If someone demanded proof that 1+1=2 is true what would you do? Would you get an object and place it next to another object and then count them? One might argue that that doesn’t prove that 1+1=2 because 1+1=2 is a necessary truth and you cannot get necessity from experience (as per the history of philosophy). If one thought that all knowledge comes from experience one might then, like Quine, think that it is possible that we could have experience that dis-confirmed mathematics. For instance David Rosenthal has argued that if we ever had irrefutable counting evidence (i.e. widespread, re-created and independently confirmed) that 1+1=2 were false then we would have to admit that it was false and so mathematic is contingent on experience. Yikes!
Or would you appeal to the Peano axioms, which include the claim that 0 is a number and that it has a successor denoted by S(0) and then define addition as
(for all a) a+0=a
(for all a and b) a+S(b)=S(a+b)
So then it is easy to show that 1+1=2 as follows
But then one might worry about the successor relation. It might seem as though we have simply assumed addition in the definition of the successor relation (i.e. it tacitly assumes that S(a)=(a+1)).
Of course we can show that S(a)=(a+1) by the definition of addition above. So, let S(0)=1 then a+1=a+S(0) by the definition of addition we get a+S(0)=S(a+0) and since a+0=a we get S(a) so (a+1)=S(a). But then we seem to have assumed addition by stipulating that S(0)=1 (as we actually did in the initial proof of 1+1=2…worse and worse!)
Is the claim that when you have nothing and add a thing you then get only the one thing (i.e. S(0)=1) supposed to be a truth that we just apprehend with pure reason? A self-evident truth that is ‘clear and distinct’? Or something that experience has trained us to believe? Is there any non-question begging reason to prefer either of these?