What is Mathematical Philosophy? (1)

A non-academic friend, Michael Ezra, asked me what mathematical philosophy is, and so I said I'd try and explain; or, at least, explain how I think of it. This is the first post. In the second, I will try and give some examples to illustrate.
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1. Explaining what mathematical philosophy is

First, I see it as analogous to mathematical physics or mathematical economics. In physics, one want to understand how physical processes---things moving around, heating up and cooling, etc.---work, and in economics one wants to understand how economies, trade, firms, etc., tick.

Mathematics is introduced in these domains, obviously. For example, we formulate the laws of nature like this:

$\nabla \cdot B = 0$
$R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab} = \frac{8 \pi G}{c^4} T_{ab}$

Here, the physical quantities are mathematical fields. (Functions on spacetime to some abstract space, such as $\mathbb{R}^n$ or a Hilbert space. In a fancier geometric setting, physical fields are sections of a fibre bundle.) What the exact role of mathematics here is is controversial. Clarifying its role is intimately tied up with debates about the Indispensability Argument and the nature of applied mathematics.

Second, in philosophy one wants to do something, but what this something is is pretty controversial. Well, look at some philosophical problems or puzzles: these can usually be expressed in a way that seems very intuitive, and non-mathematical. For example,

• How do I know I'm not a brain in a vat, and what I take to be the case, isn't?
• Why are some patterns of reasoning valid, and others invalid?
• Why should I think the future will resemble the past?
• If I say "My current statement now is false", is my statement true or false?
• Are moral statements like descriptions of facts, or more like expressions of my tastes and attitudes?
• We learn about the numbers, 0, 1, 2, and so on, as children. How to add them and multiply them and apply them to counting things around us. Are these numbers entities of some kind? Or just marks on paper?
• If I could have worn a different jumper today, does that mean there is another possible world in which I am wearing that jumper?
• Suppose, when you were asleep, God picked up all the matter in the universe and moved it 1 metre in some direction relative to space. There is no noticeable difference. Does this imply that space doesn't exist?
• Captain Kirk is beamed down to the planet. But the transporter malfunctions, and two copies of Captain Kirk materialize on the surface. Which one is the real Captain Kirk? 

2. "Über-theory" and "Meta-theory"

On one view, which I call the über-theoretic view, what this something that philosophy is doing is concerned, in a very general way, with:

how everything hangs together.

Quoting Wiflrid Sellars,

The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term. (Sellars, 1962, "Philosophy and the Scientific Image of Man").

So, examples of über-theoretic questions are:

• What kinds of things are there? 
• Are there, say, abstract things; or merely possible things; or fictional things?
• What are properties and relations? 
• Are there possible worlds? 
• What is space and time? Are there spacetime points? 
• What is causation? 
• What are facts, and states of affairs? Are facts "composed" of constituents? 
• Is reality organized into levels of dependence? Does it, or could it, have a "bottom level"? 
• Are there moral facts and properties?

On another view, which I call the meta-theoretic view, philosophy is concerned with,

understanding theories, representations and concepts. 

How do theories, representations and concepts relate to each other and to the things of which they are theories and which they represent? How should we analyse the concept of representation itself? We are concerned with concepts such as:

• existence
• identity
• abstractness
• structure
• possibility
• necessity
• meaning
• reference
• truth
• consequence
• infinity
• part-of.

We may attempt to analyse such concepts (i.e., provide "if and only if" definitions, which are analytic, and avoid counterexamples); we may attempt to "explicate" such concepts; we may attempt merely to relate such concepts to others, emphasizing their conceptual interdependence.

I make no argument here as to whether über-theory and meta-theory are exhaustive classifications, or non-overlapping. (But I think they are overlapping.) Here is an earlier post of mine on über-theory and meta-theory.

So, as I see it:

Mathematical philosophy consists in trying to examine über-theoretic questions and/or meta-theoretical questions by using mathematical methods. 

I would identify Bertrand Russell as the classic figure here, particularly his Principles of Mathematics (1903), which I mentioned also a few months ago shortly after defending the achievements of analytic metaphysics. There were antecedents, of course - e.g., Frege, Bolzano, Leibniz. But Russell has a special significance. Perhaps Russell is to modern mathematical philosophy roughly what Albert Einstein is to modern mathematical physics. (This is not, of course, to diminish the significance of, say, Newton, Maxwell, Lorentz and Poincare!)

After Russell, Rudolf Carnap was the practitioner par excellence of the second, meta-theoretic, approach, while David Lewis was the practitioner par excellence of the first, über-theoretic approach. (In addition, W.V. Quine, Hilary Putnam and Saul Kripke are very important, for both approaches.)

3. Hannes Leitgeb on mathematical philosophy

In mid 2010, Professor Hannes Leitgeb set up the Munich Center for Mathematical Philosophy (MCMP), at the Ludwig-Maximilians University, in Munich, with support from the Alexander von Humboldt Foundation. Shortly before this, he gave an interview to The Reasoner magazine (April 2010 issue). I quote from Hannes:

I just realized I had never considered before whether there was any common thread that runs through the whole of my work. If there is one, then it is on the more methodological side really: I like to apply mathematical methods in order to solve philosophical problems. I call this ‘mathematical philosophy’. Very occasionally one has some cool mathematical theorem, and one then looks for the right sort of problem to which it could be applied. But in the great majority of cases one simply comes across a philosophical theory or argument or thesis or maybe even just a clever example, and some mathematical structure presents itself—well, ‘presents itself’ after a lot of work!

I was lucky enough to be able to work alongside Hannes and others for a year and half at MCMP before moving to Oxford at the end of 2012. It really is a very intellectually stimulating environment (and also extremely welcoming and friendly, because of Hannes's incredible levels of goodwill, hard work and decency.)

4. Why should mathematics play any role at all?

One might wonder where all the mathematics comes in!

In the case of über-theory, one might initially wonder why mathematics might be relevant at all. Well, as it turns out, explaining what properties and relations are does immediately relate to mathematics, because the extension of a property is a set, and the extension of a relation is a set of ordered tuples. And the theory of sets and ordered tuples is a part of mathematics---some would say, the foundational branch. If one is interested in space and time, then our best theories of space and time are highly mathematicized theories: to understand such things, one needs to know about manifolds, co-ordinate charts, tensor fields, fibre bundles, topology, and so on. Similar points can be made in connection with causation, modality and other topics.

In the case of meta-theory, it is clearer, because meta-theory and logic are so intimately related; and logic and mathematics are intimately related. Meta-theory is relates closely to semantic theory (broadly understood), and in semantics one is concerned with all kinds of semantic relationships between syntactical entities (for example, connectives, names, predicates, variables, intensional operators) and what they denote, or refer to, or mean, etc. Probably the most important mathematical tool in meta-theory is the notion of a model, and the methods of model theory. And model theory is a branch of mathematics.

5. There is no mathematical substitute for philosophy

Surely one can't definitively solve philosophical problems using mathematics. Isn't that some kind of cheating? Or trick?

On this matter, Saul Kripke once wrote,

There is no mathematical substitute for philosophy.

I think Kripke is right. The only way to clarify Kripke's aphorism this is to look at some examples. Fortunately, I will have some examples to show you!

But, briefly, consider a slightly different approach: this might be called the applied logic approach to philosophy. Although the philosopher cannot solve their problems outright, they can relate certain doctrines---e.g., metaphysical doctrines---to others, by relations of implication, consistency, inconsistency, and so on. Roughly, things like:

Doctrines D1 and D2 together imply doctrine D3.
Doctrines D1, D2 and D3 are jointly inconsistent.

A number of possibilities arise which bring in mathematics. First, even the proper formulation of doctrines D1, D2 and D3 may require mathematical language; and, second, establishing connections like this may, in practice, require more than merely reasoning from say D1 and D2 to D3. In a sense, it is D1 and D2 plus mathematics which implies D3; and, third, even the logical relationships that one eventually arrives at may themselves sometimes be contested, and one might be prepared to consider non-classical logics: understanding these---particularly their semantics---brings in more mathematics.

So, even if the optimum output of some philosophical inquiry is to have clarified logical relationships between certain metaphysical doctrines, mathematics intrudes in a number of ways.

In the second post, I intend to give some examples from my own work.

[UPDATES (24th/25th March): I have updated this to include mention of an interview with Hannes Leitgeb. I've have moved some bits of text around, and added some more explanation. to make the organization clearer.]