Hmm, funny you should treat “I don’t believe in [Mathematical Object Y]” as Platonism. I generally characterise my ‘syntacticism’ (wh. I intend to explain more fully when I understand it the hell myself) as a “Platonic Formalism”; it is promiscuously inclusive of Mathematical Objects. If you can formulate a set of behaviours (inference rules) for it, then it has an existing Form—and that Form is the formalism (or… syntax) that encapsulates its behaviour. So in a sense, uncountable cardinals don’t exist—but the theory of uncountable cardinals does exist; similarly, the theory of finite cardinals exists but the number ‘2’ doesn’t.
This is of course bass-ackwards from a map-territory perspective; I am claiming that the map exists and the territory is just something we naïvely suppose ought to exist. After all, a map of non-existent territory is observationally equivalent to a map of manifest reality; unless you can observe the actual territory you can’t distinguish the two. Taking as assumption that the observe() function always returns an object Map, the idea that there is a territory gets Occamed out.
There is a good reason why I should want to do something so ontologically bizarre: by removing referents, and semantics, and manifest reality; by retaining only syntax, and rejecting the suggestion that one logic really “models” another, we finally solve the problems of Gödel (I’m a mathematician, not a philosopher, so I’m allowed to invoke Gödel without losing automatically) and the infinite descent when we say “first order logic is consistent because second-order logic proves it so, and we can believe second-order logic because third-order logic proves it consistent, and...”. When all you are doing is playing symbol games stripped of any semantics, “P ∧ ¬P” is just a string, and who cares if you can derive it from your axiomata? It only stops being a string when you apply your symbol games to what you unknowingly label as “manifest reality”, when you (essentially) claim that symbol game A (Peano arithmetic) models symbol game B (that part of the Physics game that deals with the objects you’ve identified as pebbles).
Platonism is not a means of excluding a mathematical object because it’s not one of the Forms; it is a means of allowing any mathematical object to have a Form whether you like it or not. I don’t believe in God, but there still exists a Form for a mathematical object that looks a lot like “a universe in which God exists”. It’s just that conceiving of a possible world only makes an arrow from your world to its, not an arrow in the reverse direction, hence why “A perfect God would have the quality of existence” is such a laughable non-starter :) What if someone broke out of a hypothetical situation in your room right now?
(I’m a mathematician, not a philosopher, so I’m allowed to invoke Gödel without losing automatically)
As a fellow mathematician, I want to point out that it doesn’t mean you win automatically, either. Just look at Voevodsky’s recent FOM talk at the IAS.
Well, of course I don’t win automatically. It’s just that there’s a kind of Godwin’s Law of philosophy, whereby the first to invoke Gödel loses by default.
Hmm, funny you should treat “I don’t believe in [Mathematical Object Y]” as Platonism. I generally characterise my ‘syntacticism’ (wh. I intend to explain more fully when I understand it the hell myself) as a “Platonic Formalism”; it is promiscuously inclusive of Mathematical Objects. If you can formulate a set of behaviours (inference rules) for it, then it has an existing Form—and that Form is the formalism (or… syntax) that encapsulates its behaviour. So in a sense, uncountable cardinals don’t exist—but the theory of uncountable cardinals does exist; similarly, the theory of finite cardinals exists but the number ‘2’ doesn’t.
This is of course bass-ackwards from a map-territory perspective; I am claiming that the map exists and the territory is just something we naïvely suppose ought to exist. After all, a map of non-existent territory is observationally equivalent to a map of manifest reality; unless you can observe the actual territory you can’t distinguish the two. Taking as assumption that the observe() function always returns an object Map, the idea that there is a territory gets Occamed out.
There is a good reason why I should want to do something so ontologically bizarre: by removing referents, and semantics, and manifest reality; by retaining only syntax, and rejecting the suggestion that one logic really “models” another, we finally solve the problems of Gödel (I’m a mathematician, not a philosopher, so I’m allowed to invoke Gödel without losing automatically) and the infinite descent when we say “first order logic is consistent because second-order logic proves it so, and we can believe second-order logic because third-order logic proves it consistent, and...”. When all you are doing is playing symbol games stripped of any semantics, “P ∧ ¬P” is just a string, and who cares if you can derive it from your axiomata? It only stops being a string when you apply your symbol games to what you unknowingly label as “manifest reality”, when you (essentially) claim that symbol game A (Peano arithmetic) models symbol game B (that part of the Physics game that deals with the objects you’ve identified as pebbles).
Platonism is not a means of excluding a mathematical object because it’s not one of the Forms; it is a means of allowing any mathematical object to have a Form whether you like it or not. I don’t believe in God, but there still exists a Form for a mathematical object that looks a lot like “a universe in which God exists”. It’s just that conceiving of a possible world only makes an arrow from your world to its, not an arrow in the reverse direction, hence why “A perfect God would have the quality of existence” is such a laughable non-starter :) What if someone broke out of a hypothetical situation in your room right now?
As a fellow mathematician, I want to point out that it doesn’t mean you win automatically, either. Just look at Voevodsky’s recent FOM talk at the IAS.
Well, of course I don’t win automatically. It’s just that there’s a kind of Godwin’s Law of philosophy, whereby the first to invoke Gödel loses by default.