On Proofs (1930-1939)

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories." (Stefan Banach, cca. 1930)

"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)

"If one operates in an arbitrary abstract number field rather than in the continuum of complex numbers, then the fundamental theorem of algebra, which asserts that every complex polynomial in one variable can be [uniquely] decomposed into linear factors, need not hold. Hence the general prescriptíon in algebraic work: See if a proof makes use of the fundamental theorem or not. In every algebraic theory, there is a more elementary part that is independent of the fundamental theorem, and therefore valid in every field, and a more advanced part for whích the fundamental theorem is indispensable. The latter part calls for the algebraic closure of the field. In most cases, the fundamental theorem marks a crucial split; its use should be avoided as long as possible." (Hermann Weyl, "Topologie und abstrakte Algebra als zwei Wege mathematischen Verständnisses", Unterrichtsblätter für Mathematik und Naturwissenschaften 38, 1932)

"Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs." (David Hilbert, "Die Grundlagen der Mathematik, 1934-1939)

"The search for the most general conditions of validity of a determined statement, if it is ready to reveal its causal proof, doesn’t succeed without a constant reworking of the implemented notions. (Georges Bouligand, "La causalite des theories mathématiques", Actualités Scientifiques et Industrielles 184, 1935)

"To square a circle means to find a square whose area is equal to the area of a given circle. In its first form this problem asked for a rectangle whose dimensions have the same ratio as that of the circumference of a circle to its radius. The proof of the impossibility of solving this by use of ruler and compasses alone followed immediately from the proof, in very recent times, that π cannot be the root of a polynomial equation with rational coefficients." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of the semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid Kantorovich, "On one class of functional equations", 1936)