User:Shapeyness/sandbox/Explanatory indispensability argument

The explanatory indispensability argument is an altered form of the Quine–Putnam indispensability argument in the philosophy of mathematics. It claims that we should believe in mathematical objects such as numbers because they are indispensable to scientific explanations of empirical phenomena.

Philosophy of mathematics
Platonism vs nominalism

Epistemic access

An indispensability argument is an argument in which the conclusion is supported by the claim that its truth is indispensable or necessary for a certain purpose.

Origin of the argument
The explanatory indispensability argument is an altered form of the Quine–Putnam indispensability argument first raised by W. V. Quine and Hilary Putnam in the 1960s and 1970s. The Quine–Putnam indispensability argument supports the conclusion that mathematical objects exist with the idea that mathematics is indispensable to our best scientific theories. It relies on the view, called confirmational holism, that scientific theories are confirmed as wholes, and that the confirmations of science extend to the mathematics it makes use of.

The reliance of the Quine–Putnam argument on confirmational holism is controversial, and it has faced influential challenges from Penelope Maddy and Elliott Sober. The argument has also been criticized for failing to specify the way in which mathematics is indispensable to science; according to Joseph Melia, we only need to believe in mathematics if it is indispensable in the right way. Specifically, it needs to be indispensable to scientific explanations for it to be as strongly justified as theoretical entities such as electrons. This claim by Melia arose through a debate with Mark Colyvan in the early 2000s over the argument, with Colyvan claiming that mathematics enhances the explanatory power of science. Inspired by this debate, Alan Baker developed an explicitly explanatory form of the indispensability argument, which he termed the enhanced indispensability argument. He was also motivated by the objections against confirmational holism; his formulation aimed to replace confirmational holism with an inference to the best explanation.

Among Baker's influences was Hartry Field, who has been credited with being the first person to draw a connection between indispensability arguments and explanation. Baker cited Field as originating an explanatory form of the argument, although Sorin Bangu states that Field merely alluded to such an argument without fully developing it, and Russell Marcus argues he was discussing explanation within the context of the original Quine–Putnam indispensability argument rather than suggesting a new explanatory indispensability argument. According to Marcus, Colyvan's discussion of explanatory power was also initially on its role within the Quine–Putnam indispensability argument, with the explanatory indispensability argument originating with Baker. Others credit Colyvan and Baker equally with the origination of the argument. [cite this, also maybe specify who says this] "Colyvan and Baker have recently initiated a new kind of indispensability argument that turns on the explanatory contributions from mathematics ... Since the publication of his book, Colyvan has emphasized the special way the existence of mathematical entities contributes to superior scientific explanations of physical phenomena. Along with similar work by Baker, this has led to the articulation of a new explanatory indispensability argument for platonism.1 Although Colyvan had mentioned this prominently in his book (Colyvan 2001, p. 7), the connection between explanatory power and the existence of mathematical entities was not put at the center of his exposition." - Pincock 2011, p.203

"Colyvan and Baker can be credited with advancing these new indispensability arguments in their strongest form. In his 2001 book The Indispensability of Mathematics, Colyvan presented a general indispensability argument that he traced back to Quine and Putnam. This argument was then refined by Baker to focus more directly on explanatory considerations" - Pincock 2023, p.61

Lyon 2012 p. 572, maybe useful for characterising things too

(The Stanford Encyclopedia of Philosophy credits the argument to Baker while the Internet Encyclopedia of Philosophy credits it to both Colyvan and Baker.)

'''[Busch & Sereni argue that it is fundamentally different to the QPIA argument and cannot be considered Quinean - maybe include? as an explanatory footnote after part about IBE?]'''

Overview
A standard formulation of the explanatory indispensability argument is given as follows:


 * We ought rationally to believe in the existence of any entity which plays an indispensable explanatory role in our best scientific theories.
 * Mathematical objects play an indispensable explanatory role in science.
 * Therefore, we ought rationally to believe in the existence of mathematical objects.

Individual examples in support of this argument come in the form of an inference to the best explanation, where the fact to be explained (explanandum) is the conclusion, and the things doing the explaining (explanans) are premises.[Bangu 2012 p.152]

The argument is premised on the idea that inference to the best explanation, which is often used to justify theoretical entities such as electrons, can provide a similar kind of support for mathematical objects. It also requires that there are genuinely mathematical explanations in science. For explanations to be genuinely mathematical, it is not enough that they are expressed with the help of mathematics. Instead, the mathematics must be playing an essential part in the explanatory work. [Vineberg p. 237, check this isn't a better source to use here] Given the argument's reliance on the existence of such explanations, much of the discussion on it has focused on evaluating specific case studies to assess if they are genuinely mathematical explanations or not.

Notes to self

Explain / discuss IBE (should distinction between epistemic / ontic explanation go here? Or in objections section?)

Maybe talk about target of argument (scientific realists) [Paseau & Baker, Baron 2020 p.19] and motivation from epistemological problem

Periodical cicadas
The most influential case study is the example of periodical cicadas provided by Baker. Periodical cicadas are a type of insect that usually have life cycles of 13 or 17 years. It is hypothesized that this is an evolutionary advantage because 13 and 17 are prime numbers. Because prime numbers have no non-trivial factors, this means it is less likely that periodic predators and other competing species of cicada can synchronize with periodic cicadas' life cycles. Baker argues that this is an explanation in which mathematics, specifically number theory, plays a key role in explaining an empirical phenomenon.

A number of non-mathematical explanations have been proposed for the length of periodical cicadas' life cycles. For example, a prominent alternative explanation claims that prime-numbered life cycles could have emerged from non-prime life cycles due to developmental delays. This hypothesis is supported by the fact that there are many other species of cicada that have non-prime life cycles, and that developmental changes with 4-year periods have often been observed in periodical cicadas. Some philosophers have also argued that the concept of primeness in the case study by Baker can be replaced with a non-numeric concept of "intersection-minimizing periods", although Baker has argued that this would reduce the generality and depth of the explanation. Others, such as Chris Daly and Simon Langford, argue that using years as a unit of measurement rather than months or seasons is arbitrary; Baker and Colyvan argue that years are an appropriate unit of time for biological development and are the unit used by biologists.

The case study has also been criticized for assuming that periodical cicadas have had predators with periodic life cycles in their evolutionary history. Baker has responded to this worry by arguing that it would be impossible to provide direct evidence that periodical cicadas have had periodic predators because "periodicity is not something that can be gleaned from the fossil record". However, he has attempted to make the claim more plausible by arguing that ecological constraints could have restricted the range of the cicadas' possible life cycles, lessening the requirements on periodic predators for the case study to remain mathematically sound. This problem can also be avoided by focusing on other ways in which the prime life cycles could be explanatorily relevant, such as avoidance of competing species of cicada or periodic migration of predators.

Bee honeycomb
Another prominent case study suggested by Aidan Lyon and Colyvan is the hexagonal structure of bee honeycomb. Lyon and Colyvan contend that the hexagonal structure of bee honeycomb can be explained by the mathematical proof of the honeycomb conjecture, which states that hexagons are the most efficient regular tiling of the plane. The explanation goes that there is an evolutionary pressure for honeybees to conserve wax in the construction of there combs, so the efficiency of the hexagonal grid explains why it is selected for.

The explanation based on the honeycomb conjecture is potentially incomplete because the proof is a solution to a tiling problem in two dimensions, and disregards the 3D structure of comb cells. Furthermore, many mathematicians do not see the proof of the honeycomb conjecture as an explanatory proof as it employs concepts outside of geometry to establish a geometrical result, although Baker argues that the proof need not be explanatory for the theorem to feature in genuine explanations in science. It is also controversial amongst philosophers whether the subject matter of geometry is purely mathematical, or whether it concerns physical space and structures, leading them to question if the explanation is truly mathematical.

There are also non-mathematical explanations for the honeycomb case study. Darwin believed that the hexagonal shape of bee combs was the result of tightly packed spherical cells being pushed together and pressed into hexagons, with bees fixing breakages with flat surfaces of wax further contributing to a hexagonal shape. More modern presentations hold that the shape of honeycomb is due to the flow of molten wax during the construction process.

Others
Another key example is the Seven Bridges of Königsberg, which concerns the impossibility of crossing each of the historical seven bridges in the Prussian city of Königsberg a single time in a continuous walk around the city. The explanation was found by Leonhard Euler in 1735 when he considered whether such a journey was possible. Euler's solution involved abstracting away from the concrete details of the problem to a mathematical representation in the form of a graph, with nodes representing landmasses and lines representing bridges. He reasoned that for each landmass, unless it is a starting or ending point, there must be a path to both enter and exit it. Therefore, there must be at most two nodes in the graph with an uneven number of lines connected to them for such a journey to be possible. But this is not the case for the graph representing the seven bridges in Königsberg, so it is mathematically impossible to cross all seven without crossing over one of the bridges multiple times.

The existence at any particular time of antipodal points on the Earth's surface with equal temperature and pressure has been cited as another example. According to Colyvan, this is explained by the Borsuk–Ulam theorem, which entails that for any physical property that varies continuously across the surface of a sphere, there are antipodal points on that sphere with equal values of that property. In response to this example, Baker has argued that it is a prediction rather than an explanation because antipodal points with equal pressure and temperature have not already been measured. Mary Leng also questions whether it is appropriate to model temperature or pressure as continuous functions across individual points on the Earth's surface.

A key class of mathematical explanations is solutions to optimization problems, which includes the cicada and bee honeycomb case studies. In these cases, a certain feature is explained by showing that it is mathematically optimal. Such explanations are important in evolutionary biology, as mathematical demonstrations of optimality may help to explain why a given trait has been selected for, but also appear in other areas of science such as physics, engineering and economics. Some examples from evolutionary biology are sunflowers' seeds being arranged in a spiral pattern because it produces the densest packing of seeds, and marine predators engaging in Lévy walks because they minimize the average energy consumption required to find prey.

A number of case studies draw from dynamical systems. Marc Lange, for example, argues that the fact that double pendulums always have four or more equilibrium configurations can be explained by the configuration space of the system forming the surface of a torus, which must have at least four stationary points. Lyon and Colyvan point to the use of phase spaces and the Poincaré map to explain the behaviour of a Hénon–Heiles system, such as the stability of a star's orbit through a galaxy. Other examples proposed by Colyvan include geometrical explanations for Lorentz contraction and gravitational lensing. Baker and Melia have objected to the geometrical aspects of these explanations, which could be interpreted physically instead of mathematically.

Some examples are drawn from outside science. For example, widely discussed cases include the explanation for why 23 strawberries cannot be divided equally amongst three people, why it is impossible to square the circle, and why it is impossible to untie a trefoil knot. However, it is unclear to what extent each of these cases are mathematical explanations of physical facts rather than either purely physical or purely mathematical explanations.

Mathematical explanation
The philosophical study of mathematical explanations in science dates back to Aristotle, but within analytic philosophy, the first philosopher to deal with the issue was Mark Steiner, beginning in the 1970s. [include quick description of Steiner's view?] Since then, there has been increasing attention on the question, and the explanatory indispensability argument has coincided with a greater emphasis on issues within the philosophy of mathematical practice more generally. As a result, a number of competing theories of mathematical explanation have arisen, some favouring the argument and others contradicting it. [cite to SEP maybe]

Notes to self

Maybe start this by talking about how both sides of the debate came to agree a theory of explanation was needed to assess cases instead

Theories of mathematical explanation

Maybe useful sources: Wójtowicz 2020, Saatsi 2016, Baron 2016

Talk about general features of case studies here? e.g. modal strength

Objections
The main response to the explanatory indispensability argument, adopted by philosophers such as Melia, Daly, Langford, and Saatsi, is to deny there are genuinely mathematical explanations of empirical phenomena, instead framing the role of mathematics as representational or indexical. According to this response, if mathematics features in scientific explanations, its role is just to help pick out physical facts instead of contributing to the explanatory power of the explanation. Saatsi, and others including Jonathan Tallant and Davide Rizza, have rephrased case studies such as the periodic cicada example to remove reference to mathematical entities in an attempt to provide the true non-mathematical versions of these explanations. Defenders of the explanatory indispensability argument typically argue that the non-mathematical explanations provided are less general and modally weaker than mathematical explanations. They also argue that such explanations contradict scientific practice because scientists often accept the mathematical explanations as genuine scientific explanations.

[  Other things that could be included: these paraphrases may not be nominalistically acceptable (Paseau & Baker), defenders of EIA need to show that there is a connection between modal strength and ontological commitment (Bueno & French), also that there is a connection between generality and the quality of the explanation (Falguera & Martínez-Vidal Ch. 3, p.42), weaseling (Bueno & French, p.165)   ]

Others, particularly mathematical fictionalists like Mary Leng and Stephen Yablo, have accepted that mathematics plays a genuinely explanatory role in science but argue it can play this role even if mathematical objects do not exist. They point to the use of idealizations like point masses that are used in scientific explanations but are not viewed as literally real. Leng argues that the explanatory power of mathematics can be explained by structural similarities between mathematical theory (viewed fictionally) and features of the real world. Yablo appeals to the expressive power of figurative language, claiming it shows that literally untrue statements can often convey more than literally true statements. Colyvan has challenged these types of responses by arguing that fictional or metaphorical language cannot play a role in genuine explanations: "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."

An objection advanced by Bangu states that the explanatory indispensability argument begs the question because it is circular. Bangu argues that examples like the periodic cicada case aim to explain statements that already contain mathematical content, namely the primeness of the cicadas' life cycles. But an inference to the best explanation assumes that the statement being explained is true, so inclusion of mathematical concepts such as primeness assume the truth of the mathematics in question. Baker has responded to this objection by arguing that the statements being explained in such case studies can be reformulated to remove reference to mathematical entities, leaving mathematics indispensable only to the explanation itself and not the thing being explained.

Notes on sources

 * Baron 2016 (a) - implications of access problem for explanatory indispensability argument - still need to read Baron's other 2016 paper
 * Drekalovic 2019 - probably not much more useful in this - still need to read other papers by Drekalovic
 * Lange 2013 - mathematical explanations are non-causal - modal strength of mathematical explanations - exploration of these ideas and how they tie into the distinction between distinctively mathematical explanations vs ordinary scientific explanations - these ideas are further explored in Lange 2016 - still need to read Lange 2021
 * Lyon - discussion of Steiner's view of mathematical explanation & Baker's criticisms + program explanations (§3) - objections from Colyvan and Daly & Langford are covered in §4
 * Mancosu - Steiner's views on mathematical explanation + Baker's criticism (§5.1) - intra-mathematical explanation, e.g. explanatory vs non-explanatory proofs (§5.3)
 * Mancosu et al. - Explanatory Indispensability Arguments section: Final two paragraphs talk about differing theories of mathematical explanation either supporting or contradicting EIA - probably need to check through the rest of the article to see if there is anything else useful
 * Marcus Ch. 7 - §Two Concepts of Explanation & §Epistemic Explanation and the Explanatory Indispensability Argument cover different theories of explanation and epistemic vs ontic explanation
 * Molinini (also discussed in Pataut paper in Synthese special issue) - discussion of Baker's idea that use of stronger mathematical apparatus can reduce nominalistic commitments, meaning mathematics contributes to nominalistic parsimony
 * Synthese special issue - Baron and de Bianchi discuss indispensable use of idealizations in science, Molinini and Sereni discuss the problem of equivalent mathematical explanations, which is also discussed by Hunt, Hunt and Busch & Morrison discuss IBE and whether it supports mathematical explanations equally to other scientific explanations (parity premise), Panza & Sereni and Galinon (still need to read, very technical/complex papers), Liggins and Plebani discuss nominalists' possible use of grounding to undermine EIA
 * Paseau & Baker - §5.4 The Audience for EIA - §6.3 restrictions to IBE, discusses epistemic explanation vs ontic explanation, eleatic arguments, "makes no difference" arguments - §7.4 discusses Hellman's modal structuralism, probably don't need to include here - everything else is pretty much covered already
 * Pincock 2011 - Is the explanation mathematical? Is it the best explanation? Replacement test + Comparison test can answer these questions - Melia & indexing (already covered in this article) - ways mathematics can contribute to the explanatory power (all from 10.1) - still need to read the rest of ch.10, also other Pincock sources
 * Saatsi 2016 - thick vs thin explanatory role (~= ontic vs epistemic) - different theories of explanation - ontic vs epistemic theories of explanation - nothing useful left in the other Saatsi sources
 * Tarziu (a) - discussion of how to distinguish genuine mathematical explanations from merely mathematicised explanations, i.e how to tell if mathematics is explanatorily relevant - presents the importance view of explanatory relevance + different degrees of explanatory relevance - looks at Saatsi, Daly & Langford's criticisms of the cicada case study - very brief mention of ontic vs epistemic explanation - probably not much useful left here for this article
 * Tarziu (b) - connection between explanation and understanding (probably not too relevant for EIA) - discussion of Steiner's view of mathematical explanation & Baker's criticisms - ....still need to finish reading §4 onwards
 * Tarziu (c) - §5. Lange’s and Pincock’s accounts of mathematical explanation - explains their accounts, argues their piecemeal/case-driven approach is question-begging in the context of EIA because the cases chosen can always be questioned by nominalists - this discussion is then slightly widened in §6 - §7 is about possible ways of reliably identifying genuine scientific explanations (intuition, understanding, expert testimony)
 * Vineberg - discussion of IBE and different principles that could underly it + what type of principle could be useful for EIA (both in terms of making the argument work and also in keeping it dialectically tenable) + possible analogy between causal and non-causal explanation (§Explanation and inference onwards) - discusses Jansson and Saatsi's arguments against Konigsberg bridges case study (§Mathematical explanation and difference makers)
 * Wakil & Justus - claim that biological optimality examples make a fallacious jump from mathematics explaining the adaptiveness of a trait (A-question) to mathematics explaining the existence of that trait (E-question); they further claim that EIA requires mathematics to explain E-questions to work - probably not space for this idea, but could maybe be added in a note
 * Wójtowicz - probably not much more useful here - interesting discussion (§4 onwards) on formalised, purely-syntactic mathematics required for QPIA (regimentation into formal language) vs informal, semantic proofs required by EIA that provide explanatory power, but unlikely to fit into this article - very brief mention of the question of whether EIA requires an explanatory proof, or whether a theorem itself can be explanatory (without needing an explanatory proof)