Why I'm going to be a philosopher of science

This semester, I entered the orals stage of my Ph.D. programme. Sometime in the next 7-14 months, I will spend 90 minutes locked in a room with five faculty members, who will happily interrogate my understanding of a list of classical and recent works of philosophy. Should I pass, I move on, more or less immediately, to the final (and interminable) thesis-writing stage. Should I fail, I can try again before the end of my fourth year (next May); if I fail at that point, or don't manage to get a second try scheduled before the end of the year, I get kicked out of school.

Despite the immense pressure, what's nice is that I get to choose what's on the list. At least, to an extent. I pick the general area, and about 2/3 of the list is standard for that general area; I get to decide what goes on the remaining 1/3. So, roughly speaking, everything on the list are things I'm interested in. I don't have to read any metaphysics or Plato or Scotus if I don't want to.

But I've had some problems picking the general area. I enjoy both philosophy of math and philosophy of science immensely, and while I don't think that they should be seen as two distinct areas of philosophy, trying to fit them both into a single orals list has proven impossible. On Thursday I met with one of my professors to get some help with figuring out the list. She gave me an ultimatum: Within two weeks, I have to decide whether I will be a philosopher of science who knows a bit about philosophy of math, or a philosopher of math who knows a bit about philosophy of science. She recommended I think about potential thesis projects and look through the major philosophy of math and philosophy of science journals to decide.

While I had some thesis projects in mind, they were hopelessly vague, and this is part of the problem I was having putting together a list. (One thing that's especially great about this particular professor: she never, ever lets me get away with any sort of hand-waving.) So I went and looked through journals and spent an hour writing down rough versions of thesis projects. Most of the projects I came up with were either impossibly difficult or, upon closer inspection, would be incredibly boring to actually work on in detail. The philosophy of math projects that looked both feasible and interesting came in two distinct flavours:

1. The relationship between mathematics and society, especially the relationship between ethico-political values and `good mathematics'.
   
2. The epistemological significance of certain features of the community of mathematicians, such as the underrepresentation of women and racial and ethnic minorities.
   

Philosophers of science are interested in these two things. But -- critically, and counterintuitively -- philosophers of math generally aren't. Philosophers of math, almost uniformly in this country, work on formal logic, the metaphysics of math (do numbers exist?), and foundations. Furthermore, what I discovered in the journals was that, to the extent that someone works on such things, they're published in philosophy of science and history of mathematics journals, and not published in philosophy of math journals. So, to work on one of these projects, I would have to read a lot of philosophy of science literature, and most of the philosophy of math literature would be completely irrelevant.

So I've made my decision. Strictly speaking, I will be a philosopher of science. Like most philosophers of science, I'll have a certain amount of expertise and affection for one science in particular -- mathematics -- and this will be reflected in what I write and teach. But I won't, strictly speaking, be a philosopher of math. My interests and research projects simply aren't the sort of things philosophers of math work on, at least in the English-speaking part of the world.

Zalta's logical truths that are not metaphysically necessary

In this post, I want to summarise the argument of Zalta's paper `Logical and analytic truths that are not necessary' for non-logicians who still know a little bit about modal logic.

The basic idea is as follows: Zalta identifies certain sentences which, first, are logical truths, in that they are true under all interpretations; but, second, they are not metaphysically necessary, in that they are not true at all possible worlds under at least one interpretation.

Zalta's argument focusses on definite descriptions. I want to use a simpler language: a standard sentential language L with a single unary operator @. If p is a wff of L, then informally, @p means p is true at the actual world.

More specifically, give L countably many primitive sentential constants p1, p2, p3, ..., and build the wffs of L using the standard logical operators ^, v, -, -> and the unary operator @ on the pi. A world W is a total function {pi : i in N} -> {T,F}, ie, an assignment of truth values. An interpretation A is an ordered pair , where D is a set of worlds and W0 is in D. Informally, W0 is the actual world under A. `p is true at W under A' is defined in the usual recursive way for the standard logical operators. `@p is true at W under A' is true if p is true at W0 under A. For purposes of order of operations, @ is stipulated to be of the lowest order, ie, it is evaluated after all other operators when there is ambiguity. `p is true under A' is true if p is true at W0 under A. `p is necessary under A' is true if, for all W in D, p is true at W.

p is logically true if, for all interpretations A, p is true under A. p is necessarily true if, for all interpretations A, p is necessary under A. Zalta goes to great pains to show that his semantics are entirely standard for languages which involve rigid designators; the only difference here is that the language is much simpler than Zalta's primary example.

For some q, let Q be the sentence

@q -> q

Informally, Q says that, if q is actually true, then q is true.

1. Q is logically true.

Q is logically true iff for all interpretations A, Q is true under A. Choose an interpretation Q. Q is true under A iff Q is true at W0 under A. Q is true at W0 iff either @q is false at W0 or q is true at W0. @q is false at W0 iff q is false at W0. So Q is logically true iff q is false at W0 or q is true at W0. So Q is logically true iff either q is false or q is true at W0. Since W0 is total, either q is false or q is true at W0. Hence Q is logically true.

2. Q is not necessary.

Q is necessary iff for all interpretations A, Q is necessary under A. Q is necessary under A iff, for all W in D, Q is true at W. Q is true at W iff either @q is false at W or q is true at W. @q is false at W iff q is false at W0. Hence Q is necessary iff for all interpretations A and all worlds W in D either q is false at W0 or q is true at W.

Hence Q is not necessary iff there is some interpretation A with some W in D such that q is true at W0 and q is false at W.

Let A be an interpretation such that q is true at W0 and false at all other W in D and let W != W0. Then q is true at W0 and q is false at W. Hence Q is not necessary.

3. There are logical truths which are not necessary.

The `counterintuitive' gap arises because of a difference in scope between necessity and logical truth. Logical truths must be true at all actual worlds under all interpretations. It does not look at any worlds besides the actual world in each interpretation. Hence the actuality operator @ is inert with respect to logical truth: @p is logically true iff p is logically true.

It is with respect to necessity that @ is doing some work. Necessity does not just consider the actual world of each interpretation. It considers the full `multiverse' of each interpretation. Necessity is actually much broader than logical truth, in the sense that it takes more into consideration. Indeed, necessity implies logical truthhood, but as 3 shows, the converse is not true. In my experience, metaphysicians think exactly the opposite -- that logical truthhood implies necessity, but not vice-versa.

This version of Zalta's example show why metaphysicians should want necessity to be the broader notion in a very clear way. Q said that, if q is actually true, then q is true. If this were necessary and q were actually true, then q would itself be necessary. That is, if logical truth implies necessity, then rigid designation makes every true sentence necessary. Modality collapses, and every `multiverse' consists of a single possible world.

If modal metaphysicians wish to retain the equipment of rigid designation or actualism and any robust notion of possibility, then they must deny the inference from logical truth to necessity, at least with respect to statements that might involve rigid designation or reference to the actual world.

Yet, as Zalta points out, this creates a serious problem for metaphysical methodology. That a modal statement is self-contradictory -- is logically false -- does not imply that it is a necessary falsehood -- necessarily not the case. It implies only that it is not necessarily the case.

Zalta's paper is `Logical and analytic truths that are not necessary', J Phil (Feb. 1998: 85.2), pp 57-74. JSTOR link

Three distinctions for an epistemology of mathematics

In The fate of knowledge, Helen Longino makes a distinction between three senses of the knowledge -- that is, three different usages of the term `knowledge'. These three senses of knowledge can help us tease apart some notions that have been tangled together in Analytic epistemology of mathematics for the last 130 years or so.

Necessary and possible

Necessity and possibility are modal notions. Modal notions show up in many different domains of enquiry: we have logical modalities, physical modalities, mathematical modalities, epistemological modalities, metaphysical modalities, and so on. It seems like unqualified modalities should be either logical or metaphysical. Zalta has argued that some logical impossibilities are metaphysical possibilities, though I suspect many metaphysicians around here assume the negation of this.

When made in mathematics, this distinction seems to be related to the truth status of mathematical propositions and the theories built up out of those propositions: Under `what circumstances' are mathematical propositions true? `All of them', so mathematical truths are `necessary'. It's therefore tied to at least one view -- and more likely a family of views -- of the content of mathematical knowledge.

A priori and a posteriori

These notions distinguish two sorts of justification or warrant relations. When I know (a priori) that 2+2=4, I have warrant for this belief. This warrant is different from the (a posteriori) warrant I have for the belief that it's a little cold in my house right now or that I'm being appeared to redly. The distinction does not refer to the doxastic processes by which I came to have these beliefs. In a famous paper, `What numbers could not be', Benacerraf argues that realists about mathematics cannot give a suitable epistemology for mathematics because their views are not compatible with any `causal theory of knowledge'. But the causal story about how I came to have my belief that 2+2=4 is not the same as the warranting story about how I came to be warranted in my belief that 2+2=4. A causal theory of knowledge may be confusing doxastic and warranting relations.

More generally, when I have a priori warrant, a certain relation obtains between my belief that 2+2=4, the proposition (or sentence or whatever) that 2+2=4, and possibly some other things (perhaps the Platonic numbers 2, 4, and the addition relation; perhaps the abstract natural number structure; perhaps something else entirely). This relation is the warrant or justification relation. I am said to have knowledge (at least in part) by standing in this relation. A priori and a posteriori therefore refer to the second, relational, sense of mathematical knowledge.

Analytic and synthetic

Until the eighteenth century, analysis and synthesis were two different and complementary mathematical methods. Analysis was the method of `breaking down' ideas, while synthesis was the method of `building up' complex ideas from simpler ones. In Ancient geometry, for example, one first analysed the relations between the given geometrical objects, and then assembled the simple relations thereby discovered into a rigorous proof of the desired claim. By the late nineteenth century, the terms were adjectives -- analytic and synthetic -- and not nouns. They were identified with logical relations, which in turn were bundled together with a priori and a posteriori. This is seen most clearly in Carnap's most important books, Der logische Aufbau der Welt and Logische Syntax der Sprache. These were the basis for Ayer's Language, truth, and logic, which provided the framework within which most Anglophone philosophers have been working for the past sixty years. The transition between these two uses of the terms is identified most prominently with Kant, whose usage is, (in)famously, neither the same as a priori/a posteriori nor easy to understand. Until Kant scholars provide us with a better understand of the use of the distinction during the eighteenth century, it is probably better to stick with the pre-Enlightenment methodological understanding of analysis and synthesis.

With this sense of the distinction in place, each of these terms refers to either one of two different knowledge-producing processes or one of two complementary parts of a single knowledge-producing process. This corresponds to Longino's third sense of knowledge.

As an epistemologist, I am unusual in that I most interested in the third sense: what are the processes by which mathematical knowledge is produced? Most contemporary philosophers of mathematics are primarily interested in the first sense, the content of mathematical knowledge, and the closely related metaphysical problems. That is, they want to understand what mathematics is about. A few philosophers have approached epistemology from the second side, and attempted to give an account of how mathematical beliefs can be warranted. For realists, these generally involve either a Platonic-Goedelian direct intuition of mathematicals or a quasi-Aristotelean abstraction of mathematical knowledge from perception. Kantian apriorist, Millean empiricist, and Fregean logicist views also show up occasionally, but are nowhere near as prominent in the literature. Obviously these five are related to views about the processes by which mathematical knowledge is produced.

Generally speaking, no philosopher of mathematics is at all happy with the proposals offered by any other philosopher of mathematics, and often a given philosopher of mathematics is not all that happy with his (the subdiscipline is ridiculously male-dominated) own views either.