The goal of Mark Povich’s Rules to Infinity is to develop a novel account of distinctively mathematical explanations and, along the way, to elaborate, defend, and extend the neo-Carnapian programme of normativism (Thomasson 2020) to philosophy of mathematics and philosophy of science.

Let me focus on his account of distinctively mathematical explanations first. These are explanations of natural phenomena where the mathematics bestows upon the explanandum a kind of necessity that mere effects of causes do not possess (p. 2). Consider Lange’s (2013) trefoil knot case. Terry failed to untie his knot. What explains this failure? The ‘empirical fact that Terry’s knot is a trefoil knot and the mathematical (knot theoretic) fact that the trefoil knot is distinct from the unknot (i.e., mathematically cannot be untied)’ (2013, p. 28). There ‘are no admissible moves of twisting, lifting, or crossing strands without cutting them […] that can transform the trefoil knot into the unknot’ (2013, p. 28). This mathematical impossibility, along with the fact that Terry’s knot is in fact a trefoil knot, jointly explain his failure. This is a distinctively mathematical explanation of a natural fact.

In chapters 2, 3, and 4, Povich develops his novel account of distinctively mathematical explanations: the narrow ontic counterfactual account (NOCA). Chapter 2 lays out three desiderata that any philosophical account of distinctively mathematical explanations should satisfy, and argues that Lange’s (2013) and Baron’s (2016, 2019) accounts do not satisfy one or more of them: First, the modal desideratum: A distinctively mathematical explanation should explain why the explanandum is modally robust. Terry could not have succeeded in untying his trefoil knot. Second, the distinctiveness desideratum: A distinctively mathematical explanations should be distinguishable from cases where mathematics merely features in an explanation without bestowing necessity. Third, the directionality desideratum: A distinctively mathematical explanation should run in one direction only. Just as Bromberger’s (1966) flagpole case shows that on the deductive-nomological model of scientific explanation, the height of the flagpole is explained by the length of its shadow, Craver and Povich (2017) have argued that explanations such as those in the trefoil knot case can be problematically ‘reversed’, and a good account of distinctively mathematical explanations must rule this out. According to Craver and Povich, the directionality constraint comes from the fact that reversed mathematical explanations intuitively strike us as non-explanatory. It would be very strange, for instance, to claim that Terry’s failure to untie his knot is what explains why the trefoil knot is mathematically distinct from the unknot.

Chapter 3 argues, contra Reutlinger (2014) and Batterman and Rice (2014), that renormalization group explanations are not distinctively mathematical explanations, but are a kind of (not necessarily causal) ontic explanation. Chapter 4 opens with a nice counter-mathematical from Jimi Hendrix: ‘if 6 turned out to be 9, I don’t mind. I don’t mind’. This is the essential chapter, in which Povich develops NOCA. NOCA falls under Povich’s generalized ontic conception of explanation, which claims that all explanations, causal and non-causal, count as explanations in virtue of allowing us to answer what-if-things-had-been-different questions (w-questions) about the explanandum (p. 94). That is, an explanation represents an ontic relation of counterfactual dependence holding between explanandum and explanans (p. 3).

NOCA claims that an explanation is a distinctively mathematical explanation just in case an empirical fact (weakly) necessarily depends counterfactually only on a mathematical fact (p. 97). This means that were the relevant mathematical fact different, the empirical fact would have been different too. The ‘(weakly) necessarily’ qualifier means that this counterfactual dependence must hold at every world where the empirical fact obtains, rather than at every possible world.

Why is this qualifier needed? The directionality desideratum requires that NOCA count the forward explanandum as a distinctively mathematical explanation while ruling out the reversed one. The problem is that both seem to follow from the same mathematical fact that the trefoil knot is distinct from the unknot. If Terry has a trefoil knot, it follows that he cannot untie it. But it equally follows that if Terry untied his knot, it was not a trefoil knot. Both the forward explanandum (Terry failed to untie his trefoil knot) and the reversed explanandum (the knot Terry untied is not a trefoil knot) are thus consequences of the same mathematical fact. A bare counterfactual account cannot distinguish them, since counterfactuals are evaluated world by world, and there will be worlds where the forward explanandum fails to counterfactually depend on the mathematical fact (for instance, a world where Terry is hit by a bus before attempting to untie his knot) and worlds where the reversed explanandum does (p. 99).

Povich’s solution recasts the trefoil knot case in terms of states of affairs rather than events (p. 100). An event is something that happens, like a glass falling off a table. A state of affairs is a matter of how things stand, like the glass being broken on the floor. In the trefoil knot case, the event is Terry’s failing to untie his knot, and the relevant state of affairs is Terry’s trefoil knot being distinct from the unknot. This distinction matters because states of affairs and events behave differently when we ask whether their counterfactual dependence on a mathematical fact is stable across worlds. When the case is recast this way, the forward and reversed explananda come apart. The forward state of affairs is that Terry’s trefoil knot is distinct from the unknot. The reversed state of affairs is that Terry’s untieable knot is not a trefoil knot.

Now consider what happens when we ask, at every world where each of these facts obtains, whether the relevant counterfactual dependence holds. The counterfactual is: were the trefoil knot isotopic to the unknot, Terry’s knot would have been isotopic to the unknot too. This counterfactual holds at every world where Terry has a trefoil knot because if the trefoil knot were isotopic to the unknot, then anything that is a trefoil knot would inherit that property, including Terry’s knot. The forward explanandum thus weakly necessarily counterfactually depends on the mathematical fact.

The same is not true of the reversed explanandum. There are worlds where Terry’s untieable knot is not a trefoil knot, but where the relevant counterfactual fails to hold: worlds, for instance, where Terry unties a different kind of knot entirely, one that has nothing to do with the trefoil. In those worlds, the fact that Terry’s untieable knot is not a trefoil knot does not counterfactually depend on the mathematical fact about the trefoil. The reversed explanandum therefore does not weakly necessarily counterfactually depend on the mathematical fact (p. 101). This is how NOCA satisfies the directionality desideratum. NOCA’s counterfactual dependence needs an underlying ontic relation too, and Povich’s preferred candidate is instantiation: the concrete object instantiates the abstract mathematical object and inherits its properties, which is naturally asymmetric and explains why the dependence is weakly necessary.

That’s the first half of the book. The second half, minus the interlude of chapter 7 (based on Povich and Dan Burnston’s forthcoming work on a ‘fully inferential theory’ of the content of mathematical models in science), is devoted to developing and defending meta-ontological deflationism. Chapter 5 ‘deflates’ NOCA by showing how modal normativism (as developed by Amie Thomasson 2020) can be extended to NOCA in a way that allows NOCA to block explanatory indispensability arguments for Platonism while maintaining its ontic bite. Chapter 6 defends a broadly inferentialist account of the conceptual content of mathematics and contains his defence of the compatibility of normativism, semantic deflationism, and inferentialism with truth-conditional semantics. Chapter 8 compares his mathematical normativism to Warren’s (2020) conventionalism, Fieldian fictionalism (Field 2022), and Linnebo’s (2018) brand of neo-Fregeanism. Povich argues that his mathematical normativism and Warren’s conventionalism are mostly compatible, that Field is wrong to think conventionalism is equivalent to fictionalism, and that the normativist can accept neo-Fregean abstraction principles with analyticity alone.

The most exciting sections of the second half, for readers interested in meta-ontology, philosophy of language, and neo-pragmatist approaches, are in chapters 5 and 6. Let me focus on those now. Traditionally, distinctively mathematical explanations have been employed in enhanced indispensability arguments for mathematical Platonism. Though Povich maintains the existence of distinctively mathematical explanations and maintains a generalized ontic conception of explanation, he will ‘normativistically’ deflate any arguments for Platonism that could follow from his view.

Modal normativism holds that necessity claims do not describe modal facts but rather express conceptual or semantic rules, or their consequences. ‘All bachelors are unmarried’ is not a description of a modal reality but an object-language expression of a rule governing the term ‘bachelor’. Povich extends this to mathematical necessity: claims like ‘2 + 2 = 4’ express semantic rules governing mathematical terms, and counter-mathematicals are ‘counterconceptual’ statements expressing the consequences of revising the semantic rules for our mathematical terms (p. 129). On this reading, ‘were the trefoil knot isotopic to the unknot, Terry’s trefoil knot would have been isotopic to the unknot’ is interpreted counter-conceptually: it expresses what would follow if we adopted revised semantic rules governing ‘trefoil knot’. The conclusion of chapter 5 is that the Platonistic language of NOCA, in terms of objects, instantiation, and mathematical necessity, can be retained while being stripped of any substantive ontological commitment to mathematical Platonism.

In chapter 5.5, Povich argues that deflating NOCA carries no cost to explanatory power or to ontic status. The explanatory power thread is handled via Woodward’s (2003) w-question criterion, according to which an explanation’s power is measured by the range of what-if-things-had-been-different questions it can answer. Since the normativist can accept or deny every counterfactual the Platonist can, the two views are explanatorily equivalent. The more philosophically loaded thread concerns ontic status, and it is here that the chapter’s most interesting move occurs. Povich responds to Kuorikoski’s (2021) same-object condition objection: Platonic accounts cannot satisfy the Woodwardian requirement that counterfactual reasoning concerns the same object under different conditions, since there is no stipulation-independent way to distinguish changing a mathematical object’s properties from contemplating an altogether different object. The normativist’s escape route is to locate the persisting object not in the mathematical realm, but in the term or concept, individuated syntactically (p. 151). An intervention on the concept ‘trefoil knot’, via interventions on brains, social conventions, or evolutionary history, changes its governing semantic rules while leaving the syntactic item intact, thereby satisfying the same-object condition. The resulting dependence relation Povich calls ‘counterconceptual causal’ explanation, and his conclusion is striking: distinctively mathematical explanation just is counter-conceptual causal explanation (p. 152). This move is elegant, but syntactic individuation of concepts is doing a lot of philosophical work. The claim that a concept genuinely persists through revision of its governing rules will be resisted from multiple directions. There are many philosophers who will maintain that, at least to some degree, meaning change constitutes concept change. The mathematical normativist needs more to speak to these parties.

Chapter 6.3 is among the most illuminating passages in the book for readers coming from philosophy of language and the broadly neo-pragmatist tradition. Its organizing move is to distinguish normativism as a functional thesis, truth-conditional semantics as a semantic thesis, and inferentialism as a meta-semantic thesis. This tripartite division of labour dissolves a cluster of prima facie objections: that normativism must give up a compositional theory of meaning, that a homogeneous semantics for mathematical and non-mathematical discourse requires Platonism, and so on. Each of these objections, Povich argues, trades on a conflation of levels. Once semantics, meta-semantics, and function are properly distinguished, and truth-conditions carefully separated from truthmakers, the apparent inconsistencies dissolve.

I want to question this wedge Povich drives between truth-conditions and truthmakers. He claims that the semanticist’s truth-conditions are not necessarily the same thing as the metaphysician’s truthmakers, and that it is a ‘pernicious confusion’ to demand that they coincide (p. 187). I think his arguments here are forceful and well motivated. But they prompt a question that this section does not fully settle. If mathematical claims are expressions of semantic and conceptual rules, rather than descriptions of a mind-independent domain, what distinguishes a true mathematical claim from a false one? The answer seems to be that a true claim correctly expresses a conceptual rule actually in force. But if so, conceptual-rules-actually-in-force  are playing something like the role truthmakers are supposed to play. A related pressure concerns the normativist’s strategy for resisting collapse into old-school conventionalism. The worry is that conventionalism renders mathematical necessity merely contingent. If ‘2 + 2 = 4’ is true only because of our conceptual rules, it could have been false had we adopted different rules. The normativist response is a rigidifying move: when we evaluate counter-conceptual conditionals, we hold fixed our own conceptual standards, so that ‘were 2 + 2 = 5 our rule, 2 + 2 would equal 5’ comes out false when assessed from our standpoint. This secures necessity, but one might press on what licenses this asymmetry. Why are we rather than the imagined community entitled to hold our standards fixed? The natural answer is that because we are the ones asking the question and deploying our concepts, we get to rigidify our conceptual rules. But is this just pragmatic stipulation or does it quietly reintroduce a privileged standpoint from which some rules are the genuinely correct ones, which edges back toward the Platonism that this view was designed to avoid? These are sketches of objections rather than fully fleshed-out arguments, raised in the spirit of identifying where the neo-Carnapian normativist’s deepest commitments lie and where future work from the normativist camp could profitably focus.

These are questions Rules to Infinity itself helps to open up, which is a mark in its favour. It is an ambitious and unificatory book. Povich brings together a detailed technical account of mathematical explanation, a carefully argued meta-ontological deflationism, and a sophisticated philosophy of language, and shows that these fit together into a coherent and well-motivated whole. Readers across all three areas will find much to engage with.

Jason DeWitt
Ohio State University
dewitt.197@osu.edu

References

Baron, S. (2016). ‘Explaining Mathematical Explanation’, Philosophical Quarterly, 66, pp. 548–80.

Baron, S. (2019). ‘Mathematical Explanation by Law’, British Journal for the Philosophy of Science, 70, pp. 683–717.

Batterman, R. and Rice, C. (2014). ‘Minimal Model Explanations’, Philosophy of Science, 81, pp. 349–76.

Bromberger, S. (1966). ‘Why Questions’, in R. G. Colodny (ed.), Mind and Cosmos, University of Pittsburgh Press, pp. 86–111.

Craver, C. and Povich, M. (2017). ‘The Directionality of Distintinctively Mathematical Explanations’, Studies in History and Philosophy of Science A, 63, pp. 31–38.

Field, H. (2022). ‘Conventionalism About Mathematics and Logic’, Noûs, 57, pp. 815–31.

Kuorikoski, J. (2021). ‘There Are No Mathematical Explanations’, Philosophy of Science, 88, pp. 189–212.

Lange, M. (2013). ‘What Makes a Scientific Explanation Distinctively Mathematical?’ British Journal for the Philosophy of Science, 64, pp. 485–511.

Linnebo, Ø. (2018). Thin Objects: An Abstractionist Account, Oxford University Press.

Reutlinger, A. (2014). ‘Why Is There Universal Macrobehavior? Renormalization Group Explanation as Noncausal Explanation’, Philosophy of Science, 81, pp. 1157–70.

Thomasson, A. (2020). Norms and Necessity, Oxford University Press.

Warren, J. (2020). Shadows of Syntax: Revitalizing Logical and Mathematical Conventionalism, Oxford University Press.

Woodward, J. (2003). Making Things Happen, Oxford University Press.