Abstract
The paper begins with an exposition of Aristotle’s own philosophy of mathematics. It is claimed that this is based on two postulates. The first is the embodiment postulate, which states that mathematical objects exist not in a separate world, but embodied in the material world. The second is that infinity is always potential and never actual. It is argued that Aristotle’s philosophy gave an adequate account of ancient Greek mathematics; but that his second postulate does not apply to modern mathematics, which assumes the existence of the actual infinite. However, it is claimed that the embodiment postulate does still hold in contemporary mathematics, and this is argued in detail by considering the natural numbers and the sets of ZFC.
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Notes
- 1.
Actually reductionism is only one form of nominalism, which is the general view that mathematical entities do not exist. Nominalism is defended in Azzouni [3]. It is often connected to the so-called fictionalist view according to which mathematics is similar to literary fictions such as Shakespeare’s Hamlet. Azzouni discusses this view in his 2015, and it is also discussed in section 7 of the present paper.
- 2.
Andrew Gregory stressed the importance of the approximation problem to me in conversation. It is one of the factors, which have led him to prefer Platonism to Aristotelianism regarding abstract entities. The approximation problem is also mentioned by Azzouni in section 2 of his 2015. Azzouni regards it as a strong argument in favour of nominalism.
- 3.
This was pointed out to me by Silvio Maracchia and Anne Newstead. See Maracchia [25].
- 4.
Exactly when the actual infinity entered mathematics in a significant way is an interesting but difficult historical question. The years between 1500 and 1800 saw the rise of algebra, analytic geometry and calculus. Did these developments introduce the actual infinite? Ladislav Kvasz has suggested plausibly (personal communication) that the actual infinite is already in Descartes. Jeremy Gray’s contribution to the present volume has some points relevant to this question. Gray says [20]: “At no stage did Apollonius, or any other Greek geometer, generalise a construction by speaking of points at infinity”. However, he goes on to show that points at infinity were introduced by Desargues in 1639. Perhaps this could be considered as an example of the actual infinity, though this is not clear. In what follows, however, I will not discuss this question in detail, but limit myself to the developments from the 1860s, which gave rise to contemporary mathematics.
- 5.
The change from the potential to the actual infinite is also discussed in Stillwell [29], particularly section 3. Stillwell’s chapter also gives more details about the set theoretic matters which are discussed briefly in the remainder of this section.
- 6.
For a detailed and informative account of the similarities and differences between Aristotle’s theory of the continuum, and those of modern mathematics, see Newstead [26].
- 7.
Cf. Wittgenstein [30]: “The world is the totality of facts, not of things”.
- 8.
What follows is my own attempt to answer Chihara’s objection. Maddy’s rather different answer to the objection is to be found on pp 150–154 of her 1990 book which gives an overall account of her views on the philosophy of mathematics at that time. It is interesting that she describes her position as [24, p. 158]: “more Aristotelian than Platonistic”.
- 9.
In his [17], Franklin discusses how to deal with “large infinite numbers” from an Aristotelian point of view. His approach is somewhat different from the one developed here.
- 10.
The point of view of this paragraph is rather similar to that of Avigad in his 2015. Avigad writes [2]: “What I am advocating is a view of mathematics as a linguistic artefact, something we have designed, and continue to design, to help us get by in the world”.
- 11.
I am here assuming Frege’s view that if a referring expression in a sentence lacks reference, then the proposition expressed by that sentence lacks a truth value.
- 12.
I had a discussion with Lakatos on this point in the late 1960s when I was doing my PhD with him. Lakatos thought that the continuum hypothesis would be decided one day, and, when I asked him why, he replied that this was because of the growth of mathematics. Assuming human civilisation continues, mathematics will undoubtedly continue to grow and to develop new concepts, which will be successfully applied to the material world. However, the direction of this future development is uncertain, and it may not consist of further development and successful application of the theory of transfinite cardinals, but rather of the development and successful application of some completely new concepts and theories. Thus the continuum hypothesis may never be decided.
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Acknowledgements
I read an earlier draft of this paper at a seminar in La Sapienza, Roma on 16 February 2015, and I am very grateful for the comments I received on that occasion. Particularly useful were some comments from three experts on Aristotle (Silvio Maracchia, Diana Quarantotto, and Monica Ugaglia) who helped me with a number of points. After the seminar, I had a very useful discussion with Carlo Cellucci concerned mainly with the problem of applying the Aristotelian approach to ZFC, and the possible use of ideas of fictionalism and if, then-ism. This led to many improvements in section 7 of the paper. In addition I was fortunate to receive extensive comments from quite a number of people to whom I sent a copy of an earlier draft. These included: Ernie Davis, James Franklin, Andrew Gregory, Ladislav Kvasz, Penelope Maddy, Anne Newstead, Alex Paseau, and Brian Simboli. Their input was very helpful in preparing the final version of the paper.
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Gillies, D. (2015). An Aristotelian approach to mathematical ontology. In: Davis, E., Davis, P. (eds) Mathematics, Substance and Surmise. Springer, Cham. https://doi.org/10.1007/978-3-319-21473-3_8
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