Here is a collection of interesting quotes on science and mathematics. The initial collection was a subset of the quotes on JonathanBorwein's quotation page. Numerous other quotes have subsequently been added. They are listed alphabetically by the surname of the author.
DISCLAIMER: We have carefully rechecked the references, but cannot guarantee that they are correct in all cases. Further, we do not necessarily endorse the views expressed in these quotes -- they are presented only for general interest in the field.
He is like the fox, who effaces his tracks in the sand with his tail. -- Niels Abel (1802-1829), regarding Gauss' mathematical writing style, quoted in G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, 1992, pg. 177.
Despite the narrative force that the concept of entropy appears to evoke in everyday writing, in scientific writing entropy remains a thermodynamic quantity and a mathematical formula that numerically quantifies disorder. When the American scientist Claude Shannon found that the mathematical formula of Boltzmann defined a useful quantity in information theory, he hesitated to name this newly discovered quantity entropy because of its philosophical baggage. The mathematician John Von Neumann encouraged Shannon to go ahead with the name entropy, however, since no one knows what entropy is, so in a debate you will always have the advantage. -- From The American Heritage Book of English Usage, pg. 158.
The new availability of huge amounts of data, along with the statistical tools to crunch these numbers, offers a whole new way of understanding the world. Correlation supersedes causation, and science can advance even without coherent models, unified theories, or really any mechanistic explanation at all. There's no reason to cling to our old ways. It's time to ask: What can science learn from Google? -- Chris Anderson (Editor-in-Chief of Wired), "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete", available at Wired, 23 Jun 2008.
Others might, of course, have quite different experiences of the causes and conditions of insight, and also of the Internet. But I'd bet that my experiences with both are not uncommon. So what should be done? A first reaction -- to largely banish the Internet from my intellectual life -- feels both difficult (like most I am at least a low-level addict) and counterproductive: information is, after all, crucially important, and the Internet is a unsurpassable tool for discovering and assembling it. -- Anthony Aguirre, "The Enemy of Insight?", 8 Jan 2010, available at Online article.
BOOK: A format for conveying information consisting of a single continuous piece of text, written on an isolated theme or telling a particular story, averaging around 100,000 words in length and authored by a single individual. Books were printed on paper between the mid-15th and early 21st century but more often delivered electronically after 2012. The book largely disappeared during the mid-21st entry as it became clear that it had only ever been a narrow instantiation, constrained by print technology, of texts and graphics of any form that could flow endlessly into others. Once free from the shackles of print technology, new story-telling modes flowered in an extraordinary burst of creativity in the early 21st century. Even before that the use of books to explain particular subjects (see textbook) had died very rapidly as it grew obvious that a single, isolated voice lacked authority, wisdom and breadth. -- Alun Anderson, "If You Don't Change the Way You Think, You Risk Extinction," 8 Jan 2010, available at Online article.
From this we can prove further that a sphere of the size attributed by Aristarchus to the sphere of the fixed stars would contain a number of grains of sand less than 10,000,000 units of the eight order of numbers [or 10^(56+7) = 10^(63)]. -- Archimedes, from the "The Sand Reckoner", in Robert Maynard Hutchins, ed., Great Books of the Western World, Encyclopedia Britannica, Inc., Chicago, 1952, vol. 11, pg. 520-526.
[T]o suggest that the normal processes of scholarship work well on the whole and in the long run is in no way contradictory to the view that the processes of selection and sifting which are essential to the scholarly process are filled with error and sometimes prejudice. -- Kenneth Arrow, from E. Roy Weintraub and Ted Gayer, "Equilibrium Proofmaking," Journal of the History of Economic Thought, Dec 2001, pg. 421-442.
Sometime in the 1970s Paul Turan spent part of a summer in Edmonton. I wanted to meet him so went there. He was a few days late so I had arrived a couple of days earlier. A group went to the airport to meet him, and stopped at a coffee shop before going to the university. It was very hot so I offered to stay in the car and keep the windows down. I said I did not drink coffee. Turan then told the joke about mathematicians being machines which turn coffee into theorems, and then added: "You prove good theorems. Just think how much better they would be if you drank coffee". I have heard the statement attributed to Renyi by more than one Hungarian, but this was somewhat later. Turan just stated it. -- Richard Askey, "The Definitive Version of `Erdos and Coffee'", as told to the Historia Mathematica e-list, 3 Feb 2005.
The history of mathematics is full of instances of happy inspiration triumphing over a lack of rigour. Euler's use of wildly divergent series or Ramanujan's insights are among the more obvious, and mathematics would have been poorer if the Jaffe-Quinn view had prevailed at the time. The marvelous formulae emerging at present from heuristic physical arguments are the modern counterparts of Euler and Ramanujan, and they should be accepted in the same spirit of gratitude tempered with caution. -- Michael Atiyah et al., "Responses to 'Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics," by A. Jaffe and F. Quinn," Bulletin of the American Mathematical Society, vol. 30, no. 2 (Apr 1994), pg. 178-207.
The common situation is this: An experimentalist performs a resolution analysis and finds a limited-range power law with a value of D smaller than the embedding dimension. Without necessarily resorting to special underlying mechanistic arguments, the experimentalist then often chooses to label the object for which she or he finds this power law a fractal. This is the fractal geometry of nature. -- David Avnir et al, Hebrew University, from "Is the Geometry of Nature Fractal?", in Science, 2 Jan 1998, pg. 39-40.
And yet since truth will sooner come out of error than from confusion. -- Francis Bacon (1561-1626), from "The New Organon (1620)", in James Spedding, Robert Ellis and Douglas Heath, ed., The Works of Francis Bacon, 1887-1901, vol. 4, pg. 149.
The quantitative aspect is obvious: why should we be clever enough to fathom the Theory of Everything? We know of mathematical theorems which are undemonstrable in principle and others that would take our fastest computers the entire age of the Universe to decide. Why should the Theory of Everything be simpler than these? -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 203.
A more interesting problem is the extent to which the brain is qualitatively adapted to understand the Universe. Why should its categories of thought and understanding be able to cope with the scope and nature of the real world? Why should be Theory of Everything be written in a 'language' that our minds can decode? Why has the process of natural selection so over-endowed us with mental faculties that we can understand the whole fabric of the Universe far beyond anything required for our past and present survival? -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 203.
Yet, at that time, there was no evident connection with any problems of physics and in 1900 Sir James Jeans, when commenting to a colleague upon the areas of mathematics that were most fruitful for the physicist to know, asserted that "we may as well cut out group theory, that is a subject which will never be of any use in physics." On the contrary, it is the systematic classification of symmetry and its canonization into the subject of group theory which forms the basis of so much of modern fundamental physics. Nature likes symmetry and so groups form a fundamental part of its description. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 211.
That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 222.
In practice, the intelligibility of the world amounts to the fact that we find it to be algorithmically compressible. We can replace sequences of facts and observational data by abbreviated statements which contain the same information content. These abbreviations we often call 'laws of Nature.' If the world were not algorithmically compressible, then there would exist no simple laws of nature. Instead of using the law of gravitation to compute the orbits of the planets at whatever time in history we want to know them, we would have to keep precise records of the positions of the planets at all past times; yet this would still not help us one iota in predicting where they would be at any time in the future. This world is potentially and actually intelligible because at some level it is extensively algorithmically compressible. At root, this is why mathematics can work as a description of the physical world. It is the most expedient language that we have found in which to express those algorithmic compressions. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 231-232.
In many ways, the computational paradigm has an affinity for the quantum picture of the world. Both are discrete; both possess dual aspects like evolution and measurement (compute and read). But greater claims could be made for the relationship between the quantum and the symmetries of nature. Half a century of detailed study by physicists has wedded the two into an indissoluble union. What might be the status of the computational paradigm after a similar investment of thought and energy? -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 241-242.
The scope of Theories of Everything is infinite but bounded; they are necessary parts of a full understanding of things but they are far from sufficient to reveal everything about a Universe like ours. In the pages of this book, we have seen something of what a Theory of Everything might hope to teach us about the unity of the Universe and the way in which it may contain elements that transcend our present compartmentalized view of Nature's ingredients. But we have also learnt that there is more to Everything than meets the eye. Unlike many others that we can imagine, our world contains prospective elements. Theories of Everything can make no impression upon predicting these prospective attributes of reality; yet, strangely, many of these qualities will themselves be employed in the human selection and approval of an aesthetically acceptable Theory of Everything. There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 245-246.
There is no reason why life has to evolve in the Universe. Such complex step-by-step processes are not predictable because of their very sensitive dependence upon the starting conditions and upon subtle interactions between the evolving state and the ambient environment. All we can assert with confidence is a negative: if the constants of Nature were not within one percent or so of their observed values, then the basic buildings blocks of life would not exist in sufficient profusion in the Universe. Moreover, changes like this would affect the very stability of the elements and prevent the existence of the required elements rather than merely suppress their abundance. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 121.
Somehow the breathless world that we witness seems far removed from the timeless laws of Nature which govern the elementary particles and forces of Nature. The reason is clear. We do not observe the laws of Nature: we observe their outcomes. Since these laws find their most efficient representation as mathematical equations, we might say that we see only the solutions of those equations not the equations themselves. This is the secret which reconciles the complexity observed in Nature with the advertised simplicity of her laws. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 138.
On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting - the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings. -- John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 11-12.
Looking over the past 150 years -- at the tiny garden at Brno, the filthy fly room at Columbia, the labs of the New York Botanical Garden, the basement lab at Stanford, and the sun-drenched early gatherings at Cold Spring Harbor -- it seems that the fringes, not the mainstream, are the most promising places to discover revolutionary advances. -- Paul Berg and Maxine Singer (biologists), in "Inspired Choices", Science, 30 Oct 1998, pg. 873-874, available at Online article.
The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right. -- David Berlinski, A Tour of the Calculus, Pantheon Books, 1995, quoted in "Vignettes: Changing Times" in Science, 28 Feb 1997, pg. 1276.
Mathematicians are like pilots who maneuver their great lumbering planes into the sky without ever asking how the damn things stay aloft. ... The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen. ... The existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself. -- David Berlinski, review of The Pleasures of Counting by T. W. Korner, Cambridge, 1996, in The Sciences, Jul/Aug 1997, pg. 37-41.
Several years ago I was invited to contemplate being marooned on the proverbial desert island. What book would I most wish to have there, in addition to the Bible and the complete works of Shakespeare? My immediate answer was: Abramowitz and Stegun's Handbook of Mathematical Functions. If I could substitute for the Bible, I would choose Gradsteyn and Ryzhik's Table of Integrals, Series and Products. Compounding the impiety, I would give up Shakespeare in favor of Prudnikov, Brychkov And Marichev's of Integrals and Series ... On the island, there would be much time to think about waves on the water that carve ridges on the sand beneath and focus sunlight there; shapes of clouds; subtle tints in the sky... With the arrogance that keeps us theorists going, I harbor the delusion that it would be not too difficult to guess the underlying physics and formulate the governing equations. It is when contemplating how to solve these equations - to convert formulations into explanations - that humility sets in. Then, compendia of formulas become indispensable. -- Michael Berry, "Why Are Special Functions Special?", available at Physics Today, Apr 2001.
"While you're trying to understand a difficult theorem, it's not fun," said Biederman, professor of neuroscience in the USC College of Letters, Arts and Sciences. "But once you get it, you just feel fabulous." The brain's craving for a fix motivates humans to maximize the rate at which they absorb knowledge, he said. "I think we're exquisitely tuned to this as if we're junkies, second by second." -- Irving Biederman, 2006, from Online article
Hardy asked "What's your father doing these days. How about that esthetic measure of his?" I replied that my father's book was out. He said, "Good, now he can get back to real mathematics". -- Garret Birkoff, discussing G. D. Birkhoff's Aesthetic Measures (1933), quoted in "Towering Figures, 1890-1950", by David E. Zitarelli, MAA Monthly Aug-Sept 2001, pg. 618.
The first [axiom] said that when one wrote to the other (they often preferred to exchange thoughts in writing instead of orally), it was completely indifferent whether what they said was right or wrong. As Hardy put it, otherwise they could not write completely as they pleased, but would have to feel a certain responsibility thereby. The second axiom was to the effect that, when one received a letter from the other, he was under no obligation whatsoever to read it, let alone answer it, because, as they said, it might be that the recipient of the letter would prefer not to work at that particular time, or perhaps that he was just then interested in other problems. ... The third axiom was to the effect that, although it did not really matter if they both thought about the same detail, still, it was preferable that they should not do so. And, finally, the fourth, and perhaps most important axiom, stated that it was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name; otherwise there would constantly arise quarrels and difficulties in that now one, and now the other, would oppose being named co-author. -- Harald Bohr, "Hardy and Littlewood's Four Axioms for Collaboration", quoted from the preface of Bella Bollobas' 1988 edition of Littlewood's Miscellany.
Anyone who is not shocked by quantum theory has not understood a single word. -- Niels Bohr, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
Now, if the law of forces were known, and the position, velocity and direction of all the points at any given instant, it would be possible for a mind of this type to foresee all the necessary subsequent motions and states, and to predict all the phenomena that necessarily followed from them. ... We cannot aspire to this, not only because our human intellect is not equal to the task, but also because we do not know the number, or the position and motion of each of thee points. -- Roger Boscovich, 1758, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 63.
This revelation [that math is important] comes from reading a stack of magazines about the future, about computers and artificial intelligence, cars and planes, food production and global warming. And I have come to the conclusion that Mr. Kool was right. ... Math has something to do with calculations, formulas, theories and right angles. And everything to do with real life. Mathematicians not only have the language of the future (they didn't send "Taming of the Shrew" into space, just binary blips) but they can use it to predict when Andromeda will perform a cosmic dance with the Milky Way. It's mathematicians who are designing the intelligent car that knows when you're falling asleep at the wheel or brakes to avoid an accident. It can predict social chaos and the probability of feeding billions. It even explains the stock market and oil prices. -- Paulette Bourgeoism, from "The Numbers Game," The Globe and Mail 13 Jul 2000, pg. A14.
This revelation [that math is important] comes from reading a stack of magazines about the future, about computers and artificial intelligence, cars and planes, food production and global warming. And I have come to the conclusion that Mr. Kool was right. ... Math has something to do with calculations, formulas, theories and right angles. And everything to do with real life. Mathematicians not only have the language of the future (they didn't send "Taming of the Shrew" into space, just binary blips) but they can use it to predict when Andromeda will perform a cosmic dance with the Milky Way. It's mathematicians who are designing the intelligent car that knows when you're falling asleep at the wheel or brakes to avoid an accident. It can predict social chaos and the probability of feeding billions. It even explains the stock market and oil prices. -- Paulette Bourgeoism, from "The Numbers Game," The Globe and Mail 13 Jul 2000, pg. A14.
I feel so strongly about the wrongness of reading a lecture that my language may seem immoderate. ... The spoken word and the written word are quite different arts. ... I feel that to collect an audience and then read one's material is like inviting a friend to go for a walk and asking him not to mind if you go alongside him in your car. -- Sir Lawrence Bragg, quoted in Science, 5 Jul 1996, pg. 76.
Mathematics is the language of high technology. Indeed it is, but I think it is also becoming the eyes of science. -- Tom Brzustowski, NSERC President, addressing the MITACS NCE annual general meeting, 6 Jun 2000.
And it is one of the ironies of this entire field that were you to write a history of ideas in the whole of DNA, simply from the documented information as it exists in the literature - that is, a kind of Hegelian history of ideas -- you would certainly say that Watson and Crick depended on Von Neumann, because von Neumann essentially tells you how it's done. But of course no one knew anything about the other. It's a great paradox to me that this connection was not seen. Of course, all this leads to a real distrust about what historians of science say, especially those of the history of ideas. -- Nobelist Sidney Brenner, as told to Lewis Wolpert, discusing von Neumann's essay on "The General and Logical Theory of Automata", from My life in Science, pg. 35-36.
where almost one quarter hour was spent, each beholding the other with admiration before one word was spoken: at last Mr. Briggs began "My Lord, I have undertaken this long journey purposely to see your person, and to know by what wit or ingenuity you first came to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it out before, when now being known it appears so easy." -- Henry Briggs, describing his first meeting with Napier whom he had traveled from London to Edinburgh to meet; from H. W. Turnbull, The Great Mathematicians, Methuen, 1929.
Philosophical theses may still be churned out about it, ... but the question of nonconstructive existence proofs or the heinous sins committed with the axiom of choice arouses little interest in the average mathematician. Like 0l' Man River, mathematics just keeps rolling along and produces at an accelerating rate "200,000 mathematical theorems of the traditional handcrafted variety ... annually." Although sometimes proofs can be mistaken -- sometimes spectacularly -- and it is a matter of contention as to what exactly a "proof" is -- there is absolutely no doubt that the bulk of this output is correct (though probably uninteresting) mathematics. -- Richard C. Brown, discussing constructivism and intuitionism in Are Science and Mathematics Socially Constructed? World Scientific, 2009, pg. 239. The inset quote is from P. J. Davis and R. Hersh, The Mathematical Experience, Houghton Mifflin, Boston, 1981, pg. 24.
So to summarise, according to the citation count, in order of descent, the authors are listening to themselves, dead philosophers, other specialists in semiotic work in mathematics education research, other mathematics education research researchers and then just occasionally to social scientists but almost never to other education researchers, including mathematics teacher education researchers, school teachers and teacher educators. The engagement with Peirce is being understood primarily through personal engagements with the original material rather than as a result of working through the filters of history, including those evidenced within mathematics education research reports in the immediate area. The reports, and the hierarchy of power relations implicit in them, marginalise links to education, policy implementation or the broader social sciences. -- Tony Brown, from "Signifying 'students', 'teachers' and 'mathematics': a reading of a special issue," available at Springer, 28 May 2008.
I will be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors, but this may occasionally be of use for the actual advancement of science. -- Constantin Caratheodory, speaking to an MAA meeting in 1936.
When I was a young student in the United States, I met Zygmund and I had an idea how to produce some very complicated functions for a counter-example and Zygmund encouraged me very much to do so. I was thinking about it for about 15 years on and off, on how to make these counter-examples work and the interesting thing that happened was that I realised why there should be a counter-example and how you should produce it. I thought I really understood what was the background and then to my amazement I could prove that this "correct" counter-example couldn't exist and I suddenly realised that what you should try to do was the opposite, you should try to prove what was not fashionable, namely to prove convergence. The most important aspect in solving a mathematical problem is the conviction of what is the true result. Then it took 2 or 3 years using the techniques that had been developed during the past 20 years or so. -- Lennart Carleson, 1966, from 1966 IMU address on his proof of Luzin's 1913 conjecture that the Fourier series of every square integrable function converges a.e. to the function.
Rigour is the affair of philosophy, not of mathematics. -- Bonaventura Cavalieri (1598-1647).
The idea that we could make biology mathematical, I think, perhaps is not working, but what is happening, strangely enough, is that maybe mathematics will become biological! -- Gregory Chaitin, 2000, from Chaitin interview.
The message is that mathematics is quasi-empirical, that mathematics is not the same as physics, not an empirical science, but I think it's more akin to an empirical science than mathematicians would like to admit. -- G. Chaitin, from "The Creative Life: Science vs. Art" Online article.
Mathematicians normally think that they possess absolute truth. They read God's thoughts. They have absolute certainty and all the rest of us have doubts. Even the best physics is uncertain, it is tentative. Newtonian science was replaced by relativity theory, and then---wrong!---quantum mechanics showed that relativity theory is incorrect. But mathematicians like to think that mathematics is forever, that it is eternal. Well, there is an element of that. Certainly a mathematical proof gives more certainty than an argument in physics or than experimental evidence, but mathematics is not certain. This is the real message of Godel's famous incompleteness theorem and of Turing's work on uncomputability. -- G. Chaitin, from "The Creative Life: Science vs. Art" Online article.
You see, with Godel and Turing the notion that mathematics has limitations seems very shocking and surprising. But my theory just measures mathematical information. Once you measure mathematical information you see that any mathematical theory can only have a finite amount of information. But the world of mathematics has an infinite amount of information. Therefore it is natural that any given mathematical theory is limited, the same way that as physics progresses you need new laws of physics. -- G. Chaitin, from "The Creative Life: Science vs. Art" Online article.
Mathematicians like to think that they know all the laws. My work suggests that mathematicians also have to add new axioms, simply because there is an infinite amount of mathematical information. This is very controversial. I think mathematicians, in general, hate my ideas. Physicists love my ideas because I am saying that mathematics has some of the uncertainties and some of the characteristics of physics. Another aspect of my work is that I found randomness in the foundations of mathematics. Mathematicians either don't understand that assertion or else it is a nightmare for them...: -- G. Chaitin, from "The Creative Life: Science vs. Art" Online article.
A proof is a proof. What kind of a proof? It's a proof. A proof is a proof. And when you have a good proof, it's because it's proven. -- Jean Chretien, Canadian Prime Minister, explaining Canada's conditions for determining if Iraq has complied, 5 Sep 2002, CBC article.
When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong. -- Arthur C. Clarke, "Hazards of Prophecy: The Failure of Imagination", in his book Profiles of the Future, 1962. See also Wikipedia article.
The only way of discovering the limits of the possible is to venture a little way past them into the impossible. -- Arthur C. Clarke, "Hazards of Prophecy: The Failure of Imagination", in his book Profiles of the Future, 1962. See also Wikipedia article.
Any sufficiently advanced technology is indistinguishable from magic. -- Arthur C. Clarke, "Hazards of Prophecy: The Failure of Imagination", in his book Profiles of the Future, 1962. See also Wikipedia article.
Because amid this unprecedented surge in connectivity, we must also recognize that these technologies are not an unmitigated blessing. These tools are also being exploited to undermine human progress and political rights. Just as steel can be used to build hospitals or machine guns, or nuclear power can either energize a city or destroy it, modern information networks and the technologies they support can be harnessed for good or for ill. The same networks that help organize movements for freedom also enable al-Qaida to spew hatred and incite violence against the innocent. And technologies with the potential to open up access to government and promote transparency can also be hijacked by governments to crush dissent and deny human rights. -- Hillary Rodham Clinton, "Remarks on Internet Freedom," 21 Jan 2010, available at Online article.
Ask Dr. Edward Witten of the Institute for Advanced Study in Princeton, New Jersey what he does all day, and it's difficult to get a straight answer. "There isn't a clear task," Witten told CNN. "If you are a researcher you are trying to figure out what the question is as well as what the answer is. ... You want to find the question that is sufficiently easy that you might be able to answer it, and sufficiently hard that the answer is interesting. You spend a lot of time thinking and you spend a lot of time floundering around." -- CNN article about Ed Witten, available at CNN article, 27 Jun 2005.
Dear brother: I have often been surprised that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the cause; viz. that though Reason is feasted, Imagination is starved; while Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert. To assist Reason by the stimulus of Imagination is the design of the following production. -- Samuel Taylor Coleridge, in a letter to his brother the Reverend George Coleridge, available at: Coleridge works
Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry -- an aberration which is happily almost impossible -- it would occasion a rapid and widespread degeneration of that science. -- Auguste Comte, Cours de Philosophie Positive, 1830, quoted in Jon Fripp, Michael Fripp and Deborah Fripp, Speaking of Science: Notable Quotes on Science, Engineering and the Environment, LLH Technology Publishing, Eagle Rock, VA, 2002, pg. 14.
Internet surfing completely absorbs me in the flux and flow of the present moment, in contrast to reading a book, or learning a machine, or studying with a teacher. These enterprises demand sustained "linear thinking," even as their substrata can jump from one place to another: habitual sustained periods of focus are necessary. But my students don't think they "need" to read a whole book to respond to any given challenge; they can simply go to the Internet with their query and a search engine will "think outside the box" for them. This has made me despondent about a general degradation, around me, in people's habituation to focused linear thinking. -- Tony Conrad, "A Question With(out) an Answer," 8 Jan 2010, available at Online article.
The lesson is not to trust the numbers too much. If math were a guy, math would be a pompous guy, the sort who's absolutely always sure about everything and never apologizes when he's wrong. And the fact is, math isn't actually ever wrong, not technically. Math is a perfectly logical and intelligent guy. He just sometimes makes the wrong assumptions. -- Libby Copeland, from Washington Post, 25 Apr 2008.
A research policy does not consist of programs, but of hiring high-quality scientists. When you hire someone good, you've made your research policy for the next 20 years. -- Vincent Courtillot (Chief CNRS advisor), quoted in "New CNRS Chief Gets Marching Orders", Science, 18 July, 1997, pg. 308.
The term Dual is not new. But surprisingly the term Primal, introduced around 1954, is. It came about this way. W. Orchard-Hays, who is responsible for the first commercial grade L.P. software, said to me at RAND one day around 1954: "We need a word that stands for the original problem of which this is the dual." I, in turn, asked my father, Tobias Dantzig, mathematician and author, well known for his books popularizing the history of mathematics. He knew his Greek and Latin. Whenever I tried to bring up the subject of linear programming, Toby (as he was affectionately known) became bored and yawned. But on this occasion he did give the matter some thought and several days later suggested Primal as the natural antonym since both primal and dual derive from the Latin. It was Toby's one and only contribution to linear programming: his sole contribution unless, of course, you want to count the training he gave me in classical mathematics or his part in my conception. -- George B. Dantzig, "Reminiscences About the Origin of Linear Programming", in Arthur Schlissel, ed., Essays in the History of Mathematics, American Mathematical Society, Mar 1984, pg. 10.
The average man identifies mathematical ability with quickness in figures. "So you are a mathematician. Why, then you have no trouble with your tax return!" What mathematician had not at least once in his career been so addressed? There is, perhaps, unconscious irony in these words, for are not most professional mathematicians spared all trouble incident to excessive income? -- Tobias Dantzig, Number: The Language of Science, Plume Books, New York, 2007, pg. 25.
We begin to understand why humanity so obstinately clung to such devices as the abacus or even the tally. Computations which a child can now perform required then [in the middle ages, prior to the adoption of modern decimal arithmetic] the services of a specialist, and what is now only a matter of a few minutes meant in the twelfth century days of elaborate work. -- Tobias Dantzig, Number: The Language of Science, Plume Books, New York, 2007, pg. 27.
The greatly increased facility with which the average man today manipulates number has been often taken as proof of the growth of the human intellect. The truth of the matter is that the difficulties then experienced [in the middle ages, prior to the adoption of modern decimal arithmetic] were inherent in the numeration in use, a numeration not susceptible to simple, clear-cut rules. The discovery of the modern positional numeration did away with these obstacles and made arithmetic accessible even to the dullest mind. -- Tobias Dantzig, Number: The Language of Science, Plume Books, New York, 2007, pg. 27.
Today, when positional numeration has become a part of our daily life, it seems that the superiority of this method, the compactness of its notation, the ease and elegance it introduced in calculations, should have assured the rapid and sweeping acceptance of it. In reality, the transition, far from being immediate, extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code. -- Tobias Dantzig, Number: The Language of Science, Plume Books, New York, 2007, pg. 33.
During the three years which I spent at Cambridge my time was wasted, as far as the academical studies were concerned, as completely as at Edinburgh and at school. I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense. -- Charles Darwin, from Autobiography of Charles Darwin, available at the Online copy.
[H]uman minds, at least, are much more than mere observers. We do more than just watch the show that nature stages. Human beings have come to understand the world, at least in part, through the processes of reasoning and science. In particular, we have developed mathematics, and by so doing have unraveled some -- maybe soon, all -- of the hidden cosmic code, the subtle tune to which nature dances. Nothing in the entire multiverse/anthropic argument ... requires that level of involvement, that degree of connection. In order to explain a bio-friendly universe, the selection processes that features in the weak anthropic principle merely requires observers to observe. It is not necessary for observers to understand. Yet humans do. Why? -- Paul Davies, Cosmic Jackpot: Why Our Universe Is Just Right for Life, Houghton Mifflin, New York, 2007, pg. 231.
Computation with Roman numerals is certainly algorithmic -- it's just that the algorithms are complicated. In 1953, I had a summer job at Bell Labs in New Jersey (now Lucent), and my supervisor was Claude Shannon (who has died only very recently). On his desk was a mechanical calculator that worked with Roman numerals. Shannon had designed it and had it built in the little shop Bell Labs had put at his disposal. On a name plate, one could read that the machine was to be called: Throback I. -- Martin Davis, following up on queries on the Historia Mathematica e-list, 12 Jan 2002.
Once the opening ceremonies were over, the real meat of the Congress was then served up in the form of about 1400 individual talks and posters. I estimated that with luck I might be able to comprehend 2% of them. For two successive weeks in the halls of a single University, ICM'98 perpetuated the myth of the unity of mathematics; which myth is supposedly validated by the repetition of that most weaselly of rhetorical phrases: "Well, in principle, you could understand all the talks." Philip J. Davis, "Impressions of the International Congress of Mathematicians", available at SIAM News, 15 Oct 1998.
Man will never reach the moon, regardless of all future scientific advances. -- Lee De Forest, Radio pioneer, 1957, quoted in Leon A. Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46.
Considerable obstacles generally present themselves to the beginner, in studying the elements of Solid Geometry, from the practice which has hitherto uniformly prevailed in this country, of never submitting to the eye of the student, the figures on whose properties he is reasoning, but of drawing perspective representations of them upon a plane. ... I hope that I shall never be obliged to have recourse to a perspective drawing of any figure whose parts are not in the same plane. -- Augustus De Morgan, quoted in Adrian Rice, "What Makes a Great Mathematics Teacher?", American Mathematical Monthly, Jun-Jul 1999, pg. 540.
In 1831, Fourier's posthumous work on equations showed 33 figures of solution, got with enormous labour. Thinking this is a good opportunity to illustrate the superiority of the method of W. G. Horner, not yet known in France, and not much known in England, I proposed to one of my classes, in 1841, to beat Fourier on this point, as a Christmas exercise. I received several answers, agreeing with each other, to 50 places of decimals. In 1848, I repeated the proposal, requesting that 50 places might be exceeded: I obtained answers of 75, 65, 63, 58, 57, and 52 places. -- Augustus De Morgan, quoted from Adrian Rice, "The Case of Augustus De Morgan", American Mathematical Monthly, Jun-Jul 1999, pg. 542.
I would hope for Humanity's future that the same is true for all my fellow highly-trained specialists. The scientific method for reaching conclusions has served us well for many generations, leading to a length and quality of life for most of us that was beyond the imagination of our ancestors. If that way of thinking were to be replaced by a blind "wisdom of the crowd" approach, which the Internet offers, then we are likely in for real trouble. For wisdom of the crowd, like its best known exemplar Google Search, gives you the mostly-best answer most of the time. -- Keith Devlin, "It All Depends on What You Mean By," 8 Jan 2010, available at Online article.
Old ideas give way slowly; for they are more than abstract logical forms and categories. They are habits, predispositions, deeply engrained attitudes of aversion and preference. Moreover, the conviction persists-though history shows it to be a hallucination that all the questions that the human mind has asked are questions that can be answered in terms of the alternatives that the questions themselves present. But in fact intellectual progress usually occurs through sheer abandonment of questions together with both of the alternatives they assume an abandonment that results from their decreasing vitality and a change of urgent interest. We do not solve them: we get over them. -- John Dewey, quoted from "The Influence of Darwin on Philosophy," Online article, 1910.
I climb the "Hill of Science,"
I "view the landscape o'er;"
Such transcendental prospect,
I ne'er beheld before!
-- Emily Dickinson, "Sic transit gloria mundi", from Dickinson poem site.
The great Arthur Eddington gave a lecture about his alleged derivation of the fine structure constant from fundamental theory. Goudsmit and Kramers were both in the audience. Goudsmit understood little but recognized it as far fetched nonsense. After the discussion, Goudsmit went to his friend and mentor Kramers and asked him, 'do all physicists go off on crazy tangents when they grow old? I am afraid.' Kramers answered, 'No Sam, you don't have to be scared. A genius like Eddington may perhaps go nuts but a fellow like you just gets dumber and dumber.' -- M. Dresden, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 112.
"Folks, the Mac platform is through -- totally." -- John C. Dvorak, PC Magazine, 1998, quoted in Leon A. Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46.
I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style. -- Freeman Dyson's review of Nature's Numbers by Ian Stewart, Basic Books, 1995, from American Mathematical Monthly, Aug-Sept 1996, pg. 612.
Godel proved that the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we got into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory. -- Freeman Dyson, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 52.
Elsewhere Kronecker said, "In mathematics, I recognize true scientific value only in concrete mathematical truths, or to put it more pointedly, only in mathematical formulas." ... I would rather say "computations" than "formulas," but my view is essentially the same. -- Harold M. Edwards, Essays on Constructive Mathematics, Springer 2005, pg. 1. Edwards comments elsewhere that his own preference for constructivism was forged by experience of computing in the fifties--"trivial by today's standards."
Imagination is more important than knowledge, for while knowledge defines everything we know and understand, imagination points to all we might yet discover and create. -- Albert Einstein, quoted in "What Life Means to Einstein: An Interview by George Sylvester Viereck," Saturday Evening Post, vol. 202 (26 Oct 1929), pg. 117.
I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. -- Albert Einstein, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 211.
There is not the slightest indication that nuclear energy will ever be obtainable. It would mean that the atom would have to be shattered at will. -- Albert Einstein, 1932, quoted in Leon A Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46.
On quantum theory, I use up more brain grease than on relativity. -- Albert Einstein, 1911, to Otto Stern, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
Those are the crazy people who are not working on quantum theory. -- Albert Einstein, 1911, referring to the inmates of an insane asylum near his office in Prague, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
I could probably have arrived at something like this myself, but if all this is true then it means the end of physics. -- Albert Einstein, referring to a 1913 breakthrough by Niels Bohr, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
Equations are more important to me, because politics is for the present, but an equation is something for eternity. -- Albert Einstein, from Helle Zeit, Dunkle Zeit, In Memoriam Albert Einstein, ed. Carl Seelig, 1956, pg. 71.
On the other hand, I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research. Only those who realize the immense efforts and, above all, the devotion without which pioneer work in theoretical science cannot be achieved are able to grasp the strength of the emotion out of which alone such work, remote as it is from the immediate realities of life, can issue. What a deep conviction of the rationality of the universe and what a yearning to understand, were it but a feeble reflection of the mind revealed in this world, Kepler and Newton must have had to enable them to spend years of solitary labor in disentangling the principles of celestial mechanics! Those whose acquaintance with scientific research is derived chiefly from its practical results easily develop a completely false notion of the mentality of the men who, surrounded by a skeptical world, have shown the way to kindred spirits scattered wide through the world and through the centuries. Only one who has devoted his life to similar ends can have a vivid realization of what has inspired these men and given them the strength to remain true to their purpose in spite of countless failures. It is cosmic religious feeling that gives a man such strength. A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people. -- Albert Einstein, New York Times Magazine, 9 Nov 1930, pg. 1-4, reprinted in Albert Einstein, Ideas and Opinions, Crown Publishers, Inc. 1954, pg. 36-40.
[H]owever, we select from nature a complex [of phenomena] using the criterion of simplicity, in no case will its theoretical treatment turn out to be forever appropriate. ... But I do not doubt that the day will come when that description [the general theory of relativity], too, will have to yield to another one, for reasons which at present we do not yet surmise. I believe that this process of deepening the theory has no limits. -- Albert Einstein, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 113.
The Poincare Conjecture says, Hey, you've got this alien blob that can ooze its way out of the hold of any lasso you tie around it? Then that blob is just an out-of-shape ball. [Grigory] Perelman and [Columbia University's Richard] Hamilton proved this fact by heating the blob up, making it sing, stretching it like hot mozzarella, and chopping it into a million pieces. In short, the alien ain't no bagel you can swing around with a string through his hole. -- Jordan Ellenberg, "Who Cares About Poincare? Million-dollar Math Problem Solved. So What?", Slate, 18 Aug 2006.
Any electronic archaeologist, sociologist or historian examining our e-lives would be able to understand, map, computer, contrast, and judge our lives in a degree of detail incomprehensible to any previous generation. Think of a single day of our lives. Almost the first thing that happens after turning off an alarm clock, before brushing teeth, having coffee, seeing a child, or opening a paper is reaching for that phone, iPhone, or Blackberry. As it comes on and speaks to us or we speak through it, it continues to create a map of almost everything in our lives. -- Juan Enriquez, "Immortality," 8 Jan 2010, available at Online article.
The term "reviewed publication" has an appealing ring for the naive rather than the realistic. ... Let's face it: (1) in this day and age of specialization, you may not find competent reviewers for certain contributions; (2) older scientists may agree that over the past two decades, the relative decline in research funds has been accompanied by an increasing number of meaningless, often unfair reviews; (3) some people are so desperate to get published that they will comply with the demands of reviewers, no matter how asinine they are. -- August Epple, from "Organizing Scientific Meetings", quoted in Science 17 Oct 1997, pg. 400.
The controversy between those who think mathematics is discovered and those who think it is invented may run and run, like many perennial problems of philosophy. Controversies such as those between idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half thousand years. I do not expect to be able to convert those committed to the discovery view of mathematics to the inventionist view. However what I have shown is that a better case can be put for mathematics being invented than our critics sometimes allow. Just as realists often caricature the relativist views of social constructivists in science, so too the strengths of the fallibilist views are not given enough credit. For although fallibilists believe that mathematics has a contingent, fallible and historically shifting character, they also argue that mathematical knowledge is to a large extent necessary, stable and autonomous. -- Paul Ernst, from "Is Mathematics Discovered or Invented?", Online article, Linguistic and Philosophical Investigations, 2007, vol. 6, no. 1.
Once humans have invented something by laying down the rules for its existence, like chess, the theory of numbers, or the Mandelbrot set, the implications and patterns that emerge from the underlying constellation of rules may continue to surprise us. But this does not change the fact that we invented the game in the first place. It just shows what a rich invention it was. As the great eighteenth century philosopher Giambattista Vico said, the only truths we can know for certain are those we have invented ourselves. Mathematics is surely the greatest of such inventions. -- Paul Ernst, from "Is Mathematics Discovered or Invented?", Online article, Linguistic and Philosophical Investigations, 2007, vol. 6, no. 1.
A centre of excellence is, by definition, a place where second class people may perform first class work. A truly popular lecture cannot teach, and a lecture that truly teaches cannot be popular. The most prominent requisite to a lecturer, though perhaps not really the most important, is a good delivery; for though to all true philosophers science and nature will have charms innumerably in every dress, yet I am sorry to say that the generality of mankind cannot accompany us one short hour unless the path is strewed with flowers. -- Michael Faraday, from J. M. Thomas, Michael Faraday and the Royal Institution: The Genius of Man and Place, Adam Hilger, Bristol, 1991.
The empirical spirit on which the Western democratic societies were founded is currently under attack, and not just by such traditional adversaries as religious fundamentalists and devotees of the occult. Serious scholars claim that there is no such thing as progress and assert that science is but a collection of opinions, as socially conditioned as the weathervane world of Paris couture. -- Timothy Ferris, The Whole Shebang: A State of the Universe(s) Report, Simon and Shuster, 1998, pg. 1.
We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work. -- Richard Feynman (1918-1988), 1966 Nobel acceptance lecture.
Another thing I must point out is that you cannot prove a vague theory wrong. ... Also, if the process of computing the consequences is indefinite, then with a little skill any experimental result can be made to look like the expected consequences. -- Richard Feynman, 1964, quoted by Gary Taubes in "The (Political) Science of Salt", Science, 14 Aug 1998, pg. 898-907.
You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. -- Richard Feynman, 1982, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
No mathematical theorem has aroused as much interest among non-mathematicians as Godel's incompleteness theorem. ... One finds invocations not only in discussion groups dedicated to logic, mathematics, computing, or philosophy, where one might expect the, but also in groups dedicated to politics, religion, atheism, poetry, evolution, hip-hop, dating, and what have you. -- Torkel Franzen, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 51.
All truths are easy to understand once they are discovered; the point is to discover them. -- Attributed to Galileo Galilei (1564-1642), although no original source is known. See: Wikiquote.
Harald Bohr is reported to have remarked, "Most analysts spend half their time hunting through the literature for inequalities they want to use, but cannot prove." -- D. J. H. Garling, in his review of Michael Steele's The Cauchy Schwarz Master Class in the American Mathematical Monthly, Jun-Jul 2005, pg. 575-579.
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others. -- Carl Friedrich Gauss (1777-1855), from an 1808 letter to his friend Farkas Bolyai (the father of Janos Bolyai).
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. -- Kurt Godel, "Some Basic Theorems on the Foundations of Mathematics and Their Implications," from Solomon Feferman, ed., Collected Works / Kurt Godel, vol. 3, Oxford University Press, 1951, pg. 304-323.
"Who ever became more intelligent," Godel answered, "by reading Voltaire?" -- Kurt Godel, quoted in Palle Yourgrau, A World Without Time, Basic Books, 2005, pg. 15.
"Only fables," he said, "present the world as it should be and as if it had meaning." Kurt Godel, quoted in Palle Yourgrau, A World Without Time, Basic Books, 2005, pg. 5.
Most working scientists may be naive about the history of their discipline and therefore overly susceptible to the lure of objectivist mythology. But I have never met a pure scientific realist who views social context as entirely irrelevant, or only as an enemy to be expunged by the twin lights of universal reason and incontrovertible observation. And surely, no working scientist can espouse pure relativism at the other pole of the dichotomy. ... In fact, as all working scientists know in their bones, the incoherence of relativism arises from virtually opposite and much more quotidian motives. Most daily activity in science can only be described as tedious and boring, not to mention expensive and frustrating. Thomas Edison was just about right in his famous formula for invention as 1% inspiration mixed with 99% perspiration. How could scientists ever muster the energy and stamina to clean cages, run gels, calibrate instruments, and replicate experiments, if they did not believe that such exacting, mindless, and repetitious activities can reveal truthful information about a real world? If all science arises as pure social construction, one might as well reside in an armchair and think great thoughts. -- Stephen J. Gould, "Deconstructing the 'Science Wars' by Reconstructing an Old Mold", Science, 14 Jan 2000, pg. 253-261.
Similarly, and ignoring some self-promoting and cynical rhetoricians, I have never met a serious social critic or historian of science who espoused anything close to a doctrine of pure relativism. The true, insightful, and fundamental statement that science, as a quintessentially human activity, must reflect a surrounding social context does not imply either that no accessible external reality exists, or that science, as a socially embedded and constructed institution, cannot achieve progressively more adequate understanding of nature's facts and mechanisms. -- Stephen J. Gould, "Deconstructing the 'Science Wars' by Reconstructing an Old Mold", Science, 14 Jan 2000, pg. 253-261.
What people forget is e-books were going strong before they were called e-books and they went on to sweep into many aspects of business and publishing. Most of this has gone unnoticed by the media. Probably because it has been a kind of backdoor revolution. To cite one example: Print law books are just about gone. People don't use them in law firms anymore. It's all electronic books or online. A revolution has occurred, but no one's noticed. -- Mark Gross, president of Data Conversion Laboratory, Wired, 25 Dec 2001.
We can clearly see that there is no bi-univocal correspondence between linear signifying links archi-writing, depending on the author, and this multireferential, multidimensional machinic catalysis. The symmetry of scale, the transversality, the pathic non-discursive character of their expansion: all these dimensions re-move us from the logic of the excluded middle and reinforce us in our dismissal of the ontological binarism we criticised previously. A machinic assemblage, through its diverse components, extracts its consistency by crossing ontological thresholds, non-linear thresholds of irreversibility, ontological and phylogenetic thresholds, creative thresholds of heterogenesis and autopoiesis. The notion of scale needs to be expanded to consider fractal symmetries in ontological terms. -- This hopelessly confused but serious piece of postmodern science scholarship is from Felix Guattari, "Chaosmosis: An Ethico-Aesthetic Paradigm", quoted in Alan Sokal and Jean Bricmont, Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science, Picador, New York, 1998, pg. 166.
This "quasi-experimental" approach to proof can help to de-emphasis a focus on rigor and formality for its own sake, and to instead support the view expressed by Hadamard when he stated, "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it" -- J. Hadamard, in E. Borel, Lecons sur la theorie des fonctions, 3rd ed. 1928, quoted in George Polya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (combined edition), New York, Wiley and Sons, 1981, pg. 2:127.
All physicists and a good many quite respectable mathematicians are contemptuous about proof. -- G. H. Hardy, 1877-1947, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Ch. 1, 1940.
The theory of numbers, more than any other science, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations. -- G. H. Hardy, quoted in Tobias Dantzig, Number: The Language of Science, Plume Books, New York, 2007, pg. 57.
The analogy is a rough one, but I am sure that it is not altogether misleading. If we were to push it to its extreme we should be led to a rather paradoxical conclusion; that we can, in the last analysis, do nothing but point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils. This is plainly not the whole truth, but there is a good deal in it. The image gives us a genuine approximation to the processes of mathematical pedagogy on the one hand and of mathematical discovery on the other; it is only the very unsophisticated outsider who imagines that mathematicians make discoveries by turning the handle of some miraculous machine. Finally the image gives us at any rate a crude picture of Hilbert's metamathematical proof, the sort of proof which is a ground for its conclusion and whose object is to convince. -- G. H. Hardy, quoted from the Preface to David Broussoud, "Proofs and Confirmation: The Story of the Alternating Sign Matrix Conjecture," available at Online article, MAA, 1999. Broussoud cites Hardy's Rouse Ball Lecture of 1928.
I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can. There are some peaks which he can distinguish easily, while others are less clear. He sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. B is now fixed in his vision, and from this point he can proceed to further discoveries. In other cases perhaps he can distinguish a ridge which vanishes in the distance, and conjectures that it leads to a peak in the clouds or below the horizon. But when he sees a peak he believes that it is there simply because he sees it. If he wishes someone else to see it, he points to it, either directly or through the chain of summits which led him to recognize it himself. When his pupil also sees it, the research, the argument, the proof is finished. -- G. H. Hardy, quoted from the Preface to David Broussoud, "Proofs and Confirmation: The Story of the Alternating Sign Matrix Conjecture", available at Online article, MAA, 1999. Broussoud cites Hardy's Rouse Ball Lecture of 1928.
Why should I refuse a good dinner simply because I don't understand the digestive processes involved? -- Oliver Heaviside (1850-1925), when criticized for his daring use of operators before they could be justified formally. See: Wikipedia article
What we observe is not nature itself, but nature exposed to our method of questioning. -- Werner Heisenberg, 1963, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths, and ultimately a reminder of our pleasure in the successful solution. ... Besides it is an error to believe that rigor in the proof is the enemy of simplicity. -- David Hilbert, in his "23 Mathematische Probleme" lecture to the Paris International Congress, 1900, from Ben Yandell, The Honors Class: Hilbert's Problems and Their Solvers, A. K. Peters, 2002.
Knowing things is very 20th century. You just need to be able to find things. -- Danny Hillis, on how Google has changed the way we think, as quoted in Achenblog, 1 Jul 2008.
Gauss's first proof of the Fundamental Theorem of Algebra, in his 1799 dissertation, was widely admired as the first wholly satisfactory proof. It relied, however, on a statement "known from higher geometry", which "seems to be sufficiently well demonstrated": If a branch of a real polynomial curve F(x,y) = 0 enters a plane region, it must leave it again. Gauss, evidently feeling more persuasion was needed, added: "Nobody, to my knowledge, has ever doubted it. But if anybody desires it, then on another occasion I intend to give a demonstration which will leave no doubt ...." According to Smale's 1981 Bulletin article (from which these quotes are taken), this "immense gap" remained even when Gauss redid this proof 50 years later, and the gap was not filled until 1920. -- Morris W. Hirsch, from Michael Atiyah et al., "Responses to 'Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics," by A. Jaffe and F. Quinn," Bulletin of the American Mathematical Society, vol. 30, no. 2 (Apr 1994), pg. 178-207.
In order to achieve a [number] system as ingenious as our own, it is first necessary to discover the principle of [digit] position. ... Nowadays, this principle seems to us to have such an obvious simplicity that we forget how the human race has stammered, hesitated and groped through thousands of years before discovering it, and that civilizations as advanced as the Greek and the Egyptian completely failed to find it. -- Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French, John Wiley, 2000, pg. 344.
The measure of the genius of Indian civilization, to which we owe our modern [decimal number] system, is all the greater in that it was the only one in all history to have achieved this triumph. ... Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own. -- Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French, John Wiley, 2000, pg. 346.
This fundamental realization [the three key principles of decimal arithmetic] therefore profoundly changed human existence, by bringing a simple and perfectly coherent notation for all numbers and allowing anyone, even those most resistant to elementary arithmetic, the means to easily perform all sorts of calculations; also by henceforth making it possible to carry out operations which previously, since the dawn of time, had been inconceivable; and opening up thereby the path which led to the development of mathematics, science and technology. -- Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French, John Wiley, 2000, pg. 346.
Now that we can stand back from the story, the birth of our modern number-system [in 4-5th Century India] seems a colossal event in the history of humanity, as momentous as the mastery of fire, the development of agriculture, or the invention of writing, of the wheel, or of the steam engine. -- Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French, John Wiley, 2000, pg. 346.
A wealthy [15th Century] German merchant, seeking to provide his son with a good business education, consulted a learned man as to which European institution offered the best training. "If you only want him to be able to cope with addition and subtraction," the expert replied, "then any French or German university will do. But if you are intent on your son going on to multiplication and division -- assuming that he has sufficient gifts -- then you will have to send him to Italy." -- Georges Ifrah, emphasizing the importance of modern decimal arithmetic (which was not in widespread use in 15thC Europe), in The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French, John Wiley, 2000, pg. 577.
In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57." ... "He really never worked on examples," Mumford observed. ... "I don't think it helped Grothendieck in the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract possible way. It's just very strange. That's the way his mind worked." -- Allyn Jackson, "Comme Appele du Neant: As If Summoned from the Void: The Life of Alexandre Grothendieck", Notices of the AMS, vol. 51, no. 10 (Nov 2004), pg. 1196.
 If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of everyone, and the receiver cannot dispossess himself of it.  Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. He who receives an idea from me, receives instruction himself without lessening mine; as he who lites his taper at mine, receives light without darkening me.  That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density at any point, and like the air in which we breathe, move, and have our physical being, incapable of confinement, or exclusive appropriation.  Inventions then cannot, in nature, be a subject of property. -- Thomas Jefferson, letter to Issac McPherson (13 Aug 1813), in The Writings of Thomas Jefferson, quoted from Lawrence Lessig, The Future of Ideas by Lawrence Lessig, Random House, 2001, pg. 94.
This is the essence of science. Even though I do not understand quantum mechanics or the nerve cell membrane, I trust those who do. Most scientists are quite ignorant about most sciences but all use a shared grammar that allows them to recognize their craft when they see it. The motto of the Royal Society of London is 'Nullius in verba': trust not in words. Observation and experiment are what count, not opinion and introspection. Few working scientists have much respect for those who try to interpret nature in metaphysical terms. For most wearers of white coats, philosophy is to science as pornography is to sex: it is cheaper, easier, and some people seem, bafflingly, to prefer it. Outside of psychology it plays almost no part in the functions of the research machine. -- Steve Jones, University College, London, from his review of How the Mind Works by Steven Pinker, quoted in The New York Review of Books, 6 Nov 1997, pg. 13-14.
We [Kaplansky and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury. -- Irving Kaplansky (1917-2006), quoted in John H. Ewing and F.W. Gehring, ed., Paul Halmos: Celebrating 50 Years of Mathematics, Springer, 1991.
The formulas move in advance of thought, while the intuition often lags behind; in the oft-quoted words of d'Alembert, "L'algebre est genereuse, elle donne souvent plus qu'on lui demande." ["Algebra is generous--it often gives more than we asked."] -- Edward Kasner, "The Present Problems of Geometry," Bulletin of the American Mathematical Society, vol. 11 (1905), pg. 285.
The basic difference between playing a human and playing a supermatch against Deep Blue is the eerie and almost empty sensation of not having a human sitting opposite you. With humans, you automatically know a lot about their nationality, gender, mannerisms, and such minor things as a persistent cough or bad breath. Years ago we had to endure chain-smokers who blew smoke our way. But Deep Blue wasn't obnoxious, it was simply nothing at all, an empty chair not an opponent but something empty and relentless. -- Garry Kasparov, "Techmate", available at Forbes, 22 Feb 1999.
This is not to say that I am not interested in the quest for intelligent machines. My many exhibitions with chess computers stemmed from a desire to participate in this grand experiment. It was my luck (perhaps my bad luck) to be the world chess champion during the critical years in which computers challenged, then surpassed, human chess players. Before 1994 and after 2004 these duels held little interest. The computers quickly went from too weak to too strong. But for a span of ten years these contests were fascinating clashes between the computational power of the machines (and, lest we forget, the human wisdom of their programmers) and the intuition and knowledge of the grandmaster. -- Garry Kasparov, "The Chess Master and the Computer," (review of Chess Metaphors: Artificial Intelligence and the Human Mind by Diego Rasskin-Gutman), New York Review of Books, 11 Feb 2010, available at Online article.
My guess is that this emerging method will be one additional tool in the evolution of the scientific method. It will not replace any current methods (sorry, no end of science!) but will complement established theory-driven science. ... The model may be beyond the perception and understanding of the creators of the system, and since it works it is not worth trying to uncover it. But it may still be there. It just operates at a level we don't have access to. -- Kevin Kelly, in response to Chris Anderson's article "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete", available at Edge, 29 Jun 2008. Anderson's article is available here: Wired.
A doctorate compels most of us to be detailed and narrow, and to carve out our own specialities, and tenure committees rarely like boldness. Later, when our jobs are safe we can be synthetic, and generalize. -- Paul Kennedy (writing about A. J. P. Taylor), "The Nonconformist", Atlantic Monthly, Apr 2001, pg. 114.
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics. -- Johannes Kepler (1571-1630), from Morris Kline, Mathematics: The Loss of Certainty, Oxford University Press, 1982, pg. 31.
The difficulty lies, not in the new ideas, but in escaping the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds. -- John Maynard Keynes (1883-1946), quoted in K. Eric Drexler, Engines of Creation: The Coming Era of Nanotechnology, Anchor, New York, 1987, available online at: Online copy.
[Isaac Newton's] peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his preeminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one's mind and apply all one's powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary---"so happy in his conjectures", said de Morgan, "as to seem to know more than he could possibly have any means of proving." -- J. M. Keynes, writing about Isaac Newton, from "Newton the Man", in James R. Newman, ed., The World of Mathematics, vol. I, Simon and Schuster, NY, 1956.
An informed list of the most profound scientific developments of the 20th century is likely to include general relativity, quantum mechanics, big bang cosmology, the unraveling of the genetic code, evolutionary biology, and perhaps a few other topics of the reader's choice. Among these, quantum mechanics is unique because of its profoundly radical quality. Quantum mechanics forced physicists to reshape their ideas of reality, to rethink the nature of things at the deepest level, and to revise their concepts of position and speed, as well as their notions of cause and effect. -- Daniel Kleppner and Roman Jackiw, quoted from "One Hundred Years of Quantum Physics" in Science 11 Aug 2000, pg. 893-898.
Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often acknowledged. Such fundamental notions as reality, space, time, and causality--notions found at the core of the scientific enterprise--all rely on particular metaphysical assumptions about the world. -- Christof Koch, in "Thinking About the Conscious Mind," a review of John R. Searle's Mind. A Brief Introduction, Oxford University Press, 2004.
How ridiculous to make evolution the enemy of God. What could be more elegant, more simple, more brilliant, more economical, more creative, indeed more divine than a planet with millions of life forms, distinct and yet interactive, all ultimately derived from accumulated variations in a single double-stranded molecule, pliable and fecund enough to give us mollusks and mice, Newton and Einstein? Even if it did give us the Kansas State Board of Education, too. -- Charles Krauthammer, "Phony Theory, False Conflict. 'Intelligent Design' Foolishly Pits Evolution Against Faith", Washington Post, 18 Nov 2005.
As I see it, the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth. Until the Great Depression, most economists clung to a vision of capitalism as a perfect or nearly perfect system. That vision wasn't sustainable in the face of mass unemployment, but as memories of the Depression faded, economists fell back in love with the old, idealized vision of an economy in which rational individuals interact in perfect markets, this time gussied up with fancy equations. ... [The central cause of the profession's failure was the desire for an all-encompassing, intellectually elegant approach that also gave economists a chance to show off their mathematical prowess. -- Paul Krugman, "How Did the Economists Get It So Wrong?", New York Times, 2 Sep 2009, from Article
Unfortunately, this romanticized and sanitized vision of the economy led most economists to ignore all the things that can go wrong. They turned a blind eye to the limitations of human rationality that often lead to bubbles and busts; to the problems of institutions that run amok; to the imperfections of markets -- especially financial markets -- that can cause the economy's operating system to undergo sudden, unpredictable crashes; and to the dangers created when regulators don't believe in regulation. -- Paul Krugman, "How Did the Economists Get It So Wrong?", New York Times, 2 Sep 2009, from Article
And Max Planck, surveying his own career in his Scientific Autobiography, sadly remarked that "a new scientific truth 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." -- Thomas Kuhn, The Structure of Scientific Revolutions, 3rd ed., Univ. of Chicago Press, 1996, pg. 151. Quoting: Max Planck, Scientific Autobiography and Other Papers, trans. F. Gaynor, New York, 1949, pg. 33-34.
Gauss could be a stern, demanding individual, and it is reported that this resulted in friction with two of his sons that caused them to leave Germany and come to the United States; they settled in the midwest and have descendants throughout the plains states. ... My wife, Paulette, and I visited several times with Charlotte and her sister Helen; they were bright, alert, and charming young women, ages 93 and 94, respectively. Their father, Gauss' grandson, had been a Methodist missionary to the region, and he had felt it unseemly to take pride in his famous ancestor (maybe there were some remnants of his father's feelings on leaving Germany); they were nevertheless happy to talk Gauss and their family. They showed us a baby spoon which their father had made out of a gold medal awarded to Gauss, some family papers, and a short biography of Gauss written by an aunt. I vividly remember Helen describing the reaction of one of her math teachers when he discovered he had a real, live, Gauss in his class. -- Jim Kuzmanovichi, quoted from Gauss article.
This diagram [the Mobius strip] can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the [human] subject. ... You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. -- This hopelessly confused but serious piece of postmodern science scholarship is from Jacques Lacan, "Of Structure As an Inmixing of an Otherness Prerequisite to Any Subject Whatever," quoted in Alan Sokal and Jean Bricmont, Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science, Picador, New York, 1998, pg. 19-20.
What is particularly ironic about this is "that it follows from the empirical study of numbers as a product of mind that it is natural for people to believe that numbers are not a product of mind!" -- George Lakoff and Rafael E. Nunez, Where Mathematics Comes From, Basic Books, 2000, pg. 81.
In recent years, there have been revolutionary advances in cognitive science--advances that have a profound bearing on our understanding of mathematics. Perhaps the most profound of these new insights are the following: (1) The embodiment of mind. ... This includes mathematical concepts and mathematical reason. (2) The cognitive unconscious. ... This includes most mathematical thought. (3) Metaphorical thought. For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in sensory-motor systems. -- George Lakoff and Rafael E. Nunez, Where Mathematics Comes From, Basic Books, 2000, pg. 5.
So our celestial science seems to be primarily instrument-driven, guided by unanticipated discoveries with unique telescopes and novel detection equipment. With our current knowledge, we can be certain that the observed universe is just a modest fraction of what remains to be discovered. Recent evidence for dark, invisible matter and mysterious dark energy indicate that the main ingredients of the universe remain largely unknown, awaiting future, serendipitous discoveries. -- Kenneth R. Lang, "Serendipitous Astronomy," Science, vol. 327, no. 59611 (Jan 2010) , pg. 39-40.
An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as of the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. -- Pierre-Simon Laplace (18-19thC French mathematician), from Will and Ariel Durant, The Story of Civilization, Simon and Schuster, New York, 11 volumes, 1975 (date of last volume), vol. 9, pg. 548.
[T]he ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius. -- Pierre Simon Laplace (18-19thC French mathematician), quoted in Georges Ifrah, The Universal History of Numbers, John Wiley, New York, 1998, pg. 361.
My larger target is those contemporaries who--in repeated acts of wish-fulfillment--have appropriated conclusions from the philosophy of science and put them to work in aid of a variety of social cum political causes for which those conclusions are ill adapted. Feminists, religious apologists (including "creation scientists"), counterculturalists, neo-conservatives, and a host of other curious fellow-travelers have claimed to find crucial grist for their mills in, for instance, the avowed incommensurability and underdetermination of scientific theories. The displacement of the idea that facts and evidence matter by the idea that everything boils down to subjective interests and perspectives is--second only to American political campaigns--the most prominent and pernicious manifestation of anti-intellectualism in our time. -- Larry Laudan, from Science and Relativism, University of Chicago Press, 1990, pg. x.
My father was a public scribe of Bejaia, where he worked for his country in Customs, defending the interests of Pisan merchants who made their fortune there. He made me learn how to use the abacus when I was still a child because he saw how I would benefit from this in later life. In this way I learned the art of counting using the nine Indian figures. ... The nine Indian figures are as follows: 9 8 7 6 5 4 3 2 1 [figures given in contemporary European cursive form]. That is why, with these nine numerals, and with this sign 0, called zephirum in Arab, one writes all the numbers one wishes. -- Leonardo of Pisa (Fibonacci), 1202, quoted in Georges Ifrah, The Universal History of Numbers, John Wiley, New York, 1998, pg. 361-362.
The dictum that everything that people do is "cultural" ... licenses the idea that every cultural critic can meaningfully analyze even the most intricate accomplishments of art and science. ... It is distinctly weird to listen to pronouncements on the nature of mathematics from the lips of someone who cannot tell you what a complex number is! -- Norman Levitt, from The Flight From Science and Reason, New York Academy of Science, quoted from Science, 11 Oct 1996, pg. 183.
There's no doubt the really big ideas in mathematics come from maybe 5 percent of the people, but you need a broad base to nourish the 5 percent and to work out all the details as they move on to more adventuresome things. ... In chemistry, people get declined, and in two months they turn around with another proposal. Mathematicians --- they get declined twice, and they fold. I think mathematicians have such a personal investment in their problems that if you turn down their proposals, they take it as if you're judging them as mathematicians. They're not as flexible and often don't seem to be able to move to another class of problems. We fund proposals, not individuals. -- D. J. Lewis of NSF, interview with Allyn Jackson, Notices of the AMS, Jun-Jul 1999, pg. 669.
[Mathematicians] have got do some demonstrations of what mathematics has accomplished for the good of society. One of the things mathematicians have done is education. For example, if mathematicians took seriously the job of training elementary and middle school teachers, they could make some claim that they really improve things. Also, science is getting so complicated, it can be done only with the help of mathematics. Is the math community willing to step up and participate? If so, they will have nonmathematicians making the case for greater funding of mathematics. It is always best to have outsiders make your case for you. Once upon a time I thought going to Capitol Hill would be effective. I don't think it will get very far if mathematicians go to Capitol Hill without the support of others. These days information technology and biology and medicine are the themes that echo well with the president and Congress. -- D. J. Lewis of NSF, interview with Allyn Jackson, Notices of the AMS, Jun-Jul 1999, pg. 672.
Mathematical proofs like diamonds should be hard and clear, and will be touched with nothing but strict reasoning. -- John Locke, from The Mathematical Universe by William Dunham, John Wiley, 1994.
Edison, Feynman, Land, and Newton all from their boyhood had intense curiosity, an enthusiasm or zeal for discovery and understanding. Each of them was able to take seriously hypotheses that others thought to be implausible (or had not thought about at all). All four possessed a kind of intellectual arrogance that permitted them to essay prodigious tasks, to undertake to solve problems that most of their contemporaries believed to be impossible. And each of them had quite extraordinary powers of concentration. ... I think what lies at the heart of these mysteries is genetic, probably emergenic. The configuration of traits of intellect, mental energy, and temperament with which, during the plague years of 1665-6, Isaac Newton revolutionized the world of science were, I believe, the consequence of a genetic lottery that occurred about nine months prior to his birth, on Christmas day, in 1642. -- David T. Lykken, "The Genetics of Genius", in Andrew Steptoe, ed., Genius and the Mind: Studiees of Creativity and Temperament, Oxford University Press, 1998.
About H.E. Smith: In the book "Elementary Number Theory" (Chelsea, New York, 1958. An English translation of vol. 1 of the German book Vorlesungen ueber Zahlentheorie), p.31, the author, Edmund Landau, mentions the question whether the infinite series $\sum \mu(n)/n$ converges (TEX notation; \mu is the Moebius function). After giving a reference to the answer in Part 7 of the same V.u.Z, and without saying what the answer is, Landau writes: "Gordan used to say something to the effect that 'Number Theory is useful since one can, after all, use it to get a doctorate with.' In 1899 I received my doctorate by answering this question." -- This is taken from Alexander Macfarlane, Ten British Mathematicians of the Nineteenth Century, 1916, pg. 63-64. A copy of the book is available on the Project Gutenberg website: Gutenberg
[Henry J. S. Smith] was a brilliant talker and wit. Working in the purely speculative region of the theory of numbers, it was perhaps natural that he should take an anti-utilitarian view of mathematical science, and that he should express it in exaggerated terms as a defiance to the grossly utilitarian views then popular. It is reported that once in a lecture after explaining a new solution of an old problem he said, "It is the peculiar beauty of this method, gentlemen, and one which endears it to the really scientific mind, that under no circumstances can it be of the smallest possible utility." I believe that it was at a banquet of the Red Lions that he proposed the toast, "Pure mathematics; may it never be of any use to anyone." -- This is taken from Alexander Macfarlane, Ten British Mathematicians of the Nineteenth Century, 1916, pg. 63-64. A copy of the book is available on the Project Gutenberg website: Gutenberg
And Bloomberg can also flash a hard-edged candor. At the breakfast with business leaders, he scoffed at a question about whether the schools' emphasis on math and reading testing was taking away from the "richness" of education in subjects such as art and music. "Well, I don't know about the 'richness of education,' " he said, his voice thick with sarcasm. "In my other life, I own a business, and I can tell you, being able to do 2-plus-2 is a lot more important than a lot of other things." -- From Alec MacGillis, "With Bloomberg on Stage, Harsher Light on Giuliani", Washington Post, 6 Aug 2007, pg. A01.
Mathematics requires both intuitive work (e.g., Gromov, Thurston) and precision (J. Frank Adams, J.-P Serre). In theological terms, we are not saved by faith alone, but by faith and works. -- Saunders Mac Lane, from Michael Atiyah et al., "Responses to 'Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics," by A. Jaffe and F. Quinn," Bulletin of the American Mathematical Society, vol. 30, no. 2 (Apr 1994), pg. 178-207.
You ought to know that the ratio of the diameter of the circle to its circumference is unknown, nor will it ever be possible to express it precisely. This is not due to any shortcoming of knowledge on our part, as the ignorant think. Rather, this matter is unknown due to its nature, and its discovery will never be attained. -- Moses Maimonides (Jewish theologian, 1135-1204), anticipating the fact that pi cannot be expressed as the ratio of two integers (a fact later proved in 1768), from his Commentary to the Mishnah, 1168, quoted by George Anastaplo, "A Timely Recapitulation, With Some Help From Socrates, Plato, and Aristotle," 13 Aug 2007, available at Online article.
A generation ago, quants turned finance upside down. Now they're mapping out ad campaigns and building new businesses from mountains of personal data. ... These slices of our lives now sit in databases, many of them in the public domain. From a business point of view, they're just begging to be analyzed. But even with the most powerful computers and abundant, cheap storage, companies can't sort out their swelling oceans of data, much less build businesses on them, without enlisting skilled mathematicians and computer scientists. The rise of mathematics is heating up the job market for luminary quants, especially at the Internet powerhouses where new math grads land with six-figure salaries and rich stock deals. Tom Leighton, an entrepreneur and applied math professor at Massachusetts Institute of Technology, says: "All of my students have standing offers at Yahoo! and Google." Top mathematicians are becoming a new global elite. It's a force of barely 5,000, by some guesstimates, but every bit as powerful as the armies of Harvard University MBAs who shook up corner suites a generation ago. -- "Math Will Rock Your World," Business Week, 23 Jan 2006, available at Online article.
In 1965 the Russian mathematician Alexander Konrod said "Chess is the Drosophila of artificial intelligence." However, computer chess has developed as genetics might have if the geneticists had concentrated their efforts starting in 1910 on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies. -- From John McCarthy's review of Kasparov versus Deep Blue by Monty Newborn (Springer, 1996) in Science, 6 Jun 1997, pg. 1518.
Today's outcome may end the interest in future chess matches between human champions and computers, according to Monty Newborn, a professor of computer science at McGill University in Montreal. Professor Newborn, who helped organize the match between Mr. Kasparov and Deep Blue, said of future matches: "I don't know what one could get out of it at this point. The science is done." ... "If you look back 50 years, that was one thing we thought they couldn't do," he said. "It is one little step, that's all, in the most exciting problem of what can't computers do that we can do." -- Dylan Loeb McClain, from a report of the defeat of world champion Vladimir Kramnik by Deep Fritz in "Once Again, Machine Beats Human Champion at Chess", New York Times, 5 Dec 2006.
Anticipatory plagiarism occurs when someone steals your original idea and publishes it a hundred years before you were born. -- Robert Merton, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 102.
My friend was an undergraduate at Princeton in the early 1960's and took a course in logic from Church. It was well known that Church's lectures followed his book exactly, so none of them took notes -- they just sat there following along with the book. Church would write everything on the board from memory. At one point they noticed that his only deviation from the book was that he left out a comma in one sentence. However, after writing a few more sentences, Church stood back from the board, stared at it, and then added the missing comma. -- Victor Miller, writing about mathematician/computer scientist Alonzo Church, email in editor's possession, 13 Mar 2010.
If I can give an abstract proof of something, I'm reasonably happy. But if I can get a concrete, computational proof and actually produce numbers I'm much happier. I'm rather an addict of doing things on the computer, because that gives you an explicit criterion of what's going on. I have a visual way of thinking, and I'm happy if I can see a picture of what I'm working with. -- John Milnor, quoted in Who Got Einstein's Office? by Ed Regis, Addison-Wesley, 1986, pg. 78.
The ingenious number-system, which serves as the basis for modern arithmetic, was used by the Arabs long before it reached Europe. It would be a mistake, however, to believe that this invention is Arabic. There is a great deal of evidence, much of it provided by the Arabs themselves, that this arithmetic originated in India. -- J. F. Montucla, 1798, quoted in Georges Ifrah, The Universal History of Numbers, John Wiley, New York, 1998, pg. 361.
I got into a research project which can be very simply described as concerned with the realization of the "Nash program" (making use of words made conventional by others that refer to suggestions I had originally made in my early works in game theory). ... In this project a considerable quantity of work in the form of calculations has been done up to now. Much of the value of this work is in developing the methods by which tools like Mathematica can be used with suitable special programs for the solution of problems by successive approximation methods. -- John Nash, from Harold W. Kuhn and Sylvia Nasar, ed., The Essential John Nash, Princeton Univ. Press, 2001, pg. 241.
For those who had realized big losses or gains, the mania redistributed wealth. The largest honest fortune was made by Thomas Guy, a stationer turned philanthropist, who owned 54,000 [pounds] of South Sea stock in April 1720 and sold it over the following six weeks for 234,000. Sir Isaac Newton, scientist, master of the mint, and a certifiably rational man, fared less well. He sold his 7,000 [pounds] of stock in April for a profit of 100 percent. But something induced him to reenter the market at the top, and he lost 20,000. "I can calculate the motions of the heavenly bodies," he said, "but not the madness of people." -- Isaac Newton, quoted by Christopher Reed in "The Damn'd South Sea", Harvard Magazine, May-Jun 1999.
The orbit of any one planet depends on the combined motions of all the planets, not to mention the actions of all these on each other. To consider simultaneously all these causes of motion and to define these motions by exact laws allowing of convenient calculation exceeds, unless I am mistaken, the forces of the entire human intellect. -- Isaac Newton, 1687, from G. Lake, T. Quinn and D. C. Richardson, "From Sir Isaac to the Sloan Survey: Calculating the Structure and Chaos Due to Gravity in the Universe,"Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1997, pg. 1-10.
There is no reason anyone would want a computer in their home. -- Ken Olson, President, Chairman and Founder, Digital Equipment Corp. [now defunct], 1977, quoted in Leon A Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46.
Consider a precise number that is well known to generations of parents and doctors: the normal human body temperature of 98.6 Farenheit. Recent investigations involving millions of measurements reveal that this number is wrong; normal human body temperature is actually 98.2 Farenheit. The fault, however, lies not with Dr. Wunderlich's original measurements - they were averaged and sensibly rounded to the nearest degree: 37 Celsius. When this temperature was converted to Farenheit, however, the rounding was forgotten and 98.6 was taken to be accurate to the nearest tenth of a degree. Had the original interval between 36.5 and 37.5 Celsius been translated, the equivalent Farenheit temperatures would have ranged from 97.7 to 99.5. Apparently, discalculia can even cause fevers. -- John Allen Paulus, in A Mathematician Reads the Newspaper Basic Books, quoted in Science, 18 Aug 1995, pg. 992.
2000 was a banner year for scientists deciphering the "book of life"; this year saw the completion of the genome sequences of complex organisms ranging from the fruit fly to the human. ... Genomes carry the torch of life from one generation to the next for every organism on Earth. Each genome--physically just molecules of DNA--is a script written in a four-letter alphabet. Not too long ago, determining the precise sequence of those letters was such a slow, tedious process that only the most dedicated geneticist would attempt to read any one "paragraph"--a single gene. But today, genome sequencing is a billion-dollar, worldwide enterprise. Terabytes of sequence data generated through a melding of biology, chemistry, physics, mathematics, computer science, and engineering are changing the way biologists work and think. Science marks the production of this torrent of genome data as the Breakthrough of 2000; it might well be the breakthrough of the decade, perhaps even the century, for all its potential to alter our view of the world we live in. Elizabeth Pennisi, from "Breakthrough of the Year: Genomics Comes of Age", Science 22 Dec 2000, pg. 2220-2221.
A new scientific truth usually does not make its way in the sense that its opponents are persuaded and declare themselves enlightened, but rather that the opponents become extinct and the rising generation was made familiar with the truth from the very beginning. -- Max Planck, quoted in F. G. Major, The Quantum Beat, Springer, 1998, preface.
My morale has never been higher than since I stopped asking for grants to keep my lab going. -- Robert Pollack, Columbia Professor of biology, speaking on "The Crisis in Scientific Morale", 19 Sep 1996 at GWU's symposium Science in Crisis at the Millennium, quoted from Science, 27 Sep 1996, pg. 1805.
Should we teach mathematical proofs in the high school? In my opinion, the answer is yes... Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics' contribution to general culture. -- George Polya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (combined edition), New York, Wiley and Sons, 1981, pg. 2:126.
A mathematical deduction appears to Descartes as a chain of conclusions, a sequence of successive steps. What is needed for the validity of deduction is intuitive insight at each step which shows that the conclusion attained by that step evidently flows and necessarily follows from formerly acquired knowledge (acquired directly by intuition or indirectly by previous steps) ... I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. -- George Polya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (combined edition), New York, Wiley and Sons, 1981, pg. 2:128.
[I]ntuition comes to us much earlier and with much less outside influence than formal arguments which we cannot really understand unless we have reached a relatively high level of logical experience and sophistication. Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. ... In the first place, the beginner must be convinced that proofs deserve to be studied, that they have a purpose, that they are interesting. -- George Polya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (combined edition), New York, Wiley and Sons, 1981, pg. 2:128.
The purpose of a legal proof is to remove a doubt, but this is also the most obvious and natural purpose of a mathematical proof. We are in doubt about a clearly stated mathematical assertion, we do not know whether it is true or false. Then we have a problem: to remove the doubt, we should either prove that assertion or disprove it. -- George Polya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (combined edition), New York, Wiley and Sons, 1981, pg. 2:129.
Where ... the ENIAC is equipped with 18,000 vacuum tubes and weights 30 tons, computers in the future may have only 1,000 vacuum tubes and weigh only 1.5 tons. -- Popular Mechanics, 1949, quoted in Leon A. Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46.
[A] certain impression I had of mathematicians was ... that they spent immoderate amounts of time declaring each other's work trivial. -- Richard Preston, from his prize winning article "The Mountains of Pi", New Yorker, 9 Mar 1992.
In [A. Church's] lectures he was painstakingly careful. There was a story that went the rounds. If Church said it's obvious, then everybody saw it a half hour ago. If Weyl says it's obvious, von Neumann can prove it. If Lefschetz says it's obvious, it's false. -- Princeton Oral History Project, available at Online article.
Just what does it mean to prove something? Although the Annals will publish Dr Hales's paper, Peter Sarnak, an editor of the Annals, whose own work does not involve the use of computers, says that the paper will be accompanied by an unusual disclaimer, stating that the computer programs accompanying the paper have not undergone peer review. There is a simple reason for that, Dr. Sarnak says--it is impossible to find peers who are willing to review the computer code. However, there is a flip-side to the disclaimer as well--Dr. Sarnak says that the editors of the Annals expect to receive, and publish, more papers of this type--for things, he believes, will change over the next 20-50 years. Dr. Sarnak points out that maths may become "a bit like experimental physics" where certain results are taken on trust, and independent duplication of experiments replaces examination of a colleague's paper. -- From "Proof and Beauty", Economist article, 31 Mar 2005.
Why should the non-mathematician care about things of this nature? The foremost reason is that mathematics is beautiful, even if it is, sadly, more inaccessible than other forms of art. The second is that it is useful, and that its utility depends in part on its certainty, and that that certainty cannot come without a notion of proof. Dr. Gonthier, for instance, and his sponsors at Microsoft, hope that the techniques he and his colleagues have developed to formally prove mathematical theorems can be used to "prove" that a computer program is free of bugs-and that would certainly be a useful proposition in today's software society if it does, indeed, turn out to be true. -- From "Proof and Beauty", Economist article, 31 Mar 2005.
No man can worthely praise Ptolemye ... yet muste ye and all men take heed, that both in him and in all mennes workes, you be not abused by their autoritye, but evermore attend to their reasons, and examine them well, ever regarding more what is saide, and how it is proved, than who saieth it, for autorite often times deceaveth many menne. -- Robert Record, medieval textbook writer in his cosmology text, "The Castle of Knowledge", 1556, quoted from Oxford Figures, Oxford University Press, 2000, pg. 47.
The Internet enables far wider participation in front-line science; it levels the playing field between researchers in major centres and those in relative isolation, hitherto handicapped by inefficient communication. It has transformed the way science is communicated and debated. More fundamentally, it changes how research is done, what might be discovered, and how students learn. -- Martin Rees, "A Level Playing Field," 8 Jan 2010, available at Online article.
The first instance of "the proof is left as an exercise" occurred in `De Triangulis Omnimodis' by Regiomontanus, written 1464 and published 1533. He is quoted as saying "This is seen to be the converse of the preceding. Moreover, it has a straightforward proof, as did the preceding. Whereupon I leave it to you for homework." -- Regiomontanus, quoted in Science, 1994.
Caution, skepticism, scorn, distrust and entitlement seem to be intrinsic to many of us because of our training as scientists. ... These qualities hinder your job search and career change. -- Stephen Rosen (former astrophysicist, now Director, Scientific Career Transitions Program, New York City, quoted in Science, 4 Aug 1995, pg. 637.
Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate. -- Bertrand Russell, from "Recent Work on the Principles of Mathematics", in International Monthly, July 1901, pg. 83-101; also Bertrand Russell, Collected Papers, vol. 3, pg. 366; revised version in Newman's World of Mathematics, vol. 3, pg. 1577.
If my teachers had begun by telling me that mathematics was pure play with presuppositions, and wholly in the air, I might have become a good mathematician, because I am happy enough in the realm of essence. But they were overworked drudges, and I was largely inattentive, and inclined lazily to attribute to incapacity in myself or to a literary temperament that dullness which perhaps was due simply to lack of initiation. -- George Santayana, Persons and Places, 1945, pg. 238-239.
Renyi would become one of Erdos's most important collaborators. ... Their long collaborative sessions were often fueled by endless cups of strong coffee. Caffeine is the drug of choice for most of the world's mathematicians and coffee is the preferred delivery system. Renyi, undoubtedly wired on espresso, summed this up in a famous remark almost always attributed to Erdos: "A mathematician is a machine for turning coffee into theorems." ... Turan, after scornfully drinking a cup of American coffee, invented the corollary: "Weak coffee is only fit for lemmas." Bruce Schechter, My Brain is Open (Schechter's biography of Erdos), Simon and Schuster, 1998, pg. 155.
I don't like it, and I'm sorry I ever had anything to do with it. Erwin Schrodinger, about the probability interpretation of quantum mechanics, from Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory", New York Times, 27 Dec 2005.
A NASA employee's explanation for the loss of a laptop, recorded in a recent report by the U.S. Government Accountability Office documenting equipment losses of more than $94 million over the past 10 years by the agency: ... "This computer, although assigned to me, was being used on board the International Space Station. I was informed that it was tossed overboard to be burned up in the atmosphere when it failed." -- From Science, vol. 317, no. 5838, 3 Aug 2007, pg. 579.
LONG BEACH, CALIFORNIA--Scientists have been scrutinizing gravity since the time of Newton, but they've had difficulty measuring the power of its pull. Now, thanks to a clever device, physicists have the most precise measurement yet. ... "[It] should have been obvious" that previous measures of big G were off, says physicist Randy Newman of the University of California, Irvine. The new result, announced this week at the American Physical Society meeting, sets big G tentatively at 6.67423 plus or minus 0.00009 x 10^(-11) m^3/(kg s^2). "It's one of the fundamental constants," Gundlach says. "Mankind should just know it. It's a philosophical thing." -- Charles Seife, "Gravity Turntable Sets New Record," ScienceNow 5 May 2000, available at Online article
Nobody contends that all of science is wrong, or that it hasn't compiled an impressive array of truths about the natural world. Still, any single scientific study alone is quite likely to be incorrect, thanks largely to the fact that the standard statistical system for drawing conclusions is, in essence, illogical. "A lot of scientists don't understand statistics," says Goodman. "And they don't understand statistics because the statistics don't make sense." -- Tom Siegfried, "Odds Are, It's Wrong," ScienceNews, vol. 177, no. 7 (27 Mar 2010), pg. 26, available at Online article.
He designed and built chess-playing, maze-solving, juggling and mind-reading machines. These activities bear out Shannon's claim that he was more motivated by curiosity than usefulness. In his words `I just wondered how things were put together.' -- From Claude Shannon's Obituary.
This skyhook-skyscraper construction of science from the roof down to the yet unconstructed foundations was possible because the behaviour of the system at each level depended only on a very approximate, simplified, abstracted characterization at the level beneath. This is lucky, else the safety of bridges and airplanes might depend on the correctness of the "Eightfold Way" of looking at elementary particles. -- Herbert A. Simon, The Sciences of the Artificial, MIT Press, 1996, pg. 16.
More than fifty years ago Bertrand Russell made the same point about the architecture of mathematics. See the "Preface" to Principia Mathematica: "... the chief reason in favour of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question allows us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reason rather for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." Contemporary preferences for deductive formalisms frequently blind us to this important fact, which is no less true today than it was in 1910. -- Herbert A. Simon, quoting Bertrand Russell, in The Sciences of the Artificial, MIT Press, 1996, pg. 16.
Numbers are not the only thing that computers are good at processing. Indeed, only a cursory familiarity with fractal geometry is needed to see that computers are good at creating and manipulating visual representations of data. There is a story told of the mathematician Claude Chevalley, who, as a true Bourbaki, was extremely opposed to the use of images in geometric reasoning. He is said to have been giving a very abstract and algebraic lecture when he got stuck. After a moment of pondering, he turned to the blackboard, and, trying to hide what he was doing, drew a little diagram, looked at it for a moment, then quickly erased it, and turned back to the audience and proceeded with the lecture. It is perhaps an apocryphal story, but it illustrates the necessary role of images and diagrams in mathematical reasoning-even for the most diehard anti-imagers. The computer offers those less expert, and less stubborn than Chevalley, access to the kinds of images that could only be imagined in the heads of the most gifted mathematicians, images that can be coloured, moved and otherwise manipulated in all sorts of ways. -- Nathalie Sinclair, 2004, from M. Carlson and C. Rasmussen, ed., Making the Connection: Research and Practice in Undergraduate Mathematics, MAA, 2008.
For Poincare, ignoring the emotional sensibility, even in mathematical demonstrations "would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility". -- Nathalie Sinclair, quoting Henri Poincare's "Mathematical Creation" (1956), in James R. Newman, ed., The World of Mathematics, Simon and Schuster, pg. 2047.
Brought up on the refined diet of music, mathematics and aesthetics, Chandrasekhar's own writing is probably the most appropriate mirror of his personality. I quote: "When Michelson was asked towards the end of his life, why he had devoted such a large fraction of his time to the measurement of the velocity of light, he is said to have replied 'It was so much fun'." Prof. Chandrasekhar goes on to some length to explain the term quoting even the Oxford Dictionary -- "fun" means "drollery", what Michelson really meant, Chandrasekhar asserts is "pleasure" and "enjoyment" -- evidently "fun" in the colloquial sense, a concept, so familiar in our so called ordinary life has no place in Chandrasekhar's dictionary... -- Bikash Sinha, in "Aesthetics and Motivations in Arts and Science", Online article.
By the turn of [the 21st] century, we will live in a paperless society. -- Roger Smith, Chair of General Motors, 1986, quoted in Leon A. Kappelman, "The Future is Ours", Communications of the ACM, March 2001, pg. 46.
The Internet synchronizes the thinking of global scientific communities. Everyone gets the news about the new papers at the same time every day. Gossip spreads just as fast on blogs. Announcements of new experimental results are video-cast through the Internet as they happen. -- Lee Smolin, "We Have become Hunter Gatherers of Images and Information," 8 Jan 2010, available at Online article.
Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity. -- Mary Somerville (1780-1872), in James Roy Newman, The World of Mathematics, vol. 4, Dover, 2000.
Rather, [scientists] cling to the dogma imposed by the long post-Enlightenment hegemony over the Western intellectual outlook, which can be summarized briefly as follows: that there exists an external world, whose properties are inde-pendent of any individual human being and indeed of humanity as a whole; that these properties are encoded in "eternal" physical laws; and that human beings can obtain reliable, albeit imperfect and tentative, knowledge of these laws by hewing to the "objective" procedures and epistemological strictures prescribed by the (so-called) scientific method. -- Alan Sokal, tongue-in-cheek "critique" of modern science, from his famous postmodern science parody-hoax, "Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity," Social Text, Spring-Summer 1996, pg. 217-252, also available at Sokal hoax.
In this way the infinite-dimensional invariance group erodes the distinction be-tween the observer and observed; the pi of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone. -- Alan Sokal, from his famous postmodern science parody-hoax, "Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity," Social Text, Spring-Summer 1996, pg 217-252, also available at Sokal hoax.
These aspects of exploratory experimentation and wide instrumentation originate from the philosophy of (natural) science and have not been much developed in the context of experimental mathematics. However, I claim that e.g. the importance of wide instrumentation for an exploratory approach to experiments that includes concept formation also pertain to mathematics. -- Hendrik Sorenson, "How Experimental is Experimental Mathematics?", manuscript, 2008.
Mathematics has been developing responses to the ubiquity of error for hundreds of years, resulting in formal logic and the mathematical proof. Computation is similarly highly error-prone, but recent enough to still be developing equivalent standards of openness and collective verification. An essential response is reproducibility of results: the release of code and data that generated the computational findings we'd like to consider as a contribution to society's stock of knowledge. This subjects computational research to the same standards of openness as filled by the role of the proof in mathematics. -- Victoria Stodden, "Cogitamus, Ergo Sum? The Difference Between Knowing the Name of Something and Knowing Something," 8 Jan 2010, available at Online article.
The early study of Euclid made me a hater of geometry. -- James Joseph Sylvester, 1814-97, quoted in D. MacHale, Comic Sections, Dublin 1993.
Universities are finally losing their monopoly on higher learning, as the web inexorably becomes the dominant infrastructure for knowledge serving both as a container and as a global platform for knowledge exchange between people. ... Meanwhile on campus, there is fundamental challenge to the foundational modus operandi of the University--the model of pedagogy. Specifically, there is a widening gap between the model of learning offered by many big universities and the natural way that young people who have grown up digital best learn. -- Dan Tapscott, "The Impending Demise of the University", available at Edge, 4 June 2009.
The old-style lecture, with the professor standing at the podium in front of a large group of students, is still a fixture of university life on many campuses. It's a model that is teacher-focused, one-way, one-size-fits-all and the student is isolated in the learning process. Yet the students, who have grown up in an interactive digital world, learn differently. Schooled on Google and Wikipedia, they want to inquire, not rely on the professor for a detailed roadmap. They want an animated conversation, not a lecture. They want an interactive education, not a broadcast one that might have been perfectly fine for the Industrial Age, or even for boomers. These students are making new demands of universities, and if the universities try to ignore them, they will do so at their peril. -- Dan Tapscott, "The Impending Demise of the University", available at Edge, 4 June 2009.
The broadcast model might have been perfectly adequate for the baby-boomers, who grew up in broadcast mode, watching 24 hours a week of television (not to mention being broadcast to as children by parents, as students by teachers, as citizens by politicians, and when then entered the workforce as employees by bosses). But young people who have grown up digital are abandoning one-way TV for the higher stimulus of interactive communication they find on the Internet. In fact television viewing is dropping and TV has become nothing more than ambient media for youth--akin to Muzak. Sitting mutely in front of a TV set--or a professor--doesn't appeal to or work for this generation. They learn differently best through non-sequential, interactive, asynchronous, multi-tasked and collaborative. -- Dan Tapscott, "The Impending Demise of the University", available at Edge, 4 June 2009.
If universities want to adapt the teaching techniques to their current audience, they should, as I've been saying for years, make significant changes to the pedagogy. And the new model of learning is not only appropriate for youth--but increasingly for all of us. In this generation's culture is the new culture of learning. ... The professors who remain relevant will have to abandon the traditional lecture, and start listening and conversing with the students--shifting from a broadcast style and adopting an interactive one. Second, they should encourage students to discover for themselves, and learn a process of discovery and critical thinking instead of just memorizing the professor's store of information. Third, they need to encourage students to collaborate among themselves and with others outside the university. Finally, they need to tailor the style of education to their students' individual learning styles. -- Dan Tapscott, "The Impending Demise of the University", available at Edge, 4 June 2009.
The digital world, which has trained young minds to inquire and collaborate, is challenging not only the lecture-driven teaching traditions of the university, but also the very notion of a walled-in institution that excludes large numbers of people. Why not allow a brilliant grade 9 student to take first-year math, without abandoning the social life of his high school? Why not deploy the interactive power of the internet to transform the university into a place of life-long learning, not just a place to grow up? -- Dan Tapscott, "The Impending Demise of the University", available at Edge, 4 June 2009.
The work then proceeds in a manner unique to science. Because practitioners publish their work electronically, through the e-print archives at the Los Alamos National Laboratory in New Mexico, the entire community can read a paper hours after its authors finish typing the last footnote. As a result, no one theorist or even a collaboration does definitive work. Instead, the field progresses like a jazz performance: A few theorists develop a theme, which others quickly take up and elaborate. By the time it's fully developed, a few dozen physicists, working anywhere from Princeton to Bombay to the beaches of Santa Barbara, may have played important parts. -- Gary Taubes, from "String Theorists Find a Rosetta Stone", Science, 23 Jul 1999, pg. 513.
The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology, and all of them the physicist can more or less easily read and adequately solve. -- D'Arcy Wentworth Thompson, John Tyler Bonner, On Growth and Form, Oxford University Press, 1992, pg. 7.
So the Internet causes scientific knowledge to become obsolete faster than was the case with the older print media. A scientist trained in the print media tradition is aware that there is knowledge stored in the print journals, but I wonder if the new generation of scientists, who grow up with the Internet, are aware of this. Also, print journals were forever. They may have merely gathered dust for decades, but they could still be read by any later generation. I can no longer read my own articles stored on the floppy discs of the 1980's. Computer technology has changed too much. Will information stored on the Internet become unreadable to later generations because of data storage changes, and the knowledge lost? -- Frank J. Tipler, "Will the Great Leveler Destroy Diversity of Thought?", 8 Jan 2010, available at Online article.
If you have a great idea, solid science, and earthshaking discoveries, you are still only 10% of the way there. -- David Tomei, LXR Biotechnology Inc., quoted in Wade Roush, "On the Biotech Pharm, a Race to Harvest New Cancer Cures", Science, 7 Nov 1997, pg. 1039-1040.
I often wonder, when reading descriptions of the scientific process by sociologists, if this is how an atom would feel if it could read a quantum mechanics textbook. -- James Trefil, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 110.
Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise. -- J. W. Tukey (1962, page 13), "The Future of Data Analysis", Annals of Mathematical Statistics, 1962, pg. 13.
Science is a differential equation. Religion is a boundary condition. -- Alan Turing, (1912-1954), letter to Robin Gandy, 1954; reprinted in Andrew Hodges, Alan Turing: the Enigma, Vintage edition, 1992, pg. 513.
A coded message, for example, might represent gibberish to one person and valuable information to another. Consider the number 14159265... Depending on your prior knowledge, or lack thereof, it is either a meaningless random sequence of digits, or else the fractional part of pi, an important piece of scientific information. -- Hans Christian von Baeyer, Information: The New Language of Science, Weidenfeld and Nicolson, 2003, pg. 11.
Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so bersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. [Mathematicians are a kind of Frenchman: whatever you say to them they translate into their own language, and right away it is something entirely different.] -- Johann Wolfgang von Goethe, Maximen und Reflexionen, no. 1279, pg. 160 (of the Penguin classic edition).
There exists today a very elaborate system of formal logic, and specifically, of logic applied to mathematics. This is a discipline with many good sides but also serious weaknesses. ... Everybody who has worked in formal logic will confirm that it is one of the technically most refactory parts of mathematics. The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or the complex number, that is with mathematical analysis. Yet analysis is the technically most successful and best-elaborated part of mathematics. Thus formal logic, by the nature of its approach, is cut off from the best cultivated portions of mathematics, and forced onto the most difficult mathematical terrain, into combinatorics. -- John von Neumann, 1948, quoted in L. Blum, P. Cucker, M. Shub and S. Smale, Complexity and Real Computation, Springer-Verlag, New York, 1998.
[A] thrill which is indistinguishable from the thrill I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues representing 'Day', 'Night', 'Evening', and 'Dawn' which Michelangelo has set over the tomb of Guiliano de'Medici and Lorenzo de'Medici. -- G. N. Watson, 1886-1965, commenting on formulas of Ramanujan, quoted from J. M. Borwein, P. B. Borwein and D. H. Bailey, "Ramanujan, Modular Equations, and Approximations to Pi", American Mathematical Monthly, Mar 1989, pg. 201-219.
I think there is a world market for maybe five computers. -- Thomas J. Watson, CEO of IBM, 1943, quoted in Leon A. Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46. [DHB: This oft-cited quote is likely a garbling of a statement made by Watson to the IBM stockholders meeting in 1953, with "20" instead of "5". See: IBM FAQ.]
This "telephone" has too many shortcomings to be seriously considered as a means of communication. The device is inherently of no value to us. -- Western Union internal memo, 1876, quoted in Leon A. Kappelman, "The Future is Ours," Communications of the ACM, March 2001, pg. 46.
The problems of mathematics are not problems in a vacuum. There pulses in them the life of ideas which realize themselves in concreto through our human endeavors in our historical existence, but forming an indissoluble whole transcending any particular science. -- Hermann Weyl, in "David Hilbert and His Mathematical Work", Bulletin of the American Mathematical Society, vol. 50 (1944), pg. 615.
The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. 'Mathematizing' may well be a creative activity of man, like language or music, of primary originality, whose Historical decisions defy complete objective rationalisation. -- Hermann Weyl, in "Obituary: David Hilbert 1862-1943", RSBIOS, vol. 4 (1944), pg. 547-553; and American Philosophical Society Year Book, 1944, pg. 392.
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong. -- Hermann Weyl, (1885-1955), from American Mathematical Monthly, Nov 1992.
The war became more and more bitter. The Dominican Father Caccini preached a sermon from the text, "Ye men of Galilee, why stand ye gazing up into heaven?" and this wretched pun upon the great astronomer [Galileo]'s name ushered in sharper weapons; for, before Caccini ended, he insisted that "geometry is of the devil," and that "mathematicians should be banished as the authors of all heresies." The Church authorities gave Caccini a promotion. -- Andrew Dickson White (American historian), A History of the Warfare of Science with Theology in Christendom, chap. 3, sec. 3, available at: White.
There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain. -- Alfred North Whitehead, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 88.
I would not go so far as to say that to construct a history of thought without a profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming -- and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings. -- Alfred North Whitehead, quoted in John D. Barrow, New Theories of Everything, Oxford University Press, 2007, pg. 202.
In all likelihood, our post-modern habit of viewing science as only a paradigm would evaporate if we developed appendicitis. We should look for a medically trained surgeon who knew what an appendix was, where it was, and how to cut it out without killing us. Likewise, we should be happy to debate the essentially fictive nature of, let us say, Newton's Laws of Gravity unless and until someone threatened to throw us out of a top-storey window. Then the law of gravity would seem very real indeed. -- A. N. Wilson, God's Funeral, Norton, 1999, pg. 178, quoted in Richard C. Brown, Are Science and Mathematics Socially Constructed?, World Scientific, 2009, pg. 207.
One major barrier to entry into new markets is the requirement to see the future with clarity. It has been said that to so foretell the future, one has to invent it. To be able to invent the future is the dividend that basic research pays. -- Eugen Wong, "An Economic Case for Basic Research," by Eugen Wong, Hong Kong University of Science and Technology, quoted in Nature, 16 May 1996, pg. 178-179.
In 1901, I said to my brother Orville that man would not fly for 50 years. Ever since I have ... avoided predictions. -- Wilbur Wright, quoted in Leon A. Kappelman, "The Future is Ours", Communications of the ACM, March 2001, pg. 46.