Insider perspectives: Mathematicians on Mathematics

Revised & Expanded May 2023. First published November 1998.

This article provides a selection of quotes, written mostly by mathematicians, that convey especially clearly essential aspects of mathematics and its culture. Comments are collected in the endnotes.

Contents
1. The Essence of Mathematics
2. The ‘Why’ of Mathematics
3. The ‘How’ of Mathematics
4. Tension in the Teaching and Learning of Mathematics
5. Doing Mathematics
6. Motivating the Required Effort
7. People in Mathematics
8. The Place of Anthropology and Historiography
9. Mathematical Humour

1. The Essence of Mathematics

Quotes by Courant, Felix Klein, Alain Connes (Fields Medalist 1982), John von Neumann, and Heinrich Hertz

1. “The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. Any one or another of these aspects of mathematics can be at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the ‘concrete’ ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual ‘reality’. In brief, the flight into abstract generality must start from and return again to the concrete and specific.” — Richard Courant (mathematician), from Mathematics in the Modern World, Scientific American Vol.211 No.3, pp.41-49, 1964

2. “In the history of the development of mathematics, three different processes of growth now change places, now run side by side independent of one another, now finally mingle. Plan A [‘axiomatic’] is based upon a more particularistic conception of science which divides the total field into a series of mutually separated parts and attempts to develop each part for itself, with a minimum of resources and with all possible avoidance of borrowing from neighboring fields. Its idea is to crystallize out each of the partial fields into a logically closed system. Plan B [‘big picture’] lays the chief stress upon the organic combination of the partial fields, and upon the stimulation which these exert one upon another. Plan B prefers, therefore, the methods which open an understanding of several fields under a uniform point of view. Its ideal is the comprehension of the sum total of mathematical science as a great connected whole. There is still a third Plan C [‘computational’], algorithmic, which, along side of and within the processes of development A and B, often plays an important role as a quasi-independent, onward-driving force, inherent in the formulas, operating apart from the intention and insight of the mathematician at the time, often indeed in opposition to them.” — Felix Klein (mathematician), Concerning the General Structure of Mathematics, pp.78-79, from Elementary Mathematics from an Advanced Standpoint, Vol.1, Arithmetic, Algebra, Analysis, 1908

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3. “Mathematics is the backbone of modern science and a remarkably efficient source of new concepts and tools to understand the reality in which we participate. The concepts themselves are the result of a long process of distillation in the alembic of human thought.” – Alain Connes, (mathematician, Fields Medalist 1982, publications), from Advice to the Beginner, 2006

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4. “There are two fundamental sources of bare facts for the mathematician, that is, there are some real things out there to which we can confront our understanding. These are, on the one hand the physical world which is the source of geometry, and on the other hand the arithmetic of numbers which is the source of number theory. Any theory concerning either of these subjects can be tested by performing experiments either in the physical world or with numbers.” – Alain Connes (mathematician) from Non-Commutative Geometry, 2000

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5. “Mathematics as a science, can also consider the possible connections of things and thus can exist quite independently of the facts of natural science or of metaphysics.” Felix Klein – [Klein, 1908b, 63]

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6. Mathematical ideas originate in empirical facts. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative [subject] governed by almost entirely aesthetic motivations than to anything else and in particular to an empirical science. But there is grave danger that the subject will develop along the line of least resistance, that the stream, so far from its [empirical] source, will separate into a multitude of insignificant tributaries, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical sources or after much abstract inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached, then the only remedy seems to be a rejuvenating return to the source: a re-injection of more or less directly empirical ideas. — John von Neumann, (mathematician, computer scientist), from The Mathematician, in The Works of the Mind, Vol.1, pp.180-196, ed. R. Haywood, 1947.

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7. Applications should always accompany theory in the teaching of mathematics! The greatest mathematicians, as did Archimedes, Newton, and Gauss, always united theory and applications in equal measure. The living thing in mathematics, its most important stimulus, its effectiveness in all directions, depends entirely upon the applications, i.e. upon the mutual relations between those purely logical things and all other domains. The needs of mathematics instruction require precisely a certain many-sidedness of the individual teacher, a comprehensive orientation in the field of pure and applied mathematics in the broadest sense. – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.15-16]

8. “We cannot study this wonderful theory [Maxwell’s electromagnetic theory] without at times feeling as if an independent life and a reason of its own dwelt in these mathematical formulas; as if they were wiser than we were, wiser even than their discoverer; as if they gave out more than had been put into them.” — Heinrich Hertz (physicist), 1889 (monograph 1890), conference in Heidelberg, referring to Maxwell’s theory of elecromagnetism. (Source: Shour, 2021)

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   2. The ‘Why’ of Mathematics

   Quotes by Yuri Manin, Herstein, Stillwell, Felix Klein, and Richard Hamming, Sir Lawrence Bragg (physicist) and Ralph Waldo Emerson (philosopher)
   

9. Good proofs are those that make us wiser. When I was very young I was extremely interested in the fact that Gauss found seven or eight proofs of the quadratic reciprocity law. What bothered me was why he needed seven or eight proofs. Every time I gained some more understanding of number theory I better understood Gauss’ mind. Of course he was not looking for more convincing arguments — one proof is sufficiently convincing. The point is, that proving is the way we are discovering new territories, new features of the mathematical landscape. Any proof is a way to get from one place to another place: when you go different ways you see different things.” – Yuri Manin, 1998, Interview, Berlin Intelligencer.

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10. The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them. — Sir Lawrence Bragg (physicist), Beyond Reductionism

11. Very often in mathematics the crucial problem is to recognize and to discover what are the relevant concepts; once this is accomplished the job may be more than half done. — I.N. Herstein (mathematician), Topics in Algebra

12. “One of the disappointments experienced by most mathematics students is that they never get a course on mathematics. They get courses in calculus, algebra, topology, and so on, but the division of labor in teaching seems to prevent these different topics from being combined into a whole. In fact, some of the most important and natural questions are stifled because they fall on the wrong side of topic boundary lines. Algebraists do not discuss the fundamental theory of algebra because “that’s analysis”, and analysts do not discuss Riemann surfaces because “that’s topology,” for example. Thus, if students are to feel they really know mathematics by the time they graduate, there is a need to unify the subject.” – John Stillwell, 1989, preface to Mathematics and Its History

13. “[You should seek] the mutual connection between problems in the various fields, a thing which is not brought out sufficiently in the usual lecture course. The real goal of your academic study [should be] the ability to draw (in ample measure) from the great body of knowledge put there before you, a living stimulus for your teaching.” – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.2]

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14. The value of a principle is the number of things it will explain. — Ralph Waldo Emerson (philosopher)

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15. The purpose of computation is insight, not numbers. — Richard Hamming (mathematician)
    

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    3. The ‘How’ of Mathematics

16. “Nothing is in the intellect that was not first in the senses.” (“Nihil in intellectu quod non prius in sensu” (Latin) – Peripatetic axiom. Old Empiricist Aphorism

17. “Premature abstraction falls upon deaf ears, whether they belong to mathematicians or to students.” — Morris Kline, Mathematical Thought from Ancient to Modern Times (Kline/1972)
    

18. By producing examples and by observing the properties of special mathematical objects, one could hope to obtain clues as to the behavior of general statements which have been tested on examples. — S.M. Ulam, Adventures of a Mathematician

19. A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one. — Paul R. Halmos (mathematician)

20. In any area [of knowledge] the number of ideas is limited. Everything else is a variation on the theme. In mathematics the number of ideas is not large. Everything that is achieved is obtained from the basic (or fundamental) concepts which are applied with some degree of variations. Mastering these basic concepts in one field of mathematics helps to distinguish and use them in other fields.” — I.M. Gelfand, from Introduction to Geometry

21. Theory is the general; experiments are the soldiers. — Leonardo da Vinci (mathematician)

22. The experiment serves two purposes, often independent one from the other: it allows the observation of new facts, hitherto either unsuspected, or not yet well defined; and it determines whether a working hypothesis fits the world of observable facts. — Rene J. Dubos

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23. “[Ramanujan] worked by induction from numerical examples far more than the majority of modern mathematicians; all of his congruence properties of partitions, for example, were discovered in this way. With his memory, his patience, and his power of calculation he combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling…” – G.H. Hardy, Ramanujan

24. “There is a strong and constructive interplay between computation, heuristic reasoning, and conjecture. It is said that Gauss, who was an excellent computationalist, needed only to work out a concrete example or two to discover, and then prove, the underlying theorem.” – Carl Pomerance, Computational Number Theory, Princeton Companion to Mathematics

25. Gauss, one of the first of the modern mathematicians to insist upon rigor, would privately engage in laborious computation and the development of numerous specific examples, for example counting through tables of factors containing million of primes (Chernac, 1811, Burckhardt, 1816) before conjecturing the celebrated Prime Number Theorem, that the number of primes up to N asymptotically approaches [Bullynck, 2010, pp.198-203]. In his public presentations, Gauss often gave no trace of his private explorations, to the extent that no less a mathematician than Abel deplored: ‘[Gauss] is like the fox, who erases his tracks in the sand with his tail.’ [Kleiner, 2007, p.143], to which Gauss is reported to have replied ‘No self respecting architect leaves the scaffolding in place after completing his building.’[Veisdal, 2020 Sep 3].

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26. “Many of the important classical theorems in number theory were discovered as a by-product of the production and inspection of tables [of factors].” [D.H. Lehmer, 1969] in Computer Technology Applied to the Theory of Numbers (chapter)

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    4. Tension in Teaching and Learning Abstract Mathematics

27. “The manner of instruction [that works best in the schools] can be designated by the words intuitive and genetic, i.e. the entire structure is gradually erected on the basis of familiar, concrete things, in marked contrast to the customary logical and systematic method at the university. … To the words intuitive and genetic, we can add a third word, applications.” – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.6-7]

28. “The presentation in the schools should be psychological and not systematic. The teacher must be a diplomat. He must take account of the psychic processes in the child in order to grip their interest, and he will only succeed if he presents things in a form intuitively comprehensible. [Indeed] one should take it to heart, that in all instruction, even in the university, mathematics should be associated with everything that is seriously interesting to the pupil at that particular stage of his development and that can in any way be brought into relation with mathematics.” – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.3-4]

    
    “For beginners, pure theory is sterile unless it can be seen to be used for something.” — Ralph P. Boas, Invitation to Complex Analysis, 1987

29. “I have adopted a parallel mode of development in which the abstract concepts are introduced only as they are needed to understand the computations. We never introduce a concept without first justifying its importance and relating it to something already in the students’ sphere of interest. In this way, the students see the abstract part of the text as a natural outgrowth of the computational part. We often begin with the application and show how the necessity to solve a real world problem forces us to develop certain mathematical tools. This kind of development is more interesting for the student, and more realistic. Engineers or mathematicians will be required to do exactly this kind of thinking when they get into the workplace.” – Richard C. Penney, [Penney, 1998]

30. “The purpose of proof is to remove doubt and convey insight, not to belabor the obvious. I use the word ‘proof’ to mean an argument that I hope my intended audience will find convincing. … Students differ from logicians in their power of skepticism, and logicians differ among themselves from one generation to the next. It seems unlikely that any fixed, unalterable, absolute meaning can possibly be attached to the concept of proof. What a proof is depends on who and when you are.” [Simmons, 1987, p.2-3]

31. Only professional mathematicians learn anything from proofs. Other people learn from explanations. – Ralph P. Boas, Jr.

32. “Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse [for the mathematician] is that its perception should be instantaneous, to see at a moment’s glance the whole architecture and all its ramifications. Clinging stubbornly to the logical sequence inhibits visualization of the whole, and yet this logical structure must predominate or chaos would result.” [Emil Artin, A Review of Bourbaki’s Algebra, Bulletin of the American Mathematical Society, 1953, p.474]

33. “Many have thought that one could, or indeed that one must, teach all mathematics deductively throughout, by starting with a definite number of axioms and deducing everything from these by means of logic. This method… does not correspond to the historical development of mathematics. Mathematics has grown like a tree, which sends its roots deeper and deeper at the same time and rate that its branches and leaves are spreading upward. It progresses according to the demands of science itself, and of prevailing interests, now in one direction toward new knowledge, now in the other through the study of fundamental principles. Our standpoint today with regard to foundations is different from that of the investigators of a few decades ago; what we today would state as ultimate principles and the latest truths will be still more meticulously analyzed and referred back to something still more general. As regards fundamental investigations in mathematics, there is no final ending, and therefore no first beginning which we could offer as an absolute basis for instruction. – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.15]

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34. “[Mathematics] is a beautiful subject whose qualities of elegance, order, and certainty have exerted a powerful attraction on the human mind for many centuries. Despite this, many students emerge from their [mathematics] courses with mixed feelings of confusion and relief. Why? One of the reasons is that they have been ground down by complicated trivialities and offered little compensating insight into the ideas that really matter. They have been bombarded with innumerable nit-picking definitions and also with elaborate, boring “step-reason, step-reason” proofs of statements that in most cases are obvious to begin with.”[Simmons, 1987, p.2-3]

35. Very early in our mathematical education—in fact in junior high school or early in high school itself—we are introduced to polynomials. For a seemingly endless amount of time we are drilled, to the point of utter boredom, in factoring them, multiplying them, dividing them, simplifying them. Facility in factoring a quadratic becomes confused with genuine mathematical talent. — I.N. Hernstein (mathematician), Topics in Algebra

36. “The root of the problem is slavish adherence to the doctrine of Deductive Reasoning. This is the notion that knowledge is somehow not legitimate or genuine until it has been organized into an elaborate formal system of theorems that are carefully deduced from a small number of axioms or ‘self-evident truths’ stated at the beginning. Deductive Reasoning is an interesting idea that educated people ought to know something about, just as they should know something about representative government, the internal combustion engine, and other human inventions. Mathematics [should be] considered [either] for its own sake and for the sake of its use as an indispensable tool in science and engineering, [but] not as a vehicle for teaching deductive reasoning.[Simmons, 1987, p.2-3]

37. “I strive for simplicity and clearness rather than for rigor and for logical exactness. Defining terms that are intuitively clear always causes difficulties for inexperienced readers. I have become convinced from my own experience as a student that the presence of a large number of unfamiliar terms greatly increases the difficult of a book, and therefore I have attempted to practice the greatest economy in this respect.” – I.M. Yaglom, Geometric Transformations, 1962, p.5-6

38. “[While] it is generally agreed that the axiomatic method provides the most elegant and efficient technique for the study of modern or ‘abstract’ algebra, [it should be remembered that] the axiomatic method is an organizing principle and not the substance of the subject. A survey of algebraic structures is liable to promote the misconception that mathematics is the study of axiom systems of arbitrary design.” (Allen Clark), [Clark, 1971, v]

39. When physicists write articles, they generally start them with a paragraph saying, “Up until now, this has been thought to be the case. Now, so-and-so has pointed out this problem. In this article, we are going to try to suggest a resolution of this difficulty.” On the other hand, I have seen books of mathematics, not just articles, but books, in which the first sentence of the preface was, “Let H be a nilpotent subgroup of …” These books are written in what I would call a lapidary style. The idea seems to be that there should be no word in the book that is not absolutely necessary, that is inserted merely to help the reader to understand what is going on. I think this is getting much better, but I think a lot more has to be done. There is still too much mathematics written which is not only not understandable to experimental or theoretical physicists, but is not even understandable to mathematicians who are not the graduate students of the author. — Steven Weintraub, from Mathematics: The Unifying Thread in Science, Notices of the American Mathematical Society, Vol 33, 1986, pp.716-733

40. I believe that the teaching of linear algebra has become too abstract. This subject is as useful and central and applicable as calculus. It has a simplicity which is too valuable to be sacrificed. Linear algebra allows and even encourages a very satisfying combination of both elements of mathematics — abstraction and application. We hope to treat linear algebra in a way that concentrates more on understanding — we try to explain rather than deduce. In every case the underlying theory has to be there; it is the core of the subject, but it can be motivated and reinforced by examples. Our goal is to prepare for the applications, and that preparation can only come by understanding the theory. – Gilbert Strang [Strang, 1976]

41. “[I] would put the function concept at the very center of instruction because, of all the concepts of the mathematics of the past two centuries (1700-1900), this one plays the leading role wherever mathematical thought is used.” – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.4]
    

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    5. Doing Mathematics

42. The secret of science is to ask the right questions, and it is the choice of problem more than anything else that marks the man of genius in the scientific world. — Sir Henry Tizard in C.P. Snow, A Postscript to Science and Government

43. It is by logic we prove, it is by intuition that we invent. — Henri Poincare, Mathematical definitions in education (1904)

44. “The secret to being dull is to say everything.” – Voltaire

45. The only way to learn mathematics is to do mathematics — Paul Halmos (mathematician, Hilbert Space Problem Book

46. The difference between a text without problems and a text with problems is like the difference between learning to read a language and learning to speak it. — Freeman Dyson (physicist), Disturbing the Universe

47. You can see a lot just by looking. — Yogi Berra

48. “By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race… By the aid of symbolism, we can make transitions almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain. It is a profoundly erroneous truism, that we should cultivate the habit of thinking about what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.” – Alfred North Whitehead, from Introduction to Mathematics, 1911

49. “[In] a meeting dedicated to Al-Khorezmi, Donald Knuth started his talk with a funny statement. In his opinion the primary importance of computers for the mathematical community is that [it created a subculture for] those people who were interested in mathematics but had an algorithmic sort of mind. I take this argument quite seriously. I have a great suspicion that for example Euler today would spend much more of his time on writing software because he spent so much of his time e.g. in efforts in calculating tables of moon positions. And I believe that Gauss as well would spend much more time sitting in front of the screen.” – Yuri Manin, Interview, “Good Proofs are Proofs that Make us Wiser.”, Berlin Intelligencer, 1998, pp.1-6

50. Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long as relations do not change. — Henri Poincare (mathematician)

51. “Often in mathematics, understanding comes from generalisation, instead of considering the object per se one tries to find the concepts which embody the power of the object.” [Alain Connes, 2000, Non-Commutative Geometry]

52. The whole of science is nothing more than a refinement of everyday thinking. — Albert Einstein, Physics and Reality

53. All perception of truth is the detection of an analogy. — Henry David Thoreu, Journal

    
    

54. “Proofs and Refutations” is a method that follows “a very general heuristic pattern of mathematical discovery” but which was however only discovered it seems in the 1840s [by Seidel 1847], after the “infallibilist conceit” of Euclidean rationalism in mathematics was challenged by the discovery of non-Euclidean geometries. “Yet even today it seems paradoxical to many people”, such is the authoritarian nature of the Euclidean, deductive approach to mathematics, and the inductive approach to science. See Imre Lakatos, Proofs and Refutations, 1976, pp.127-128, 139, 142-144

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55. The Proofs and Refutations methodology follows “a simple pattern of the growth of informal mathematical theories, which consists of the following stages:
    1. primitive conjecture,
    2. proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures or lemmas),
    3. ‘global’ counterexamples (to the primitive conjecture) emerge,
    4. proof re-examined: the guilty lemma to which the global counterexample is a local counterexample is spotted. This guilty lemma may have previously remained hidden or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem (the improved conjecture) supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.
       These four stages constitute the essential kernel of proof analysis. Further stages are:

    5. proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance
    6. the hitherto accepted consequences of the original and now refuted conjecture are checked;
    7. counterexamples are turned into new examples – new fields of inquiry open up.”

    – Imre Lakatos, Proofs and Refutations, 1976, pp.127-128

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56. Learn to reason forward and backward on both sides of a question. — Thomas Blandi

57. My mind rebels at stagnation. Give me problems, give me work, give me the most obstruse cryptogram, or the most intricate analysis, and I am in my own proper atmosphere. — Sherlock Holmes, The Sign of the Four

58. “Data! data! data!” he cried impatiently. “I can’t make bricks without clay.” — Sherlock Holmes

59. “I’ve yet to see any problem, however complicated, which when you looked at it the right way, didn’t become still more complicated.” — Poul Anderson, 1957, “Call Me Joe” (source)

60. Sixty minutes of thinking of any kind is bound to lead to confusion and unhappiness. — James Thurber

61. There’s a mighty big difference between good, sound reasons and reasons that sound good. — Burton Hillis

62. Error is a hardy plant; it flourisheth in every soil. — Martin Farquhar Tupper, Proverbial Philosophy [1838-1842]

63. Give me a fruitful error anytime, full of seeds, bursting with its own corrections. You can keep your sterile truth for yourself! — Vilfredo Pareto

64. Every prospector drills many a dry hole, pulls out his rig, and moves on. — John L. Hess

65. Failure is instructive. The person who really thinks learns quite as much from his failures as from his successes. — John Dewey

66. If you don’t learn from your mistakes, there’s no sense in making them. — Herbert V. Prochnow

67. Damn it, if the machine can detect an error, why can’t it locate the position of the error and correct it? — Richard W. Hamming (mathematician)

68. When I was young I observed that nine out of every ten things I did were failures, so I did ten times more work. — George Bernard Shaw

69. The road to wisdom?---Well it's plain and simple to express:
    Err
    and err
    and err again
    but less
    and less
    and less.
    -- Piet Hein, ``The Road to Wisdom,'' Grooks (1966)
    

70. Think for yourself. — Title of a song by George Harrison, Rubber Soul

71. There is always a right and a wrong way, and the wrong way always seems the more reasonable. — George Moor, The Bending of the Bough.

72. It took me so long to find out. — John Lennon and Paul McCartney, Day Tripper, single

73. Think and you won’t sink. — B.C. Forbes, Epigrams

74. He thinks things through very carefully before going off half-cocked. — General Carl Spaatz, in Presidents Who Have Known Me, George E. Allen

75. Think before you think! — Stanislaw J. Lec, Unkempt Thoughts

76. You know my methods, apply them! — Sherlock Holmes, The Hound of the Baskervilles

77. One machine can do the work of fifty ordinary men. No machine can do the work of one extraordinary man. — Elbert Hubbard, Roycraft Dictionary and Book of Epigrams
    

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    6. Motivating the Required Effort

    from the chapter and exercise headings of Professor Joseph A. Gallian’s text Contemporary Abstract Algebra, 3rd edition, 1994, D.C. Heath & Co.

78. “The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying.” — Jean Dieudonne

79. “I needed to study. You have to neglect things if you intend to get what you want done. There’s no question about this.” – Richard Hamming, “You and Your Research”, a talk at Bell Labs, 7 March, 1986

80. Problems worthy of attack
    prove their worth by hitting back.
    -- Piet Hein, ``Problems,'' Grooks (1966)
    

81. It looked absolutely impossible. But it so happens that you go on worrying away at a problem in science and it seems to get tired, and lies down and lets you catch it. — William Lawrence Bragg (Note: Bragg, at age 24, won the Nobel Prize for the invention of x-ray crystallography. He remains the youngest person ever to receive the Nobel Prize.)

82. Perhaps the most valuable result of all education is the ability to make yourself do the thing you have to do, when it ought to be done, whether you like it or not. — Thomas Henry Huxley, Technical Education

83. Work now or wince later. — B.C. Forbes, Epigrams

84. It is a great nuisance that knowledge can only be acquired by hard work. — W. Somerset Maugham

85. All wish to possess knowledge, but few, comparatively speaking, are willing to pay the price — Juvenal

86. The dictionary is the only place where success comes before work. — Arthur Brisbane

87. The good Lord made us with two ends—one to sit on and one to think with. How well you succeed in life depends on which one you use. — Isaac Dworetsky

88. You know it ain’t easy, you know how hard it can be. — John Lennon and Paul McCartney, The Ballad of John and Yoko.

89. When you feel how depressingly
    slowly you climb,
    it's well to remember that
    Things Take Time.
    -- Piet Hein, ``T.T.T.,'' Grooks (1966)
    

90. If but the will be firmly bent, no stuff resists the mind’s intent. — Oliver St. John Gogarty, The Image-Maker

91. The greater the difficulty, the more glory in surmounting it. Skillful pilots gain their reputation from storms and tempests. — Epicurus

92. We can work it out. — single, December 1965, John Lennon and Paul McCartney

93. He who labors diligently need never despair; for all things are accomplished by diligence and labor. — Menander of Athens

94. Difficulties strengthen the mind, as labor does the body. — Seneca

95. The more we do, the more we can do. — William Hazlitt

96. Seeing much, suffering much, and studying much are the three pillars of learning. — Benjamin Disraeli

97. For those who keep trying failure is temporary. — Frank Tyger

98. Failure is the path of least persistence. — Author unknown

99. With every mistake we must surely be learning. — George Harrison, While My Guitar Gently Weeps, The Beatles (White Album)

100. We are an intelligent species and the use of our intelligence quite properly gives us pleasure. In this respect the brain is like a muscle. When it is used we feel very good. Understanding is joyous. — Carl Sagan, Broca’s Brain

101. “You don’t understand mathematics, you just get used to it.” – Von Neumann (attributed, perhaps apocryphal?)
     Source: http://gowers.wordpress.com/2007/09/13/how-should-logarithms-be-taught/#more-5
     Comment by Jeremy Henty

102. If at first you do succeed—try to hide your astonishment. — Harry F. Banks

103. If at first you don’t succeed, try, try, again. Then quit. There’s no use being a damn fool about it. — W.C. Fields

104. If at first you don’t succeed — that makes you about average. –Bradenton, Florida Herald

105. It don’t come easy! — (title of song), Ringo Starr, May 1971

106. Yes I get by with a little help from my friends. — John Lennon and Paul McCartney, Sgt. Pepper’s Lonely Hearts Club Band

107. If I rest, I rust. — Martin Luther

108. There is no substitute for hard work. — Thomas Alva Edison, Life

109. When things go wrong, as they sometimes will,
     When the road you're trudging seems all up hill,
     When care is pressing you down a bit,
     Rest, if you must---but don't you quit.
     Often the goal is nearer than
     It seems to a faint and faltering man,
     Often the struggler has given up
     When he might have captured the victor's cup.
     -- Unknown
     

110. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: it is not a subject which strikes everyone; the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it. — Carl Friedrich Gauss (mathematician)

111. If you think you can or can’t, you are right. — Henry Ford

112. Example is the school of mankind, and they will learn at no other. — Edmund Burke, On a Regicide Peace

113. No pain, no gain. — Unknown

114. Nothing worthwhile comes easy. Work, continuous work and hard work, is the only way to accomplish results that last. — Hamilton Holt

115. Work is the greatest thing in the world, so we should always save some of it for tomorrow. — Don Herald

116. Not one student in a thousand breaks down from overwork. — William Allan Neilson

117. Minus times minus is plus.
     The reason for this we need not discuss.
     -- W.H. Auden
     

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     7. People in Mathematics

118. “Read Euler, Read Euler. He is the master of us all.” – Pierre Simon Laplace

119. “The study of Euler’s works will remain the best school for the different fields of mathematics and nothing else can replace it.” – Carl Gauss
     [Dunham, 1999] – Euler, the Master of Us All, Willian Dunham, 1999 (Online)
     [Sandifer, 2007] – How Euler Did It, C. Edward Sandifer, 2007 (Online, 40 monthly columns on MAA Online)
     [Sandifer, 2015] – How Euler Did Even More, C. Edward Sandifer, 2015 (Online)

120. “Professor Felix Klein was a distinguished investigator. But he was also an inspiring teacher. With the rareness of genius, he combined familiarity with all the fields of mathematics and the ability to perceive the mutual relations of these fields; and he made it his notable function, as a teacher, to acquaint his students with mathematics, not as isolated disciplines, but as an integrated living organism. He was profoundly interested in the teaching of mathematics in the secondary schools, both as to the material which should be taught, and as to the most fruitful way it should be presented. … He endeavored to reduce the gap between the school and the university … to the end that mathematical education should be a continuous growth.” – Hedrick & Noble, Preface to the English Edition of Felix Klein’s Elementary Mathematics from an Advanced Standpoint (1908, 1924)

121. Sir Michael Atiyah was considered one of the world’s foremost mathematicians. Atiyah’s approach to mathematics was based primarily on the idea of finding new horizons and opening up new perspectives. Even if the idea was not validated by the mathematical criterion of proof at the beginning, “the idea would become rigorous in due course, as happened in the past when Riemann used analytic continuation to justify Euler’s brilliant theorems.” – Alain Connes and Joseph Kounehir, from Sir Michael Atiyah, a Knight Mathematician, 2019

122. “Some people may sit back and say, I want to solve this problem, and they sit down and say, ‘How do I solve this problem?’. I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.” – Sir Michael Atiyah, from [Connes, 2019]

123. Alexander Grothendeick

124. Paul Erdos

125. Cedric Villani

126. The logical point of view alone appears to interest [Hilbert]. Being given a sequence of propositions, he finds that all follow logically from the first. With the foundations of this first proposition, with its psychological origin, he does not concern himself. [But] Logic remains barren unless fertilised by intuition. [What is this intuition?] … To make geometry [what it is] something other than pure logic is necessary. [And] to describe this `something’ we have no word other than intuition.” — H. Poincare, A review of Hilbert’s Foundations of geometry (1902)

127. Galois (1811-1832) at seventeen was making discoveries of epochal significance in the theory of equations, discoveries whose consequences are not yet exhausted after more than a century. — E.T. Bell, Men of Mathematics

128. I really love my subject. — J.J. Sylvester (mathematician)

129. I never got a pass mark in math. The funny thing is I seem to latch on to mathematical theories without realizing what is happening. No indeed, I was a pretty poor pupil at school. — M.C. Escher

130. It is true that Fourier has the opinion that the principal object of mathematics is the public utility and the explanation of natural phenomena; but a scientist like him ought to know that the unique object of science is the honor of the human spirit and on this basis a question of the theory of numbers is worth as much as a question about the planetary system. — C.J. Jacobi (mathematician)

131. Fresh out of graduate school, he [Michael Aschbacher (mathematician)] had just entered the field, and from that moment he became the driving force behind my program. In rapid succession he proved one astonishing theorem after another. Although there were many other contributors to this final assault, Aschbacher alone was responsible for shrinking my projected 30-year timetable to a mere 10 years. — Daniel Gorenstein (mathematician), Scientific American, describing part of the 25-year effort, by hundreds of mathematicians, that led to the classification of the finite simple groups.

132. He [Gauss] lives everywhere in mathematics — E.T. Bell, Men of Mathematics

133. Gauss’s 1801 book, Disquisitiones Arithmeticae, marks the dividing line between mathematics as a universal science and mathematics as a union of special disciplines, between the geometer as a universal savant in the sense of the 18th century and the specialized mathematician of modern times. As is typical for a man of transition, Gauss does not belong to either category, he was universal and specialized. The struggle raged within him and made him suffer. — Res Jost, from Mathematics and Physics since 1800: discord and sympathy, in The Fairy Tale about the ivory tower, essays and lectures, pp. 219-240, ed. Hepp, Hunziker, Kohn

134. For [Emil] Artin, to be a mathematician meant to participate in a great common effort, to continue work begun thousands of years ago, to shed new light on old discoveries, to seek new ways to prepare the developments of the future. Whatever standards we use, he was a great mathematician. — Richard Brauer (mathematician), Bulletin of the American Mathematical Society

135. A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics. — G.H. Hardy (mathematician)
     

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     8. Anthropology/Historiography of Mathematics

136. The true method of foreseeing the future of mathematics is to study its history and its actual state — H. Poincare, Science and Method, 1908

137. “The character of mathematical thinking and argument is strongly affected — indeed is almost essentially determined — by the dynamics of the specific social, mostly professional environments in which it is carried” [Hoyrup, 2017]

     ²¹

138. The core of this case-study aims to elaborate the point that informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. The dialogue form reflects the dialectic of the story; it is meant to contain a sort of rationally reconstructed or ‘distilled’ history. The real history will chime in in the footnotes, most of which are to be taken, therefore, as an organic part of the story. – Imre Lakatos, Proofs and Refutations (PDF), 1976, p.5 (Review here, and the analogy with programming as a dialectical logic.[Benthall/2014])

     ²²

139. The diversification of mathematics [in the 20th century] was connected first of all with external social phenomena: the rapid growth of the scientific community in general and the ground-breaking discoveries in physics. In my opinion, the mathematics of the last 100 years did not produce anything comparable to quantum theory or general relativity in terms of the resulting change of our total world perception. But I do believe that without the mathematical language physicists couldn’t even say what they were seeing. This interrelation between physical discoveries and mathematical ways of thinking, the mathematical language, in which these discoveries can only be expressed, is absolutely fantastic. In this sense, the 20th century certainly will be regarded as a century of great breakthroughs. — Yuri Manin, Interview, “Good Proofs are Proofs that Make us Wiser.”, Berlin Intelligencer, 1998, pp.1-6

140. [We may say that] transition[s] to [M]athematics occurred [in history] when pre-existent and previously independent mathematical practices and techniques were wielded by specialist practitioners who were organized professionally and linked in a network of communication. Such professional groups fall into two main types: in one type knowledge is transmitted within an apprenticeship-system of ‘learning by doing under supervision’ [sub-scientific mathematics]. The other type involves some kind of school [in which] teaching is separate from actual work. In the former type, those who transmit are actively involved in the practical activities of their trade; they will tend to train exactly what is needed, and the understanding they will try to communicate will be that of practical procedures. [In the latter type, the] school teaching of mathematical skill is bound to a writing system extensive enough to carry a literate culture. [While] teachers in the school type may well have as their aim to impart knowledge for practice, the mathematical understanding that they teach will concentrate on inner connections of the topic, i.e. on mathematical explanations.” – Jens Hoyrup, 2017, Perspectives on an Anthropology of Mathematics

141. “Recently, historians have been concerned to recover how the ancients thought about mathematics. In order to recover ancient mathematical concepts, historians have gone back to the original sources and retranslated them in a way which tries to stay as faithful as possible to the texture of the original vocabulary and syntax. To use translators’ jargon, the mid-twentieth century translations [Neugebauer, etc.] are domesticating, in that they aim to make ancient mathematics familiar and comfortable. The newer translations, on the other hand, are alienating, in that they try to maintain the intellectual distance between the sources and us.” – Eleanor Robson [Robson/2005]

142. “There are many possible barriers to the reading of a text in a foreign language, and the purpose of a scholarly translation is as I understand it to remove all barriers having to do with the foreign language itself, leaving all other barriers intact. – Reviel Netz [Netz/2004: 3]
     

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     9. Mathematical Humour

143. Mathematical humour: A set of mathematical equivoques, Ken Suman

Appendix: Mathematicians’ Pages

1. Alain Connes – Noncommutative Geometry
2. John Baez – n-Category Cafe
3. Terence Tao
4. Doron Zeilberger
5. Don Knuth
6. Steven Strogatz
7. Nick Higham
8. Rob Hyndman
9. Larry Wasserman – Normal Deviate
10. Peter Woit
11. Edsgar W. Dijstra
12. Vladik Kreinovich

References

1. [Gallian, 1994] – Contemporary Abstract Algebra, 3rd edition, 1994, D.C. Heath & Co. Many of the quotes in the later sections are taken from the chapter and exercise headings of Professor Gallian’s text.
2. [Strang, 1976] – Linear Algebra and Its Applications, Gilbert Strang, 1976, 1st ed, 1980 2e, 1988 3e, 2006 4e (Online)
3. [Penney, 1998] – Linear Algebra: Ideas and Applications, Richard C. Penny, 1998 1st ed, 2004 2e, 2008 e3, xxxx 4e, 2020 5e (Online)

Footnotes