If we did not use integers as a basis, what would we be using?

[Hygro]

Integers are too deeply ingrained in how we are programmed to think about math for me to figure out an alternative that doesn't involve counting. You'd have to rebuild math from scratch to answer that.

Be the river.

 

[Formaldehyde]

Be the river.

That would be a classical fluid dynamics problem.

You aren't going to get very far with differential equations without numbers. And you aren't going to get very far with numbers without natural numbers.

The mathematical system of natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system first written down by the mathematician Peano in 1889. He chose the axioms (see Peano axioms), in the language of a single unary function symbol S (short for "successor"), for the set of natural numbers to be:

There is a natural number 0.

Every natural number a has a successor, denoted by Sa.

There is no natural number whose successor is 0.

Distinct natural numbers have distinct successors: if a ≠ b, then Sa ≠ Sb.

If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers ("Induction axiom").

 

[Perfection]

Hartry Field did.

 

[Formaldehyde]

As soon as the Department of Mathematics becomes subordinate to the Department of Philosophy again, let me know.

Until then, I'm sticking to the axiomatic basis of mathematics. It has worked quite well so far.

 

[Perfection]

No one is trying to take away your precious numbers. Don't be so paranoid.

 

[Hygro]

That would be a classical fluid dynamics problem.

You aren't going to get very far with differential equations without numbers. And you aren't going to get very far with numbers without natural numbers.

Get out the car

 

[Borachio]

Yes. A very nice story. I enjoyed that.

 

[Formaldehyde]

Yes. That is EXACTLY my problem. I can't get out of the math/science car.

 

[Hygro]

glad you two liked it.

 

[gangleri2001]

We already have an example of this I think. One of the things that surprised Spanish conquerors of the Aztecs was their notion of space-time continuum being equal to numbers, as atested in their calendar and religious thought. That's why pre-columbian nahuatl didn't have a number above 400: because they classified divine space and time that way. So I'd go with that, space or perhaps even space and time even though Iadmit I haven't read enough on aztec mathematical thought.