3 ms·

Philosophers of math aren't confused by this. What they are asking is whether "mathematical truth" has any connection with truth. Formalists more or less deny t

Philosophers of math aren't confused by this. What they are asking is whether "mathematical truth" has any connection with truth. Formalists more or less deny this, Platonists affirm it. As a side note, correspondence with reality is a contentious definition of truth. It's just one theory of what truth is. I don't really know what to say about your claims about meaninglessness, except that I think they're false. But I don't even know where to start in arguing that. And while I don't think this is much of an argument, I don't think there are really any philosophers, mathematicians, or logicians who have worked out any theory along those lines.

catnaroek 11y ago
> I don't really know what to say about your claims about meaninglessness, except that I think they're false. Syntax has no meaning outside of the rules for manipulating it. If no rule applies to a particular expression, then it has no meaning whatsoever. And, in any sensible formal system, like typed lambda calculi with Church-style semantics, expressions only have meaning in contexts where their free variables are bound.
- hyperpape 11y ago
  It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx. The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it. This is much like the mistake that Russell criticized in On Denoting, with the difference that he discussed definite descriptions. But the error is the same: he attacked opponents who thought that "the present king of france is bald" was either meaningless, or required some sort of bizzaro existence for the present King of France. And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.
  - catnaroek 11y ago
    > It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx. This is a very low insult - to assume I don't understand the distinction between “F(a)”, where “a” appears free, and “Ex. F(x)”, where “x” appears bound. I understand logical quantifiers and variable binding just fine, thanks. > The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it. Another very low insult - to assume that I would conflate “false” with “meaningless”. “There is a natural number that is both even and odd” is false. “There is a myppit inside the quxxit” is meaningless unless we first define: (0) what a “myppit” is, (1) what a “quxxit” is, (2) what specific “quxxit” we're talking about. The one who's making a mistake is you, because “F” isn't the same as “F(x)”, but rather “λx.F(x)”. In a context where the variable “x” isn't bound, the the latter is meaningful, but the former is not. In other words, the judgment “Γ ⊢ F(x) : Prop”, asserting that “F(x)” is indeed a proposition in the context “Γ”, shouldn't be derivable if “Γ” doesn't contain “x”. But the judgment “Γ ⊢ λx.F(x) : T → Prop”, asserting that “F” is a predicate ranging over “T”s, is derivable from “Γ, x:T ⊢ F(x) : Prop”, and “Γ, x:T” is a context containing “x”. > And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France. Unfortunately, “Bald(king)“ isn't the same as “E!person. King(person) ∧ Bald(person)”.
    - hyperpape 11y ago
      Regarding insults, you started by suggesting that philosophers of math were ignorant of the distinction between truth as either proof theoretic derivation from axioms, or truth in a model and truth simpliciter, I don't think you have room to worry about who is being insulting. Philosophers typically would be introduced to these concepts in advanced undergraduate logic or early on in grad school. As for the rest of it, I'll leave with one final thought: as a meta-philosophical point, we should rarely attribute meaninglessness to statements unless we are forced to. And it's easy enough to show how to interpret existentials as meaningful even in the absence of a referent. You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.

catnaroek 11y ago

> I don't really know what to say about your claims about meaninglessness, except that I think they're false. Syntax has no meaning outside of the rules for manipulating it. If no rule applies to a particular expression, then it has no meaning whatsoever. And, in any sensible formal system, like typed lambda calculi with Church-style semantics, expressions only have meaning in contexts where their free variables are bound.

hyperpape 11y ago
It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx. The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it. This is much like the mistake that Russell criticized in On Denoting, with the difference that he discussed definite descriptions. But the error is the same: he attacked opponents who thought that "the present king of france is bald" was either meaningless, or required some sort of bizzaro existence for the present King of France. And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.
- catnaroek 11y ago
  > It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx. This is a very low insult - to assume I don't understand the distinction between “F(a)”, where “a” appears free, and “Ex. F(x)”, where “x” appears bound. I understand logical quantifiers and variable binding just fine, thanks. > The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it. Another very low insult - to assume that I would conflate “false” with “meaningless”. “There is a natural number that is both even and odd” is false. “There is a myppit inside the quxxit” is meaningless unless we first define: (0) what a “myppit” is, (1) what a “quxxit” is, (2) what specific “quxxit” we're talking about. The one who's making a mistake is you, because “F” isn't the same as “F(x)”, but rather “λx.F(x)”. In a context where the variable “x” isn't bound, the the latter is meaningful, but the former is not. In other words, the judgment “Γ ⊢ F(x) : Prop”, asserting that “F(x)” is indeed a proposition in the context “Γ”, shouldn't be derivable if “Γ” doesn't contain “x”. But the judgment “Γ ⊢ λx.F(x) : T → Prop”, asserting that “F” is a predicate ranging over “T”s, is derivable from “Γ, x:T ⊢ F(x) : Prop”, and “Γ, x:T” is a context containing “x”. > And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France. Unfortunately, “Bald(king)“ isn't the same as “E!person. King(person) ∧ Bald(person)”.
  - hyperpape 11y ago
    Regarding insults, you started by suggesting that philosophers of math were ignorant of the distinction between truth as either proof theoretic derivation from axioms, or truth in a model and truth simpliciter, I don't think you have room to worry about who is being insulting. Philosophers typically would be introduced to these concepts in advanced undergraduate logic or early on in grad school. As for the rest of it, I'll leave with one final thought: as a meta-philosophical point, we should rarely attribute meaninglessness to statements unless we are forced to. And it's easy enough to show how to interpret existentials as meaningful even in the absence of a referent. You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.