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Books > Science & Mathematics > Mathematics > Philosophy of mathematics
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The Logic of Number (Hardcover)
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The Logic of Number (Hardcover)
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In The Logic of Number, Neil Tennant defines and develops his
Natural Logicist account of the foundations of the natural,
rational, and real numbers. Based on the logical system free Core
Logic, the central method is to formulate rules of natural
deduction governing variable-binding number-abstraction operators
and other logico-mathematical expressions such as zero and
successor. These enable 'single-barreled' abstraction, in contrast
with the 'double-barreled' abstraction effected by principles such
as Frege's Basic Law V, or Hume's Principle. Natural Logicism
imposes upon its account of the numbers four conditions of
adequacy: First, one must show how it is that the various kinds of
number are applicable in our wider thought and talk about the
world. This is achieved by deriving all instances of three
respective schemas: Schema N for the naturals, Schema Q for the
rationals, and Schema R for the reals. These provide
truth-conditions for statements deploying terms referring to
numbers of the kind in question. Second, one must show how it is
that the naturals sit among the rationals as themselves again, and
the rationals likewise among the reals. Third, one should reveal
enough of the metaphysical nature of the numbers to be able to
derive the mathematician's basic laws governing them. Fourth, one
should be able to demonstrate that there are uncountably many
reals. Natural Logicism is realistic about the limits of logicism
when it comes to treating the real numbers, for which, Tennant
argues, one needs recourse to geometric intuition for deeper
starting-points, beyond which logic alone will then deliver the
sought results, with absolute formal rigor. The resulting program
enables one to delimit, in a principled way, those parts of number
theory that are produced by the Kantian understanding alone, and
those parts that depend on recourse to (very simple) a priori
geometric intuitions.
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