Category Theory I

"It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation […]. The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether […]" (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The theory [of categories] also emphasizes that, whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these new objects as an integral part of their definition. The pursuit of this program entails a simultaneous consideration of objects and their mappings (in our terminology, this means the consideration not of individual objects but of categories). This emphasis on the specification of the type of mappings employed gives more insight onto the degree of invariance of the various concepts involved." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The notion of an abstract group arises by consideration of the formal properties of one-to-one transformations of a set onto itself. Similarly, the notion of a category is obtained from the formal properties of the class of all transformations y : X → Y of any one set into another, or of continuous transformations of one topological space into another, or of homomorphisms, of one group into another, and so on." (Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56, 1950)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)

"The point is simply that when explaining the general notion of structure and of particular kinds of structures such as groups, rings, categories, etc., we implicitly presume as understood the ideas of operation and collection." (Solomon Feferman, "Categorical foundations and foundations of category theory", 1975)

"[…] it would be technically possible to give a purely category-theoretic account of all mathematical notions expressible within axiomatic set theory, and so formally possible for category theory to serve as a foundation for mathematics insofar as axiomatic set theory does." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)

"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", 1983)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

"Category theory has developed classically, beginning with definitions and axioms and proceeding to a long list of theorems. Category theory is not topology (and so will not be described here), but it can be used to understand some of the relationships that exist among classes of topological spaces. It can be used to bring unity to diversity. [...] the theory of categories is not complete, it may not be completable, but it is a step forward in understanding foundational questions in mathematics." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

Set Theory I

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"Since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic. This method was prepared long ago (not least by Frege’s profound investigations); it has been most successfully explained by the acute mathematician and logician Russell. One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole." (David Hilbert, "Axiomatisches Denken" ["Axiomatic Thinking"], [address] 1917)

"It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum." (Hermann Weyl, "Reimanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie", 1925)

"To say that mathematics in general has been reduced to logic hints at some new firming up of mathematics at its foundations. This is misleading. Set theory is less settled and more conjectural than the classical mathematical superstructure than can be founded upon it." (Willard van Orman Quine, "Elementary Logic", 1941)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)

"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)

"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)

"At the basis of the distance concept lies, for example, the concept of convergent point sequence and their defined limits, and one can, by choosing these ideas as those fundamental to point set theory, eliminate the notions of distance." (Felix Hausdorff)

Saunders Mac Lane - Collected Quotes

"In a metamathematical sense our theory provides general concepts applicable to all branches of abstract mathematics, and so contributes to the current trend towards uniform treatment of different mathematical disciplines. In particular, it provides opportunities for the comparison of constructions and of the isomorphisms occurring in different branches of mathematics; in this way it may occasionally suggest new results by analogy." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation […]. The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether […]" (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The invariant character of a mathematical discipline can be formulated in these terms. Thus, in group theory all the basic constructions can be regarded as the definitions of co- or contravariant functors, so we may formulate the dictum: The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The theory [of categories] also emphasizes that, whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these new objects as an integral part of their definition. The pursuit of this program entails a simultaneous consideration of objects and their mappings (in our terminology, this means the consideration not of individual objects but of categories). This emphasis on the specification of the type of mappings employed gives more insight onto the degree of invariance of the various concepts involved." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas.” (Saunders MacLane, "Of Course and Courses", The American Mathematical Monthly, Vol. 61, No. 3, March, 1954)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)

"Just as each generation of historians must analyse the past again, so in the exact sciences we must in each period take up the renewed struggle to present as clearly as we can the underlying ideas of mathematics." (Saunders Mac Lane, "Hamiltonian Mechanics and Geometry", The American Mathematical Monthly Vol. 77 (6), 1970)

"Many cumbersome developments in the standard treatments of mechanics can be simplified and better understood when formulated with modern conceptual tools, as in the well-known case of the use of the 'universal' definition of tensor products of vector spaces to simplify some of the notational excesses of tensor analysis as traditionally used in relativity theory" (Saunders Mac Lane, "Hamiltonian Mechanics and Geometry", The American Mathematical Monthly Vol. 77 (6), 1970)

"Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows." (Saunders Mac Lane, "Categories for the Working Mathematician", 1971)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)