"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid V Kantorovich, "On one class of functional equations", 1936)
"This abstracting of common experience is one of the principal sources of the utility of mathematics and the secret of its scientific power. The world that impinges on the senses of all but introverted solipsists is too intricate for any exact description yet imagined by human beings. By abstracting and simplifying the evidence of the senses, mathematics brings the worlds of science and daily life into focus with our myopic comprehension, and makes possible a rational description of our experiences which accords remarkably well with observation." (Eric T Bell, "The Development of Mathematics", 1940)
"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)
"Proof serves many purposes simultaneously […] Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics […] Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems." (Reuben Hersh, "The Mathematical Experience", 1981)
"People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like something else (association); and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. Personally, I put a lot of effort into ‘listening’ to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations." (William P Thurston, "On proof and progress in mathematics", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)
"While mathematicians now recognize that there is some freedom in the choice of the axioms one uses, not any set of statements can serve as a set of axioms. In particular, every set of axioms must be logically consistent, which is another way of saying that it should not be possible to prove a particular statement simultaneously true and false using the given set of axioms. Also, axioms should always be logically independent - that is, no axiom should be a logical consequence of the others. A statement that is a logical consequence of some of the axioms is a theorem, not an axiom." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)
"Fractals' simultaneous chaos and order, self-similarity, fractal dimension and tendency to scalability distinguish them from any other mathematically drawable figures previously conceived." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)
"One trick to seeing beauty in mathematics is to nurture this sense of 'odd as it may seem' while at the same time understanding the subject well enough to know that oddities arise despite our attempts to set the subject on a simple, straightforward footing."
"Abstraction is indeed an essential element of concept formation and mathematics is the discipline that has investigated the process of abstraction in greater depth than anywhere else. [...] Mathematics is simultaneously the most abstract and the most concrete of disciplines." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
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