The philosophical analysis of mathematical explanation concerns itselfwith two different, although connected, areas of investigation. Thefirst area addresses the problem of whether mathematics can play anexplanatory role in the natural and social sciences. The second dealswith the problem of whether mathematical explanation occurs withinmathematics itself. Accordingly, this entry surveys the contributionsto both areas, it shows their relevance to the history of philosophy,mathematics, and science, it articulates their connection, and pointsto the philosophical pay-offs to be expected by deepening ourunderstanding of the topic.

• 1. Mathematical explanation in the empirical sciences
  • 1.1 Some historical remarks
  • 1.2 Theories of explanation
  • 1.3 Mathematical models and idealization
  • 1.4 Explanatory indispensability arguments
• 2. Mathematical explanation in mathematics
  • 2.1 Some historical remarks
  • 2.2 Models of mathematical explanation
• 3. Some connections to other debates
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Mathematical explanation in the empirical sciences

Nearly all of our most successful empirical sciences employ a greatdeal of mathematics. In addition, scientists often emphasize the valueof explaining some phenomenon that they have discovered. It is naturalto wonder, then, if mathematics is well-suited to contribute to theexplanation of natural phenomena and what these contributions mightbe. In the philosophy of science most accounts of explanation identifyan explanation with an appropriate description of a cause (see Salmon1984, Cartwright 1989, Woodward 2003, Strevens 2008, and Beebee,Hitchcock & Menzies 2010 for an overview). Nearly everyone canadmit that mathematical tools are an excellent means of tracking orrepresenting causes. For example, mathematics can be used to explainwhy Halley’s comet’s orbit has a period of 75 years. Muchof the debate about mathematical explanation in the empirical scienceshas focused on more contentious cases: what role might mathematicsplay in non-causal explanations, if there are any, and how might thesecases challenge this or that account of causal explanation (Reutlinger& Saatsi 2018)?

One kind of case that has been emphasized aims to explain thepossibility or impossibility of some process. For example, why can wenot divide our 23 strawberries equally among three friends (Lange2013), or why can we arrange 81 stamps into a 9 by 9 array? Thelegitimate explanations seem to be that 23 is not divisible by 3without remainder and that \(9 \times 9 = 81.\) Neither mathematicalfact is a cause of the feature of the process in question, so we seemto have a non-causal explanation where mathematics is part of theexplanation. The possibility or impossibility of other processes areexplained by other areas of mathematics that investigate structures orformal features of natural systems. For example, why can we not make acircuit of the bridges of Königsberg that involves crossing eachof the bridges exactly once (Pincock 2007, Molinini 2012)? This isexplained by the abstract structure of the bridge network. Why is itpossible for there to be stable planetary orbits? One explanation thathas been proposed appeals to the dimensions of space-time (Woodward2003).

Another kind of non-causal mathematical explanation deals with astriking or surprising feature of a phenomenon, where that feature canbe identified through a mathematical analysis of the situation. Thefeature may be tied to a minimization process, or be especiallyresilient or stable for what is arguably a mathematical reason.Perhaps the most discussed case is the length in years of thelife-cycle of three species of periodic cicada: why are these lengthseither 13 or 17 years (Baker 2005, 2017)? An explanation is that 13and 17 are prime numbers and that prime numbered life-cycles confer arelative fitness advantage in avoiding predators and competition forscarce resources like food. Other broadly evolutionary cases includethe hexagonal shape of honeycomb cells (Lyon & Colyvan 2008,Räz 2017, Wakil & Justus 2017) and the pattern of seeds in asunflower (Lyon 2012). There is an extensive literature on how theseoptimality explanations might work in biology and economics (Potochnik2007, Rice 2015, 2021). However, this explanatory contribution frommathematics can be found in other domains as well. For example, why dosoap films obey Plateau’s laws (Lyon 2012, Pincock 2015a)? Thiscan be explained through a process of surface minimization, subject toconstraints. The mathematics of the situation is central to thecharacter that the laws take on. Other cases turn on a mathematicalanalysis of the stability or instability of some process. For example,why do the so-called Kirkwood gaps appear in our solar system’sasteroid belt (Colyvan 2010)? Occupying some spatial regions isunstable, so that an asteroid that starts in such a region isoverwhelmingly likely to leave it. Similar analyses explain patternsin the rings of Saturn or the collapse of an engineered structure likea bridge. (See also Ashbaugh, Chicone & Cushman 1991, Colyvan2001, Lipton 2004, Baker 2015a and Lange 2017 for a range of otherexamples.)

The rest of section 1 will consider some of the history of the debatesabout non-causal mathematical explanations (section 1.1) and theirsignificance for various theories of scientific explanation (1.2). Thesection then turns to two other debates that are closely related tothese features of these explanations: how mathematical models mayexplain despite their highly idealized character (1.3) and how theexplanatory role of mathematics in science could support a platonisticinterpretation of pure mathematics (1.4).

1.1 Some historical remarks

Does mathematics help explain the physical world or does it actuallyhinder a grasp of the physical mechanisms that explain the how and whyof natural phenomena? It is not possible here to treat this topic inits full complexity but a few remarks will help the reader appreciatethe historical importance of the question.

Aristotle describes his ideal of scientific knowledge in“Posterior Analytics” in terms of, among other things,knowledge of the cause:

We suppose ourselves to possess unqualified scientific knowledge of athing, as opposed to knowing it in the accidental way in which thesophist knows, when we think that we know the cause on which the factdepends as the cause of the fact and of no other, and further, thatthe fact could not be other than it is. (BWA, 111, Post. An. I.1, 71b5–10)

The causes [aitia] in question are the four Aristotelian causes:formal, material, efficient, and final. Nowadays, translators andcommentators of Aristotle prefer to translate aition [aitia] as‘explanation[s]’, so that the theory of the four causesbecomes an account of four types of explanations. For instance, hereis Barnes’ translation of the passage quoted earlier: “Wethink we understand a thing simpliciter (and not in the sophisticfashion accidentally) whenever we think we are aware both that theexplanation because of which the object is is its explanation, andthat it is not possible for this to be otherwise.” (CWA, 115,Post. An. I.1, 71b 5–10)

But how do we obtain scientific knowledge? Scientific knowledge isobtained through demonstration. However, not all logically cogentproofs provide us with the kind of demonstration that yieldsscientific knowledge. In a scientific demonstration “thepremisses must be true, primary, immediate, better known and prior tothe conclusion, which is further related to them as effect tocauses.” (BWA, 112, Post. An. I.1, 71b 20–25) InBarnes’ translation: “If, then, understanding is as weposited, it is necessary for demonstrative understanding in particularto depend on things which are true and primitive and immediate andmore familiar than and prior to and explanatory of theconclusion.” (CWA, 115, Post. An. I.1, 71b 20–25)

Accordingly, in “Posterior Analytics” I.13, Aristotledistinguished between demonstrations “of the fact” anddemonstrations “of the reasoned fact”. Although both arelogically cogent only the latter mirror the causal structure of thephenomena under investigation, and thus provide us with knowledge. Wecan call them, respectively, “non-explanatory” and“explanatory” demonstrations.

In Aristotle’s system, physics was not mathematized althoughcausal reasonings were proper to it. However, Aristotle also discussedextensively the so-called mixed sciences, such as optics, harmonics,and mechanics, characterizing them as “the more physical of themathematical sciences”. There is a relation of subordinationbetween these mixed sciences and areas of pure mathematics (see Dear2011). For instance, harmonics is subordinated to arithmetic andoptics to geometry. Aristotle is in no doubt that there aremathematical explanations of physical phenomena:

For here it is for the empirical scientist to know the fact and forthe mathematical to know the reason why; for the latter have thedemonstrations of the explanations, and often they do not know thefact, just as those who consider the universal often do not know someof the particulars through lack of observation. (CWA, vol. I, 128,Post. An. I.13, 79a1–79a7)

However, the topic of whether mathematics could give explanations ofnatural phenomena was one on which there was disagreement. As thedomains to which mathematics could be applied grew, so also did theresistance to it. One source of tension consisted in trying toreconcile the Aristotelian conception of pure mathematics, asabstracting from matter and motion, with the fact that both physics(natural philosophy) and the mixed sciences are all conversant aboutnatural phenomena and thus dependent on matter and motion. Forinstance, an important debate in the Renaissance, known as theQuaestio de Certitudine Mathematicarum, focused in large part onwhether mathematics could play the explanatory role assigned to it byAristotle (Mancuso 1996, ch. 1). Some argued that lacking causality,mathematics could not be the ‘explanatory’ link in theexplanation of natural phenomena (see also sections 1.2 and 1.3).

By the time we reach the seventeenth century and the Newtonianrevolution in physics, the problem reappears in the context of achange of criteria of explanation and intelligibility. This has beenbeautifully described in an article by Y. Gingras (2001). Gingrasargues that “the use of mathematics in dynamics (as distinctfrom its use in kinematics) had the effect of transforming the verymeaning of the term ‘explanation’ as it was used byphilosophers in the seventeenth century” (2001, 385). WhatGingras describes, among other things, is how the mathematicaltreatment of force espoused by Newton and his followers – atreatment that ignored the mechanisms that could explain why and howthis force operated – became an accepted standard forexplanation during the eighteenth century. After referring to theseventeenth and eighteenth centuries’ discussions on themechanical explanation of gravity, he remarks:

This episode shows that the evaluation criteria for what was to countas an acceptable ‘explanation’ (of gravitation in thiscase) were shifting towards mathematics and away from mechanicalexplanations. Confronted with a mathematical formulation of aphenomenon for which there was no mechanical explanation, more andmore actors chose the former even at the price of not finding thelatter. This was something new. For the whole of the seventeenthcentury and most of the eighteenth, to ‘explain’ aphysical phenomenon meant to give a physical mechanism involved in itsproduction. … The publication of Newton’sPrincipia marks the beginning of this shift wheremathematical explanations came to be preferred to mechanicalexplanations when the latter did not conform to calculations. (Gingras2001, 398)

Among those who resisted this confusion between “physicalexplanations” and “mathematical explanations” wasthe Jesuit Louis Castel. In “Vrai système de physiquegénérale de M. Isaac Newton” (1734), he discussedPrincipia’s proposition XIII of Book III (onKepler’s law of areas). He granted that the propositionconnected mathematically the inverse square law to the ellipticity ofthe course of the planets. However, he objected that “the one isnot the cause, the reason of the other” (Castel 1734, 97) andthat Newton had not provided any physical explanation, only amathematical one. Indeed “physical reasons are necessary reasonsof entailment, of linkages, of mechanism. In Newton, there is none ofthis kind.” (Castel 1734, 121)

Some contemporary discussions bear close proximity to these worries.Consider Morrison’s bookUnifying Scientific Theories(2000). One of the major theses of the book is that unification andexplanation often pull in different directions and come apart(contrary to what is claimed by unification theories of explanation).One of the examples discussed in her introduction reminds us ofCastel’s objections:

Another example is the unification of terrestrial and celestialphenomena in Newton’sPrincipia. Although influenced byCartesian mechanics, one of the most striking features of thePrincipia is its move away from explanations of planetarymotions in terms of mechanical causes. Instead, the mathematical formof force is highlighted; the planetary ellipses discovered by Keplerare “explained” in terms of a mathematical description ofthe force that produces those motions. Of course, the inverse-squarelaw of gravitational attraction explains why the planets move in theway they do, but there is no explanation of how this gravitationalforce acts on bodies (how it is transported), nor is there any accountof its causal properties. (Morrison 2000, 4)

Using several case studies (Maxwell’s electromagnetism, theelectroweak unification, etc.), Morrison argues that the mathematicalstructures involved in the unification “often supply little orno theoretical explanation of the physical dynamics of the unifiedtheory” (Morrison 2000, 4). In short, the mathematical formalismfacilitates unification but does not help us explain the how and whyof physical phenomena.

We have to close these historical remarks here, although it would beinteresting to pursue these questions in a more systematic way intothe nineteenth and the twentieth centuries (see however Dorato 2017for a wide-ranging claim concerning “explanatory switches”at crucial junctions in the history of physics).

The aim of the above was to prepare the ground for showing how incontemporary discussions in philosophy of science, to which we nowturn, we are still confronted with such issues.

1.2 Theories of explanation

Two legacies of the Aristotelian tradition surveyed in section 1.1 arethat explanations require causes and that providing an explanationrequires giving an argument that turns on laws. Debates in thephilosophy of science since the 1960s have shown how one can privilegeone legacy over another (Salmon 1989). Hempel’sdeductive-nomological analysis of explanation requires that anexplanation be a deductively valid argument from true premises, whereat least one premise is a scientific law (Hempel 1965). Hempel andother empiricists in this broad tradition are wary of making causescentral to explanation. This is apparent even in some ofHempel’s critics such as Kitcher, who emphasizes the unifyingpower of explanations. For Kitcher, an explanation is an instance of adeductive argument scheme, where the schemes to adopt are identifiedon the basis of global features of the claims we accept (Kitcher 1989,see section 2.2.2 for additional discussion).

By contrast, Salmon’s work has persuaded many philosophers ofscience that explanations need only provide causal information aboutthe explanatory target (Salmon 1984, 1989). For Salmon and others inthis tradition, explanations do not require laws and need not even bearguments. One development of this approach preserves Salmon’semphasis on causal mechanisms as a special sort of process. Theso-called “new mechanists” endorse a broader notion ofcausal mechanism than Salmon allowed for, and identify an explanationof some target with a mechanism that produces it (Machamer, Darden& Craver 2000). Other approaches to causal explanation includeDavid Lewis’ counterfactual analysis of causation (Lewis 2004)and Woodward’s interventionist theory (Woodward 2003, 2021a).Despite their differences, Lewis and Woodward allow for causalexplanations in the absence of mechanisms. This makes their approachesto explanation easier to generalize to non-causal cases.

Philosophical discussions of non-causal mathematical explanation canbe classified based on how they are related to these debates aboutscientific explanation more generally. One position argues for theneed to restore something like Hempel’s emphasis on laws orKitcher’s claims about unification (Baron 2019). Anotherposition generalizes from an account of causal explanation so that itcan include these mathematical cases (Saatsi & Pexton 2013,Reutlinger 2016). Yet a third position is pluralist about explanation,and argues that explanations come in a variety of distinct sorts thatcannot be fit into one or the other of these two options (Pincock2018, 2023).

Lange’s extensive discussions of non-causal explanations can beseen as a valiant attempt to preserve the law-based approach toscientific explanation that goes back to Hempel (and Aristotle) (Lange2013, 2017). Lange’s work on laws emphasizes how to identifyclaims with the right kind of modal strength to contribute toexplanations (Lange 2009). Consider the contrast between “Allgold cubes are less than 1 cubic mile in volume” and “AllUr-235 cubes are less than 1 cubic mile in volume”. The formerstatement is contingent, while the latter statement has some degree ofnecessity. This allows the latter statement to contribute to anexplanation. Lange’s approach to mathematical explanationextends this point so that mathematical claims can function inexplanations in a distinctive way due to their special degree ofnecessity. This allows mathematics to contribute to what Lange callsexplanations by constraint that show how some outcome is guaranteed toarise (Lange 2017, ch. 2). This is how Lange treats the strawberrydivision case and also cases like the bridges of Königsberg.Other mathematical explanations turn on the dimensions of thequantities involved or the statistical features of some process (Lange2017, ch. 5, 6). In each of these types of cases, the modal characterof the mathematical claim allows it to explain just as the modalcharacter of ordinary scientific laws allow them to explain.

Unsurprisingly, many philosophers have challenged Lange’sproposals in ways that are reminiscent of how Salmon objected toHempel’s deductive-nomological account (Pincock 2015a,Reutlinger 2017b, Saatsi 2018). For example, Craver and Povich objectthat, in the absence of a causal constraint on explanation,Lange’s proposals lack any suitable worldly basis, and so counttoo many representations of some target as explanatory (Craver &Povich 2017, Lange 2018a). As a mechanist about explanation quitegenerally, Craver seems inclined to dismiss the possibility ofnon-causal mathematical explanations (Craver 2014). His co-authorPovich has offered a more constructive proposal for these cases thatallows for explanations with a variety of worldly or ontic bases(Povich 2020, 2021). Povich deploys a version of what amounts to a newconsensus for handling non-causal explanations: if a proposedexplanation relates to the right counterfactuals in the right way,then it is legitimate (Woodward 2018, Rice 2021). However, there is asyet no agreement on what sorts of counterfactual tests are sufficientfor explanation (see Lange 2021a for some objections to thisapproach).

One option is to treat a mathematical claim like a law that governssome situation, and to credit the mathematics with explanatory powerwhenever it allows us to assess a range of counterfactual scenarios.This is Reutlinger’s proposal, which deliberately loosens therequirements that Woodward places on interventions (Reutlinger 2016,2017a, 2018, Reutlinger, Colyvan & Krzyzanowska 2022). Forexample, in the bridges of Königsberg case, the mathematicalclaim indicates that a circuit of the bridges would be possible in thecounterfactual scenario where some of the actual bridges were absent.Reutlinger concludes that the mathematical claim thus explains what isgoing on in the actual world by indicating what makes a difference tothe feature of interest. Arguably, though, this approach is tooliberal. Suppose we ask why we have \(81\) stamps. It is amathematical truth that \(9 \times 9 = 81,\) and so we can arrange ourstamps in a \(9 \times 9\) array. So if we could not arrange ourstamps in a \(9 \times 9\) array, then we would not have \(81\)stamps. But the truth that \(9 \times 9 = 81\) does not explain why wehave \(81\) stamps.

Another option is to treat a mathematical claim like a cause. Then theclaim will be explanatory when a “counter-mathematical”that supposes that this claim is false winds up making a difference tothe target in question. This sort of counter-mathematical involvesimpossible worlds where necessary truths come out false. This is theoption that Povich takes (Povich 2020, 2021). Another family ofproposals along these lines has been developed by Baron incollaboration with Colyvan and Ripley (Baron, Colyvan & Ripley2017, 2020). Baron et al. draw on David Lewis’ procedure forevaluating counterfactuals tied to causation: consider the scenariothat arises through a miraculous change that is just enough of achange to make the antecedent of the counterfactual true. In addition,Baron et al. require that the features of the natural world that areimplicated in this shift in the mathematics be changed in acorresponding way. For example, for the cicada case, the centralmathematical claim is that prime periods minimize intersections whencompared to non-prime periods. So among the years 12, 13, 14, 15, 16,17, and 18 (that are identified by ecological constraints), the primes13 and 17 stand out as comparatively more fit. Baron et al. considerthe consequences of supposing that 13 is not prime. If 13 was notprime, they argue, then it would have factors besides 1 and 13, and sohaving a 13-year life cycle would not confer any relative fitnessadvantage. Thus, in this impossible world, the cicadas would not haveevolved a 13-year life cycle. This is meant to show that 13’sbeing prime makes a difference to the evolution of 13-year life cyclecicadas. (See also section 2.2.1 for a parallel debate for puremathematics.)

There are pressing questions for these proposals about the nature ofimpossible worlds and our epistemic access to them (Kasirzadeh 2021a).Another sort of objection has been raised by Baron himself in workthat develops another account of mathematical explanation (Baron2020). Baron, like Baker and Lange, aims to identify a special classof genuinely or distinctively mathematical explanations of naturalphenomena (Baker 2005, 2009a, Lange 2013). What is special about thesecases is that the mathematics explains, but not by representing ordescribing some non-mathematical explainers such as causes or otherworldly difference makers. Baron’s general worry is that simplyusing countermathematicals fails to distinguish explanations thatemploy mathematics from these genuinely mathematical explanations. Wecan see this using our original stamp case: why can we arrange ourstamps in a \(9 \times 9\) array? Because we have \(81\) stamps and\(9 \times 9 = 81.\) This case passes Baron et al.’scountermathematical test, for were \(9 \times 9\) not equal to \(81\),then our \(81\) stamps could not be arranged in a \(9 \times 9\)array. However, the mathematical claim here seems to be simplytracking the non-mathematical features of the stamps, and so Baron andothers would not want to count this as a special sort of mathematicalexplanation.

Baron concludes that some additional requirements must be imposedbeyond the truth of the relevant countermathematical. Here Baronreaches back to Kitcher’s idea that explanations are instancesof a special sort of argument scheme (Kitcher 1989), where the schemesare found through a process of appropriately unifying the claims thatwe accept (for objections to this proposal see Pincock 2023 and Povichforthcoming). Other recent work on non-causal mathematicalexplanations also seems to be returning to some of the originalsources of these debates. For example, Lange argues that the best wayto make sense of the explanatory power of pure mathematics in theempirical sciences is to adopt an Aristotelian interpretation of puremathematics (Lange 2021b).

Lyon proposed another way of relating mathematical explanations tocausal explanations by adapting Jackson and Pettit’s notion of a“program explanation” (Jackson & Pettit 1990). Aprogram explanation does not invoke a property that causes the outcomeof interest. Instead, the explanation appeals to a property \(A\) thatguarantees the presence of some member of a family of properties,where some such property \(B\) causes the outcome of interest. As Lyonsummarizes his proposal, “An explanation of an empirical fact ismathematical – i.e., it has mathematics doing explanatory work– if the explanation is a program explanation that usesmathematics in a way that is indispensable to the program” (Lyon2012, 568). One concern with this proposal is that it includes caseswhere the mathematics merely represents some causally relevantproperty, as with the stamps case noted above: programming is tooindiscriminate a relation to avoid this worry (Saatsi 2012).

A sweeping way of dealing with the apparent tension between causes andmathematics has been pursued by some ontic structural realists(Ladyman & Ross 2007, French 2014). They identify the fundamentalmetaphysical structure of the world with a mathematical structure. Ifone adopted this kind of structural realism, then the explanatorypower of mathematics in the empirical sciences would receive asatisfying analysis. In fundamental physics, scientists would beworking with the fundamental mathematical structure directly, and soexplanations there would be essentially mathematical. Innon-fundamental domains such as biology or economics, scientists wouldbe investigating features of the world that are ultimately grounded insome mathematical structure. So again it would make sense for many ofthe explanations in non-fundamental sciences to be mathematical.Causal explanations would then turn out to be perfectly consistentwith more fundamental mathematical explanations. Few philosophers arewilling to adopt such a metaphysical position in order to resolvequestions about mathematical explanation, although at least onephysicist has defended this approach (Tegmark 2014).

Another, less metaphysical, solution to these difficulties is toretain an account of how causal explanations work and to simplysupplement it with a distinct account of how various kinds ofnon-causal explanations arise (Pincock 2015a, Pincock 2018, withcriticisms from Knowles 2021a). This sort of explanatory pluralism isalso reminiscent of one aspect of the Aristotelian tradition. Onechallenge for the explanatory pluralist is to make sense of the valuethat scientists ascribe to explanations: how can there be some specialvalue in having an explanation if explanations come in different kindsthat have nothing in common? One response to this challenge is thatgrasping an explanation produces scientific understanding, but thenature and value of this scientific understanding remains a subject ofactive debate (Rice & Rohwer 2021). Another response is to defenda restrictive form of explanatory monism. It may turn out that thismonism about explanation is so restrictive that there are nomathematical explanations in science (Zelcer 2013, Kuorikoski 2021).This is reminiscent of one side of the Renaissance debate noted insection 1.1.

Another sort of pluralism arises from supposing that causalexplanations involve the mechanisms championed by the new mechanists,and then allowing for other sorts of explanations that work indifferent ways. One type of case that has received extensivediscussion is so-called “topological” explanation. Theseexplanations appeal to structural or network-based features to explainan aspect of a system (Kostic 2020, 2023, Ross 2021). One position isthat topological explanations are non-causal, non-mechanicalexplanations that are based on a different kind of explanatorilyrelevant feature. For example, Kostic and Khalifa argue that anon-ontic approach that privileges scientist’s explanatory goalsis needed to make sense of topological explanation (Kostic &Khalifa 2021, 2022). Another position is that an appropriatelyflexible notion of mechanism can count genuine topologicalexplanations as mechanical explanations (Bechtel & Abrahamsen2010, Bechtel 2020, Huneman 2010, 2018, Brigandt 2013, Green et al.2018). As with the debates arising from Woodward’s andLewis’ counterfactual approaches to causal explanation, the mainquestion is what counts as a mechanism and how non-mechanisms can beexplanatory (Janson 2018, Janson & Saatsi 2019, Andersen 2020, Jhaet al. 2022).

More recently, some authors have tried to restore some kind ofexplanatory monism by arguing that all scientific explanations turn onnon-representational, expressive elements (McCullough-Benner 2022,Hunt forthcoming). If this was right, then there would be nodifficulty making sense of mathematical explanations so long asmathematics can be seen to perform whatever expressive function anauthor identifies. The viability of an expressive approach thus turnson questions about the interpretation of pure mathematics that areconsidered in sections 1.4 and 2.2.

1.3 Mathematical models and idealization

One assumption of the Aristotelian tradition that is oftenunquestioned in the work summarized in section 1.2 is that whateverprovides the explanation (i.e. the explanans) must be true.Philosophical investigations of scientific models and how these modelsmay explain have convinced many that an explanans need not be true.The argument for this conclusion is straightforward: Scientific modelsexplain and scientific models are not true. So, truth is not requiredfor explanation (Bokulich 2011; see also Cartwright 1983, Morrison2015, Rice 2018 and Yablo 2020).

A traditional response to this argument is that even though models arenot true, a model can only explain if it generates some truths aboutthe target of explanation (Colyvan 2010, see section 1.4 for morediscussion). That is, a model explains only when it represents itstarget to be a certain way. The debate thus turns on the options formaking sense of how models represent, especially when those models aremathematical, and if the representational aspects of models aresufficient to make sense of model-based explanation. One proposal isthat a model explains when the model represents a target system to bea certain way and also represents something else that explains why thesystem is that way. For example, a causal model of outcome \(E\) needsto represent \(E\) and some cause \(C\) of \(E\). However, there is noconsensus on what it takes for a model to represent something.

As a model is distinct from its target, and the model and target areoften composed of different materials, it is natural to conclude thatstructural relations are central to what a model represents. However,it is hard to maintain that a model represents a target just in casethere is a structural relation between the model and target(Suárez 2010, 2015). For example, a model is isomorphic toitself, and so it stands in a structural relation to itself. But we donot want to say that a model represents itself. There are alsomodel-target relations that lack any clear structuralcharacterization. For example, a model may represent the solar system,and yet contain only two point particles moving on trajectories thatfail to stand in any non-trivial structural relation to anything inthe actual solar system. So it seems that standing in a structuralrelation is neither necessary nor sufficient for a model to representa target.

One response to these problems is that a model represents a targetwhen agents claim that a structural relation obtains between the modeland the target, where that relation may be quite selective and involvereinterpretations of various elements of the model (Pincock 2012, ch.2, Frigg & Nguyen 2020). For example, the projection relating amap to some country may be fairly complicated, and involve variousconventions for what symbols on the map indicate about the country.Some authors associate the relations that agents establish betweenmodel and target with inferential principles (Bueno & Colyvan2011, Bueno & French 2018). So according to these variousproposals, a model explains a feature of a target when either somerepresentational relationship or inferential licenses from model totarget have been established by agents, and these connectionsgenuinely explain that feature (e.g. they are causes of the feature).For example, a suitably interpreted map can explain the impossibilityof train travel between two cities by accurately representing thetrain network that fails to link those cites.

The different approaches to explanation surveyed in section 1.2 canthen be used to identify explanatory models and what they explain. Amechanist about explanation can allow that mathematical models explainby representing mechanisms, while difference-making views will requirean explanatory model to represent difference-making, i.e. how changingfactor \(X\) will go along with a change in outcome \(Y\). All ofthese proposals will argue that scientific models do not need to betrue in order to explain. All that needs to occur is for the model toprovide some truths by representing the right things about the target.So the presence of falsehoods that the model also provides about thetarget does not stand in the way of the model’s explanatorypower.

This approach to explanatory models and idealization has been calledinto question by Batterman. One argument from Batterman is that thereare explanatory models that do not explain in virtue of an element ofthe model representing some explanatorily relevant factor such as acause or more exotic non-causal difference maker. Instead, in suchcases, “while we have a genuine mathematical explanation ofphysical phenomena, there is no appeal to the existence ofmathematical entities or their properties. Instead, the appeal isto a mathematical idealization resulting from a limitoperation that relates one model … to another”(2010, 7–8). The case that Batterman is discussing here involvesan operation (known as taking the thermodynamic limit) that transformsa “finite statistical mechanical model” into a“continuum thermodynamic model”. This is central to theexplanation of the universality of some features of phase transitionsthat include liquid/gas transitions and magnetization. The featuresare universal in the sense that they arise across systems with verydifferent microphysical features, and so seem especially puzzling.

One point that Batterman is making here is that mathematicaloperations that connect models can be significant for a mathematicalexplanation of an empirical phenomena (see also Batterman & Rice2014, Batterman 2019, Batterman 2021). The defenders of traditionalapproaches to model explanation often focus on cases where a singlemathematical structure is used to explain. However, the basic ideas ofthe traditional approach can be extended to deal with the explanatorysignificance of some mathematical operations. For example, onemathematical model may be transformed into another mathematical modelthrough a mathematical operation. If this operation reflects somethingof explanatory significance, then the two models and the operationconnecting them may be central to the explanation. Some idealizationsare associated with these operations, as in the case where an ocean istreated as infinitely deep or a planet is modeled as a point particle.In such cases, the operations function by changing or removing theinterpretation of the elements of the model.

Batterman also develops another point that poses a more significantchallenge to traditional approaches to model explanation. This is thatthe “mathematical idealization” that results from thisoperation, and that is tied to one of these models, is essential tothe whole explanation. In the ocean case, there is no temptation tosay that the ocean being infinitely deep in one model is explanatorilyrelevant to the character of the waves on the surface of the actualocean. All that this idealization turns on is that the depth is abovesome threshold. Other cases can be handled using similar“Galilean” idealizations that eliminate the falsehood fromthe genuine explanation (Weisberg 2007). But in Batterman’scases, such as the phase transition case, he is clear that he takesthe idealization to be essential to the explanation: “Thesenontraditional idealizations play essential explanatory rolesinvolving operations or mathematical processeswithoutrepresenting the system(s) in question” (2010, 23). If thispoint is accepted, then these cases would undermine the scope of thetraditional approach.

In the philosophy of physics there has been an extensive discussion ofhow essential these idealizations are to the explanations in question(Belot 2005, Bokulich 2008, Norton 2012, Lange 2015a, Franklin 2018,Sullivan 2019, Strevens 2019, Rodriguez 2021. See also Easwaran et al.2021). Some critics of Batterman have argued that these cases can bedealt with using explanations that avoid these idealizations or thattreat these idealizations in the manner that we treated the infinitelydeep ocean. Other critics of Batterman have argued that a moreselective approach to what these models represent allows one to admitthat the idealizations are essential to generating the explanation,but that they are not literally to be included in the explanationitself. For example, a counterfactual approach to these cases wouldidentify the explanation with some counterfactuals that are generatedby the model. Batterman and others, in turn, have responded that allof these criticisms fail to do justice to the phenomena in question orwhat scientists say about their explanations (Morrison 2018, Batterman2019, McKenna 2021).

Other alternatives to a traditional approach to mathematical modelingand explanation have been developed using other sorts of cases astheir primary motivations (Rizza 2013, Berkovitz 2020, Kasirzadeh2021b, McKenna 2022). One theme of this work is a generalization ofBatterman’s point that scientific explanations often involvemany models whose representational relation to the explanatory targetis more involved than what is usually allowed. For example, Kasirzadehconsiders a case with two mathematical models, with different spatialscales, of a process of skin color pattern formation (Kasirzadeh2021b). Biologists asked for an explanation of how the two processesrelated to one another. Kasirzadeh argues that this explanationrequired additional “bridge mathematics” over and abovethe mathematics found in the original two models. The additionalmathematics contributed to the explanation by characterizing how themicroscopic processes gave rise to the unexpected macroscopicstructures. McKenna goes further and argues for the importance ofcases where “models cannot be stitched together in purelymathematical terms” (McKenna 2022). In McKenna’s main casevarious models of sea ice permeability are developed for the purposesof large-scale climate modeling. No single mathematical model of seaice proved to be adequate to supply the right parameters to thelarge-scale model. Instead, different models of sea ice were used inconjunction with high-resolution empirical data about samples of seaice formations. The significance of these cases for the explanationsthat arise from these modeling techniques is likely to be a subject ofongoing debate.

1.4 Explanatory indispensability arguments

Many philosophers are interested in non-causal mathematicalexplanations in science because they seem to support an explanatoryindispensability argument for a platonist interpretation of puremathematics. Colyvan and Baker have been the most ardent defenders ofsuch an argument (Colyvan 2001, 2010, Baker 2005, 2009a, 2022). In his2001 book Colyvan presented a general indispensability argument forthe existence of mathematical entities like the natural numbers:

1.

We ought to have ontological commitment to all and only thoseentities that are indispensable to our best scientific theories;

2.

Mathematical entities are indispensable to our best scientifictheories.

Therefore:

3.

We ought to have ontological commitment to mathematical entities(Colyvan 2001, 11).

This notion of ontological commitment was first articulated by Quine(Quine 2004, Putnam 2010). These commitments reflect what one shouldbelieve exists. Premise 1 is tied to a naturalistic approach to thesebeliefs that claims they should be determined by the character of ourbest scientific theories. For Quine, one’s ontologicalcommitments are settled by the best regimentation of one’sscientific theories into first-order logic, where what makes aregimentation the best is determined by ordinary scientific criterialike consistency and simplicity. Some of Colyvan’s cases in hisbook invoked the explanatory contribution that mathematical entitiesmake to our best theories. If we suppose that one aim of science is toexplain, then a regimentation may be the best in part because itaffords explanations of various scientific phenomena.

An explanatory version of this indispensability argument became moreprominent after Melia’s exchange with Colyvan (Melia 2000, 2002,Colyvan 2002). Melia argued that indispensable quantification overmathematical entities was not sufficient for ontological commitment.Any such commitments could be canceled by a “weaseling”maneuver that added “but I do not accept the existence of anymathematical entities.” For example, one could use numbers tocount how many apples and pears one has and conclude that there aremore apples than pears. But Melia would then add that he rejected theexistence of natural numbers, thereby canceling that commitment.Colyvan replied that such an addition was incoherent when themathematical entities were explaining something, and Melia agreed:“Were there clear examples where the postulation of mathematicalobjects results in an increase in the same kind of utility as thatprovided by the postulation of theoretical entities, then it wouldseem that the same kind of considerations that support the existenceof atoms, electrons and space-time equally supports the existence ofnumbers, functions and sets” (Melia 2002, 75–76). Thedebate about indispensability and platonism then largely turned to theevaluation of cases.

Baker took up Melia’s challenge by reformulating the argument sothat the explanatory question became central. Baker also introducednew cases like the cicada case where the parallel between atoms andnumbers was meant to be clearer. In Baker’s formulation, theargument is:

1′.

We ought rationally to believe in the existence of any entitythat plays an indispensable explanatory role in our best scientifictheories.

2′.

Mathematical objects play an indispensable explanatory role inscience.

3′.

Hence, we ought rationally to believe in the existence ofmathematical objects (Baker 2009a, 613, premises renumbered).

The emphasis on explanation in premise 1′ is of courseconsistent with Quine’s process of regimentation. For example,we opt for the theory of cicadas that best explains their character.If we then work out the best regimentation of this theory, we willfind that it will entail that prime numbers exist. All premise1′ maintains, then, is that one should endorse whatever theseontological commitments turn out to be. However, a second way tosupport premise 1′ is available: one could appeal to inferenceto the best explanation and its use to support scientific realismabout unobservable entities like electrons. Baker sometimes ties theappeal of his indispensability argument to scientific realism:“A crucial plank of the scientific realist position involvesinference to the best explanation (IBE) to justify the postulation inparticular cases of unobservable theoretical entities … theindispensability debate only gets off the ground if both sides takeIBE seriously, which suggests that explanation is of key importance inthis debate” (Baker 2005, 225). The appeal to IBE avoids theQuinean process of regimentation by directing our attention to someexplanatory target such as the length of the life-cycles of somespecies of cicada. If we are scientific realists, then we accept theuse of IBE in support of our claims about the existence of variousentities. So if we find that the best explanation also includesabstract objects like prime numbers, then we should also accept theirexistence.

Three worries about this explanatory indispensability argument can befruitfully distinguished. The first worry is that the argument issomehow circular, begs the question or else fails to correctlyidentify the basis for our knowledge of the existence of mathematicalentities. The point was forcefully presented by Steiner in 1978.Steiner argues for the existence of mathematical explanations and alsoclaims to know of the existence of abstract, mathematical entities.However, “no explanatory argument can establish the existence ofmathematical entities” (1978b, 20). The reason for this isattributed to Morgenbesser: “We cannot say what the world wouldbe like without numbers, because describing any thinkable experience(except for utter emptiness) presupposes their existence”(1978b, 19–20). The point seems to be that we must be able tocompare mathematical and non-mathematical explanations of some targetin order to get an explanatory argument going. But if the targetalways “presupposes” the existence of some mathematicalentities, then this comparison is not possible. Bangu has developedthis point by noting how many of the cases discussed have targets thatare mathematical in character, as with the prime periods of thecicadas (Bangu 2008, 2012, see Baker 2021a for a response). So, theworry continues, mathematical explanations are only indispensable inscience if we have used mathematical entities to characterize thetarget phenomena. Pincock has a somewhat similar concern: if thetargets are characterized in weak mathematical terms, then only weakmathematical theories will be needed to explain these targets, andthese theories can be easily supplied with a nominalisticinterpretation that preserves these theories’ explanatory power(Pincock 2012, see Baker 2015b for a reply to these worries).

Another worry accepts that there are in some sense mathematicalexplanations in science. However, premise 2′ is rejected becausethese explanations fail to involve the existence of any mathematicalobjects. Saatsi has developed this criticism by claiming thatmathematics only explains by representing some non-mathematicalfeatures of the physical world (Saatsi 2007, 2011). On this reading,premise 2′ requires the existence of some“distinctively” or “genuinely” mathematicalexplanations, but there are no such explanations. As Saatsi puts theworry, “what really matters for the indispensability argument– all that matters! – is whether or not mathematics playsthe kind of explanatory role that we should take as ontologicallycommitting” (Saatsi 2016, 1051). Until the defenders of theargument clarify what distinctively mathematical explanations are andhow they involve mathematical objects, it seems that premise 2′is in trouble. Other versions of this objection may be found in Daly& Langford 2009, Rizza 2011, Tallant 2013, Liggins 2016, Busch& Morrison 2016, Barrantes 2019 and Boyce 2021 (see also Panza& Sereni 2016 for a helpful overview of these debates).

A third kind of worry about premise 2′ is developed bymathematical fictionalists like Yablo and Leng (Leng 2010, 2021, Yablo2012, 2020). Fictionalists accept the existence of distinctivelymathematical explanations and yet argue that these explanations do notpresuppose the existence of any mathematical objects. For example,Leng argues that “we can generate mathematical explanations ofphysical phenomena that do not appeal to any abstract mathematicalobjects, but instead only require modal truths about what followslogically from our mathematical assumptions, together with therecognition that the assumptions of our mathematical theories are truewhen interpreted as about the physical system under examination”(Leng 2021, 10437). Leng can thus endorse the very same unifiedderivation of the features of Baker’s cicadas, and yet refrainfrom accepting the existence of mathematical objects. While Saatsitakes the physical features of the system to be the genuineexplainers, Leng uses those same features to interpret themathematical theories that are doing the explanatory work. Either way,premise 2′ of the indispensability argument comes out false.

Colyvan and Baker’s strategy for supporting premise 2′ haslargely involved cases where it appears that mathematical objects playa role in the explanation that is analogous to what unobservableentities like electrons play in other explanations. For example, inthe cicada case, the prime numbers afford a unified derivation of thetarget of the explanation. This strategy would be most effectiveagainst representational approaches to mathematical explanation. Thebasic idea is that scientists value these explanations, and so anyreinterpretation of them in non-mathematical terms risks privilegingan unmotivated philosophical theory over some legitimate scientificpractice. As Baker and Colyvan put the point in a reply directed atDaly and Langford (2009): “Commitments to philosophical theoriessuch as nominalism, a causal theory of explanation, or the‘indexing’ view of mathematical applications are not goodreasons for rejecting well-supported scientific and mathematicalclaims” (Baker & Colyvan 2011, 332).

A second strategy that Colyvan has pursued is to challenge critics torecast these explanations in non-mathematical terms. The refusal to dothis involves an “easy road” to nominalism that Colyvanthinks is untenable: “when some piece of language is deliveringan explanation, either that piece of language must be interpretedliterally or the non-literal reading of the language in questionstands proxy for the real explanation” (Colyvan 2010, 300). Thisstrategy is most effective against fictionalists. It involves aconception of scientific explanation that requires that every genuineexplanation be presentable in literal, non-metaphorical language: tosay why something is the case, we must literally say what isresponsible for what, and so it must be in principle possible to avoidfictional or metaphorical tools. If fictionalists are right about puremathematics, then mathematics is simply such a tool, and so theyshould be able to sketch a non-mathematical version of theexplanations at issue. The fictionalist response is to deny thisconception of explanation.

Yet another kind of criticism of premise 2′ accepts both theexistence of mathematical explanations and that these explanationsinvolve mathematical objects, but maintains that these explanationsare dispensable from our best science. That is, either the bestregimentation of our scientific theories will avoid quantificationover mathematical objects or no appeal to IBE will actually supportadopting such an explanation. This sort of criticism can be tracedback to Field’s pioneeringScience without numbers(Field 1980). There Field contrasts “intrinsic”explanations with “extrinsic” explanations. He claims thatall mathematical explanations are extrinsic and that for everyextrinsic explanation of some target, there is a superior, intrinsicexplanation of that very target (Field 1980, 43–44; see Marcus2013). The explanations championed by Colyvan, Baker and otherssuggest that it is not clear that a mathematical explanation is alwaysextrinsic or that a non-mathematical explanation that is intrinsic issuperior in all respects. Consider again the non-causal explanation ofthe impossibility of traversing the bridges of Königsberg or theevolutionary explanation of the prime periods of the cicada. Whilesome non-mathematical derivation of these targets is surely available,this does not settle whether or not these derivations should count asexplanations or what their explanatory virtues might be.

Perhaps the most promising defense of premise 2′ would be toprovide a positive account of distinctively mathematical explanationsthat would clarify how endorsing such explanations commits one to theexistence of some mathematical objects. The recent literature on thisissue again seems to lead to a kind of standoff. Consider, forexample, Baron’s “Pythagorean” proposal for theseexplanations (Baron forthcoming). Baron defines a Pythagorean to besomeone who not only believes in mathematical objects as abstractentities, but who claims that some of the intrinsic properties ofthese abstract entities are also possessed by concrete entities. Thisis possible because the salient intrinsic properties of themathematical entities are structural properties that are found in theconcrete world whenever the concrete entities are arranged in theright structure. These shared, structural properties and theirnecessary mathematical relations thus enable mathematical truths toexplain features of physical systems such as the bridges or cicadas.In addition, Baron is clear that “Structural properties on myaccount make indispensable reference to abstract objects” (Baronforthcoming, 25). So one defense of premise 2′ involves adoptingBaron’s Pythagoreanism.

Other accounts of distinctively mathematical explanation support therejection of premise 2′. For example, Lange’s modalinterpretation of distinctively mathematical explanations leads him toendorse what he calls an “Aristotelian realist”interpretation of pure mathematics in terms of a special kind ofabstract property, without the recourse to any abstract objects (Lange2021b, see also Franklin 2008). According to Lange, the best way tomake sense of the explanatory power of mathematical truths is tosuppose that “mathematics concerns mathematical propertiespossessed by physical systems” (Lange 2021b, 50). As with Baron,these mathematical properties can help to explain why these physicalsystems have some other properties. For Lange the main benefit of suchan Aristotelian interpretation is that the salient modal features ofthe physical systems arise from the presence of the mathematicalproperties. This helps to clarify in what sense the mathematical truthand the properties it invokes may be explanatorily prior to sometarget property. Crucially, though, for Lange this interpretation ofpure mathematics eliminates the need to invoke abstract objects. IfLange is right, then, premise 2′ of the explanatoryindispensability argument is false. (See also Knowles & Saatsi2021, Knowles 2021a, Knowles 2021b and Baker 2022 on additionalchallenges to this premise.)

One diagnosis of the problems with indispensability arguments is thatthe conclusion of the argument concerns the interpretation of puremathematics while the premises of the argument consider howmathematics is used in science. Perhaps, then, a platonist would bebetter served by focusing on explanatory considerations that arisewithin the practice of pure mathematics. This is the focus of section2.

2. Mathematical explanation in mathematics

Much mathematical activity is driven by factors other thanestablishing that a certain theorem is true. In many casesmathematicians are unsatisfied by merely knowing that a mathematicalfact holds and reprove it, while also claiming explanatory benefitsfor the new proof. This type of explanatory activity appears withinmathematics itself (see the Preamble) and thus one often speaks of“internal” or “intra-mathematicalexplanations” (Baron, Colyvan & Ripley 2020, Betti 2010,Mancosu 2008). The expression “internal mathematicalexplanation” covers a wide range of different phenomena: aninternal mathematical explanation could amount to the recasting of anentire area of mathematics or it might aim at providing explanatoryproofs for specific theorems. The variety of these mathematicalexplanatory activities has been investigated in D’Alessandro (2020),Hafner and Mancosu (2005), Lange (2018b), and Sandborg (1997, Ch.1).

Amongst these different explanatory activities, most of the attentionhas been focused on proofs which not only provethat atheorem is true, but also showwhy it is true. While theremight not be agreement on specific instances, many mathematiciansoften claim that certain proofs have an explanatory power and thatothers do not. These claims are found throughout the history ofmathematics and the philosophy of mathematics (see Lange 2015c, 2016,2017 (Ch. 7–9) and Mancosu 2001). In the words of Bouligand:

Many theorems can be given different demonstrations. The mostinstructive are of course those that let one understand the deepreasons of the results that one is establishing. (Bouligand, 1932, 6,Mancosu’s translation)

And the real algebraic geometer Gregory Brumfiel draws a starkcontrast between two different types of proofs to be found in realalgebraic geometry, i.e., what he calls transcendental proofs (i.e.,proofs based on transfer theorems that infer the truth of a statementfor all real closed fields from its truth on a specific real closedfield, say the real numbers) vs. a type of proof that holds uniformlyfor all real closed fields. The first type of proof is rejected byBrumfiel as non-explanatory and, by contrast, the latter provideexplanatory benefits. In his words:

In this book we absolutely and unequivocally refuse to give proofs ofthis […] type [transcendental proofs]. Every result is proveduniformly for all real closed ground fields. Our philosophicalobjection to transcendental proofs is that they might logically provea result, but they do not explain it, except for the special case ofreal numbers. (Brumfiel 1979, 166)

As the previous examples show, explanatory proofs could be of severaltypes and explain in different manners. A recent debate has focused onthe issue of whether proofs by induction are explanatory. On the onehand, Lange (2010) argues that proofs by induction are notexplanatory. His argument relies on the use of a form of upward anddownward induction from a fixed number k, with k≠1. According toLange, if proofs by ordinary mathematical induction are explanatory,so are proofs by upward and downward induction from a fixed number k,with k≠1. But if so, then the typical asymmetry of explanations,which also holds of mathematical explanations, is not respected: for acertain property P, P(1) is part of the explanation of P(k) and P(k)is part of the explanation of P(1). Baker (2009b) rejectsLange’s argument by arguing against the explanatory equivalencebetween proofs by ordinary mathematical induction and proofs by upwardand downward induction. Hoeltje et al. (2013) reject what they see asan unacknowledged assumption in Lange’s argument, namely that auniversal sentence explains its instances. Dougherty (2017)’sline of attack is based on Lange’s need to presuppose aproblematic notion of identity of proofs, which he questions using analternative criterion of identity spelled out using two equivalentcharacterizations (the first appealing to the language of homotopytype theory and the second using algebraic representatives to proofs).Both Baldwin (2016) and Lehet (2019) defend the explanatory value ofinduction in mathematics: while Baldwin offers positive considerationsas to why inductive arguments are explanatory, Lehet dwells oninductive definitions which – she argues – might be casesof explanations in mathematics.

The distinction between explanatory and non-explanatory proofs hasalso been applied to other types of proofs, for instance proofs thatexplain by using diagrams (see D’Alessandro 2020, Brown 1997),or proofs that explain by drawing on analogies (see Lange 2017).Significant philosophical activity has focused on those proofs thatexplain by revealing the reasons, or the grounds, why a theorem istrue. As stressed by Lange (2021c), in this context the word“ground” should not be understood as connected to therecent literature on metaphysical grounding (e.g., see Correia andSchnieder (2002)). We should rather think of the notion of“conceptual grounding”, as developed by, e.g. Smithson(2020). This notion of ground has an illustrious pedigree inphilosophers and mathematicians such as Bolzano (see Kitcher 1975,Mancosu 1999 and Sebestik 1992) or Cournot (see Mancosu 1999), andrecent contributions have stressed its value in the mathematical realm(see Betti 2010, Detlefsen 1988, Jansson 2017, Pincock 2015b,Poggiolesi and Genco 2023). Indeed, just as in the scientificliterature it is widely accepted that causal explanations track acausal relation in the world and explain by revealing the causes ofwhy a certain fact holds, it seems reasonable to accept that (at leastcertain) mathematical explanatory proofs track a grounding relation inthe mathematical realm and thus explain by mentioning the grounds orreasons why a theorem is true.

2.1 Some historical remarks

Since contributions in analytic philosophy to the study ofmathematical explanations date back only to Steiner 1978a, one mightsuspect that the topic was a byproduct of the Quinean conception ofscientific theories (see Resnik & Kushner, 1987, 154). Oncemathematics and natural science were placed on the same footing, itbecame possible to apply a unified methodology to both areas. Thus, itmade sense to look for explanations in mathematics just as in naturalscience. However, this historical reconstruction would be mistaken.Mathematical explanations of mathematical facts have been part ofphilosophical reflection since Aristotle. We have already seen insection 1.1 the distinction Aristotle drew between demonstrations“of the fact” and demonstrations “of the reasonedfact”. Both are logically rigorous but only the latter provideexplanations for their results. Aristotle had also claimed thatdemonstrations “of the reasoned fact” occur inmathematics. Only these demonstrations can be called“explanatory” demonstrations, and some of thesedemonstrations will be mathematical proofs.

Aristotle’s position on explanatory proofs in mathematics wasalready challenged in ancient times. Proclus, in his “Commentaryon the first book of Euclid’s Elements”, informs us onthis point. He reports: “Many persons have thought that geometrydoes not investigate the cause, that is, does not ask the question‘Why?’” (Proclus 1970, 158–159; for more onProclus on mathematical explanation see Harari 2008). Proclus himselfsingles out certain propositions in Euclid’s“Elements”, such as I.32, as not being demonstrations“of the reasoned fact”. Euclid I.32 states that the sum ofthe internal angles of a triangle is equal to two right angles. If thedemonstration were given by a scientific syllogism in the Aristoteleansense, the middle of the syllogism would have to provide the‘cause’ of the fact. But Proclus argues thatEuclid’s proof does not satisfy these Aristotelian constraints,for the appeal to the auxiliary lines and exterior angles is not‘causal’:

What is called “proof” we shall find sometimes has theproperties of a demonstration in being able to establish what issought by means of definitions as middle terms, and this is theperfect form of demonstration; but sometimes it attempts to prove bymeans of signs. This point should not be overlooked. Althoughgeometrical propositions always derive their necessity from the matterunder investigation, they do not always reach their results throughdemonstrative methods. For example, when [from] the fact that theexterior angle of a triangle is equal to the two opposite interiorangles it is shown that the sum of the interior angles of a triangleis equal to two right angles, how can this be called a demonstrationbased on the cause? Is not the middle term used here only as a sign?For even though there be no exterior angle, the interior angles areequal to two right angles; for it is a triangle even if its side isnot extended. (Proclus 1970, 161–2)

In addition, Proclus also held that proofs by contradiction were notdemonstrations “of the reasoned fact”. The rediscovery ofProclus in the Renaissance was to spark a far-reaching debate on thecausality of mathematical demonstrations referred to above as theQuaestio de Certitudine Mathematicarum (see section 1.1 for more onthis debate). The first shot was fired by Alessandro Piccolomini in1547. Piccolomini’s aim was to disarm a traditional claim to theeffect that mathematics derives its certainty on account of its use of“scientific demonstrations” in the Aristotelean sense(such proofs were known as “potissimae” in theRenaissance). Since “potissimae” demonstrations had to becausal, Piccolomini attacked the argument by arguing that mathematicaldemonstrations are not causal. This led to one of the most interestingepistemological debates of the Renaissance and the seventeenthcentury. Those denying the “causality” of mathematicaldemonstrations (Piccolomini, Pereyra, Gassendi etc.) argued byproviding specific examples of demonstrations from mathematicalpractice (usually from Euclid’s Elements) which, they claimed,could not be reconstructed as causal reasonings in the Aristoteliansense. By contrast, those hoping to restore “causality” tomathematics aimed at showing that the alleged counterexamples couldeasily be accommodated within the realm of “causal”demonstrations (Clavius, Barrow, etc.). Interestingly, both positionsin the debate assumed that mathematical proofs could be syllogized(Mancosu & Mugnai 2023). The historical developments have beenpresented in detail in Mancosu 1996 and Mancosu 2000.

What is more important here is to appreciate that the basic intuition– the contraposition between explanatory and non-explanatorydemonstrations – had a long and successful history that hasinfluenced both mathematical and philosophical developments wellbeyond the seventeenth century. For instance, Mancosu 1999 shows thatBolzano and Cournot, two major philosophers of mathematics in thenineteenth century, construe the central problem of philosophy ofmathematics as that of accounting for the distinction betweenexplanatory and non-explanatory demonstrations. In the case of Bolzanothis takes the form of a theory of Grund (ground) and Folge(consequence). Kitcher 1975 was the first to read Bolzano aspropounding a theory of mathematical explanations (see Betti 2010 andRoski 2017 for recent contributions). In the case of Cournot this isspelled out in terms of the opposition between “ordrelogique” and “ordre rationelle” (see Cournot 1851).In Bolzano’s case, the aim of providing a reconstruction ofparts of analysis and geometry, so that the exposition would use only“explanatory” proofs, also led to major mathematicalresults, such as his purely analytic proof of the intermediate valuetheorem.

In conclusion to this section, we should also point out that there isanother tradition of thinking of explanation in mathematics thatincludes Mill, Lakatos, Russell and Gödel. These authors aremotivated by a conception of mathematics (and/or its foundations) ashypothetico-deductive in nature and this leads them to construemathematical activity in analogy with how explanatory hypotheses occurin science (see Mancosu 2001 for more details). Related to inductivismare Cellucci 2008, 2017, which emphasize the connection betweenmathematical explanation and discovery.

2.2 Models of mathematical explanation

From the above, it should be obvious that both philosophers andmathematicians have appealed to the notion of explanation withinmathematics and that amongst the different contexts in which suchexplanatory activity appears, proofs play a special role. But whatdistinguishes a proof that explains from one that doesn’t? Howshould one proceed in providing an account of explanatory proofs? Itis here that two possibilities emerge. On the one hand, one can followa top-down approach where one starts with a general model ofexplanatory proof and then tries to see how well it accounts for thepractice. On the other hand, one can embrace a bottom-up approachwhere one begins by avoiding, as much as possible, any commitment to aparticular theoretical framework. Only afterwards, one attempts toprovide a taxonomy of recurrent types of mathematical explanatoryproofs and tries to see whether these patterns are heterogeneous orcan be subsumed under a general account.

Supporters of the bottom-up approach include Hafner and Mancosu(2005), Mancosu (2008) and Lange (2015b, 2015c, 2017, 2018b). Probablythe main characteristic of their investigations is the extremely richand varied set of examples considered; precisely in virtue of thisvariety, Lange argues that there is no general pattern characterizingexplanatory proofs; at most one can claim that there are differentclasses of explanatory proofs. Lange proposes several salient featuresof mathematical theorems, which (in different contexts) areresponsible (in those contexts) for the distinction betweenexplanatory and non-explanatory proofs. Among them, he discussesextensively symmetry and simplicity. As for simplicity, it amounts tothe requirement for a proof of a simple result “exploits somesimilar, simple feature of the setup” (see Lange 2017, 257). Asfor symmetry, it is a property that arises when dealing withmathematical results that display some striking symmetry: for theproof to count as explanatory, it needs to show how such symmetryfollows from a similar symmetry in the set-up of the problem. Langedefends these properties by using cases studies drawn fromprobability, real analysis, number theory, complex numbers andgeometry, among other areas of mathematics. One of the mostrepresentative (see Lange 2017, 239–242) is the proof ofd’Alembert’s theorem to the effect that in a polynomialequation ofn-th degree in the variablex and havingonly real coefficients, the nonreal roots always come in pairs (anynon-real root and its complex conjugate will both satisfy theequation). What explains this symmetry? A non-explanatory proof can begiven by algebraic manipulations but this does not reveal the reasonfor the result which, according to Lange, is the fact that the axiomsof complex arithmetic are invariant under substitution ofifor -i. Bueno and Vivanco (2019) points out that it isunclear why what Lange isolates as the symmetric feature of the proof(which makes it explanatory) is a symmetry at all. They suggest thatthis proof turns on an appropriate feature of the relevantstructure.

Other instances of the bottom-up approach may be found in Paseau(2010), Arana & Mancosu (2012), Colyvan, Cusberg & McQueen(2018), D’Alessandro (2021) and Ryan (2021). Each articleconsiders an aspect of mathematical practice and tries to address iton its own terms.

Top-down approaches take their start from a general theory ofmathematical explanation and then explore how well the practice fitsthe model. A typical example of a top-down approach in mathematicalexplanation is Kitcher’s unificationist theory, to be discussedbelow. But one can also apply this description to overarching views onthe nature of explanation. While there are several examples one couldmention, here we present an influential proposal that finds its originin Kim (1994). In order to classify the different accounts ofscientific explanation Kim uses the contraposition between‘explanatory internalism’ and ‘explanatoryexternalism.’ Whereas for ‘explanatory internalism’explanations are activities internal to an epistemic corpus (a theoryor set of beliefs), an ‘explanatory externalist’ will lookfor some systematic pattern of objective dependence relations whichexplanations track or can be identified with. We divide the presentsection in two subsections that follow this division: one will bededicated to the presentation of the externalist, or ontic, accountsof explanatory mathematical proofs, while the other to theinternalist, or epistemic, ones. Among the externalist accounts, wewill discuss Steiner’s theory, several counterfactual theoriesof mathematical explanation, and some other proposals. Amonginternalist accounts we will discuss Kitcher’s theory, togetherwith two novel ones proposed by Frans (2021) and Inglis &Mejía-Ramos (2019).

2.2.1 Externalist models of mathematical explanation

Amongst the several existing contemporary externalist models ofexplanatory proofs, the oldest and probably most well-known isSteiner’s account. Steiner aims at finding criteria that couldcharacterize explanatory proofs. After having discussed severalpossible criteria, such as abstractness, generality, andvisualizability, Steiner rejects them all in favor of the idea that“to explain the behavior of an entity, one deduces the behaviorfrom the essence or nature of the entity” (Steiner 1978a, 143).Although such an idea could seem prima facie intuitive andinteresting, it turns out to be quite problematic. First, it leads tothe notorious difficulties linked to the concepts of essence oressential property; moreover, such concepts risk having littletraction in a mathematical context since all mathematical truths areregarded as necessary. Hence, instead of talking of“essence,” Steiner speaks of “characterizingproperties” by which he means “a property unique to agiven entity or structure within a family or domain of such entitiesor structures,” where he takes the notion of a family asundefined. In other words, for Steiner the difference betweenexplanatory and non-explanatory proofs lies in the characterizingproperties, which are found only in the former but not by the latter.However, this is not all: an explanatory proof needs to begeneralizable as well. Varying the relevant feature (and hence acertain characterizing property) in such a proof needs to give rise toan array of corresponding theorems, which are proved – andexplained – by an array of “deformations” of theoriginal proof.

There have been two extensive critical discussions of Steiner’saccount. The first was provided by Resnik and Kushner (1987) whoargued that the distinction between explanatory and non-explanatoryproofs is context-dependent. The second, which also offers acounterexample to the theory based on a case of explanation from realanalysis recognized as such in mathematical practice, has beendeveloped by Hafner and Mancosu (2005). There have also been attemptsto improve Steiner’s model. The work developed by Weber andVerhoeven (2002) can for example be seen as an attempt to improveSteiner’s notion of deformation. Indeed, while Steiner suggeststhat explanation concerns an array of related proofs and theorems,although maintaining that each proof is an explanation of theindividual theorem, Weber and Verhoeven (2002) start with what makespairs of proofs – P1 and P2 – count as explanatory. Inparticular, they focus on explaining why, while a certain class ofobject \(x\) has a property \(Q\) (proof P1), another class of objects\(y\) enjoys property Q’ (proof P2). Here P1 and P2 use the sameaxioms and the same logical rules, but while P1 uses a characterizingproperty of \(x\), but not of \(y\), P2 uses a characterizing propertyof \(y\), but not of \(x\). A final attempt to enrich Steiner’saccount is proposed by Salverda (2017) who tries to adapt thisapproach to an internalist perspective on explanations of the sortdiscussed in section 2.2.2.

In the field of causal explanations, a dominant perspective has beenformulated in counterfactual terms. Although there has long been aresistance in the use of counterfactuals to account for explanationsin mathematics (see Lange 2017, 88, 2022), many authors adopt thisapproach, perhaps due to the attractiveness of a unified theory ofexplanation that promises to hold in both causal and non-causalcontexts.

According to a counterfactual account, the evaluation of whether amathematical fact \(F\) explains another mathematical fact \(G\) boilsdown to the evaluation of the following two counterfactuals:

CF1:

if \(F\) had not been the case, \(G\) would not have been thecase,

CF2:

if \(G\) had not been the case, \(F\) would not have been thecase.

The first counterfactual, CF1, needs to be true: it directly accountsfor the explanatory power of the relation between \(F\) and \(G\). Bycontrast, the second counterfactual, CF2, needs to be false since itserves to ensure that the relation between \(F\) and \(G\) isasymmetric, i.e., it shows that it is not the case that \(G\) explains\(F\).

Once the counterfactuals are specified, a theory for counterfactualsneeds to make clear what truth-conditional account of counterfactualsis adopted. Here (at least) two options naturally emerge. On the onehand, one can evaluate a counterfactual using possible worldssemantics; for example, Lewis’s closeness-based semantics, whichtrivializes for mathematical counterfactuals (see Lewis 1973, andStalnaker 1968), has recently been extended to avoid thesetrivialities (see for example Nolan 2001 and Priest 2002). Thisextension, which considers both possible and impossible words, can beused to evaluate the truth value of counterfactuals CF1 and CF2. Onthe other hand, one can also try to adapt the standard tools ofstructural equation modelling (see Pearl 2000) to evaluate the truthvalue of CF1 and CF2. In this case, one interprets mathematical factsas variables which can either take the value 1 or the value 0,according to whether the propositions they represent are either trueor false. While the variable that denotes \(F\) is anexogenous variable – its values are determined byfactors outside the model – the variable that denotes \(G\) isendogenous – its values are determined by the value ofother variable(s), in our case \(F\). In order to test whether thecounterfactual CF1 is true, one needs tointervene on thevalue of the variable assigned to \(F\) and check whether this changeaffects the value of the variable assigned to \(G\). As for the truthvalue of CF2, its falsity, and thus the asymmetry required by theexplanatory relation, is built into the nature of endogenousvariable.

Both Reutlinger et al. (2022) and Baron et al. (2020) support acounterfactual theory of explanation that is mainly discussed in thepossible worlds’ semantics framework. More precisely, whileReutlinger et al. defends the value of a monist theory of explanation,Baron et al. (2020) exemplify the counterfactual approach tomathematical explanations with a real case of explanatory proof.

Gijsbers (2017) develops a counterfactual account of explanatoryproofs which relies on the structural equation framework, but wherethe notion of “intervention” à la Woodward (2003),cannot be employed in the mathematical context. As Woodwardemphasizes, an intervention is a causal change to the value of avariable. Instead Gijsbers introduces the idea of a“quasi-interventionist” theory of mathematicalexplanation: in this theory, quasi-interventions reveal asymmetrieswhich are inherent not in the mathematical proofs, but in themathematical practice (see Gijsbers, 2017, 59). In other words,asymmetries are no longer accounted for in an objective, but rather ina more subjective way that is tied to the features of the practice inquestion.

In a sense Gijsbers’ model is complementary to the one proposedby Frans and Weber (2014). Indeed, while Gijsbers accounts for theexplanatory power of proofs in counterfactual terms, without using thenotion of intervention, Frans and Weber account for the explanatorypower of proofs with a mechanistic model of explanation that directlygeneralizes on Woodward’s notion of an intervention.

The use of counterfactuals to model mathematical explanations has beencriticized by Kasirzadeh (2021) and Lange (2022). While Kasirzadehquestions whether the explanans of an explanatory proof can bemeaningfully varied in a mathematical context, as the counterfactualaccounts would require, Lange argues instead that the counterfactualaccount is rather threatened by the existence of too manynon-trivially true countermathematicals. Both Kasirzadeh and Langeemphasize that the capacity to answerwhat-if-things-had-been-different questions does not correlate withexplanatory power in mathematics. Finally, note that also in Jansson(2018) one might find criticisms on the use of the structuralequations’ framework to model dependence relations other thancausation, and thus arguably also dependence relations in amathematical context.

Not all externalist models for mathematical explanations aremodifications of Steiner’s theory or conveyed in counterfactualterms. Pincock (2015b) for example proposes to classify a proof asexplanatory when it invokes more abstract kind of entities than thetopic of the theorem it proves. Wilhelm (2021) and Poggiolesi(forthcoming) contain different proposals for the analysis ofexplanatory proofs that are similar in perspective to Pincock’sapproach. In their cases the determination of the explanatory power ofdifferent proofs requires a formalization of the proofs in logicalsystems. While for Wilhelm the explanatory power of a proof comes fromthe balance between the simplicity and the depth of the formalizedproof, Poggiolesi distinguishes an explanatory proof from anon-explanatory one in that only in the (formalized version of the)former one can witness an increase of conceptual complexity from theassumptions to the theorem the proof aims to establish.

2.2.2 Internalist models of mathematical explanation

In a paper of 1974, Friedman posed a challenge for any coherentaccount of scientific, and thus presumably also mathematical,explanation: he argued that any such account needed to show howexplanation generates understanding. “I don’t see how thephilosopher of science can afford to ignore such concepts as‘understanding’ and ‘intelligibility’ whengiving a theory of the explanation relation” (Friedman 1974, 8).While externalist, or ontic, accounts do not directly concernthemselves with Friedman’s challenge, they do not deny the linkbetween explanation and understanding. However, they simply do notpose understanding as a defining characteristic of explanation. Bycontrast, internalist, or epistemic, accounts are those which directlyaddress this challenge.

In the philosophy of science, one of the main conceptions ofscientific understanding is the unificationist model which argues thatexplanations provide understanding by unifying different phenomena.Although the idea is undoubtedly intuitively appealing, the keyquestion is whether the notion of unification can be made more preciseso that we can distinguish between what an explanation is and what isnot. Friedman (1974) is an early attempt to do this, although hisformulation was quickly shown to suffer from several technicalproblems (see Kitcher 1976). Kitcher is, on the other hand, the mainsupporter of the unificationist approach. His proposal consists inlooking at unification as the reduction of the number of argumentpatterns used in providing explanations while being as comprehensiveas possible in the number of phenomena explained:

Understanding the phenomena is not simply a matter of reducing the“fundamental incomprehensibilities” but of seeingconnections, common patterns, in what initially appeared to bedifferent situations. Here the switch in conception from premiseconclusion pairs to derivations proves vital. Science advances ourunderstanding of nature by showing us how to derive descriptions ofmany phenomena, using the same patterns of derivation again and again,and, in demonstrating this, it teaches us how to reduce the number oftypes of facts that we have to accept as ultimate (or brute). So thecriterion of unification I shall try to articulate will be based onthe idea that E(K) is a set of derivations that makes the besttradeoff between minimizing the number of patterns of derivationemployed and maximizing the number of conclusions generated. (Kitcher1989, 432)

Let us make this a little bit more formal. Let us start with a set Kof beliefs assumed to be consistent and deductively closed (informallyone can think of this as a set of statements endorsed by an idealscientific community at a specific moment in time; Kitcher 1981, 75).A systematization of K is any set of arguments that derive somesentences in K from other sentences of K. The explanatory store overK, E(K), is the best systematization of K (Kitcher here makes anidealization by claiming that E(K) is unique). Corresponding todifferent systematizations we have different degrees of unification.The highest degree of unification is that given by E(K). But accordingto what criteria can a systematization be judged to be the best? Thereare three factors: the number of patterns, the stringency of thepatterns, and the set of consequences derivable from theunification.

Two remarks are in order when it comes to Kitcher’s proposal.First, his account of theoretical unification is mainly thought of forthe general question of scientific explanation, although he sees asone of the virtues of his viewpoint to be extendable to mathematics aswell. Secondly, Kitcher’s model is not meant to address thelocal question of what distinguishes an explanatory proof from onethat does not explain (as all other accounts do); it rather provides anovel perspective on the global question of how to systemize a wholebody of knowledge that has explanatory value. The application ofKitcher’s model to explanatory proofs has been explored in twoopposite directions. On the one hand, Hafner and Mancosu (2008) testedKitcher’s model with three different methods to prove theoremsabout real closed fields (see Brumfiel 1979); the authors showed thatthe model makes predictions about the explanatory power of thesemethods which contradicts judgments coming from the mathematicalpractice (See also Pincock 2015b). On the other hand, Frans (2021) notonly reassesses the value of unificatory understanding, which it is atype of explanatory understanding, for mathematics; additionally, heshows through a plethora of different examples, ranging fromPythagoras’ theorem to the theorem that states that sum of thefirst n integers equals n(n+1)/2, that proofs can contribute tounificatory understanding.

A novel internalist account has recently been proposed by Inglis andMejía-Ramos (2019), who apply Wilkenfeld’s functionalmodel of understanding (see Wilkenfeld (2014)) to the mathematicalcase. Wilkenfield’s approach consists in reversingFriedman’s perspective: while Friedman demanded thatphilosophers clarify how explanations, suitably defined, generateunderstanding, Wilkenfield defines explanations as those things thatgenerate understanding. By doing so, Wilkenfield moves the burden ofclarification from the notion of explanation to that of understandingand how it is generated: this move– he argues– has recently become tenable as philosophical accounts ofunderstanding have become more and more sophisticated.

In Inglis & Mejía-Ramos (2019) the conception ofunderstanding adopted is that of Kelp (2016), along with amodalmodel of the generation of understanding (see Atkinson andShiffrin 1968). With these two elements at hand, Inglis &Mejía-Ramos identify three properties that any mathematicalexplanatory proof is likely to have: (i) an explanatory proof woulddirect the reader’s attention to its conceptually importantsection; (ii) it would reorganize the new and existing informationinto coherent new schema; (iii) it would reduce the chances of workingmemory capacity to be exceeded.

Other internalist accounts of explanatory proof have been developed byDelarivière, Frans & Kerkhove (2017), Dutilh Novaes (2018),and Lehet (2021).

3. Some connections to other debates

A number of fruitful studies have recently appeared connectingmathematical explanation to mathematical beauty, purity of methods,understanding in mathematics, mathematical style, and mathematicaldepth. We simply refer to one or two such background studies andencourage the reader to explore the bibliography of the studiesreferred to. The most extensive studies connecting mathematical beautyand explanation are Giaquinto (2016) and Lange (2016). The notion ofpurity of method has long been of interest to mathematicians andphilosophers (see Detlefsen and Arana (2011) and Arana and Mancosu(2012)). Among the most recent contributions on purity andmathematical explanation are Skow (2015), Lange (2015b), Ryan (2021)and Arana (2023). The connection between mathematical explanation andunderstanding has been discussed in Molinini (2011), Cellucci (2014),and Delariviére et al. (2017). For connections betweenmathematical depth and mathematical explanation see Lange (2015c).Moreover, theorists of style in mathematics and science haveemphasized the importance of explanatory arguments for characterizingstyle (see Mancosu (2021) for an overview).

The issues that have shaped the debates about mathematical explanationreviewed in sections 1 and 2 also arise in ongoing debates in thephilosophy of mind and moral theory. For the philosophy of mind, onepuzzle is how appealing to mental properties can explain human actionseven though a human is a physical entity. If non-mental, physicalproperties are apt to explain any physical event or pattern ofphysical events, then it seems that mental properties are dispensableor “epiphenomenal”. For moral theory, a series ofquestions arise about how moral properties relate to the presumablynon-moral features of the physical world. In terms of explanation, itseems like there is no explanatory work for moral properties to do, atleast with respect to physical events. However, our ordinary practicesfrequently appeal to these properties in putative explanations. So, aswith the philosophy of mind, it seems that we must either revise ourexplanatory practices or else find a place for these properties in amore comprehensive conception of reality.

Kim’s exclusion argument is a prominent driver of these debatesin the philosophy of mind (Kim 2005). Kim argues that the existence ofmental properties requires that these properties provide some genuinecontribution to the explanation of physical events. However, Kimmaintains that mental properties are excluded from this contributionby the causal closure of the physical, i.e. every physical event has apurely physical explanation. One response to Kim is that the rightconception of causal explanation makes space for mental properties toexplain (Shapiro & Sober 2007, Woodward 2021b). The“explanatory autonomy” of the mental can thus be obtainedin a way that parallels similar generalizations of causal explanationto allow for genuine mathematical explanations of physical phenomena(section 1.2). Pluralists about mathematical explanation can develop adifferent response to exclusion arguments: if explanations come indifferent kinds, then an explanation of one kind does not stand in theway of an explanation of another kind (Batterman 2021). Baker (2022)has pursued a different kind of response that compares Dennett’sintentional stance with a “mathematical stance” thatenables mathematical explanations of physical phenomena.

Harman and Street have advanced explanatory challenges to moralproperties that can be fruitfully compared to criticisms of theexplanatory indispensability argument for mathematical platonism(Harman 1977, Street 2006). While Harman focuses on the explanation ofmoral observations (e.g. that some action is wrong), Street emphasizesa broader concern with explaining other phenomena such as theprevalence of some moral judgments (e.g. that murder is wrong). Forboth, the challenge is that the best explanation does not involvemoral properties. That is, moral properties are explanatorilydispensable for the targets in question. As Sinclair and Leibowitzemphasize, this argument, and the responses to it, parallel debatesabout the explanatory dispensability of mathematical objects (Sinclair& Leibowitz 2016). One innovation in the debate about moralproperties is Enoch’s argument that it is sufficient for moralproperties to be indispensable to practical deliberation. If thisnon-Quinean condition for ontological commitment is granted, then itmay be feasible to identify new forms of explanatory indispensabilityfor mathematical objects. For some investigations into how sucharguments may or may not extend to mathematics, see Leng (2016), Baker(2016), Enoch (2016) and Clark-Doane (2020).

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