“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

"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.

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Hilary Putnam8

American philosopher 1926–2016

Walter F. Buckley (1922–2006) American sociologist

Source: Sociology and modern systems theory (1967), p. 47.

“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.

“Concrete presents, besides a coefficient of thermal expansion, a coefficient of hygroscopic dilation which is more appreciable. While concrete is kept wet… it expands; and as it dries, it contracts, in inverse proportion to the hygroscopic degree of the ambient air. …The shrinkage …will vary with the porosity of the concrete and with the thickness used in the structural member. …Shrinkage is much greater in structures consisting of thin elements subjected to intemperateness of dry climate, and disappears or becomes negligible in underground structures”

Eduardo Torroja (1899–1961) Spanish architect

e.g., foundations
Philosophy of Structures (1958)

“When the Löwenheim-Skolem theorem is applied to particular formal systems, we obtain as special cases: Every group, field, ordered field, etc., has a countable subsystem of the same type. A more spectacular result follows from applying the theorem to set theory (a system which we shall later formalize): There is a countable collection of sets, such that if restrict the membership relation to these sets alone, they form a model for set theory (more precisely all the true statements of set theory are true in this model). In particular, within this model which we may denote by M, there must be an uncountable set. This paradox, that a countable model can contain an uncountable set, is explained by noting that to say a set is uncountable merely asserts the nonexistence of a one-one mapping of the set with the set of integers. The "uncountable" set in M set actually has only countably many members in M, but there is no one-one correspondence \underline {within} M of this set with the set of integers.”

Paul Cohen (1934–2007) American mathematician

Set theory and the continuum hypothesis, pp. 19–20 https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA19 <br class="br">Set Theory and the Continuum Hypothesis (1966)

“We have travelled far from the standpoint which identifies the real with the concrete. Even the older philosophy found it necessary to admit exceptions; for example, time must be admitted to be real, although no one could attribute to it a concrete nature.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

III, p.33
Science and the Unseen World (1929)

“Here we refer to the ideology of an age or of a concrete historico-social group, e.g. of a class, when we are concerned with the characteristics and composition of the total structure of the mind of this epoch or of this group. Although they have something in common, there are also significant differences between them.”

Karl Mannheim (1893–1947) Hungarian sociologist

Ideology and Utopia (1929)
Context: In general there are two distinct and separable meanings of the term "ideology" — the particular and the total.
The particular conception of ideology is implied when the term denotes that we are sceptical of the ideas and representations advanced by our opponent. They are regarded as more or less conscious disguises of the real nature of a situation, the true recognition of which would not be in accord with his interests. These distortions range all the way from conscious lies to half-conscious and unwitting disguises; from calculated attempts to dupe others to self-deception. This conception of ideology, which has only gradually become differentiated from the common-sense notion of the lie is particular in several senses. Its particularity becomes evident when it is contrasted with the more inclusive total conception of ideology. Here we refer to the ideology of an age or of a concrete historico-social group, e. g. of a class, when we are concerned with the characteristics and composition of the total structure of the mind of this epoch or of this group. Although they have something in common, there are also significant differences between them.

“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

“Concepts of uncertainty and information processing are used to integrate the diverse organization design/structure literatures. This approach more fully explicates the concept of congruence which lies at the heart of contingency ideas. The review suggests a contingency approach to design which develops a feasible set of structural alternatives from which the organization can choose.”

David A. Nadler (1948–2015) American organizational theorist

Source: "Information Processing as an Integrating Concept in Organizational Design." 1978, p. 613: Abstract

“This compositional variety is mediated by a highly redundant set structure, a second-order all-combinatorial set; each set form is hexachordally equivalent to or totally disjunct from fifteen other set forms, so that one-third of all the available set forms belong to a collection of sets which are hexachordally aggregate forming, that is, hexachordally identical.”

Milton Babbitt (1916–2011) American composer

On Igor Stravinsky's Movements. Perspectives of new music, Spring-Summer 1964

“I will say without reservations that from my point of view there can be no abstractions. Any shape or area that has not the pulsating concreteness of real flesh and bones, its vulnerability to pleasure or pain is nothing at all. Any picture that does not provide the environment in which the breath of life can be drawn does not interest me.”

Mark Rothko (1903–1970) American painter

letter to Clyfford Still, undated; as quoted in Mark Rothko : A Biography (1993), James E. B. Breslin / and Abstract Expressionism, Creators and Critics, ed. Clifford Ross, Abrams Publishers New York 1990, p. 170
after 1970, posthumous

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Hilary Putnam8

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