"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)
"In modern mathematics set theory plays the role of the frame, the skeleton which joins all its parts into a single whole but cannot be seen from the outside and does not come in direct contact with the external world. This situation can be truly understood and the formal and contentual aspects of mathematics combined only from the 'linguistic' point of view regarding mathematics." (Valentin F Turchin, "The Phenomenon of Science: a cybernetic approach to human evolution", 1977)
"To use set theory in the way it is used by modern mathematics, however, it is not at all necessary to force one's imagination and try to picture actual infinity. The 'sets' which are used in mathematics are simply symbols, linguistic objects used to construct models of reality. The postulated attributes of these objects correspond partially to intuitive concepts of aggregateness and potential infinity; therefore intuition helps to some extent in the development of set theory, but sometimes it also deceives. Each new mathematical (linguistic) object is defined as a 'set' constructed in some particular way. This definition has no significance for relating the object to the external world, that is for interpreting it: it is needed only to coordinate it with the frame of mathematics, to mesh the internal wheels of mathematical models. So the language of set theory is in fact a metalanguage in relation to the language of contentual mathematics, and in this respect it is similar to the language of logic. If logic is the theory of proving mathematical statements, then set theory is the theory of constructing mathematical linguistic objects." (Valentin F Turchin, "The Phenomenon of Science: a cybernetic approach to human evolution", 1977)
"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements." (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)
"The axiom of choice has many important consequences in set theory. It is used in the proof that every infinite set has a denumerable subset, and in the proof that every set has at least one well-ordering. From the latter, it follows that the power of every set is an aleph. Since any two alephs are comparable, so are any two transfinite powers of sets. The axiom of choice is also essential in the arithmetic of transfinite numbers." (R Bunn, "Developments in the Foundations of Mathematics, 1870-1910", 1980)
"It is a remarkable empirical fact that mathematics can be based on set theory. More precisely, all mathematical objects can be coded as sets (in the cumulative hierarchy built by transfinitely iterating the power set operation, starting with the empty set). And all their crucial properties can be proved from the axioms of set theory. [...] At first sight, category theory seems to be an exception to this general phenomenon. It deals with objects, like the categories of sets, of groups etc. that are as big as the whole universe of sets and that therefore do not admit any evident coding as sets. Furthermore, category theory involves constructions, like the functor category, that lead from these large categories to even larger ones. Thus, category theory is not just another field whose set-theoretic foundation can be left as an exercise. An interaction between category theory and set theory arises because there is a real question: What is the appropriate set-theoretic foundation for category theory?" (Andreas Blass, "The interaction between category theory and set theory", Mathematical Applications of Category Theory, 1983)
"Set theory is peculiarly important [...] because mathematics can be exhibited as involving nothing but set-theoretical propositions about set-theoretical entities." (David M Armstrong, "A Combinatorial Theory of Possibility", 1989)
"And you should not think that the mathematical game is arbitrary and gratuitous. The diverse mathematical theories have many relations with each other: the objects of one theory may find an interpretation in another theory, and this will lead to new and fruitful viewpoints. Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole." (David Ruelle, "Chance and Chaos", 1991)
"Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole. Mathematics is a great kingdom, and that kingdom belongs to those who see." (David Ruelle, "Chance and Chaos", 1991)
"In abstract mathematics, special attention is given to particular properties of numbers. Then those properties are taken in a very pure" (and primitive) form. Those properties in pure form are then assigned to a given set. Therefore, by studying in details the internal mathematical structure of a set, we should be able to clarify the meaning of original properties of the objects. Likewise, in set theory, numbers disappear and only the concept of sets and characteristic properties of sets remain." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Set theory has a dual role in mathematics. In pure mathematics, it is the place where questions about infinity are studied. Although this is a fascinating study of permanent interest, it does not account for the importance of set theory in applied areas. There the importance stems from the fact that set theory provides an incredibly versatile toolbox for building mathematical models of various phenomena." (Jon Barwise & Lawrence Moss, "Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena", 1996)
"Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. […] However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs." (Sun-Joo Shin, "Situation-Theoretic Account of Valid Reasoning with Venn Diagrams", [in "Logical Reasoning with Diagrams"], 1996)
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