I recently had a discussion around why only certain concepts are considered “mathematical”. For example, why is “number” considered mathematical and “color” (in the everyday sense we understand it), not? A mathematician friend shared the very insightful idea that, at the foundation of our mathematics, we require formalisms that are intertwined with concept. It led me to question some very fundamental ideas about mathematics.
First, I thought, what is a concept? A rough definition might be that it is an idea about the essential quality of an object or set of objects. For example, consider a bowl of fruit. This is a collection of objects that has a number of essential qualities that we might agree on: color, ripeness, shape. It also might have qualities that are considered more subjective, such as how appetizing the fruit is. One thing we are very likely to agree on, though, is how many fruit are in the bowl. Here we perceive a fundamental mathematical concept - that of “number.” And this concept is not unique to this bowl of fruit: we might find that the number of fruit in the bowl is coincidentally the same as the number of coins we have in our pocket, in which case these two radically different collections, otherwise incomparable, contain this shared quality, which is quite remarkable.
This is very interesting to think about in its own right. Not only is the “number” shared across different collections, but it seems almost everyone will agree about this quality. It also requires intermediary concepts that are even more universal, like “collection” and “object” and “quantity.” The fact that we all perceive these concepts reveals something deeply shared by us as humans and how we perceive the world.
But concept alone is not enough for mathematics. To begin with, in order to communicate this idea to others, or to arrange it among other ideas in our mind, we must label the concept. So we must associate it with a symbol, for example the word, “number” or “color.” But even a label is not enough for mathematics, it seems - you and I may agree that certain objects have the quality “green” and even may make statements about “green” that we agree upon, such as the fact that green has an earthy quality. But we cannot build a system of mathematics around this, at least in the way we understand modern mathematics.
Why not? Firstly, mathematics is a science and science must be rigorous. And perhaps because mathematics deals with concept and not the physical, it is extremely subtle and requires the utmost rigor. One cannot get very far in mathematics without being precise - for example, I can have the concepts of “round things” and “pointy things” but I am very limited in the statements I can make from these. It is a very crude mathematics if one wants to consider it mathematics at all.
But if I distill these ideas into the more precise formulations of “circle” and “triangle,” I have a much stronger foundation upon which I can build a powerful mathematical system. Or perhaps another analogy is that precise definitions allow us to inspect and traverse the nuanced world of concept at a much more detailed level. Indeed, because mathematics deals with the conceptual, one requires and can achieve a level of precision that cannot be found in the physical world.
So, in order to get anywhere in mathematics, we require formalizations, which is essentially the result of distilling ideas to a very precise conceptual essence. This is not to say that all of mathematics is formal - in fact the process of mathematics is very creative, informal, and experiential, and often devoid of concept or labeling. But in order to progress in mathematics, individually or collaboratively, the result of our work must be formalized.