Author's Introduction
- When René Descartes was 31 years old, in 1627, he began to write a manifesto on the proper methods of philosophising. He chose the title Regulae ad Directionem Ingenii, or Rules for the Direction of the Mind. It is a curious work. Descartes originally intended to present 36 rules divided evenly into three parts, but the manuscript trails off in the middle of the second part. Each rule was to be set forth in one or two sentences followed by a lengthy elaboration. The first rule tells us that ‘The end of study should be to direct the mind to an enunciation of sound and correct judgments on all matters that come before it,’ and the third rule tells us that ‘Our enquiries should be directed, not to what others have thought … but to what we can clearly and perspicuously behold and with certainty deduce.’ Rule four tells us that ‘There is a need of a method for finding out the truth.’
- But soon the manuscript takes an unexpectedly mathematical turn. Diagrams and calculations creep in. Rule 19 informs us that proper application of the philosophical method requires us to ‘find out as many magnitudes as we have unknown terms, treated as though they were known’. This will ‘give us as many equations as there are unknowns’. Rule 20 tells us that, ‘having got our equations, we must proceed to carry out such operations as we have neglected, taking care never to multiply where we can divide’. Reading the Rules is like sitting down to read an introduction to philosophy and finding yourself, an hour later, in the midst of an algebra textbook.
- The turning point occurs around rule 14. According to Descartes, philosophy is a matter of discovering general truths by finding properties that are shared by disparate objects, in order to understand the features that they have in common. This requires comparing the degrees to which the properties occur. A property that admits degrees is, by definition, a magnitude. And, from the time of the ancient Greeks, mathematics was understood to be neither more nor less than the science of magnitudes. (It was taken to encompass both the study of discrete magnitudes, that is, things that can be counted, as well as the study of continuous magnitudes, which are things that can be represented as lengths.) Philosophy is therefore the study of things that can be represented in mathematical terms, and the philosophical method becomes virtually indistinguishable from the mathematical method.
Author's Conclusion
- Mathematics has therefore soldiered on for centuries in the face of intractability, uncertainty, unpredictability and complexity, crafting concepts and methods that extend the boundaries of what we can know with rigour and precision. In the 1930s, the American theologian Reinhold Niebuhr asked God to grant us the serenity to accept the things we cannot change, the courage to change the things we can, and the wisdom to know the difference. But to make sense of the world, what we really need is the serenity to accept the things we cannot understand, courage to analyse the things we can, and wisdom to know the difference. When it comes to assessing our means of acquiring knowledge and straining against the boundaries of intelligibility, we must look to philosophy for guidance.
- Great conceptual advances in mathematics are often attributed to fits of brilliance and inspiration, about which there is not much we can say. But some of the credit goes to mathematics itself, for providing modes of thought, cognitive scaffolding and reasoning processes that make the fits of brilliance possible. This is the very method that was held in such high esteem by Descartes and Leibniz, and studying it should be a source of endless fascination. The philosophy of mathematics can help us understand what it is about mathematics that makes it such a powerful and effective means of cognition, and how it expands our capacity to know the world around us.
- Ultimately, mathematics and the sciences can muddle along without academic philosophy, with insight, guidance and reflection coming from thoughtful practitioners. In contrast, philosophical thought doesn’t do anyone much good unless it is applied to something worth thinking about. But the philosophy of mathematics has served us well in the past, and can do so again. We should therefore pin our hopes on the next generation of philosophers, some of whom have begun to find their way back to the questions that really matter, experimenting with new methods of analysis and paying closer attention to mathematical practice. The subject still stands a chance, as long as we remember the reasons we care so much about it.
Author Narrative
- Jeremy Avigad is a professor in the department of philosophy at Carnegie Mellon University in Pittsburgh. He is associated with Carnegie Mellon's interdisciplinary programme in pure and applied logic.
Notes
- I found this paper a bit of a muddle. Is it talking about metaphilosophy - how philosophy is to be conducted, and should this method be modeled on that of mathematics, or is it about the philosophy of mathematics and the conduct of mathematicians?
- It covers quite a lot of ground in a seemingly random manner, including AI, Incompleteness theorems (Kurt Gödel) and Chomskyan linguistics (Noam Chomsky).
- There's a reference to "Aronson (Polina) & Duportail (Judith) - The quantified heart".
- As the paper was issued in 2018, it's a bit dated with respect to AI, but is probably worth a second reading.
Comment:
- Sub-Title: "Is it possible that, in the new millennium, the mathematical method is no longer fundamental to philosophy?"
- For the full text see Aeon: Avigad - Principia.
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- Blue: Text by me; © Theo Todman, 2026
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