28 Nov 2008

Limits and Numerical Value

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Benjamin Robins realized that the varying quantity of the infinitesimal does not need to be considered as finally reaching the fixed quantity as its final value, although this last value "'is considered as the quantity to which the varying quantity will at last or ultimately become equal'" (Boyer quoting Robins 230d).

Robins' conception of the limit drew the criticism that like Achilles and the Tortoise, the value will never overtake its target. But this objection confuses physical distance with numerical value.

The question as to whether the variable Sn reaches the limit S is furthermore entirely irrelevant and ambiguous, unless we know what we mean by reaching a value and how the terms "limit" and "number" are defined independently of the idea of reaching. Definitions of number, as given by several later mathematicians, make the limit of an infinite sequence identical with the sequence itself. Under this view, the question as to whether the variable reaches its limit is without logical meaning. Thus the infinite sequence .9, .99, .999 . . . is the number one, and the question, "Does it ever reach one?" is an attempt to give a metaphysical argument which shall satisfy intuition. Robins could hardly have had such a sophisticated view of the matter, but he apparently realized . . . that any attempt to let a variable "reach" a limit would involve one in the discussion as to the nature of 0/0. Thus he is hardly to be criticized for his restriction.

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.