Are you a platonist? Test yourself

Whether your own philosophy of mathematics is platonism or not, can be determined easily by using the following test. Let us consider the twin prime number sequence (two primes are called twins, if their difference is 2): $$(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), \dots$$

It is believed (as conjectured in 1849 by Alphonse de Polignac) that there are infinitely many twin pairs (the famous Twin Prime Conjecture), yet the problem remains unsolved up to day. Do you believe that the twin prime conjecture must be either true, or false, and that this does not depend on us, humans? Imagine, we are moving along the natural number system: $$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, \dots$$

And we meet twin pairs from time to time: $$(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), \dots$$

It seems there are only two possibilities:

1. 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),
2. 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 the natural number system as a specific independent “world”, very much 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 (following Rashevsky [1973]) that the natural number system contains not only some information about real things of the 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?