Version: v1, Published online: 1998
Retrieved December 05, 2020, from https://www.rep.routledge.com/articles/thematic/logicism/v-1
2. Connection with developments in arithmetic, algebra and analysis
The thesis that mathematics is a part of logic could not have been maintained before logic had undergone the deep transformation which took place in the late nineteenth century, most especially through the work of Frege. Before that time, actual mathematical reasoning could not be carried out under the recognized logical forms of argument. The new logic, however, made it possible to represent standard mathematical reasoning in the form of purely logical derivations. It is equally true that no such claim could have been made except for a deep transformation that had taken place in mathematics itself during the nineteenth century; and a brief consideration of this may help to shed some light on a plausible way to construe the logicist thesis.
From the time of Aristotle, it had been usual to regard mathematics as concerned with ‘discrete’ and ‘continuous’ quantity: whole numbers on the one hand; continuous magnitudes on the other. In the seventeenth and early eighteenth centuries it was generally supposed that our knowledge of mathematical truths was exclusively concerned with ‘relations of ideas’, and that – even if these ideas were obtained by some process of abstraction from empirical things’ – this character explained how it is that the knowledge in question is certain and independent of experienced ‘facts’ (such a view was common, for instance, to the rationalist Leibniz and the empiricist Hume). Kant had presented arguments to subvert this position: he argued, rather convincingly, not just that logic (as it then existed) was insufficient to warrant the reasoning of mathematicians, but also that such reasoning demanded attention to more than ‘ideas’, or (rather) concepts – that a kind of mental visualization of the objects falling under mathematical concepts was an indispensable ingredient.
Among mathematicians – although within a fairly small circle – confidence in either the Kantian or the pre-Kantian traditional view of the nature of mathematical knowledge was disturbed by the development, in the early nineteenth century, of new forms of geometry: these seemed to show that neither mere consideration of ‘relations of ideas’ (that is, of such concepts as ‘point’, ‘line’, ‘distance’ and so on), nor the appeal to ‘intuition’, could decide the truth of propositions of geometry (see Geometry, philosophical issues in). At the same time, attempts were begun to work out new foundations for analysis – that is, the theory of the real (and complex) numbers, functions thereof, and limiting processes – that should be independent of any ‘extrinsic’ appeal to an ‘intuition’ of continuous magnitude. Again, in parallel (and partly in connection) with the preceding, systematic procedures were developed for the introduction of ‘new objects’ – or ‘ideal objects’ – in several branches of mathematics: for example, ‘infinitely distant points’ and ‘points with imaginary coordinates’ in geometry; real and imaginary numbers (alongside the rational numbers) in analysis; and, perhaps most decisively for our subject, ‘ideal factors’ of algebraic integers in the newly developing subject of algebraic number theory. In this last context in particular, Dedekind achieved fundamental new results with the help of concepts that involved the treatment of classes of (already introduced) objects as new objects (‘modules’, ‘ideals’, ‘orders’, ‘number fields’, among others) which could themselves be made subject both to general mathematical investigation and to actual calculation.
With all of this work in the background, Dedekind was led to examine the foundations of the theory of the whole numbers themselves; and he arrived at the conviction that this theory, together with algebra and analysis which are further derived from it, is ‘a part of logic’ – in saying which, he tells us, ‘I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought’ ( 1901: 31; emphasis added). The way he develops his theory allows us to conclude that he regards both the concept of number and the theorems of arithmetic, as well as the concepts and theorems of the other mathematical theories he mentions, as resulting from certain fundamental abilities of the mind: the ability to form classes (his word is ‘systems’) of objects of thought (which themselves then have the status of such objects); to correlate or map such systems onto one another; and to ‘create’ new objects to ‘represent’ ones already present (‘to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing’); and he tells us that without this ability, ‘no thinking is possible’.
This version of the logicist thesis – namely, that mathematics (or that part of it which is claimed to belong to logic) requires no other source, either of its concepts or of its knowledge, than what is presupposed by all systematic thought – may be taken as applicable to the doctrines of all who have made that claim, whatever epistemological or metaphysical differences may attach to their further elaboration of what those ‘presuppositions of all thought’ might be.
Stein, Howard. Connection with developments in arithmetic, algebra and analysis. Logicism, 1998, doi:10.4324/9780415249126-Y061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/logicism/v-1/sections/connection-with-developments-in-arithmetic-algebra-and-analysis.
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