I always been a lover of ‘math for the sake of math’ and I found annoying to ask ‘but what is this useful for?’ when learning about a new concept.
It might seem weird, but to a pure mathematician ‘apply’ sounds like ‘spoil’. Applications are a kind of low rank pursue for a mathematician, something ‘easy’ and very much unexciting. I happily embarked this line of reasoning very early in my studies, giggling about the superb degree of purity my career was going to have.
That’s also something that the general public, the muggles, seem to get. Mathematics is all about abstraction, and abstraction means getting far from reality. The more abstract we mathematicians soar, the more we enjoy it, refreshing ourselves with pristine, unspoilt Platonic ideas.
The starting point of this reflection is the existential question of math, that is: is this of any utility whatsoever?
It’s actually a causality issue: if we stopped pursuing very abstract and very theoretical mathematics, will we really miss any practical application? When is abstract too much?
I often struggled to find a single example of application for more math than not. And even if the charm of it still has a big effect on me, it started to be not enough. Of course you start to deal with these questions when you get in touch with really whimsical stuff like ‘pointless spaces’, whose very name hints to something really difficult to apply to anything whatsoever.
So, why all the fuss?
I guess the answer lies more in the form than in the substance. A good example to illustrate my point comes from category theory. It was conceived as a good taxonomical tool for algebraic discourse, to formalize general abstract nonsense. Yet, it turns out, categorical concepts pop out everywhere. ‘Adjoints are everywhere’ said someone. Would you ever see them if no one ever defined what a functor is?
In the same fashion, a big chunk of mathematics might be justified just by appealing to its form. Topology is useful because topological concepts are indeed ubiquitous in other mathematical tools. Brouwer’s fixed point theorem makes a lot of sense when stated about morphism of topological spaces, and when proved using the classical algebraic topology argument. Can you wonder how quirky would it sound if stated without any reference to topology?
This made-up example is actually what happened with Abel-Ruffini’s theorem: Ruffini concocted an unbearably long proof about the unsolvability of quintics (so long, almost no one was brave enough to read it all). Fast forward less than 50 years later, and Abel’s proof is neat and short: why? Because it used powerful concepts from the new-born science of abstract algebra, which made much more evident what all the question was about: the structure of the symmetric groups .
The moral is, a good part of math is simply there to make other chunks look reasonable . Category theory is the royal example: as put by Freyd, ‘the purpose of category theory is to show that what is trivial is trivially trivial’.
This insight leads us to a much deeper one, that mathematics is actually a language. What I mean by language is a set of symbols and rules on how to assemble them to convey meaningful messages.
Clearly mathematics has a language , yet I’m arguing here mathematics is itself a language.
The main observation is that mathematics is highly hierarchical and fractal-like. Higher mathematics is of course ‘made of’ lower mathematics (e.g. you need linear algebra to grasp tensor algebra), but at the same time any significantly developed mathematical theory finds itself mirrored in some other, either completely and rigorously so or just partially (e.g. the duality between geometry and algebra). Undoubtedly, finding similarities between different areas of mathematics is considered as an highly desirable, elegant and fruitful achievement .
The symbols of mathematics are its own concepts, which should be intended in a broadly and elastic sense: ‘group’ is a concept, and so is the subject of topology as a whole. A better word is ideas: groups embody the idea of modeling symmetries algebraically, while topology is the idea of studying a space by defining what is ‘near’ to a given point . Theories are ideas, too: e.g., Morse theory is the idea that singular points of a manifold must tell something about its topology.
The rules of the language of mathematics are simply any meaningful way to put together mathematical ideas. This is too quite blurry, so let’s make some examples: singular homology theory is a composed idea, which is made from the idea of probing a (topological) space with maps from simplices and the idea of building an algebraic gadget out of this process. Both ideas can be generalized separately, respectively to get fundamental groups (we study maps from spheres) and homology theory (we study the same algebraic idea applied to different constructions, e.g. cubes).
Of course homology theory is also an idea itself. This the power of mathematics as a language: any composed idea can become itself a ‘simple’ idea upon which we can build more complex ideas, and so on. I believe this explains both why abstraction is so powerful and how mathematicians can work on increasingly advanced topics as easily (or as hardly) as an undergraduate works on linear algebra: both are just surfing the wave of recursive complexity.
Until this point, I seem to have described not mathematics but a wider generalization of it: thought. We need to ensure our language is tied down to a formal, rigorous system (or deontology), so that a ‘successful’ idea is one which can be morphed into a provable statement, or at least to a statement we can judge logically. So this distinguishes the idea of considering the zeroes of Riemann zeta function as eigenvalues of a suitable (self-adjoint) Hamiltonian and actually proving the Riemann Hypothesis.
This view, moreover, explains my previous claim that some math is just about math, just as some parts of English are just about grammar (like the word ‘grammar’ itself). It does say something about why it seems so abstract, too: its composition rules produce the fractal structure of the mathematical edifice, thus moving quickly into ever involved concepts and long chains of generalizations. Mathematics has the ability to summon a whole theory by just observing a particular property in an objects: algebraic sets satisfies the properties of a lattice of closed sets? Behold as topology rushes in! Suddenly, you’re speaking about compactness and separability in a context which was mostly ‘polynomials and ring algebra’.
To draw a fictional comparison, picture a (spoken) language in which entire debates are condensed in a single word, and then proceed to be used in new debates. Clearly meanings add up and you start to feel dizzy as a ten-words conversation spirals out in a twenty-volumes reference to previous discussions. In some sense we do this in everyday language, but in a lot less meticulous way as in math: nobody (actively) discusses the validity of Euclid’s fifth axiom anymore while the same can’t be said about communist theories (notice both where ‘clarified’ around the same time!). In a sense, the rigour imposed upon mathematical ideas makes the whole edifice solid and trustworthy. This is a luxury not even hard sciences have.
This makes mathematics extremely unwordly, because it is various strata of meaning above ‘real stuff’, yet phenomenally powerful. Mathematicians routinely handle behemoth ideas by hiding it under an even more gargantuan pile of, let’s face it, abstraction. This can be exploited to reflect a similar feature of reality: things are simple in theory, not so much in practice.
This is something that is not extraneous to science, intended as the human endevour of modeling reality with math: by its very definition, scientists never argue to have a perfect model of reality, just a working, ‘good enough’ one. As science progresses, so do the accuracy of models. And we can only do this by building up on previous models, using smaller and smaller discrepancies from the old ones to guide the introduction of a new one. Naturally, models tend to get less straightforward with each iteration, as to capture a phenomenon more faithfully you’ll need to consider more complex interactions, higher order effects, and nitty-gritty details. Hence to handle complex situations, we need to be able to work with complex theories.
Wrapping up then, yes, we’ll miss a lot if we stop pursuing pure math. The feeling of dissatisfaction with more and more abstract math is a symptom of something else: calling applied math sour grapes. Instead of facing the daunting task of modeling complex tasks, we prefer to turn around and pretend applied math is some trivial and inferior endeavour.
 Another moral is that concepts in mathematics can’t be thought, yet alone taught, as independent chunks. They need to be properly motivated (and historical background is great at this), and inserted in their rightful position in the mathematical edifice.
 Math has languages on formal and informal levels. On one hand, every mathematical proposition can be regarded as written down in a formal system of some sort; while on the other hand, mathematicians use a common language made up of naming conventions, common notations, canonical subdivisions of disciplines, and a very distinctive prosaic style.
 We could go as far as saying any mathematical progress could be decomposed in a vertical component (‘going deeper’ into the subject or ‘building higher’) and an horizontal one, linking the subject to other areas. This picture fits nicely with the informal entity of the ‘mathematical edifice’.
 I’m referring here to the definition of a topology using neighbourhoods. Other definitions also embody specific ideas about which aspect of ‘being a (topological) space’ should be fundamental. The very fact we have strikingly different yet equivalent definitions is highly interesting, and makes topology a useful and strong theory. In the fractal analogy, topology exhibits a lot of self-similarity.