Recent discussions of how axioms are extrinsically justified have appealed to abductive considerations: on such accounts, axioms are adopted on the basis that they constitute the best explanation of some mathematical data, or phenomena. In the first part of this paper, I set out a potential problem caused by the appeal made to the notion of mathematical explanation and suggest that it can be remedied once it is noted that all the justificatory work is done by appeal to the theoretical virtues. In the second part of the paper, I appeal to the theoretical virtues account of axiom justification to provide an argument that judgements of theoretical virtuousness, and therefore of extrinsic justification, are subjective in a substantive sense. This tells against a recent claim by Penelope Maddy that such justification is “wholly objective”.
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The justification we provide for non-axiomatic mathematical beliefs is likely to be motley: we take some mathematical beliefs to be justified on the basis that we (or, more likely, someone else) possess a proof or proof-sketch, some may be non-deductively justified whilst others may be non-inferentially justified.
For discussion of potential examples of such disagreements, see Barton, Ternullo and Venturi ms, pp. 6–8; Clarke-Doane (2013, pp. 473–474).
Of course, it’s unlikely that in scientific practice, observation statements are strictly deduced from the theories accepted.
This sort of explanationist account (offered as an account not of extrinsic justification, but axiom justification simpliciter) is spelled out in more detail in work by Barton, Ternullo and Venturi (Barton, Ternullo & Venturi ms). I take the considerations raised against such accounts in the next section to tell against any account of axiom justification that places explanation at its heart.
Thanks to an anonymous reviewer for pressing me to be clearer about the degree to which the explanatory account of axiom justification captures Russell and Gödel’s regressive method.
There is, of course, a prior bifurcation into explanations within mathematics and mathematical explanations of non-mathematical facts.
However, Baron et al. (2017) do some of the groundwork for an account of mathematical explanation that appeals to counterfactuals, leaving the door open to a genuinely monist account of explanation.
Barton, Ternullo and Venturi suggest additional theoretical virtues (though they do not use this name to refer to them): theoretical completeness (where a theory T1 is more theoretically complete than another theory T2iff there is a sentence implied by T1 that is not implied by T2 and not vice versa), predictive power (in terms of a candidate axiom having consequences that are already provable without the candidate axiom)—and there could well be more.
Maddy’s Thin Realist accepts what set theory says about sets but demurs from attempting to answer any further questions about what sets are like. These questions are not settled by set-theoretic practice and so, the story goes, for the properly naturalistic philosopher of mathematics they do not require settling. Maddy holds that a descriptive account of mathematical practice is consistent with both Thin Realism and what she calls Arealism. The Arealist agrees with the nominalist that we lack good reason to think that sets exist—but thinks that this is so because “well-developed methods of confirming existence and truth aren’t even in play” (Maddy 2011: p. 89). Maddy holds that Arealism should be thought of as distinct from standard forms of nominalism because such positions appeal to “a priori prejudice against abstract objects” or “preconceptions about what knowledge must be like” (ibid: 97). I’m inclined to think that whether Maddy’s Arealist is a kind of nominalist will bottom out as a terminological dispute. If nominalism is the view that we lack good reason to believe in mathematical objects, then the Arealist is a nominalist in a sense that the Thin Realist (as well as, obviously, the Platonist) is not. If nominalism, however, must incorporate substantive views about knowledge, or the nature of abstract objects, then the Arealist is not a nominalist. I see no substantive reason for holding that one or other of these candidate meanings of ‘nominalism’ is to be preferred to the other. Thanks to an anonymous reviewer for pushing me to be clearer on how the relationship between nominalism and Maddy’s Arealism is to be understood.
There are open questions about exactly how these two kinds of objectivity are to be spelled out and understood. The formulations above are sufficient for capturing the distinction appealed to in the following three sections: in Sect. 4.5 I turn in more detail to the question as to how exactly we ought to understand, in particular, the strong form of objectivity.
As should be clear from the below, none of the argument requires accepting any of Kuhn’s substantive claims about scientific progress, etc.
Although we should be careful to not infer from the fact that all of us would discount a weighting that (for example) prioritised simplicity above all other virtues to the conclusion that the discounting of such a weighting stems directly from the scientific matter being dealt with.
The focus of recent critical discussion of the no neutral algorithm claim has concerned whether or not there is any algorithm for theory choice that satisfies certain plausible criteria (in short, whether Kuhn thought he saw a vast bounty of potential algorithms when instead there is a lack). Okasha (2011) argues, co-opting Arrow’s impossibility theorem from social choice theory, that there is no such algorithm. See Bradley (2017) and Marcoci and Nguyen (2019) for replies.
An anonymous reviewer raises a potential consequence of such consensus: namely, if one thinks that agreement qua consensus (amongst some subset of agents) can in and of itself be reason to think that this agreement tracks truth, then there is at least one potential sense in which consensus about virtue-weighting and that particular weighting being connected, in the right way, to the relevant set-theoretic facts. For this sort of argument to be made good on, one would have to make the case both that the relevant sociological thesis is true (that such consensus exists) and that the purported connection between consensus and truth can be made good on, two matters that are beyond our current scope.
Consider, as an analogy, Field’s informal argument for thinking that mathematics is conservative over physical theories (put loosely, that if a nominalistic statement is derivable from a scientific theory with both mathematical and nominalistic content, then it could have been derived from the nominalistic content of the theory alone—that adding mathematical content to a nominalistic scientific theory doesn’t introduce any new derivable nominalistic consequences). Field notes that “it would be extremely surprising if it were to be discovered that standard mathematics implied that there are at least 1060 non-mathematical objects in the universe, or that the Paris Commune was defeated […] good mathematics is conservative” (Field 2016: p. 13). Field’s thought here is that it is constitutive of mathematics, or at least of good mathematics, that it doesn’t (by itself) have consequences for non-mathematical affairs like the Paris Commune. Similarly, some set theory that implied that one particular weighting function was to preferred over another would be bad set theory—weighting functions are not in its proper domain. One should be careful, here: there are some stronger claims in the vicinity that are plausibly false. Some badly-conceived weighting functions might be ruled out for mathematical or logical reasons and, in this sense, mathematical facts can have bearing on the choice of weighting functions. However, barring some extremely surprising result that there is in fact only one mathematically possible weighting function that meets a set of criteria, mathematical facts alone are insufficient to single out a particular weighting—and it is this that one would require to get the result that the weighting function is found ‘in the math’ rather than ‘in us’. One final consideration (raised by an anonymous reviewer) is that one should leave open the possibility that a weighting function is determined by some non-set-theoretic but also non-subject-sensitive facts—that is, some facts from some other scientific domain. This suggestion does seem to complicate the bifurcation between the weighting function being located either in some facts about us as subjects or in some set-theoretic facts: however, if one is convinced by the argument above (that it is constitutive of good set theory, of good cosmology, etc. that it is silent on weighting functions), then one should be similarly convinced that it is constitutive of other branches of scientific endeavour that they are silent on how virtues ought to be weighted in the foundations of set theory.
In Sect. 5.1. I suggest that the falsity of this claim about a universal norm would constitute a good explanation for some facts about set-theoretic practice.
Saatsi (2017) makes a similar point regarding the use of explanationist considerations to choose between rival metaphysical theories (pushing back against the kind of anti-exceptionalism about theory choice in philosophy that is put forward by Williamson). There is an interesting sense in which the claim about the lack of empirical feedback—at least when it comes to the use of the theoretical virtues in set theory—might be pushed back against, however. As an anonymous reviewer notes, for some of those invested in pursuing the program of extending ZFC, there are constraints on the program being well-founded. That is, the program must involve conjectures such that (if these conjectures fall a certain way), the program is ended. This clearly has similarities with the kind of feedback we may get regarding scientific theories when they generate a prediction that fails to pan out. Whether or not the set-theoretic constraint therefore counts as properly empirical seems like another semantic choice-point regarding broadening the extension of a term.
Barton, Ternullo and Venturi mention weighting as one of the questions left open by their explanatory account, asking: “is it possible to come up with a way of assigning different theories [meaning collections of axioms] weights and comparing them satisfactorily?” (Barton, Ternullo and Venturi ms: 28). Although they mention assigning weights to theories (or collections of axioms), it seems as though what we want to assign weights to is the various good making features of axiom collections (the theoretical virtues) and using this weight-assignment to produce an overall assessment of the rival collections. Their open question, then, is slightly different to the question pressed by the weighting objection. Their question concerns whether or not it’s possible to produce some means of weighing up axiom collections that instantiate the various virtues to different extents: in Kuhnian vocabulary, whether or not one could produce an algorithm, neutral or otherwise. The weighting argument isn’t an argument that there could be no algorithm: it’s an argument, in Kuhn’s words, that there can be no neutral algorithm. So, a positive answer to Barton, Ternullo and Venturi’s open question is fully consistent with subjectivism—the subjectivist about axiom justification merely presses that whatever weighting we produce will come from “in us” rather than “the math”.
This isn’t, of course, to say that it’s altogether straightforward to show that, for each of the theoretical virtues, a (collection of) axiom(s) instantiating that particular virtue is subject-insensitive—just that the falsity of the weak objectivity claim isn’t a consequence of the weighting argument.
Some comparative claims will, practically speaking, require no weighting of virtues—but those that are going on in active set-theoretic debate will.
One might worry that taking ‘being theoretically virtuous’ and ‘being mathematically deep’ to be functionally equivalent terms occludes a possible response on behalf of a Maddy-influenced objectivist. If mathematical depth is the good-making, justification-conferring property of axioms, then the problem of weighting doesn’t arise: as there’s nothing to weigh depth against. On this line of response, the weighting problem is a consequence of the theoretical virtues account of extrinsic justification, but it’s too quick to sign Maddy up to this account. However, I think it’s clear that in Maddy (2011), ‘mathematical depth’ is a term used to refer to a variety of good-making, justification-conferring properties of axioms (and concepts, etc.): Maddy herself says that she “lumped a number of different notions together under a broad umbrella of ‘depth’” (Ernst et al. 2015b: p. 245). I think a case can be made that using the terminology of ‘depth’ (when, at the very least, judgements about depth will involve judgements about fruitfulness, unifying power, etc.) occludes more than it helps: its aesthetic connotations, ironically enough, making subjectivism about such judgements look more tempting than it ought initially to do.
Thanks to an anonymous reviewer for raising this worry.
Thanks to an anonymous reviewer for suggesting this sort of approach to demarcating weak and strong objectivity.
Of course, if we had a compelling argument that the theoretical virtues are truth-conducive (either simpliciter or in local set-theoretic contexts), then this might constitute the beginnings of a case that the mere fact that a collection of axioms instantiates the virtues to various extents speaks (defensibly) in favour of its acceptance.
We should stop short of building into the notion of a cognitively flawless agent (regarding some proposition P) the idea that such agents only believe truly (regarding that proposition). One could be cognitively flawless regarding some proposition, receive exclusively misleading evidence regarding P, and therefore believe falsely regarding P (but not as a consequence of some cognitive flaw).
An anonymous reviewer expresses the worry that if we can sensibly understand some set-theorist who advocates for the acceptance of ZFC without extension as a cognitively flawless agent, then there might be some sense in which absolute undecidability collapses into ordinary undecidability. However, according to the notion of an agent being cognitively flawless set out above, no actually existing set-theorist is going to count as a cognitively flawless set-theorist (even the best amongst us, sadly, are not logically omniscient, for example). At most, some actual agents are going to be approximations of cognitively flawless agents (where this might, but need not, involve such actual agents having the same set of beliefs as cognitively flawless agents). What might be behind the thought that some actual set-theorists may be cognitively flawless, however, is the notion that some actual set-theorists might be epistemically blameless. I would be a better cognitive agent were I aware of all the logical relations between propositions (and therefore, I fail to be a flawless agent in the sense that is relevant here), but it would be inappropriate to blame me for not being aware of these logical relations, in virtue of the fact that I can do nothing to bring it about that I have such awareness.
Thanks to an anonymous reviewer for this interesting suggestion about explaining the possibility of absolute undecidability.
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Thanks to Alex Franklin, Eleanor Knox, audiences and reading groups in London, and two anonymous reviewers for this journal for helpful comments on various earlier versions of this material.
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Heron, J. Set-theoretic justification and the theoretical virtues. Synthese (2020). https://doi.org/10.1007/s11229-020-02784-z
- Set theory
- Axiom choice
- Penelope Maddy
- Theoretical virtues
- Abductive inference