Comment author:VKS
17 March 2012 03:35:25PM
27 points
[-]

All math pays rent.

For all mathematical theorems can be restated in the form:

If the axioms A, B, and C and the conditions X, Y and Z are satisfied, then the statement Q is also true.

Therefore, in any situations where the statements A,B,C and X,Y,Z are true, you will expect Q to also be verified.

In other words, mathematical statements automatically pay rent in terms of changing what you expect. (Which is) the very thing it was required to show. ■

In practice:

If you demonstrate Pythagoras's Theorem, and you calculate that 3^2+4^2=5^2, you will expect a certain method of getting right angles to work.

If you exhibit the aperiodic Penrose Tiling, you will expect Quasicrystals to exist.

If you demonstrate the impossibility of solving to the Halting Problem, you will not expect even a hypothetical hyperintelligence to be able to solve it.

If you understand why you can't trisect an angle with an unmarked ruler and a compass (not both used at the same time), you will know immediately that certain proofs are going to be wrong.

and so on and so forth.

Yes, we might not immediately know where a given mathematical fact will come in handy when observing the world, but by their nature, mathematical facts tell us exactly when to expect them.

## Comments (245)

OldAll math pays rent.

For all mathematical theorems can be restated in the form:

If the axioms A, B, and C and the conditions X, Y and Z are satisfied, then the statement Q is also true.

Therefore, in any situations where the statements A,B,C and X,Y,Z are true, you will expect Q to also be verified.

In other words, mathematical statements automatically pay rent in terms of changing what you expect. (Which is) the very thing it was required to show. ■

In practice:

If you demonstrate Pythagoras's Theorem, and you calculate that 3^2+4^2=5^2, you will expect a certain method of getting right angles to work.

If you exhibit the aperiodic Penrose Tiling, you will expect Quasicrystals to exist.

If you demonstrate the impossibility of solving to the Halting Problem, you will not expect even a hypothetical hyperintelligence to be able to solve it.

If you understand why you can't trisect an angle with an unmarked ruler and a compass (not both used at the same time), you will know immediately that certain proofs are going to be wrong.

and so on and so forth.

Yes, we might not immediately know where a given mathematical fact will come in handy when observing the world, but by their nature, mathematical facts tell us exactly when to expect them.