By L. Smith
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Additional resources for Algebraic Topology, Gottingen 1984
29 Bernstein describes ‘Finitismus’ as a familiar foundational stance that was opposed to set theory since the very beginnings of that theory in Cantor’s and Dedekind’s work. Somewhat indiscriminately, Kronecker and Hermite, Borel and Poincaré, Richard and Lindelöf, and then also Brouwer and Weyl (Weyl’s work Das Kontinuum is cited, i. , Weyl 1918 ) are viewed as members of the ﬁnitist movement. Bernstein’s paper was published before Hilbert and Bernays viewed ‘ﬁnite Mathematik’ as foundationally signiﬁcant for proof theory.
20 above), 32 That Dirichlet and Dedekind held to such a principle is stated in the Vorwort to Dedekind 1888 . For Kronecker’s position, see Kronecker 1887 , 338–339. Hilbert himself alludes to Weierstraß’s adherence to such a principle in his ‘Feriencurs’ for 1896, pp. 6–7; see Hallett and Majer 2004 , 154. This adherence is also stated in Hurwitz’s Mitschrift of Weierstraß’s lectures ‘Einleitung in die Theorie der analytischen Funktionen’ from 1878. Cantor (Cantor 1883a, 553) states the view as follows: Das eigentliche Material der Analysis wird, ausschliesslich, dieser Ansicht zufolge, von den endlichen, realen, ganzen, Zahlen gebildet und alle in der Arithmetik und Analysis gefundenen oder noch der Entdeckung harrenden Wahrheiten sollen als Beziehungen der endlichen ganzen Zahlen untereinander aufzufassen sein; .
Ausarbeitung of the 1921/22 lectures, p. 4a; p. ) That recognition has to be obtained on the basis of ﬁnitist logic; so Hilbert argues that we have to extend our considerations in a diﬀerent direction in order to go beyond elementary number theory: We have to extend the domain of objects to be considered; i. , we have to apply our intuitive considerations also to ﬁgures that are not number signs. 18 Introduction Thus we have good reason to distance ourselves from the earlier dominant principle according to which each theorem of pure mathematics is in the end a statement concerning integers.