Mathematical existence: does it really matter?

The existence of mathematical entities is one of the main topic of philosophy of math in the twentieth century.

Carlo Cellucci explicitly questions the topic in his "Indiscrete Variations on Gian-Carlo Rota’s Themes" (in: E. Damiani, O. D’Antona, V. Marra, F. Palombi (eds.), From Combinatorics to Philosophy. The Legacy of G.-C. Rota, Springer, New York, 2009.). Cellucci notes that there are four possible answers to this question:

(i) Mathematical objects exist;
(ii) they do not;
(iii) We don’t know;
(iv) The question is irrelevant to mathematical practice or meaningless.

Cellucci states that (iv) is a very sensible answer (it was given in XVII and XVIII century by many thinkers - e.g. Descartes, Locke, Hume).

Also G.C. Rota reply to this question according to (iv). As cellucci puts it, "he states that it does not matter whether mathematical items exist, and probably it makes little sense to ask the question. If someone proved beyond any reasonable doubt that mathematical items did not exist, that would not affect the truth of any mathematical statement. Discussions concerning the existence of mathematical objects are motivated by deep-seated emotional cravings for permanence which are of psychiatric rather than philosophical interest".

For Cellucci moves beyond (iv) and argues for the thesis that mathematical objects are 'hypotheses'. In particular, Cellucci states that "a more satisfactory account of the nature of mathematical objects can be given stating that mathematical objects are hypotheses tentatively introduced to solve mathematical problems. A mathematical object is the hypothesis that a certain condition is satisfable" and "if, in the course of reasoning, the condition turns out to be satisfable, we say that the object `exists', if it turns out to be unsatisfable, we say that it `does not exist'. Thus speaking of `existence' is just a metaphor". Therefore, "there is no more to mathematical existence than the fact that mathematical objects are hypotheses tentatively introduced to solve mathematical problems. Such hypotheses are in turn a problem to be solved, it will be solved by introducing other hypotheses, and so on. Thus solving a mathematical problem is a potentially infinite task".
Cellucci's thesis is explicitely related to Plato's view (Plato, Republic, VI 510 c 2-5), according to which mathematicains hypothesize mathematical objects and they also hypothesize properties of such objects. "Therefore mathematical hypotheses concern either mathematical objects or their properties".
For, hypotheses characterize identity of mathematical entities (and "identity is characterized by the hypothesis, and characterized differently by different hypotheses") but say nothing about their existence. In turn, as Cellucci stresses, hypotheses don't characterize the identity of mathematical items completely and conclusively but only partially and provisionally: their identity is always open to new determinations. It's so because mathematical objects are open to interactions with other objects, from which new properties may arise.
Moreover Cellucci notes that 'hypotheses' in this sense are different from fictions:
1) fictions are explicitly not real, hypotheses are aimed at reality. "They are meant to provide an adequate approach to a still unknown or not perfectly known reality, although, as in the case of proofs by reductio ad absurdum, they eventually may fail to provide an adequate approach. The determination of reality given by a hypothesis is provisional, but one can try to make it stable showing that the hypothesis is plausible, that is, compatible with the facts of experience".
2) fictions are merely thinkable, hypotheses are supposed to be possible: "they must agree with the facts of experience. Only then hypotheses can be said to provide an adequate approach to reality. On the contrary, fictions are not supposed to be possible, and hence are not rejected if they do not agree with the facts of experience".

So, according to this view, hypotheses provide an only partial and provisional characterization of mathematical objects and new determinations are possible through interactions between hypotheses and experience: "from the interactions of the given mathematical objects with other objects, new facts may emerge which may suggest to modify or completely replace the hypothesis by which the identity of the given mathematical objects had been characterized. This is a potentially endless process, for mathematical objects are always open to interactions with other objects".

One of the most stimulating points of Cellucci's view is the fact that if mathematical items are hypotheses, is there a difference between mathematical and physical entities? Or between mathematical and certain chemical items?

'Mavericks' vs 'foundationalists' and the search for a third way into philosophy of mathematics

Paolo Mancosu, in his recent work 'The philosophy of mathematical practice' (OUP, 2008), raises - pp. 3-7 - a stimulating issue about philosophy of mathematics, that is the emergence of two different traditions in philosophy of math, namely the 'maverick' tradition and the 'foundationalist' tradition.

One of the most interesting aspects of Mancosu's analysis is the fact that he seems to aim at constructing a 'third way' into philosophy of math (pp. 18-20), trying to see into its future.
Mancosu characterizes the two traditions in the following way:

- 'Foundationalist'
The analytic philosophy of mathematics

- 'Maverick' (according to Mancosu: Lakatos, Aspray, Kitcher, Gillies, Cellucci, Hersh, Tymoczko, Davis, Kline, Grosholz, Breger, van Kerkhove, van Bengedem, Krieger, Corfield, Ferreiros, Gray):
a) anti-foundationalism, i.e. there is no certain foundation for mathematics; mathematics is a fallible activity;
b) anti-logicism, i.e. mathematical logic cannot provide the tools for an adequate analysis of mathematics and its development;
c) attention to mathematical practice: only detailed analysis and reconstruction of large and significant parts of mathematical practice can provide a philosophy of mathematics worth its name.

On one side, according to Mancosu, "there is no question that the ‘mavericks’ have managed to extend the boundaries of philosophy of mathematics", but is seems that they "were throwing away the baby with the bathwater"(p. 6) and that, in the end, they have "not managed to substantially redirect the course of philosophy of mathematics" (p. 5).
On the other side, "the aspects of mathematical practice we investigate are absolutely vital to an understanding of mathematics" and "having ignored them has drastically impoverished analytic philosophy of mathematicshas", but this does not mean "that we think that the achievements of this tradition should be discarded or ignored as being irrelevant to philosophy of mathematics. [...] We do not dismiss the analytic tradition in philosophy of mathematics but rather seek to extend its tools to a variety of areas that have been, by and large, ignored" (p. 18).
Therefore, Mancosu seems to aim at moving beyond these two traditions, and tries to construct a 'third way' which, as he puts it, is "calling for an extension to a philosophy of mathematics that will be able to address topics that the foundationalist tradition has ignored" (p. 18) .
The starting point of this new direction into philosophy of math suggested by Mancosu is not "to make any invidious comparisons but only to provide a fair account of what the previous traditions have achieved and why we think we have achieved something worth proposing to the reader. We are only too aware that we are making the first steps in a very difficult area and we hope that our efforts might stimulate others to do better" (pp. 19-20).
So, roughly, that's the way the philosophy of mathematics of the future, according to Mancosu, should follow.