Gottlob Frege, answered from the texts and cited to the page.
Arithmetic is reducible to logic. Numbers are not psychological entities, nor empirical generalisations, nor Kantian intuitions; they are objective logical objects, defined by purely logical means. The cardinal number belonging to a concept F is the extension of the concept "equinumerous with F"; two concepts F and G are equinumerous if and only if there exists a one-one correspondence between the objects falling under F and the objects falling under G (Hume's Principle, Grundlagen §63).1
The formal derivation of arithmetic from logic is the work of the Grundgesetze (1893, 1903). The failure of Basic Law V — the principle that the value-range of F equals the value-range of G if and only if F and G are coextensive — under Russell's paradox is a failure of this particular axiom, not of the logicist program as such.
The proper diagnosis is that the unrestricted comprehension of value-ranges is too strong; restrictions that preserve the derivation of arithmetic while avoiding the paradox are the neo-logicist program (Wright, Hale, Boolos).234
So that here there is complete agreement. Certainly we do not say that one number is wider than another, in the sense in which the extension of one concept is wider than that of another; but then it is also quite impossible for a case to occur where the extension of the concept "equal to the concept F" would be wider than $1 the extension of the concept "equal to the concept G".The Foundations of Arithmetic, pp. 105–106
From this it can be seen that the proposition "every object which falls under F stands in the relation ¢@ to an object falling under G" is, in the case where there is no object falling under F, true; for in that case the first proposition "a falls under F" ""a falls under G" and "no object falling under F stands to a in the relation 6" cannot, whatever @ may be, both be true together.The Foundations of Arithmetic, pp. 108–110
Construction (lg): - - - - - - - - - - 586 e.. -¥.bl'"\ s p bl'"\s e..-¥.PI'"\ s pl'"\ s js (670 (588):: - - - - - - -............... SM e.. -¥.bl'"\s p..-¥.bl'"\s 233. Analysis p..-¥.e l'"\ s Using (670), we now prove the proposition js p = e p l'"\ s (o js (Ille): ·-·-·-·-·-·-·- · ps e.. bl'"\s pl'"\s p.. -¥.bl'"\s a..-¥.(e..e) l'"\s p..Basic Laws of Arithmetic, pp. 573–594
In practice, this law amounts to the claim that one can always replace an equality with the corresponding universally quantified concept-script expression in any concept-script pro:p,osition (i.e.. · as the argument of ·any function g). Note that the presence of the Roman letter ' g · renders this much more general than the modern version of the indiscerni bility of identicals.Basic Laws of Arithmetic, pp. 652–653
The Grundgesetze logicist program is the central project of Frege's mathematical-philosophical career and the failure of Basic Law V under Russell's paradox is the central crisis of Frege's intellectual life. The "Way Out" Appendix to Vol II (Law V′, October 1902) was proposed in haste; Frege discovered in 1906 that Law V′ does not preserve even the theorem that 0 ≠ 1. In the late unpublished writings of 1924 Frege abandoned logicism altogether and proposed that arithmetic, like geometry, rests on geometric intuition (the kindergeometrical Quelle der Erkenntnis). The contemporary neo-logicist program (Wright 1983, Hale and Wright 2001, Boolos in his collected logic papers) revives the spine of the project: Frege's Theorem establishes that Hume's Principle alone, taken as an axiom in second-order logic, entails the Peano axioms — so arithmetic can be derived from a single abstraction principle, even if not from purely logical principles narrowly construed. The Caesar Problem (whether Hume's Principle defines what numbers are) remains the principal philosophical worry. Russell's type theory and Quine's NF set theory are the rival paradox-avoidance frameworks; the philosophical Frege of the neo-logicists is closer to the Grundlagen than to the late capitulation.