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.
In “What is Evidence?” I wrote:1
Cihan Baran replied:2
I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations,” and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.
Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared—in contrast to my stored memory that 2 + 2 was supposed to equal 4. Moreover, when I visualized the process in my own mind, it seemed that making xx and xx come out to xxxx required an extra x to appear from nowhere, and was, moreover, inconsistent with other arithmetic I visualized, since subtracting xx from xxx left xx, but subtracting xx from xxxx left xxx. This would conflict with my stored memory that 3 - 2 = 1, but memory would be absurd in the face of physical and mental confirmation that xxx - xx = xx.
I would also check a pocket calculator, Google, and perhaps my copy of 1984 where Winston writes that “Freedom is the freedom to say two plus two equals three.” All of these would naturally show that the rest of the world agreed with my current visualization, and disagreed with my memory, that 2 + 2 = 3.
How could I possibly have ever been so deluded as to believe that 2 + 2 = 4? Two explanations would come to mind: First, a neurological fault (possibly caused by a sneeze) had made all the additive sums in my stored memory go up by one. Second, someone was messing with me, by hypnosis or by my being a computer simulation. In the second case, I would think it more likely that they had messed with my arithmetic recall than that 2 + 2 actually equalled 4. Neither of these plausible-sounding explanations would prevent me from noticing that I was very, very, very confused.3
What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.
There was a time when I had no idea that 2 + 2 = 4. I did not arrive at this new belief by random processes—then there would have been no particular reason for my brain to end up storing “2 + 2 = 4” instead of “2 + 2 = 7.” The fact that my brain stores an answer surprisingly similar to what happens when I lay down two earplugs alongside two earplugs, calls forth an explanation of what entanglement produces this strange mirroring of mind and reality.
There’s really only two possibilities, for a belief of fact—either the belief got there via a mind-reality entangling process, or not. If not, the belief can’t be correct except by coincidence. For beliefs with the slightest shred of internal complexity (requiring a computer program of more than 10 bits to simulate), the space of possibilities is large enough that coincidence vanishes.4
Unconditional facts are not the same as unconditional beliefs. If entangled evidence convinces me that a fact is unconditional, this doesn’t mean I always believed in the fact without need of entangled evidence.
I believe that 2 + 2 = 4, and I find it quite easy to conceive of a situation which would convince me that 2 + 2 = 3. Namely, the same sort of situation that currently convinces me that 2 + 2 = 4. Thus I do not fear that I am a victim of blind faith.5
1See Map and Territory.
2Comment: http://lesswrong.com/lw/jl/what_is_evidence/f7h.
3See “Your Strength as a Rationalist” in Map and Territory.
4For more on belief formation and beliefs of fact, see “Feeling Rational” and “What Is Evidence?” in Map and Territory. For more on belief complexity, see “Occam’s Razor” in the same volume.
5If there are any Christians reading this who know Bayes’s Theorem, might I inquire of you what situation would convince you of the truth of Islam? Presumably it would be the same sort of situation causally responsible for producing your current belief in Christianity: We would push you screaming out of the uterus of a Muslim woman, and have you raised by Muslim parents who continually told you that it is good to believe unconditionally in Islam.
Or is there more to it than that? If so, what situation would convince you of Islam, or at least, non-Christianity? And how confident are you that the general kinds of evidence and reasoning you appeal to would have been enough to dissuade you of your religion if you had been raised a Muslim?