Mathematical Foundations

Throughout primary and high school, mathematics is studied as a series of somewhat unrelated ideas which are used to describe different aspects of the world. We learn the natural numbers to "quantify" things, fractions to describe what it means to "split up" an object into pieces, and calculus to describe "the continuum" on which motion exists.

Eventually a transition happens, often in the second or third year of an undergraduate degree, in which students are expected to present arguments with greater rigour. This is because in order to study mathematics at a high level of abstraction, such as group theory, topology, measure theory or category theory, one has to have a strong ability to manipulate mathematical objects based on only the abstract properties that have been specifically assigned to them. No longer can one reliably deduce facts about mathematics by pure intuition. This then allows studying mathematics in a more foundational way, which exposes connections between some of the fields of mathematics studied at high school.

An interested student may then pose the questions: How does the mathematics they have already studied thus far fit into a more rigorous and formal system? Can it be axiomatised in the same way that say, abstract algebra is?

Unfortunately, such a student is never really presented with an answer. In fact, in many occasions the professors teaching them couldn't give an answer when pressed. For some reason, we see it as acceptable that a student can graduate from an undergraduate degree in pure mathematics and never be taught the set theory axioms which underpin the mathematics they study.

The optimist in me hopes these kinds of things will change, and more time and attention will be taken to mathematics foundations both in teaching and in research. The pessimist in me however believes that to a certain extent, satisfying answers are not given to these questions because satisfying answers do not exist. Has anyone seriously studied the formal definition of the real numbers in terms of Cauchy sequences and thought "wow, this is such a beautiful and elegant way to model the continuum"? I doubt it, in fact, in most cases the definition is presented and then quickly followed by "let's not worry about the details" and properties of real numbers such as commutativity, the least upper bound property, and closure under certain roots are taken on faith. Or for instance in category theory, where the set of objects of a category is referred to as a "collection" without any rigorous explanation for what that means, and a handy wavy note about the fact that it can't be a set is made when a student goes to the effort of recreating Russel's paradox in this context (is it really that hard to just teach about classes properly?). If we do not study the formalisms in depth, how do we know that the most sensible choices have been made?

To be clear, I am not claiming here that the formalisations are wrong, or even that the axioms are poor choices, but rather than many mathematicians consider these decisions to be like an arbitration from God, which they cannot and will not doubt, and this stagnates our understanding of mathematics, especially when students are not being taught these formalisations.

While you may disagree with his approach, mathematicians like Norman Wildberger are often criticised for his objections to axioms that are taken for granted. I also have my objections and disagreements with Wildberger's positions, but I have more respect for him than (many... most?) mathematicians who are simply unwilling to pose such questions, or unwilling to answer them despite claiming to have a satisfactory answer. This kind of questioning of foundations leads to interesting and alternative ways of doing existing mathematics, which justifies its value.

It seems to my eye that the mathematical community needs a big splash of water to the face: FUNDAMENTALS MATTER! and not just in the "oh someone else has thought about this" kind of way. If you believe the foundations on which mathematics is based are strong and solid, then you should be able to answer basic questions about how those fundamentals work.