Abstract
Invoking a form of quantifier variance promises to let us explain mathematicians’ freedom to introduce new kinds of mathematical objects in a way that avoids some problems for standard platonist and nominalist views. In this paper I’ll note that, despite traditional associations between quantifier variance and Carnapian rejection of metaphysics, Siderian realists about metaphysics can naturally be quantifier variantists. Unfortunately a variant on the Quinean indispensability argument concerning grounding seems to pose a problem for philosophers who accept this hybrid. However I will charitably reconstruct this problem and then argue for optimism about solving it.
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Notes
By this I mean that, for each such quantifier sense there is some possible language such that all applications of the standard syntactic introduction and elimination rules for the existential quantifier within that language are truth preserving. However, that does not mean that one can form a single language containing both quantifier senses and then apply the introduction and elimination rules to prove the equivalence of these senses. See Warren (2014), among others, on this point.
That is, these variant quantifier senses need not be interpretable only as ranging over some subset of the objects which exist in the fundamental quantifier sense, in the way that we might say the “all’ in a typical utterance of “all the beers are in the fridge” restricts a more generous quantifier sense to only range only over objects in the speakers house.
See, for example, Hirsch (2011), Eklund (2009) and Chalmers’ characterization of Quantifier Variance as (roughly) the idea that, “there are many candidate meanings for the existential quantifier (or for quantifiers that behave like the existential quantifier in different communities), with none of them being objectively preferred to the other.” (Chalmers 2009).
See the argument that (even from Sider’s point of view) we don’t actually speak a language with Sider’s maximally joint carving quantifier sense in most philosophical contetexts (including discussions of metaphysics and ontology).
Note that saying some kinds of objects (e.g., cities, numbers) might not exist in the sense relevant to the Sider’s fundamental ontology room doesn’t amount to saying that these objects ‘don’t really exist’. It is entirely compatible with truthful assertion that these objects literally exist in the course of daily life (and while studying ethics or the metaphysics, of money and gender, or writing philosophy of mathematics papers like this one)—much as acknowledging that rabbits don’t exist on the the (relatively) more natural and joint-carving quantifier sense employed by fundamental physics is compatible with saying rabbits literally exist in most ordinary contexts, including biology seminars. When outside the fundamental physics/ontology room, our position on such objects seems much more naturally expressed by saying that rabbits/holes/cities/numbers might not be fundamental than that they don’t really exist.
Also note that (as discussed in REDACTED) using quantifier variance does not require one to accept that normal English employs verbally different expressions corresponding to at least two different quantifier senses (a metaphysically natural and demanding one and a laxer one), so that it might be true to say things bad-sounding things like “composite objects exist but they do not really exist” in certain contexts. With regard to any particular context we can fully agree with David Lewis that, “The several idioms of what we call ‘existential’ quantification are entirely synonymous and interchangeable. It does not matter whether you say ‘some things are donkeys’ or ‘there are donkeys’ or ‘donkeys exist’...whether true or whether false all three statements stand or fall together.” (Lewis 1990).
See, for example Balaguer (2001).
We can imagine this structure being formed by adding objects which behave like a layer of classes over our original mathematical universe, and then note that the result must be larger than the original universe for Cantorian reasons.
One appealing way to do this is to explain knowledge of mathematical structures which mathematicians seem to have categorical conceptions of (stateable in something like the language of second order logic), like the natural numbers and the real numbers, by saying mathematicians axiom choice acted like an ontologically inflationary stipulative definition. One could take the same attitude towards the hierarchy of sets if you were willing to accept a conception of the hierarchy of sets that was categorical with respect to height as well as width. However, I argue in REDACTED there are set theory specific reasons (relating to the Buralli-Forti paradox) for treating this case differently and avoiding arbitrariness in a different way by taking a potentialist approach to set theory along lines of Putnam (1967), Linnebo (2013), Hellman (1994). I further suggest that the resulting limited, independently motivated departure from Benacerraf’s desideratum of treating mathematical talk similarly to ordinary object talk is not so bad.
It seems that the nominalist must either unattractively say that mathematicians statements are literally false (Recall Lewis saying, “I am moved to laughter at the thought of how presumptuous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways, and abjure countless errors, now that philosophy has discovered that there are no classes?” (Lewis 1991)), or say that mathematical statements have a different logical form from claims which ordinary speakers treat similarly (e.g, apparent existence claims about holes and countries).
Despite these advantages, many questions have been raised about Quantifier Variance and the Quantifier Variance explanation of mathematicians’ freedom. For example, worries have been raised about whether the Quantifier Variantist can say something attractive about the following. What would happen if mathematicians simultaneously adopted a pair of internally consistent, but incompatible, conceptions of pure mathematical structures? What would happen if mathematicians’ adopted a conception of some mathematical structure which imposed undue constraints on the total size of the universe (e.g., a logically coherent collection of axioms describing a purported mathematical structure which imply that the total universe contains at most 100 things?). I articulate some of my preferred answers to these standard worries see REDACTED.
See, for example, Field’s remarks at the beginning of Field (1980).
Technically appeal to the metaphysical semantics lets Sider eliminate the ‘in virtue of’ notion above from his theory, and restate completeness as follows, “New completeness: Every sentence that contains expressions that do not carve at the joints has a metaphysical semantics.” Sider’s examples of such a metaphysical semantics often have the form of a truth theory ‘Sentence S of L is true in L iff \(\phi\)’ (where \(\phi\) is a sentence involving only fundamentalia). But he writes “Metaphysical semantics are not required by definition to take any particular form. They must presumably be compositional in some sense (since they must be explanatory and hence cast in reasonably joint-carving terms, and must contend with infinitely many sentences). But this still allows considerable variation.” (Sider 2011).
Note that this difference in terminology doesn’t reflect a commitment by Sider to only use the most fundamental quantifier sense when doing philosophy or even metaphysics. He also accepts that one sometimes does philosophy using less fundamental quantifier senses. It is merely a pure terminological difference.
As before, this new structure would not already have been instantiated within the original mathematical universe because it must be strictly larger than that universe by Cantor’s diagonal argument that the power set of a set is always larger than the original set.
Note that this same argument would let us derive a contradiction from the assumption that all possible human-describable structures are instantiated by fundamentalia (the new structure created would be such a possible structure). But if (contra Sider) we can’t actually refer to the fundamental quantifier sense (or if the reference trick above wouldn’t count as an acceptable conception of a mathematical structure, we don’t get an outright contradiction from the assumption that all mathematical structures we could posit are fundamental. However, we still suffer from the issue of arbitrariness.
Admittedly, it’s not too hard to answer grounding and paraphrase challenges as it applies to pure mathematical statements, and certain kinds of simple applied mathematical challenges. For they can simply re-purpose nominalists’ existing logical regimentations of these statements as stories about grounding. For example in Hellman (1994) Hellman nominalistically regiments each pure mathematical sentence \(\phi\) about the natural numbers, with (simplifying slightly) a claim about logical necessity of the form \(\Box\)[If there are objects with the intended structure of the natural numbers i.e., objects related by some relations as per the axioms of second order Peano Arithmetic (which categorically describe the natural numbers), then \(\phi\) holds of these objects.] The proponent of QVEMF can accept the surface logical form of S but instead take Hellman’s paraphrase strategy to show how the truth of S (and the existence of any mathematical objects it quantifiers over) can be seem as systematically grounded in the nominalistic fact \(\phi\). And Berry (2018b) suggests a way of conceptually simplifying these paraphrases.
However, it’s much less clear that one can adequately ground statements of applied mathematics (especially ones that make complex claims involving magnitudes like length and charge or probabilities). One might think that whatever blocks classic nominalists from systematically (nominalistically) paraphrasing contemporary physical theories involving objective probability (and the like), will also block Quantifier Variantist from providing an adequate grounding for such claims.
c.f. Sider’s Completeness thesis: “Completeness Every nonfundamental truth holds in virtue of some fundamental truth.”.
For example, one could also resist this argument by rejecting claim 4, the Purity thesis that fundamental facts must only involve fundamental notions. Saying the above would clearly involve some divergence from the Siderian framework above. For claim 4 was Sider’s purity thesis. But perhaps we can separately motivate this rejection by considering the puzzles about what grounds grounding facts (Fine 2010) which can be avoided by taking grounding facts to be fundamental.
If we reject Sider’s purity thesis we can say that mathematical objects are not metaphysically fundamental (and ground the truth of internal statements about then in facts about pure logical possibility as Hellman suggests), but still ground scientific facts in facts involving mathematical objects however the platonist would. For example we can ground scientific facts about mass in facts involving, say, real numbers and a mass/mass ratio relation relating objects/pairs of objects to real numbers just as standard platonist would. However this option would still seem to require treating some pure mathematical objects as special (by saying fundamental physical magnitude facts are/are grounded in relations to these objects), if we are to avoid massive redundancy and the return of the issues for plenitudinous platonism noted above. Thus I don’t see it providing many advantages over the Agnostic Platonist option discussed below.
See section 3.4 of Hellman (1994) and note Hellman’s explicit comment that complete success in this project would not suggest that we could remove quantification over mathematical objects from our best scientific theories.
Admittedly this style of response to the Revenge of Quinean indispensability challenge is somewhat (structurally) similar to a Melia’s defense of nominalism against Quine’s classic indispensability challenge in Melia (1995). Melia motivates the idea that “we should not always believe in the entities our best physical theory quantifies over” because quantifying over mathematical objects is just a tool to let finite creatures like ourselves express claims which less limited creatures would express by asserting infinite conjunctions/disjunctions of nominalistic sentences (e.g., a claim about the number of planets around the average star might abbreviate an infinite disjunction of nominalistic descriptions of specific universe states).
And an influential line of criticism maintains that if we accept Melia’s proposal—or in any other way drop the requirement that someone engaged in ontology state their best total theory of the world without quantifying over any objects they want to deny exist (in the sense relevant to the ontology room)—then we get a scenario where ‘anything goes’ as regard to ontology. That is, we loose any concrete grip we may hope to have had on how to settle ontological questions—and thereby perhaps any grip on what questions of traditional ontology mean. I’m not sure whether this criticism ultimately works against classic nominalists like Melia. For the the inference from, ‘ If \(\lnot P\) then we don’t have a coherent and fruitful grip on the project of philosophical ontology’ to ‘P’ can seem like a case of unjustified wishful thinking.
But even if this argument cut ice against a nominalist like Melia, we should note that the quantifier variantist who rejects mathematical fundamentalia and demands for finitary grounding has special tools for answering it which the nominalist does not. For, philosophers are already independently working on a theory of grounding and formal constraints on when one thing can be said to be grounded in another, and the Quantifier Variantist can say that this prevents it from being the case that ‘anything goes’ with respect to grounding.
Relatedly, one might argue that philosophers giving a traditional ‘hard-road’ response to the Quinean Indispensiblity challenge (by providing a paraphrase for their best scientific theories) are likely to try to write paraphrases using notions Quine would accept. Thus, they are likely to use only first order logical quantification, and to avoid use of modal vocabulary (like a notion of logical possibility/coherence. In contrast, when QVEMF theorist attempts to explain how applied mathematical facts could be grounded in facts about relations between non-mathematical objects plus facts about logical possibility, they are free to use modal notions like logical coherence/possibility. For a menu of options for how such modal vocabulary can be useful see chapter 3 of Hellman (1996).
See Berry (2018a) for an argument that these access worries about logical coherence are solvable.
Perhaps grounding considerations motivate thinking that some collection of mathematical objects which are sufficiently plentiful and richly structured to do certain work in applied mathematics exist fundamentally. But, as has often been remarked (For example, see Clarke-Doane 2014) indispensability considerations don’t seem to justify belief in any particular mathematical structure as different mathematical structures seem capable of doing the same work in regimenting/grounding our physical theories.
Indeed, one might argue as follows. Applied mathematics hasn’t seemed to motivate a unique choice of which mathematical structures exist, because (from a traditional platonist point of view) the total collection of mathematical objects must do two jobs. It must make sense of applied mathematics and everything we could study in pure mathematics. Given this goal, it has seemed natural to consider both, e.g., both a free standing real number structure and a copy of the real numbers within various larger structures, like the hierarchy of sets (containing objects for pure mathematical study), as candidates for mathematical reference within our best physical theories. And there’s no uniquely natural choice of a collection of mathematical objects which does both jobs.
However the agnostic platonist does not expect fundamental mathematical objects to do both these jobs. (As noted above) they can take the truth of existence claims about pure mathematical objects to be grounded in something like facts about logical possibility. Thus it seems more plausible that whatever aspects of our best physical theories make appeal to some fundamental mathematical objects indispensable (if such there are) should suggest a unique most natural collection of mathematical structures to take to be grounding fundamental.
See McSweeney (2019) for defense of an analogous agnostic realist perspective on logical concepts.
Thanks to an anonymous reviewer for pointing this out.
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Berry, S. Quantifier Variance, Mathematicians’ Freedom and the Revenge of Quinean Indispensability Worries. Erkenn 87, 2201–2218 (2022). https://doi.org/10.1007/s10670-020-00298-1
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DOI: https://doi.org/10.1007/s10670-020-00298-1