Abstract
Platonists in mathematics endeavour to prove the truthfulness of the proposal about the existence of mathematical objects. However, there have not been many explicit proofs of this proposal. One of the explicit ones is doubtlessly Baker’s Enhanced Indispensability Argument (EIA), formulated as a sort of modal syllogism. We aim at showing that the purpose of its creation – the defence of Platonist viewpoint – was not accomplished. Namely, the second premise of the Argument was imprecisely formulated, which gave space for various interpretations of the EIA. Moreover, it is not easy to perceive which of the more precise formulations of the above-mentioned premise would be acceptable. For all these reasons, it is disputable whether the EIA can be used to defend Platonist outlook. At the beginning of this century, Baker has shown that the so-called Quine-Putnam Indispensability Argument can not provide “full” platonism - a guarantee of the existence of all mathematical objects. It turns out, however, that the EIA has a similar disadvantage.
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Notes
Baker (2009, p. 613).
It is well known that there are skeptical objections to this viewpoint. Nevertheless, this paper will not deal with the justifiability of this premise. Also, whenever the term science is used henceforth, it will imply the natural sciences.
Sereni (2014).
Molinini (2014).
Intuitively, we can state that a mathematical entity x, which is a part of a mathematical theory M, has an indispensable explanatory role in a scientific theory T if a physical phenomenon P described by the theory T cannot be explained without using the entity x.
Molinini (2014).
Euler (1750).
SO(n) is the subgroup of O(n) (O(n) is the group of real orthogonal n × n matrices. Every orthogonal matrix has determinant either +1 or − 1) formed by the orthogonal n × n matrices with determinant +1. In is the identity matrix of size n.
There are two such examples in Molinini (2014).
The possibility that, in the context of the EIA, statements (2a) and (2b) may be accepted as de re or de dicto statements was considered in Drekalović and Žarnić (2018).
Baker (2009, p. 614).
Goles et al. (2001, p. 33).
Baker (2009, pp. 615–19).
If the notion of indispensability is understood as suggested in the footnote 5, it cannot, therefore, be stated that if the choice of a mathematical object O for explaining a physical phenomenon P is not arbitrary it then follows that the object O has an indispensable explanatory role for the P. Indeed, it appears that the facts described by the empirical sciences do not allow an unlimited freedom to the researcher when it comes to the choice of mathematical theory by which to explain a physical phenomenon. For example, it is most likely that the case of cicada cannot be explained by means of the objects taken from the functional analysis, such as functional series. Nevertheless, it does not imply that there is no mathematical entity from another mathematical theory that cannot be chosen for such an explanation and the choice of which is conditioned by the nature of the phenomenon to be explained by it. The before mentioned Molinini’s examples illustrate very well the possibility of the multiple mathematical explanation of the same phenomenon.
It is clear that only one good example would be sufficient to refute nominalism. However, the purpose of creating the EIA would not appear to be fulfilled by that.
Let us remind that, for instance, cardinality of only one set of mathematical objects, the set of numbers belonging to the interval (a,b); a,b ∈ R, equals C. In other words, that set makes uncountable many mathematical objects.
For instance, in medical science or zoology.
At this point, some further clarification should be provided. When speaking of indispensability of mathematical entities for explaining physical phenomena, one must bear in mind the fact that it is an area where several sciences overlap, with their respectively different methodologies of proving propositions. On the one hand, there is mathematics, where justification of a proposition is to be found only in formal proof. On the other hand, there exists a mathematical explanation of the facts in empirical science, such as, for example, biology, in which statements are often justified by inductive conclusions (which is not the same as mathematical induction), entirely unlike mathematical formalism. Finally, on the third hand, the question of existence of mathematical objects and their possible relevance to the explanatory role of mathematics in the empirical science is a matter of philosophy. The well-grounded argument is that every discussion of existence of mathematical objects is primarily a philosophical question, as the feature “exists” is not a mathematical feature the same as “is divisible by 3” or “is less that 37”. Thus, it is not to be expected that a strict mathematical methodology should be exclusively used when discussing the proof of existence of mathematical objects. However, it appears that the philosophical analysis, being carried out along such border-line fields, must not allow for vague, random and arbitrary justification of its own propositions. On the contrary, it must be based and structured in such a way so as to demonstrate the highest possible exactness by means of the methods it uses.
Molinini (2014).
A proper subset S1 of a set S, denoted S1⊊ S, is a subset that is strictly contained in S and so necessarily excludes at least one member of S.
The term domain of the argument is used in this text in an entirely non-formal sense. Let us mention, however, that it is also possible to interpret it formally in the context of this topic. Indeed, an arbitrary platonist argument can be perceived as strictly formal, as a function which, for example, maps a set (a domain) that is a subset of the set of all mathematical objects, into a singleton {“exists”}. The meaning of the informal term domain would thus overlap with the formal meaning. We could deal with this function and its features in the same way as with the features of any other function.
The stronger version of the EIA (∃∀) proposes the existence of mathematical objects which have a role in every respective explanation of the situation with cicada, whereas the weaker version of the EIA (∀∃) proposes that every suitable explanation of this phenomenon includes a need for some mathematical objects (see Baker (2009, p. 616) and Hunt (2015)). When compared to the stronger, the weaker version allows for a possibility to use more mathematical objects, depending on the explanation. Nevertheless, the set of objects that can be used for explaining a phenomenon is a finite set that cannot have a larger cardinality than the set of explanations of the phenomenon in question.
I am grateful to one of the anonymous referees for drawing my attention to the paper Baker (2003) which raises a similar issue, considering the idea of full platonism, but in the context of the (original) Quine-Putnam Indispensability Argument.
The basic difference between the IA and the EIA is that in the second argument, in contrast to the first, the role of mathematical objects in science is not merely reduced to quantification, but rather insists on the condition of an indispensable explanatory role of mathematical objects in science.
Baker (2003).
Baker (2003, p. 53).
Collections M1, M2, ..., Mk may be disjoint, or some of them may have common elements.
Baker (2003, pp. 64–65).
On one hand, there are a number of scientific theories in which in some way the mathematical theory has not been used. However, to find, among them, examples in which mathematical objects play an explanatory role in science is not easy. In addition to the aforementioned example with cicada in biology, there are, in the literature, only a few examples in which mathematical objects play such a role. Namely, there are other examples such as the honeycomb, the bridges of Konigsberg, and the asteroid belt around Jupiter. For more see Pincock (2012).
Baker (2003, p. 65).
Quine (1986).
Baker (2003, pp. 51–52).
Although it is not common to talk about the discovery of mathematical theories, this term is rather used when it comes to the emergence of scientific theories, there are arguments that show that this term is quite appropriate in mathematics in the Platonistic context as well. For more see Drekalović (2015).
Baker (2003, p. 52).
As it is known, George Boole laid the foundations of this theory in the 1930s. Nevertheless, the application of Boolean algebra in the field of computing got started after more than one century since its foundation.
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Drekalović, V. Is the Enhanced Indispensability Argument a Useful Tool in the Hands of Platonists?. Philosophia 47, 1111–1126 (2019). https://doi.org/10.1007/s11406-018-0033-3
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DOI: https://doi.org/10.1007/s11406-018-0033-3