Softening the Blow of Mathematical Fictionalism

Mathematical fictionalists are representationalists about mathematical discourse.  Not (necessarily) in the sense that they think that the meaning of mathematical sentences is to be understood in terms of reference or truth-making relations—mathematical fictionalists might be, and often are, deflationists about truth and reference—but rather in the sense that they take mathematical discourse to describe mathematical objects.  If someone claims there are 88 narcissistic numbers in base ten then the content of this claim has to do with the way it stands with a domain of things.  Mathematical discourse is descriptive rather than, say, expressivist.  Mathematical fictionalists also think that mathematical objects don’t exist, and hence the claims that there are 88 narcissistic numbers in base ten, which mathematicians accept, are, strictly and literally speaking, false.

This is usually enough to put people off mathematical fictionalism; even if the arguments for the position seem well founded enough, the conclusion that mathematical claims are, strictly and literally speaking, false (or trivially true if, e.g., they make universal negative claims such as “there are no positive integers x, y and z such that x3 + y3 = z3”, or form the antecedent of a conditional claim) will be too much to bear.  That modus ponens from fictionalism to the falsehood of mathematical claims will always seem like a modus tollens against fictionalism.

But the function of a discourse—the reason that we go in for this kind of discourse in the first place—needn’t determine the meaning of that discourse.  So, a discourse can be representational, in the above sense, but it might exist in order to serve a purpose other than representing the way things are with the world.  There are many things we can do with language, other than picture the world as being a particular way.

The “blow” of fictionalism isn’t really a blow at all, so long as we think the following things:  Mathematical discourse is representational, but the point or purpose of engaging in mathematics is not to describe or picture how things stand with a realm of mathematical objects.  Moreover, mathematicians have objective standards of rightness and wrongness that determine which mathematical statements are correct or incorrect.  So although mathematical claims may be strictly false in the sense that they do not picture how things stand with a realm of abstract objects, this is wholly orthogonal to the goals of mathematics.  Here’s the litmus test: When you describe fictionalism replace every instance of “is true” with “correctly describes how things stand with a realm of abstract objects” and “is false” with “does not correctly describe how things stand with a realm of abstract objects”.  How bad does fictionalism sound now?

Your Mother and Modal Epistemology

Here is a problem with modal conditions on knowledge, as traditionally understood.  Some of the beliefs we form pertain to things that our own existence is ontologically dependent on.  Consider the following scenario:  

Two people, Timothy and Titus, look at a photograph of a woman and form the belief  She existed at some point.  Neither, let us suppose, know anything about the person in the photograph, however, as it happens, she is the mother of Timothy.

Were the belief false, Timothy would not exist.  As a result, in the closest world(s) in which the belief is false Timothy fails to form the belief, and there are no close worlds in which Timothy believes that proposition in which it is not true.  So Timothy’s belief is both sensitive and safe (according to traditional construals of sensitivity and safety), and necessarily so.  Yet, depending on contingent background facts about the photograph, there are situations in which Titus’s belief fails to be sensitive or safe.  So, according to traditional accounts of safety and sensitivity, the epistemic status of Timothy and Titus’s beliefs are different, but, according to common sense, this is not the case.

More on Sensitivity and Closure

Traditionally, sensitivity theorists deny closure.  In the last post I suggested that some anti-sceptical beliefs (e.g. I am not a BIV) which are often taken to be non-sensitive, are in fact sensitive, when the sensitivity condition is parsed so as to take into account belief-forming methods. This though doesn’t mean that there might not be some, more elaborately contrived, beliefs in which closure would fail, even given a version of sensitivity that takes into account belief-forming methods.

But closure failure could be avoided if we adopted a disjunctive account of knowledge, whereby knowledge is either sensitively formed true belief, or belief that is soundly inferred from a sensitively formed true belief.  Oftentimes disjunctive explanations (or disjunctive proofs) are seen as being less explanatory than non-disjunctive counterparts (as they are less unifying), but there is some virtue in this disjunctive account of knowledge.  It does justice to the holistic nature of our beliefs.  Beliefs, taken individually, may lack sensitivity or responsiveness to the world, but may constitute knowledge because of the way they are apperceptively integrated into a wider whole, of which some parts are appropriately responsive to the world.  It is well-known that coherentist constraints on knowledge, taken on their own, leave out the important thought that beliefs that constitute knowledge must in some sense be responsive to reality (“frictionless spinning in the void” and all that); but modal constraints on knowledge, taken on their own, may also leave out the important role that coherence-making relationships have with respect to knowledge.  It may be that both kinds of consideration must be built into an account of knowledge, but that neither can be understood in terms of the other.  In which case, a disjunctive account of knowledge would be in order.

Sensitivity and Closure

Kelly Becker, in his book Epistemology Modalized, gives a nice modal account of knowledge:

S knows that p iff:

1. p is true
2. S believes that p
3. S’s belief that p is formed by a belief-forming process or methodw that produces a high ratio of true beliefs in the actual world and throughout close possible worlds (reliability condition).
4. If p were false, S would not believe that p via the methodn S actually uses in forming the belief that p (sensitivity condition). (Epistemology Modalized, p.88)

Methodsw are individuated very narrowly, but not so narrowly as to include specific belief contents.  Specific belief contents are however included in methodsn.  Becker individuates a methodw as the narrowest specific-content-neutral method or process that is causally operative in belief formation.  An example of a methodw might be forming beliefs about which people are in the vicinity based on quick looks in at least dim lighting.  A methodn on the other hand might be something like If what I am looking at now has short legs and floppy ears (and such and so other features) then it’s a dachshund.

Becker also makes a serious and interesting case against closure under known entailment, and he takes it, as epistemologists generally do, that sensitivity is incompatible with closure:

The sensitivity component of our theory somehow predicts this result – I do not know not-[sceptical hypothesis] because, if it were false, I would believe it anyway. (Epistemology Modalized, p.120)

But in fact, it isn’t clear that his account of sensitivity is incompatible with closure.  Take a standard BIV case.  I believe I am not a BIV, yet 4 holds: if I was a BIV I would not believe that I was not a BIV via the methodn I actually use in forming the belief that I am not a BIV.  My method, after all, involves coming to know ordinary propositions about the world around me by interacting with it, and inferring from these ordinary propositions that I am not a BIV.  Since brains in vats cannot employ the same kinds of methods that embodied humans do, my belief that I am not a BIV is sensitive according to Becker’s analysis.

How to Eschew Metaphysics

Here’s how Blackburn describes pragmatism:

You will be a pragmatist about an area of discourse if you pose a Carnapian external question: how does it come about that we go in for this kind of discourse and thought?  What is the explanation of this bit of our language game?  And then you offer an account of what we are up to in going in for this discourse, and the account eschews any use of the referring expressions of the discourse; any appeal to anything that a Quinian would identify as the values of the bound variables if the discourse is regimented; or any semantic or ontological attempt to ‘interpret’ the discourse in a domain, to find referents for its terms, or truth-makers for its sentences … Instead the explanation proceeds by talking in different terms of what is done by so talking.  It offers a revelatory genealogy or anthropology or even a just-so story about how this mode of talking and thinking and practising might come about, given in terms of the functions it serves.  Notice that it does not offer a classical reduction, finding truth-makers in other terms.  It finds whatever plurality of functions it can lay its hands upon. [Simon Blackburn, Expressivism, Pragmatism and Representationalism: 75]

I'm interested in the claim often made by pragmatists, such as Simon Blackburn or Huw Price, that they are eschewing metaphysics, in contrast to platonists, fictionalists, error theorists and the like. Pragmatic accounts of a discourse provide a genealogy, or some consanguineous account, of why it is we go in for this way of talking and, as it may happen, this account may be metaphysically deflationary.  So it may be that the motivation for talking about, say, mathematical objects, does not involve representing how things stand with a domain of mathematical objects.  If there is some such story—if we can account for the uses of mathematical talk, without invoking mathematical objects—then we have an ontologically deflationary pragmatic account of mathematical discourse.

But so far, what’s been said about pragmatic accounts of mathematical discourse is open for the fictionalist to adopt.  The difference between the fictionalist (who is apparently engaged in metaphysics) and the pragmatist (who apparently eschews metaphysics) is that the fictionalist claims that mathematical talk is, strictly speaking, false, whereas the pragmatist does not.

Fictionalists and pragmatists then agree in methodology: provide an account of the usefulness of mathematical (or moral, or possible worlds) discourse that makes the existence of mathematical (or moral, or modal) objects orthogonal to the practice.  Their point of divergence is not methodological or ontological, but semantic: whether one opts for fictionalism or pragmatism depends on what one takes the meaning of existential quantification to be.  Here, the pragmatist reads the pragmatic purpose of quantification over mathematical objects back into the semantics of quantification over mathematical objects, and the fictionalist does not.  A truism: people can engage in ontological disputes.  There is something at stake between someone who claims that the Higgs boson exists and someone who claims that it does not, or between someone who claims that God exists and someone who claims that he does not.  The interlocutors in these debates are in disagreement over what the world is like.  So, sometimes at least, quantificational talk is used to express disagreements about what the world is like.  Ultimately then, the difference between the fictionalist and the pragmatist lies in what they take the meaning of existential quantification to be.  Fictionalists take existential quantification to be univocal: it always expresses claims about what the world is like.  Pragmatists (are committed to) taking existential quantification to be multivocal: within discourses whose purpose is to describe the world existential quantification expresses claims about what the world is like; within discourses whose purpose is not to describe the world, existential quantification does not express claims about what the world is like. (Note that the point of divergence is not, or need not be, over semantic minimalism. The person engaged in metaphysics need not couch what he is doing in terms of finding truth-makers or referents to be relata in substantive relations of truth or reference to given sentences; he can simply couch what she is doing in terms of whether such and such objects exist. Hartry Field is a case in point.)  

The take-away claim: whether one gets to eschew metaphysics depends on whether existential quantification is univocal or multivocal.

Kant and the Necessary 3-Dimensionality of Space

Here’s a thought.  Kant took it to be necessary that space was 3-dimensional.  Bracketing the possibility that space is transcendentally ideal for the moment—of course, you might think I’m bracketing the most interesting thing here—most people who are paid to think about these things now reject the necessary 3-dimensionality of space on the grounds that spaces with more dimensions are possible, and the standard argument for this is the following.  A 3-dimensional space can be modelled as $\mathbb{R}^3$ = $\mathbb{R} \times \mathbb{R} \times \mathbb{R} $ with each n-tuple representing a point in 3-dimensional space.  Methods of this sort allow for higher-dimensional spaces to be represented, since extending or generalizing the model to represent higher-dimensional spaces is quite straightforward. 4 dimensional space is represented as $\mathbb{R}^4$, 5-dimensional space as $\mathbb{R}^5$, and so on.

But why think a thing like this constitutes grounds for taking higher-dimensional spaces to be metaphysically possible?  Why think that because a model of 3-dimensional space can be extended in this sort of way (and remain coherent), higher-dimensional spaces themselves are possible, or even coherent?  Why think that because (i) there is a space which can be represented using $\mathbb{R}^3$, and (ii) there is nothing in consistent about $\mathbb{R}^4$, that (iii) there could be a space that is represented by $\mathbb{R}^4$?  Now, there may be other good reasons to think that (iii) is true, but the standard argument looks to be enthymematic at best.

[Cross-posted at Kant and Laws]

Is naturalism coherent?

Here is how Huw Price characterises naturalism in a recent book, although I think it’s a characterisation that many philosophers would endorse:

What is philosophical naturalism?  Most fundamentally, presumably, it is the view that natural science properly constrains philosophy, in the following sense.  The concerns of the two disciplines are not simply disjointed, and science takes the lead where the two overlap.  At the very least, then, to be a philosophical naturalist is to believe that philosophy is not simply a different enterprise from science, and that philosophy should defer to science, where the concerns of the two disciplines coincide. [Expressivism, Pragmatism and Representationalism: 3]

But in what sense is it possible for philosophy to defer to science?  One way we might think this could go is in the following scenario: we have in our possession, say, both a successful scientific theory which posits backwards causation, and an a priori philosophical argument that backwards causation is impossible.  Deferring to science—which is an essential trait of naturalism, as understood above—involves accepting the scientific theory and rejecting the philosophical argument as (somehow) unsound.  But there is a problem with thinking of this as philosophical deference to science (as opposed to some other kind of deference to science).  Consider the maxim: When a claim of a successful scientific theory conflicts with the conclusion of an a priori argument, reject the conclusion of the a priori argument in favour of the claim of the successful scientific theory.  This is, on any reasonable measure, a philosophical dictum rather than the claim of a scientific theory.  In which case, someone who follows the maxim is being guided by a philosophical dictum rather than the claim of a scientific theory.  Moreover—although I’m not really arguing for this latter claim here—it is plausible that any adjudicative maxim of this kind would be philosophical rather than scientific per se; and in that case, it wouldn’t make sense to say that philosophy could defer to science.