The natural structure of mathematics

[DavidLaPierre]

As I understand it, mathematics is a study that is founded on the principles of logic and set theory and branches out into specialized fields. I was just wondering about what that structure would look like. It seems like there might be significantly different ways to do this since there is often more that one way to prove something. How do you see this?
Here's a rough idea of what I mean. This isn't meant to be complete, and definitely isn't totally accurate. But, it's 5 am here and I need to get some sleep
Thanks for the input everyone.


I. Fundamentals
a. Logic
b. Set Theory
II. Arithmetic
a. Addition
b. Subtraction
c. Multiplication
d. Division
e. Exponentiation
III. Numbers
a. Integers & Number Theory
b. Rational Numbers
c. Irrational Numbers
d. Transcendent Numbers
e. Imaginary Numbers
f. Complex Numbers
IV. Geometry & Trigonometry
a. Two Dimension
i. Flat Space
ii. Curved Space
b. Higher Dimensions
V. Calculus
a. Differentiation
b. Integration

[delooper]

Various types of set theories can provide a foundation for mathematics but there are also people who prefer alternative foundations so I don't think it's right to say mathematics is founded on the principles of logic and set theory. More than anything I think foundations are largely useful as pedagogical devices, as they make learning a subject easier, as they're more like little machines that you get to open and inspect, part-by-part.

If anything, I'd turn what you're trying to do on its head. I'd say mathematics is the "mental reasoning space" that's useful for finding answers to practical problems. Sometimes the problems we deal with in reality are beyond the everyday scales or modes where simple guess-work is practical. So we need things that are more well-developed and systematically thought-out. That's where math starts.

So the structure of mathematics very much looks like the structure of human thoughts about abstract problems, tending towards the more objective (although not always quantifiable) variety.

[fishfry]

Here's Dave Rusin's excellent clickable index of mathematics. It has the entire subject organized by area, with detailed writeups and references about every major branch.

http://www.math.niu.edu/~rusin/known-ma ... thmap.html

[z4lis]

I'm not certain that subdividing mathematics into a hierarchy of more and more advanced material, each thing depending on the last thing, branching out into little independent trees, is the best way to view the subject. While we've already labelled and divided knowledge into various fields (algebra, geometry, analysis, etc.) there's not a clear cut way of figuring out what depends on what. It's an interconnected and organic mesh of ideas whose "proper order" is, as someone already pointed out, really of a pedagogical nature. And many of the little branches reconnect back together. For instance, someone studying Lie groups needs to know a good deal of geometry, analysis, and abstract algebra. Homology theory demands both algebra and topology. At the moment, I'm reading a book on probability on trees and networks that requires I know analysis, group theory, and geometry. Using your list as an example, basic multivariable calculus requires some geometry, but talking about curved spaces and the like requires calculus. Which way should the arrows run? We could subdivide the subjects into smaller and small chunks of material, but how close to just having the "parts" of our tree just be individual theorems are we willing to go? And how do we choose which theorems to connect to which theorems? Most have multiple proofs!

Just some things to think about.

[Desiato]

Various types of set theories can provide a foundation for mathematics but there are also people who prefer alternative foundations so I don't think it's right to say mathematics is founded on the principles of logic and set theory. More than anything I think foundations are largely useful as pedagogical devices, as they make learning a subject easier, as they're more like little machines that you get to open and inspect, part-by-part.

Sadly and frankly, that's not true. I dare say that anyone that tries starting to learn mathematics with rigorous foundations is doomed to not get anywhere.

Note that what most mathematicians would consider among "foundations" is set and model theory, and that's a vast and extremely abstract field that is pretty much impossible to study without a more or less solid idea how math works before even starting to learn it.

In fact, I know several mathematicians with a master's degree (well, technically, an older German semi-equivalent called "Diplom") who have no clue about this topic that goes beyond "AoC is scary".

[delooper]

Sadly and frankly, that's not true. I dare say that anyone that tries starting to learn mathematics with rigorous foundations is doomed to not get anywhere.

It's odd you would say that because most of the non-US trained mathematicians I know very much learn mathematics from this bottom-up foundational approach. Some aspects are learned non-linearly, of course. But a set-theoretic introduction to analysis tends to happen in many Canadian universities, for example. Take Alberta, U. Calgary, Waterloo, Toronto, as examples. It's popular some places in the States as well -- the textbook by Apostol was used at UCLA for some time, for example. And you do get places quite quickly this way.

Note that what most mathematicians would consider among "foundations" is set and model theory, and that's a vast and extremely abstract field that is pretty much impossible to study without a more or less solid idea how math works before even starting to learn it.

In fact, I know several mathematicians with a master's degree (well, technically, an older German semi-equivalent called "Diplom") who have no clue about this topic that goes beyond "AoC is scary".

What is interpreted as "foundational" is really a fairly broad ill-defined part of mathematics in a sense. For many people, just having a set-theoretic definition of the real numbers, together with proofs of all their main properties is a standard "foundation". For that, books like Apostol suffice.

[capefeather]

I certainly did not start with foundational set theory... However, my first analysis class did start with taking about a dozen given properties of numbers and building the calculus from that. My first algebra class was similar, taking the eight field axioms and deriving properties of fields and linear maps and such. I'm guessing this is what delooper is talking about. Then again, most math classes pretty much start out that way.

[fishfry]

My sense is that the vast majority of working mathematicians don't care about set theory. If ZFC turned out to be inconsistent, they'd just keep on doing their work, secure in the knowledge that eventually someone will patch up the foundations.

Not only that, but these days there are people actively working on alternative foundations. Category theory is one approach. Complexity theory is another. Who knows what the foundations of math will be 100 or 200 years from now.

Newton invented calculus two hundred years before Cantor discovered that there are uncountable sets. These days we think of real analysis as logically depending on set theory; but it doesn't have to be. In the future it might not be.

But even if you don't believe that; what's unarguably true is that a lot of working mathematicians simply don't give the whole issue any thought. They're working in very specialized fields. They are working on reading or writing specialized papers. They don't think about foundations at all.