Mathematics is About the World: How Ayn Rand's Theory of Concepts Unlocks the False Alternatives Between Plato's Mathematical Universe and Hilbert's Game of Symbols by Knapp Robert

Author:Knapp, Robert [Knapp, Robert]
Language: eng
Format: epub
Published: 2014-12-02T22:00:00+00:00

Final Comments

Dedekind’s work, and Cantor’s use of Cauchy sequences, properly interpreted, exhibited the relationship of irrational numbers to rational numbers and gave meaning to the operations of arithmetic as they apply to irrational numbers. Restored, over Dedekind’s dead body, to its proper referential context, Dedekind cuts, in particular provide a systematic way, a modern implementation of the fundamental insights of Eudoxus, to utilize rational numbers to characterize irrational numbers and to establish the domain of real numbers as comprising both rational numbers and irrational numbers.55

Cauchy’s earlier work had established the definitions of limits and continuity that were later formalized in the well-known “epsilon-delta” approach to continuity and limits. Cauchy’s work completes the theory of approximation by establishing the proper definition of a limit: One says, for example, that a function y = f(x) has a limit y = y0 at x = x0 if, no matter how close one needs to approximate y0, one can guarantee the required precision by selecting an x sufficiently close to x0.56These insights are essential and fundamental. But they are not sufficient to understand the way that limiting processes relate to the world that they measure. So they left a gap, a gap that Dedekind and Cantor rushed to fill later in the century.

Properly interpreted, both Dedekind cuts and Cauchy sequences provide valuable tools. As I have shown in my positive treatment, Cauchy sequences, in particular, provide infinite mathematical precision by providing a system of approximations, a system that presupposes that all specific precision requirements are finite, but provides a contingency for any specific precision requirement that might ever arise.

At this point in the discussion, we still stand at the threshold of mathematical abstraction. But as one proceeds to study higher mathematics, the same issues resurface repeatedly. For example to solve differential equations one needs to extend the theory of approximation to include mathematical functions. Yet the essential principles and required understandings are encountered at the beginnings of the subject. The principles, though their application may require specialized knowledge, remain the same.

1 Jeremy Gray, Plato’s Ghost the Modernist Transformation of Mathematics, 2008, Princeton, Princeton University Press, Apparently Newton held this view: On page 134, Gray refers, in passing, to the “Newtonian view that the real numbers were ratios of quantities.” See also: Penelope Maddy, Realism in Mathematics (Oxford, Clarendon Paperbacks, 1992), p 89, for an alternative viewpoint from a “set theoretical realist” perspective: “Knowledge of numbers is knowledge of sets, because numbers are properties of sets.”

2 As a young student, I recall the explanation that 2/7 + 3/7 = 5/7 because two of something plus three of something is five of something. For a discussion from a

similar perspective, see Ronald Pisauturo and Glenn D. Marcus, The Intellectual Activist, September 1994, Vol 8, No 5, “The Foundation of Mathematics, Part II”, p 10.

3 Aristotle, Physics, edited by Jonathan Barnes, Revised Oxford Translation (Princeton University Press, 1984), Book III, Section 6, at 206b, lines 12–13

4 Ayn Rand, Introduction to Objectivist Epistemology Expanded Second Edition, April 1979, paperback

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