There are several concepts of beauty in mathematics. One which many, including mathematicians, subscribe to is the lazy tautological notion. The astonishment about symmetries in nature and connecting that to group theory, even if this is primarily a circular ascription. The excitement about knots in a tile or any geometrical representation of abstract concepts. The worst is forcing the Fibonacci sequence into any spiral object. Equally bad, making a case for Pollock paintings through fractals, as if there is some divine concept behind dripping paints on a canvas.
No matter how much some of these wrong ideas, i.e. Fibonacci for instance, have been debunked, there is still a strong urge to hold onto them, maybe to feel a connection between tangible and intangible or a connection with non-mathematicians or maybe in principle one knows the process of creating mathematical structures is far beyond only truth evaluation. For any reason, strong sentiment reveals a deeper desire. I call this type of association both the first group of correct examples and second group of wrong examples intentionally lazy, because it hinders a profound aesthetic practice mathematics is concerned with and conscious mathematicians are aware of; i.e. the abstract practice of value assignment to mathematical structures beyond truth value, or in Poincaré’s words, the self-suffering “intellectual beauty”, the one that is worth all painful labours.
Are values only limited to aesthetic ones? Although that is an important question and clarification needs to be made, here I attempt to make a case for its “existence”, that “there is” an element of “taste” behind the scenes that does not appear in the discussions of AI maths, and it has been an impactful factor in shaping mathematics.
We insist on not claiming mathematicians are only and only bearers of truth. As a social experiment, ask any mathematician (excluding logicians) to describe the law of excluded middle or the importance of a choice function or describe even casually axiomatic set theory. In another experiment, ask those who use proof assistants or researchers working on AI maths to describe the conditions under which the Curry-Howard correspondence is defined. In both experiments, you do not hear immediate correct answers.
What do they mean by “truth” then? They predominantly mean there is a verified construction: either mathematics literature or software libraries, and they aim to work within those true statements or packages and produce truth values within this construct. They hardly think from first principles even when formalising a proof.
Paradoxically, maybe much of their progress, creativity, and insights come from shortcutting and ignoring first principles. Years of training, contemplating, succeeding, failing and reading mathematics, give one an eye for detecting problems which they consider worthwhile and even sometimes wrong. We all have seen at least an incident where an established mathematician looks at a proof and with an immediate visceral reaction or even disgust says: “this does not look right”, and sometimes it turns out to be a correct guess. (Of course, they are wrong too, but having a wrong unverified guess is not surprising, the reverse is, and it is not always a 50/50 intuitive guess.)
We have heard words such as “elegant” or “simple” or “style” or “dry” or “compact” or “ugly” in the literature. These terms are the judgement layer after truth evaluation, and that is what precisely distinguishes mathematicians. Give two proofs of the same theorem to one, and ask them to pick a proof, and keep questioning them about their choice; they might first declare that they choose one over another because it is useful or because they need the technique; and if you continue questioning them, you hear words such as “simple”, or “elegant” or “compact” or “beautiful”. In their selection, they consider other factors apart from “correctness” and even “utility”, they choose based on their taste; and maybe good mathematicians are those with refined tastes.
What do they consider “elegant”? What do they mean by “simple”? I do not intend to give an exhaustive list or to explore their meanings. First, I do not think it is straightforward and second, I am only establishing the case for its “existence”. But take “simplicity”, which many mention as the reason behind their choice (Quine, for instance, considered it as the main virtue for deciding between competing theories or proofs). But on the definition of simplicity, they diverge. Some mean the number of lines in a proof, some mean the simplicity of the mathematical argument, fewer citations to other lemmas, some mean the statement of the proof, and some equate simplicity with a notion of cost efficiency or computational complexity. But they all have a notion of “simplicity” despite different definitions.
I can clearly see reactions to the essay at this stage with hand-wavy gestures claiming that AI agents will eventually develop a taste, and it will happen soon. I do not object to that, and I do not think humans develop their taste because of the involvement of some divinity, but if one goes beyond handwavy arguments, one should answer these questions: Why do mathematicians develop different styles and different tastes? Why does one prove an open conjecture and another one come up with new foundational structures? Why do they ask different questions along the way of proving the same theorem? Why are some styles more elegant and compact than others? Most importantly, what do they mean by “elegant”, “beautiful”, “simple” and other values or virtues: aesthetic or else?
These are important questions that need serious intellectual reexamination and attention. To answer them, one needs to first admit that “there is” an element of value judgement beyond truth, and “taste” plays a role, and in Putnam’s words, and to deny such is intellectual dishonesty.
P.S.
[1] I use “virtue” and “value” liberally and synonymously. For the purpose of establishing “existence”, I do not make any distinction as the attempt to do so made the essay unnecessarily long.
[2] No use of LLM. Only books, online dictionaries and papers.
[3] More on aesthetic judgement here.
Received 10 August 2026, revised 2 October 2026.
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