(February 25, 2013 at 11:56 am)ChadWooters Wrote: Whereas I think that is your mistake. The best example which I can think of is the distinction between things required for the universe to exist, like mathematical certainties, and universal constants, like the speed of light. One can conceive of a universe in which one or all of the known physical constants have different values, however, you cannot conceive of a universe in which 1+1=3. Knowledge of constants (inherencies) can come from empirical observation alone, but to acquire empirical knowledge of mathematical certainties (contingencies), say by setting apples in a row, there must be an a priori knowledge, knowledge that could be obtained by pure reason alone. Many mathematical concepts, like higher dimensions, are developed before they find real world application.
Not quite. These mathematical certainties and universal constants are required for this universe to exist. That is to say, if these the universe were something other than what it is, these constants and certainties would be different as well - leading to our understanding or rationality and mathematics being different as well. The concept of apriori knowledge doesn't enter here because someone, somewhere had to teach you and demonstrate to you that 1+1=2 - otherwise, you would not have known it. Further, I am not a mathematician, but my understanding is that it is an axiomatic field, i.e. the knowledge and principles applicable in a field of mathematical study would depend upon the axioms assumed to be true. Therefore, it would be possible for us to conceive of a body of mathematics where 1+1 = 3.
