Quotations

"For each successive class of phenomena, a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature."

  - Henry John Stephen Smith, as quoted in Nature, Volume 8, page 450 (1873).


"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, indeed to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius' calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

 - Carl Friedrich Gauss, as quoted in Carl Friedrich Gauss: Werke, Volume 8, page 298; and as quoted in Robert Edouard Moritz's  book Memorabilia Mathematica or The Philomath's Quotation Book, quotation #1215 (1914).


"How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments."

 - David Hilbert, as quoted in Robert Edouard Moritz's book Memorabilia Mathematica or The Philomath's Quotation Book (1914).


"Civilization advances by extending the number of important operations which we can perform without thinking about them."

 - Alfred North Whitehead, from his book An Introduction to Mathematics (1911).


"A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it would ever be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful."

 - John von Neumann, from “The Role of Mathematics in the Sciences and in Society" a 1954 address to Princeton alumni, published in John von Neumann : Collected Works edited by A. H. Taub (1963); also quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths by R. Schmalz (1993).


"Insight must precede application."

 - Max Planck, as quoted by the Max Planck Society and used as its motto.


"As an answer to those who are in the habit of saying to every new fact, 'What is its use?', [Benjamin] Franklin says, 'What is the use of an infant?'"   

 - Michael Faraday, as quoted by I. Bernard Cohen in his article "Faraday  and Franklin's "Newborn Baby"" in Proceedings of the American Philosophical Society, Volume 131, Number 2 (1987). 


“Only yesterday the practical things of today were decried as impractical, and the theories which will be practical tomorrow will always be branded as valueless games by the practical man of today.”

 - William Feller, from his book The Nature of Probability Theory (1968).


"It is the fate of the scientist to face the constant demand that he show his learning to have some “practical use.”  Yet it may not be of interest to him to have such a “practical use” exist; he may feel that the delight of learning, of understanding, of probing the universe, is its own reward."

  - Isaac Asimov, from his article "Of what use?", an introduction to the book The Greatest Adventure: Basic Research That Shapes Our Lives (1974).


"... throughout the whole history of science most of the really great discoveries which had ultimately proved beneficial to mankind had been made by men and women who were driven not by the desire to be useful but merely by the desire to satisfy their curiosity. ... Curiosity, which may or may not eventuate in something useful, is probably the outstanding characteristic of modern thinking. It is not new. It goes back to Galileo, Bacon, and to Sir Isaac Newton, and it must be absolutely unhampered. Institutions of learning should be devoted to the cultivation of curiosity and the less they are deflected by considerations of immediacy of application, the more likely they are to contribute not only to human welfare but to the equally important satisfaction of intellectual interest which may indeed be said to have become the ruling passion of intellectual life in modern times.”

 - Abraham Flexner, from his article “The usefulness of useless ideas”, Harper’s Magazine, Issue 179, 1939.


“[Hans Christian] Oersted would never have made his great discovery of the action of galvanic currents on magnets had he stopped in his researches to consider in what manner they could possibly be turned to practical account; and so we would not now be able to boast of the wonders done by the electric telegraphs. Indeed, no great law in Natural Philosophy has ever been discovered for its practical implications, but the instances are innumerable of investigations apparently quite useless in this narrow sense of the word which have led to the most valuable results.” 

 - William Thompson (a.k.a. Lord Kelvin), from his “Introductory lecture to the course of natural philosophy”, published in Silvanus Phillips Thompson’s book The Life of Lord Kelvin (1910).


“I would undervalue [scholastic] grades based on knowing things, and find ways to reward curiosity. In the end, it's the people who are curious who change the world.” 

 - Neil deGrasse Tyson, as quoted in the book Priceless Thoughts on Knowledge by Folorunsho Mejabi (2016).       


“I am interested in mathematics only as a creativeart." - G. H. Hardy, from his book A Mathematician's Apology (1941).


“Mathematicians pride themselves on being useless.”

 - Michio Kaku, from his YouTube (Internet) video “Michio Kaku: Is God a Mathematician?”


"The reward of a thing well done is to have done it."

 - Ralph Waldo Emerson, from his Essays: Second Series, 1844; and as quoted by Robert Katz in a conversation with Michael Grossman in the 1970s.


"A century ago, [Einstein's] general relativity had no obvious “impact” ... . It didn’t even have a clear goal, except intellectually."

 - Philip Ball, from his article "There’s no space for today’s young Einsteins" in The Guardian (12 February 2016).


"When Einstein finalized his theory of gravity and curved spacetime in November 1915, ending a quest which he began with his 1905 special relativity, he had little concern for practical or observable consequences."  

 - Clifford M. Will, from his article "Einstein's relativity and everyday life: What good is fundamental physics to the person on the street?" in the PhysicsCentral website of the American Physical Society, as it appeared on 6 November 2016.


"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory."

 - Mark Kac, from Chapter 2 of his autobiography Enigmas of Chance (1985).   


"There seem to be two kinds of discovery. In one kind, the goal is given first, and then the mind goes from the goal to the means, that is, from the question to the solution. In the other kind, the mind goes from the means to the goal, that is, the mind first discovers a fact and then seeks a use for it. In mathematics, and elsewhere,  most significant discoveries are of the second kind. As Hadamard has put it, 'Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.' An outstanding example in mathematics is the exhaustive study of the conics by the Greeks, and then, some two -semitism.       thousand years later, Kepler's stunning application of the Greek findings to the movement of the planets in the solar system."

- Howard Eves, from his book Mathematical Circles Squared, pages 167 and 168 (1972).


"... we find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies - such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later."

- Arthur Koestler, from his book The Sleepwalkers(1959).


From the 3rd century onwards, orthodox Christianity, based on a Hebrew story and worshipping the Jew Jesus, also led many campaigns of anti-semitism. 

  - Ivor Grattan-Guinness, from his book The Rainbow of Mathematics: A History of the Mathematical Sciences (2000).


‘’It [ non-Euclidean geometry ] would be ranked among the most famous achievements of the entire [nineteenth] century, but up to 1860 the interest was rather slight.”

 - Ivor Grattan-Guinness, from his book The Rainbow of Mathematics: A History of the Mathematical Sciences (2000).


"Newton's epoch-making works (1669, 1671) were offered to the Royal Society and Cambridge University Press but, incredible as it now seems, were rejected for publication."

 - John Stillwell, from his book Mathematics and Its History (2010).


"New theories, methods, or interventions that challenge dogma, people in power, or "the way we've always done it" can run into resistance, ridicule, and minds shut tight. Barbara McClintock knew what that was like. A distinguished scientist, she discovered genetic transposition ("jumping genes"), a stunning advance that earned her dismissive ridicule and ostracism from colleagues for two decades. ... History furnishes all too many examples of those labeled lightweights, fools, fanatics, true believers, pseudoscientists, quacks, frauds, or heretics: Galileo Galilei, Muhammad ibn  Zakariya al-Razi, Ruth Sager, Ignaz Semmelweis, to name a few. ... Their unwavering loyalty to what they believe to be true, their persistence in looking for evidence to support their vision, and their determination to hold onto their belief despite the resistance it arouses inspire us." 

 -  Kenneth S. Pope and Melba J. T. Vasquez, from their book Ethics in Psychotherapy and Counseling: A Practical Guide (2016).


“An old error is always more popular than a new truth.”  

 - German proverb.


"Many new mathematical concepts, even though logically acceptable, meet forceful resistance after their appearance and achieve acceptance only after an extended period of time."

 - Michael J. Crowe, from his article "Ten ‘Laws’ concerning patterns of change in the history of mathematics" in Historia Mathematica (May of 1975).


“Katalin Karikó began her innovative work on mRNA in the 1970’s but suffered ridicule and repeated rejection for decades. At the University of Pennsylvania in the 1990’s, Karikó's work to apply mRNA “to fight disease was deemed too radical, too financially risky to fund. She applied for grant after grant, but kept getting rejections, and in 1995, she was demoted from her position at UPenn,” as CNN.com put it. Today, it’s the basis for the first two, highly-effective covid-19 vaccines, by Pfizer/BioNTech and Moderna.”


 - Joan Michelson, from her article “Who Is ‘Credible’? Women Innovators Are Different”, Forbes, 8 March 2021.


“An Israeli scientist [Dan Schectman] who suffered years of ridicule and even lost a research post for claiming to have found an entirely new class of solid material was awarded the 2011 Nobel Prize for chemistry last week for his discovery of quasicrystals. … "People just laughed at me," Shechtman recalled in an interview this year with Israeli newspaper Haaretz, noting how Linus Pauling, a colossus of science and double Nobel laureate, mounted a frightening "crusade" against him, saying: "There is no such thing as quasicrystals, only quasi-scientists."  After telling Shechtman to go back and read the textbook, the head of his research group asked him to leave for "bringing disgrace" on the team. … Dan Shechtman had to fight a fierce battle against established science. His battle eventually forced scientists to reconsider their conception of the very nature of matter.”

 - From the article “Ridiculed work wins Nobel for Israeli” in the China Daily (9 October 2011).


"The mathematician Georg Cantor spoke of a law of conservation of ignorance. A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held."

 - Morris Kline, from his book Mathematics: The Loss of Certainty (1980).


"The human understanding when it has once adopted an opinion ... draws all things else to support and agree with it. And though there be a greater number and weight of instances to be found on the other side, yet these it either neglects or despises, or by some distinction sets aside and rejects ...  This mischief insinuate[s] itself into philosophy and the sciences; in which the first conclusion colors and brings into conformity with itself all that come after."

 - Francis Bacon, from his book Novum Organum(1620), and as quoted in Kenneth S. Pope and Melba J. T. Vasquez's book Ethics in Psychotherapy and Counseling: A Practical Guide (2016).


"Perhaps the only thing that saves science from invalid conventional wisdom that becomes effectively permanent is the presence of mavericks in every generation - people who keep challenging convention and thinking up new ideas ... ."     

 - David M. Raup, from his book The Nemesis Affair: A Story of the Death of Dinosaurs and the Ways of Science (1999).


“Thus dramatic changes or revolutions in a field of science are often made by outsiders or "trespassers" who do not define their expertise as mastery of the old methods or by newcomers who are not yet beholden to the old ways.”

 - David Ellerman, from his book Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics, (1995).


"More than 350 years after the Roman Catholic Church condemned Galileo, Pope John Paul II is poised to rectify one of the Church's most infamous wrongs -- the persecution of the Italian astronomer and physicist for proving the Earth moves around the Sun."

 - Alan Cowell, from his article "After 350 Years, Vatican Says Galileo Was Right: It Moves" in The New York Times, 31 October 1992.


"Yet Gauss, universally regarded as the foremost mathematician of his day, did not publicize his findings. ...  In an 1829 letter to a confidant, Gauss observed that he had no plans 'to work up my very extensive researches for publication, and perhaps they will never appear in my lifetime, for I fear the howl of the Boeotians if I speak my opinion out-loud.' ... Obviously Gauss had little regard for the reception of the mathematical community for his new ideas."

 - William Dunham, as quoted in his book Journey Through Genius (1990).


"Of all of Gauss' creations, the most revolutionary in concept and the most weighty in affecting the course of mathematics and science is non-Euclidean geometry. ... Since Gauss did not make known many of his brilliant creations, other mathematicians [such as Lobatchevsky and Bolyai] worked toward and arrived at the same results independently. ... In the case of non-Euclidean geometry, Gauss had an additional reason for not publishing his work. ... [A]nyone professing to present another geometry conflicting with Euclid's would undoubtedly have been judged insane not only by the general public, which was ignorant of all geometry, but by mathematicians, scientists, and philosophers. ... There are limits to allowable heterodoxy, even in mathematics, and these limits are soon reached in every era."

 - Morris Kline, from his book Mathematics and the Physical World (1959).


"For thirty years or so after the publication of Lobatchevsky's and Bolyai's works all but a few mathematicians ignored the non-Euclidean geometries. ... The mere fact that there can be alternative geometries was in itself a shock."

 - Morris Kline, from his book Mathematics: The Loss of Certainty (1980).


“His [Nikolai Lobachevsky’s] remarkable discoveries [non-Euclidean geometry], along with those of an even more neglected mathematician, Janos Bolyai, are now recognised as the start of a gigantic revolution in human thought about geometry and the nature of physical space. But it is ever the fate of pioneers to be misrepresented and misunderstood. Ideas that should have been hailed for their originality are routinely denounced as nonsense, and their originators receive little recognition. Hostility is more likely; think of evolution and climate change. I sometimes feel that the human race doesn’t deserve its great thinkers. When they show us the stars, prejudice and lack of imagination drag us all back into the mud.”

 - Ian Stewart, from his book Significant Figures(2017).


"One may say without great exaggeration that Grassmann invented linear algebra ... Linear algebra has become a part of the mainstream of mathematics, though Grassmann gets scant credit for it ... His achievements in mathematics were virtually unrecognized, and it has taken a century for their importance to become visible."

 - Desmond Fearnlet-Sander, from his article "Hermann Grassmann and the Creation of Linear Algebra" published in The American Mathematical Monthly, Volume 86 (1979).


"Grassmann [1809-1877] invented what is now called exterior algebra. This was joined to Hamilton's quaternions by Clifford in 1878. ... Clifford algebras are used today in the theory of quadratic forms and in relativistic quantum mechanics. Clifford algebras appear together with Grassmann's exterior algebra in differential geometry. ... It is interesting to see just how many leading mathematicians failed to recognise that the mathematics Grassmann presented would become the basic foundation of the subject in 100 years time."

 - John O'Connor and Edmund Robertson, from their article "Hermann Günter Grassmann", The MacTutor History of Mathematics Archive website, University of St Andrews (2005). 


"The fame of the creator of a new mathematical concept has a powerful, almost a controlling, role in the acceptance of that mathematical concept, at least if the new concept breaks with tradition. Compare the reception accorded [William Rowan] Hamilton's Lectures on Quaternions (1853) with that of [Hermann] Grassmann's Ausdehnungslehre (1844). Both are among the classics of mathematics, yet the work of the former author, who was already famous for empirically confirmed results, was greeted with lavish praise in reviews by authors who had not read his book, whereas the book of Grassmann, an almost unpublished high school teacher, received but one review (by its author!) and found, before it was used for waste paper in the early 1860's, only a handful of readers. Or consider the fate of [Nikolai] Lobachevsky and [János] Bolyai, whose publications remained as unknown as their authors until, thirty years after their publications, some posthumously published letters of the illustrious [Carl Friedrich] Gauss led mathematicians to take an interest in non-Euclidean geometry."

 - Michael J. Crowe, from his article "Ten “Laws” concerning patterns of change in the history of mathematics" in Historia Mathematica (May of 1975).


  "The first person to really come to understand the problem of the parallels was Gauss. He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four. By 1813 he had made little progress and wrote:

In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics... 

  "However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.

  "Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate. Farkas Bolyai taught his son, János Bolyai, mathematics but, despite advising his son not to waste one hour's time on that problem of the problem of the fifth postulate, János Bolyai did work on the problem.

  "In 1823 János Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded ... out of nothing I have created a strange new world. However it took Bolyai a further two years before it was all written down and he published his strange new world as a 24 page appendix to his father's book, although just to confuse future generations the appendix was published before the book itself.

  "Gauss, after reading the 24 pages, described János Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order. However in some sense Bolyai only assumed that the new geometry was possible. He then followed the consequences in a not too dissimilar fashion from those who had chosen to assume the fifth postulate was false and seek a contradiction. However the real breakthrough was the belief that the new geometry was possible. Gauss, however impressed he sounded in the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had discovered all this earlier but had not published. Although this must undoubtedly have been true, it detracts in no way from Bolyai's incredible breakthrough.

  "Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication. Lobachevsky's attempt to reach a wider audience had failed when his paper was rejected by [the mathematician Mikhail Vasilyevich] Ostrogradski.

  "In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his momentous work. He published "Geometrical investigations on the theory of parallels" in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky's work. The publication of an account in French in Crelle's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary."

 -  John O'Connor and Edmund Robertson, from their article "Non-Euclidean geometry", The MacTutor History of Mathematics Archive website, University of St Andrews (1996).


"By and large, the reaction to Bolyai and Lobachevskii's ideas during their lifetimes was one of neglect and hostility, and they died unaware of the success their discoveries would ultimately have. ... [Their work] was dismissed with scorn, as if it were self-evident that it was wrong: so wrong that it would be a waste of time finding the error it surely contained, so wrong that the right response was to heap ridicule upon its authors or simply dismiss them without comment."

 - Jeremy Gray, from his article "Geometry" in the book The Princeton Companion to Mathematics(2010).


"Georg Cantor (1845 - 1918), the creator of transfinite set theory, is one of the most imaginative and controversial figures in the history of mathematics. ... Because his views were unorthodox, they stimulated lively debate and at times vigorous denunciation. Leopold Kronecker considered Cantor a scientific charlatan, a renegade, a "corrupter of youth," but Bertrand Russell described him as one of the greatest intellects of the nineteenth century. David Hilbert believed Cantor had created a new paradise for mathematicians, though others, notably Henri Poincare, thought set theory and Cantor's transfinite numbers represented a grave mathematical malady, a perverse pathological illness that would one day be cured. ... Cantor's creation of transfinite set theory was an achievement of major consequence in the history of mathematics."

 - Joseph Dauben, from his book Georg Cantor: His Mathematics and Philosophy of the Infinite, pages 1 and 6 (1990).


"Resistance to irrationals [i.e., irrational numbers] continued for thousands of years. In the late nineteenth century, when the gifted German mathematician Georg Cantor did groundbreaking work to put them on firmer footing, his former teacher, a crab named Leopold Kronecker who "opposed" the irrationals, violently disagreed with Cantor and sabotaged his career at every turn. Cantor, unable to tolerate this, had a breakdown and spent his last days in a mental institution."

 - Leonard Mlodinow, from his book Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (2001).


"Mathematicians, let it be known, are often no less illogical, no less closed-minded, and no less predatory than most men. Like other closed minds they shield their obtuseness behind the curtain of established ways of thinking while they hurl charges of madness against the men who would tear apart the fabric."  

 - Morris Kline, from his book Mathematics in Western Culture (1953).


"It all comes down to the adage that it is the mathematical community that decides what is worthwhile. And that decision is based on intrinsic merit but also on fashion. This may seem odd to the uninitiated, but mathematics has vogues, cliques, and prejudices, just as art surely, music, or literature, or the clothing industries do."

 - Steven G. Krantz, from his book The Proof is in the Pudding: The Changing Nature of Mathematical Proof (2011).


"One often sees it stated that Frege's work was worthless because of the inconsistency pointed out by [Bertrand] Russell. In fact this is far from the truth and one must view Frege as the person who made one of the most important contributions to the foundations of mathematics that has ever been made. In fact in many ways Russell is correct when he wrote in his History of Western Philosophy: 'In spite of the epoch-making nature of [Frege's] discoveries, he remained wholly without recognition until I drew attention to him in 1903.' ... [Joan] Weiner writes in [her book Frege: Past Masters, 1999]: 'Gottlob Frege's writings have had a profound influence on contemporary thought. His revolutionary new logic was the origin of modern mathematical logic - a field of import not only to abstract mathematics, but also to computer science and philosophy.' "

 - John O'Connor and Edmund Robertson, from their article "Friedrich Ludwig Gottlob Frege", The MacTutor History of Mathematics Archive, University of St Andrews (2002). 


"Gregor Mendel is accorded a special place in the history of genetics. ... Poignantly overshadowing the creative brilliance of Mendel's work is the fact that it was virtually ignored for thirty-four years. Only after the dramatic rediscovery in 1900 - sixteen years after Mendel's death - was Mendel rightfully recognized as the founder of genetics."

 - Daniel L. Hartl and Vitezslav Orel, from their article "What did Gregor Mendel think he discovered?", Genetics, Volume 131, Genetics Society of America (1992).


“The moral of this story is that modesty is not always a virtue. Maxwell and Mendel were both excessively modest. [Gregor] Mendel’s modesty setback the progress of biology by fifty years. [James Clerk] Maxwell’s modesty setback the progress of physics by twenty years. It is better for the progress of science if people who make great discoveries are not too modest to blow their own trumpeots. If Maxwell had had an ego of Galileo or Newton, he would have made sure that his work was not ignored.”

 - Freemen Dyson, from his essay "Why Maxwell’s Theory is so hard to understand".


“Science is not a heartless pursuit of objective information. It is a creative human activity.”

 - Stephen Jay Gould, from his book Ever Since Darwin (1977).


“A society’s competitive advantage will come not from how well its schools teach the multiplication and periodic tables, but from how well they stimulate imagination and creativity.” 

―  Albert Einstein, from Walter Isaacson’s book Einstein: His Life and Universe (2007).


“Because the academic career puts a young person in a sort of compulsory situation to produce scientific papers in impressive quantity, a temptation to superficiality arises that only strong characters are able to resist.”

 - Albert Einstein, from John Stachel’s book “Einstein from ‘B’ to’Z’ (2002).


"Sometimes scientists have too much invested in the status quo to accept a new way of looking at things. This was certainly true when Albert Einstein‘s 1905 paper on “special relativity” first challenged the British conception of ether. ... In France, Einstein was largely ignored until he visited in 1910. In the U.S., a few understood it, but, in general, relativity was ridiculed as “totally impractical and absurd.” In Britain, his theories met with resistance, because relativity was seen as a direct challenge to the widely accepted theory of ether. ... The scientific status quo was so firmly entrenched that in Britain, Einstein’s groundbreaking theory fell on deaf ears, ... ."

 - Matthew Wills, from his article "Why No One Believed Einstein, JSTOR Daily, August 19, 2016.


"Some truly revolutionary scientific theories may take years or decades to win general acceptance among scientists. This is certainly true of plate tectonics, one of the most important and far-ranging geological theories of all time; when first proposed, it was ridiculed, but steadily accumulating evidence finally prompted its acceptance, with immense consequences for geology, geophysics, oceanography, and paleontology. And the man who first proposed this theory was a brilliant interdisciplinary scientist, Alfred Wegener. ... Reaction to Wegener's theory was almost uniformly hostile, and often exceptionally harsh and scathing ... ."

  - from the article "Alfred Wegener (1880-1930)" at the website of the University of California Museum of Paleontology.


"Nikola Tesla was one of the greatest electrical inventors who ever lived. ...  Ridiculed by his contemporaries, his ideas frequently appeared in works of science fiction. ... Tesla was so far ahead of his time that many of his ideas are only appearing today. His legacy can been seen in everything from microwave ovens to MX missiles."

 - from the abstract "Who Was Nikola Tesla?" for The Public Broadcasting Service (PBS) program Tesla: Master of Lightning, which first aired on PBS in December of 2000.


"A scientist whose work was so controversial he was ridiculed and asked to leave his research group has won the [2011] Nobel Prize in Chemistry. Daniel Shechtman, 70, a researcher at Technion-Israel Institute of Technology in Haifa, received the award for discovering seemingly impossible crystal structures [quasicrystals] in frozen gobbets of metal that resembled the beautiful patterns seen in Islamic mosaics. ... In an interview this year with the Israeli newspaper, Haaretz, Shechtman said: "People just laughed at me." He recalled how Linus Pauling, a colossus of science and a double Nobel laureate, mounted a frightening "crusade" against him. After telling Shechtman to go back and read a crystallography textbook, the head of his research group asked him to leave for "bringing disgrace" on the team."

 - Ian Sample, from his article "Nobel Prize in chemistry for dogged work on 'impossible' quasicrystals" in The Guardian, 5 October 2011.


"When Ludwig Boltzmann [1844 - 1906] first established statistical mechanics, he was ridiculed by his colleagues Ernst Mach, Wilhelm Ostwald, and others for his atomistic ideas, and it took three decades until experiments made the reality of atoms and molecules concrete enough to convince the community."

 - Christoph von der Malsburg, William A. Phillips, and Wolf Singer, from their book Dynamic Coordination in the Brain: From Neurons to Mind(2010).


"Chandrasekhar developed a theory about the nature of stars for which he would be awarded a Nobel Prize 53 years later, in 1983. His theory challenged the common scientific notion of the 1930s that all stars, after burning up their fuel, became faint, planet-sized remnants known as white dwarfs. ... Initially his theory was rejected by peers and professional journals in England. The distinguished astronomer Sir Arthur Eddington publicly ridiculed his suggestion that stars could collapse into such objects, which are now known as black holes. ... Today, the extremely dense neutron stars and black holes implied by Chandrasekhar’s early work are a central part of the field of astrophysics."

 - from the The University of Chicago's press release "Subrahmanyan Chandrasekhar", 22 August 1985.


"Few things inhibit the undertaking of a new venture more than the fear of ridicule."

 - Robert Katz, as quoted in Paul Dickson's book The Official Rules [28] (2014).


"The obstacles mainly were in the very beginning, in the late '60s, when we proposed the idea that tumors need to recruit their own private blood supply. That was met with almost universal hostility and ridicule and disbelief by other scientists. ...  A lot of people would walk out of my presentations. There were many critics, very great experts who kept saying this couldn't be. ... We had ten years of really tough ridicule. I was sometimes very upset."

 ― Judah Folkman, from an interview on June 18, 1999, published in the online-article "Judah Folkman Interview" by the Academy of Achievement on 21 September 2010. (Judah Folkman (1933 - 2008) founded angiogenesis research, a field of biology which revolutionized biomedical research and clinical drug development. He created a novel approach to understanding and treating many diseases, including cancer.)


"In 1972, a British scientist [John Yudkin] sounded the alarm that sugar – and not fat – was the greatest danger to our health. But his findings were ridiculed and his reputation ruined. How did the world’s top nutrition scientists get it so wrong for so long?" 

 - Ian Leslie, from his article "The sugar conspiracy" in The Guardian (7 April 2016).


"Ignaz Semmelweis (born Semmelweis Ignác Fülöp; 1 July 1818 – 13 August 1865) was a Hungarian physician of German extraction now known as an early pioneer of antiseptic procedures. ... Semmelweis discovered that the incidence of puerperal fever (also known as "childbed fever") could be drastically cut by the use of hand disinfection in obstetrical clinics. ... The so-called Semmelweis reflex — a metaphor for a certain type of human behaviour characterized by reflex-like rejection of new knowledge because it contradicts entrenched norms, beliefs or paradigms — is named after Semmelweis, whose perfectly reasonable hand washing suggestions were ridiculed and rejected by his contemporaries."

 - from the article "Ignaz Semmelweis" at the Wikipedia website on 15 August 2015.


"In 1891, Dr. William B. Coley [1862 - 1936] injected streptococcal organisms into a patient with inoperable cancer. He thought that the infection he produced would have the side effect of shrinking the malignant tumor. He was successful, and this was one of the first examples of immunotherapy. ... William Coley's intuitions were correct: Stimulating the immune system may be effective in treating cancer.  ...  But he was a man before his time, and he met with severe criticism. Despite this criticism, however, Coley stuck with his ideas, and today we are recognizing their potential value."

 - Edward F. McCarthy, from his article "The toxins of William B. Coley and the treatment of bone and soft-tissue sarcomas" in The Iowa Orthopaedic Journal, Volume 26, US National Library of Medicine, National Institutes of Health (2006).


"Biologist Lynn Margulis died on November 22nd[2011]. Her major work was in cell evolution ...  Her ideas were generally either ignored or ridiculed when she first proposed them; [her theory of] symbiosis in cell evolution is now considered one of the great scientific breakthroughs."

 - John Brockman, from his article "Lynn Margulis 1938-2011" on the Edge.org website, 23 November 2011. 


"In Western Australia, a young doctor, working in relative isolation, pursued another hypothesis, the possibility that these chronic stomach ailments [such as ulcers and gastritis] were caused by a microscopic corkscrew-shaped organism called Helicobacter pylori. For over a decade, Dr. Barry Marshall endured the hostility and ridicule of a medical establishment deeply invested in the received wisdom that peptic ulcers were a chronic condition requiring a lifetime of treatment. ... Today his discovery is recognized as one of the greatest breakthroughs in medicine since the polio vaccine."

 - from the Academy of Achievement article "Barry Marshall: Profile", http://www.achievement.org/autodoc/page/mar1pro-1, February 26, 2010. (In 2005, Barry Marshall was awarded The Nobel Prize in Physiology or Medicine.)


"As you well know, for many years lots of people, especially various pure mathematicians, claimed that our work was useless. But, despite their discouraging and sometimes arrogant comments, we always knew that non-Newtonian calculus has considerable potential for application in science, engineering, and mathematics. -- And we were right!!"

 - Michael Grossman, from his letter to Robert Katz on 21 July 2014. 


"After a long period of silence in the field of non-Newtonian calculus introduced by Grossman and Katz [Non-Newtonian Calculus] in 1972, the field experienced a revival with the mathematically comprehensive description of the geometric calculus by Bashirov et al. ["Multiplicative calculus and its applications", 2008], which initiated a kick-start of numerous publications in this field."

 - Mustafa Riza and Bugce Eminaga, from their 2015 article "Bigeometric Calculus and Runge Kutta Method" [215].


"There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems. This very original piece of mathematics will surely expose a number of missed opportunities in the history of the subject."

 - Ivor Grattan-Guinness,  from his review of Non-Newtonian Calculus [101] (1977).


"Of course, calculus does not stop at Newton/Leibniz's work in physics or mathematics. There are many new developments in later generations. ... Calculus later developed some alternative non-Newtonian parallel calculus forms, such as the Multiplicative Calculus [Geometric Calculus], Bigeometric Calculus."

 - From an online article (in Chinese) about Newton, Leibniz, and the history of calculus; as it appeared on 25 September 2018 [451] (2018).


“[Stefan] Banach wrote "One can imagine that the ultimate mathematician is one who can see analogies between analogies." Newtonian calculus analogizes change and non-Newtonian calculus analogizes Newtonian calculus.”

 - Peter Carr, from his post, 3 May 2019, on the YouTube video “Michael Grossman talks about the Bigeometric Calculus (a multiplicative non-Newtonian calculus)”. [485] 


"[B]y far the most usual way of handling phenomena so novel that they would make for a serious rearrangement of our preconceptions is to ignore them altogether, or to abuse those who bear witness for them."

  - William James, as quoted in the book Pragmatism: A Series of Lectures by William James, 1906-1907(2008).


“Almost all of the important papers in economics and finance were rejected for publication.” (This assertion is corroborated in the 1994 article “How Are the Mighty Fallen: Rejected Classic Articles by Leading Economists”, written by Joshua S. Gans and George B. Shepherd, and “inspired” by Kenneth J. Arrow.) [490])

 - Peter Carr, from his e-mail to Michael Grossman on 16 May 2019.


"I also realised this about scientific accomplishment: if it's too early, it will be ignored; if it's too late, it's too late."

 - Allan Cormack (Nobel Laureate), from Imagining the Elephant: A Biography of Allan MacLeod Cormack by Christopher L. Vaughan (2008). (Allan Cormack won a 1979 Nobel Prize for co-inventing X-ray computed tomography, i.e., CAT scanning.)


“I had fallen victim to the fallacy of the 'growing edge:’ the belief that only the very frontier of scientific advance counted; that everything that had been left behind by that advance was faded and dead.” -  Isaac Asimov, from his book Adding a Dimension: Seventeen Essays on the History of Science (1964).


"If an area begins to grow, it has to be because some clump of people feel that there's something it offers them." - Joseph Ford, as quoted in James Gleick's book Chaos: Making a New Science (2011).


"I'm pleased that, towards the end of his life, [Benoit Mandelbrot] received due recognition, because it took a long time for the mathematical community to understand something that must have been obvious to him: fractals were important. They were a game changer, opening up completely new ways to think about many aspects of the natural world. But for a long time it was not difficult to find professional research mathematicians who stoutly maintained that fractals and chaos were completely useless and that all of the interest in them was pure hype. This attitude persisted into the current century, when fractals had been around for at least twenty-five years and chaos for forty. That this attitude was narrow-minded and unimaginative is easy to establish, because by that time both areas were being routinely used in branches of science ranging from astrophysics to zoology. It was clear that the critics hadn’t deigned to sully their lily-white hands by picking up a random copy of Natureor Science and finding out what was in it."

 -  Ian Stewart, from the article "The influence of Benoît B. Mandelbrot on mathematics", edited by Michael F. Barnsley and Michael Frame, in Notices of the American Mathematical Society, October of 2012. 


Benoit Mandelbrot survived Nazi-occupied France to become one of the most creative thinkers of the 20th century. ... He coined the term “fractal” in 1975, from the Latin word “fractus,” meaning broken or shattered, to better measure rough shapes and irregular surfaces, from graphs of the stock market to coastlines. ... [H]is fractal sets have turned out to have a fabulous number of applications in many additional fields, including mathematics, economics, the sciences and the arts. ... IBM employed him for decades as a researcher (“I was in an industrial laboratory because academia found me unsuitable,” Mandelbrot explained at the time). ... Mandelbrot’s new ideas were laughed at widely when first developed ... Mandelbrot was likewise seen as making little sense in his adopted homeland, since French mathematics was governed for decades by the accomplished Bourbaki group, led by André Weil — brother of the philosopher Simone Weil — and other trendsetters. To such intellectuals, Mandelbrot was a visibly freakish phenomenon. ... Mandelbrot proved to be a uniquely serious innovator, a Kepler of the century past.

  - Benjamin Ivry, from his article "Benoit  Influenced Art and Mathematics" in The Jewish Daily Forwardof 23 November 2012.


"[The author Amir D. Aczel] ... has skipped over what I consider the pernicious effects of Bourbaki. In France, where Bourbaki ruled the roost for a while, applied mathematics received a sharp slap in the face; more than that: a body blow. In the USA, the New Math, located in the high clouds of abstraction, drew its breath of life from Bourbaki. After expending the energies of enthusiasts and spending tens of millions in cash, after abusing the patience of teachers, parents, and a goodly proportion of professional mathematicians, the New Math was finally acknowledged to be an abject and unmitigated failure. (And the failure was predictable, according to some.)

 - Philip J. Davis, from his review "Departed Glories: Bourbaki" of Amir D. Aczel's book The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed, in SIAM NEWS (Society for Industrial and Applied Mathematics) on 12 January 2006.


“The phrase [“New Math”] is often used now to describe any short-lived fad which quickly becomes highly discredited.”

 - from the article “New Math” at the Wikipedia website on 8 February 2019.


"Psychologically, the teaching of abstractions first is wrong. Indeed, a thorough understanding of the concrete must precede the abstract."

 - Morris Kline, from his book Why Johnny Can't Add: The Failure of the New Math (1974).


"In addition, the teaching of theories from axioms, or some close imitation of them such as the basic laws of an algebra, is usually an educational disaster. ... The teaching of axioms should come after conveying the theory in a looser version."

 - Ivor Grattan-Guinness, from his book The Rainbow of Mathematics: A History of the Mathematical Sciences (2000).


"For unrestricted abstraction tends also to divert attention from whole areas of application whose very discovery depends on features that the abstract point of view rules out as being accidental."

 - Mark Kac, as quoted in Richard Hamming's book Methods of Mathematics Applied to Calculus, Probability, and Statistics (2012).


"Throughout the centuries there were men who took first steps down new roads armed with nothing but their own vision. Their goals differed, but they all had this in common: that the step was first, the road new, the vision unborrowed, and the response they received -- hatred. ... The first motor was considered foolish. The first airplane was considered impossible. The power loom was considered vicious. Anesthesia was considered sinful. But the men of unborrowed vision went ahead. They fought, they suffered and they paid. But they won."  

 - Ayn Rand, from her book The Fountainhead (1943).


"The history of innovation and progress of all kinds is made up mostly of failures ... and any great successful revolution you hear of was almost certainly proposed and rejected many times before it found any support in the world at all. You’ll find very few big ideas that were adopted with immediate open arms and unconditional love by those in power."

 - Scott Berkun, from his speech on 24 March 2010 at Ideas Economy: Innovation 2010, an event at the University of California Berkeley organized by the The Economist newspaper (2010).


“The history of breakthroughs is a tale of persistence against rejection. Much of what makes a successful innovator is the ability to persuade and convince conservative people of the merits of the ideas, a very different skill from creativity itself.”

 - Scott Berkun, from the website ScottBerkun.com as posted on 26 March 2013.


‘’It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.”

 - Theodore Roosevelt, from his speech Citizenship in a Republic at the Sorbonne in Paris, France, on 23 April 23 1910.


“The human mind treats a new idea the same way the body treats a strange protein; it rejects it.”

 - Peter B. Medawar (Nobel Laureate), from his book The Art of the Soluble (1967).


"Scientists sometimes boast by implication when they criticize or minimize the achievements of others."

 - Stanislaw Ulam, from his book Adventures of a Mathematician (1976).


"Like all revolutionary new ideas, the subject [of space exploration] has had to pass through three stages, which may be summed up by these reactions: (1) 'It's crazy - don't waste my time.' (2) 'It's possible, but it's not worth doing.' (3) 'I always said it was a good idea.'"

 - Arthur C. Clarke, from his book Report on Planet Three and Other Speculations (1972).


"Every great advance in science has issued from a new audacity of imagination."

 - John Dewey, as quoted in the book The Quest for Certainty: Gifford Lectures, a series of lectures by John Dewey in 1929.


"I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset."

 - Harish-Chandra, as quoted by Robert P. Langlands from his article "Harish-Chandra: 11 October 1923-16 October 1983" in Biographical Memoirs of Fellows of the Royal Society (1985).


"Imagination is more important than knowledge."

 - Albert Einstein, from an interview with George Sylvester Viereck, published in The Saturday Evening Post magazine on 26 October 1929.


"The moving power of mathematical invention is not reasoning but imagination."

 - Augustus De Morgan, as quoted in Robert Perceval Graves' book Life of Sir William Rowan Hamilton, Volume 3 (1889). 


"Many who have never had an opportunity of knowing any more about mathematics confound it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination."

 - Sofia Kovalevskaya, as quoted in her book Sónya Kovalévsky: Her Recollections of Childhood, translated from Russian by Isabel F. Hapgood (1895).


“In every live deductive theory novelties are constantly appearing, due to the introduction of new definitions by ingenious and original investigators. These new combinations are not implicitly contained in the postulates, but reflect the intelligent interest of some creative human agent.”

 - E. V. Huntington, from his article “The Method of Postulates” in the journal Philosophy of Science, Volume IV, Number 4, October of 1937.


"Science goes where you imagine it.”

  - Judah Folkman, as quoted in the article "Folkman Looks Ahead" by Claudia Kalb in Newsweek magazine on 18 February 2001. 

"The formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advance in science."  

 - Albert Einstein and Leopold Infeld, from their book The Evolution of Physics (1938). 

"It seems plausible that people who need to study functions from this point of view might well be able to formulate problems more clearly by using [bigeometric] calculus instead of [classical] calculus."

 - Ralph P. Boas, Jr., from his review [47] of Bigeometric Calculus: A System with a Scale-Free Derivative (1984).


"It is known that non-Newtonian  calculus models real life problems more accurately."

 - R. C. Mittal, from the ResearchGate website on 12 November 2014 [218].


"In the classical calculus the line is used as a standard against which other curves are compared. In the following remarks, Roger Joseph Boscovich (1711 - 1787) may have been the only person to anticipate a fundamental idea in non-Newtonian calculus: a nonlinear curve may be used as a standard against which other curves (including lines) can be compared. 

    " 'But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes. 

   " 'We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.' " (Roger Joseph Boscovich was a scientist/mathematician of remarkable imagination whose insightful and visionary achievements have yet to be fully recognized.)

 - Jane Grossman, Michael Grossman, and Robert Katz, from their book The First Systems of Weighted Differential and Integral Calculus (1980) [9]. (The Boscovich quotation also appears in the article [37] by Raymond W. K. Tang and William E. Brigham, and in the article [308] by J. F. Scott.)


“ ... the norm of biological growth - the standard to which all actual instances of growth must be referred - is exponential growth.”

 - Peter Medawar (Nobel laureate); from his 1982 book Plato’s Republic.


“It is departure from exponential growth that calls for comment and explanation, just as with departure from uniform motion in a straight line.”

 - Peter Medawar  (Nobel laureate); from his 1957 book The Uniqueness of the Individual.


"To someone working in the world of finance, 'constant rate of change' means 'exponential growth'."                      

  - Eric Gaze (Bowdoin College); from his keynote speech at the 27th International Conference on Technology in Collegiate Mathematics (ICTCM) in March of 2015. [224]


“It has long been recognized that biological growth is multiplicative.”

 - Peter Medawar (Nobel laureate); from his 1957 book The Uniqueness of the Individual.


"In mathematics the art of asking questions is more valuable than solving problems."      - Georg Cantor, from his doctoral dissertation (1867).


"In an era of technological triumph, a physicist-turned-philosopher [Thomas Kuhn] dared to argue that science routinely rewards conformists and rejects real innovators. ...  Kuhn’s “normal” scientists are what in corporations would be called “company men.” They’re loyal to the reigning scientific paradigm. ... Normal scientists are problem solvers and paper shufflers, not revolutionaries. They don’t challenge the paradigm as Galileo and Darwin challenged traditional astronomy and biology. Kuhn’s normal scientists rarely consider alternative paradigms, ... ."

 - Keay Davidson, from his review "Groupthink" of Thomas Kuhn’s book The Structure of Scientific Revolutions, in the CALIFORNIA magazine (Cal Alumni Association, University of California Berkeley) in January-February of 2008.


"Uncomprehension; resistance; anger; acceptance. Those who had promoted [the concept of] chaos longest saw all of these."

 - James Gleick, from his book Chaos: Making a New Science (2011).


"As a result, new ideas in science are often met with resistance, anger, and ridicule. ... to propose a startling new scientific theory is to risk exposing yourself to attack for being misguided or just plain inadequate."

 - Leonard Mlodinow, from his book The Upright Thinkers (2015).


"Even after a great idea finally is fully-formed and clearly expressed, it is not always readily accepted by the scientific community. Mlodinow echoes Thomas Kuhn’s idea that revolutionary theories are seldom generally accepted until the existing scientific cohort dies off."

 - James A. Broderick, from his review of Leonard Mlodinow's book The Upright Thinkers at the Rhapsody in Books Weblog (World Wide Web) on 23 August 2015.


“But the present generation will probably behave just as badly if another Darwin should arise, and inflict upon them that which the generality of mankind most hate—the necessity of revising their convictions.”

 - Thomas Henry Huxley, in the book The Life and Letters of Charles Darwin by Charles Darwin’s son Francis Darwin (1898).


“Now, my uncle [Szolem Mandelbrot], who was a [brilliant] mathematician given to strong opinions, was very scornful of some of his peers. He said that they were very, very good, but they were just theorem-provers. They have an extraordinary arsenal of techniques, remember many previous results, and put them together in new ways. But they don't have the creativity to ask new questions. So in mathematics there has been historically this more or less sharp distinction between those who are best known for asking questions and those who are best known for proving theorems that others have conjectured. The greatest mathematician in my private pantheon has been Henri Poincaré. Altogether a very great man, he started many branches of mathematics from scratch, but he acknowledged himself that he didn't prove any difficult theorem and cared about proofs less than about concepts." - Benoit Mandelbrot, from the interview "A Radical Mind" with Benoit Mandelbrot, conducted by Bill Jersey on 24 April  2005, posted at PBS's NOVA Online on 01 October 2008, and broadcast on PBS's NOVA television-program Hunting the Hidden Dimension on 24 August 2011.  

" I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts.  ... The principle point is that ... mathematicians could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts beyond those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity."

 - Eugene Wigner, from his article "The unreasonable effectiveness of mathematics in the natural sciences," in the journal Communications in Pure and Applied Mathematics, Volume 13, Number 1 (February 1960). 

“Every discovery I made while at IBM fell well outside the scope of any university department. ... [M]y work on the distribution of galaxies would not be published until it became known and understood, but could not become known and understood until it was published. ... The stigma attached to being, to a degree, a vanity-press book is erased by large sales ... and wide influence."

 - Benoit Mandelbrot, from "A maverick's apprenticeship: The Wolf Prize for Physics", an essay he wrote on receiving the 1993 Wolf Prize for Physics (1993).


“This finding raises concerns regarding whether peer review is ill-suited to recognize and gestate the most impactful ideas and research. ... The rejection of the 14 most-cited articles in our dataset also suggests that scientific gatekeeping may have problems with dealing with exceptional or unconventional submissions.” 

 -  Kyle Siler, Kirby Lee, and Lisa Bero, from their article “Measuring the effectiveness of scientific gatekeeping” in Proceedings of the National Academy of Sciences of the United States of America. [491]


People often reject creative ideas, even when espousing creativity as a desired goal. ...  Furthermore, this bias against creativity interfered with participants’ ability to recognize a creative idea. These results reveal a concealed barrier that creative actors may face as they attempt to gain acceptance for their novel ideas.”

 - Jennifer S. Mueller, Shimul Melwani, and Jack A. Goncalo, from their article “The bias against creativity: why people desire but reject creative ideas” in Psychological Science. [492]    


"Some insights are resisted with such intensity that it may take decades before they're widely accepted among scientists. ... Are you ready to have the top scientists in your field criticizing your work in journals and dissing you at meetings? Because history shows that the deeper your idea cuts into the heart of a field, the more your peers are likely to challenge you. Human nature being what it is, what ought to be reasoned discussion may turn personal, even nasty. ... Progress is made when good scientists keep working -- and keep supporting what they believe is true -- despite the criticism."

 - Anne Sasso, from her article "Audacity, Part 5: Rejection and Ridicule" in the magazine Science (American Association for the Advancement of Science), 11 June 2010. 

"A new scientific innovation does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it."

 - Max Planck, from his book The Philosophy of Physics (1936).


"Flying in the face of the Establishment with unconventional ideas and methods ... is highly esteemed in academia --  until somebody actually does it."

 - Edward Tenner, from his article "Benoit Mandelbrot the Maverick, 1924-2010" in The Atlantic magazine (16 October 2010).


NOTE. The six books on non-Newtonian calculus and related matters by Jane Grossman, Michael Grossman, and Robert Katz are indicated below, and are available at some academic libraries, public libraries, and booksellers such as Amazon.com. On the World Wide Web, each of the books can be read and downloaded, free of charge, at HathiTrust, Google Books, and the Digital Public Library of America.  



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