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?
I have been reading Jason Stanely’s paper on names and rigid designation from the Oxford Companion to the Philosophy of Language in the course of doing some research for my frigidity v. rigidity axe-grinding. It is an interesting and informative, though technical, introduction to issues about rigidity and I will come back to its relation to frigidity in a later post… but one thing caught my attention early on. He says,
consider Kripke’s class of strongly rigid designators (Kripke, 1980, p. 48). This class contains the rigid designators of necessary existents. That is, this class contains all and only those designators d of an object x which exists in all possible worlds, which designate the same thing in all possible worlds (viz. x). For example, the descriptive phrase “the result of adding two and three” is a strongly rigid designator, since its actual denotation, namely the number five, exists in all possible worlds, and the phrase denotes that number with respect to all possible worlds.
Is it really the case that ‘the number five exists in all possible worlds’? Isn’t there a possible world where fictionalism about math is true? In that world 2+2=4 is not true because ‘2’ stands for an existing object, viz. The Number Two, it is true because ‘in the story we tell about mathematics’ ‘2’ stands for The Number Two in just the same way that ‘Santa wears a red suit’ is true, not because ‘Santa’ picks out some guy who wears a red suit but because ‘in the story about Santa’ ‘Santa’ picks out a guywho wears a red suit. Maybe fictionalism about math isn’t actual, but surely it’s possible, isn’t it?
We can give the same kind of argument for any proposed ‘strongly rigid’ designator. Take God for instance. It take it that Atheism is a legitimate possibility for the actual world. That is, it migt actually turn out to be the case that there is no God. Of course it might also turn out to be the case that there isn’t one. Each of these seems to me to be a metaphysical possibility, not merely an epistemic possibility. If so then there is a possible world where God does not exist (it may or may not be the actual world). Isn’t this some reason to prefer, when faced with the possiblility of a proof of necessary existence, to take my view (fix it) rather than Williamson’s (accept it)? That is, isn’t there an issue here about whether there are any ‘strongly rigid’ designators?