Truth Pluralism and Mathematics

When I started out as a graduate student writing a thesis defending mathematical nominalism, my naïve view was that mathematical claims (the correct ones) were true in a different way to empirical claims (the correct ones). Before long though, under the influence of Tarski, the basic model theory I was teaching in Logic 1, etc. I came to reject that view. Truth had to do with satisfaction in the Tarskian sense which required a domain of objects to do the satisfying. Recently though I came across this passage from Huw Price:

[W]e need to distinguish the notion of keeping track as something that we do within the assertoric language game – a notion constituted, within the game, by the fact that the normative structures always hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player – from a notion that we might employ from outside the game, in saying that in at least some of its versions its function is to aid the players in keeping track of their physical environment. … [T]here’s a temptation to call both kinds of external constraint ‘truth’, but we shouldn’t make the mistake of thinking we’re dealing with two aspects of sub-species of a single notion of truth. Both notions may be useful, for various theoretical purposes, but we shouldn’t confuse them’ (Expressivism, Pragmatism and Representationalism, 191)

I’m beginning to think that my naïve view might just be the right one. Any assertoric discourse [Could there be a non-assertoric discourse? I’m not sure.] with standards of correctness and incorrectness will need a truth predicate for the sorts of expressive purposes deflationists get excited about. (We need a truth predicate to express commitments without having to state them explicitly. E.g. if a theory \(\Gamma\) entails infinitely many things I can say ‘Everything \(\Gamma\) entails is true’ but I can’t possibly explicitly assert everything entailed by \(\Gamma\)  Hence truth predicates are indispensable for certain expressive purposes.) Mathematical discourse seems like a good candidate for a discourse that’s governed by internal rather than external standards. In empirical matters the world gets to answer back: we bump up against the world, probe it, test it experimentally. But mathematical investigations don’t involve interactions with mathematical objects, they involve proofs. Of course the results of these investigations can surprise us or be counterintuitive. But none of that requires an external world of mathematical objects; only that the normative structures constituted within the game ‘hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player’. Objectivity doesn’t require objects. Moreover, as I’ve argued before there’s good reason to think that the norms of correctness and incorrectness of mathematical discourse are internal, rather than external, in the senses above.

Horwich and Wittgenstein's Metaphilosophy

In a recent book, Wittgenstein’s Metaphilosophy, Paul Horwich defends (as the title might lead one to expect) a Wittgensteinian metaphilosophy which opposes the idea that philosophical thinking can lead us to surprising, substantive metaphysical results, and accords good philosophy the more deflationary role of dissipating confusions. This is a view I find attractive and plausible. According to Horwich, Wittgenstein’s metaphilosophy involves the claim that within the domain of philosophy:

There are no surprising discoveries to be made of facts, inaccessible through the methods of science, yet discoverable ‘from the armchair’ by means of some blend of pure thought, contemplation, and conceptual analysis. (Wittgenstein’s Metaphilosophy, 1–2)

Philosophical problems arise from 'bewitchment of our intelligence by means of language’. This typically involves overgeneralisation: applying some principle, which is locally valid in some domains, where it doesn’t belong:

[A] common way for us to be mislead by language, according to Wittgenstein, is that when we see a noun appearing as the subject of a true sentence, we expect there to be a thing to which the noun refers. This expectation comes from reflection on countless statements such as

- Neptune is a planet

- Boston has subways

- Plato taught philosophy

which suggest a universal underlying semantic structure for all sentences of that simple syntactic type: the subject, a noun phrase, picks out a particular object in the world, and the rest of the sentence denotes some property or characteristic attributed to that object. (ibid. 11)

Now, all this sounds great for the nominalist. Here is a nice way to diagnose how philosophers erroneously arrive at platonism: we expect, by overgeneralising on some paradigmatic cases, that whenever nouns feature in true sentences there are things to which the nouns refer. But mathematical language isn’t like this—that isn’t it purpose, mathematical standards of correctness and incorrectness haven’t to do with accurately depicting an abstract mathematical realm, etc.—we have simply overgeneralised from the paradigmatic representational use of language. Having diagnosed this overgeneralisation we are free to see that platonism is unmotivated, and adopt nominalism accordingly.

This isn’t what Horwich does though—he’s a staunch anti-nominalist. Why? I think the answer lies not in Horwich’s philosophy, but in the “pre-philosophical” convictions he brings to it. Horwich talks about this deflationary metaphilosophy allowing us to retain the ‘naïve idea that numbers are abstract objects [and] our naïve aspiration to discover what is true about them’ (ibid. 16). Horwich was a pre-philosophical platonist, so his belief that philosophy could not produce surprising, substantive metaphysical results leads him to believe that it cannot overturn platonism. I, on the other hand, was a pre-philosophical nominalist; so the self-same belief that philosophy cannot produce surprising, substantive metaphysical results leads me to think that the existence of mathematical objects can’t be discovered by philosophical means.

I wonder if the lesson here is that the platonism-nominalist debate is irresolvable by philosophical means. I wonder, but I’m not sure; and that’s because even some of our pre-philosophical convictions can arise from the sorts of confusions that good philosophy can dissolve. In this case, I think that attention to mathematical-linguistic practice could plausibly show that it’s not in the business of depicting an extant mathematical realm; but that’s a story for another day.

The norms of mathematical discourse and inquiry

David Lewis once (influentially) commented that it would be ludicrous to expect mathematicians to change their ways on the basis of philosophical arguments that mathematical objects don’t exist. Why he thought mathematical practices would have to be emended in the light of ontological facts about the existence of mathematical objects, I’m not sure.

Here’s a thought experiment to make explicit your own implicit commitments about this. Imagine that, instead of a philosopher, an infallible oracle told the world that mathematical objects don’t exist. Would mathematics professors be obliged to hand in their resignations? Would their discipline have been exposed as a sham?

I think the answer to these questions is a, very obvious, “no”, and I suspect that almost everyone would agree. But notice what that means. If we don’t accept that mathematical practices ought to change in light of word from an infallible oracle that mathematical objects don’t exist, then we must also accept that the norms governing mathematical discourse are not representational, in the robust sense of that word as pertaining to mapping, tracking or picturing how things stand with a domain of mathematical objects. The standards of correctness and incorrectness in mathematics do not derive from mathematical objects, but from standards internal to the game (or perhaps “game”) of mathematics itself.

Call this view normative nominalism. But if one is committed to normative nominalism (as a “no” answer to the above questions would reveal), then what could possibly be the motivation for platonism?

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’.

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.

Two Kinds of Indispensability Argument

Continuing on the theme of nominalism and pragmatism…

The Putnam of yore took it that mathematical objects exist and is credited along with Quine as being an early proponent of the indispensability argument.  There are though two very different kinds of indispensability argument that Putnam made.  The first runs like this:

[Q]uantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.  This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes. [Philosophy of Logic: 347]

Why be a platonist?  Because, according to the argument, nominalism is inconsistent with physics.  One big problem with the argument is that there is plenty that gets quantified over in the sciences that we don’t take to exist; especially idealised versions of physical systems, the stock examples being frictionless surfaces, continuous fluids and the like.  So nominalism’s being “inconsistent with physics” in this sense isn’t a big deal, since the (clearly true) claims that fluids are not continuous, that there are no frictionless planes etc. are also “inconsistent with physics”.  (Penelope Maddy in Naturalism in Mathematics and Mary Leng in Mathematics and Reality both make this kind of point.)

Putnam also made a very different kind of indispensability argument, often conflated with the first, that goes like this:

I believe that the positive argument for realism has an analogue in the case of mathematical realism.  Here too, I believe, realism is the only philosophy that doesn’t make a success of science a miracle.  [Philosophy of Logic: 73]

There is an important shift from looking flatly to what entities are quantified over in our best scientific theories to looking at what quantification over these entities can be used to achieve.  This pragmatic spin is in fact necessary because quantification over mathematical objects is not indispensable simpliciter (if such a notion even makes sense), but indispensable for certain ends.  We could do without quantification over mathematical objects; we might just also have to do without iPhones, air travel and so on, if we did.  As Sellars famously said (in ‘A Semantical Solution to the Mind-Body Problem’) ‘[c]learly human beings could dispense with all discourse, though only at the expense of having nothing to say’.  So, if the indispensability of quantification over mathematical objects is supposed to be a problem for nominalism, it must be because talk of mathematical objects must be made use of to achieve certain ends; in which case what is at issue are mathematical practices.  The best explanation for the success of mathematical practices must involve the existence of mathematical objects, or so the thought goes.  But here’s the kicker: mathematical objects, because they are acausal, changeless and not subject to any events, cannot be invoked to explain any practices.  So the best explanation of the success of science needn’t invoke mathematical objects.

Horwich vs. Nominalism

In a recent exchange with Huw Price (Expressivism, Pragmatism and Representationalism) Paul Horwich makes an 11-point case against nominalism (he calls this a case against naturalism, but it seems to be nominalism in particular that is being targeted).  The argument is quite condensed, but it provides a nice summary of why (I think) so many philosophers embrace some form of platonism (or 'anti-nominalism' for those who prefer the apophatic formulation) as well as why (I think) the case against nominalism goes awry.

Firstly, there is the purported motivation for nominalism (for 'naturalism' read 'nominalism'):

1. Naturalism rests on the impression that non-natural facts would be intolerably weird. 

2. That impression has three sources: first, the singular practical and explanatory importance of naturalistic facts; second the very broad scope of the naturalistic order – the striking range and diversity of the facts that it demonstrably encompasses; and, third, the feeling that reality must 'surely' be fundamentally uniform – so all facts must be naturalistic. [EPR: 124-5]

This theme is continued:

6. The committed naturalist will not be greatly perturbed by the accusation that [his defence of naturalism is] ad hoc, contrived and intrinsically implausible.  For he will reason that although such defects may indeed be present, and are indeed unwelcome in themselves, they are a price well worth paying for the wonderfully simple metaphysics that naturalism provides. [EPR: 125]

Now, perhaps some nominalists really are motivated by metaphysical simplicity, and for some reason take simple pictures of reality to be intrinsically more plausible than complicated ones; but nominalists don't, as a matter of habit, primarily motivate their view  by appealing to any metaphysical claims of this kind.  On the contrary, the most prominent  defenders of nominalism, such as Field or Leng, object to platonism on epistemological grounds.  Knowledge of abstract (and hence acausal) objects is problematic because the existence or non-existence of mind-independent abstract objects can make no difference to any grounds one might have for believing in them.  Since the existence of abstract objects has no consequences for anything could possibly take place it seems impossible in principle to (i) provide justificatory grounds for belief in abstract objects, or even (ii) provide some hardcore externalist model of belief in abstract objects that could explain how these beliefs could count as knowledge, even in the absence of justification.  (Note that this applies to indispensability arguments: the indispensability of quantification over mathematical objects in scientific theories does not depend on the existence of a domain of abstract mathematical objects—it is a function of more mundane things, such as the complexity of the concrete systems being modelled and the expressive resources available to the agents doing the modelling.)

The crux of the matter however seems to lie in the pro-case for platonism:

4. Note, to start with, that it's prima facie extremely implausible that amongst the facts we recognise, some are non-natural – for example, that there are numbers […]  An unbiased consideration of such facts will indicate that they aren't naturalistic.  For it's as plain as day (to anyone not 'in the grip of a theory') that they aren't spatio-temporally located, aren't engendered by facts of physics and don't enter into causal/explanatory relations with other facts. 

7. But this apology for naturalism is inaccurate in two related respects.  In the first place, what is given up for its sake is not justly described as 'local theoretical simplicity'.  For what must be denied are data – epistemologically basic convictions.  It is blindingly obvious to us … that Julius Caesar wan't a number. […] And no less obviously false are certain implications of every one of the sceptical 'error theories' (i.e. denials of existence) and strained reductive analyses aimed at safeguarding naturalism from the threats posed by numbers…  

8. And, in the second place, the norm of simplicity, as it is deployed in science, is not in fact a licence to reject recalcitrant data … A scientist is obliged to respect all relevant data, and when they don't conform to a simple pattern, that reality must be accepted. [EPR: 125-6]

The thought that facts about an abstract domain of numbers are simply data, I would hazard a guess, is an important motivation for contemporary platonists, and explains why arguments for nominalism are often simply written off on the grounds that they entail an unacceptable conclusion.  But are we right to see these as data?  Horwich seems to hold that everyone is (or was, at some point) a pre-theoretical platonist, and adopted nominalism for the sake of metaphysical simplicity.  I suggested before that the second part of this claim is false, but the first part is also, at least to some extent, inaccurate.  I for one was a pre-theoretical nominalist: I didn't realise that anyone believed in the existence of numbers until I took a class in metaphysics, and when I made mathematical claims I didn't take the purpose of this practice to reside in describing a domain of abstract objects.  It would be interesting to see some stats on the topic, but at the very least, a number of people don't take facts about an abstract domain of numbers to be data (I've met a few).  So, not everyone has epistemologically basic convictions about the existence of mathematical objects, and to treat these as data is question-begging within the context of this debate.

Besides being question-begging in the current context, there is a kind of hard-line stance against the appeal to "epistemologically basic convictions" of this sort—on the grounds that it is unduly conservative: insulating views from criticism—as articulated, with characteristic understatement, by Kant:

To appeal to ordinary common sense when insight and science run short, and not before, is one of the subtle discoveries of recent times, whereby the dullest windbag can confidently take on the most profound thinker and hold his own with him.  So long as a small residue of insight remains, however, one would do well to avoid resorting to this emergency help.  And seen in the light of day, this appeal is nothing other than a call to the judgement of the multitude; applause at which the philosopher blushes, but which the popular wag becomes triumphant and defiant. [Prolegmonena: 4:259]

Now, Horwich isn't exactly a dull windbag or a popular wag (nor, for that matter, was Reid); and, in any given debate, something will be treated as data, if only temporally, since possessing some common commitments is a precondition of debate in the first place.  So what sorts of things ought one allow as data in the debate between platonists and nominalists?  A plausible supposition is that the division between platonists and nominalists coincides with a division over the kind of things that one allows as data.  Platonists will take the data to be true sentences about mathematical objects, whilst nominalists will take the data to be mathematical practices.  I take it that the latter view is the correct one (and usefully non-question-begging in the context of this debate): what must be accounted for are facts about how (e.g.) solving differential equations and carrying out measurement procedures allows us to track features of and make predictions about concrete systems. This would explain why nominalists (like me) are drawn to nominalism: it is very hard to see how the existence of abstract objects would be required to explain any practice. The take-home claim (that I've not really defended here with any rigour): there are deep connections between nominalism and pragmatist methodology.