On David Hilbert

"Hilbert’s mathematics may be a pretty game with formulas, more amusing even than chess; but what does it have to do with knowledge, since its formulas should admittedly have no contentual significance by virtue of which they would express intuitive truths?" (Hermann Weyl, "Philosophie der Mathematik und Naturwissenschaften", 1927)

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined non-tentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920." (Eric T Bell, "The Development of Mathematics", 1940)

"Historically speaking, topology has followed two principal lines of development. In homology theory, dimension theory, and the study of manifolds, the basic motivation appears to have come from geometry. In these fields, topological spaces are looked upon as generalized geometric configurations, and the emphasis is placed on the structure of the spaces themselves. In the other direction, the main stimulus has been analysis. Continuous functions are the chief objects of interest here, and topological spaces are regarded primarily as carriers of such functions and as domains over which they can be integrated. These ideas lead naturally into the theory of Banach and Hilbert spaces and Banach algebras, the modern theory of integration, and abstract harmonic analysis on locally compact groups." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The theory of rings and ideals was put on a more systematic and axiomatic basis by Emmy Noether, one of the few great women mathematicians [...] Many results on rings and ideals were already known [...] but by properly formulating the abstract notions she was able to subsume these results under the abstract theory. Thus she reexpressed Hilbert's basic theorem [...] as follows: A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis. In this reforumulation she made the theory of invariants a part of abstract algebra." (Morris Kline, "Mathematical Thought From Ancient to Modern Times", 1972)

"The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. [...] Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa." (C West Churchman et al, "Thinking for Decisions Deduction Quantitative Methods", 1975)

"At the most elemental level, reality evanesces into something called Schröedinger's Wave Function: a mathematical abstraction which is best represented as a pattern in an infinite-dimensional space, Hilbert Space. Each point of the Hilbert Space represents a possible state of affairs. The wave function for some one physical or mental system takes the form of, let us say, a coloring in of Hilbert Space. The brightly colored parts represent likely states for the system, the dim parts represent less probable states of affairs." (Rudy Rucker, "The Sex Sphere", 1983) 

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"[...] all the laws of algebra correspond to projective coincidences, and von Staudt showed that all the required coincidences follow from the theorems of Pappus and Desargues. Then in 1899 David Hilbert showed that all laws of algebra except the commutative law for multiplication follow from the Desargues theorem. And in 1932 Ruth Moufang showed that all except the commutative and associative laws follow from the little Desargues theorem. Thus the Pappus, Desargues, and little Desargues theorems are mysteriously aligned with the laws of multiplication!" (John Stillwell, "The Four Pillars of Geometry", 2000)