"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87 (2), 1980)
"Thus statistics should generally be taught more as a practical subject with analyses of real data. Of course some theory and an appropriate range of statistical tools need to be learnt, but students should be taught that Statistics is much more than a collection of standard prescriptions." (Christopher Chatfield, "The Initial Examination of Data", Journal of the Royal Statistical Society A Vol. 148, 1985)
"[…] calling upon the needs of rigor to explain the development of mathematics constitutes a circular argument. In actual fact, new standards of rigor are formed when the old criteria no longer permit an adequate response to questions that arise in mathematical practice or to problems that are in a certain sense external to mathematics. When these are treated mathematically, they compel changes in the theoretical framework of mathematics. It is thus not by chance that mathematical physics and applied mathematics have generally been formidable stimuli to the development of pure mathematics." (Umberto Bottazzini, "The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass", 1986)
"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth." (National Research Council, "Everybody Counts", 1989)
"'Technique' is a term used equally by chess players and mathematicians to describe sequences of moves which are standard, familiar and unoriginal. Once upon a time, the particular technique was an invention, a new discovery, but no longer. The precise sequence of moves required may never have been played before in the history of the world, yet no new ideas, no originality and no imagination are demanded, at least of the experienced player. (To the learner, of course, the most mundane sequences will appear novel and require original thought.)" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)
"Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality." (Michael Aschbacher,"Highly complex proofs and implications", 2005)
“When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze.” (William P Thurston, [preface to (John H Hubbard, “Teichmüller Theory and Applications”,Vol. 1, 2006)])
"Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress." (Edward Frenkel, "Love and Math", 2014)
"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)
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