Math Hierarchy: Fundamentals, Axioms & Exponentiation

[Nick R]

If one were to do a bottom up study of math how would it go?

Set theory
.
.
.
?

Is set theory the most fundamental, or does it build on the axioms of some other ideas?

Also, in what section of mathematics is the concept of vectors established? What about "euclidean space" or "other space" established?

Are things like exponentiation arbitrarily defined or is there something more to it that is developed somewhere?

 

[Dickfore]

I would say mathematical logic, a subfield of which is set theory.

 

[Tac-Tics]

If one were to do a bottom up study of math how would it go?

Like a roly poly, logic curls up into a ball when you poke at it. You end up with a handful of mathematical objects which can each be defined in terms of the other. Functions are just sets in set theory. But you could just as easily to define sets in terms of functions functions (category theory does something similar to this).

In first-order logic, OR can be defined in terms of NOT and AND. AND can be defined in terms of NOT and OR. NOT can be defined in terms of IMPLICATION and FALSE.

Any Turing-complete system (ie: a computer or any sufficiently powerful logic) is isomorphic to any other.

For bread and butter math, naive set is good enough, as long as you don't poke at it.

 

[HallsofIvy]

There is no such hierarchy. Yes, you can start with "sets" and build up mathematics from that. But you can also start with other things, such as, say, group theory, defining the objects in some abstract manner (when you have a set of objects with operations defined on them, it is always possible to define the operations first and then take the objects as "given" by the operations) and then derive set theory from that. There are simply too many interconnections between mathematics to set up any "natural" hierarchy.

 

[Nick R]

So basically, the only way a "bottom up" hierarchy exists is if you personally think there is an approach that is more "well motivated" by real/physical examples...

Is it sometimes true that, people concerned with "pure mathematics" may often choose an approach to things that is completely unmotivated by real examples, suggesting that this approach is representative of "what's really going on" even though the motivated approach and the unmotivated approach are logically identical, and the former just makes more sense?

 

[rasmhop]

So basically, the only way a "bottom up" hierarchy exists is if you personally think there is an approach that is more "well motivated" by real/physical examples...

People don't think of it as a bottom up hierarchy. If you've already shown that (1) and (2) are equivalent then you can just choose with whatever you find more convenient at the time. For many mathematical objects there are many definitions, and you just pick whatever suits your problem (assuming of course you have shown them identical).

Is it sometimes true that, people concerned with "pure mathematics" may often choose an approach to things that is completely unmotivated by real examples, suggesting that this approach is representative of "what's really going on" even though the motivated approach and the unmotivated approach are logically identical, and the former just makes more sense?

If they are logically equivalent, then the mathematician should be able to choose the one he prefers whether he gives "real world motivation" or not. If he prefer to work with the one you call unmotivated, then it becomes motivated by the fact that it's useful (the motivation is the usefulness rather than analogy). There is no need to give preferential treatment to one point of view. Intuition can be hard to explain, but if he gives vague statements like an approach representing what's really going on, then that probably just means that he has better intuition about this approach. This is enough motivation as it will allow him to more easily understand it (whether it comes from a physical object or not).

 

[Werg22]

There is no such hierarchy. Yes, you can start with "sets" and build up mathematics from that. But you can also start with other things, such as, say, group theory, defining the objects in some abstract manner (when you have a set of objects with operations defined on them, it is always possible to define the operations first and then take the objects as "given" by the operations) and then derive set theory from that. There are simply too many interconnections between mathematics to set up any "natural" hierarchy.

I don't know whether I agree. Yes you can definitely start with other fundamental objects and build set theory from that. But I thought it was the vast consensus in the mathematical community that modern day math should ultimately have ZFC as its starting point.

 

[HallsofIvy]

That is a convention, not a consensus on the structure of mathematics.

 

[Werg22]

That is a convention, not a consensus on the structure of mathematics.

Well yea, but math is man made anyways. The conventions define the subject.