Things have been quite around here lately, mostly because I have been lost in GTA IV and finishing my dissertation. Well I am making some progress (on both 🙂 ) and so will try to get to some of the comments around here.
In answering the comments on the Refutatioin of Rationalism I started thinking about Quine’s indispensibility argument for the empirical justification of mathematics. The argument starts from Quine’s claim that we are ontologically committed to the things which we quantify over in our best theories. The indispensibility of mathematics to physics means that we are committed to the existence of numbers (but not, obviously, to their non-physical existence). So Quine went on to argue that, since our theories all get confirmed or disconfirmed together as a group, the empirical confirmation of physics is empirical confirmation for mathematics. In this way mathematics is empirically justified.
One problem with this argument is that it depends on confirmation holism. That is, it depends on the claim that all of our theories are confirmed or discomfirmed together. None ‘face the tribunal of experience alone’. I then started thinking about how Rosenthal’s version of this argument avoids this commitment and so is a better argument. Sadly Rosenthal has never published this argument (I heard it in a seminar on Quiene and Sellers he gave) so I will try to recreate it as best as I remember.
The basic idea is: if we ever had empirical evidence that some truth of arithmatic was false we would have to admit that it was false. But if so then mathematics is empirically justified. To make the case he asks us to entertain the following scenerio. Suppose that you had two pens of sheep; one with 6 and one with 7 sheep. Now suppose that you counted the sheep individually in each pen (and got 6 and 7) and then you counted all of the sheep and got 14. Suppose you did it again. 1. 2. 3. 4. 5. 6. Yep six sheep in that pen. 1. 2. 3. 4. 5. 6. 7. Yep seven sheep in that pen. Then all the sheep. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Suppose that this was repeated by all of your friends with the same results. Suppose that it was on the news and tested scientifically and confirmed. Suppose that this phenomenon was wide spread, observable, and repeatable.
If this were the case we would be forced to admit that 7+6=14 is true therefore mathematics is empirically justified.