For a physicist mathematics is not just a tool by means of which
phenomena can be calculated, it is the main source of concepts and
principles by means of which new theories can be created.
-- F. Dyson
For a physicist mathematics is not just a tool by means of which
phenomena can be calculated, it is the main source of concepts and
principles by means of which new theories can be created.
-- F. Dyson
The introduction of numbers as coordinates ... is an act of violence...
-- H. Weyl, Philosophy of Mathematics and Natural Science
1949
An engineer, a chemist and a mathematician are staying in three adjoining
cabins at an old motel. First the engineer's coffee maker catches fire.
He smells the smoke, wakes up, unplugs the coffee maker, throws it out
the window, and goes back to sleep. Later that night the chemist smells
smoke too. He wakes up and sees that a cigarette butt has set the trash
can on fire. He says to himself, "Hmm. How does one put out a fire?
One can reduce the temperature of the fuel below the flash point, isolate
the burning material from oxygen, or both. This could be accomplished
by applying water." So he picks up the trash can, puts it in the shower
stall, turns on the water, and, when the fire is out, goes back to sleep.
The mathematician, of course, has been watching all this out the window.
So later, when he finds that his pipe ashes have set the bed-sheet on fire,
he is not in the least taken aback. He says: "Aha! A solution exists!"
and goes back to sleep.
1+1 = 3, for large values of 1
PI=
3.141592653589793238462643383279502884
19716939937510582097494459230781640628
6208998628034825342117067982148086513...
Euler E=
2.718281828459045235360287471352662497
75724709369995957496696762772407663035
3547594571382178525166427427466391932...
Each problem that I solved became a rule which served afterwards to
solve other problems
-- R. Decartes
Say what you know, do what you must, come what may.
-- S. Kovalevsky
God made the integers, all else is the work of man.
-- L. Kronecker
There is no branch of mathematics, however abstract, which may not some
day be applied to phenomena of the real world.
-- N. Lobatchevsky
THEOREM: Every natural number can be completely and unambiguously
identified in fourteen words or less.
PROOF:
1. Suppose there is some natural number which cannot be unambiguously
described in fourteen words or less.
2. Then there must be a smallest such number. Let's call it n.
3. But now n is "the smallest natural number that cannot be unambiguously
described in fourteen words or less".
4. This is a complete and unambiguous description of n in fourteen words,
contradicting the fact that n was supposed not to have such a description!
5. Since the assumption (step 1) of the existence of a natural number that
cannot be unambiguously described in fourteen words or less
led to a contradiction, it must be an incorrect assumption.
6.Therefore, all natural numbers can be unambiguously described in fourteen
words or less!
THEOREM: 1=2
PROOF:
1. Let a=b.
2. Then a^2 = ab,
3. a^2 + a^2 = a^2 + ab,
4. 2 a^2 = a^2 + ab,
5. 2 a^2 - 2 ab = a^2 + ab - 2 ab,
6. and 2 a^2 - 2 ab = a^2 - ab.
7. Writing this as 2 (a^2 - a b) = 1 (a^2 - a b),
8. and cancelling the (a^2 - ab) from both sides gives 1=2.
I married a widow, who had an adult stepdaughter. My father, a widow
and who often visited us, fell in love with my stepdaughter and married her.
So, my father became my son-in-law and my stepdaughter became my stepmother.
But my wife became the mother-in-law of her father-in-law. My stepmother,
stepdaughter of my wife had a son and I therefore a brother, because
he is the son of my father and my stepmother. But since he was in the same
time the son of our stepdaughter, my wife became his grandmother and I became the
grandfather of my stepbrother. My wife gave me also a son. My stepmother,
the stepsister of my son, is in the same time his grandmother, because he is
the son of her stepson and my father is the brother-in-law of my child, because his
sister is his wife. My wife, who is the mother of my stepmother, is therefore
my grandmother. My son, who is the child of my grandmother, is the grandchild
of my father. But I'm the husband of my wife and in the same time the grandson of
my wife. This means: I'm my own grandfather.
I never could make out what those damned dots meant.
-- Lord Randolph Churchill (1849-1895)
Brittish conservative politician, referring
to decimal points.
Theorem: the square root x of 2 is irrational.
Proof: x=n/m with gcd(n,m)=1 implies 2=n^2/m^2 which is
2 m^2=n^2 so that n must be even and n^2 a multiple of 4.
Therefore m is even. This contradicts gcd(n,m)=1.
Of all escapes from reality, mathematics is the most
successful ever. It is a fantasy that becomes all the
more addictive because it works back to improve the
same reality we are trying to evade. All other escapes-
sex, drugs, hobbies, whatever - are ephemeral by comparison.
The mathematician's feeling of triumph, as he forces
the world to obey the laws his imagimation has created,
feeds on its own success. The world is premanently
changed by the workings of his mind, and the certainty
that his creations will endure renews his confidence as no
other pursuit.
-- Gian-Carlo Rota
Mathematics consists of proving the most obvious thing in the least
obvious way.
-- G. Polya
If the entire Mandelbrot set were placed on an ordinary sheet of paper,
the tiny sections of boundary we examine would not fill the width of
a hydrogen atom. Physicists think about such tiny objects; only
mathematicians have microscopes fine enough to actually observe them.
-- J. Eving
A positive integer n is called a perfect number if it is equal to
the sum of all of its positive divisors, excluding n itself.
Examples are 6=1+2+3, 28=1+2+4+7+14. An integer k is an even perfect
number if and only if it has the form 2^(n-1)(2^n-1) and 2^n-1 is
prime. In that case 2^n-1 is called a Mersenne prime and n must be
prime. It is unknown whether there exists an odd perfect number.
The following is a transcript of an interchange between defence
attorney Robert Blasier and FBI Special Agent Roger Martz on
July 26, 1995, in the courtroom of the O.J. Simpson trial:
Q: Can you calculate the area of a circle
with a five-millimeter diameter?
A: I mean I could. I don't...math I don't ...
I don't know right now what it is.
Q: Well, what is the formula for the area of a circle?
A: Pi R Squared
Q: What is pi?
A: Boy, you ar really testing me. 2.12... 2.17...
Judge Ito: How about 3.1214?
Q: Isn't pi kind of essential to being a scientist
knowing what it is?
A: I haven't used pi since I guess I was in high school.
Q: Let's try 3.12.
A: Is that what it is? There is an easier way to do...
Q: Let's try 3.14. And what is the radius?
A: It would be half the diameter: 2.5
Q: 2.5 squared, right?
A: Right.
Q: Your honor, may we borrow a calculator?
[pause]
Q: Can you use a calculator?
A: Yes, I think.
Q: Tell me what pi times 2.5 squared is.
A: 19
Q: Do you want to write down 19? Square millimeters, right?
The area. What is one tenth of that?
A: 1.9
Q: You miscalculated by a factor of two, the size, the
minimum size of a swatch you needed to detect EDTA
didn't you?
A: I don't know that I did or not. I calculated a little
differently. I didn't use this.
Q: Well, does the area change by the different method of
calculation?
A: Well, this is all estimations based on my eyeball. I
didn't use any scientific math to determine it.
--David Blatner, the joy of pi
Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer,
You take one down, and pass it around,
Aleph-null bottles of beer on the wall.
An integer 2^n-1 is called a Mersenne number. If it is prime,
it is called a Mersenne prime. In that case, n must be prime.
Known examples are n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107,
127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941,
11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091,
756839, 859433, 1257787, 1398269, 2976221, 3021377. It is not
known whether there are infinitely many Mersenne primes.
The more ambitious plan may have more chances of success
-- G. Polya, How To Solve It
You have hit a fortune from the fortune collection 'math'
containing mathematical fortune entries. Latest update: June 6 2001
This fortune got started in October 1999 and is maintained in
http://www.dynamical-systems.org
The file contains mathematical quotations, jokes, annectotes,
even some mathematical results with short proofs.
To install this file:
1) 'strfile math' produces a file 'math.dat' .
2) Copy both files 'math' and 'math.dat' into '/usr/share/games/fortunes'
or wherever your fortune program keeps the fortunes. You might check
first, if not a newer version of this file or an other file named
'math' already resides there. The 'fortune' program itself is usually in
'/usr/games/fortune'. If you are not system administrator, put the files
'math' and 'math.dat' into a private directory like for example '~/lib'
and replace 'fortune math' by 'fortune ~/lib/math'.
Examples to use this fortune:
fortune math random entries from this file
fortune -m PI= PI with 100 digits
fortune -m E= E with 100 digits
fortune -m Mandelbrot Mandelbrot set (generated with pbmtoascii)
You might want to erase this first entry of the fortune before
installation.
Dean, to the physics department. "Why do I always have to give you guys so
much money, for laboratories and expensive equipment and stuff. Why couldn't
you be like the maths department - all they need is money for pencils, paper
and waste-paper baskets. Or even better, like the philosophy department.
All they need are pencils and paper."
What, in effect are the conditions for the construction of formal
thought? The child must not only apply operations to objects - in
other words, mentally execute possible actions on them - he must
also 'reflect' those operations in the absence of the objects which
are replaced by pure propositions. Thus 'reflection' is thought
raised to the second power. Concrete thinking is the representation
of a possible action, and formal thinking is the representation of
a representation of possible action... It is not surprising,
therefore, that the system of concrete operations must be completed
during the last years of childhood before it can be 'reflected' by
formal operations. In terms of their function, formal operations do
not differ from concrete operations except that they are applied to
hypotheses or propositions whose logic is an abstract translation of
the system of 'inference' that governs concrete operations.
-- Jean Piaget
Never speak more clearly than you think.
-- Jeremy Bernstein
It has been said that the First World War was the chemists' war because
mustard gas and chlorine were empolyed for the first time, and that the
Second World War was the physicists war, because the atom bomb was
detonated. Similarly, it has been argued that the Third World War would
be the mathematicians' war, because mathematics will have control over
the next great weapon of war - information.
-- Simon Singh, in 'The code book'
This reminds me of the Hilbert story, which I learned from
my teacher Franz Rellich in Goettingen:
When Hilbert - who was old and retired - was asked at a
party by the newly appointed Nazi-minister of education:
“Herr Geheimrat, how is mathematics in Goettingen, now
that it has been freed of the Jewish influences” he
replied: "Mathematics in Goettingen? That does not EXIST
anymore".
-- Jurgen Moser, in Dynamical Systems-Past and Present,
Doc. Math. J. DMV I p. 381-402, 1998
There is no royal road to geometry.
Euclid