Topics Mathematics

Question Would you agree that numbers are synthetic truths rather than analytic truths? This is because I can imagine a universe, whenever I walk 2 meters foward, space itself 'bends' so I end up 3 meters ahead of where I started. In this universe, when 2 is added you end up with 3. 2+2=5 (or maybe 6).

Accepted:
November 13, 2005

Comments

It is true that we can imagine a universe in which when you walk forward 2 meters, you end up 3 meters ahead of where you started. However, I would say that in that universe, 2+2 is still equal to 4, but addition does not describe how one's position changes when one walks forward. Inhabitants of that universe might invent a new mathematical operation, "walk-addition", to describe how one's position changes when one walks forward, but that would be a different operation from the operation of addition.

In fact, in our universe we have done something similar with geometry. Einstein discovered that Euclidean geometry does not accurately describe our universe. We didn't conclude that Euclid was wrong, we just concluded that we needed a different kind of geometry to describe our universe.

The point is that mathematical objects and operations are not defined in terms of their applications to the physical world. They are abstractions, and although those abstractions may be motivated by things we observe in the physical world, their definitions are independent of the physical world. For more on this, see question 276.