|Consistency, philosophy, logic: within a system, consistency may be demonstrated, but not beyond the boundaries of this system, since the use of the symbols and the set of possible objects are only defined for this system.|
Within mathematics, and only there applies that the mathematical objects, which are mentioned in consistent formulas, exist (Hilbert, Über das Unendliche, 1926). See also falsification, verification, existence, well-formed.
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Consistency/Bigelow/Pargetter: a way to guarantee that a description is consistent is to show that something meets this description.
Definition Principle of instantiation/Bigelow/Pargetter: we can call this the principle of instantiation (instantiation principle).
Contradiction-free/Bigelow/Pargetter: is essential for mathematics, for other areas it is more like housekeeping.
Consistency/Hilbert: precedes existence. A mathematical proof exists only if it is non-contradictory.
Consistency/FregeVsFormalism/FregeVsHilbert/Bigelow/Pargetter: Existence precedes the consistency. Consistency requires the existence of a consistently described thing. If it exists, the corresponding description is consistent. If it does not exist, how do we guarantee consistency?
Frege/Bigelow/Pargetter: thinks here epistemically, in terms of "guarantees". But his view can be extended: if there is no object, there is no difference between a consistent and a contradictory description.
Frege/Bigelow/Pargetter: pro Frege: this is the basis for modern mathematics. This is also the reason why quantum theory is so important: it provides examples of everything that mathematicians wish to investigate (at least until recently).
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990