Abstract
In ‘The varieties of indispensability arguments’ Marco Panza and Andrea Sereni argue that, for any clear notion of indispensability, either there is no conclusive argument for the thesis that mathematics is indispensable to science, or the notion of indispensability at hand does not support mathematical realism. In this paper, I shall not object to this main thesis directly. I shall instead try to assess in a naturalistic spirit a family of objections the authors make along the way to the use of indispensability premises in indispensability arguments.
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Notes
As expounded by Quine, and more recently defended in Colyvan (2001).
Thus I shall not discuss the claim that we ought to have an ontological commitment to all (and perhaps only) those entities that are indispensable to our best scientific theories, or any other version of the crucial premise iii in Panza and Sereni’s reconstruction of IAs.
Those claims are made in various places in Sect. 4.2.1 of ‘The varieties of indispensability arguments’. Those claims bear some similarity to the anti-IA strategies analyzed in Baker (2003), as the authors notice (e.g. in fn 47), but they are different and do not especially hinge on the multiplicity of foundational frameworks for mathematics.
See ‘The varieties of indispensability argument’, Sect. 3.5. Roughly, the proposed definition of indispensability says that a mathematical theory T is indispensable to a scientific theory S in order to accomplish a given task in an appropriate way if and only if all instances of S that accomplish the task in the appropriate way make use of some instance of T.
A method of resolution is pure if it avoids the use of concepts that are alien to the proper domain of the problem to be solved, and it is impure otherwise. For more on purity of methods, see Detlefsen and Arana (2011).
Say, some use of Fermat’s last theorem to predict that a certain planet will not hit the Earth.
By the end of Sect. 4.2.1, the authors write:
It can be argued that [set theory’s] role amounts, globally, to the role that could be played, piecemeal and conjunctively, by other mathematical theories, also conceived as autonomous. (‘The varieties of indispensability arguments’, Sect. 4.2.1)
I take the claim that mathematical statements need to receive some form of (mathematical) justification prior to their use in application as unproblematic and acceptable even to a die-hard instrumentalist. The assumption is very similar to a suggestion the authors make within the context of the discussion of explanatory indispensability in Sect. 4.2 of ‘The varieties of indispensability arguments’.
Think of the case of someone who is not confident about the consistency of ZFC. Then she cannot justify the use of B to make a scientific prediction, while she might have been able to do so with A, given her confidence in arithmetic.
This formulation would need serious refinements, among other reasons because many notions of interpretations might be used. For more on these notions of interpretability and useful insights about the relation between interpretation function and knowledge transfer, see Walsh (2014), which discusses related issues in depth within the context of a critical assessment of the neo-logicist program.
Regarding the authors’ other suggestion that the mathematical role of set theory reduces to the role of an unifying framework, so that set theory’s role itself could be played just as well by classical theories as far as applications are concerned, let me mention that not everyone agrees with this conception of set theory. In particular, Dehornoy (1996) has argued that set theory’s role cannot be reduced to its foundational role. On a par with classical theories, Dehornoy argues, set theory has its own methods, with fruitful applications in topology—which in turn plays an important role in modern physics.
The authors write:
For (finitary) arithmetic things look even simpler. Indeed, to show that there cannot be any sound genuine IAs involving arithmetic in which ‘Q’s’ is replaced by ‘theories’, and \({\mathfrak {L}}\) is replaced by \({\mathfrak {L}}_{\beta }\) or \({\mathfrak {P}}_{\beta }\), it is enough to notice that natural numbers can be replaced with numerical quantifiers in any scientific statement where they occur, without missing or diminishing the descriptive and predictive power of this statement. (‘The varieties of indispensability arguments’, Sect. 4.1.1. my emphasis)
See, e.g., Buss (1994) for relevant technical results.
Panza and Sereni write:
Consider, for example, any version of RealAn of real analysis and suppose that \(S_i\) is an instance of a scientific theory S that has recourse to an instance of RealAn. It is highly plausible to admit that there is another instance \(S_j\) of S [...] that has not recourse to RealAn, insofar as it replaces it with an appropriate theory of rational numbers [...] which accomplishes the same descriptive or predictive task as \(S_i\) in such a way that no difference between the two, with respect to the accomplishment of this task, can be appreciated on the basis of our capacity of discerning their descriptive and/or predictive power (on the basis of which we assign to these theories the epistemic property \(P^E\)). (‘The varieties of indispensability arguments’, 4.2.1)
I am not sure how this line of argument extends to predictive indispensability.
The example is taken from Maddy (1992, pp. 281–282).
Accepting this much is not incompatible with accepting the possibility that further reasons could later convince us to finally take such a model of continuous space-time to be descriptively false, as Maddy (1992) contemplates (citing Feynman).
They cannot be replaced, that is, until someone has shown how to dispense with them without epistemic loss.
Eleatism is the doctrine according to which the ‘eleatic principle’ is true. I borrow the ‘eleatic principle’ from Colyvan (2001). The principle reads:
An entity is to be counted as real if and only if it is capable of participating in causal processes. (Colyvan 2001, p. 40)
If mathematical entities are acausal, as probably most philosophers believe, then obviously their role in explanation is not a causal role.
This is how I read their analysis of the explanation of the life-cycle of Cicadas in the last section of their paper.
The quote is found in Sect. 4.2.2 and continues:
which is perfectly independent of the existence and properties of any sort of number, too. (‘The Varieties of Indispensability Arguments’, 4.2.2)
In particular, the use of mathematics in Baker (2005) does not reduce to reference to arbitrary numbers in a conventional unit of time measurement. The explanation using co-primeness tells us something more essential about certain abstract relations between time periods and seems prima facie to give these abstract relations some explanatory role. More generally, the fact that dimensionless quantities play a role in a variety of physical laws and scientific explanations is of course of more significance for the indispensability of numbers in explanations than the mere use of numbers.
Again: not necessarily a ‘causal’ role, which would partially beg the question in favor of the dispensability thesis. Moreover, it seems to me that the strength of IA does not depend upon accepting the metaphysics of causality. For lack of space, I am assuming without argument that the thesis that whatever plays a crucial role in our best explanations cannot be treated by us as fictional does not depend on a specifically causal conception of explanation. Thus I am assuming that to put so-called ‘enhanced’ IAs to work any bona fide mathematical scientific explanation will do. On criteria to distinguish what counts as bona fide mathematical scientific explanations, see below.
Typically, disagreement about whether physical explanations must be causal explanations and whether to play an essential role in such explanation is to take part in the causal process will tend to make it more difficult to agree on the thesis that mathematics are essential to scientific explanations.
See Baker (2005, p. 234).
The philosophical problem of the indispensability of mathematics to science is normative in the sense that it is all a debate about epistemic norms, e.g. what ought to count or not as a genuine scientific explanation, what can be dispensed with in science without significantly impairing it, etc.
What I call ‘radical naturalism’ here is the view that philosophy must give way to science everywhere empirical science is possible. What I call ‘minimal naturalism’ is the less radical thesis that philosophy is in continuity with science, a science among other sciences with no special legislative power. Moreover if, following Huw Price, we define ‘object naturalism’ as the view that ‘all genuine knowledge is scientific knowledge’ whereas ‘subject naturalism’ is the apparently weaker view that ‘philosophy needs to begin with what science tells us about ourselves’ (Price 2004, p. 73), let us remark that ‘subject naturalism’ is enough to motivate an empirical inquiry about what we—as a scientific community—take as counting as scientific explanations.
Here is perhaps one such example of anecdotal evidence. Very recently, the French neurobiologists Thomas Boraud and François Gonon have argued in public debate that the theoretical difficulties presently facing neurobiologists are caused by their lack of mathematical sophistication (see Boraud and Gonon 2013). Their idea is that there are a number of explanations that could now be within our reach in neurobiology, but that neurobiologists fail to provide them because of their lack of mathematical knowledge, especially in handling multivariate statistical methods.
Published by Springer, the editorial presentation reads:
The Journal of Mathematical Neuroscience (JMN) publishes research articles on the mathematical modeling and analysis of all areas of neuroscience, i.e., the study of the nervous system and its dysfunctions. The focus is on using mathematics as the primary tool for elucidating the fundamental mechanisms responsible for experimentally observed behaviours in neuroscience at all relevant scales, from the molecular world to that of cognition. The aim is to publish work that uses advanced mathematical techniques to illuminate these questions.
This journal is itself affiliated with the Society for Mathematical Psychology.
Published by Taylor and Francis. In the presentation of the aim and scope of the journal one can read:
Because the Journal of Mathematical Sociology is addressed primarily to sociologists it is anticipated that most articles will be oriented toward a mathematical understanding of emergent complex social structures rather than to an analysis of individual behavior. These structures include, for example, informal groups, social networks, organizations, and global systems.
It may be true that what scientists regard as an explanation might sometimes differ from the philosopher’s view of the matter. It might also happen that what scientists regard as an explanation quickly evolves over time. I take it as an advantage of the present suggestion that it does not rely only on the philosopher’s biased opinion about what counts as a good explanation to assess the role of mathematics in scientific explanations, and I do not regard the fact that it will yield different results as science and scientists’ conceptions evolve over time as a defect of the method. Moreover, I think that, even if it is true that scientists might be wrong about the ontological or various metaphysical conceptions they bring along their scientific activities, there is a sense in which we cannot, or should not, take them to be in the same way collectively wrong about what they take the criteria of the aims and success of the scientific activity to be—unless we can oppose strong evidence. See the above remarks about minimal naturalism.
Personal communication. Thanks to Kevin Boyack for performing the relevant queries on his map of science and for allowing me to use his data and maps.
See the pdf version of this paper online to see the colored version of the map. I have provided a quick clarification of my use of the term ‘similar’ in the caption below Fig. 1.
For one, it would of course be necessary to check the typical contexts of occurrences of the string ‘mathematic’ in the titles, abstract of papers and books, titles of journals, and see if we have even remotely succeeded in capturing the core idea of a set of papers where mathematics are ‘essentially involved’ in explanation. After all, for all I have said, the results may well all be papers explicitly devoted to ‘metaphorical explanations’, or ‘untrue but short explanations’, etc. However, I see no a priori reasons why an empirical inquiry along these lines could not be conducted in a convincing way.
I conflate the two issues since they are basically equivalent. In one direction, the indispensability of mathematics in science is ordinarily understood as implying that mathematics do not merely play an auxiliary role in scientific explanation but that there are mathematical explanations proper to science. In the other direction, if there are properly mathematical explanations in science, then we cannot dispense with mathematics in some scientific explanations.
References
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Galinon, H. Naturalizing indispensability: a rejoinder to ‘The varieties of indispensability arguments’. Synthese 193, 517–530 (2016). https://doi.org/10.1007/s11229-015-0978-8
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DOI: https://doi.org/10.1007/s11229-015-0978-8