The land of easy mathematics where he who works adds up and he who retires subtracts.
Mathematics possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture.1903 The Principles of Mathematics.
The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. ... Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.george pólya
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.Richard Courant in: The Australian Mathematics Teacher, Volumes 39-40, Australian Association of Mathematics Teachers, 1983, p.3
The principles of logic and Mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and Mathematics are analytic propositions or tautologies.alfred jules ayer
Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than Mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.kenneth e. iverson
Magic is that which it is; it is by itself, like the Mathematics; for it is the exact and absolute science of Nature and its laws . Magic is the science of the Ancient Magi: and the Christian religion, which has imposed silence on the lying oracles, and put an end to the prestiges of the false Gods , itself reveres those Magi who came from the East, guided by a Star , to adore the Saviour of the world in His cradle.albert pike
The study of Mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.Alfred North Whitehead
The significant contribution to Mathematics we owe to the Arabs was to absorb Greek and Hindu Mathematics, preserve it, and ultimately, through events we have yet to look in, transmit it to Europe. (...) In Spain the Arabs were constantly attacked and finally conquered in 1492 by the Christians; this ended the mathematical and scientific activity in the region.Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1990, p.197
Herbert A. Simon's scientific output goes far beyond the disciplines in which he has held professorships: political science, administration, psychology and information sciences. He has made contributions in the fields of science theory, applied Mathematics, statistics, operations research, economics and business and public administration (and), in all areas in which he has conducted research, Simon has had something of importance to say."The Royal Swedish Academy of Sciences (1978) "Nobel Prize announcement of the Royal Academy of Sciences in Sweden, 16 okt. 1978"
The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in Mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution.Kenneth E. Iverson (1979) Notation as a Tool of Thought. §5.4
The terms synthesis and analysis are used in Mathematics in a more special sense than in logic. In ancient Mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid , XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."Florian Cajori, A History of Mathematics (1893). p. 30
What is the use of aesthetics if they can neither teach how to produce beauty nor how to appreciate it in good taste? It exists because it behooves rational human beings to provide reasons for their actions and assessments. Even if aesthetics are not the Mathematics of beauty , they are the proof of the calculation.Franz Grillparzer , in A Primer on Environmental Decision-Making: An Integrative Quantitative Approach, p. 184
The land of easy Mathematics where he who works adds up and he who retires subtracts.Núria Añó, 2066. Beginning the age of correction