Mathematics as MethodA Conversation with Alexander V. Gheorghiu

Bryan Kam in conversation with Alex, assistant professor and a New Frontiers Fellow in the School of Electronics and Computer Science at the University of Southampton.

As you'll hear in this podcast, my meeting with Alex Gheorghiu was random and fortuitous. In this podcast we discuss whether and how mathematics and logic relate to reality, why Buddhist thought challenges Western categories, and what Gödel's incompleteness theorem might mean for how we understand the world.

Alex traces his intellectual development from teenage mathematical realism—the belief that mathematics describes the fundamental structure of reality—to his current anti-realist position. Through studying algebra and analysis during his degree, he came to the view that these mathematical tools are cultural constructs rather than discoveries about an objective reality "A model is just a model in the way that a map is never the land itself."

Alex is also a Zen practitioner. We explored the famous Zen koan of Master Joshu, to the question of whether a dog has Buddha-nature. He responds "mu"—which neither affirms it nor denies it, but rather rejects the question. This exemplifies a philosophical move that transcends binary thinking, similar to how the Daodejing presents the Dao as preceding both unity and duality. We discuss how Chinese philosophy, lacking the Indo-European grammatical structures that equate existence and predication, developed fundamentally different approaches to how categories work.

Through Michael Dummett's anti-realist philosophy, we explore how meaning emerges from use rather than correspondence to reality. This challenges millennia of Western philosophical assumptions about categories and definitions.

The ancient tension between Parmenides (static being) and Heraclitus (dynamic becoming, which I've written about here) continues to shape philosophy today. We examine how Plato attempted to reconcile these positions through his theory of forms, and why this synthesis may have taken Western philosophy down a particular path—one that privileges nouns over verbs, objects over processes, and abstract categories over lived experience.

Eugene Wigner's famous question—why mathematics works so unusually well in describing nature—dissolves when viewed through an anti-realist lens. If mathematics is a human tool rather than a discovery of reality's structure, its effectiveness becomes less mysterious and more a reflection of how we've shaped our tools to solve our problems.

Alex shares his vision for bringing Gödel's incompleteness theorem into public consciousness the way physics has done with black holes. Having just won the 2025 Graham Hoare Prize for his essay, he argues that this "small technical result" has profound implications for how we understand the limits of formal systems and human knowledge itself.

Alex Gheorghiu is an assistant professor at the University of Southampton and honorary fellow at University College London, working in logic with interests spanning philosophy of mathematics, theories of language, and the relationships between reasoning and reality. He's currently developing a mathematical account of Dummett's philosophy and working to make logic and mathematics accessible to wider audiences.

Bryan Kam hosts the Clerestory podcast and is writing Neither/Nor, exploring how conceptual and experiential ways of knowing can inform both individual flourishing and our approach to philosophical problems.

Recorded at Drake & Morgan, London, where philosophical work happens with "consistently low" productivity but high engagement.

On Thursday 7th August, I walked around Bethnal Green with Christopher Daniel, who organises Long Now London.

We walked through East London, discussing architecture, philosophy, and the importance of long-term thinking. We explored the evolution of our own projects, especially Long Now London and Bryan's discussion group Through a Glass Darkly. We delved into broader themes like societal change, the impact of technology, and embodied experience versus conceptual abstraction. We wandered the streets, thinking about history, our own personal stories, and how to create meaningful and sustainable communities.

The Bertrand Russell quote I butchered:“Work is of two kinds: first, altering the position of matter at or near the earth's surface relatively to other such matter; second, telling other people to do so. The first kind is unpleasant and ill paid; the second is pleasant and highly paid.”

• See some photos I took of East London over on Substack
• My piece on Heraclitus
• My piece on Dependent Arising
• Jamie Stantonian on the Uffington White Horse
• Anab from Superflux whom Chris mentioned a few times.
• I wrongly called one of the Greek schools of medicine "Dogmatist" when I should have said "Rationalist"; Sextus opposes both dogmatism and rationalism. The three schools of medicine in 2nd century AD were "Empirical," "Rationalist," and "Methodist."

In this podcast, I discuss the draft of the academic paper on Neither/Nor which I wrote about here, with my coauthor, developmental psychologist Dr. Isabela Granic.

This paper has been a lot of work, and we discussed the writing right after finishing a major draft. In the paper and in our discussion, we emphasise the six principles of Neither/Nor. The article currently has an editor at Nature assessing the submission.

1. Two Modes of Knowing: The first principle identifies the two distinct ways we learn about life — through concepts and experiences. These modes are complementary, and we consider them trainable skills. Neither should be privileged over the other.

2. Commitment to Oscillation: Rather than selecting a definitive standpoint, we advocate for a dynamic process of oscillating between the conceptual and experiential skills, allowing us to adapt to the strengths of each mode.

3. Process Over Static Entities: Our understanding should prioritise processes rather than fixed entities, recognising the ever-changing nature of knowledge as it evolves through interaction with the living world. Categories are useful, but we can’t allow them to become too static.

4. Trial and Error Learning: Engage with the world through continuous experimentation, using trial and error to iteratively refine our understanding and approach — this is at the heart, for us, of adaptive learning.

5. Social Construction of Knowledge: A recognition that all knowledge is inherently social. Our reason for wanting to know is always social. Knowledge itself is always culturally embedded. This is empowering, and allows to contextual flexibility, not relativism — some ways are more effective than others.

6. Historicity: Understanding that knowledge and its acquisition grow from historical context gives us a holistic understanding of how changes occur in personal, cultural, and scientific pursuits.

Would you like to read the paper?

If so, you can request a copy of the pre-print here.

Thanks for listening. If you enjoyed the conversation, please share it with someone! Please also consider supporting me on Patreon or Ko-Fi.

Best,

Bryan

The Six Principles of Neither/NorWould you like to read it?