(October 11, 2018 at 11:42 am)Jörmungandr Wrote:(October 10, 2018 at 9:57 pm)polymath257 Wrote: In terms of overall philosophy, I am closest to Hume. Kant made some mistakes in thinking space and time are synthetic a priori (they are not). The list of topics you gave can be largely addressed via the scientific method. To the extent they cannot be so, they are literally meaningless.
For a good explanation of the term synthetic a priori which also relates its use to metaphysics and mathe, see Synthetic A Priori Knowledge.
And I disagree with Kant. Mathematics is analytic, not synthetic in nature. Once we have the axioms and rules of inference, the rest of math follows. So, 2+2=4 is a result derived from a formal system in which 2, +, 4, and = are symbols with rules of inference from some axioms.
For this particular result, we have the following axioms:
1. If x is a number, x' is a number.
2. 0 is a number.
3. x+0=x for all numbers x.
4. x+y' =(x+y)' for all numbers x and y
5. x=(x) for all numbers x.
Rules of deduction:
1. If x=y and y=z, then x=z.
2. if x=y, then any time x appears in a statement, it can be replaced by y.
and the following definitions:
0'=1, 1'=2, 2'=3, 3'=4.
Then,
1 is a number by axioms 1 and 2.
2 is a number from axiom 1. So are 3 and 4.
2+2 = 2+1' by definition of 2.
2+1' =(2+1)' by axiom 4.
2+2=(2+1)' by the rule of deduction 1.
(2+1)' = (2+0')' by definition of 1.
2+2 = (2+0')' by rule of deduction 1.
(2+0')' = ( (2+0)' )' by axiom 4.
2+2 = ( (2+0)' )' by rule of deduction 1.
( (2+0)' )' = ( (2)' )' by axiom 3.
2+2 = ((2)')' by rule of deduction 1.
((2)')' = (2')' by axiom 5
2+2=(2')' by rule of deduction 2.
(2')'=3' by definition of 3 and rule of deduction 2.
2+2=3' by rule of deduction 1.
3'=4 by definition of 4.
2+2 =4 by rule of deduction 1.
Any system that has those 5 axioms, those 2 rules of deduction, and those definitions will have 2+2=4. This is an analytic consequence of the axioms, rules of deduction, and definitions.