Phil. of maths

Parsons’s Mathematical Thought: Sec. 15, Mathematical modality

Chapter 3 of Mathematical Thought and Its Objects is called “Modality and structuralism”. Before turning to discuss modal structuralism in Secs. 16 and 17, Parsons discusses what kind modality it might involve. Setting aside epistemic modalities as not to the present purpose, he considers (i) physical (or natural) necessity, (ii) metaphysical necessity (truth in all possible worlds), (iii) mathematical necessity, (iv) logical necessity (meant in a narrow sense that can be explicated model-theoretically).

Parsons argues that we don’t want to spell out a modal structuralism in terms of (i) natural modalities: “it demands too much to ask that the structures considered in mathematics be physically possible; indeed, in the case of higher set theory, there is every reason to believe that they are not physically possible.” I’ll buy that.

Second, Parsons argues that logical possibility — in the sense explicated via the idea of there being a suitable model — reveals itself as itself a mathematical notion, given that models are (at least typically) mathematical entities. So(?), “It is very doubtful that a generous notion of logical possibility would be distinguishable in a principled way from … mathematical possibility.”

But there is surely something rather odd here. For the idea, to repeat, is that we explicate “it is logically possible that P” (in the generous sense of allowed-at-least-by-considerations-of-logical-form, that runs beyond metaphysical possibility) in terms of there being a mathematical model on which P can be interpreted as true. It seems we don’t have a modality in the explanans here. Indeed, Parsons himself remarks on the common view that a mathematical truth (falsehood) is necessarily true (false): and on that view the very idea of a kind of “mathematical possibility” distinguished from plain truth evaporates.

So I’m left puzzled when Parsons concludes that the two runners for the kind of modality that might be involved in a modal structuralism are metaphysical modality and mathematical modality: for I just don’t have a grip on the latter notion.

(Relatedly, Parsons reads Putnam as holding that “it is mathematically possible that there should be no sets of uncountable rank, although it is a theorem of ZF that there are such sets”. Again, I really just don’t know how to construe that “mathematically possible” if that is supposed to be neither epistemic nor equivalent to “true”.)

Parsons’s Mathematical Thought: Sec 14, Structuralism and application

We’re considering the schematic idea that an ordinary arithmetical statement is elliptical for something generalizing over structures, along the lines of

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),

where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N – {0}) to be “simply infinite”, and A(N, 0, S) is appropriately correlated with the ordinary statement. (Parsons, you’ll recall, associates such a view with Dedekind. That doesn’t seem historically correct. But let that pass.)

Does this “eliminative structuralist” view have a problem accounting for the application of numbers as cardinals? Recall Frege’s remark: “It is applicability alone that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of necessity.” And Frege further takes it that an account of numbers should start from their use in counting (so a structuralist understanding that explains the nature of arithmetical truths prior to explaining their application is going wrong). But, Parsons argues, our structuralist in fact can resist that further thought.

I’m not sure I fully have the measure of Parsons thinking here. Part of the trouble is that he slips back into talking of numbers as objects (e.p. pp. 74–75), while I thought the attraction of the eliminative structuralism was to get rid of numbers as a special kind of object. But I take it the thought is something like this. Counting some objects involves putting them into one-one correspondence with an initial segment of some paradigm simply infinite system (of numerals, say). That involves setting up some external relations between some members of the relevant simply infinite system, over an above the internal relations which constitute their being a such a system. But now, via the Dedekind categoricity theorem, we see that these external relations will engender a one-one correspondence with an isomorphic initial segment of any simply infinite system. So, in counting, we automatically get an implicit generalization over simply infinite systems — which is what, according to the eliminative structuralist, talk of numbers amounts to. Hence, as Frege wanted, even on the structuralist view, we do after all have an essential connection between numbers and their application in counting.

That, I think, does deal with the supposed general problem. Now, Dummett has raised a more specific problem — roughly, defining a simply infinite system doesn’t tell us whether its initial element is to be treated as 0 or 1 (or indeed, I suppose, 42). But Parsons (rightly in my view) doesn’t find this worry a telling one for the structuralist. He can regard it as just a matter of pragmatic convention whether, in applications, we start counting at 0 or 1, depending on how much we care about having a number for empty collections.

One final comment on this section. Having quieted worries about the structuralist view, Parsons remarks that as well as the natural number 3, we have the integer 3, the rational 3, the real number 3 and the complex number 3 (not to mention more exotic constructions). And the structuralist can say that the use of “3” each time signifies not the same entity but the same structural role, a point congenial to his general account of the significance of number words. But, contra Parsons, I don’t see that the multiple use of “3” counts at all against the Fregean view that numbers are specific objects. The Fregean can just say that there are here a number of different terms, (“natural number 3”, “rational number 3”, etc.) with different objects as reference — with the common elements of the referring terms justified by the likeness of the role of the denoted objects in the respective families.

Parsons’s Mathematical Thought: Sec 13, Nominalism and second-order logic

A general comment before proceeding. Parsons himself says that this book has been a very long time in the writing. And I suspect that what we are reading is in fact a multi-layered text with different passages added at different times, without the whole being finally reorganized and rewritten from beginning to end. This does make for a bumpy read, with the to-and-fro of argument not always ideally well signalled.

Anyway, Sec. 13 falls into two parts, both related to nominalist takes on second-order logic. First, Parsons offers some remarks on the Fieldian project of using mereology to do the work of second-order logic. The key thought is this. For mereology to do all the work Field wants, it needs an (impredicative) comprehension principle: “Given a predicate of individuals that is true of at least one individual, there is a sum of just the individuals of which the predicate is true, and moreover, the admissible predicates will be closed under quantification over all individuals, including those very sums.” (Cf. the principle “Cs” in Field’s “On Conservativeness and Incompleteness”.) But what entitles Field to such a strong comprehension principle? Well, Parsons notes that it’s not clear that Field can offer any direct a priori argument (but then, I wonder, would he want to?). The justification will be that “the comprehension principle is a hypothesis justified by its consequences in systematizing the geometrical basis of physics”. But then “Field’s view, on this reading, puts him in a position in which we have found other formulations of nominalism: making the justification of mathematics turn on some hypothesis about the physical world, which is more vulnerable to refutation than the mathematics.”

But how troubled will a Fieldian be by that complaint? Suppose we decide that our physical theory of the world doesn’t require such a strong comprehension principle (we can get away with recognizing a less wide-ranging plurality of regions). That’s not at all implausible, actually, given that (nearly) all the mathematics required for physics can be reconstructed in a weak second-order arithmetic like ACA_0 with only predicative comprehension. Then the Fieldian response will (surely?) be just to demote the full mathematical apparatus of the classical reals from its status in Science without Numbers as a supposedly justified tool for getting more nominalistically acceptable consequences out of our best physics. It is no longer so justified. In that sense, for the Fieldian, the “justification” of a bit of mathematics is wrapped up with our hypotheses about the physical world, and Parsons’s complaint will seem question-begging. [Or am I missing something here?]

The second part of Sec. 13 considers Boolos’s attempt to make second-order logic ontologically tame by giving a plural reading to the second-order quantifiers. The thought under scrutiny is that plural quantification is ontologically innocent because, in plurally quantifying over Fs, we are just committing ourselves to Fs (not to sets or to Fregean concepts). Parsons’s discussion [or again, am I missing something here?] initially advances familiar sorts of worries about this claim of innocence. But Parsons does make one point towards the end of the section that I find very congenial (i.e. I’ve argued similarly myself!).

Consider (say) the range of second-order arithmetics that Simpson discusses in SOSOA. As we advance through theories with stronger and stronger comprehension principles, then — on a standard platonist construal — we are countenancing more and more sets of numbers. If we reconstrue the second-order quantifiers plural-wise, then, as we go from theory to theory, we are countenancing more and more …. well, more what? It is tempting to say “pluralities”. And indeed it is convenient to give an informal gloss of the plural reading using talk of pluralities. But — if this isn’t to smuggle back reference to pluralities-as-single-entities, i.e. sets — this convenient way of talking needs to be eliminable (cf. Linnebo’s nice article on plural logic). So how do we eliminate it here? We might, I suppose, trade in talk of countenancing more and more pluralities for talk of allowing more and more different ways we can take numbers together: but this seems tantamount to re-instating Fregean concepts as the values of the second-order variables — which is fine by me, but then the supposed ontological gain of interpreting the second-order quantifiers via plurals is lost.

The question then is this: if we accept the pluralist’s contention that we can treat second-order numerical quantifiers as ontologically committing just us to numbers, period, then how are we to think of the surely varying commitments we take on with varying strengths of comprehension principle. As Parsons puts it, “If there is no enlargement of ontological commitment [my emphasis] as one passes to less restricted versions of the comprehension schema, then perhaps that speaks against the importance of the notion.”

Parsons’s Mathematical Thought: Sec. 12, Nominalism

This is a short and rather insubstantial section, which I’m just taking separately to get out of the way, because the next section is weighty (and one of the longest in the book).

Parsons understands ‘nominalism’ Harvard-style — no surprise there, then! — to mean the rejection of abstract entities and the eschewing of (ineliminable) modality. What hope, then, for giving a response to the potential-vacuity problem for eliminative structuralism about arithmetic (say) which meets nominalist constraints? We can’t, by hypothesis, go modal: so what to do?

Well, as the physical world actually is (or so we might well now believe), there are in fact enough physical things — e.g. space time points — and suitable physical orderings on them to give us physically realized ‘simply infinite’ structures. But Parsons is unhappy with this way of meeting the vacuity worry, and for familiar reasons: “[S]hould it be taken as a presupposition of elementary mathematics that the real world instantiates a mathematical conception of the infinite? This would have the consequence that mathematics is hostage to the future possible development of physics.”

But (although I have no particular nominalist sympathies myself), I’m not sure how worried the nominalist eliminative structuralist should be about giving such hostages to fortune. As things are, given how we believe the world actually to be, he can reasonably continue to speak with the vulgar and treat arithmetical claims as true or false. Even if the worst happens, so we come to believe the world is ultimately grainy and finite in all respects, it’s not that ‘school-room’ arithmetic is going to get undermined. At most, it is the idealizing rounding out of school-room arithmetic which insists on an infinitude of numbers. And if it should emerge that the rounding out, construed the eliminative-structuralist way, collapses in vacuity — well, formal arithmetic can still be played as an intriguingly entertaining game. It’s just that then, after all, the nominalist eliminative structuralist who is relying on physical realizations for structures can no longer readily construe idealized arithmetic’s claims as true or false, and so the nominalist has to sound a bit more revisionary. But, he’ll say, so what? (Parsons says “a great deal of the historically given mathematics would have to be jettisoned in this case” — but that’s too quick. Talk of ‘jettisoning’ covers over a slide. For no longer thinking of arithmetic as construable as literally true by the eliminative structuralist manoeuvre is not the same as throwing arithmetic into the trash-can, as any fictionalist will insist.)

What about the other line that offered to the nominalist at the end of Sec. 11? — i.e. sidestep the vacuity problem by going modal in an anodyne way (“interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted”). Well, again Parsons sees trouble, this time arising from the fact that there might be physical limitations in how big a proof-token could be, and so a theory could count as (nominalistically) consistent — because no proof of an inconsistency could be tokened — even if we can show that there is a process which, given world enough and time, would produce an inconsistency. But again, I’m not sure that the obstreperous nominalist couldn’t swallow that too.

At the end of this section, Parsons revisits the question of how to frame an eliminative structuralism for arithmetic. He looked at a move from a set-theoretic formulation to a more ‘logical’, second-order formulation. But could we go first-order, in a way more congenial no doubt to those of nominalist inclinations? The trouble is, of course, that we won’t get categoricity (whatever we build into the axioms), so the eliminative structuralist who goes first-order runs up against the intuition that the natural numbers have a unique structure. But how secure, in fact, is that intuition? Parsons raises that excellent question (too often passed over in silence), but only to shelve it until Ch. 8. So we’ll have to return to that later.

Parsons’s Mathematical Thought: Secs 8 – 11

Back, after rather a gap, to Charles Parsons’s book and on to the first half of his second chapter, “Structuralism and nominalism”.

(Sec. 8) Parsons says that he himself thinks that “something close to the structuralist view is true”. But structuralist in what sense? It is often said, perhaps in a Bourbachiste spirit, that mathematics is the study of structures. But — as Parsons stresses — that leaves it wide open what picture we should adopt of the ontology of mathematical objects. He is more concerned with structuralism(s) with more ontological bite — something along the lines suggested by “the objects of mathematics are positions in structures, [and] have no identity or features outside of a structure” (to quote from Michael Resnik’s well-known 1981 Nous paper).

(Sec. 9) But what are structures? The usual modern mathematical story sees these as sets (or classes) with distinguished elements, equipped with relations and/or functions. So it looks as though an account of mathematical objects as positions in structures already presupposes familiar kinds of objects (sets, classes) to build structures out of, and explaining their nature in structuralist terms threatens circularity. But Parsons puts this worry on hold for the moment.

(Sec. 10) So go with the set-theoretic conception of structure, just pro tem, and consider as an exemplar Dedekind’s treatment of the natural numbers. Dedekind defines what it is for a set N, with distinguished element 0, and a mapping S: N -> N – {0} to be “simply infinite”. Abbreviate those (categorical) conditions Ω(N, 0, S). With some effort, an ordinary statement of arithmetic can be correlated with a version A(N, 0, S) whose primitives are again N, 0, S. And on one reading of Dedekind — the eliminative reading — the suggestion is that the ordinary statement can be treated as elliptical for

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S).

This is ‘eliminative’ in that a statement apparently about one kind of thing, numbers, is treated as in fact a disguised generalization about other kinds of things. The suggestion neatly sidesteps “multiple reduction” problems for more straightforward attemps to reduce arithmetic to set theory. But (on the face of it) it faces the worry that if there are no simply infinite systems then any ordinary arithmetical statement comes out as vacuously true and arithmetic is inconsistent. True, that first worry won’t be pressing if we already buy into a background universe with enough sets, but it will become more urgent when we try to repeat the trick and give an eliminative structuralist account of them. And there’s a related second worry. Ω(N, 0, S) will involve quantification over sets, as indeed will a typical A(N, 0, S) as we give explicit definitions of e.g. recursive arithmetical functions. Do we want really want a structuralist account of a particular familiar kind of mathematical object, numbers, to tells us that we’ve been generalizing about some other rather less familiar kind of object all along? (Parsons wonders: Maybe we need to generalize over structures to state structuralism as a general thesis: but does a structuralist account of a particular kind of object have to similarly generalize over structures?)

(Sec. 11) Well, we can sidestep the second of those worries, and the worries of Sec. 9, perhaps, by trading in an explicitly set-theoretic presentation of Dedekind’s eliminative structuralism for a version couched in second-order logical terms. We get a new second-order definition of being simply infinite, Ω'(N, 0, S), a new correlate of an ordinary arithmetical claim, A‘(N, 0, S), and correspondingly a new suggestion that the ordinary statement can be treated as elliptical for

For any N, 0, S, if Ω'(N, 0, S) then A’(N, 0, S).

where now ‘any N‘ and ‘any S‘ are treated as second-order. If we are relaxed enough about second-order quantification, we might find this easier to swallow that the previous version (though that’s quite a big “if”). However, this kind of ‘if-thenism’ is still threatened by the possibility of vacuity. What to do?

One option is to read the conditional as stronger-than-material, e.g. by discerning a governing modal operator. But that opens up another set of problems. What kind of modality is involved here? Can we e.g. give a modest possibility-as-consistency reading? Perhaps “we interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted.” The suggestion is to be pursued critically in Sec. 12.

OK, so much by way of brisk summary of these sections (I didn’t find them entirely easy to follow, but I hope I’ve fairly represented the way the discussion develops). I don’t think I have much to add by way of commentary: in fact, the dialectic so far is a pretty familiar one.

Parsons’s Mathematical Thought: Sec. 7

Back in Sec. 1, Parsons says “Roughly speaking, an object is abstract if it is not located in space and time and does not stand in causal relations.” In the last section of the first chapter, he returns to question of characterizing abstract objects, and suggests a distinction among them between pure abstract objects (e.g. pure sets) and those which “have an intrinsic relation to the concrete” — Parsons calls the latter quasi-concrete.

As a paradigm example of the quasi-concrete, Parsons takes the example of sentence types: “what a sentence [type] is is a matter of what physical inscriptions are or would be its tokens”. (Actually, just as an aside, I suppose we might wonder whether sentence types might be a counter-example to the claim that abstract objects lack temporal location. We might ask: did the sentence type “the cat is on the mat” really exist in 2000 BC before anyone spoke English?)

But how should we generalize from this case? Parsons writes “What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete ‘representation'”. The scare quotes are there in Parsons — and you can see why. Should we really say, for example, that a sentence token is a representation of its type? Your first response might be: the token isn’t about the type, so isn’t a representation of it. But, reading on, it becomes clear that Parsons doesn’t mean representation but representative. And then, yes, we might say that the token is a representative of the type. Parsons also writes “Although sets in general are not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership.” (no scare quotes!). Again, we might say the spoon in my coffee cup is a representative of the set of cutlery (though not a representation).

How clear is the idea of “having a concrete representative”? You might have supposed that the Earth’s equator is a candidate for belonging with sentence types as tangled with the concrete. But does the equator have a concrete representative? Could it? What about that old Fregean example, the direction of a line. Of course there can be physical lines with that direction; but it doesn’t seem quite natural to me to say a particular line is a representative of the direction. (We might say the equator or a direction could have a representation, painted on the ground!)

Parsons’s discussion here thus seems to me to be rather undercooked. To be sure, it is plausible to say that some abstract objects are more purely abstract than others, but I don’t think he has given a sharp characterization of the phenomenon.

But let’s go, for the moment, with his notion of the quasi-concrete. Then he raises the question, are numbers quasi-concrete? We might be tempted to say yes, suggesting that the number five, for example, has the concrete representatives like: ||||| . Parsons makes two Fregean points against this. First, to take that block as representative, we have already to take it as a set or sequence of strokes (rather than as a single grid, for example). So the representative here is not strictly concrete but itself quasi-concrete. Perhaps then we can say that numbers are quasi-quasi-concrete (meaning they have quasi-concrete representatives). But second, that can’t be the whole story, as numbers can number anything, including the purely abstract. (Parsons says he is going to return to talk about this in Chapter 6, so I’ll say no more for the moment.)

Parsons’s Mathematical Thought: Sec. 6, ‘Being and existence’

At the outset of this section, Parsons writes that one point at which “reservations about standard first-order logic as the universal measure of ontology can affect the notion of mathematical object is the ancient question whether reference to objects is necessarily reference to objects that exist.”

A comment before proceeding. Note that Parsons had earlier (Secs 1 to 4) proposed that (1) “speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification” to make serious, and indeed true, statements. And defending that view about, so to speak, the measure of what objects we are committed to falls short of saying that (2) standard first-order logic is the universal measure of ontology in general. Resisting the more sweeping claim is quite consistent with accepting Parsons’s initial Fregean claim about objects. Not that I’m suggesting that Parsons thinks otherwise. I’m just emphasizing that if (e.g. as a Fregean) you are not persuaded by Parsons Sec. 5 suggestions, and hold that we are committed to entities that are not objects, then you can accept formulation (1) without accepting (2).

Anyway, what of reference to objects that is not reference to objects that exist? Parsons discusses Meinongian views in some detail (this is one of the longest sections in the book). Here’s part of his final summary of the discussion.

We are left with the question whether the “true” meaning of the existential quantifier is [i] the permissive Meinongian one [allowing quantification over objects that do not exist], [ii] existence that allows freely for abstract objects but that rules out impossibilia, or [iii] something like actuality. The logic based concept of object does not decide between these alternatives, although, once it has been set forth, the case for [iii] is weakened. But in order to understand the notions of object and existence in mathematics we have to put more flesh on the bare form given by formal logic. We need to fill out the logic-based conception by looking at cases. … [C]onsiderations proper to mathematics will not lead us to favour [i] over [ii]. General as the notion of object in mathematics is, there is still a constraint of possibility, coherence, or consistency that objects postulated in Meinongian theories are allowed to violate.

The talk here of having to “fill out the logic-based conception” might initially seems surprising given what has gone before. But, though he is not entirely clear, I assume that what Parsons means is simply this: the Fregean thesis is that objects are just whatever are we have to construe terms that behave in the right sorts of way in true sentences as referring to. So, to fill out that general template view about objects, we have to say what kinds of sentences we do in fact accept as being true. If we e.g. take statements like “Sherlock Holmes is more famous than any living detective” and “There’s a fictional detective who is more famous than any living detective” at face value as true claims then (the suggestion goes) we have to accept (i) the Meinongian line that there are objects that do not exist. If we paraphrase away apparent talk of fictional objects and the like, but accept that there are true mathematical statements talking of numbers, sets, etc., then (ii) we are not committed to non-existent objects, but have to accept that there are abstract objects which aren’t “actual”. If we insist on also paraphrasing away apparent straight talk of numbers (e.g. construing it as governed by an operator “in the arithmetical fiction …”), then perhaps (iii) we may only be committed to actual objects.

Parsons is sceptical about whether we have any need “to admit into the range of our quantifiers such objects as the golden mountain, the round square, Pegasus and Sherlock Holmes”, though it is not his concern to argue for this here. But he does argue that “considerations proper to mathematics” don’t give any impetus for preferring the Meinongian views (i) over (ii). Mathematics doesn’t countenance impossibilia like the round square, or present itself as fictional discourse. As to (iii), I assume Parsons thought is that a critic of our common-or-garden standards of mathematical truth on the basis of a metaphysical repudiation of abstract objects is (in danger of) getting things upside down, at least by the lights of the truth-first, “logic-based conception” of objects, according to which we don’t have a handle on the notion of an object except via a prior grip on the notion of truth for the relevant object-referring statements.

If this reading of Parsons is right, then I agree with him.

Parsons’s Mathematical Thought: Sec. 5

Parsons has been proposing the view that “speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification”. And the focus so far has been on first-order quantification. But what about generalizations about properties, the sort of generalization involved in familiar mathematical statements like the induction principle for arithmetic, or the separation axiom in set theory? Should we construe those as involving generalization over something like Frege’s “unsaturated” concepts, entities which aren’t objects? Or is the commitment here just to more objects? I’ll try to outline some of Parsons’s discussion (though I did not find it always easy to construe).

One way of perhaps resisting the Fregean line arises from noting that we can easily parlay quantification into predicate position into just more quantification into subject position (or so it seems). Suppose, using Parsons’s notation, we use ‘(Ox)Fx‘ to denote some object corresponding to the Fregean concept expressed by ‘F…’. And suppose we use ‘$’ for an appropriate copula (‘has’ if the object is a property/quality, ‘is a member of’ if the object is a set, etc.) Then we have Ft if and only if t $ (Ox)Fx. And so, given a context when we are minded to quantify into the position held by ‘F‘ we could instead first nominalize and then quantify into the position held by the singular term ‘(Ox)Fx‘ instead. It seems then that we can treat quantification over properties (as we might initially put it) as just more quantification over a kind of object. This after all seems common mathematical practice, as when we familiarly regiment second-order arithmetic as a theory of numbers and sets of numbers.

Still, at least two objections to the nominalizing strategy as an across-the-board way of eliminating ‘direct’ quantification into predicate position readily suggest themselves (as Parsons notes). First, the claim that Ft if and only if t $ (Ox)Fx is, itself, intended as a generalization, to express which we need to generalize into predicate position in a way that can’t be nominalized away. And second, that generalization in any case has to be restricted or else or we could instantiate with the predicate ‘¬x $ x‘, and paradox ensues.

However, that’s not yet game set and match to the Fregean. Can’t the force of the first objection be turned by adding the device of semantic ascent to our armoury? We can, for example, generalize about the possibility of nominalization by saying that for any predicate ‘F’ (and term ‘t’), ‘Ft‘ is true if and only if ‘t $ (Ox)Fx‘ is true.

Ah, it will be protested, the device of semantic ascent still doesn’t really allow us fully to capture what we want to say by means of quantifications over properties. Compare for example the familiar thought that the content of the full informal arithmetic induction axiom is not captured by semantically ascending and saying that all instances of the first-order schema are true. Reply: that familiar thought is true, if we confine the instances to a fixed language. But suppose we treat the schema in an open-ended way, available to be instantiated however we extend our language (as Parsons puts it, “In practice, in any language in which we talk about natural numbers, we are prepared to affirm induction for any predicate of that language”). Then, by treating the schema as open-ended we arguably recapture the intended sweep of the informal axiom still without taking on ontological commitments to Fregean concepts.

And as to second objection against the nominalizing strategy, the threat of paradox only arises if we take the reference of ‘(Ox)Fx‘ as an object that is, so to speak, already in the original domain of objects (i.e. of subjects of predication). But we could take the moral here to be that objects segregate into different types, the references of nominalized predicates being of a different type to the references of common-or-garden singular terms.

So where does this take us? Parsons summarizes: “the present discussion does show that considerations about predication do not lead inevitably to our taking second-order logic as our canonical framework and admitting, as values of our second-order variables, entities that are not objects.”

Three comments about all this. First, about semantic ascent and the open-ended nature of our commitment e.g. to the induction schema. Just why do we stand prepared to take on all-comers and instantiate the schema with any novel predicate we care to extend our language with? Kreisel suggested long since that we accept the instances of the induction schema because we already accept the full second-order induction axiom. I think there are issues about that claim (which I can’t pursue here and now). But the claim is a familiar one that many have found persuasive. And a fuller defence of the idea that we can avoid taking second-order quantifications at face value would require Parsons to say more about this.

Second, about avoiding paradox on the nominalizing strategy. The Fregean might well riposte that saying that the way to go is to segregate objects into different types just sounds like theft of Frege’s key insight rather than an alternative story. After all, speaking with the vulgar, the Fregean will say that what he is arguing for is precisely a distinction among “entities” between saturated and unsaturated types, between objects and concepts. So he has a principled type story to tell. And, he will add, once the distinction is made in the right way, the temptation to pursue the nominalizing strategy, putting all the work of unifying propositions into a copula, should evaporate. And what is the alternative principled story supposed to be?

Third, I’m left unclear exactly how Parsons thinks about the relationship between the two ways of avoiding second-order quantification that he discusses (i.e. the routes via nominalization and ascent). He does say that “The laws of logic have a certain dialectical character, in that the method of nominalization and the method of semantic ascent can both be used to state them, and neither can completely displace the other.” I’ve wrestled with this a bit, and I don’t have a clear grasp of the point. (And helpful comments on that here would be welcome!)

Parsons’s Mathematical Thought: Secs. 1-4

Right, as promised, time to make a start commenting on Charles Parsons’s long awaited Mathematical Thought and Its Objects (CUP, 2008).

For those who haven’t had a copy in their hands, this is a pretty substantial volume (pp. xx + 378). Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years; but glancing ahead the material indeed seems to be reworked into a continuous book. The nine chapters are divided into 55 sections numbered continuously through the book, and those divisions will be very handy here: I’ll aim to comment on small groups of sections (from one to three or four) at a time. From what I’ve seen so far, the book needs and repays slow reading.

Chapter 1 is entitled Objects and Logic. And the claim to be defended is that “Speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification to make serious statements”. Thus construed, the idea of objects in general is loosened from ties with the idea of actuality (Kant’s Wirklichkeit) — where this has something to do with “act[ing] on our senses or at least producing effects which may cause sense-perceptions as near or remote consequences” (to quote Frege). Talk of objects is also loosened from ties with ideas of intuitability (whatever that Kantian idea comes to: things are left pretty murky at this stage, but then Parsons is going to talk a lot about intuition later in the book). Consequently, endorsing the logical conception of an object will “defuse too-high expectations of what the existence of objects of some mathematical type such as numbers would entail.” The suggestion is that those who are inclined to deny abstract objects, or find them puzzling, are illegitimately(?) imposing requirements on being an object that go beyond those captured in the logical conception.

Now, I’m entirely sympathetic to the Fregean line Parsons is following here. He says that “its most important advocates in more recent times are Carnap and Quine”. But I would have added Dummett’s name to the list, starting with his early paper on nominalism: and Dummett initiated the most sophisticated development of the Fregean line in the hands of Crispin Wright in his Frege’s Conception of Numbers as Objects, and then particularly Bob Hale’s Abstract Objects (neither of which Parsons mentions here).

I’m not sure, though, in quite what spirit Parsons is proposing “the view that the most general notion of object has its home in formal logic”.

Actually, as an aside, I’d remark that that surely isn’t the happiest way of summing up the view. After all, suppose we translate back from first-order logical notation into a disciplined core fragment of English — the sort of regimented English whose sentences are equivalent to the content of the logical wffs (and indeed the sort of English which we use in giving determinate content to the artificial language in the first place). Then here too we will find the core devices of singular terms, predication, identity and quantification. And the Quinean will presumably say that our commitments to objects are revealed equally well by rendering our theory of the world into the idioms of this disciplined core of ordinary language. Or if that’s not exactly right, because we can never quite discipline English enough (e.g. we can’t quite ensure that “It is not the case that …” always expresses propositional negation), then this is not, so to speak, a deep failing of the vernacular. Formal languages don’t magically do what ordinary language can’t do: they just do ordinary things like use singular terms and quantify in tidier ways. So turning to “formal logic” doesn’t really give us a different take on the general notion of object. Surely Parsons spoke better when he expressed the position he is proposing as the view that “speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification” to make serious, and indeed true, statements.

But to continue, as I said, I’m not sure in quite what spirit this view is being advanced. The fully Fregean line would be to insist that objects are what are referred to by singular terms in true sentences, and a singular term is whatever walks, quacks, and swims like a singular term in a disciplined way. We can’t first pick out a class of genuine objects and then locate the genuine singular terms as those that refer to them: it goes the other way about (e.g. from identifying true sentences by the appropriate mathematical criteria, via identifying the singular terms in those sentences by their compositional behaviour, to insisting that those singular terms functioning in truths refer to mathematical objects).

But suppose you rejected that line. You might still think, in a Quinean spirit, that such is the mess and conversational plasticity in our various ordinary ways of talking that to determine when we are committed to objects of one kind or another, the best thing to do is to see how things look when we regiment our claims into a well-understood disciplined core discourse of singular terms, predication, identity and quantification — the apparatus formalized in first-order logic.

A couple of Parsons’s remarks suggest the stronger and more contentious Fregean line. But then it is perhaps odd that he doesn’t more explicitly argue for it, and engage with the Dummett/Wright/Hale defence.

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