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6 D L, d R R 6 Mathematical Models in Science: A Debate about Ontology
Marco Fahmi
Introduction
Some philosophers of mathematics have claimed, after Quine, that the use of mathematical models is indispensable in scientific language and have concluded from this that we ought to believe in the existence of some mathematical entities. This argument is what is commonly referred to as the indispensability argument for mathematical realism.
A recent debate on the indispensability argument has centred on the role that mathematical models play in science. In an effort to refute Quine, some have argued that mathematical models are only used in science as instruments; that is, they are mere devices that describe physical phenomena. Regardless of whether they are indispensable or not, they are nonetheless not ontologically committing.
Supporters of the indispensability argument have opposed the instrumentalist view and claimed that mathematical models play another, more fundamental role, in science. In particular, they contend that more than describe, mathematical models also explain certain physical phenomena. Mathematical models, they conclude, cannot be treated instrumentally and ought to be endowed with ontological rights.
I examine in this paper the debate on the explanatory role of mathematical models in science. I critically assess certain assumptions that they explanatory account relies on and argue that they are untenable. I conclude that one ought to side with the instrumentalist view of mathematical models and thus, reject any claims that the use of mathematics in science warrants claims about a mathematical ontology. Finally, I suggest an anti-realist account of why mathematical models appear to play an explanatory role in science when they, in fact, do not.
Mathematical Models as Descriptive Frameworks: Balaguers Account
There are several anti-realist accounts of the use of mathematics in science that one may espouse to reject mathematical realism. Some of these accounts are enshrined in a global antirealist framework a la van Fraassen while others are not. In fairness to the proponents of Quines indispensability argument, I shall focus on two anti-realist accounts that are limited in their scope to the mathematics while agnostic towards (and, therefore, compatible with) Quines more general doctrines of holisms and naturalism. The two accounts claim that mathematics play a strictly descriptive role. The first account has, I believe, certain merits though it ultimately remains ambiguous about how we should understand the use of mathematical models whereas the second has been tackled full-on by mathematical realists with the counter-claim that mathematical models also play an explanatory role in science. I shall briefly describe the two accounts and then on to the issue of whether mathematical models can be explanatory.
2.1 Balaguers Descriptive Account
The first account of mathematical models as descriptive tools is proposed by Balaguer. Balaguer rejects the indispensability argument by advancing two separate counterarguments against the alleged existence of mathematical entities: first, he claims that the existence of causally inert objects such as mathematical entities is implausible and, second, that the role of mathematical models in science is always descriptive.
We shall only be interested in the second counterargument, though it should be noted that Balaguers first counterargument has come under attack (rightly, I think) because it crucially relies on a principle of causal efficiency ADDIN EN.CITE Balaguer19981031361103Balaguer, Mark1998Platonism and Anti-Platonism in MathematicsNew YorkOxford University Press(Balaguer, 1998:136); one that has been rejected by some mathematical realists, as lacking ADDIN EN.CITE Colyvan199834034Colyvan, Mark1998Is Platonism a bad bet?Australasian Journal of Philosophy761115–9March 1998(Colyvan, 1998).
This principle, it turns out, is also crucial for Balaguers second counterargument. According to him, the use of mathematical models in scientific language can be accurately described in instrumental terms as:
(TA) Empirical theories use mathematical-object talk only in order to construct theoretical apparatuses (or descriptive frameworks) in which to make assertions about the physical world. ADDIN EN.CITE Balaguer1998103 emphasis in original1371103Balaguer, Mark1998Platonism and Anti-Platonism in MathematicsNew YorkOxford University Press(Balaguer, 1998:137 emphasis in original)
(TA), Balaguer claims, is true because science uses mathematics strictly as a description of certain features or behaviours of the physical world. The role of the mathematical machinery is to clarify and simplify our scientific theories by using the machinery. And, most importantly, it does not confer to mathematical models any ontological rights.
But could ontological rights be derived from the indispensable use of mathematics in science, as Quine contends? Balaguer believes not. He explains that mathematics is so widely (but not indispensably) used in science because there is a homomorphism, i.e. a structural similarity, between certain physical structures and certain mathematical structures ADDIN EN.CITE Balaguer19981031381103Balaguer, Mark1998Platonism and Anti-Platonism in MathematicsNew YorkOxford University Press(Balaguer, 1998:138).
While I have great sympathy with this explanation, we are, of course, not out of the woods yet. Claiming the existence of a homomorphism without giving an anti-realist account of what a homomorphism simply begs the question. Balaguer, unfortunately, is silent on this matter. He only motivates his view by asserting that all mathematical applications in science, including indispensable ones, can be accounted for by (TA) ADDIN EN.CITE Balaguer19981031121103Balaguer, Mark1998Platonism and Anti-Platonism in MathematicsNew YorkOxford University Press(Balaguer, 1998:112).
He claims, for example, that statements such as the physical system S is forty degrees Celsius, which in appearance quantifies over mathematical entities, is really about the resemblance that exists between a certain physical structure and a mathematical structure ADDIN EN.CITE Balaguer1998103138-1391103Balaguer, Mark1998Platonism and Anti-Platonism in MathematicsNew YorkOxford University Press(Balaguer, 1998:138-139). No word is given about what a resemblance is and whether its existence ought to be taken literally or not. I shall return to this point later on.
Notwithstanding, Balaguer asserts that the denial of (TA) is untenable. For, if the Platonist tries to deny (TA), this would entail that mathematical entities play some sort of causal role in science and this would contradict the accepted premise that mathematical entities are acausal:
The only way to deny (TA) is to maintain that the reason we refer to mathematical objects in empirical science is that they are important components, in some sense, of the facts that empirical science is ultimately concerned with But weve already seen [] that this view is untentable [because mathematical entities are causally inert]. ADDIN EN.CITE Balaguer19981031381103Balaguer, Mark1998Platonism and Anti-Platonism in MathematicsNew YorkOxford University Press(Balaguer, 1998:138)
So, again, the untenability of denying (TA) falls back on the principle of causal efficacy. In section 4, we shall encounter reasons to avoid this principle or to, at least, divorce the descriptive view of mathematical models from it. This is, in a way, what Lengs representational account does.
2.2 Lengs Representational Account
Lengs antirealist account of the role of mathematical models in science exploits a dichotomy that Quines indispensability argument generates in the ontology of mathematical entities.
According to Quine, the one and only reason why we should believe in the existence of some mathematical entity or another is because of its indispensable invocation in scientific language. But, as pointed out by Parsons ADDIN EN.CITE Parsons198690790Parsons, Charles1986Quine on the Philosophy of MathematicsHahn, Lewis EdwinSchilpp, Paul ArthurThe Philosophy of W.V. QuineLa Salle, Ill.Open Court396-4031st(Parsons, 1986), some mathematical entities can never be invoked in scientific language simply because the realm of mathematical structures is far richer than the realm of physical structures. Quines agreed and declared that such entities are part of what he termed recreational mathematics, a mathematics that has no direct applications in science and, thus, one which is not privy to any ontological rights ADDIN EN.CITE Quine198685400785Quine, Willard Van Orman1986Reply to Charles ParsonsHahn, Lewis EdwinSchilpp, Paul ArthurThe Philosophy of W.V. QuineLa Salle, Ill.Open Court396-4031st(Quine, 1986:400).
Leng latches on Quines subtle distinction and claims that even on Quines reading no mathematical model deserves ontological rights as it is not truly indispensable. A proper understanding of the role of mathematics in science, she argues, clearly shows that all mathematical models are recreational and, thus, cannot lead us to believe in their existence.
Her opening salvo is to reiterate Sobers objection to the indispensability argument ADDIN EN.CITE Sober19935345-46053Sober, Elliott1993Mathematics and indispensabilityThe Philosophical Review102135-57January 1993(Sober, 1993:45-46): that when deciding between competing scientific hypotheses, their mathematical underpinnings are never seriously put in the question. Indeed, all candidate hypotheses make liberal use of mathematical models. The claim that the existence of mathematical entities is the result of scientific empirical confirmation is, therefore, highly exaggerated.
Leng pushes this point further. She claims that when a scientific theory fails its mathematical component is never rejected. At the most, it is judged inadequate and simply replaced by a more adequate mathematical model ADDIN EN.CITE Leng20021054120105Leng, Mary2002What's wrong with indispensabiliy? (Or, the case for recreational mathematics)Synthese131395-417(Leng, 2002:412). She illustrates her argument by an example:
Consider the paradigm case of a mathematical theory which did not do what was expected of it: Catastrophe Theory. This area of mathematics was heralded as The most important development since calculus (Newsweek), but its initial promise proved to be a great deal of hot air. The result? Catastrophe Theory became a much less popular area of research, but no one would claim that the mathematics of Catastrophe Theory had been falsified by its magnificent scientific failures. ADDIN EN.CITE Leng20021054070105Leng, Mary2002What's wrong with indispensabiliy? (Or, the case for recreational mathematics)Synthese131395-417(Leng, 2002:407)
Leng claims that science is rife with examples of mathematical models that were abandoned when they do not fit the bill. But this is no fault, according to Leng; it is what scientists do with mathematics: when they quantify over mathematical entities, they do not imply ontological commitment and only use the mathematical models recreationally.
Leng concludes from this that if we take Quines distinction between applied and recreational mathematics seriously, then it would soon become apparent that all mathematical models fall in the latter category and are, thus, without any ontological rights ADDIN EN.CITE Leng20021054110105Leng, Mary2002What's wrong with indispensabiliy? (Or, the case for recreational mathematics)Synthese131395-417(Leng, 2002:411).
But while Lengs rejection of mathematical realism based on applications in science is a prima facie convincing, two worries soon arise. First, given that Lengs account is clearly an outgrowth of Sobers original point against indispensability, it is consequently vulnerable to any objections to Sober. Sobers argument has come under attack ADDIN EN.CITE Colyvan200642742Colyvan, Mark2006Scientific Realism and Mathematical Nominalism: A Marriage Made in HellCheyne, ColinWorrall, JohnRationality and Reality: Conversations with Alan MusgraveSpringer(Colyvan, 2006) for embracing a contrastive view of empiricism which, the argument goes, is fundamentally incompatible with Quines empiricism.
This is a methodological point of contention that I shall not take up here as it does not concern the role of mathematical models in science. But suffice to note that Leng owes us an adequate response.
The second worry, which it shares with Balaguers account, is about the exhaustiveness of the representational role of mathematical model in science. This is, precisely, Colyvans starting point in his defence of the indispensability argument.
The Explanatory Role of Mathematical Models
While Colyvan concedes that mathematical models are sometimes used in science to model physical phenomena, he denies that this is all that mathematics is for. He argues that a complete account of mathematics in science must also acknowledge the explanatory role of mathematical models. In his view mathematical model do more than just represent certain physical phenomena; they can sometimes explain them! If this is right then, Colyvan concludes, such representational account as Lengs and Balaguers accounts are ill-suited to accommodate this aspect of mathematics:
On [Lengs] account of the relationship between mathematics and science, mathematics provides nothing more than a convenient set of representational tools. But such an account seems to seriously understate the role of mathematics in science mathematics is more than a mere representational tool and the modelling picture is wrong. After all, if mathematics is contributing directly to explanations, it is hard to see how any scientific realist can accept the explanations yet deny the truth of the mathematics. ADDIN EN.CITE Colyvanforthcoming44§4744Colyvan, MarkforthcomingMathematical Recreation Versus Mathematical KnowledgeLeng, MaryPaseau, AlexPotter, MichaelMathematical KnowledgeOxfordOxford University Press(Colyvan, forthcoming:4)
I will come to disagree with Colyvan that mathematical models can contribute to any scientific explain. Let us start, though, by examining what and how mathematics is supposed to explain physical phenomena.
Colyvan supports this explanatory account with a host of examples that, according to him, show how only the mathematics can be the sought-after explanation. Consider, for instance, his antipodal weather patterns example:
We discover at some time t0 there are two antipodal points p1 and p2 on the earths surface with exactly the same temperature and barometric pressure. What is the explanation of this coincidence? ADDIN EN.CITE Colyvan20016949169Colyvan, Mark2001The indispensability of MathematicsNew YorkOxford University Pressviii, 172019513754X (acid-free paper)Mathematics Philosophy.(Colyvan, 2001:49)
Colyvan argues that, while historical data and our knowledge of earths weather patterns may give us a detailed causal explanation of how each point, separately, came to have that particular temperature and barometric pressure, it does not explain why these points are antipodal or why there should be any such points at all.
Colyvan contends that the explanation of this latter fact resides in a corollary of the Borsuk-Ulam theorem in topology. The corollary states that (if we assume that earth is a perfect sphere and temperature and barometric pressure are continuous functions) there are always two antipodal points on the planets surface that have the same temperature and barometric pressure. In his words:
[The phenomenon] is due to a theorem of algebraic topology that states that for any time t there are antipodal points on the surface of the earth that simultaneously have the same temperature and barometric pressure. This theorem, or more correctly the proof of this theorem, provides the missing part of the causal explanation. It guarantees that there will be two such antipodal points at any time, and, furthermore, the explanation makes explicit appeal to non-causal entities such as continuous functions and spheres. ADDIN EN.CITE Colyvan20016949169Colyvan, Mark2001The indispensability of MathematicsNew YorkOxford University Pressviii, 172019513754X (acid-free paper)Mathematics Philosophy.(Colyvan, 2001:49)
A quibble, here. The above is, of course, slightly misleading. The Borsuk-Ulam corollary is not about planets and weather patterns. It is about spheres and functions. Yet this raises an immediate question of what exactly is the subject-matter of the explanation: is it spheres or planets? If it is not the latter then what warrants tying spheres and planets together? Perhaps Baker may be expressing similar worries when he criticises Colyvans examples ADDIN EN.CITE Baker2005104227-2280104Baker, Alan2005Are there genuine mathematical explanations of physical phenomena?Mind114454223-238April 2005(Baker, 2005:227-228) because the geometric aspect of his examples makes it ambiguous what is being explained.
But there are more problems with Colyvans account.
Why Mathematical Models Do Not Explain
For the explanatory account to go through, mathematical models ought to be truly explanatory. That is, we would expect them to satisfy at least three uncontroversial conditions: (1) the mathematical explanation has to be true (2) the mathematical explanation has to be somehow bottom-level or at least irreplaceably by some non-mathematical explanation and (3) the mathematical explanation has to conform to the assumed theory of explanation.
It is relatively straightforward to defend the three conditions. If the mathematical explanation is not a true explanation, then the explanatory account fails. Alternatively, if the mathematical explanation can be reduced to some other non-mathematical explanation, then the mathematical explanation does not require the ontological commitment to mathematical entities. Finally, if the explanation does not conform to a theory of explanation that is acceptable to a scientific realist then it is clearly question begging.
Now, how do we know that some mathematical explanation is true? For example, take Colyvans claim the Borsuk-Ulam corollary explains earths antipodal weather patterns. How do we know that it is a true explanation? Indeed, an infinite number of mathematical models can play the role that the corollary plays. Take a slightly modified version of the corollary, call it B-U*, B-U* is identical to Borsuk-Ulam in everyway except that it also makes certain claims about constructible sets or whatever. B-U*, according to Colyvans account, would also be a true explanation of earths antiposal weather. But how two (indeed, an infinity of) different explanations be true?
In general, there is no way to determine whether a particular mathematical model is, on its own, a true explanation of physical phenomena. Indeed, there is little sense in calling a mathematical model as a true explanation. Given the richness of mathematics, we can construct a mathematical model for any physical phenomenon we might care to examine and call it true so as it matches the available empirical data. Take for example a pre-Uranus astrophysicist who is interested in explaining the orbital distance of the planets in the solar system. This astrophysicist may erroneously come to believe that Bodes law is a true explanation of why some planet is at a such and such distance from the sun. Not only that, later when the existence Uranus is empirically confirmed, the discovery might even further confirm the predictive power of Bodes law. But this is all in error. What what we know today is that Bodes law does not explain anything.
Had our astrophysicists failed to make further planetary discoveries, we would have no means to distinguish the supposedly true Borsuk-Ulam explanation from the patently false Bode one. Freedman summarises this point well:
I sometimes have a nightmare about Kepler. Suppose a few of us were transported back in time to the year 1600, and were invited by the Emperor Rudolph II to set up an Imperial Department of Statistics in the court of Prague. Despairing of those circular orbits, Kepler enrolls in our department. We teach him the general linear model. Least squares, dummy variables, everything. He goes back to work, fits the best circular orbit for Mars by least squares, puts in a dummy variable for the exceptional observation and publishes. And thats the end, right there in Prague at the beginning of the 17th century. ADDIN EN.CITE Freedman19851333597133Freedman, D.1985Statistics and the scientific methodMason W.Fienberg S.Cohort Analysis in Social Research: Beyond the identification problemNew YorkSpringer-Verlag(Freedman, 1985:359) cited in ADDIN EN.CITE Humphreys20041321331132Humphreys, Paul2004Extending Ourselves: Computational science, empiricism, and scientific methodNew YorkOxford University Press(Humphreys, 2004:133)
The moral is, divorced from empirical evidence, no mathematical explanations can be falsified. What has come to be shown as false, in the case of Bodes law, is not the Bode equation itself but that the use the equation has been put to (governing the the distribution of planets in the solar system.)
This, I believe, gives us good reason to deny that the mathematical facts, alone, explain directly.
Bottom-Level Explanations in Science
Now, Colyvans main thesis is that mathematical models or structures are genuine bottom-level explanations of physical phenomena, i.e. one that cannot be replaced by some more fundamental non-mathematical explanation. As I have tried to show above, there are problems with the genuine claim. Now I wish to contest the bottom-level claim.
This is, of course, an important premise for Colyvan. Without it, he cannot claim that mathematical explanation leads us to believe in the existence of mathematical entities. One may question whether there is such a thing as a bottom-level explanation in science. See, e.g., ADDIN EN.CITE Musgrave1999116137116Musgrave, Alan1999Explanation, Description and Scientific RealismEssays on Realism and RationalismAmsterdam-AtlantaRodopi(Musgrave, 1999:13). Colyvan does not give us an argument that mathematical explanations are.
But even if mathematical explanations were not bottom-level, couldnt they, nonetheless, possess some explanatory power? Musgrave does not think so:
are all scientific theories explanatory in the sense that they have at least some explanatory uses (figure in some explanatory derivations)? A [] likely candidate for laws which have no explanatory uses are numerical formula which merely summarize facts, such as Bodes Law or Balmers formula. One can derive the mean distance of Uranus from Bodes Law together with the information that Uranus is the seventh planet from the sun: but this does not explain why Uranus has the mean distance that it does, nor can I think of any explanatory deductions in which Bodes Law figures essentially. ADDIN EN.CITE Musgrave199911657116Musgrave, Alan1999Explanation, Description and Scientific RealismEssays on Realism and RationalismAmsterdam-AtlantaRodopi(Musgrave, 1999:5)
So, according to Musgrave, unless there is something that is essentially Borsuk-Ulam-like about earth, then the Borsuk-Ulam corollary cannot any explanatory use except as a mere summary of facts. That is the corollary is, here, merely an instrument. Colyvan does not give us reason to believe that earth is Borsuk-Ulam-like (even if that might well be the case) and, consequently, he does not warrant the move from earth and its weather patterns to its mathematical explanation.
Overarching point here, one that has to do with scientific practice, is that it is uncharacteristic for scientists to take mathematical explanations, even coincidental ones such as Bodes law, at face value. More typically, they will try to find intrinsic physical explanations for the phenomena under examination whenever they are able to. Some astrophysicists, for instance, believe now that Bodes law might be due to the orbital resonance pattern of planets which sounds a lot more bottom level and a lot less mathematical than a mere equation.
Finding an Adequate Theory of Explanation
Let us look at Colyvans theory of explanation. In bringing out his examples, Colyvan endorses what he terms as a broad-minded view of causation and explanation so that they may appeal to everyone:
For obvious reasons I wish to be as broad-minded as possible about both causation and explanation. I will assume only that an explanation must be enlightening it must make the phenomena being explained less mysterious. ADDIN EN.CITE Colyvan20016947169Colyvan, Mark2001The indispensability of MathematicsNew YorkOxford University Pressviii, 172019513754X (acid-free paper)Mathematics Philosophy.(Colyvan, 2001:47)
Mathematical explanations are, according to him, legitimate and even plausible because they make certain physical phenomena less mysterious. Furthermore, these explanations are appealing because they make phenomena less mysterious.
But there are two objections that can be raised here. The first (mild) objection is this: how can an explanation positing the existence of platonic abstract, acausal entities make a physical phenomenon less mysterious? Arguably, if we take Colyvans view of explanation seriously, the introduction of these entities and their apparent indispensability can only deepen the mystery.
The second objection is against this alleged broad-minded view of explanation which frames scientific investigation as the venue of solving mysteries. This is what we expect from scientific explanations then, clearly, they fail miserably. Scientific explanations inevitably and invariably end up referring to some unexplainable brute fact of the world. And this fact is no less mysterious or puzzling that the phenomenon to be explained in the first place ADDIN EN.CITE Musgrave199911687116Musgrave, Alan1999Explanation, Description and Scientific RealismEssays on Realism and RationalismAmsterdam-AtlantaRodopi(Musgrave, 1999:8). Colyvans broad-minded view of explanation ought to appeal to no one.
6.1 Bakers Defence
In contrast to Colyvan, Baker is enthusiastic about the explanatory role of mathematics in science but does not rely on this mistaken notion of explanation. He tries to gauge the adequacy and merit of mathematical explanations against the leading theories explanation:
What needs to be checked is that the mathematical component of the explanation is explanatory in its own right, rather than functioning as a descriptive or calculational framework for the overall explanation. This is difficult to do without having in hand some substantive general account of explanation. ADDIN EN.CITE Baker20051042340104Baker, Alan2005Are there genuine mathematical explanations of physical phenomena?Mind114454223-238April 2005(Baker, 2005:234)
He then goes on to argue that none of the three leading accounts of explanation can be used to reject mathematical explanations outright. Namely, he claims that the causal account clearly begs the question since mathematical explanations are, by definition, non-causal ADDIN EN.CITE Baker20051042340104Baker, Alan2005Are there genuine mathematical explanations of physical phenomena?Mind114454223-238April 2005(Baker, 2005:234). He claims that the pragmatic account accepts mathematical explanations unequivocally and that the deductive-nomological account accepts mathematical explanations if we extend the laws of nature to include mathematics ADDIN EN.CITE Baker20051042350104Baker, Alan2005Are there genuine mathematical explanations of physical phenomena?Mind114454223-238April 2005(Baker, 2005:235).
Baker concludes from this that no matter what theory of explanation we adopt, if we commit to the principle of inference to the best explanation then cases such as earths antipodal weather points give us good reason to favour mathematical explanations and, consequently, endorse mathematical realism:
I have argued that there are genuine mathematical explanations of physical phenomena... If this is right, then applying inference to the best explanation [] yields the conclusion that numbers exist. ADDIN EN.CITE Baker20051042360104Baker, Alan2005Are there genuine mathematical explanations of physical phenomena?Mind114454223-238April 2005(Baker, 2005:236)
I disagree with Baker, however, because of what I perceive as two important difficulties in his argument. First, Baker does not give us reason to include mathematics as part of the laws of nature, except for observing hat some mathematical formulae have the general structure of natural laws ADDIN EN.CITE Baker20051042350104Baker, Alan2005Are there genuine mathematical explanations of physical phenomena?Mind114454223-238April 2005(Baker, 2005:235). This is misleading and begs the question. Many statements that refer to entities that do not exist have the form of general laws of nature (all winged horses have wings; the mass of a winged horse is proportional to its energy, and so on). Besides, if mathematical antirealists believe that mathematical entities do not exist, then admitting mathematical statements as laws of nature clearly begs the question. The argument is, precisely, about whether or not we ought to believe in the existence of such entities and cannot assume their existence by admitting mathematical statements into the laws of nature.
Secondly, Bakers appeal to the principle of the inference to the best explanation is gratuitous. The question is not whether non-mathematical explanations are second-best to mathematical explanation. The whole premise of Colyvans examples and his explanatory account is that there is no other explanation for the physical phenomenon. The only explanation (not the best explanation) is a mathematical explanation.
But even if it were true (which it is not) that a non-mathematical explanation is being compared to a mathematical explanation then Baker gives no reason to judge the mathematical explanation better. In fact, the non-mathematical explanation would be compatible with all three theories of explanation without having to admit mathematicsin the laws of nature and, therefore, technically superior to the mathematical explanation.
A Nominalistic Explanation of the Antipodal Weather Patterns
If the position I have argued against is wrong then the explanatory role of mathematical models in science ought to be rejected. The burden is, therefore, on the mathematical realist to show that mathematical models play more than a representational role when they are invoked in mathematical theories.
This instrumentalist view, however, does not resolve the issue of how it is that mathematics comes to be a useful representational device. In particular, if it is not the models themselves who offer scientific explanations in Colyvans examples and others then what is that does the explaining?
I suggest, here, that the physical explanation underlies the physical phenomenon. These explanations may not bottom-level or less mysterious, but they do not require the input of mathematical models. The reason why mathematical models appear to explain is the subtle equivocation that happens when mathematical entities are confounded with physical entities. As we have seen in the antipodal weather patterns example, this happened when earth was compared to sphere.
What ought to be said in the first place, is that earth is, following Musgrave, Borsuk-Ulam-like. But since the Borsuk-Ulam corollary is true for any sphere, then it suffices to say that the explanation is that earth is sphere-like. By merely asserting that earth is spherical (and that temperature and pressure and continuous function-like) we obtain that earth satisfies the Borsuk-Ulam corollary via straightforward deductive steps.
In other words, while this is not immediately obvious, it is simply the particular shape of the planet what does the explanatory work here: an impure function that maps earth to a perfect sphere and not the mathematical devices of topology. Is this explanation true? Empirical evidence suggests that (to a degree) earth is, indeed, spherical. Thus, unlike Bodes law, it reflects a genuine relationship between earth and spheres. This is tantamount to earlier Balaguers claim that there is a homomorphism between certain physical structures (in this case earth) and mathematical structures (sphere).
Can we offer an anti-realist account of this homomorphism? I believe that it can be done. In particular, one may adopt a form of resemblance nominalism that would deflate platonistic talk of resemblance between earth and a sphere to a mere relationship between earth and other earth-like entities. Of course, like all of its competitors, resemblance nominalism is saddled with its share of difficulties (see for ADDIN EN.CITE Rodriguez-Pereyra20021181118Rodriguez-Pereyra, Gonzalo2002Resemblance Nominalism: A Solution to the Problem of UniversalsOxfordClarendon(Rodriguez-Pereyra, 2002) a sustained defence), but nothing in this proposal hinges on the details of an anti-realist account of properties.
Conclusion
I endorse the view that mathematical structures are used in science as models or descriptive devices. Scientists do not take the ontological commitments of those models seriously and simply judge them to be adequate or inadequate depending on how accurately they represent the physical phenomenon under investigation.
I reject the claim that mathematical models also play an explanatory role in science. I take issue with the validity of the explanatory framework that this account assumes and I raise doubt that mathematical models can be empirically shown to be genuine explanations of physical phenomena.
I suggest that what may be explanatory in mathematics is the impure function that connects our physical structures to certain mathematical structures. And that this impure function, what may be termed a homomorphism between physical and mathematical structures, is what is truly at play. More work needs, however, to show that this homomorphism does not implicitly require an appeal to abstract entities.
If this is right, then scientific realists who sympathise with Quines doctrines need not be mathematical realists as the mathematical models are never genuinely put to indispensable use in science.References
ADDIN EN.REFLIST Armstrong, D. M. (1989). A Combinatorial Theory of Possibility. Cambridge: Cambridge University Press.
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114(454), 223-238.
Balaguer, M. (1998). Platonism and Anti-Platonism in Mathematics. New York: Oxford University Press.
Colyvan, M. (1998). Is Platonism a bad bet? Australasian Journal of Philosophy, 76(1), 115119.
Colyvan, M. (2001). The indispensability of Mathematics. New York: Oxford University Press.
Colyvan, M. (2006). Scientific Realism and Mathematical Nominalism: A Marriage Made in Hell. In C. Cheyne & J. Worrall (Eds.), Rationality and Reality: Conversations with Alan Musgrave: Springer.
Colyvan, M. (forthcoming). Mathematical Recreation Versus Mathematical Knowledge. In M. Leng, A. Paseau & M. Potter (Eds.), Mathematical Knowledge. Oxford: Oxford University Press.
Freedman, D. (1985). Statistics and the scientific method. In M. W. & F. S. (Eds.), Cohort Analysis in Social Research: Beyond the identification problem. New York: Springer-Verlag.
Humphreys, P. (2004). Extending Ourselves: Computational science, empiricism, and scientific method. New York: Oxford University Press.
Leng, M. (2002). What's wrong with indispensabiliy? (Or, the case for recreational mathematics). Synthese, 131, 395-417.
Musgrave, A. (1999). Explanation, Description and Scientific Realism. In Essays on Realism and Rationalism. Amsterdam-Atlanta: Rodopi.
Parsons, C. (1986). Quine on the Philosophy of Mathematics. In L. E. Hahn & P. A. Schilpp (Eds.), The Philosophy of W.V. Quine (1st ed., pp. 396-403). La Salle, Ill.: Open Court.
Quine, W. V. O. (1986). Reply to Charles Parsons. In L. E. Hahn & P. A. Schilpp (Eds.), The Philosophy of W.V. Quine (1st ed., pp. 396-403). La Salle, Ill.: Open Court.
Rodriguez-Pereyra, G. (2002). Resemblance Nominalism: A Solution to the Problem of Universals. Oxford: Clarendon.
Sober, E. (1993). Mathematics and indispensability. The Philosophical Review, 102(1), 35-57.
Various versions of the principle have been defended by Armstrong and others. See, for example, ADDIN EN.CITE Armstrong19891067-81106Armstrong, David M.1989A Combinatorial Theory of PossibilityCambridgeCambridge University Press(Armstrong, 1989:7-8)
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