Sentence examples for arithmetical principles from inspiring English sources

Unlike formal semantics and set theory, there may not be any obvious arithmetical principles that give rise to contradiction.

More precisely, they attempted to construct a derivation of arithmetical principles from definitions of arithmetical concepts, using only logical laws.

Examples are discussed in The Problems of Philosophy (1912a) where Russell states that propositions with the highest degree of self-evidence (what he here calls "intuitive knowledge") include "those which merely state what is given in sense, and also certain abstract logical and arithmetical principles, and (though with less certainty) some ethical propositions" (1912a, 109).

Poincaré insists on the non-invariance of mathematical reasoning with respect to its content and advances, so to speak, a local conception of reasoning (for example, the mathematical induction principle for arithmetical reasoning).

For this purpose he introduced his principle of the permanence of equivalent forms, a principle connecting results in arithmetical algebra to those in symbolical algebra.

Of course solving arithmetical problems in arithmetic is in some cases practically impossible.

Dedekind wanted "a purely arithmetical and perfectly rigorous foundation for the principles of infinitesimal analysis" [Emphases added].[8] Once again we see the presumption at work: in laying a foundation for the theory of real numbers, one must avoid any recourse to geometrical intuition.

In first-order formalizations of arithmetic, this is formulated as a scheme: for each first-order arithmetical formula of the language of arithmetic with one free variable, one instance of the induction principle is included in the formalization of arithmetic.

Arithmetical logicism is secured if Hume's Principle can be shown to be analytic in an appropriate sense.

This principle has two parts: (1) General results in arithmetical algebra belong to the laws of symbolical algebra.

Then Kurt Gödel proved that there exist arithmetical statements that are undecidable in Peano Arithmetic (Gödel 1931).