„Only if mathematical rigor is adhered to, can systems problems be dealt with effectively, and so it is that the systems engineer must, at least, develop an appreciation for mathematical rigor if not also considerable mathematical competence.“

Источник: A Mathematical Theory of Systems Engineering (1967), p. 3.

Взято из Wikiquote. Последнее обновление 3 июня 2021 г. История
A. Wayne Wymore фото
A. Wayne Wymore13
American mathematician 1927 - 2011

Похожие цитаты

„Mathematics brought rigor to Economics. Unfortunately, it also brought mortis“

—  Kenneth E. Boulding British-American economist 1910 - 1993

Attributed to Kenneth Boulding in: Peter J. Dougherty (2002) Who's afraid of Adam Smith?: how the market got its soul. p. 110
1990s and attributed

Richard Courant фото
George Dantzig фото
A. Wayne Wymore фото

„[The process of system design is]… consisting of the development of a sequence of mathematical models of systems, each one more detailed than the last.“

—  A. Wayne Wymore American mathematician 1927 - 2011

A. Wayne Wymore (1970) Systems Engineering Methodology. Department of Systems Engineering, The University of Arizona, p. 14/2; As cited in: J.C. Heckman (1973) Locating traveler support facilities along the interstate system--a simulation using general systems theory. p. 43.

Oliver Heaviside фото

„Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion?“

—  Oliver Heaviside electrical engineer, mathematician and physicist 1850 - 1925

[Oliver Heaviside (1850-1927) - Physical mathematician, http://teamat.oxfordjournals.org/content/2/2/55.extract, https://www.gwern.net/docs/science/1983-edge.pdf, Teaching mathematics and its applications, Oxford Journals, 2, 2, 55-61, 1983, DA Edge]
This quote cannot be found in Heaviside's corpus, Edge provides no reference, the quote first appears around the 1940s attributed to Heaviside without any references. The quote is actually a composite of a modified sentence from Electromagnetic Theory I https://archive.org/details/electromagnetict02heavrich/page/8/mode/2up (changing 'dinner' to 'eat'), a section header & later sentence from Electromagnetic Theory II https://archive.org/details/electromagnetict02heavrich/page/4/mode/2up, and the paraphrase of Heaviside's views by Carslaw 1928 https://www.gwern.net/docs/math/1928-carslaw.pdf ("Operational Methods in Mathematical Physics"), respectively:
"Nor is the matter an unpractical one. I suppose all workers in mathematical physics have noticed how the mathematics seems made for the physics, the latter suggesting the former, and that practical ways of working arise naturally. This is really the case with resistance operators. It is a fact that their use frequently effects great simplifications, and the avoidance of complicated evaluations of definite integrals. But then the rigorous logic of the matter is not plain! Well, what of that? Shall I refuse my dinner because I do not fully understand the process of digestion? No, not if I am satisfied with the result. Now a physicist may in like manner employ unrigorous processes with satisfaction and usefulness if he, by the application of tests, satisfies himself of the accuracy of his results. At the same time he may be fully aware of his want of infallibility, and that his investigations are largely of an experimental character, and may be repellent to unsympathetically constituted mathematicians accustomed to a different kind of work."
"Rigorous Mathematics is Narrow, Physical Mathematics Bold And Broad. § 224. Now, mathematics being fundamentally an experimental science, like any other, it is clear that the Science of Nature might be studied as a whole, the properties of space along with the properties of the matter found moving about therein. This would be very comprehensive, but I do not suppose that it would be generally practicable, though possibly the best course for a large-minded man. Nevertheless, it is greatly to the advantage of a student of physics that he should pick up his mathematics along with his physics, if he can. For then the one will fit the other. This is the natural way, pursued by the creators of analysis. If the student does not pick up so much logical mathematics of a formal kind (commonsense logic is inherited and experiential, as the mind and its ways have grown to harmonise with external Nature), he will, at any rate, get on in a manner suitable for progress in his physical studies. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. There is no end to the subtleties involved in rigorous demonstrations, especially, of course, when you go off the beaten track. And the most rigorous demonstration may be found later to contain some flaw, so that exceptions and reservations have to be added. Now, in working out physical problems there should be, in the first place, no pretence of rigorous formalism. The physics will guide the physicist along somehow to useful and important results, by the constant union of physical and geometrical or analytical ideas. The practice of eliminating the physics by reducing a problem to a purely mathematical exercise should be avoided as much as possible. The physics should be carried on right through, to give life and reality to the problem, and to obtain the great assistance which the physics gives to the mathematics. This cannot always be done, especially in details involving much calculation, but the general principle should be carried out as much as possible, with particular attention to dynamical ideas. No mathematical purist could ever do the work involved in Maxwell's treatise. He might have all the mathematics, and much more, but it would be to no purpose, as he could not put it together without the physical guidance. This is in no way to his discredit, but only illustrates different ways of thought."
"§ 2. Heaviside himself hardly claimed that he had 'proved' his operational method of solving these partial differential equations to be valid. With him [Cf. loc. cit., p. 4. [Electromagnetic Theory, by Oliver Heaviside, vol. 2, p. 13, 1899.]] mathematics was of two kinds: Rigorous and Physical. The former was Narrow: the latter Bold and Broad. And the thing that mattered was that the Bold and Broad Mathematics got the results. "To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical enquiries." Only the purist had to be sure of the validity of the processes employed."
Apocryphal

„Only by a study of the development of mathematics can its contemporary significance be understood.“

—  George Frederick James Temple British mathematician 1901 - 1992

100 Years of Mathematics: a Personal Viewpoint (1981)
Контексте: The professional mathematician can scarcely avoid specialization and needs to transcend his private interests and take a wide synoptic view of the whole landscape of contemporary mathematics. His scientific colleagues are continually seeking enlightenment on the relevance of mathematical abstractions. The undergraduate needs a guidebook to the topography of the immense and expanding world of mathematics. There seems to be only one way to satisfy these varied interests... a concise historical account of the main currents... Only by a study of the development of mathematics can its contemporary significance be understood.

Bertrand Russell фото

„Only mathematics and mathematical logic can say as little as the physicist means to say.“

—  Bertrand Russell logician, one of the first analytic philosophers and political activist 1872 - 1970

The Scientific Outlook (1931)
1930s
Контексте: Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.

Lucio Russo фото

„Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.“

—  Lucio Russo Italian historian and scientist 1944

2.4, "Discrete Mathematics and the Notion of Infinity", p. 45
The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (2004)

Eduard Jan Dijksterhuis фото
Hans Reichenbach фото
L. E. J. Brouwer фото
John Von Neumann фото
Paul Klee фото

„To emphasize only the beautiful seems to me to be like a mathematical system that only concerns itself with positive numbers.“

—  Paul Klee German Swiss painter 1879 - 1940

Diary entry (March 1906), # 759, in The Diaries of Paul Klee, 1898-1918; University of California Press, 1968
1903 - 1910

Auguste Comte фото

„Mathematical Analysis is… the true rational basis of the whole system of our positive knowledge.“

—  Auguste Comte French philosopher 1798 - 1857

Bk. 1, chap. 1; as cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), p. 224
System of positive polity (1852)

Bertrand Russell фото

„If a "religion" is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.“

—  Bertrand Russell logician, one of the first analytic philosophers and political activist 1872 - 1970

John D. Barrow, Between Inner and Outer Space: Essays on Science, Art and Philosophy (Oxford University Press, 2000, ISBN 0-192-88041-1, Part 4, ch. 13: Why is the Universe Mathematical? (p. 88). Also found in Barrow's "The Mathematical Universe" http://www.lasalle.edu/~didio/courses/hon462/hon462_assets/mathematical_universe.htm (1989) and The Artful Universe Expanded (Oxford University Press, 2005, ISBN 0-192-80569-X, ch. 5, Player Piano: Hearing by Numbers, p. 250
Misattributed

David Deutsch фото
John D. Barrow фото

„If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.“

—  John D. Barrow British scientist 1952

The Artful Universe (1995)
Контексте: If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.<!-- Ch. 5, p. 211

Связанные темы