantifier apply? In this sense most things exist -- and this to me feels kinda obviously correct. Debates are mostly confusions over the domain: are mathematical objects in the physical domain? NO. And no one thinks they are. '2' has no spatio-temporal properties. If we just drop the idea of a privileged quantifier, everything becomes much clearer. There is no "really real", or "really exists!". There are just enumerable domains of objects of which we can form claims.

pdonis 5y ago
> Your observation here really, I think, solves your own problem. It's not my problem; I am not a mathematical Platonist, so I have no issue at all with the idea that any claim about mathematical objects "existing" is simply in a different domain from claims about physical objects "existing". However, for someone who is a mathematical Platonist, there is a problem, because mathematical Platonism appears to be claiming that "exist" is not domain-specific. For example, the article under discussion formalizes the "existence" claim as "Ex Mx" (I don't have the backwards "E" on my keyboard), where "Mx" is the predicate "x is a mathematical object"--i.e., it separates the "existence" claim (the quantifier) from the "domain" claim (the predicate). This only makes sense if the quantifier is not domain-specific but general--i.e., if the quantifier is the same one for all possible domains, so that the existence claim is an absolute one. Which, as I noted in my previous comment, and as you appear to agree, is simply not how existential quantifiers in mathematics are supposed to work.
- mjburgess 5y ago
  Maybe the issue isnt the platonists but the physicalists. I suspect platonists would be happy with their own domain. The issue is physicalists want a privileged quantifier E_physical, and they want a hierarchy of quantifiers where everything reduces to E_physical. Ie., E_mathematical is "really just" a subset of E_physical. The worry of the physicalist is that Ex means x has spatiotemporal properties; so '2' has to "be somewhere".