"A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection." (Bertrand Russell. International Monthly, Vol. 4, 1901)
"Let us subjugate a collection of objects taking into account their qualities and differences each from another, then we are lead, in our mathematical perspective, to the study of integers and their connecting operations, that is, we are lead to Number Theory. If we, however, disregard the qualities of each individual object and only account for the difference between two objects insofar that they are different, then we are lead to investigations which are concerned with the position, the order, the choosing of these objects. This branch of mathematics is called Combinatorics." (Eugen Netto, "Lehrbuch der Combinatorik", 1901)
"[...] hence upon the principle that even for an infinite totality of sets there are always mappings that associate with every set one of its elements, or, expressed formally, that the product of an infinite totality of sets, each containing at least one element, itself differs from zero. This logical principle cannot, to be sure, be reduced to a still simpler one, but it is applied without hesitation everywhere in mathematical deduction." (Ernst Zermelo, 1904)
"If one intends to base arithmetic on the theory of natural numbers as finite cardinals, one has to deal mainly with the definition of finite set; for the cardinal is, according to its nature, a property of a set, and any proposition about finite cardinals can always be expressed as a proposition about finite sets. In the following I will try to deduce the most important property of natural numbers, namely the principle of complete induction, from a definition of finite set which is as simple as possible, at the same time showing that the different definitions [of finite set] given so far are equivalent to the one given here." (Ernst Zermelo, "Ueber die Grundlagen der Arithmetik", Atti del IV Congresso Internazionale dei Matematici, 1908)
"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35" (904), 1912)
"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)
"The word is of Arabic origin. ‘Al’ is the Arabic article the, and ‘gebar’ is the verb to set, to restitute." (Tobias Dantzig & Joseph Mazur, “Number: The Language of Science”, 1930)
"The principle underlying sampling is that a set of objects taken at random from a larger group tends to reproduce the characteristics of that larger group: this is called the Law of Statistical Regularity. There are exceptions to this rule, and a certain amount of judgment must be exercised, especially when there are a few abnormally large items in the larger group. With erratic data, the accuracy of sampling can often be tested by comparing several samples. On the whole, the larger the sample the more closely will it tend to resemble the population from which it is taken; too small a sample would not give reliable results." (Lewis R Connor, "Statistics in Theory and Practice", 1932)
"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." (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)
"Topology begins where sets are implemented with some cohesive properties enabling one to define continuity." (Solomon Lefschetz, "Introduction to Topology", 1949)
"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)
"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)
"A classification is a scheme for breaking a category into a set of parts, called classes, according to some precisely defined differing characteristics possessed by all the elements of the category." (Alva M Tuttle, "Elementary Business and Economic Statistics", 1957)
"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)
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