There is this interesting stereotype that mathematicians deal with the cold, hard truth. Yet, even the most technical mathematics, deep down, is highly warm and playful. Most mathematicians long for the moment a proof finally clicks. This is something beyond its correctness: when it clicks, it also feels right and looks right. It usually comes with a sense of elegance or beauty, be it because of some surprising connection or for remarkable simplicity. It is common that in those cases one wonders—after the fact and forgetting the long struggle that led there—how come no one has seen it before.

You don’t need to be an expert mathematician to experience it. It is not that different from solving a complex riddle. It can also happen at school. I still recall my first encounter with this sensation. Our mathematics teacher in the first year of secondary school made us construct number sets and discover their algebraic properties in a rather hands-on way, starting from the Naturals, via Peano axioms. It was like a drug, the pleasure of all these simple bits miraculously working together to produce these complex structures. It felt alive and at the time was rather crazy to believe that things that seemed so complex were so naturally emerging from such simple and clear rules. It hooked me, leading me to explore cellular automata and fractals and so many fascinating things, things that even a kid can get into at a superficial but very hands-on level.

But if one never had the opportunity to experience it, it is really hard to explain or show. I would push it as far as to say that it is like certain forms of art. It takes a trained eye to recognize it, it can feel elitist for this, and it surely has an enormous subjective component, with different people arguing for their different, personal aesthetic.

Some push it even further

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. – G.H Hardy, A Mathematician’s Apology (1941)

It is an interesting reflection to have, especially in this moment where utility has become the prime driver of scientific research and educational institutions have become basically education industries.

Coming back to this post. This whole rant was prompted by an essay in AEON, How should we define mathematical beauty? I think it is well worth reading. In particular, the following quote is what set this flow of consciousness in motion

Do you consider this a beautiful proof? Over the past few months, I have asked many senior mathematicians the very same question, and whether they have certain criteria or a rule-book for deciding and discovering beautiful mathematical structures. The answers were not straightforward and were surprisingly distinct. But most mentioned that a beautiful piece of mathematics should be simple. ‘What do you mean by simple?’ I asked many times, and that was one point of divergence. Simplicity does not evoke the same meaning for mathematicians, even those who work in the same field. Some mean the number of lines in a proof; some mean its self-sufficiency, such as requiring few citations to other lemmas; some mean the simplicity of the proof idea or even of the theorem statement.

Despite the long conversation and the many reads to prepare for the Good Mathematics Problem episode of It’s Not Just Numbers, I am still longing for clarity in my thoughts. And while this discussion may seem irrelevant to most, I came to believe that it is a deeply central reflection on the state of our discipline.

The development of mathematics and the culture around it are both deeply social activities, and the meaning of truth as well as what makes problems interesting and results beautiful are an important and inseparable underlying drive. As it is social and taste-driven, mathematics is constantly changing in relation to changes in its surrounding environment, and it can be easily influenced by current trends—and hopefully also influences them in a positive (whatever this means) way.

I find this to be the key to understand the societal usefulness of mathematics, more than just its (arguably very numerous) applications.

If beauty is this subjective and taste-driven, though, it is hard to avoid a deeper question: what is mathematics, really? Is it a body of eternal truths we discover, or something we collectively construct?

There’s a famous anecdote about Srinivasa Ramanujan who claimed that his insights came to him in dreams from the goddess Namagiri. Whether or not you believe in divine inspiration, there’s something poetic about the idea that mathematics is not just a product of the mind, but some sort of spiritual connection to a divine perfect world.

And indeed, mathematics can be perceived as a realm of eternal, immutable and perfect truths: Plato’s hyperuranion, the world of perfect forms, where numbers and shapes exist independently of human thought.

This is one of many views though. For another perspective, I have always been a huge proponent of exploring mathematics not as a discipline but as a language, and recently found out that I am not alone.

Like light behaves both as a wave and as a particle, mathematics could be both the Platonic idea and something far more human: a language we invented to make sense of the world, shaped by our needs, our intuitions, and our desire for beauty.

Quoting its abstract, Jeremy Avigad’s essay Mathematics and language

proposes that we view mathematics as a system of conventions and norms that is designed to help us make sense of the world and reason efficiently.

From this perspective, mathematical truth isn’t an empirical fact we trip over in the woods; it’s an artifact of a language we have collectively engineered to help us deliberate reliably with one another. Truth, in this sense, has the “dignity of a rule” (Wittgenstein).

I needed quite some time to understand what this even means, but I think it is a way to say that it functions as a transcendental norm: not something we discover within experience but something that structures the very possibility of there being experience at all.

I found this as interesting as it is confusing: mathematics as a human language that is not discovered but that itself allows to experience a certain type of reality and discovery. Feels tautological, I’d really like the help of a philosopher here.

This also connects to something I mentioned above: I don’t think the evolution of mathematics was inevitable. I would argue that it was (and is) driven by human curiosity and the pursuit of deeper understanding, sometimes necessity.

Seen in this way, mathematics becomes a deeply human creation. Then its truths may not be eternal in the Platonic sense. More likely they would be contingent: shaped by the axioms we choose, the problems we consider important and the beauty we find along the way. This doesn’t make them any less true, at least in a “logical” sense, but it does make them a lot more human. And does raise the question: What is truth?

A question we touched upon in Degrees of Freedom, but on which I have even less of a clear philosophical grounding. However, this is steering the discussion too much from my original intent. I’ll get back to it at some other point in the future.

The other half of the reason this post came to be, is the rabbit hole I fell into when I discovered Avigad’s adventures in the philosophy of mathematics, particularly in relation to formal proofs and computer (or LLM) assisted mathematics.

Leaving the meaning of truth unanswered, we can still try to reason about its role, especially in relation to beauty. If we agree with Avigad that truth is a statement in a linguistic framework we use to communicate, then simply knowing that a statement is true cannot be the ultimate goal.

In Varieties of mathematical understanding, Avigad refers to a comment he received that directly addresses this fact:

[the goal of modern mathematics is] to get a conceptual and rigorous understanding of which mathematical statements are true. Both adjectives are important.

When a computer churns out a massive, brute-force calculation to prove a theorem, some people instinctively feel cheated and complain that

there is nothing to understand here, it’s only a computation.

While I may partly disagree on this view, I do share that view that we don’t just want the answer, we want the ’lightbulb’ moment. Avigad quotes the Xena project here, where it is argued that

Ideas are the purely artistic part of mathematics.

Maybe I am wrong, but it seems to me that this craving for conceptual understanding comes from the fact that such understanding is what can be shared, taught, and importantly debated. This, at least to me, is the best part of our job. In this respect, a massive computation is a black box but a beautiful idea can be a bridge between people: it is conceptual understanding that ends up revealing connections, unifying disparate fields, and opening up new avenues of research. Insight and feeling play a way more central role than truth in getting to it.

This does not even try to touch upon another important part of what mathematics is about, which is asking the “right” questions… Again, with a big asterisk on what “right” means. This slowly but steadily brings us closer to the final point I want to discuss. What is the role of automation in mathematics?

The advent of computers could have been a threat to mathematics, but it ended up mostly being an ally. Allowing easier typesetting, easier long-distance collaborations, easier ways to share our ideas and to connect with each other, and powerful means of computation. It is enough to think about the use of Mathematica or python in aiding symbolic computations and the enormous impact of numerical methods in our day to day lives. Algorithms may be deemed a computer-sciencey thing, but integrating algebraic, differential, or difference equations sits at the root of many technologies we use every day, from the GPS in our phones to the weather forecast.

I am ready to bet that many people felt excited, many depressed, and even more oscillated in between, much like today. Now, large corporations and smaller venture-capital funded startups are trying to leverage the complexity of mathematical thinking and the power of proof-assistants like lean or rocq, to “solve mathematics” and show how good they are. After all, if they can achieve such a feat, (at least) their perceived value will be enormous.

As I wrote multiple times in my recent posts, I struggle to be so negative. I hope that the overuse of AI will be scaled down, that models will keep getting smaller and more specialized, and that there will be more investment in open infrastructures. So that we will have excellent mathematical tools that are cheap to run instead of massive models in polluting datacenters that are only accessible by the elite. It will be swimming against the tide of the current capitalist movement and there is no way to predict what will happen really. I think the way to move forward, though, at least in our mathematical niche, is in the hands of the mathematics community. I honestly believe that it is within our collective reach. And it is now the time to not be passive and agree on our common values and ambitions to steer the discussion in a positive direction.

So, as (again!) Avigad was asking in a recent essay, is mathematics obsolete? I would say far from it, but like everything in life, it will probably have to evolve. Like it evolved many times in the past 2000+ years, in many different ways. We just need to be active to work together to steer where and how.

I think the advent of mechanized mathematics, with the increased energy in automatic proofs, especially in lean, is a huge opportunity for new forms of collaboration. Like many have been showing so far. I also tried to argue that the human factor in these developments is unavoidable and central, as figuring out what is interesting and what is meaningful is a very social and human activity. I really hope I am right, and I am more and more convinced that I am not alone in thinking this way.

In mathematics in the age of AI, Avigad seems to head in a similar direction, perhaps with more enthusiasm for embracing AI than I have here. I think such tension is itself part of the conversation that would be worth having. And a shared collective dialogue, like the one that has been growing recently, is exactly how I think we should figure out where we want to head collectively.

Mathematics is a language, a tool, and an art. In all cases, it is a way of seeing the world (real or of ideas), of making sense of its patterns, and of finding beauty in its structure. And while we may need to adapt how we do mathematics to a changing world, this won’t change why we do it. Am I scared about change? Maybe. But I am way more curious to see how and energetic to try and steer it in a positive direction.