Infinity and Concept-Determination

Abstract

This chapter takes up a particular important aspect of Wittgenstein’s philosophy of mathematics; he highlights the difference between what he calls ‘the determination of a concept’ and ‘the discovery of a fact of nature’. Wittgenstein himself brings this distinction to bear, amongst other topics, on investigations of the notion of infinity in mathematics and philosophy; I stress its importance for obtaining a clear overview of the historical development of mathematical ideas in general and of numbers in particular. Beginning with the distinction in question (concept-determination vs. fact-discovery), the chapter first clarifies matters by looking at how Wittgenstein used it in his treatment of Freudian notions in the philosophy of mind. Wittgenstein introduces us to the strange notion of ‘painless toothache’. How can we at all make sense of such an apparently nonsensical idea? … The answer is surprisingly relevant for our understanding of both psychology and numbers, as this chapter makes clear. Returning to the main thread and looking at the notion of infinity, the chapter then investigates two specific ways this latter notion has been treated historically as part of mathematics. One of these ways, due to Georg Cantor, is relatively well-known, as perhaps is Wittgenstein’s own criticism of it. The other is less well-known; we might describe it as ‘infinity as-∞’ as opposed to (Cantor’s) ‘infinity-as- \(\aleph\)’. It was developed in the seventeenth century by the mathematician John Wallis. Infinity-as-∞, we might say, did not ‘catch on’ with the mathematical community in the way Cantor’s infinity-as- \(\aleph\) certainly has (in spite of Wittgenstein’s strictures, we should perhaps add). Such ‘catching on’ (or not) is important in the overall development of mathematics over the centuries and millennia, and the chapter goes on to highlight this importance in general and by means of examples. The first example taken up is that of the ancient Greek determination/discovery of incommensurable or irrational numbers, a cause celebre for intellectual thought at the time. The story is well-known to historians of ideas; its relevance to Wittgenstein’s thought and the philosophy of mathematics itself is brought to the fore here. The story then moves to a consideration—deriving from ‘irrational numbers’—of Wittgenstein’s concerns with what we might call ‘infinity in the small’; here I offer some exegetical remarks about some puzzling things Wittgenstein says about ‘Dedekind’s Theorem’, a contemporary foundation for real numbers in present-day mathematical analysis. The common thread here, once again, is Wittgenstein’s distinction adumbrated above between concept-determination and fact-discovery. The importance of this distinction gets a final emphasis, and the history of number development a satisfying coda, by means of another example: that of the development of the infinitesimal calculus. Again, some of this story is well-known to historians of mathematics: apparent logical difficulties with Newton’s and Leibniz’s development of the calculus (highlighted in particular by Bishop George Berkeley) were sorted-out in the nineteenth century by such mathematicians as Cauchy, Bolzano and Weierstrass. Or so the conventional wisdom would have it. However, there is another way of telling this story, as I explain to end the chapter. Non-Standard Analysis, developed by the logician Abraham Robinson in the second half of the twentieth century, effectively (as Robinson himself points out) re-establishes Leibnizian ideas and enables another way of telling the story of conceptual development, one that itself goes a long way towards vindicating the importance of Wittgenstein’s distinction for our view of mathematics. So by the end of Chap. 3, we have a clear sight of the philosophical problem of mathematical applicability as well as a mix of historical underpinning and conceptual methodology in place enabling the overview I claim to be necessary for its solution. The details of history and philosophy are important; this is an aspect that does not come across well in summary, but mentioning it underlines how what we are striving for—an overview of the philosophical terrain—comes about. Before applying the overview, however, the book extends it further by looking at, and dealing with, what might be thought—has been thought—a serious objection to Wittgenstein’s philosophy of mathematics in general: This comes in Chap. 4. Chapter 3 argues for due importance to be given to the notion of mathematics as proceeding via determination of concepts rather than common-or-garden discovery of facts. This connects with what was highlighted earlier in Chap. 2 as something shared by Field and Wittgenstein—the idea that mathematics is to do more with rules of inference than with assertions of mathematical truths. The difficulty, now to be dealt with in Chap. 4, lies in the way such a focus away from discovery might seem to lead to some kind of (unacceptable) conventionalism about mathematics and logic.