9 Apr 2016 (c) 2016

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,

*
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,"
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.

*
[1] 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. [2] 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. [3] 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.
[4] 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.

*
[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.

*
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.