Rigor Quotes (29 quotes)
[about Fourier] It was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius.
As quoted in P. Davis and R. Hersh The Mathematical Experience (1981).
Almost every major systematic error which has deluded men for thousands of years relied on practical experience. Horoscopes, incantations, oracles, magic, witchcraft, the cures of witch doctors and of medical practitioners before the advent of modern medicine, were all firmly established through the centuries in the eyes of the public by their supposed practical successes. The scientific method was devised precisely for the purpose of elucidating the nature of things under more carefully controlled conditions and by more rigorous criteria than are present in the situations created by practical problems.
Personal Knowledge (1958), 183.
At times the mathematician has the passion of a poet or a conqueror, the rigor of his arguments is that of a responsible statesman or, more simply, of a concerned father, and his tolerance and resignation are those of an old sage; he is revolutionary and conservative, skeptical and yet faithfully optimistic.
— Max Dehn
Address (18 Jan 1928) at the University of Frankfurt am Main, Germany. Trans. by Abe Schenitzer, and published in 'The Mentality of the Mathematician: A Characterization', The Mathematical Intelligencer (1983), 5, No. 2. As quoted in Michael Fitzgerald and Ioan James, The Mind of the Mathematician (2007), 6.
Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.
'Mathematical Problems', Bulletin of the American Mathematical Society (Jul 1902), 8, 441.
Exercise in the most rigorous thinking that is possible will of its own accord strengthen the sense of truth and right, for each advance in the ability to distinguish between correct and false thoughts, each habit making for rigour in thought development will increase in the sound pupil the ability and the wish to ascertain what is right in life and to defend it.
In Anleitung zum mathematischen Unterricht in den höheren Schulen (1906), 28.
I do not hope for any relief, and that is because I have committed no crime. I might hope for and obtain pardon, if I had erred, for it is to faults that the prince can bring indulgence, whereas against one wrongfully sentenced while he was innocent, it is expedient, in order to put up a show of strict lawfulness, to uphold rigor… . But my most holy intention, how clearly would it appear if some power would bring to light the slanders, frauds, and stratagems, and trickeries that were used eighteen years ago in Rome in order to deceive the authorities!
In Letter to Nicolas-Claude Fabri de Peiresc (22 Feb 1635). As quoted in translation in Giorgio de Santillana, The Crime of Galileo (1976), 324.
I have come to the conclusion that the exertion, without which a knowledge of mathematics cannot be acquired, is not materially increased by logical rigor in the method of instruction.
In Jahresbericht der Deutschen Mathematiker Vereinigung (1898), 143.
I learned the inebriation of research, the practice of rigor, and the art of disappointment.
In 'J. Michael Bishop: Biographical', website of nobelprize.org.
If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago. ... This time, however, it is not only based on intuition, but also on experiments of great precision and sophistication, and on a rigorous and consistent mathematical formalism.
In The Tao of Physics (1975), 19.
If there is something very slightly wrong in our definition of the theories, then the full mathematical rigor may convert these errors into ridiculous conclusions.
Feynman Lectures on Gravitation, edited by Brian Hatfield (2002), 21.
If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
In Letter to a friend, as quoted in George Finlay Simmons, Calculus Gems (1992), 188.
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.
In Electromagnetic Theory (1892), Vol. 2, 5.
It [analysis] lacks at this point such plan and unity that it is really amazing that it can be studied by so many people. The worst is that it has not at all been treated with rigor. There are only a few propositions in higher analysis that have been demonstrated with complete rigor. Everywhere one finds the unfortunate manner of reasoning from the particular to the general, and it is very unusual that with such a method one finds, in spite of everything, only a few of what many be called paradoxes. It is really very interesting to seek the reason.
In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
From a letter to his professor Hansteen in Christiania, Oslo in Correspondence (1902), 23 . In Umberto Bottazzini and Warren Van Egmond, The Higher Calculus (1986), 87-88.
It was not alone the striving for universal culture which attracted the great masters of the Renaissance, such as Brunellesco, Leonardo da Vinci, Raphael, Michelangelo and especially Albrecht Dürer, with irresistible power to the mathematical sciences. They were conscious that, with all the freedom of the individual fantasy, art is subject to necessary laws, and conversely, with all its rigor of logical structure, mathematics follows aesthetic laws.
From Lecture (5 Feb 1891) held at the Rathhaus, Zürich, printed as Ueber den Antheil der mathematischen Wissenschaft an der Kultur der Renaissance (1892), 19. (The Contribution of the Mathematical Sciences to the Culture of the Renaissance.) As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 183.
Kirchhoff’s whole tendency, and its true counterpart, the form of his presentation, was different [from Maxwell’s “dramatic bulk”]. … He is characterized by the extreme precision of his hypotheses, minute execution, a quiet rather than epic development with utmost rigor, never concealing a difficulty, always dispelling the faintest obscurity. … he resembled Beethoven, the thinker in tones. — He who doubts that mathematical compositions can be beautiful, let him read his memoir on Absorption and Emission … or the chapter of his mechanics devoted to Hydrodynamics.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 30, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 187. From the original German, “Kirchhoff … seine ganze Richtung war eine andere, und ebenso auch deren treues Abbild, die Form seiner Darstellung. … Ihn charakterisirt die schärfste Präcisirung der Hypothesen, feine Durchfeilung, ruhige mehr epische Fortentwicklung mit eiserner Consequenz ohne Verschweigung irgend einer Schwierigkeit, unter Aufhellung des leisesten Schattens. … er glich dem Denker in Tönen: Beethoven. – Wer in Zweifel zieht, dass mathematische Werke künstlerisch schön sein können, der lese seine Abhandlung über Absorption und Emission oder den der Hydrodynamik gewidmeten Abschnitt seiner Mechanik.” The memoir reference is Gesammelte Abhandlungen (1882), 571-598.
Man has never been a particularly modest or self-deprecatory animal, and physical theory bears witness to this no less than many other important activities. The idea that thought is the measure of all things, that there is such a thing as utter logical rigor, that conclusions can be drawn endowed with an inescapable necessity, that mathematics has an absolute validity and controls experience—these are not the ideas of a modest animal. Not only do our theories betray these somewhat bumptious traits of self-appreciation, but especially obvious through them all is the thread of incorrigible optimism so characteristic of human beings.
In The Nature of Physical Theory (1936), 135-136.
Mathematical rigor is like clothing; in its style it ought to suit the occasion, and it diminishes comfort and restrains freedom of movement if it is either too loose or too tight.
In Differential Equations: With Applications and Historical Notes (1972), ix.
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?
As quoted by Charles Melbourne Focken in Dimensional Methods and Their Applications (1953), 17.
Measurement has too often been the leitmotif of many investigations rather than the experimental examination of hypotheses. Mounds of data are collected, which are statistically decorous and methodologically unimpeachable, but conclusions are often trivial and rarely useful in decision making. This results from an overly rigorous control of an insignificant variable and a widespread deficiency in the framing of pertinent questions. Investigators seem to have settled for what is measurable instead of measuring what they would really like to know.
'Patient Care—Mystical Research or Researchable Mystique/', Clinical Research (1964), 12, no. 4, 422.
Real science exists, then, only from the moment when a phenomenon is accurately defined as to its nature and rigorously determined in relation to its material conditions, that is, when its law is known. Before that, we have only groping and empiricism.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (reprint 1999), 74.
Rigor is the gilt on the lily of real mathematics.
In 'The Ancients Versus the Moderns, a New Battle of the Books', The Mathematics Teacher (Oct 1958), 51, No. 6, 423.
The great artifice of regarding small deviations from the truth as being the truth itself is at the same time the foundation of wit, where the whole thing would often collapse if we were to regard these deviations in a spirit of philosophical rigor.
Aphorism from Georg Christoph Lichtenberg and R.J. Hollingdale (trans.) 'Notebook A: 1765-1770', The Waste Books (1990), 4. Also seen translated as, “The great trick of regarding small departures from the truth as the truth itself—on which is founded the entire integral calculus—is also the basis of our witty speculations, where the whole thing would often collapse if we considered the departures with philosophical rigour,” for example, as quoted in FractalVision: Put Fractals to Work For You (1992), 5, citing Aphorisms: 1764-1799.
The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
In History of Mathematics (3rd Ed., 1901), 468.
The only way in which to treat the elements of an exact and rigorous science is to apply to them all the rigor and exactness possible.
Quoted in De Morgan, Trigonometry and Double Algebra (1849), Title page.
The productive research worker is usually one who is not afraid to venture and risk going astray, but who makes a rigorous test for error before reporting his findings.
In The Art of Scientific Investigation (1957), 59.
The sociological context of the times [affects education]. Some people call it television culture—you’re supposed to be able to get everything in 30 seconds, a sort of quiz-show attitude.
Recognizing in education a declining “dedication to rigorous thinking and the fact that things are really hard to understand.” In interview, Rushworth M. Kidder, 'Grounded in Space Science', Christian Science Monitor (22 Dec 1989).
There is no rigorous definition of rigor.
In Mathematics: The Loss of Certainty (1982), 315.
True rigor is productive, being distinguished in this from another rigor which is purely formal and tiresome, casting a shadow over the problems it touches.
From address to the section of Algebra and Analysis, International Congress of Arts and Sciences, St. Louis (22 Sep 1904), 'On the Development of Mathematical Analysis and its Relation to Certain Other Sciences,' as translated by M.W. Haskell in Bulletin of the American Mathematical Society (May 1905), 11, 417.
Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly, mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.
As quoted in Hermann Weyl, Unterrichtsblätter für Mathematik und Naturwissenschaften (1932), 38, 177-188. As translated by Abe Shenitzer, in 'Part I. Topology and Abstract Algebra as Two Roads of Mathematical Comprehension', The American Mathematical Monthly (May 1995), 102, No. 7, 453.
In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
(1987) -- 
