Sunday, June 24, 2018

Dedekind on ontology

[A footnote to this essay.]

Footnote re the irrationals:

Dedekind stressed the distinction of category  between cut and number  in 1888; against the view of his friend Heinrich Weber  that “the irrational number is nothing other than the cut itself”, he explained that “as I prefer it, to create something New distinct from the cut, to which the cut corresponds.  We have the right to grant ourselves such power of creation”,  and cuts corresponding to both rational and irrational numbers were examples.
-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 87

A seemingly slight, even pedantic distinction;  but like many another such, it might have its point.   Cf. my astonished delight in junior high-school, upon meeting the distinction between  x (the thing itself) and ‘x’ (the name of x) -- already adequately foreshadowed in Alice in Wonderland, but encountered now in a new context.  Likewise the difference between  x and {x} (the singleton-set of x).

In the case of an algebraic number like √2, a simple number staring you in the face out of a hypotenuse  versus the infinite train of rational pilgrims (never quite arriving at their destination) of a Dedekind cut,  one is reminded of the variety of definitions of something so familiar as a tangent:  the slope of a curve (at a point); the closest linear approximation to the curve (at that point); versus the distressing definition in Loomis & Sternberg as an infinite equivalence-class of curves (through that point).

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