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Jean Paul Van Bendegem NON-REALISM, NOMINALISM AND Jean Paul Van Bendegem NON-REALISM, NOMINALISM AND STRICT FINITISM THE SHEER COMPLEXITY OF IT ALL 1. Introduction The contributions to this volume are meant to be critical appraisals of the book by Philip Hugly and Charles Sayward. This does not exclude, however, to say a few things first about those points where I fully agree with the authors: (a) “What makes for the sense of a serious issue is, then, some kind of thinking which takes place outside of mathematics – some non- mathematical [the authors’s emphasis] thinking. What makes for the sense of a serious issue is philosophical (idem) thinking” (p. 134). It produces a rather good feeling to see two philosophers, well acquainted with logical and mathematical thinking, to claim so resolutely that mathematics on its own, what we are used to call ‘pure’ mathematics, is not capable on its own to resolve deep philosophical issues, such as whether or not numbers exist; in their own words, “whether numerical expressions are terms of reference is a key question (perhaps the key question) in the philosophy of mathematics” (p. 282). As I will emphasize further on in this paper, I share this view that I prefer to formulate in these words: it requires arguments and/or proofs to show that mathematics on its own can decide a philosophical issue or, to put it otherwise, the burden of proof is on those thinkers that claim that pure mathematics is ontologically committal.1 (b) One of my basic philosophical beliefs or attitudes is the dictum: “the weaker your ontology, the better your worldview.” I like to consider myself as an ontological minimalist – just put into the world the stuff you 1 I have to add straight away that the question of the burden of proof is a very difficult and tricky problem. The claim that the burden of proof is on this or that person is therefore precisely what it means: a claim that itself requires argumentative support In: Philip Hugly and Charles Sayward, Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic, edited by Pieranna Garavaso (Poznań Studies in the Philosophy of the Science and the Humanities, vol. 90), pp. 343-365. Amsterdam/New York, NY: Rodopi, 2006. 344 Jean Paul Van Bendegem “really” need and leave out all other things – and therefore any view about mathematical objects that rejects Platonism or some similar ontological approach sounds good if not actually right to me. However, and I will return to this point later, I am somewhat reluctant about Hugly’s and Sayward’s rejection of nominalism. So, thesis one – “In extra-mathematical statements of number, arithmetical expressions do not function as referential expressions” (p. 41) – is a thesis I have doubts about, as I will try to show. Ontological minimalism might prove to be compatible, as I believe, with certain forms of nominalism. (c) Related to the previous points, I sympathize with the authors’ view that applied mathematics comes first, so to speak, in comparison with pure mathematics. In their words: “our calculations result in statements” (p. 36). I equally side with them when they remark that the distinction between numbers and numerals is an important one and that numerals come first2: “for that person [i.e., the non-realist] clarity in the philosophy of mathematics is best achieved by seeing how the signs [my emphasis] of mathematics are actually used outside of mathematics” (p. 207). Furthermore it also leads to a very nice answer to both the success of mathematics in the “real” world (pp. 235-240) and how to solve the problem of the necessity of mathematical knowledge (and us having access to that knowledge). (d) Quite apart from the previous considerations, I would like to remark that at first I was somewhat amazed and, after reading the chapter, deeply impressed to find a serious discussion of Wittgenstein’s view on mathematics as it is to be found in the Tractatus (thus not the Wittgenstein of the Remarks on the Foundations of Mathematics). Frankly, it was my belief (up to now, of course) that this view was so exceptional and difficult to defend that hardly any philosopher would still be interested to deal with it. What Hugly and Sayward have done here is, at least, to show how to read the early Wittgenstein in such a way that it becomes a real contender for a philosophy of mathematics (and definitely a non-realist one at that). These introductory, positive comments make clear that I will not quarrel with the authors about thesis two: “ The meanings of the signs of arithmetic are fixed not merely by their systematic interconnections as constructed in that theory, but only by that combined with their application in extra-mathematical statements of number” (p. 41). In fact, the idea of the systematicity, which is mentioned in this quote, as an 2 Although I have to add that perhaps here I am interpreting Hugly and Sayward a little bit too much in the direction of my favorite views, probably more strongly than they themselves would be willing to do. .
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