Many success stories in philosophy can usefully be viewed as disambiguations or formalisations.

Disambiguation

Wittgenstein wrote that “philosophy is a battle against the bewitchment of our intelligence by means of language”. Ordinary language developed to work in ordinary contexts. When we deal with philosophically tricky issues, however, ordinary language rarely coincides with the underlying concepts in a one-to-one mapping. Sometimes ordinary language will use two different words for the same concept. This case rarely leads to problems. But when instead ordinary terms are ambiguous between two or more meanings, this is fertile ground for confusion. A lot of good philosophy disambiguates between these meanings to dissolve apparent paradoxes.

Phrases that have been disambiguated include:

• “The present king of France is not bald” (Quantification vs reference, Negation scope ambiguity)
• “Some cat is feared by every mouse” (Universal quantifier scope)
• “If P is true, then P cannot be false” (Modal scope ambiguity)
• “I decided to do that of my own free will” (Could-have-been-otherwise vs unconstrained)
• “We should expect humans to behave selfishly” (Is vs ought)

Formalisation

Sometimes people find my purported success stories mathematical rather than philosophical. I’ve even been accused of lumping the whole of mathematics into philosophy. I see why this intuition is compelling. Logic, the analysis of computability, Bayesianism and so on just look mathsy. It seems natural to cluster them with maths rather than philosophy. And that definitely makes sense in some contexts.

Here, I’m trying to understand how philosophy works, and what it can do for us when it’s successful. In that context, I claim, these stories should be clustered with philosophy. We should look beyond superficial patterns, like what the work looks like on the printed page, and instead ask: what kind of cognitive work is being done?

Now is a good time to ask: what do we call mathematics? In primary school, you might get away with defining mathematics as that which deals with quantity or number. But modern mathematics goes far beyond that. Wikipedia tells us: “Starting in the 19th century, when the study of mathematics increased in rigour and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics”.

I want to emphasise that whenever something is sufficiently formal, we tend to call it mathematical. Mathematics uses the form of strings to manipulate them according to perfectly precise rules. (I hope this is uncontroversial. I take no view on whether mathematics is only formalism).

Before we knew how to reason about the trajectories of medium-sized objects, we speculated and used vague verbiage. Since classical mechanics was solved, we use coordinates and derivatives. Object trajectories have been mathematised. But nothing about the subject matter of trajectories has changed, or (I claim) was distinctive in the first place. Formalisation is just what is looks like to fully solve a conceptual problem. Once we fully understood trajectories, they “became part of mathematics”.

Here’s another example. Logic has nothing to do with quantity or number, but is often called mathematical, and ‘$\rightarrow$’ and ‘$\neg$’ are said to be mathematical symbols. Sider (Logic for Philosophy) writes: “Modern logic is called “mathematical” or “symbolic” logic, because its method is the mathematical study of formal languages. Modern logicians use the tools of mathematics (especially, the tools of very abstract mathematics, such as set theory) to treat sentences and other parts of language as mathematical objects.” But logic is just culmination of a long-standing project: to distinguish good from bad arguments. Formal logic means we have succeeded fully. We have wholly clarified certain kinds of deductive reasoning.

I don’t mean to claim that all of mathematics should be clustered with philosophy. I just mean the initial mathematisation of a previously informal area of study. Once the formal cornerstones have been laid, philosophy really does hand off to mathematics. My rough picture of intellectual progress is the following:

1. Confusion reigns. People get lost in vague verbiage, and there is no standard way to adjudicate disagreements.
2. Much work is done in the service of clarification. Ultimately, maximal clarification is achieved through formalisation.
3. With a formal system at hand, people go to town with it, proving things left and right, extending the system, and so on.
4. We begin to view this area of study as mathematical or even part of mathematics.

Stage (1) is what most people think philosophy looks like. I say: it’s philosophy when it’s still failing. Stage (2) is successful philosophy (or at least one kind it). But the philosophical nature of the contribution in (2) is often forgotten in the subsequent wave of mathematical enthusiasm for steps (3) and (4).

I hope I’ve now built the intuition enough to move on to the success stories that people have found most counter-intuitive.

With the analysis of computability, the philosophical work of clarification was to formalise the notion of effective calculability with a Turing machine. This allowed mathematical work to be done with the formal notion. In this case, Turing did step (3) immediately, in the same paper, he went on to prove many results about Turing machines. So Turing’s paper is, in some sense, first some philosophy, then some mathematics. Wikipedia tells us that Hilbert’s problems ranged greatly in precision. Some of them are propounded precisely enough to enable a clear affirmative or negative answer, while others had to be substantially clarified. The Entscheidungsproblem was more philosophical because it involved significant work of clarification. And it’s a particularly cool story, because the precisification proposed by Turing turned out to (i) gain virtually universal approval and (ii) have wide philosophical significance and applicability.

In the case of the development of probability theory, it’s emphatically not the case that, pre-Pascal, people were disagreeing on a point of mathematics. They were much more deeply confused. They just had no appropriate notion of probability or expected value, and were trying to cobble together solutions to particular problems using ad-hoc intuitions. Because Pascal launched probability theory, it seems only natural to view his first step as part of probability theory. But in an important sense the first step is very different. It’s much more philosophical.