tags:

Some thoughts and comments on various topics

John Bartlett wrote one of the most famous collections of quotations. His book, Bartlett’s Familiar Quotations, was published in 1855 and is still published today, or is available on-line.

Today I thought I would share some quotes about mathematics, theory, and science.

I hope you enjoy them and like some of them. My favorite is ${\dots}$

Quotes—Computer Science

Only math nerds would call ${2^{500}}$ finite.
—Leonid Levin

When a professor insists computer science is X but not Y, have compassion for his graduate students.
—Alan Perlis

Computers are useless. They can only give you answers.
—Pablo Picasso

All parts should go together without forcing. You must remember that the parts you are reassembling were disassembled by you. Therefore, if you can’t get them together again, there must be a reason. By all means, do not use a hammer.
—IBM Manual, 1925

The biggest difference between time and space is that you can’t reuse time.
—Merrick Furst

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.
—John von Neumann

The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.
—Ronald Graham

Quotes—Mathematics

The infinite we shall do right away. The finite may take a little longer.
—Stanislaw Ulam

Complete disorder is impossible.
—Theodore Motzkin

A set is a Many that allows itself to be thought of as a One.
—Georg Cantor

The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don’t understand about the link between addition and multiplication.
—Brian Conrey

If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
—David Hilbert

Quotes—Problem Solving

Try a hard problem. You may not solve it, but you will prove something else.
—John Littlewood

If you don’t work on important problems, it’s not likely that you’ll do important work.
—Richard Hamming

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

A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.
—Stefan Banach

It isn’t that they can’t see the solution. It is that they can’t see the problem.
—Gilbert Chesterton

I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated.
—Poul Anderson

The open secret of real success is to throw your whole personality at a problem.
—George Polya

“Obvious” is the most dangerous word in mathematics.
—Eric Bell

In mathematics the art of proposing a question must be held of higher value than solving it.
—Georg Cantor

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
—Hermann Weyl

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.
—Stanley Gudder

For every complex problem there is an answer that is clear, simple, and wrong.
—Henry Mencken

Proof is an idol before whom the pure mathematician tortures himself.
—Arthur Eddington

Open Problems

December 28, 2010 11:24 am

Fantastic Collection! Personally I really like:
– Only math nerds would call {2^{500}} finite.
– Computers are useless. They can only give you answers.
– The infinite we shall do right away. The finite may take a little longer.
– If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
– It isn’t that they can’t see the solution. It is that they can’t see the problem.
– Proof is an idol before whom the pure mathematician tortures himself.

2. December 28, 2010 11:38 am

Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book.
— Alan M. Turing

December 28, 2010 11:51 am

One of the big misapprehensions about mathematics that we perpetrate
in our classrooms is that the teacher always seems to know the answer to
any problem that is discussed. This gives students the idea that there
is a book somewhere with all the right answers to all of the interesting
questions, and that teachers know those answers. And if one could get
hold of the book, one would have everything settled. That’s so unlike the
true nature of mathematics.
— Leon Henkin

December 28, 2010 12:28 pm

We should explain, before proceeding, that it is not our object to consider this program with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution…. If we take \$n\$ simple equations containing \$n-1\$ variables, \$n\$ being a number unlimited in magnitude, the case becomes still more obvious, as the same three cards might take the place of thousands or millions of cards.
— Ada Augusta Byron King, Countess of Lovelace, translator’s notes for Luigi F. Menabrea, ”Sketch of the Analytical Engine invented by Charles Babbage, Esq.” (1843)

The problem is that we attempt to solve the simplest questions cleverly, thereby rendering them unusually complex. One should seek the simple solution.
— Anton Chekhov

There is always an easy solution to every human problem—neat, plausible, and wrong.
— H. L. Mencken

Each problem that I solved became a rule which served afterwards to solve other problems.
— René Descartes

If a problem has no solution, it may not be a problem, but a fact—not to be solved, but to be coped with over time.
— Shimon Peres

Keep it simple. If you cannot explain how your circuit works to a retarded chimpanzee, you will probably have difficulty explaining why, in fact, it does not work at all.
— Bergeson’s Second Law of Circuit Design

The moment a man begins to talk about technique that’s proof that he is fresh out of ideas.
— Raymond Chandler

I believe there exist only two kinds of modern mathematics books: one which you cannot read past the first page
and one which you cannot read past the first sentence.
— C. N. Yang

When you come to a fork in the road, take it.
— Yogi Berra

Anything that, in happening, causes itself to happen again, happens again.

The secret to productivity is getting dead people to do your work for you.
— Robert J. Lang (2009)

5. December 28, 2010 12:39 pm

“Computer science is an empirical discipline.[…] Each new machine that is built is
an experiment. […] Each new program that is built is an experiment.”
– Allen Newel1 and Herbert A. Simon (1975 ACM Turing Award Lecture).

“MMIX is a polyunsaturated, 100% natural computer.”
– Donald Knuth

6. December 28, 2010 12:51 pm

As an adolescent I aspired to lasting fame, I craved factual certainty, and I thirsted for a meaningful vision of human life — so I became a scientist. This is like becoming an archbishop to meet girls.

Matt Cartmill
Professor of Biological Anthropology and Anatomy
Duke University
in American Scientist, 76, 453, 1988.

7. December 28, 2010 1:33 pm

1. “In mathematics you don’t understand things. You just get used to them.”
2. “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”
3. “You insist that there is something a machine cannot do. If you tell me precisely what it is a machine cannot do, then I can always make a machine which will do just that.”
John von Neumann

December 28, 2010 3:28 pm

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” – John von Neumann

December 28, 2010 4:02 pm

I was surprised not to see any quotes from Dijkstra – a source of some of the most colourful quotes in Computer Science – so here’s one:

Computer Science is no more about computers than astronomy is about telescopes.

• December 28, 2010 5:33 pm

Ok! It is well-known quote from Dijkstra. Unfortunately, it is not very helpful 😦 And moreover, it seems to me that for practical astronomy knowledge about telescopes is very important 😉

December 29, 2010 7:29 pm

It’s just misleading, try rephrasing it as ‘CS is about computers in the same way that astronomy is about telescopes’. They’re both the tool, not the end goal.

• December 30, 2010 11:07 am

Science is a tool, not the end goal also 🙂
For example, see Paul D. Ackerman, Psychology as a Science
http://www.creationism.org/csshs/v05n1p13.htm

10. December 28, 2010 4:05 pm

“‘The virtue of a logical proof is not that it compels belief but that it suggests doubts.’ The proof tells us where to concentrate our doubts.”

The inner comment is by Henry George Forder, the outer one by Morris Kline.

For “logical proof”, scientists can substitute “experiment”; engineers can substitute “mechanism”; computer scientists can substitute “algorithm.”

11. December 28, 2010 4:30 pm

“The history of mathematics is crowned with glorious achievements but is also a record of calamities. The loss of truth is certainly a tragedy of the first magnitude, for truths are man’s dearest possessions and a loss of even one is cause for grief. The realization that the splendid showcase of human reasoning exhibits a by no means perfect structure but one marred by shortcomings and vulnerable to the discovery of disastrous contradictions at any time is another blow to the stature of mathematics. But these are not the only grounds for distress. Grave misgivings and cause for dissension among mathematicians stem from the direction which research of the past one hundred years has taken. Most mathematicians have withdrawn from the world to concentrate on problems generated within mathematics. They have abandoned science. This change in direction is often described as the turn to pure as opposed to applied mathematics. ”

– Morris Kline, “Mathematics. The Loss of Certainty”

“Space Rn, for n> 3 is a mathematical fiction only. However, a very ingenious fiction, which helps to mathematically understand the complex phenomena.”

– Y.S. Bugrov, S.M. Nikolskii, Elements of linear algebra and analytic geometry

“If the knowable World is cracked and we actually can not eliminate the cracks, we should not hide them.”

– Pavel Florensky, The Pillar and Ground of the Truth: An Essay in Orthodox Theodicy in Twelve Letters

• December 29, 2010 6:51 am

Stop being fidgety about the “loss of truth”, since the world is potentially continuous it’s no wonder that all our discrete models fail to match it exactly.
The mathematical discourse about continuity IS discrete and finite (ever seen a paper with an uncountable or even countable infinity of words/symbols?).
This also holds if actually The World is Digital because for all practical purposes an extremely large finite number incommensurate with our coding capabilities counts as well as infinity see Doron Zeilberger.

• December 29, 2010 5:24 pm

Feel the force…!
— George Lucas, Star Wars

Hilbert’s goal of achieving perfect certainty by the laying of firm foundations died with Gödel’s work, but the problem of complexity would have killed his dreams with equal finality fifty years later. We finally ask if there are further crises still to be faced. One possibility is the discovery of a contradiction in a mathematical argument whose complexity is beyond any yet contemplated. One might imagine that the contradiction is the result of a mistake that is too deep for us to be able to locate it, even with the aid of computers. This may seem farfetched, but a somewhat similar problem has already arisen in computer chess programs, which occasionally make moves for which the best chess grandmasters can find no rationale. The computer can, of course, only declare that the said move yielded the highest score out of billions of combinations that it had considered. This does not imply that the move is indeed the best in the given position, because the method of scoring positions is derived from human advice. If such a scenario materializes, we may finally have to admit to limits on what our species can aspire to in the mental realm, as well as in other types of activity.

– Brian Davies, Whither Mathematics?
http://www.ams.org/staff/jackson/comm-davies.pdf

December 28, 2010 4:35 pm

“If builders built buildings the way programmers wrote programs, the first woodpecker to come along would destroy civilization.” — Harry Weinberger

13. December 28, 2010 11:35 pm

Problems worthy of attack,
prove their worth by hitting back.
-Piet Hein

• December 30, 2010 4:41 pm

Thanks for recalling the wonderful Piet Hein! Here’s another of his Grooks:

We shall have to evolve
problem-solvers galore–
since each problem they solve
creates ten problems more.

December 29, 2010 12:21 am

“In mathematics you don’t understand things. You just get used to them.” – John von Neumann

“You wake me up early in the morning to tell me that I’m right? Please wait until I’m wrong.” – John von Neumann

“You can tell whether a man is clever by his answers. You can tell whether a man is wise by his questions.” – Naguib Mahfouz

“I have had my results for a long time: but I do not yet know how I am to arrive at them.” – Karl Friedrich Gauss

“There is no sense being precise when you don’t even know what you’re talking about.” – John von Neumann

“Every society honors its live conformists and its dead troublemakers.” – Mignon McLaughlin

“The important thing is not to stop questioning; curiosity has its own reason for existing.” – Albert Einstein

“We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.” – Carl Friedrich Gauss

“Arithmetic is the art of counting up to twenty without taking off your shoes.” – Mickey Mouse 😀

December 29, 2010 6:56 am

At school in the last century, my Math teacher proclaimed that there was only one subject (Maths) all the rest was book keeping.
I wonder if anyone knows where that came from, or did he make it up. He wasn’t a famous mathmetician, except to me of course.

December 30, 2010 3:10 pm

Well, it’s rather strongly reminiscent of the famous remark of Ernest Rutherford, “All science is either physics or stamp collecting.”

16. December 29, 2010 7:30 am

Three by Saunders Mac Lane:

Mathematical ideas do not live fully until they are presented clearly, and we never quite achieve that ultimate clarity.

Analysis is full of ingenious changes of coordinates, clever substitutions, and astute manipulations. In some of these cases, one can find a conceptual background. When so, the ideas so revealed help us understand what’s what. We submit that this aim of understanding is a vital aspect of mathematics.

A prevalent theme in Mac Lane’s writings is that the mathematical conception of “naturality” and the scientific conception of “physicality” are in essence the same conception; that this is the reason for what Wigner’s famous essay called The Unreasonable Effectiveness of Mathematics in the Natural Sciences; and that this accounts for shared perceptions of beauty by mathematicians and scientists.

Regrettably, my BiBTeX files contain no snappy aphorism by Mac Lane or anyone else that expresses this duality of naturality & physicality concisely … suggestions are welcome.

Dick and Ken, thanks for sustaining this weblog, which so beautifully embodies Mac Lane’s principle that “Mathematical ideas do not live fully until they are presented clearly.”

December 29, 2010 8:56 am

«In science nothing that is provable ought to be believed without proof».

Richard Dedekind — «What are numbers and what should they be»

«It seems absurd to suppose that I could have ‘done better’. I have no linguistic or artistic ability, and very little interest in experimental science. I might have been a tolerable philosopher, but not one of a very original kind. I think that I might have made a good lawyer; but journalism is the only profession, outside academic life, in which I should have felt really confident of my changes. There is no doubt that I was right to be a mathematician,
if the criterion is to be what is commonly called success.»

G. H. Hardy — «A mathematician’s apology»

December 29, 2010 9:42 am

My preferred are

1) Only math nerds would call {2^{500}} finite.
—Leonid Levin

2) Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.
—John von Neumann

3) The biggest difference between time and space is that you can’t reuse time.
—Merrick Furst

4) The essence of mathematics is not to make simple things complicated, but to make complicated things simple.
—Stanley Gudder

December 29, 2010 10:48 am

While this one isn’t math related, I like the saying that “Experience is a hard teacher, because she give you the test first and then the lesson!”

December 29, 2010 1:03 pm

I liked these two the most inspiring:

Computers are useless. They can only give you answers.
—Pablo Picasso

The biggest difference between time and space is that you can’t reuse time.
—Merrick Furst

21. December 29, 2010 1:28 pm

[On mathematical models in physics] First, they’re pretty disgraceful. Second, they work extremely well…One of the faults of mathematicians is: when physicists give them an equation, they take it absolutely seriously. —Henry McKean

22. December 29, 2010 1:47 pm

“Obvious” is the most dangerous word in mathematics.
—Eric Bell

very nice!
I fully agree!

23. December 29, 2010 2:24 pm

I’ve commented this once before, but let me re-raise a famous quotation by Einstein:

Raffiniert ist der Herrgott, aber boshaft ist Er nicht.
“God is tricky, but not nasty”, or as I render his clarification to Oswald Veblen, “Nature hides her secrets as the perquisite of lofty birth, but not through nasty trickery.”

That is to say, in physics one’s intuition will not be wantonly deceived. At least within our “Hubble Bubble”, nastiness is limited by the reasonableness of the parameters. However, mathematics is ruled by the principle that anything that can go, goes. Thus we must expect to face every possible nastiness, whether connected to the Riemann Hypothesis, complexity barriers, or everything. The great beauty that we extract is there, but so is the nastiness—or at least nastiness that has short enough descriptions for us to encounter it.

24. December 29, 2010 7:00 pm

He correctly realizes that difficulties will occur with regular graphs, but doesn’t seem to be aware of strongly regular graphs. My guess is that nonisomorphic SR graphs with the same parameter set will be counter-examples to his algorithm.

– A reviewer of my paper “Polynomial Time Algorithm for Graph Isomorphism Testing”; this is template for any such paper! Be sure to use this template, if and when any editor asks you for a review (reading of the paper is not necessary for that!).

Reviewers/PC (program committee) members want to minimize their work (for instance, by giving scores, but no justifications), while trying to reject papers that compete with their own papers, and accepting papers from their friends. They want to reject unacceptable papers that would embarrass them. Finally, they want to get the prestige of being in the PC.

– Jon Crowcroft, S. Keshav, and Nick McKeown, Scaling the Academic Publication Process to Internet Scale. A proposal to remedy problems in the reviewing process. Communications of the acm, January 2009, vol. 52, no. 1, p.28.

25. December 29, 2010 7:02 pm

Being on jury duty all this week has given me ample time (albeit in small random blocks) to comb through my STEM quotation database.

The themes of von Neumann’s essay The Mathematician (1948) are echoed and amplified by Terry Tao’s essay What Is Good Mathematics? (2007) in these two lovely passages:

The concept of mathematical quality is a high-dimensional one and lacks an obvious canonical total ordering. I believe this is because mathematics is itself complex and high-dimensional and evolves in unexpected and adaptive ways.

… This diverse and multifaceted nature of “good mathematics” is very healthy for mathematics as a whole, as it allows us to pursue many different approaches to the subject and exploit many different types of mathematical talent towards our common goal of greater mathematical progress and understanding.

… There does however seem to be some undefinable sense that a certain piece of mathematics is “on to something”, that it is a piece of a larger puzzle waiting to be explored further. And it seems to me that the pursuit of such intangible promises of future potential is at least as important an aspect of mathematical progress as the more concrete and obvious aspects of mathematical quality listed previously.

Oops… “it’s round-up time!”

Happy holidays to all! 🙂

26. December 29, 2010 7:18 pm

Well, Grisha Perelman made a world discovery without any investments …

– Vladimir Putin, Prime Minister of Russia

December 29, 2010 7:42 pm

Beware of bugs in the above code; I have only proved it correct, not tried it.
–Donald Knuth

It is not even wrong.
–Wolfgang Pauli, asked to review a paper by a young physicist

28. December 30, 2010 11:51 am

Here are more quotations from the Seattle jurors waiting room … these quotations are selected for their universal relevance to math, science, and engineering.

First we have Andy Groves (of Intel’s) maxim:

Let chaos rein, then reign-in chaos.

This maxim applies broadly in mathematics (Szemeredi’s theorem), science (the second law of thermodynamics), and engineering (the systems design process). To say nothing of enterprise!

Then we have Eisenhower’s planning maxim:

I have found that plans are useless, but planning is indispensable.

The same can be said of theorems: often they have little practical use in themselves; it is (what Dick calls) the proof technologies and insights by which they are constructed that has enduring importance.

And as a coda, a stream of quotations from Si Ramo’s The development of systems engineering (1984):

Systems Engineering is the design of the whole as distinguished from the design of the parts.

… The systems engineer harmonizes optimally an ensemble of subsystems and components—machines, communications networks, humans, space—all related by channeled flows of information …

… The principle tools of the systems engineer are the human brain, the electronic computer, and numerous mathematical analysis techniques …

… Their efforts begin with an attempt to comprehend thoroughly the problem to be solved, the tools available to solve it, and all constraints linking the parameters …

… Two major trends may be expected in system engineering. First, the capabilities of the analytical tools of the system engineer will continue to increase. … The second major trend is the increase in the complexity of systems being developed.

… We should anticipate the use of the techniques of system engineering on an even wider range of problems than any of the past.

If we substitute the profession of mathematician for the profession of systems engineer … we are led to wonder … are these two professions fundamentally different?

December 30, 2010 7:09 pm

Both by Poincare, though the second may not be quite accurate

1) Point set topology is a disease from which the human race will soon recover.
(Quoted in D MacHale, Conic Sections (Dublin 1993))

2) Later generations will regard Mengenlehre (Set theory) as a disease from which one
has recovered
(Whether or not he actually said this is a matter of debate.)

December 31, 2010 12:26 pm

today’s pseudorandom fortune from http://math.furman.edu/cgi-bin/randquote.pl is:
Wordsworth, William (1770 – 1850)
[Mathematics] is an independent world
Created out of pure intelligence.

31. January 2, 2011 11:32 am

Another quotation selected for it’s universal relevance to math, science, and engineering … from Mark Twain’s A Connecticut Yankee in King Arthur’s Court:

Intellectual “work” is misnamed; it is a pleasure, a dissipation, and is its own highest reward. The poorest paid architect, engineer, general, author, sculptor, painter, lecturer, advocate, legislator, actor, preacher, singer is constructively in heaven when he is at work; and as for the musician with the fiddle-bow in his hand who sits in the midst of a great orchestra with the ebbing and flowing tides of divine sound washing over him—why, certainly, he is at work, if you wish to call it that, but lord, it’s a sarcasm just the same.

Here Twain’s remarks (in the year 1889) are in near-perfect accord with modern cognitive science … and with Bill Thurston’s remarks on mathematical understanding too.

May all the readers of Gödel’s Lost Letter and P=NP find themselves, at least sometimes, “constructively in heaven” during 2011! 🙂

32. January 4, 2011 8:44 pm

Here is another universal quote …. this one is by Garrison Keillor, from the introduction to his collection Good Poems

What makes a poem memorable is its narrative line. A story is easier to remember than a puzzle.

Are theorems all that different? Physical laws? Engineering designs? Imaginative enterprises?

January 15, 2011 9:43 pm

“The traditional mathematician recognizes and appreciates mathematical elegance when he sees it. I propose to go one step further, and to consider elegance an essential ingredient of mathematics: if it is clumsy, it is not mathematics.”
(E. Dijkstra)

“Simplicity is prerequisite for reliability.”
(E. Dijkstra)

“No human research can be called true science unless it can be mathematically proved”
(Leonardo da Vinci)

“Perhaps the greatest paradox of all is that there are paradoxes in mathematics.”
(E. Kasner and J. Newman)