A traditional topic in philosophy is that of the nature of mathematical and deductive knowledge. For some philosophers, such as those of Cartesian Rationalism, mathematics and, more broadly, deductive knowledge is not only absolute to our minds, but also universal, and they point in the direction of mathematics being a very real (yet abstract) part of our Universe, the one we see and experience, the one we live in. To them, mathematical truths are universal truths, as they’re common to all observes and, according to them, a part of the universe as meaningful as everything else. Deductive knowledge is consistent, effective, and, when well constructed, irrefutable, thus for them it’s the best tool we have to describe our world.

On the other hand, some other philosophers such as empiricist David Hume, have argued that mathematics are purely a mathematical construct; sometimes even questioning whether logic is or not real, or merely a mechanism in our minds. In their words, it could be possible for an other intelligence out there to have an other form of mathematics or logic, and the only special thing about ours is that it’s universal to ourselves, our own mind mechanism, and we can construct no other, even if there can be. In some sense, they claim for a “bigger picture”, as sometimes said, in which our minds limit us to see only a part of it, in which only our form of mathematics makes sense, even if other forms of it are possible and or real, only not in our smaller reality.

This is just part of the much broader epistemological main debate, ongoing ever since Ancient Greece, and possibly beyond, and those are not even the only positionings, just the classical antagonists. Nevertheless, as a formally trained mathematician, this question never appeared mentioned in any of my notes, and was never dicussed in class or amongst students. In the eyes of mathematics students, maths just are, and none of them really cares about any epistemological implications, just for the same reason as why the rationalist point of view of the question has proven so popular and influential over time. Mathematical logic and results are extremely effective when applied in science to describe the world we live in, an excellent tool by which our scientifical models, especially in physics, can be consistently and precisely expressed, complementing empirical observations. Mathematical constructs prove consistent in all situations, irrefutable, which makes it not hard at all to wonder if they are a part of nature just as matter is, only in a more abstract way.

However, the fact that mathematics are not physical but a complete abstract concept makes attaching them to reality not so straightforward. In fact, when looking closely into what they are, maths being a part of reality becomes less and less convincing. For a start, mathematics is not about numbers nor the symbols through which we represent them. Instead, as I discussed in a previous article (see The Postulate Theory), mathematics is the result of combing a set of starting ideas (axioms and definitions), and a handful of rules on how to connect them, which we call mathematical logic. Everything, from the computations in a calculator to the complexity of algebraic topology, is based upon that principle, always with the same rules, and sometimes adding or dismissing some axioms or others. A good example to picture this is the story of Euclidean Geometry, Bolyái János, and how we got spherical and hyperbolic geometries, also discussed in said article. For this reason, out of construction, it is perfectly clear that mathematics are a product of the human mind, rather than a part of nature, regarless of how accurate it is in representing reality.

The question on whether logic is or not universal is way too large for our purposes here, and a question, if not for an other time, for somebody better knowing than me right now, but whether it is or not, mathematics fits better the description of a language than of a science. We start with a bunch of words (axioms and definitions) which we use a set of rules we call grammar (mathematical logic) to build consistent meaningful sentences (propositions and theorems) which we then can argue (prove), gradually becoming texts, books, or libraries. Not only that, but we use this language to express and communicate other sciences consistently and reliably with each other; in the case of mathematics, in the form of models, laws, formulas, relations, functions, and so on, which we see in all branches of science. In addition, as sciences are focused on studying and describing parts of nature, mathematics only studies and describes itself, as if it were the most selfish branch of knowledge of all.

Because of this analogy between mathematics and language, the dissociation between the Universe and maths becomes more and more clear, as it approaches more a human construct than anything else. In fact, mathematical resarch and deep studies seem completely unrelated to anything tangible at all. It’s all in the abstract, expressed solely by symbols and our imagination, which we can only share with ourselves explicitly, finding ourselves completely locked in communicating them through symbols between each other. Very often, problems and ideas begun in an attempt to describe a very tangible real problem, a practical necessity, and as the mathematical model became more and more refined, and was more and more generalised, it dettached of it’s origins, commonly reversing the matter in such a way that the original question becomes a consequence, rather than a cause, of the model. Trigonometry, for example, started as a model for the study of the geometry of angles and triangles, and now it’s defined abstractly, in such a way that it’s original properties are now consequences of the abstract, unintuitive definition.

That said, and due to our permanent power of choice in the starting point, mathematics seems to me more and more clearly a language by which we attempt to modelise what we experience in a logical manner, being logic the only set of rules that, so far, seem to be universal for all. It’s not that mathematics are part of nature because they describe it so elegantly, but instead that we descibe nature very precisely through their scope, just as language can, in our eyes, express transparently our experiences, views, and phenomena.