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.
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Notes
A similar interpretation is given in Colyvan (2001).
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.
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’.
References
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.
Baker, A. (2009). Mathematical explanation in science. The British Journal for the Philosophy of Science, 60, 611–633.
Baker, A. (2012). Science-driven mathematical explanation. Mind, 121, 243–267.
Colyvan, M. (2001). The indispensability of mathematics. Oxford: Oxford University Press.
Molinini, D. (2016). Evidence, explanation and enhanced indispensability. Synthese, 193, 403–422.
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.
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Drekalović, V., Žarnić, B. Which Mathematical Objects are Referred to by the Enhanced Indispensability Argument?. J Gen Philos Sci 49, 121–126 (2018). https://doi.org/10.1007/s10838-017-9381-0
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DOI: https://doi.org/10.1007/s10838-017-9381-0