Is nominalism self-defeating?

Here's an objection to nominalism I've heard a few times.  Sometimes the 'access problem' to abstract objects is motivated by the idea that embodied creatures adapted to a particular environment, such as ourselves, need to interact with the world in order to gain knowledge of what it's like.  If we're to learn something about a given domain of objects, at some point we will require some kind of causal interaction with at least some members of that domain of objects.  Abstract objects, such as mathematical objects, are not like this: there is no method by which we could interact with anything in a domain of abstract objects at any time.  As a result, even if abstract objects exist, there is no means by which we could come to gain knowledge of which abstract objects exist or what properties they have.  This kind of minimal causal condition for knowledge is sometimes called the (or a) "eleatic principle".  But it's sometimes said that this eleatic principle is self-defeating.  Here's Sorin Bangu in his nice book The Applicability of Mathematics in Science (pp.18-9):

My naturalist’s reaction to the reformulated [eleatic principle] challenge is to point out that it is ultimately self-defeating.  That is, the naturalist notes that one cannot even formulate the challenge without actually making appeal to mathematics: one simply can’t grasp what the new naturalized [eleatic principle] actually says unless one understands the physical theories describing the abovementioned types of interactions.  But these theories are, of course, thoroughly mathematical!  So, anyone attempting to advance a challenge of the [eleatic principle] type in naturalistically acceptable terms finds herself engaged in the self-undermining enterprise of rejecting the very (mathematical) terms which allow the (acceptable naturalistic version of the) challenge to be meaningfully formulated in the first place.

This criticism seems wrong to me, on two counts.  Firstly, it ignores responses to the indispensability argument.  The nominalist will need some response to the indispensability argument.  If this response doesn't work then the nominalist is in trouble anyway.  If it does work—whether it involves doing without reference to or quantification over mathematical objects in scientific theories, like Field, Chihara etc., or offering some account of why it's acceptable for the nominalist to continue to refer to or quantify over mathematical objects in scientific theories, like Leng—then it will work here too: that we give mathematical models of how we (concrete) creatures interact with (concrete) parts of the world will pose no special problems.  Secondly, even in lieu of a response to the indispensability argument, the eleatic principle can be used to give a sort of reductio of mathematical platonism: (i) assume mathematical platonism is true, (ii) motivate the eleatic principle, (iii) our own mathematicized theories which describe how we interact with the world show that we cannot have knowledge of mathematical objects. So the assumption we began with is unknowable and rationally self-defeating.

What I talk about when I talk about numbers

Here is a valid argument:

(1) The number of Front national MEPS is worrying. 

(2) The number of Front national MEPS is 24. 

(3) 24 is worrying.

At least it’s valid if you think, as almost all philosophers who think about mathematical language seem to, that (2) refers to a number.  Contrast (2) with

(2*) There are 24 Front national MEPS.

(2*) is a statement about Front national MEPS, but (2) and (2*) are treated as being equivalent; not in the sense that they have the same meaning (one refers only to a political party, the other refers to a number) but in the sense that given (2) we can always infer (2*) and given (2*) we can always infer (2).  We can do this because we accept the abstraction principle:

(*) There are n Fs if and only if the number of Fs is n.

I’m not sure what to make of this.  I used to think that claims like (2*) were true because they predicate a property of something real, whereas claims like (2) were literally false because they make reference to something that doesn’t really exist—a number.  Making inferences using (literally false) claims like (2) was, I thought, fine, because doing so wouldn’t lead us astray with respect to how things stood with what really existed.  Similarly we could accept (*), not as being literally true, but as being “nominalistically adequate”, i.e. unable to lead us astray with respect to how things stand with what really existed.  (Compare: we accept ‘There is a dent in the car’ not because dents really exist or because they are an extra bit of the furniture of reality over and above the car, but because saying this doesn’t lead us astray with respect to the topographical properties of the car.)  But here is a problem with this: (3) is absurd.  A convenient fiction that aids inference-making is one thing; an absurd convenient fiction that aids inference-making is something else.  This is disastrous for the platonist who thinks that numbers really exist.  For the platonist (3) is true.  But it’s also bad news for the fictionalist who accepts that (2) refers (or at least purports to refer) to a number, because although fictionalist take (3) to be false, they’re still left with a problem: (3) isn’t even nominalistically adequate.  (3) can be used to infer falsehoods about the concrete world:

(3) 24 is worrying. 

(4) The number of Tunnock’s Teacakes in a four-pack is 24. 

(5) The number of Tunnock’s Teacakes in a four-pack is worrying.

(5) is about the concrete world and is false.  Maybe the only option is to drop the claim that phrases of the form ‘The number of Fs is n refer to the number n.  In this case the ‘is’ can’t be the ‘is’ of identity; the phrase can’t mean ‘The number of Fs = n’.

Death and the continuousness of time

Sometimes philosophy is a life and death matter. Let's say that time is continuous; i.e. that it can be represented by the real number line. The real number line is dense:


\[(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R})(x <y \rightarrow (\exists z \in \mathbb{R}) (x < z <y))\]

For any two real numbers there is another real number on the line between them. This means that no two real numbers can be "touching"—there will always be another (in fact infinitely many) real numbers between them. And if time can be represented by the real number line then time is also like this; for any two points in time t_i, t_j such that t_i < t_i, there is another point in time t_k such that t_i < t_k < t_j. No two points in time can be touching—there will always be another (in fact infinitely many) points in time between them.

Now, at some time t₁ a person S is alive and at some later time t₂ S is dead. If no two points in time can be touching then there will be a period of time in which S is neither alive nor dead. This can be avoided by saying that life and death overlap; that there is a point at which S, in the manner of Schrödinger's cat, is both alive and dead, but both options seem like nonsense.

Is this a paradox? Perhaps not, or perhaps at least not a very deep one. I think the thing to say here is that the boundary between life and death is vague (though maybe there are reasons further down the line to think this could not be the case). If vagueness is the way out though, there is still a problem for those who hold an epistemic theory of vagueness. If vagueness is epistemic—if there is a definite boundary between life and death, but we just don't know exactly where it lies—then the problem just reappears.

Propositions cannot exist

What a proposition is, or is supposed to be, can be grasped through abstraction principles. An abstraction principle is something of the form:

\[\forall \alpha \forall \beta (\Sigma(\alpha) = \Sigma(\beta) \leftrightarrow \alpha \sim \beta \]

Where \(\Sigma \) is an appropriate term-forming operator and \(\sim \) an equivalence relation. In the case of propositions, the abstraction principle will be something like:

The proposition expressed by u₁ = the proposition expressed by u₂ if and only if the content of u₁ is the same as the content of u₂

where u_i are appropriate tokenings such as utterances or inscriptions. A proposition is what is expressed by a sentence, written or spoken, and, furthermore, propositions are taken to be truth-bearers: they are the sorts of things that can be true or false. All this tells us that propositions are intentional entities: they are about or of the world; they pertain to things, and so on. The problem for propositions arises when one starts to consider what intentionality consists in, or what it is for something to be about, to be of, or to pertain to the world. It’s often (rightly) said that whatever aboutness propositions or sentence tokens have must be derivative from the fundamental intentionality associated with intentional agents. But something stronger can be said. Being about something essentially requires being responsible to that thing—not, it is worth emphasising, be responsive to a thing: lumps of wax are responsive to heat but are not about heat, thoughts about things outside our light cone are not responsive to those things but are about them; something different is required. If I think about the Empty Quarter I make my thinking responsible to the Empty Quarter itself. If I think of the Empty Quarter that it is the largest expanse of sand in the world then my thinking goes wrong—is subject to negative normative assessment—if it is not the largest expanse of sand in the world, and my thinking goes right—is subject to positive normative assessment—if it is the largest expanse of sand in the world. This isn’t an accidental feature of intentionality, it’s an essential one. So the only entities that can be about things are those that can be responsible to those things. Only persons are responsible in this way, abstract objects like propositions can’t be. But propositions are defined as things which are about the world; the result being that propositions would have to possess an essential property they cannot possibly have. Propositions then, cannot exist.

Softening the Blow of Truth Fictionalism

In the last post I said that mathematical fictionalism has consequences that sound terrible, but really aren't worrisome at all, when you think about what they actually entail. Something similar could be said about alethic fictionalism, or fictionalism about truth. Consider the truth predicate.  It’s a widely held view that the purpose of having a truth predicate is to allow people to undertake commitments to certain claims without having to explicitly state those claims.  So, if a theory $\Gamma$ entails infinitely many claims that you want to endorse, you can say ‘Everything entailed by $\Gamma$ is true', rather than explicitly state every claim entailed by $\Gamma$, which would be impossible.  So the job of the truth predicate isn’t to ascribe a special property, TRUTH, to things, but to allow us to undertake commitments without having to articulate those commitments explicitly.

Nevertheless, you could consistently hold that, even though what explains why we have a truth predicate has nothing to do with ascribing the property TRUTH to things, the meaning of a predicate is always to ascribe a property to something.  And if, in addition, you held that there is no such property as TRUTH, then you would end up being a fictionalist about truth discourse.  Any claim of the form ' $\phi$ is true’ would be false, since nothing has the property of truth.  But this wouldn’t really matter, since the truth predicate would still allow us to do what it was designed to do.  (It would, however, sound like a really bad result.)

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.