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Whether your own philosophy of mathematics is Platonism or not, can be easily determined by using the following test. Let us consider the twin prime number sequence:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), (101, 103), (107,109), (137, 139), (149, 151), (179, 181), (191, 193), ..., (1787, 1789), ..., (1871, 1873), ..., (1931, 1933), (1949, 1951), (1997, 1999), (2027, 2029), ...
It is believed that there are infinitely many twin pairs (the famous twin prime conjecture), yet the problem remains unsolved up to day. Suppose, someone has proved that the twin prime conjecture is unprovable in set theory. Do you believe that, still, the twin prime conjecture possesses an "objective truth value"? Imagine, you are moving along the natural number system:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...
And you meet twin pairs in it from time to time: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), ... It seems there are only two possibilities:
a) We meet the last pair and after that moving forward we do not meet any twin pairs (i.e. the twin prime conjecture is false),
b) Twin pairs appear over and again (i.e. the twin prime conjecture is true).
It seems impossible to imagine a third possibility...
If you think so, you are, in fact, a Platonist. You are used to treat natural number system as a specific "world", very like the world of your everyday practice. You are used to think that any sufficiently definite assertion about things in this world must be either true or false. And, if you regard natural number system as a specific "world", you cannot imagine a third possibility: maybe, the twin prime conjecture is neither true nor false. Still, such a possibility would not surprise you if you would realize that the natural number system contains not only some information about real things of human practice, it also contains many elements of fantasy. Why do you think that such a fantastic "world" (a kind of Disneyland) should be completely perfect? To continue click here.
My response is that this problem CAN'T be unprovable in set theory. This may be very naïve of me, given Gödel's incompleteness theorem, but this strikes me as the kind of number theoretic proposition which MUST be decided (although I guess intuitively-concrete Diophantine equations are likewise undecidable). And if it's not decidable in some particular set theory, that set theory is WRONG (in the sense of being a bad model), because twin primes *EXIST*, at least in my intuition.
Perhaps the lesson to learn from Gödel's theorem is that even though no axiomatization of number theory can be complete, we can always progress. Perhaps the union of the provable propositions in those "conservative" axiomatizations correspond to the "intuitively true" or "really true" propositions.
But that begs the question: so why don't we use the union of those axioms.... which would get us in trouble again.
I still have to struggle with Gödel's theorem, before I feel comfortable with it. I wonder how many experts feel philosophically comfortable with it and how many just gave up and got used to it.