Which Mathematical Objects are Referred to by the Enhanced Indispensability Argument?

Discussion

Abstract

This discussion note points to some verbal imprecisions in the formulation of the Enhanced Indispensability Argument (EIA). The examination of the plausibility of alternative interpretations reveals that the argument’s minor premise should be understood as a particular, not a universal, statement. Interpretations of the major premise and the conclusion oscillate between de re and de dicto readings. The attempt to find an appropriate interpretation for the EIA leads to undesirable results. If assumed to be valid and sound, the argument warrants the rationality of the belief in an unusual variant of Platonism (partial and mutable domain admitting gaps and gluts). On the other hand, if taken as it stands, the argument is either invalid or is unsound or does not support the mathematical Platonism. Thus, the EIA in its present form cannot serve as a useful device for the Platonist.

Keywords

Platonism Enhanced Indispensability Argument Mathematical explanation de re and de dicto statements 
The Platonism in mathematics claims that mathematical objects exist. One recent attempt to defend this view is the so-called Enhanced Indispensability Argument (henceforth EIA), itself a variant of Quine-Putnam indispensability argument. The argument has been explicitly formulated by Alan Baker:
  1. (1)

    We ought rationally to believe in the existence of any entity that plays an indispensable explanatory role in our best scientific theories.

     
  2. (2)

    Mathematical objects play an indispensable explanatory role in science.

     
  3. (3)

    Hence, we ought rationally to believe in the existence of mathematical objects (Baker 2009, 613).

     

The major premise states that a belief in a variant of realism is a requirement of rationality. The notion of indispensability relation between theoretical terms and our best theories in empirical sciences has been replaced by the more narrow notion of the indispensable explanatory role. The argument’s force rests on the role of mathematics in scientific explanations.

In this paper, the terms ‘entity’ and ‘object’ will be taken to mean ‘the denotation of a term in a mind-independent and language-independent structure’.1 This paper will not deal with the justifiability of the major premise. Instead, its focus will be on the exact interpretation of the minor premise and on the assessment of the argument’s validity. More precisely, the aim of this paper is to first point to an imprecision in the formulation of the minor premise. It is this imprecision that results in a further imprecision of formulating the EIA as a whole, which questions the very meaningfulness of the EIA. This article will then propose a more precise and naturally expected formulation/interpretation of the minor premise, relying on the arguments (examples) which Baker used in order to justify it. Eventually, we will analyse the possibility of the EIA being justifiable if the minor premise is thus understood, and if the two other statements—major premise and the conclusion—are understood either as de re or de dicto statements.

1 What is the Correct Version of the Minor Premise in the EIA?

The minor premise (2) of the EIA can be interpreted either as particular (2.a) or universal (2.b) affirmative statement:
  1. (2.a)

    Some mathematical objects play an indispensable explanatory role in science.

     
  2. (2.b)

    All mathematical objects play an indispensable explanatory role in science.

     

1.1 Baker’s Understanding of the Minor Premise

In writing about the role of concrete examples of the indispensability of mathematical objects Baker seems to presuppose the particular statement–interpretation of the minor premise. He offered a single example for the indispensability of mathematical objects in explaining a specific physical phenomenon: prime numbers are indispensable for the explanation of the life cycle of the North American cicada. In his articles dealing, inter alia, with the existence of mathematical objects, he did not focus on finding some more examples of the explanatory indispensability of other mathematical objects.2 Rather he focused on proving that the choice of a mathematical theory is not arbitrary in explaining an empirical phenomenon. If Baker had understood the minor premise in the universal sense, then his proof would have been different. A single example suffices only for the proof of the particular variant of minor premise since no number of examples can establish the truth of the universal variant if the Platonic domain is infinite or contains as yet unknown mathematical objects.

1.2 Is the Relation of Explanatory Indispensability Functional?

Molinini (2016) has introduced the definition for explanatory dispensability (E-DISP) and has given two examples of alternative mathematical explanations of the same physical phenomena.

(E-DISP) A mathematical entity x is explanatorily dispensable to a scientific theory T iff it is possible to find a theory T* that: (a) does not employ the vocabulary of the mathematical theory M in which x is defined; (b) offers the same (or even more) explanatory power as T; (c) is empirically equivalent to T (Molinini 2016, 408).

The fact of the existence of alternative mathematical explanations grounds his conclusion that, in this case, “the mathematical entities that do the explanatory work are, according to (E-DISP), dispensable”.

The concept (E-INDISP) of explanatory indispensability is the contradictory concept of (E-DISP). According to (E-INDISP), the explanatory indispensability relation must be a functional relation. Thus, if there are two objects, x and x′, from different mathematical theories, M and M′, used in empirical theories, T and T′, that are equal in explanatory power and have the same observational consequences, then neither object x nor object x′ is explanatorily indispensable to these empirical theories. What Molinini defines is the ‘individual explanatory indispensability’. Nevertheless, it makes sense to introduce the notion of the ‘collective explanatory indispensability’ where some object is indispensable although no definite object is indispensable.3

If the indispensability relation is conceived as functional, then the empirical theory using the indispensable mathematical object must be strictly preferable to all other theories that are equal in explanatory power and have the same observational consequences. A non-functional relation gives a more relaxed notion of explanatory indispensability.4 It is required that no theory is strictly preferable to the one using the indispensable mathematical object. This requirement does not exclude the possibility that the two objects belonging to the vocabularies of different mathematical theories both play an explanatory indispensable role to the same scientific theory.

No matter which definition is chosen, the EIA will fail to support Platonism in the face of alternative mathematical explanations.

Suppose that explanatory indispensability is defined as a functional relation and that alternative but preferentially indistinguishable mathematical explanations have been given. In this situation, it sounds convincing that rationality requires the belief in the existence of a referent for at least one among alternative mathematical terms. The formal notation shows this more clearly. If T and T′ are alternative and preferentially indistinguishable empirical theories using mathematical theories M and M′, respectively, and t belongs to the vocabulary of M and t′ to that of M′, then
$${\text{OB}}\left( {\exists x\left( {x = ref\left( t \right)\; \vee x = ref\left( {t^{{\prime }} } \right)} \right)} \right).$$
The formula in the consequent expresses a doxastic obligation: it is obligatory, O, that it is believed, B, that there is an object which is the referent, ref, either of t or of t′. This consequent is not warranted by the EIA.

On the other hand, if the relation of explanatory indispensability is defined in a non-functional way, then in the presence of alternative and preferentially indistinguishable mathematical explanations the EIA will justify a doxastic obligation, but with the cost of an overloaded Platonic domain. In this case, both t and t′ are explanatory indispensable and so, by two applications of EIA argument, the two doxastic obligations appear, the one requiring the belief that the referent of t exists, and the other requiring the belief that the referent of t′ exists. Rationality requires beliefs to be closed under conjunction, and so it follows that we ought rationally to believe in the existence of the two mathematical objects none of which is indispensable per se. The formal notation shows this more clearly. If T and T′ are alternative and preferentially indifferent empirical theories using mathematical theories M and M′, respectively, and t belongs to the vocabulary of M and t′ to that of M′, then \({\text{OB}}\left( {\exists x\exists y\left( {x = ref\left( t \right)\; \wedge y = ref\left( {t^{{\prime }} } \right)} \right)} \right).\)

The formula says that it is obligatory, O, that it is believed, B, that both the referent of t and the referent of t′ exist. Therefore, if there are alternative and preferentially indistinguishable mathematical explanations, the Platonic domain warranted by EI-type arguments will either have a gap or will have a glut.

2 Mutable Platonic Domain

The vagueness and imprecision in the original formulation of EIA gave rise to various interpretations. Nevertheless, the analyses of EIA have opened a general question concerning the scope of a Platonic argument, namely, does it establish the existence of all or some mathematical objects? A precise formulation of a Platonic argument ought to include the determination of the set of objects to which it applies. If an argument guarantees the existence of some but not all mathematical objects, it can only justify a partial or fragmentary Platonism.

Suppose that A1, A2,…, An are all known instances of EI-type arguments (the type of explanatory indispensable arguments) at the moment m in the history of empirical science. Let these arguments guarantee the existence of mathematical objects in sets S1, S2,…, Sn, respectively. If M is a set of all mathematical objects, then either
$$\cup_{1 \le i \le n} S_{i } = M,\;{\text{or}}\;{\text{not}} .$$

In the first case, \(\cup_{1 \le i \le n} S_{i } = M\), each mathematical object plays an indispensable explanatory role in some most preferred empirical theory. The two doubtful claims follow: at the moment m the complete mathematical knowledge has been attained, and the pure mathematical theory is fictitious.

In the second case, \(\cup_{1 \le i \le n} S_{i } \ne M\), the justification is provided not for a full, but only for a partial Platonism. In addition to this, an unexpected proposition receives the status of the obligatory belief. It becomes obligatory to believe that the domain of mathematical objects is mutable. The history of ideas shows that an empirical theory most preferred at some time can later lose this status. Suppose that a mathematically explained phenomenon p receives a new and better mathematical explanation. Further, suppose that mathematical objects o1,…, on are used in explaining p at the moment m and not used in explaining any other phenomena at any time. Later, at the moment m′ a new and better mathematical explanation of the phenomenon p is found, using objects o1,…, om, each of which is different from any of previously used objects o1,…, on. If there are no other Platonic arguments besides EI-type arguments, then at the moment m′ there will be no sufficient reason to believe in the existence of objects o1,…, on, and hence this belief will cease to be rationally required. The rationality requires truthful beliefs, OBφ → φ. By EIA, it is rationally required at m to believe in the existence of objects o1,…, on. By the requirement of truthfulness, objects o1,…, on exist. If EI type arguments are the sole normative source of Platonistic conclusions, then at m′ it is not rationally required to believe in the existence of existing objects o1,…, on. Hence, the only way to preserve the coherence of beliefs is to accept that the Platonic domain is mutable.

3 From de re Premise to de dicto Conclusion

The examination of the logical form of EIA shows that either it is not valid or is unsound or does not support the intended Platonic interpretation of the conclusion. The source of the problem lies in the failure to give proper attention to the de rede dicto disambiguation.

Let Bx stand for the predicate ‘x is a best scientific theory’, Rxy for ‘x plays an indispensable explanatory role in y’, Mx for ‘x is a mathematical term’, and let ref (x) stand for the functional term ‘the (mind—and language—independent) referent of x’. Let the quantification be many-sorted with variables t, …ranging over terms, s, …over scientific theories, and x, …over all objects. Suppose that the major premise and the conclusion are de dicto statements. The logical form of the major premise is given by the following formula:
$$\left( {{\text{M}} - {\text{dd}}} \right) {\text{OB}}\left( {\forall t\forall s\left( {(Bs \wedge Rts) \to \exists x\,x = ref\left( t \right)} \right)} \right).$$
The form of the conclusion is given by:
$$\left( {{\text{C}} - {\text{dd}}} \right){\text{OB}}\left( {\exists t\left( {Mt \wedge \exists x\,x = ref\left( t \right)} \right)} \right).$$
The normativity of rationality is either internal or external: the internal rationality requires consistency and coherence, while the external rationality requires truthfulness. The minor premise is not a requirement of rationality, not a doxastic obligation, but a factual claim. Therefore, an additional deontic premise is needed for the deduction of (C − dd). Suppose that EIA is an enthymeme to be completed with an innocuous and obvious premise. The missing premise would be the requirement of external rationality, the requirement to believe all and any truths, φ → OB(φ). This is an impossible requirement and it must be rejected on the grounds of ought implies can principle. Therefore, in the de dicto interpretation of the major premise and conclusion, the EIA is invalid if taken as it stands and unsound if taken as an enthymeme.
Suppose that the major premise is a de re statement,
$$\left( {{\text{M}} - {\text{dr}}} \right) \forall t\forall s\left( {\left( {Bs \wedge Rts} \right) \to {\text{OB}}\left( {\exists x\;x = ref\left( t \right)} \right)} \right).$$
In this case, the de re conclusion does follow:
$$\left( {{\text{C}} - {\text{dr}}} \right)\;\exists t\left( {Mt \wedge OB\left( {\exists x\;x = ref\left( t \right)} \right)} \right).$$
Nevertheless, (C − dr) does not have the intended Platonic reading. The intended Platonic conclusion is (C − dd), a de dicto statement, i.e. a statement where no free variable occurs within the scope of modal operators. The Platonic conclusion (C − dd) does not follow from (M − dr) with (2.a). The conclusion which does follow is ‘there is a mathematical term of which rationality requires the belief in the existence of its referent’. This is different from and does not imply the Platonic de dicto belief with the content ‘there is a mathematical term and its referent exists’. It is coherent to believe of a mathematical term that its referent exists and still not believe that this term is a mathematical one.

The remaining two interpretative variants—from (M − dd) with (2.a) to (C − dr), and from (M − dr) with (2.a) to (C − dd)—are non sequiturs since, in the first case, the de dicto does not imply the de re statement, while the second case is similar to the variant discussed in the preceding paragraph.

This paper aimed at discussing some of the doubts which arise when it comes to EIA. We have attempted to point to unsolvable difficulties that are reasons why EIA cannot be used as the argument by means of which the Platonists can defend their viewpoint. The first difficulty, as we have shown, refers to the imprecision in the formulation of the minor premise which renders the understanding of the minor premise, as well as the entire EIA, ambiguous. Whichever of the two possible interpretations of the minor premise we accept, we inevitably encounter new problems that make EIA unacceptable. On the one hand, as we have seen, it is not clear how the truthfulness of minor premise, if understood as universal statement, can be proved. On the other hand, if we try to make some sort of correction of the formulation of minor premise, understanding it as a particular statement, we will be forced to speak about some kind of partial Platonism, which is hardly acceptable to any mathematician or philosopher. Even if we allow for such a Platonism under some hypothetical conditions, we would see that with the two remaining statements of EIA—the major premise and the conclusion—understood in either de re or de dicto sense, we cannot reach a correct logical form of the EIA.

Footnotes

  1. 1.

    A similar interpretation is given in Colyvan (2001).

  2. 2.

    Cf. Baker (2005, 2009, 2012).

  3. 3.

    Using (E-DISP) as the model, the collective type of indispensability can be defined as follows: (C-E-INDISP) A set c of mathematical entities is explanatorily indispensable to a scientific theory T iff for any theory T* which does not employ the vocabulary of any mathematical theory in which entities from c are defined, it holds that T* either has strictly lesser explanatory power than T or is not empirically equivalent to T.

  4. 4.

    Colyvan’s (2001, 77): “An entity is dispensable to a theory iff the following two conditions hold: (1) There exists a modification of the theory in question resulting in a second theory with exactly the same observational consequences as the first, in which the entity in question is neither mentioned nor predicted. (2) The second theory must be preferable to the first.” The non-functional type of indispensability can be obtained from this definition by interpreting ‘…is not preferable to…’ as ‘…is as either as good as or worse then…’, what gives the following definiens: ‘any modification of the theory in question resulting in a second theory with exactly the same observational consequences as the first, in which the entity in question is neither mentioned nor predicted, satisfies the condition that the second theory is not preferable to the first’.

Notes

Acknowledgements

Even though it is not common for an author to express gratitude to the co-author, it is however necessary to do so under present circumstances. Namely, Berislav Žarnić, professor at the University of Split, passed away on 25th May 2017, at the time when our joint paper was being under review of the Journal for General Philosophy of Science. Being his co-author, colleague and an old friend, I owe him a profound gratitude for the effort he invested in this work, feeling no less profound regret for the fact that he did not see it published.

References

  1. Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.CrossRefGoogle Scholar
  2. Baker, A. (2009). Mathematical explanation in science. The British Journal for the Philosophy of Science, 60, 611–633.CrossRefGoogle Scholar
  3. Baker, A. (2012). Science-driven mathematical explanation. Mind, 121, 243–267.CrossRefGoogle Scholar
  4. Colyvan, M. (2001). The indispensability of mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
  5. Molinini, D. (2016). Evidence, explanation and enhanced indispensability. Synthese, 193, 403–422.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Philosophy, Faculty of PhilosophyUniversity of MontenegroNikšićMontenegro
  2. 2.Research Center for Logic, Epistemology and Philosophy of Science, Faculty of Humanities and Social SciencesUniversity of SplitSplitCroatia
  3. 3.Filozofski fakultetSplitCroatia

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