In this blog post, Matthew Inglis reflects on the role of philosophy in mathematical cognition, and discusses a study which explored mathematical aesthetics. Specifically, Matthew asks whether appreciating the beauty of a mathematical proof requires, or is defined by, fully understanding it.

This blog post relates to a special issue in the Journal of Mathematical Behaviour (The Journal of Mathematical Behavior | Mathematics Education meets the Philosophy of Mathematical Practice | ScienceDirect.com by Elsevier), and was edited by Jo Eaves.

Philosophy and Mathematical Cognition

Cognitive science emerged in the mid-20th century as a response to the perceived limitations of behaviourist approaches to psychology. From its early days, the field was an explicitly interdisciplinary endeavour. Howard Gardner famously depicted “the cognitive sciences” as nodes on a hexagon: psychology, artificial intelligence, neuroscience, anthropology, linguistics and philosophy. The logo of the Cognitive Science Society remains a hexagon to this day. Their website lists the same six foundational disciplines as Gardner did, but adds a seventh, education, to the list.

So the early cognitive scientists regarded philosophy as a core component of cognitive science. It played a foundational role by framing important questions about representation, computation, and intentionality, and by shaping theoretical models of cognition. But what role does philosophy play in the branch of cognitive science focused on mathematics, mathematical cognition?

Not much. Few philosophers attend MCLS conferences or publish in the Journal of Numerical Cognition. And, of the 2786 articles in the Web of Science database which have either “numerical cognition” or “mathematical cognition” in their abstracts, just 121 (4.3%) are from the philosophy category.

A couple of years ago the Centre for Mathematical Cognition at Loughborough hosted a workshop designed to explore this situation. The event led to a special issue of the Journal of Mathematical Behaviour, which has recently been published. The issue, entitled “Mathematics Education meets the Philosophy of Mathematical Practice”, explores the intersection between two fields that have historically operated in parallel. The collection of papers brings together research from mathematics education and philosophy of mathematical practice, and attempts to demonstrate how each can inform the other. In this blog post, I discuss the contribution that my colleague George Kinnear and I made to the issue, with the hope that it demonstrates how the empirical methods of mathematical cognition can produce findings that shed light on philosophical questions. The original article can be found here.

Mathematical Aesthetics

Many mathematicians say they know a beautiful proof when they see one: Henri Poincaré famously described mathematical beauty as a “real aesthetic feeling” recognised by “all true mathematicians”. Mathematicians routinely describe proofs as elegant, inspired or profound, and such judgements shape what is valued, remembered and pursued. Education researchers have also argued that aesthetic experiences can motivate learners, support engagement, and help students to see mathematics as meaningful rather than mechanical.

However, some philosophers of mathematics have questioned whether aesthetic judgements in mathematics are truly aesthetic. One influential view, associated with Gian-Carlo Rota, suggests that calling a proof “beautiful” is really a way of talking about understanding. On this account, beauty is shorthand for enlightenment. A mathematician who claims that a proof is beautiful, according to Rota, is actually saying that they understand the proof: that they can see how it fits into the wider mathematical landscape.

If that view is right, then probably mathematics educators shouldn’t worry about helping students to appreciate mathematical aesthetics. Focusing on developing a deep understanding would be enough, as aesthetic appreciation would simply follow.

Can empirical work shed light on this issue? Specifically, was Rota correct to suggest that when we ask questions about aesthetics in mathematical contexts we get answers that in fact relate to understanding?

George Kinnear and I conducted a study investigating this question, which was recently published in the special issue of the Journal of Mathematical Behavior. We asked second-year undergraduate mathematics students to read a well-known proof: Cantor’s diagonal argument showing that the real numbers between 0 and 1 cannot be listed in a sequence. This proof is often described as a classic example of mathematical beauty.

After reading it, students completed two kinds of tasks. First, they rated the proof using descriptive words. Some words captured aesthetics, such as ‘striking’ or ‘profound’. Others related to different qualities, such as precision or usefulness. Second, we measured their understanding in three different ways:

• Self-reported understanding, where students said how well they thought they understood the proof.
• A comprehension test, with questions checking knowledge of definitions, steps, and applications of the proof’s ideas.
• A proof summary task, where students wrote a short explanation of the key ideas, later judged by experts.

This combination allowed us to compare how aesthetic judgements relate to both perceived understanding and performance-based measures. The main result was clear. Students who said they understood the proof better were more likely to judge it as being aesthetically appealing. However, students who actually demonstrated better understanding on the test or summary task were not. In other words, feeling that you understand a proof and finding it beautiful go together to some extent. Actually understanding it, as judged by performance, is unrelated to whether you think it’s beautiful.

The performance-based measures of understanding didn’t seem to be invalid. They were related to each other and to students’ wider results in their studies. Understanding was being measured in a serious way: it just was not what drove students’ aesthetic appraisals.

This finding challenges the idea that aesthetic judgements in mathematics are merely disguised assessments of understanding. If that were the case, we would expect strong links between aesthetic ratings and all our measures of understanding, not just the self-reports. Instead, the results suggest that aesthetic appraisal is a distinct psychological response. It is connected to understanding, but not reducible to it. A proof can feel inspired or striking even when its details are only partially grasped.

This aligns with broader research showing that self-assessments of learning are influenced by emotion, confidence, and general impressions, not just by what has been learned. A proof that feels pleasing or impressive may create a sense of understanding, whether or not that sense is accurate.

What does this mean for the field of mathematical cognition?

Should mathematical cognition research concern itself with topics, such as mathematical aesthetics, more traditionally associated with the philosophy of mathematics? I think so. The goal of mathematical cognition is to understand how humans comprehend and use mathematics. Many branches of the philosophy of mathematics touch on these issues: epistemologists ask how humans can come to know mathematical ideas, philosophers of mathematical practice interrogate how mathematicians work, and how mathematical cultures develop. All these topics should be of interest to the field of mathematical cognition. I hope that the Journal of Mathematical Behavior special issue demonstrates the value of regarding philosophy as a core part of mathematical cognition research.

About the Author

Matthew Inglis is Professor of Mathematical Cognition at Loughborough University. His research focuses on how people understand, evaluate, and experience mathematics, including proof comprehension, reasoning, and mathematical aesthetics.