"The object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered manifold; the law of order of this manifold must be subject to our choice; the latter is the case in both of the only conceivable kinds of manifolds, in the discrete as well as in the continuous." (Erwin Papperitz, "Über das System der rein mathematischen Wissenschaften", 1910)
"The bridging of the chasm between the domains of the discrete and the continuous, or between arithmetic and geometry, is one of the most important - nay, the most important - problem of the foundations of mathematics. [...] Of course, the character of reasoning has changed, but, as always, the difficulties are due to the chasm between the discrete and the continuous - that permanent stumbling block which also plays an extremely important role in mathematics, philosophy, and even physics." (Abraham Fraenkel, "Foundations of Set Theory", 1953)
"As the sensations of motion and discreteness led to the abstract notions of the calculus, so may sensory experience continue thus to suggest problem for the mathematician, and so may she in turn be free to reduce these to the basic formal logical relationships involved. Thus only may be fully appreciated the twofold aspect of mathematics: as the language of a descriptive interpretation of the relationships discovered in natural phenomena, and as a syllogistic elaboration of arbitrary premise." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)
"Increasingly [...] the application of mathematics to the real world involves discrete mathematics [...] the nature of the discrete is often most clearly revealed through the continuous models of both calculus and probability. Without continuous mathematics, the study of discrete mathematics soon becomes trivial and very limited. [...] The two topics, discrete and continuous mathematics, are both ill served by being rigidly separated." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Although discrete mathematics and statistics provide necessary foundations for computer engineering and social sciences, calculus remains the archetype of higher mathematics. It is a powerful and elegant example of the mathematical method, leading both to major applications and to major theories. The language of calculus has spread to all scientific fields; the insight it conveys about the nature of change is something that no educated person can afford to be without." (Mathematical Sciences Education Board, "Everybody Counts: A Report to the Nation on the Future of Mathematics Education", 1989)
"The word 'discrete' means separate or distinct. Mathematicians view it as the opposite of 'continuous'. Whereas, in calculus, it is continuous functions of a real variable that are important, such functions are of relatively little interest in discrete mathematics. Instead of the real numbers, it is the natural numbers 1, 2, 3, ... that play a fundamental role, and it is functions with domain the natural numbers that are often studied." (Edgar G Goodaire & Michael M Parmenter, "Discrete Mathematics with Graph Theory" 2nd Ed., 2002)
"Discrete mathematics is the part of mathematics devoted to the study of discrete objects. (Here discrete means consisting of distinct or unconnected elements.) [...] More generally, discrete mathematics is used whenever objects are counted, when relationships between finite (or countable) sets are studied, and when processes involving a finite number of steps are analyzed. A key reason for the growth in the importance of discrete mathematics is that information is stored and manipulated by computing machines in a discrete fashion." (Kenneth H. Rosen, "Discrete Mathematics and Its Applications" 5th Ed., 2003)
"Discrete math is about things you can count, rather than things you can measure." (Daniel Ashlock, "Fast Start Advanced Calculus", 2022)
No comments:
Post a Comment