“Mathematics… is the set of all possible self-consistent structures”

— Michio Kaku, book Hyperspace

Source: Hyperspace (1995), Ch.15 Conclusion
Context: Mathematics... is the set of all possible self-consistent structures, and there are vastly more logical structures than physical principles.

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Michio Kaku19

American theoretical physicist, futurist and author 1947

Andrew H. Van de Ven (1945) American business theorist

Robert Drazin, and Andrew H. Van de Ven. "Alternative forms of fit in contingency theory." Administrative science quarterly (1985): 514-539.

“The real significance of the Russell paradox, from the standpoint of the modal-logic picture, is this: it shows that no concrete structure can be a standard model for the naive conception of the totality of all sets; for any concrete structure has a possible extension that contains more 'sets.'”

Hilary Putnam (1926–2016) American philosopher

"Mathematics without foundations"
Source: Philosophical Papers Volume 1: Mathematics, Matter, and Method (1975, 1979)
Context: (If we identify sets with the points that represent them in the various possible concrete structures, we might say: it is not possible for all possible sets to exist in any one world!) Yet set theory does not become impossible. Rather, set theory becomes the study of what must hold in, e.g. any standard model for Zermelo set theory.

“Mathematics catalogues everything that is not self-contradictory; within that vast inventory, physics is an island of structures rich enough to contain their own beholders.”

Greg Egan book Oceanic

Source: Oceanic

“Mathematical magic combines the beauty of mathematical structure with the entertainment value of a trick.”

Martin Gardner (1914–2010) recreational mathematician and philosopher

Mathematics, Magic, and Mystery https://books.google.com/books?id=-kOFBQAAQBAJ&pg=PR11#v=onepage&q=%22Mathematical%20magic%20combines%22%23v%3Dsnippet&f=false (1956), p. ix

“Before a man can successfully plan a structure… he must study, from every possible angle, the ultimate purpose of his building. Attention must be directed to the basic structural concept before the mathematical process of calculation is undertaken.”

Eduardo Torroja (1899–1961) Spanish architect

p, 125
Philosophy of Structures (1958)

“The art of doing mathematics consists in finding that special case which contains all the germs of generality.”

David Hilbert (1862–1943) German prominent mathematician

Quoted in Constance Reid, "Hilbert" (1970)

“A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales…”

Benoît Mandelbrot (1924–2010) Polish-born, French and American mathematician

As quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

“The rules of logic are to mathematics what those of structure are to architecture.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

1900s, "The Study of Mathematics" (November 1907)

“Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.”

Joshua Girling Fitch (1824–1903) British educationalist

The Fourth Dimension simply Explained. (New York, 1910), p. 58. Reported in Moritz (1914); Also cited in: Howard Eves (2012), Foundations and Fundamental Concepts of Mathematics, p. 167

“[M]ost of our structures in the past consisted of individual members having relatively small transverse dimensions as compared with their length, and in such members stress analysis is much more simple. …[M]any modern structures consist of frameworks of this type, and frequently the word "structure" refers specifically to assemblies consisting of linear elements.”

Eduardo Torroja (1899–1961) Spanish architect

Philosophy of Structures (1958)

“Mathematics… is the set of all possible self-consistent structures”

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Michio Kaku19

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