Prologue

For the typical primary school or high school student, the following definition of mathematics would suffice:

(Naive definition) Starting from obviously true axioms, use obviously correct inference rules to derive additional truths.

In this sense, mathematics is all about discovering indisputable truths. For example, the theorems proved in geometry would be literally true statements about the physical space. Someone may argue that there is no such physical object as a geometrical point, or a geometrical line, but this is no issue because we can reply that geometrical objects are nothing more than locations in the physical space, and thus they can happily exist even if nobody can see them materialized. As for the exotic topic of complex numbers, they can be viewed as a man-made tool that sometimes comes in handy for mathematicians in describing reality.

Historically, this looked plausible until the early nineteenth century, when non-Euclidean geometries were invented. It wasn't generally believed any longer that Euclidean geometry was the true description of physical space. Curiously, it turned out to be impossible to decide which geometry, if any, was the true one. But then it raises the question: if mathematics does not necessarily describe reality, what does it describe then? After some meditation, we may adjust our previous definition as follows:

(Modern definition) Starting from obviously clear assumptions, use obviously correct inference rules to derive consequences.

Here we acknowledge that mathematics is all about discovering the logical consequences of given assumptions, where it is not required that the assumptions are actually true. It's comparable to making logical deductions based on the "facts" set out in a detective fiction.

Again, this looks plausible until we find out that different groups of mathematicians don't even agree on which logical rules should be permitted when deriving consequences. Most notably, certain groups don't accept the unconditional application of the law of excluded middle. This stems from their divergent views on the existence of mathematical objects.

What was told so far shows that defining mathematics is far more involved than expected. And we didn't even try to define the scope of mathematics, only its methodology was considered. To move things forward, the attempt to define mathematics must be accompanied by a better understanding of existing approaches to the foundations of mathematics.

One remark before we continue: in the definitions given above, "obviously true", "obviously correct", and "obviously clear" do not mean true, correct, and clear, respectively. Such a wording was used solely to indicate things that, for most people out there with a brain, would or used to appear as obvious. It is this entanglement with the obvious what makes mathematics possess the illusion of indisputability.