Mathematics

5+5=9 is not an empirical fact. However it can be proven empirically (put 5 objects and four objects together, then count the result). How is it possible for non-empirical facts to be proven empirically?

Comments

Counting things is something we actually do to get to the answers to arithmetic problems (who hasn't counted on their fingers at some point?) but we need to be careful. What if you put five drops of water together with five drops of water and count the results? Or what if you put an electron and a positron together?

We might say that putting drops of water together or combining electrons and positrons doesn't count as addition. But that's because when when we do, we don't end up with the right answer to the arithmetic question we're supposedly trying to settle empirically. What we see is that arithmetic isn't an hypothesis about what we'll find when we count in various cases. Rather, arithmetic tells us whether an empirical operation is a good model of addition, or whether what we're mumbling under our breath as we flex those fingers really counts as counting.

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