Wednesday, August 19, 2009

Is Julius Ceasar Identical to the number 2?

Suppose we got a bunch of arithmatical truths like 2+7=9 and 5+7=12 and so on. Those truths have referring expressions such as '1' and '2' and so on. Some people wonder what those expressions refer to. I guess I really don't know much about this, but I think there are a few things that people tend to accept about arithmatical truths. First, we have an incomplete axiomatization of arithmatic. Those axioms have certain structural properties that arithmatic must satisfy. But, other things satisfy those axioms as well. For example, under one interpretation of the axioms, the number 2 is identical to the set containing the set containing the null set. On another interpretation, the number 2 is identical to Julius Ceasar. Just like with other logical axioms, there is nothing in the structure that guarantees that the referring expressions refer to the some things rather than others. This is supposed to be problematic for arithmatic because it is hard to see what kind of causal relations would help to fix the referent of the referring expressions in arithmatic. So, how do we solve this problem?

Some people solve it by claiming that arithmatical expressions are quantified expressions that quantify over the things that satisfy the axioms of arithmatic. So, for example, 2+2=4 turns out to say something like for any interpretation of the referring expressions and function in '2+2=4' that satisfies the axioms of arithmatic, the function expressed by '+' on that interpretation applied to the referents of '2' and '2' on that interpretation yeilds the referent of '4' on that interpretation (or something like that). Views like this are structuralist views. I think structuralist view is mistaken. Simply put, '2+2=4' just doesn't have a quantificational structure. So, structuralist views are mistaken.

However, I wonder if structuralist views are on the right track. Suppose that we have some arithmatical axioms. We might simply supervaluate over those axioms. Thus, 2+2=4 turns out to be supertrue in virtue of the fact that on any admissiable interpretation of the terms, '2+2=4' expresses a truth. However, it does not express a quantificational truth. One happy consequence of this view is that the Julius Ceasar problem goes away. Although 2=Julius Ceasar on some interpretation, it does not on others. So, it is not supertrue that 2=Julius Ceasar. In fact for any thing whatsoever, it is not supertrue that 2= that thing. However, it is supertrue that 2=something. Supervaluationists allow for supertrue existential statements without true instances. Now, if we simply identify truth in arithmatic as supertruth, we have a solution to the Julius Ceasar problem and we can have all the supposed benefits of structuralism (epistemic access to arithmatical truths, no causal contact with any abstract entities, etc).

Like I said, I don't know much about philosophy of math. So, maybe this view has already been expressed. If it has, I would like to know. If it hasn't I'd like to know what others think of it.