How the different mathematical topics are related to each other

Question

In my actual understanding of maths, each mathematical topic is built from a set of axioms. If two mathematical topics are built from different sets of axioms, how is it sometimes possible to use these two topics to build a theorem. For example, according to Wikipedia, the Pythagorean theorem is an Euclidean geometry theorem. However, the axioms of Euclidean geometry don’t provide any definition nor properties of equalities, so that, in my opinion, the equation $a^2 + b^2 = c^2$ is not interpretable with only geometry knowledges, so that we have to use algebra to be able to manipulate this expression. Another example is that I remember my calculus professor telling us that one of the theorems he was stating was derived from topology. How the use of multiple mathematical topics based on different axioms to build a theorem can be rigorously justified?

In my searches, I found that ZFC axioms can be used to derive most modern mathematical topics. In which sense this sentence is true? Can modern mathematics be derived from ZFC axioms and from the definitions of the mathematical objects that we are manipulating (lines, equations, functions, numbers, matrices, etc.)? Can most modern mathematics be expressed in ZFC-language?

Thank you in advance for your answers.

Answer 1

Yes, most math can be formulated in ZFC, including Euclidean geometry. First it must be shown that we can build within ZFC the set of real numbers $\mathbb{R}$, and then we can consider theorems about the plane $\mathbb{R}^2$. Then, we can show that the axioms of plane geometry hold in $\mathbb{R}^2$. But they become theorems of ZFC, and no longer axioms. Then, anything you can prove from those axioms will also be valid in the ZFC model of the plane. Thus, there's no need to combine the axioms from the different axiomatic systems - ZFC alone is sufficient. For other kinds of math you may need to add more axioms to ZFC, but for most math, including classical geometry, ZFC is good enough.

Answer 2

ZFC is not necessary to answer this question, and indeed may confuse what is an important and elementary question.

Let's ask the question "What is an axiom?". Historically, the answer was "something whose truth is self-evident as to make it not require proof". We try to keep these to a minimum, because there is great advantage in proving things. But, ultimately, all math and indeed all knowledge is founded on concepts whose clarity is self-evident but cannot be proven (see e.g. "epistemological morass" or What Achilles Said to The Tortoise by Lewis Carroll).

If we accept these axioms, we then accept any theorems that can be proven by them. We can then apply these anywhere: to physics, biology, and to other branches of mathematics.

Now, in the past hundred years, a purely formal approach to axioms has developed. But, this should not be viewed as redefining the notion of axiom to mean "an arbitrary concept which we have no reason to believe is true". Rather, formalism is a program to formalize a large number of axioms and show how they themselves can be viewed as theorems coming from a much smaller set of "base" axioms. Yet even formalism needs to start with an ultimate foundation of things accepted as self-evident by logic alone; otherwise, it will be tortoises all the way down.