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?

Wright on Deflationism

I’m reading Crispin Wright’s Truth and Objectivity for a reading group at the moment, where he sums up an argument against deflationism about truth in the following way (I quote at length):

The deflationist holds that “true”, although gramatically a predicate, denotes no substantial quality of statements, or thoughts, but is merely a device of assertoric endorsement, of use to us only because we sometimes wish so to endorse a single statement, referred to in a way which doesn’t specify it’s content, or batches of statements all at once. Apart from applications of those two kinds, it is, for the deflationist, a complete explanation of the truth predicate that it satisfies the Disquotational Schema. It is a consequence of this general conception of the role of the truth predicate that it can register no norm governing assertoric discourse distinct from warranted assertibility. Yet the central place assigned to the Disquotational Schema—and thereby to the Negation Equivalence—actually clashes with that consequence, for it follows that, while normative of assertoric discourse, and indeed coincident in (positive prescriptive) normative force with warranted assertibility, “true” is nevertheless potentially extensionally divergent from warranted assertibility—and hence has to be accounted as registering a distinct such norm. Since it’s compliance or non-compliance with a norm distinct from assertoric warrant can hardly be an “insubstantial”property of a statement, and since a uniform account is possible of what it is for any particular statement so to comply, deflationism collapses. (pp. 71–2)

For reference, the Disquotational Schema is:

(DS) “P” is T is and only if P

One can’t, without engaging in a kind of doublethink, say or believe things like ‘P and it is not warrantedly assertible that P’ for some proposition P, since you can rationally assert P if and only if it is warrantedly assertible (for you) that P. But truth can be used to contrast with warranted assertibility. Take (DS) and substitute ‘It is not the case that P’ for P:

(i) ‘It is not the case that P’ is T is and only if it is not the case that P.

From (i) and (DS) one can infer:

(ii) It is not the case that P if and only if it is not the case that ‘P’ is T.

And from (i) and (ii) you get:

(iii) ‘It is not the case that P’ is T if and only if it is not the case that ‘P’ is T.

But (iii) cannot be right if T means warrantedly assertible, so ‘true’ registers a norm distinct from warranted assertibility. Deflationism is the view that ‘true’ just is a devise of assertoric endorsement, and Wright thinks that a mere devise of assertoric endorsement couldn’t register a norm distinct from warranted assertibility, so deflationism must be false.

But there is a way to register the truth norm without using the truth predicate. One can say ‘P and it is not warrantedly assertible for S that P’ or ‘¬P and it is warrantedly assertible for S that P’ where S is some person other than yourself, and in doing so can register the truth norm without using the truth predicate. What is required is that one contrasts one’s own perspective with that of another. Registering the truth norm requires some kind of I-Thou contrast. If that’s the case then ‘true’ is not what, fundamentally, allows one to register the truth norm contrasting with the norm of warranted assertibility, and deflationism is off the hook.

I think this might tap into something deep about objectivity—more specifically, our ability to see the world as being objective or to conceptualise there being objective facts that outstrip our ability to know them—as it coheres with something Robert Brandom says about objectivity. Brandom (I won’t spell out the details here) also argues that conceptualising objectivity requires I-Thou relationships. This is made explicit in paradigmatically referential of-statements like ‘He believes of this criminal that he is an innocent man.’ Understanding “of” requires navigating between one’s own perspective and that of another. Since “of” is how we refer objectively to the world, talking (and hence thinking) about the world objectively requires navigating between one’s own perspective and that of another.

Why I really hope we vote no on Thursday

Maybe I’m odd this way, but I love Britain. I love Britain because it has a kind of contrapuntal brilliance—the way the craggy summits of the Munroes perfectly compliment the gentle pastures of Cambridgeshire, and Edinburgh stands like a poised and dignified sister to exuberant, thrumming London. And I love Britain because the rest of the UK isn’t Westminster—it’s J.R.R. Tolkein, and The Wind in the Willows, and Radiohead, and Wallace and Gromit, and Stewart Lee, and Viz, and Bertrand Russell, and Ant and Dec, and my cherubic little nephew.

I believe in many of the things that have made breaking away from the UK seem attractive to a lot of people: essentially the benefits of political localism—a political class that is close to, and so responsive to, the needs and concerns of the constituents they serve. The thing is, that all these things could be achieved, without the damage involved in breaking up the union, through devo max. Not only that, devo max is the democratically mandated option; it’s what most Scottish people actually want. Far better that than the division we’ll have within Scotland if we permanently break away from the UK on the basis of a tiny majority of separatists. Moreover, it’s an option that’s on the table if and only if we vote no in the upcoming election.

This isn’t primarily why I’m voting no, because I would vote no even if devo max wasn’t on the table. It’s not primarily for economic reasons either, although I would by no means dismiss these as somehow crass or “not what’s really important”. Questions about the economy just are questions about how the most vulnerable in our society will fare. They’re also questions about how my family will fare. If the dire warnings about recession, the flight of business, the disaster of shared currency without political union and mortgage rates skyrocketing are even close to true, breaking up the union might mean losing our home.

But the fundamental reason I’m voting no isn’t economic, because I would vote no even if the economic consequences of separation weren’t so grim. It’s not economic because even if, by some miracle, we were sitting on an oil bonanza, I wouldn’t, for one moment, resent it paying for someone’s medical treatment in Yorkshire, or Liverpool, or wherever. The reason I'm voting no has to do with the value of unity itself. People within a nation state have differing political, religious and ethical convictions. A nation state is held together by bonds of mutual trust, solidarity and coöperation that somehow transcend these things. Just look to Iraq, Syria or any other truly dysfunctional states to see that these coöperative bonds are not natural necessities but deeply contingent, and deeply valuable. A united kingdom is a precious and hard-won achievement, and it would be a terrible waste to throw that away by choice. Salmond’s convinced many of us that the SNP are somehow progressive visionaries. That’s what his well-greased rhetoric is designed to suggest anyway, though the facts don’t match the bluster. Unionists are voting for unity, separatists are, in practice, voting for the opposite.  Scots have always been cosmopolitan and internationalist in outlook, and have always used Britain to make our mark in the world.  But if we break up the UK and sever the unique bonds of coöperation that link compatriots, we will, in a very real, very concrete way, be making Scotland a less inclusive, less open and more parochial place. That’s the politics of division, and that’s why I really hope we say no to it on Thursday.

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.