Maths in Philosophy

Alexander Douglas
Sep 4 · 5 min read

Before Peter Boghossian and James Lindsay discovered that their true calling was being to Ethics in Humanities Journalism what gamerbros were to Ethics in Games Journalism, they made the case, in an article for The Philosopher’s Magazine, that philosophy should be done using the methods of modern science.

We should have cancelled them then, lol jk. But seriously. Here’s the opening to their article:

If you want to be a good philosopher, don’t rely on intuition or comfort. Study maths and science. They’ll allow you access the best methods we have for knowing the world while teaching you to think clearly and analytically. Mathematics is the philosophical language nature prefers, and science is the only truly effective means we have for connecting our philosophy to reality.

My forthcoming book discusses and to some extent defends the traditional view that philosophy proceeds via intuition and aims at comfort. There’s very little science in my book, except some clinical evidence to suggest that if the world is as it appears to the senses than human life is basically hell — so let’s hope that’s not the case. But there is some mathematics in it. This is the point I want to make here: mathematics and empirical science don’t come as a package deal. To some extent they’re in tension.

In an erudite moment B&L refer back to the Great History of their discipline:

Fortunately, the idea that philosophy should be more mathematical and scientific has a strong precedent in the history of the discipline. (Spinoza, Descartes and others, for example, are known for using the “Geometric Method” in philosophy.)

Spinoza certainly presented his philosophy more geometrico. Descartes did so only upon request and with many caveats. So B&L are right about only this much: Spinoza thought that philosophy should be mathematical in some sense.

But look how the words ‘and scientific’ have crept in here. In fact Spinoza was very sceptical about the ability of empirical science to give us much substantive (pun intended) knowledge of the world, as Eric Schliesser and I have argued. In his juvenile work, the Treatise on the Emendation of the Intellect, Spinoza pointed out that the causal structures of the world are too complex for our tiny minds to get much of a handle on. Consistently he dismisses the relevance of that sort of knowledge to philosophy.

Also, I argue in my book that Spinoza was basically a Pythagorean. He didn’t think, as B&L assert without argument, that ‘Mathematics is the philosophical language nature prefers’ (how silly to talk of nature preferring a language — we should have cancelled them then, lol jk, but seriously). Spinoza thought that mathematics is reality. Or, more precisely, he thought that all real objects are abstract objects, existing as mathematical objects do: in virtue of their coherent conceivability. The senses are misleading insofar as they encourage us to think of the world as something concrete and existing independently of its conceivability.

This view makes mathematical statements literally true in a fairly clean way. Why does an existence proof — a proof that a certain object is consistent with certain axioms — guarantee the existence of an object? Because the object’s being is its conceivability. The so-called ‘access problem’ — how can our apparently concrete minds know the objects in an abstract realm? — is solved by the implication that our minds are abstract, not concrete. More on this here.

Empirical science has a harder time with mathematics. It uses it to great effect. But it has a difficult time explaining both the ontology and the epistemology of mathematics.

Take B&L’s ham-fisted discussion:

As a quirk of our base-ten number system, for example, the number 0.999…, the one that is an infinite concatenation of nines, happens to equal 1. That is, 0.999… is 1, and the two expressions, 0.999… and 1, are simply two ways to express the same thing. The proofs of this fact are numerous, easy, and accessible to people without a background in mathematics (the easiest being to add one third, 0.333…, to two thirds, 0.666…, and see what you get). This result isn’t intuitive, and — as anyone who has taught it can attest — not everyone is comfortable with it at first blush.

(As an aside, most of this is wrong or misleading. It isn’t really a matter of a ‘quirk of our base-ten number system’; you’d have the same issues with ternary or duodecimal expansions of fractions as you do with decimal expansions. Also their ‘proof’ is silly; it only works if you assume that 1/3 = 0.333… and 2/3 = 0.666…, but I thought that was exactly the thing in question.)

Really what’s being confronted in this example is the strange structure of numbers. To really grasp it you’d have to think about partitions in the number-line, and one of the reasons that’s hard to grasp is because nothing we observe with the senses appears to have that sort of structure (space sort of looks continuous, or at least dense, but any measurements we can make are consistent with a sufficiently populous network of discrete points). To think of something with the structure of the continuum you need to think abstractly — mathematically. But then what objects are you thinking about, and how do you know about them?

And mathematical thinking depends on intuition. At least that was what Descartes thought, or so I argue. You can say that mathematical thinking is thinking within an axiomatic system, according to certain rules. But how then do we decide on the axioms and the rules? Descartes saw right away that we can’t reduce mathematical thinking to the only formal system of reasoning around in his day: the syllogistic. The ultimate decisions about foundations and consequences come down to what Descartes called ‘intellectual intuitions’. These he believed to be the direct perception of mathematical objects by the mind. But this was not the same as the empirical observation on which modern science depends; it was, for Descartes, the way we know God and our own thoughts, which is certainly not empirically. Descartes marked the difference by speaking of intuitions.

Descartes’s point was that mathematics bottoms out in intuition. So, therefore, should philosophy, if it hopes to attain the level of certainty found in mathematics. The question is then how to connect up this intuitive knowledge of an abstract realm with our much-less-certain empirical knowledge of a concrete realm. Descartes had little to say about this, and his greatest disciple, Spinoza, solved it by simply eliminating the concrete realm. Following the path of intuition — that is to say, respecting the example of mathematics — he arrived at a vision that was also comforting.

The opposition B&L set up, between intuition/comfort on one side and mathematics on the other, is a false one. There is a much better case to be made for an opposition between mathematics and empirical science — or rather a naturalistic ontology based on empirical science. But then I would say that, as a proponent of Rationalism — yet another word, by the way, that B&L have tried to ruin. Anyway they’re cancelled now.

Alexander Douglas

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Lecturer in Philosophy, University of St. Andrews — personal website:

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