## 1. Mathematical explanation in the empirical sciences



Nearly all of our most successful empirical sciences employ a great
deal of mathematics. In addition, scientists often emphasize the value
of explaining some phenomenon that they have discovered. It is natural
to wonder, then, if mathematics is well-suited to contribute to the
explanation of natural phenomena and what these contributions might
be. In the philosophy of science most accounts of explanation identify
an explanation with an appropriate description of a cause (see Salmon
1984, Cartwright 1989, Woodward 2003, Strevens 2008, and Beebee,
Hitchcock & Menzies 2010 for an overview). Nearly everyone can
admit that mathematical tools are an excellent means of tracking or
representing causes. For example, mathematics can be used to explain
why Halley's comet's orbit has a period of 75 years. Much
of the debate about mathematical explanation in the empirical sciences
has focused on more contentious cases: what role might mathematics
play in non-causal explanations, if there are any, and how might these
cases challenge this or that account of causal explanation (Reutlinger
& Saatsi 2018)?



One kind of case that has been emphasized aims to explain the
possibility or impossibility of some process. For example, why can we
not divide our 23 strawberries equally among three friends (Lange
2013), or why can we arrange 81 stamps into a 9 by 9 array? The
legitimate explanations seem to be that 23 is not divisible by 3
without remainder and that \(9 \times 9 = 81.\) Neither mathematical
fact is a cause of the feature of the process in question, so we seem
to have a non-causal explanation where mathematics is part of the
explanation. The possibility or impossibility of other processes are
explained by other areas of mathematics that investigate structures or
formal features of natural systems. For example, why can we not make a
circuit of the bridges of Konigsberg that involves crossing each
of the bridges exactly once (Pincock 2007, Molinini 2012)? This is
explained by the abstract structure of the bridge network. Why is it
possible for there to be stable planetary orbits? One explanation that
has been proposed appeals to the dimensions of space-time (Woodward
2003).



Another kind of non-causal mathematical explanation deals with a
striking or surprising feature of a phenomenon, where that feature can
be identified through a mathematical analysis of the situation. The
feature may be tied to a minimization process, or be especially
resilient or stable for what is arguably a mathematical reason.
Perhaps the most discussed case is the length in years of the
life-cycle of three species of periodic cicada: why are these lengths
either 13 or 17 years (Baker 2005, 2017)? An explanation is that 13
and 17 are prime numbers and that prime numbered life-cycles confer a
relative fitness advantage in avoiding predators and competition for
scarce resources like food. Other broadly evolutionary cases include
the hexagonal shape of honeycomb cells (Lyon & Colyvan 2008,
Raz 2017, Wakil & Justus 2017) and the pattern of seeds in a
sunflower (Lyon 2012). There is an extensive literature on how these
optimality explanations might work in biology and economics (Potochnik
2007, Rice 2015, 2021). However, this explanatory contribution from
mathematics can be found in other domains as well. For example, why do
soap films obey Plateau's laws (Lyon 2012, Pincock 2015a)? This
can be explained through a process of surface minimization, subject to
constraints. The mathematics of the situation is central to the
character that the laws take on. Other cases turn on a mathematical
analysis of the stability or instability of some process. For example,
why do the so-called Kirkwood gaps appear in our solar system's
asteroid belt (Colyvan 2010)? Occupying some spatial regions is
unstable, so that an asteroid that starts in such a region is
overwhelmingly likely to leave it. Similar analyses explain patterns
in the rings of Saturn or the collapse of an engineered structure like
a bridge. (See also Ashbaugh, Chicone & Cushman 1991, Colyvan
2001, Lipton 2004, Baker 2015a and Lange 2017 for a range of other
examples.)



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


### 1.1 Some historical remarks



Does mathematics help explain the physical world or does it actually
hinder a grasp of the physical mechanisms that explain the how and why
of natural phenomena? It is not possible here to treat this topic in
its full complexity but a few remarks will help the reader appreciate
the 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 a
> thing, as opposed to knowing it in the accidental way in which the
> sophist knows, when we think that we know the cause on which the fact
> depends as the cause of the fact and of no other, and further, that
> the fact could not be other than it is. (BWA, 111, Post. An. I.1, 71b
> 5-10)
> 



The causes [aitia] in question are the four Aristotelian causes:
formal, material, efficient, and final. Nowadays, translators and
commentators of Aristotle prefer to translate aition [aitia] as
'explanation[s]', so that the theory of the four causes
becomes an account of four types of explanations. For instance, here
is Barnes' translation of the passage quoted earlier: "We
think we understand a thing simpliciter (and not in the sophistic
fashion accidentally) whenever we think we are aware both that the
explanation because of which the object is is its explanation, and
that 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 is
obtained through demonstration. However, not all logically cogent
proofs provide us with the kind of demonstration that yields
scientific knowledge. In a scientific demonstration "the
premisses must be true, primary, immediate, better known and prior to
the conclusion, which is further related to them as effect to
causes." (BWA, 112, Post. An. I.1, 71b 20-25) In
Barnes' translation: "If, then, understanding is as we
posited, it is necessary for demonstrative understanding in particular
to depend on things which are true and primitive and immediate and
more familiar than and prior to and explanatory of the
conclusion." (CWA, 115, Post. An. I.1, 71b 20-25)



Accordingly, in "Posterior Analytics" I.13, Aristotle
distinguished between demonstrations "of the fact" and
demonstrations "of the reasoned fact". Although both are
logically cogent only the latter mirror the causal structure of the
phenomena under investigation, and thus provide us with knowledge. We
can call them, respectively, "non-explanatory" and
"explanatory" demonstrations.



In Aristotle's system, physics was not mathematized although
causal reasonings were proper to it. However, Aristotle also discussed
extensively the so-called mixed sciences, such as optics, harmonics,
and mechanics, characterizing them as "the more physical of the
mathematical sciences". There is a relation of subordination
between these mixed sciences and areas of pure mathematics (see Dear
2011). For instance, harmonics is subordinated to arithmetic and
optics to geometry. Aristotle is in no doubt that there are
mathematical explanations of physical phenomena:



> 
> For here it is for the empirical scientist to know the fact and for
> the mathematical to know the reason why; for the latter have the
> demonstrations of the explanations, and often they do not know the
> fact, just as those who consider the universal often do not know some
> of 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 of
natural phenomena was one on which there was disagreement. As the
domains to which mathematics could be applied grew, so also did the
resistance to it. One source of tension consisted in trying to
reconcile the Aristotelian conception of pure mathematics, as
abstracting from matter and motion, with the fact that both physics
(natural philosophy) and the mixed sciences are all conversant about
natural phenomena and thus dependent on matter and motion. For
instance, an important debate in the Renaissance, known as the
Quaestio de Certitudine Mathematicarum, focused in large part on
whether mathematics could play the explanatory role assigned to it by
Aristotle (Mancuso 1996, ch. 1). Some argued that lacking causality,
mathematics could not be the 'explanatory' link in the
explanation of natural phenomena (see also sections 1.2 and 1.3).



By the time we reach the seventeenth century and the Newtonian
revolution in physics, the problem reappears in the context of a
change of criteria of explanation and intelligibility. This has been
beautifully described in an article by Y. Gingras (2001). Gingras
argues that "the use of mathematics in dynamics (as distinct
from its use in kinematics) had the effect of transforming the very
meaning of the term 'explanation' as it was used by
philosophers in the seventeenth century" (2001, 385). What
Gingras describes, among other things, is how the mathematical
treatment of force espoused by Newton and his followers - a
treatment that ignored the mechanisms that could explain why and how
this force operated - became an accepted standard for
explanation during the eighteenth century. After referring to the
seventeenth and eighteenth centuries' discussions on the
mechanical explanation of gravity, he remarks:



> 
> This episode shows that the evaluation criteria for what was to count
> as an acceptable 'explanation' (of gravitation in this
> case) were shifting towards mathematics and away from mechanical
> explanations. Confronted with a mathematical formulation of a
> phenomenon for which there was no mechanical explanation, more and
> more actors chose the former even at the price of not finding the
> latter. This was something new. For the whole of the seventeenth
> century and most of the eighteenth, to 'explain' a
> physical phenomenon meant to give a physical mechanism involved in its
> production. ... The publication of Newton's
> *Principia* marks the beginning of this shift where
> mathematical explanations came to be preferred to mechanical
> explanations when the latter did not conform to calculations. (Gingras
> 2001, 398)
> 



Among those who resisted this confusion between "physical
explanations" and "mathematical explanations" was
the Jesuit Louis Castel. In "Vrai systeme de physique
generale de M. Isaac Newton" (1734), he discussed
*Principia*'s proposition XIII of Book III (on
Kepler's law of areas). He granted that the proposition
connected mathematically the inverse square law to the ellipticity of
the course of the planets. However, he objected that "the one is
not the cause, the reason of the other" (Castel 1734, 97) and
that Newton had not provided any physical explanation, only a
mathematical one. Indeed "physical reasons are necessary reasons
of entailment, of linkages, of mechanism. In Newton, there is none of
this kind." (Castel 1734, 121)



Some contemporary discussions bear close proximity to these worries.
Consider Morrison's book *Unifying Scientific Theories*
(2000). One of the major theses of the book is that unification and
explanation 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 of
Castel's objections:



> 
> Another example is the unification of terrestrial and celestial
> phenomena in Newton's *Principia*. Although influenced by
> Cartesian mechanics, one of the most striking features of the
> *Principia* is its move away from explanations of planetary
> motions in terms of mechanical causes. Instead, the mathematical form
> of force is highlighted; the planetary ellipses discovered by Kepler
> are "explained" in terms of a mathematical description of
> the force that produces those motions. Of course, the inverse-square
> law of gravitational attraction explains why the planets move in the
> way they do, but there is no explanation of how this gravitational
> force acts on bodies (how it is transported), nor is there any account
> of its causal properties. (Morrison 2000, 4)
> 



Using several case studies (Maxwell's electromagnetism, the
electroweak unification, etc.), Morrison argues that the mathematical
structures involved in the unification "often supply little or
no theoretical explanation of the physical dynamics of the unified
theory" (Morrison 2000, 4). In short, the mathematical formalism
facilitates unification but does not help us explain the how and why
of physical phenomena.



We have to close these historical remarks here, although it would be
interesting to pursue these questions in a more systematic way into
the nineteenth and the twentieth centuries (see however Dorato 2017
for 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 in
contemporary discussions in philosophy of science, to which we now
turn, we are still confronted with such issues.


### 1.2 Theories of explanation



Two legacies of the Aristotelian tradition surveyed in section 1.1 are
that explanations require causes and that providing an explanation
requires giving an argument that turns on laws. Debates in the
philosophy of science since the 1960s have shown how one can privilege
one legacy over another (Salmon 1989). Hempel's
deductive-nomological analysis of explanation requires that an
explanation be a deductively valid argument from true premises, where
at least one premise is a scientific law (Hempel 1965). Hempel and
other empiricists in this broad tradition are wary of making causes
central to explanation. This is apparent even in some of
Hempel's critics such as Kitcher, who emphasizes the unifying
power of explanations. For Kitcher, an explanation is an instance of a
deductive argument scheme, where the schemes to adopt are identified
on 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 of
science that explanations need only provide causal information about
the explanatory target (Salmon 1984, 1989). For Salmon and others in
this tradition, explanations do not require laws and need not even be
arguments. One development of this approach preserves Salmon's
emphasis on causal mechanisms as a special sort of process. The
so-called "new mechanists" endorse a broader notion of
causal mechanism than Salmon allowed for, and identify an explanation
of some target with a mechanism that produces it (Machamer, Darden
& Craver 2000). Other approaches to causal explanation include
David Lewis' counterfactual analysis of causation (Lewis 2004)
and Woodward's interventionist theory (Woodward 2003, 2021a).
Despite their differences, Lewis and Woodward allow for causal
explanations in the absence of mechanisms. This makes their approaches
to explanation easier to generalize to non-causal cases.



Philosophical discussions of non-causal mathematical explanation can
be classified based on how they are related to these debates about
scientific explanation more generally. One position argues for the
need to restore something like Hempel's emphasis on laws or
Kitcher's claims about unification (Baron 2019). Another
position generalizes from an account of causal explanation so that it
can 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 that
cannot be fit into one or the other of these two options (Pincock
2018, 2023).



Lange's extensive discussions of non-causal explanations can be
seen as a valiant attempt to preserve the law-based approach to
scientific explanation that goes back to Hempel (and Aristotle) (Lange
2013, 2017). Lange's work on laws emphasizes how to identify
claims with the right kind of modal strength to contribute to
explanations (Lange 2009). Consider the contrast between "All
gold cubes are less than 1 cubic mile in volume" and "All
Ur-235 cubes are less than 1 cubic mile in volume". The former
statement is contingent, while the latter statement has some degree of
necessity. This allows the latter statement to contribute to an
explanation. Lange's approach to mathematical explanation
extends this point so that mathematical claims can function in
explanations in a distinctive way due to their special degree of
necessity. This allows mathematics to contribute to what Lange calls
explanations by constraint that show how some outcome is guaranteed to
arise (Lange 2017, ch. 2). This is how Lange treats the strawberry
division case and also cases like the bridges of Konigsberg.
Other mathematical explanations turn on the dimensions of the
quantities involved or the statistical features of some process (Lange
2017, ch. 5, 6). In each of these types of cases, the modal character
of the mathematical claim allows it to explain just as the modal
character of ordinary scientific laws allow them to explain.



Unsurprisingly, many philosophers have challenged Lange's
proposals in ways that are reminiscent of how Salmon objected to
Hempel's deductive-nomological account (Pincock 2015a,
Reutlinger 2017b, Saatsi 2018). For example, Craver and Povich object
that, in the absence of a causal constraint on explanation,
Lange's proposals lack any suitable worldly basis, and so count
too many representations of some target as explanatory (Craver &
Povich 2017, Lange 2018a). As a mechanist about explanation quite
generally, Craver seems inclined to dismiss the possibility of
non-causal mathematical explanations (Craver 2014). His co-author
Povich has offered a more constructive proposal for these cases that
allows for explanations with a variety of worldly or ontic bases
(Povich 2020, 2021). Povich deploys a version of what amounts to a new
consensus for handling non-causal explanations: if a proposed
explanation relates to the right counterfactuals in the right way,
then it is legitimate (Woodward 2018, Rice 2021). However, there is as
yet no agreement on what sorts of counterfactual tests are sufficient
for explanation (see Lange 2021a for some objections to this
approach).



One option is to treat a mathematical claim like a law that governs
some situation, and to credit the mathematics with explanatory power
whenever it allows us to assess a range of counterfactual scenarios.
This is Reutlinger's proposal, which deliberately loosens the
requirements that Woodward places on interventions (Reutlinger 2016,
2017a, 2018, Reutlinger, Colyvan & Krzyzanowska 2022). For
example, in the bridges of Konigsberg case, the mathematical
claim indicates that a circuit of the bridges would be possible in the
counterfactual scenario where some of the actual bridges were absent.
Reutlinger concludes that the mathematical claim thus explains what is
going on in the actual world by indicating what makes a difference to
the feature of interest. Arguably, though, this approach is too
liberal. Suppose we ask why we have \(81\) stamps. It is a
mathematical truth that \(9 \times 9 = 81,\) and so we can arrange our
stamps in a \(9 \times 9\) array. So if we could not arrange our
stamps 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 we
have \(81\) stamps.



Another option is to treat a mathematical claim like a cause. Then the
claim will be explanatory when a "counter-mathematical"
that supposes that this claim is false winds up making a difference to
the target in question. This sort of counter-mathematical involves
impossible worlds where necessary truths come out false. This is the
option that Povich takes (Povich 2020, 2021). Another family of
proposals along these lines has been developed by Baron in
collaboration with Colyvan and Ripley (Baron, Colyvan & Ripley
2017, 2020). Baron et al. draw on David Lewis' procedure for
evaluating counterfactuals tied to causation: consider the scenario
that arises through a miraculous change that is just enough of a
change to make the antecedent of the counterfactual true. In addition,
Baron et al. require that the features of the natural world that are
implicated in this shift in the mathematics be changed in a
corresponding way. For example, for the cicada case, the central
mathematical claim is that prime periods minimize intersections when
compared to non-prime periods. So among the years 12, 13, 14, 15, 16,
17, and 18 (that are identified by ecological constraints), the primes
13 and 17 stand out as comparatively more fit. Baron et al. consider
the consequences of supposing that 13 is not prime. If 13 was not
prime, they argue, then it would have factors besides 1 and 13, and so
having a 13-year life cycle would not confer any relative fitness
advantage. Thus, in this impossible world, the cicadas would not have
evolved a 13-year life cycle. This is meant to show that 13's
being prime makes a difference to the evolution of 13-year life cycle
cicadas. (See also section 2.2.1 for a parallel debate for pure
mathematics.)



There are pressing questions for these proposals about the nature of
impossible worlds and our epistemic access to them (Kasirzadeh 2021a).
Another sort of objection has been raised by Baron himself in work
that develops another account of mathematical explanation (Baron
2020). Baron, like Baker and Lange, aims to identify a special class
of genuinely or distinctively mathematical explanations of natural
phenomena (Baker 2005, 2009a, Lange 2013). What is special about these
cases is that the mathematics explains, but not by representing or
describing some non-mathematical explainers such as causes or other
worldly difference makers. Baron's general worry is that simply
using countermathematicals fails to distinguish explanations that
employ mathematics from these genuinely mathematical explanations. We
can see this using our original stamp case: why can we arrange our
stamps in a \(9 \times 9\) array? Because we have \(81\) stamps and
\(9 \times 9 = 81.\) This case passes Baron et al.'s
countermathematical 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 simply
tracking the non-mathematical features of the stamps, and so Baron and
others would not want to count this as a special sort of mathematical
explanation.



Baron concludes that some additional requirements must be imposed
beyond the truth of the relevant countermathematical. Here Baron
reaches back to Kitcher's idea that explanations are instances
of a special sort of argument scheme (Kitcher 1989), where the schemes
are found through a process of appropriately unifying the claims that
we accept (for objections to this proposal see Pincock 2023 and Povich
forthcoming). Other recent work on non-causal mathematical
explanations also seems to be returning to some of the original
sources of these debates. For example, Lange argues that the best way
to make sense of the explanatory power of pure mathematics in the
empirical sciences is to adopt an Aristotelian interpretation of pure
mathematics (Lange 2021b).



Lyon proposed another way of relating mathematical explanations to
causal explanations by adapting Jackson and Pettit's notion of a
"program explanation" (Jackson & Pettit 1990). A
program explanation does not invoke a property that causes the outcome
of interest. Instead, the explanation appeals to a property \(A\) that
guarantees the presence of some member of a family of properties,
where some such property \(B\) causes the outcome of interest. As Lyon
summarizes his proposal, "An explanation of an empirical fact is
mathematical - i.e., it has mathematics doing explanatory work
- if the explanation is a program explanation that uses
mathematics in a way that is indispensable to the program" (Lyon
2012, 568). One concern with this proposal is that it includes cases
where the mathematics merely represents some causally relevant
property, as with the stamps case noted above: programming is too
indiscriminate a relation to avoid this worry (Saatsi 2012).



A sweeping way of dealing with the apparent tension between causes and
mathematics has been pursued by some ontic structural realists
(Ladyman & Ross 2007, French 2014). They identify the fundamental
metaphysical structure of the world with a mathematical structure. If
one adopted this kind of structural realism, then the explanatory
power of mathematics in the empirical sciences would receive a
satisfying analysis. In fundamental physics, scientists would be
working with the fundamental mathematical structure directly, and so
explanations there would be essentially mathematical. In
non-fundamental domains such as biology or economics, scientists would
be investigating features of the world that are ultimately grounded in
some mathematical structure. So again it would make sense for many of
the explanations in non-fundamental sciences to be mathematical.
Causal explanations would then turn out to be perfectly consistent
with more fundamental mathematical explanations. Few philosophers are
willing to adopt such a metaphysical position in order to resolve
questions about mathematical explanation, although at least one
physicist has defended this approach (Tegmark 2014).



Another, less metaphysical, solution to these difficulties is to
retain an account of how causal explanations work and to simply
supplement it with a distinct account of how various kinds of
non-causal explanations arise (Pincock 2015a, Pincock 2018, with
criticisms from Knowles 2021a). This sort of explanatory pluralism is
also reminiscent of one aspect of the Aristotelian tradition. One
challenge for the explanatory pluralist is to make sense of the value
that scientists ascribe to explanations: how can there be some special
value in having an explanation if explanations come in different kinds
that have nothing in common? One response to this challenge is that
grasping an explanation produces scientific understanding, but the
nature and value of this scientific understanding remains a subject of
active debate (Rice & Rohwer 2021). Another response is to defend
a restrictive form of explanatory monism. It may turn out that this
monism about explanation is so restrictive that there are no
mathematical explanations in science (Zelcer 2013, Kuorikoski 2021).
This is reminiscent of one side of the Renaissance debate noted in
section 1.1.



Another sort of pluralism arises from supposing that causal
explanations involve the mechanisms championed by the new mechanists,
and then allowing for other sorts of explanations that work in
different ways. One type of case that has received extensive
discussion is so-called "topological" explanation. These
explanations appeal to structural or network-based features to explain
an aspect of a system (Kostic 2020, 2023, Ross 2021). One position is
that topological explanations are non-causal, non-mechanical
explanations that are based on a different kind of explanatorily
relevant feature. For example, Kostic and Khalifa argue that a
non-ontic approach that privileges scientist's explanatory goals
is needed to make sense of topological explanation (Kostic &
Khalifa 2021, 2022). Another position is that an appropriately
flexible notion of mechanism can count genuine topological
explanations as mechanical explanations (Bechtel & Abrahamsen
2010, Bechtel 2020, Huneman 2010, 2018, Brigandt 2013, Green et al.
2018). As with the debates arising from Woodward's and
Lewis' counterfactual approaches to causal explanation, the main
question is what counts as a mechanism and how non-mechanisms can be
explanatory (Janson 2018, Janson & Saatsi 2019, Andersen 2020, Jha
et al. 2022).



More recently, some authors have tried to restore some kind of
explanatory monism by arguing that all scientific explanations turn on
non-representational, expressive elements (McCullough-Benner 2022,
Hunt forthcoming). If this was right, then there would be no
difficulty making sense of mathematical explanations so long as
mathematics can be seen to perform whatever expressive function an
author identifies. The viability of an expressive approach thus turns
on questions about the interpretation of pure mathematics that are
considered in sections 1.4 and 2.2.


### 1.3 Mathematical models and idealization



One assumption of the Aristotelian tradition that is often
unquestioned in the work summarized in section 1.2 is that whatever
provides the explanation (i.e. the explanans) must be true.
Philosophical investigations of scientific models and how these models
may explain have convinced many that an explanans need not be true.
The argument for this conclusion is straightforward: Scientific models
explain and scientific models are not true. So, truth is not required
for explanation (Bokulich 2011; see also Cartwright 1983, Morrison
2015, Rice 2018 and Yablo 2020).



A traditional response to this argument is that even though models are
not true, a model can only explain if it generates some truths about
the target of explanation (Colyvan 2010, see section 1.4 for more
discussion). That is, a model explains only when it represents its
target to be a certain way. The debate thus turns on the options for
making sense of how models represent, especially when those models are
mathematical, and if the representational aspects of models are
sufficient to make sense of model-based explanation. One proposal is
that a model explains when the model represents a target system to be
a certain way and also represents something else that explains why the
system is that way. For example, a causal model of outcome \(E\) needs
to represent \(E\) and some cause \(C\) of \(E\). However, there is no
consensus on what it takes for a model to represent something.



As a model is distinct from its target, and the model and target are
often composed of different materials, it is natural to conclude that
structural relations are central to what a model represents. However,
it is hard to maintain that a model represents a target just in case
there is a structural relation between the model and target
(Suarez 2010, 2015). For example, a model is isomorphic to
itself, and so it stands in a structural relation to itself. But we do
not want to say that a model represents itself. There are also
model-target relations that lack any clear structural
characterization. For example, a model may represent the solar system,
and yet contain only two point particles moving on trajectories that
fail to stand in any non-trivial structural relation to anything in
the actual solar system. So it seems that standing in a structural
relation is neither necessary nor sufficient for a model to represent
a target.



One response to these problems is that a model represents a target
when agents claim that a structural relation obtains between the model
and the target, where that relation may be quite selective and involve
reinterpretations of various elements of the model (Pincock 2012, ch.
2, Frigg & Nguyen 2020). For example, the projection relating a
map to some country may be fairly complicated, and involve various
conventions for what symbols on the map indicate about the country.
Some authors associate the relations that agents establish between
model and target with inferential principles (Bueno & Colyvan
2011, Bueno & French 2018). So according to these various
proposals, a model explains a feature of a target when either some
representational relationship or inferential licenses from model to
target have been established by agents, and these connections
genuinely explain that feature (e.g. they are causes of the feature).
For example, a suitably interpreted map can explain the impossibility
of train travel between two cities by accurately representing the
train network that fails to link those cites.



The different approaches to explanation surveyed in section 1.2 can
then be used to identify explanatory models and what they explain. A
mechanist about explanation can allow that mathematical models explain
by representing mechanisms, while difference-making views will require
an explanatory model to represent difference-making, i.e. how changing
factor \(X\) will go along with a change in outcome \(Y\). All of
these proposals will argue that scientific models do not need to be
true in order to explain. All that needs to occur is for the model to
provide some truths by representing the right things about the target.
So the presence of falsehoods that the model also provides about the
target does not stand in the way of the model's explanatory
power.



This approach to explanatory models and idealization has been called
into question by Batterman. One argument from Batterman is that there
are explanatory models that do not explain in virtue of an element of
the model representing some explanatorily relevant factor such as a
cause or more exotic non-causal difference maker. Instead, in such
cases, "*while we have a genuine mathematical explanation of
physical phenomena, there is no appeal to the existence of
mathematical entities or their properties*. Instead, the appeal is
to a mathematical idealization resulting from a limit
*operation* that relates one model ... to another"
(2010, 7-8). The case that Batterman is discussing here involves
an operation (known as taking the thermodynamic limit) that transforms
a "finite statistical mechanical model" into a
"continuum thermodynamic model". This is central to the
explanation of the universality of some features of phase transitions
that include liquid/gas transitions and magnetization. The features
are universal in the sense that they arise across systems with very
different microphysical features, and so seem especially puzzling.



One point that Batterman is making here is that mathematical
operations that connect models can be significant for a mathematical
explanation of an empirical phenomena (see also Batterman & Rice
2014, Batterman 2019, Batterman 2021). The defenders of traditional
approaches to model explanation often focus on cases where a single
mathematical structure is used to explain. However, the basic ideas of
the traditional approach can be extended to deal with the explanatory
significance of some mathematical operations. For example, one
mathematical model may be transformed into another mathematical model
through a mathematical operation. If this operation reflects something
of explanatory significance, then the two models and the operation
connecting them may be central to the explanation. Some idealizations
are associated with these operations, as in the case where an ocean is
treated as infinitely deep or a planet is modeled as a point particle.
In such cases, the operations function by changing or removing the
interpretation of the elements of the model.



Batterman also develops another point that poses a more significant
challenge to traditional approaches to model explanation. This is that
the "mathematical idealization" that results from this
operation, and that is tied to one of these models, is essential to
the whole explanation. In the ocean case, there is no temptation to
say that the ocean being infinitely deep in one model is explanatorily
relevant to the character of the waves on the surface of the actual
ocean. All that this idealization turns on is that the depth is above
some threshold. Other cases can be handled using similar
"Galilean" idealizations that eliminate the falsehood from
the genuine explanation (Weisberg 2007). But in Batterman's
cases, such as the phase transition case, he is clear that he takes
the idealization to be essential to the explanation: "These
nontraditional idealizations play essential explanatory roles
involving operations or mathematical processes *without
representing* the system(s) in question" (2010, 23). If this
point is accepted, then these cases would undermine the scope of the
traditional approach.



In the philosophy of physics there has been an extensive discussion of
how 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 be
dealt with using explanations that avoid these idealizations or that
treat these idealizations in the manner that we treated the infinitely
deep ocean. Other critics of Batterman have argued that a more
selective approach to what these models represent allows one to admit
that the idealizations are essential to generating the explanation,
but that they are not literally to be included in the explanation
itself. For example, a counterfactual approach to these cases would
identify the explanation with some counterfactuals that are generated
by the model. Batterman and others, in turn, have responded that all
of these criticisms fail to do justice to the phenomena in question or
what scientists say about their explanations (Morrison 2018, Batterman
2019, McKenna 2021).



Other alternatives to a traditional approach to mathematical modeling
and explanation have been developed using other sorts of cases as
their primary motivations (Rizza 2013, Berkovitz 2020, Kasirzadeh
2021b, McKenna 2022). One theme of this work is a generalization of
Batterman's point that scientific explanations often involve
many models whose representational relation to the explanatory target
is more involved than what is usually allowed. For example, Kasirzadeh
considers a case with two mathematical models, with different spatial
scales, of a process of skin color pattern formation (Kasirzadeh
2021b). Biologists asked for an explanation of how the two processes
related to one another. Kasirzadeh argues that this explanation
required additional "bridge mathematics" over and above
the mathematics found in the original two models. The additional
mathematics contributed to the explanation by characterizing how the
microscopic processes gave rise to the unexpected macroscopic
structures. McKenna goes further and argues for the importance of
cases where "models cannot be stitched together in purely
mathematical terms" (McKenna 2022). In McKenna's main case
various models of sea ice permeability are developed for the purposes
of large-scale climate modeling. No single mathematical model of sea
ice proved to be adequate to supply the right parameters to the
large-scale model. Instead, different models of sea ice were used in
conjunction with high-resolution empirical data about samples of sea
ice formations. The significance of these cases for the explanations
that arise from these modeling techniques is likely to be a subject of
ongoing debate.


### 1.4 Explanatory indispensability arguments



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



1.
We ought to have ontological commitment to all and only those
entities that are indispensable to our best scientific theories;
2.
Mathematical entities are indispensable to our best scientific
theories.


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 should
believe exists. Premise 1 is tied to a naturalistic approach to these
beliefs that claims they should be determined by the character of our
best scientific theories. For Quine, one's ontological
commitments are settled by the best regimentation of one's
scientific theories into first-order logic, where what makes a
regimentation the best is determined by ordinary scientific criteria
like consistency and simplicity. Some of Colyvan's cases in his
book invoked the explanatory contribution that mathematical entities
make to our best theories. If we suppose that one aim of science is to
explain, then a regimentation may be the best in part because it
affords explanations of various scientific phenomena.



An explanatory version of this indispensability argument became more
prominent after Melia's exchange with Colyvan (Melia 2000, 2002,
Colyvan 2002). Melia argued that indispensable quantification over
mathematical 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 any
mathematical entities." For example, one could use numbers to
count how many apples and pears one has and conclude that there are
more apples than pears. But Melia would then add that he rejected the
existence of natural numbers, thereby canceling that commitment.
Colyvan replied that such an addition was incoherent when the
mathematical entities were explaining something, and Melia agreed:
"Were there clear examples where the postulation of mathematical
objects results in an increase in the same kind of utility as that
provided by the postulation of theoretical entities, then it would
seem that the same kind of considerations that support the existence
of atoms, electrons and space-time equally supports the existence of
numbers, functions and sets" (Melia 2002, 75-76). The
debate about indispensability and platonism then largely turned to the
evaluation of cases. 



Baker took up Melia's challenge by reformulating the argument so
that the explanatory question became central. Baker also introduced
new cases like the cicada case where the parallel between atoms and
numbers was meant to be clearer. In Baker's formulation, the
argument is:



1'.
 We ought rationally to believe in the existence of any entity
that plays an indispensable explanatory role in our best scientific
theories.
2'.
 Mathematical objects play an indispensable explanatory role in
science.
3'.
 Hence, we ought rationally to believe in the existence of
mathematical objects (Baker 2009a, 613, premises renumbered).


The emphasis on explanation in premise 1' is of course
consistent 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 will
find that it will entail that prime numbers exist. All premise
1' maintains, then, is that one should endorse whatever these
ontological commitments turn out to be. However, a second way to
support premise 1' is available: one could appeal to inference
to the best explanation and its use to support scientific realism
about unobservable entities like electrons. Baker sometimes ties the
appeal of his indispensability argument to scientific realism:
"A crucial plank of the scientific realist position involves
inference to the best explanation (IBE) to justify the postulation in
particular cases of unobservable theoretical entities ... the
indispensability debate only gets off the ground if both sides take
IBE seriously, which suggests that explanation is of key importance in
this debate" (Baker 2005, 225). The appeal to IBE avoids the
Quinean process of regimentation by directing our attention to some
explanatory target such as the length of the life-cycles of some
species of cicada. If we are scientific realists, then we accept the
use of IBE in support of our claims about the existence of various
entities. So if we find that the best explanation also includes
abstract objects like prime numbers, then we should also accept their
existence.



Three worries about this explanatory indispensability argument can be
fruitfully distinguished. The first worry is that the argument is
somehow circular, begs the question or else fails to correctly
identify the basis for our knowledge of the existence of mathematical
entities. The point was forcefully presented by Steiner in 1978.
Steiner argues for the existence of mathematical explanations and also
claims to know of the existence of abstract, mathematical entities.
However, "no explanatory argument can establish the existence of
mathematical entities" (1978b, 20). The reason for this is
attributed to Morgenbesser: "We cannot say what the world would
be 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 to
compare mathematical and non-mathematical explanations of some target
in order to get an explanatory argument going. But if the target
always "presupposes" the existence of some mathematical
entities, then this comparison is not possible. Bangu has developed
this point by noting how many of the cases discussed have targets that
are mathematical in character, as with the prime periods of the
cicadas (Bangu 2008, 2012, see Baker 2021a for a response). So, the
worry continues, mathematical explanations are only indispensable in
science if we have used mathematical entities to characterize the
target phenomena. Pincock has a somewhat similar concern: if the
targets are characterized in weak mathematical terms, then only weak
mathematical theories will be needed to explain these targets, and
these theories can be easily supplied with a nominalistic
interpretation 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 mathematical
explanations in science. However, premise 2' is rejected because
these explanations fail to involve the existence of any mathematical
objects. Saatsi has developed this criticism by claiming that
mathematics only explains by representing some non-mathematical
features of the physical world (Saatsi 2007, 2011). On this reading,
premise 2' requires the existence of some
"distinctively" or "genuinely" mathematical
explanations, but there are no such explanations. As Saatsi puts the
worry, "what really matters for the indispensability argument
- all that matters! - is whether or not mathematics plays
the kind of explanatory role that we should take as ontologically
committing" (Saatsi 2016, 1051). Until the defenders of the
argument clarify what distinctively mathematical explanations are and
how 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 by
mathematical fictionalists like Yablo and Leng (Leng 2010, 2021, Yablo
2012, 2020). Fictionalists accept the existence of distinctively
mathematical explanations and yet argue that these explanations do not
presuppose the existence of any mathematical objects. For example,
Leng argues that "we can generate mathematical explanations of
physical phenomena that do not appeal to any abstract mathematical
objects, but instead only require modal truths about what follows
logically from our mathematical assumptions, together with the
recognition that the assumptions of our mathematical theories are true
when interpreted as about the physical system under examination"
(Leng 2021, 10437). Leng can thus endorse the very same unified
derivation of the features of Baker's cicadas, and yet refrain
from accepting the existence of mathematical objects. While Saatsi
takes the physical features of the system to be the genuine
explainers, Leng uses those same features to interpret the
mathematical 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' has
largely involved cases where it appears that mathematical objects play
a role in the explanation that is analogous to what unobservable
entities like electrons play in other explanations. For example, in
the cicada case, the prime numbers afford a unified derivation of the
target of the explanation. This strategy would be most effective
against representational approaches to mathematical explanation. The
basic idea is that scientists value these explanations, and so any
reinterpretation of them in non-mathematical terms risks privileging
an unmotivated philosophical theory over some legitimate scientific
practice. As Baker and Colyvan put the point in a reply directed at
Daly and Langford (2009): "Commitments to philosophical theories
such as nominalism, a causal theory of explanation, or the
'indexing' view of mathematical applications are not good
reasons for rejecting well-supported scientific and mathematical
claims" (Baker & Colyvan 2011, 332).



A second strategy that Colyvan has pursued is to challenge critics to
recast these explanations in non-mathematical terms. The refusal to do
this involves an "easy road" to nominalism that Colyvan
thinks is untenable: "when some piece of language is delivering
an explanation, either that piece of language must be interpreted
literally or the non-literal reading of the language in question
stands proxy for the real explanation" (Colyvan 2010, 300). This
strategy is most effective against fictionalists. It involves a
conception of scientific explanation that requires that every genuine
explanation be presentable in literal, non-metaphorical language: to
say why something is the case, we must literally say what is
responsible for what, and so it must be in principle possible to avoid
fictional or metaphorical tools. If fictionalists are right about pure
mathematics, then mathematics is simply such a tool, and so they
should be able to sketch a non-mathematical version of the
explanations at issue. The fictionalist response is to deny this
conception of explanation.



Yet another kind of criticism of premise 2' accepts both the
existence of mathematical explanations and that these explanations
involve mathematical objects, but maintains that these explanations
are dispensable from our best science. That is, either the best
regimentation of our scientific theories will avoid quantification
over mathematical objects or no appeal to IBE will actually support
adopting such an explanation. This sort of criticism can be traced
back to Field's pioneering *Science without numbers*
(Field 1980). There Field contrasts "intrinsic"
explanations with "extrinsic" explanations. He claims that
all mathematical explanations are extrinsic and that for every
extrinsic explanation of some target, there is a superior, intrinsic
explanation of that very target (Field 1980, 43-44; see Marcus
2013). The explanations championed by Colyvan, Baker and others
suggest that it is not clear that a mathematical explanation is always
extrinsic or that a non-mathematical explanation that is intrinsic is
superior in all respects. Consider again the non-causal explanation of
the impossibility of traversing the bridges of Konigsberg or the
evolutionary explanation of the prime periods of the cicada. While
some non-mathematical derivation of these targets is surely available,
this does not settle whether or not these derivations should count as
explanations or what their explanatory virtues might be.



Perhaps the most promising defense of premise 2' would be to
provide a positive account of distinctively mathematical explanations
that would clarify how endorsing such explanations commits one to the
existence of some mathematical objects. The recent literature on this
issue again seems to lead to a kind of standoff. Consider, for
example, Baron's "Pythagorean" proposal for these
explanations (Baron forthcoming). Baron defines a Pythagorean to be
someone who not only believes in mathematical objects as abstract
entities, but who claims that some of the intrinsic properties of
these abstract entities are also possessed by concrete entities. This
is possible because the salient intrinsic properties of the
mathematical entities are structural properties that are found in the
concrete world whenever the concrete entities are arranged in the
right structure. These shared, structural properties and their
necessary mathematical relations thus enable mathematical truths to
explain features of physical systems such as the bridges or cicadas.
In addition, Baron is clear that "Structural properties on my
account make indispensable reference to abstract objects" (Baron
forthcoming, 25). So one defense of premise 2' involves adopting
Baron's Pythagoreanism.



Other accounts of distinctively mathematical explanation support the
rejection of premise 2'. For example, Lange's modal
interpretation of distinctively mathematical explanations leads him to
endorse what he calls an "Aristotelian realist"
interpretation of pure mathematics in terms of a special kind of
abstract property, without the recourse to any abstract objects (Lange
2021b, see also Franklin 2008). According to Lange, the best way to
make sense of the explanatory power of mathematical truths is to
suppose that "mathematics concerns mathematical properties
possessed by physical systems" (Lange 2021b, 50). As with Baron,
these mathematical properties can help to explain why these physical
systems have some other properties. For Lange the main benefit of such
an Aristotelian interpretation is that the salient modal features of
the physical systems arise from the presence of the mathematical
properties. This helps to clarify in what sense the mathematical truth
and the properties it invokes may be explanatorily prior to some
target property. Crucially, though, for Lange this interpretation of
pure mathematics eliminates the need to invoke abstract objects. If
Lange is right, then, premise 2' of the explanatory
indispensability argument is false. (See also Knowles & Saatsi
2021, Knowles 2021a, Knowles 2021b and Baker 2022 on additional
challenges to this premise.)



One diagnosis of the problems with indispensability arguments is that
the conclusion of the argument concerns the interpretation of pure
mathematics while the premises of the argument consider how
mathematics is used in science. Perhaps, then, a platonist would be
better served by focusing on explanatory considerations that arise
within the practice of pure mathematics. This is the focus of section
2.


## 2. Mathematical explanation in mathematics



Much mathematical activity is driven by factors other than
establishing that a certain theorem is true. In many cases
mathematicians are unsatisfied by merely knowing that a mathematical
fact holds and reprove it, while also claiming explanatory benefits
for the new proof. This type of explanatory activity appears within
mathematics itself (see the Preamble) and thus one often speaks of
"internal" or "intra-mathematical
explanations" (Baron, Colyvan & Ripley 2020, Betti 2010,
Mancosu 2008). The expression "internal mathematical
explanation" covers a wide range of different phenomena: an
internal mathematical explanation could amount to the recasting of an
entire area of mathematics or it might aim at providing explanatory
proofs for specific theorems. The variety of these mathematical
explanatory 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 attention
has been focused on proofs which not only prove *that* a
theorem is true, but also show *why* it is true. While there
might not be agreement on specific instances, many mathematicians
often claim that certain proofs have an explanatory power and that
others do not. These claims are found throughout the history of
mathematics 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 most
> instructive are of course those that let one understand the deep
> reasons of the results that one is establishing. (Bouligand, 1932, 6,
> Mancosu's translation)
> 



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



> 
> In this book we absolutely and unequivocally refuse to give proofs of
> this [...] type [transcendental proofs]. Every result is proved
> uniformly for all real closed ground fields. Our philosophical
> objection to transcendental proofs is that they might logically prove
> a result, but they do not explain it, except for the special case of
> real numbers. (Brumfiel 1979, 166)
> 



As the previous examples show, explanatory proofs could be of several
types and explain in different manners. A recent debate has focused on
the issue of whether proofs by induction are explanatory. On the one
hand, Lange (2010) argues that proofs by induction are not
explanatory. His argument relies on the use of a form of upward and
downward induction from a fixed number k, with k[?]1. According to
Lange, 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 a
certain 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) rejects
Lange's argument by arguing against the explanatory equivalence
between proofs by ordinary mathematical induction and proofs by upward
and downward induction. Hoeltje et al. (2013) reject what they see as
an unacknowledged assumption in Lange's argument, namely that a
universal sentence explains its instances. Dougherty (2017)'s
line of attack is based on Lange's need to presuppose a
problematic notion of identity of proofs, which he questions using an
alternative criterion of identity spelled out using two equivalent
characterizations (the first appealing to the language of homotopy
type theory and the second using algebraic representatives to proofs).
Both Baldwin (2016) and Lehet (2019) defend the explanatory value of
induction in mathematics: while Baldwin offers positive considerations
as to why inductive arguments are explanatory, Lehet dwells on
inductive definitions which - she argues - might be cases
of explanations in mathematics.



The distinction between explanatory and non-explanatory proofs has
also been applied to other types of proofs, for instance proofs that
explain 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 that
explain by revealing the reasons, or the grounds, why a theorem is
true. As stressed by Lange (2021c), in this context the word
"ground" should not be understood as connected to the
recent literature on metaphysical grounding (e.g., see Correia and
Schnieder (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 in
philosophers and mathematicians such as Bolzano (see Kitcher 1975,
Mancosu 1999 and Sebestik 1992) or Cournot (see Mancosu 1999), and
recent 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 scientific
literature it is widely accepted that causal explanations track a
causal relation in the world and explain by revealing the causes of
why a certain fact holds, it seems reasonable to accept that (at least
certain) mathematical explanatory proofs track a grounding relation in
the mathematical realm and thus explain by mentioning the grounds or
reasons why a theorem is true.


### 2.1 Some historical remarks



Since contributions in analytic philosophy to the study of
mathematical explanations date back only to Steiner 1978a, one might
suspect that the topic was a byproduct of the Quinean conception of
scientific theories (see Resnik & Kushner, 1987, 154). Once
mathematics and natural science were placed on the same footing, it
became possible to apply a unified methodology to both areas. Thus, it
made sense to look for explanations in mathematics just as in natural
science. However, this historical reconstruction would be mistaken.
Mathematical explanations of mathematical facts have been part of
philosophical reflection since Aristotle. We have already seen in
section 1.1 the distinction Aristotle drew between demonstrations
"of the fact" and demonstrations "of the reasoned
fact". Both are logically rigorous but only the latter provide
explanations for their results. Aristotle had also claimed that
demonstrations "of the reasoned fact" occur in
mathematics. Only these demonstrations can be called
"explanatory" demonstrations, and some of these
demonstrations will be mathematical proofs.



Aristotle's position on explanatory proofs in mathematics was
already challenged in ancient times. Proclus, in his "Commentary
on the first book of Euclid's Elements", informs us on
this point. He reports: "Many persons have thought that geometry
does not investigate the cause, that is, does not ask the question
'Why?'" (Proclus 1970, 158-159; for more on
Proclus on mathematical explanation see Harari 2008). Proclus himself
singles 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 of
the internal angles of a triangle is equal to two right angles. If the
demonstration were given by a scientific syllogism in the Aristotelean
sense, the middle of the syllogism would have to provide the
'cause' of the fact. But Proclus argues that
Euclid'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 the
> properties of a demonstration in being able to establish what is
> sought by means of definitions as middle terms, and this is the
> perfect form of demonstration; but sometimes it attempts to prove by
> means of signs. This point should not be overlooked. Although
> geometrical propositions always derive their necessity from the matter
> under investigation, they do not always reach their results through
> demonstrative methods. For example, when [from] the fact that the
> exterior angle of a triangle is equal to the two opposite interior
> angles it is shown that the sum of the interior angles of a triangle
> is equal to two right angles, how can this be called a demonstration
> based 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 are
> equal to two right angles; for it is a triangle even if its side is
> not extended. (Proclus 1970, 161-2)
> 



In addition, Proclus also held that proofs by contradiction were not
demonstrations "of the reasoned fact". The rediscovery of
Proclus in the Renaissance was to spark a far-reaching debate on the
causality of mathematical demonstrations referred to above as the
Quaestio de Certitudine Mathematicarum (see section 1.1 for more on
this debate). The first shot was fired by Alessandro Piccolomini in
1547. Piccolomini's aim was to disarm a traditional claim to the
effect that mathematics derives its certainty on account of its use of
"scientific demonstrations" in the Aristotelean sense
(such proofs were known as "potissimae" in the
Renaissance). Since "potissimae" demonstrations had to be
causal, Piccolomini attacked the argument by arguing that mathematical
demonstrations are not causal. This led to one of the most interesting
epistemological debates of the Renaissance and the seventeenth
century. Those denying the "causality" of mathematical
demonstrations (Piccolomini, Pereyra, Gassendi etc.) argued by
providing specific examples of demonstrations from mathematical
practice (usually from Euclid's Elements) which, they claimed,
could not be reconstructed as causal reasonings in the Aristotelian
sense. By contrast, those hoping to restore "causality" to
mathematics aimed at showing that the alleged counterexamples could
easily be accommodated within the realm of "causal"
demonstrations (Clavius, Barrow, etc.). Interestingly, both positions
in the debate assumed that mathematical proofs could be syllogized
(Mancosu & Mugnai 2023). The historical developments have been
presented 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-explanatory
demonstrations - had a long and successful history that has
influenced both mathematical and philosophical developments well
beyond the seventeenth century. For instance, Mancosu 1999 shows that
Bolzano and Cournot, two major philosophers of mathematics in the
nineteenth century, construe the central problem of philosophy of
mathematics as that of accounting for the distinction between
explanatory and non-explanatory demonstrations. In the case of Bolzano
this takes the form of a theory of Grund (ground) and Folge
(consequence). Kitcher 1975 was the first to read Bolzano as
propounding a theory of mathematical explanations (see Betti 2010 and
Roski 2017 for recent contributions). In the case of Cournot this is
spelled out in terms of the opposition between "ordre
logique" and "ordre rationelle" (see Cournot 1851).
In Bolzano's case, the aim of providing a reconstruction of
parts of analysis and geometry, so that the exposition would use only
"explanatory" proofs, also led to major mathematical
results, such as his purely analytic proof of the intermediate value
theorem.



In conclusion to this section, we should also point out that there is
another tradition of thinking of explanation in mathematics that
includes Mill, Lakatos, Russell and Godel. These authors are
motivated by a conception of mathematics (and/or its foundations) as
hypothetico-deductive in nature and this leads them to construe
mathematical activity in analogy with how explanatory hypotheses occur
in science (see Mancosu 2001 for more details). Related to inductivism
are Cellucci 2008, 2017, which emphasize the connection between
mathematical explanation and discovery.


### 2.2 Models of mathematical explanation



From the above, it should be obvious that both philosophers and
mathematicians have appealed to the notion of explanation within
mathematics and that amongst the different contexts in which such
explanatory activity appears, proofs play a special role. But what
distinguishes a proof that explains from one that doesn't? How
should one proceed in providing an account of explanatory proofs? It
is here that two possibilities emerge. On the one hand, one can follow
a top-down approach where one starts with a general model of
explanatory proof and then tries to see how well it accounts for the
practice. On the other hand, one can embrace a bottom-up approach
where one begins by avoiding, as much as possible, any commitment to a
particular theoretical framework. Only afterwards, one attempts to
provide a taxonomy of recurrent types of mathematical explanatory
proofs and tries to see whether these patterns are heterogeneous or
can 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). Probably
the main characteristic of their investigations is the extremely rich
and varied set of examples considered; precisely in virtue of this
variety, Lange argues that there is no general pattern characterizing
explanatory proofs; at most one can claim that there are different
classes of explanatory proofs. Lange proposes several salient features
of mathematical theorems, which (in different contexts) are
responsible (in those contexts) for the distinction between
explanatory and non-explanatory proofs. Among them, he discusses
extensively symmetry and simplicity. As for simplicity, it amounts to
the requirement for a proof of a simple result "exploits some
similar, simple feature of the setup" (see Lange 2017, 257). As
for symmetry, it is a property that arises when dealing with
mathematical results that display some striking symmetry: for the
proof to count as explanatory, it needs to show how such symmetry
follows from a similar symmetry in the set-up of the problem. Lange
defends these properties by using cases studies drawn from
probability, real analysis, number theory, complex numbers and
geometry, among other areas of mathematics. One of the most
representative (see Lange 2017, 239-242) is the proof of
d'Alembert's theorem to the effect that in a polynomial
equation of *n*-th degree in the variable *x* and having
only real coefficients, the nonreal roots always come in pairs (any
non-real root and its complex conjugate will both satisfy the
equation). What explains this symmetry? A non-explanatory proof can be
given by algebraic manipulations but this does not reveal the reason
for the result which, according to Lange, is the fact that the axioms
of complex arithmetic are invariant under substitution of *i*
for -*i*. Bueno and Vivanco (2019) points out that it is
unclear why what Lange isolates as the symmetric feature of the proof
(which makes it explanatory) is a symmetry at all. They suggest that
this proof turns on an appropriate feature of the relevant
structure.



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 article
considers an aspect of mathematical practice and tries to address it
on its own terms.



Top-down approaches take their start from a general theory of
mathematical explanation and then explore how well the practice fits
the model. A typical example of a top-down approach in mathematical
explanation is Kitcher's unificationist theory, to be discussed
below. But one can also apply this description to overarching views on
the nature of explanation. While there are several examples one could
mention, here we present an influential proposal that finds its origin
in Kim (1994). In order to classify the different accounts of
scientific explanation Kim uses the contraposition between
'explanatory internalism' and 'explanatory
externalism.' Whereas for 'explanatory internalism'
explanations are activities internal to an epistemic corpus (a theory
or set of beliefs), an 'explanatory externalist' will look
for some systematic pattern of objective dependence relations which
explanations track or can be identified with. We divide the present
section in two subsections that follow this division: one will be
dedicated to the presentation of the externalist, or ontic, accounts
of explanatory mathematical proofs, while the other to the
internalist, or epistemic, ones. Among the externalist accounts, we
will discuss Steiner's theory, several counterfactual theories
of mathematical explanation, and some other proposals. Among
internalist accounts we will discuss Kitcher's theory, together
with two novel ones proposed by Frans (2021) and Inglis &
Mejia-Ramos (2019).


#### 2.2.1 Externalist models of mathematical explanation



Amongst the several existing contemporary externalist models of
explanatory proofs, the oldest and probably most well-known is
Steiner's account. Steiner aims at finding criteria that could
characterize explanatory proofs. After having discussed several
possible criteria, such as abstractness, generality, and
visualizability, Steiner rejects them all in favor of the idea that
"to explain the behavior of an entity, one deduces the behavior
from the essence or nature of the entity" (Steiner 1978a, 143).
Although such an idea could seem prima facie intuitive and
interesting, it turns out to be quite problematic. First, it leads to
the notorious difficulties linked to the concepts of essence or
essential property; moreover, such concepts risk having little
traction in a mathematical context since all mathematical truths are
regarded as necessary. Hence, instead of talking of
"essence," Steiner speaks of "characterizing
properties" by which he means "a property unique to a
given entity or structure within a family or domain of such entities
or structures," where he takes the notion of a family as
undefined. In other words, for Steiner the difference between
explanatory and non-explanatory proofs lies in the characterizing
properties, which are found only in the former but not by the latter.
However, this is not all: an explanatory proof needs to be
generalizable as well. Varying the relevant feature (and hence a
certain characterizing property) in such a proof needs to give rise to
an array of corresponding theorems, which are proved - and
explained - by an array of "deformations" of the
original proof.



There have been two extensive critical discussions of Steiner's
account. The first was provided by Resnik and Kushner (1987) who
argued that the distinction between explanatory and non-explanatory
proofs is context-dependent. The second, which also offers a
counterexample to the theory based on a case of explanation from real
analysis recognized as such in mathematical practice, has been
developed by Hafner and Mancosu (2005). There have also been attempts
to improve Steiner's model. The work developed by Weber and
Verhoeven (2002) can for example be seen as an attempt to improve
Steiner's notion of deformation. Indeed, while Steiner suggests
that explanation concerns an array of related proofs and theorems,
although maintaining that each proof is an explanation of the
individual theorem, Weber and Verhoeven (2002) start with what makes
pairs of proofs - P1 and P2 - count as explanatory. In
particular, they focus on explaining why, while a certain class of
object \(x\) has a property \(Q\) (proof P1), another class of objects
\(y\) enjoys property Q' (proof P2). Here P1 and P2 use the same
axioms and the same logical rules, but while P1 uses a characterizing
property of \(x\), but not of \(y\), P2 uses a characterizing property
of \(y\), but not of \(x\). A final attempt to enrich Steiner's
account is proposed by Salverda (2017) who tries to adapt this
approach to an internalist perspective on explanations of the sort
discussed in section 2.2.2.



In the field of causal explanations, a dominant perspective has been
formulated in counterfactual terms. Although there has long been a
resistance in the use of counterfactuals to account for explanations
in mathematics (see Lange 2017, 88, 2022), many authors adopt this
approach, perhaps due to the attractiveness of a unified theory of
explanation that promises to hold in both causal and non-causal
contexts. 



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



CF1:
 if \(F\) had not been the case, \(G\) would not have been the
case,


CF2:
 if \(G\) had not been the case, \(F\) would not have been the
case.


The first counterfactual, CF1, needs to be true: it directly accounts
for the explanatory power of the relation between \(F\) and \(G\). By
contrast, the second counterfactual, CF2, needs to be false since it
serves to ensure that the relation between \(F\) and \(G\) is
asymmetric, i.e., it shows that it is not the case that \(G\) explains
\(F\).



Once the counterfactuals are specified, a theory for counterfactuals
needs to make clear what truth-conditional account of counterfactuals
is adopted. Here (at least) two options naturally emerge. On the one
hand, one can evaluate a counterfactual using possible worlds
semantics; for example, Lewis's closeness-based semantics, which
trivializes for mathematical counterfactuals (see Lewis 1973, and
Stalnaker 1968), has recently been extended to avoid these
trivialities (see for example Nolan 2001 and Priest 2002). This
extension, which considers both possible and impossible words, can be
used to evaluate the truth value of counterfactuals CF1 and CF2. On
the other hand, one can also try to adapt the standard tools of
structural equation modelling (see Pearl 2000) to evaluate the truth
value of CF1 and CF2. In this case, one interprets mathematical facts
as variables which can either take the value 1 or the value 0,
according to whether the propositions they represent are either true
or false. While the variable that denotes \(F\) is an
*exogenous* variable - its values are determined by
factors outside the model - the variable that denotes \(G\) is
*endogenous* - its values are determined by the value of
other variable(s), in our case \(F\). In order to test whether the
counterfactual CF1 is true, one needs to *intervene* on the
value of the variable assigned to \(F\) and check whether this change
affects the value of the variable assigned to \(G\). As for the truth
value of CF2, its falsity, and thus the asymmetry required by the
explanatory relation, is built into the nature of endogenous
variable.



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



Gijsbers (2017) develops a counterfactual account of explanatory
proofs which relies on the structural equation framework, but where
the notion of "intervention" a la Woodward (2003),
cannot be employed in the mathematical context. As Woodward
emphasizes, an intervention is a causal change to the value of a
variable. Instead Gijsbers introduces the idea of a
"quasi-interventionist" theory of mathematical
explanation: in this theory, quasi-interventions reveal asymmetries
which are inherent not in the mathematical proofs, but in the
mathematical practice (see Gijsbers, 2017, 59). In other words,
asymmetries are no longer accounted for in an objective, but rather in
a more subjective way that is tied to the features of the practice in
question.



In a sense Gijsbers' model is complementary to the one proposed
by Frans and Weber (2014). Indeed, while Gijsbers accounts for the
explanatory power of proofs in counterfactual terms, without using the
notion of intervention, Frans and Weber account for the explanatory
power of proofs with a mechanistic model of explanation that directly
generalizes on Woodward's notion of an intervention.



The use of counterfactuals to model mathematical explanations has been
criticized by Kasirzadeh (2021) and Lange (2022). While Kasirzadeh
questions whether the explanans of an explanatory proof can be
meaningfully varied in a mathematical context, as the counterfactual
accounts would require, Lange argues instead that the counterfactual
account is rather threatened by the existence of too many
non-trivially true countermathematicals. Both Kasirzadeh and Lange
emphasize that the capacity to answer
what-if-things-had-been-different questions does not correlate with
explanatory power in mathematics. Finally, note that also in Jansson
(2018) one might find criticisms on the use of the structural
equations' framework to model dependence relations other than
causation, and thus arguably also dependence relations in a
mathematical context.



Not all externalist models for mathematical explanations are
modifications of Steiner's theory or conveyed in counterfactual
terms. Pincock (2015b) for example proposes to classify a proof as
explanatory when it invokes more abstract kind of entities than the
topic of the theorem it proves. Wilhelm (2021) and Poggiolesi
(forthcoming) contain different proposals for the analysis of
explanatory proofs that are similar in perspective to Pincock's
approach. In their cases the determination of the explanatory power of
different proofs requires a formalization of the proofs in logical
systems. While for Wilhelm the explanatory power of a proof comes from
the balance between the simplicity and the depth of the formalized
proof, Poggiolesi distinguishes an explanatory proof from a
non-explanatory one in that only in the (formalized version of the)
former one can witness an increase of conceptual complexity from the
assumptions 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 coherent
account of scientific, and thus presumably also mathematical,
explanation: he argued that any such account needed to show how
explanation generates understanding. "I don't see how the
philosopher of science can afford to ignore such concepts as
'understanding' and 'intelligibility' when
giving a theory of the explanation relation" (Friedman 1974, 8).
While externalist, or ontic, accounts do not directly concern
themselves with Friedman's challenge, they do not deny the link
between explanation and understanding. However, they simply do not
pose understanding as a defining characteristic of explanation. By
contrast, internalist, or epistemic, accounts are those which directly
address this challenge.



In the philosophy of science, one of the main conceptions of
scientific understanding is the unificationist model which argues that
explanations provide understanding by unifying different phenomena.
Although the idea is undoubtedly intuitively appealing, the key
question is whether the notion of unification can be made more precise
so that we can distinguish between what an explanation is and what is
not. Friedman (1974) is an early attempt to do this, although his
formulation was quickly shown to suffer from several technical
problems (see Kitcher 1976). Kitcher is, on the other hand, the main
supporter of the unificationist approach. His proposal consists in
looking at unification as the reduction of the number of argument
patterns used in providing explanations while being as comprehensive
as possible in the number of phenomena explained:



> 
> Understanding the phenomena is not simply a matter of reducing the
> "fundamental incomprehensibilities" but of seeing
> connections, common patterns, in what initially appeared to be
> different situations. Here the switch in conception from premise
> conclusion pairs to derivations proves vital. Science advances our
> understanding of nature by showing us how to derive descriptions of
> many phenomena, using the same patterns of derivation again and again,
> and, in demonstrating this, it teaches us how to reduce the number of
> types of facts that we have to accept as ultimate (or brute). So the
> criterion of unification I shall try to articulate will be based on
> the idea that E(K) is a set of derivations that makes the best
> tradeoff between minimizing the number of patterns of derivation
> employed and maximizing the number of conclusions generated. (Kitcher
> 1989, 432)
> 



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



Two remarks are in order when it comes to Kitcher's proposal.
First, his account of theoretical unification is mainly thought of for
the general question of scientific explanation, although he sees as
one of the virtues of his viewpoint to be extendable to mathematics as
well. Secondly, Kitcher's model is not meant to address the
local question of what distinguishes an explanatory proof from one
that does not explain (as all other accounts do); it rather provides a
novel perspective on the global question of how to systemize a whole
body of knowledge that has explanatory value. The application of
Kitcher's model to explanatory proofs has been explored in two
opposite directions. On the one hand, Hafner and Mancosu (2008) tested
Kitcher's model with three different methods to prove theorems
about real closed fields (see Brumfiel 1979); the authors showed that
the model makes predictions about the explanatory power of these
methods which contradicts judgments coming from the mathematical
practice (See also Pincock 2015b). On the other hand, Frans (2021) not
only reassesses the value of unificatory understanding, which it is a
type of explanatory understanding, for mathematics; additionally, he
shows through a plethora of different examples, ranging from
Pythagoras' theorem to the theorem that states that sum of the
first n integers equals n(n+1)/2, that proofs can contribute to
unificatory understanding. 



A novel internalist account has recently been proposed by Inglis and
Mejia-Ramos (2019), who apply Wilkenfeld's functional
model of understanding (see Wilkenfeld (2014)) to the mathematical
case. Wilkenfield's approach consists in reversing
Friedman's perspective: while Friedman demanded that
philosophers clarify how explanations, suitably defined, generate
understanding, Wilkenfield defines explanations as those things that
generate understanding. By doing so, Wilkenfield moves the burden of
clarification from the notion of explanation to that of understanding
and how it is generated: this move **-** he argues
- has recently become tenable as philosophical accounts of
understanding have become more and more sophisticated. 



In Inglis & Mejia-Ramos (2019) the conception of
understanding adopted is that of Kelp (2016), along with a *modal
model* of the generation of understanding (see Atkinson and
Shiffrin 1968). With these two elements at hand, Inglis &
Mejia-Ramos identify three properties that any mathematical
explanatory proof is likely to have: (i) an explanatory proof would
direct the reader's attention to its conceptually important
section; (ii) it would reorganize the new and existing information
into coherent new schema; (iii) it would reduce the chances of working
memory capacity to be exceeded.



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


## 3. Some connections to other debates



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



The issues that have shaped the debates about mathematical explanation
reviewed in sections 1 and 2 also arise in ongoing debates in the
philosophy of mind and moral theory. For the philosophy of mind, one
puzzle is how appealing to mental properties can explain human actions
even though a human is a physical entity. If non-mental, physical
properties are apt to explain any physical event or pattern of
physical events, then it seems that mental properties are dispensable
or "epiphenomenal". For moral theory, a series of
questions arise about how moral properties relate to the presumably
non-moral features of the physical world. In terms of explanation, it
seems like there is no explanatory work for moral properties to do, at
least with respect to physical events. However, our ordinary practices
frequently appeal to these properties in putative explanations. So, as
with the philosophy of mind, it seems that we must either revise our
explanatory practices or else find a place for these properties in a
more comprehensive conception of reality.



Kim's exclusion argument is a prominent driver of these debates
in the philosophy of mind (Kim 2005). Kim argues that the existence of
mental properties requires that these properties provide some genuine
contribution to the explanation of physical events. However, Kim
maintains that mental properties are excluded from this contribution
by the causal closure of the physical, i.e. every physical event has a
purely physical explanation. One response to Kim is that the right
conception of causal explanation makes space for mental properties to
explain (Shapiro & Sober 2007, Woodward 2021b). The
"explanatory autonomy" of the mental can thus be obtained
in a way that parallels similar generalizations of causal explanation
to allow for genuine mathematical explanations of physical phenomena
(section 1.2). Pluralists about mathematical explanation can develop a
different response to exclusion arguments: if explanations come in
different kinds, then an explanation of one kind does not stand in the
way of an explanation of another kind (Batterman 2021). Baker (2022)
has pursued a different kind of response that compares Dennett's
intentional stance with a "mathematical stance" that
enables mathematical explanations of physical phenomena.



Harman and Street have advanced explanatory challenges to moral
properties that can be fruitfully compared to criticisms of the
explanatory indispensability argument for mathematical platonism
(Harman 1977, Street 2006). While Harman focuses on the explanation of
moral observations (e.g. that some action is wrong), Street emphasizes
a broader concern with explaining other phenomena such as the
prevalence of some moral judgments (e.g. that murder is wrong). For
both, the challenge is that the best explanation does not involve
moral properties. That is, moral properties are explanatorily
dispensable for the targets in question. As Sinclair and Leibowitz
emphasize, this argument, and the responses to it, parallel debates
about the explanatory dispensability of mathematical objects (Sinclair
& Leibowitz 2016). One innovation in the debate about moral
properties is Enoch's argument that it is sufficient for moral
properties to be indispensable to practical deliberation. If this
non-Quinean condition for ontological commitment is granted, then it
may be feasible to identify new forms of explanatory indispensability
for mathematical objects. For some investigations into how such
arguments may or may not extend to mathematics, see Leng (2016), Baker
(2016), Enoch (2016) and Clark-Doane (2020).