Hermann (1885 - 1955)
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. UnterrichtsblŐtter fčr Mathematik und Naturwissenschaften,
38, 177-188 (1932). Translation by Abe Shenitzer appeared in The American
Mathematical Monthly, v. 102, no. 7 (August-September 1995), p. 646.
There is no branch of mathematics,
however abstract, which may not some day be applied to phenomena of
the real world. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome
Press Inc., 1988.
Mathematical economics is
old enough to be respectable, but not all economists respect it. It
has powerful supporters and impressive testimonials, yet many capable
economists deny that mathematics, except as a shorthand or expository
device, can be applied to economic reasoning. There have even been rumors
that mathematics is used in economics (and in other social sciences)
either for the deliberate purpose of mystification or to confer dignity
upon common places as French was once used in diplomatic communications.
In J. R. Newman (ed.) The World of Mathematics, New Yorl: Simon and
"But," you might say, "none
of this shakes my belief that 2 and 2 are 4." You are quite right, except
in marginal cases -- and it is only in marginal cases that you are doubtful
whether a certain animal is a dog or a certain length is less than a
meter. Two must be two of something, and the proposition "2 and 2 are
4" is useless unless it can be applied. Two dogs and two dogs are certainly
four dogs, but cases arise in which you are doubtful whether two of
them are dogs. "Well, at any rate there are four animals," you may say.
But there are microorganisms concerning which it is doubtful whether
they are animals or plants. "Well, then living organisms," you say.
But there are things of which it is doubtful whether they are living
organisms or not. You will be driven into saying: "Two entities and
two entities are four entities." When you have told me what you mean
by "entity," we will resume the argument. In N. Rose Mathematical Maxims
and Minims, Raleigh NC:Rome Press Inc., 1988.
Newman, James R.
The most painful thing about
mathematics is how far away you are from being able to use it after
you have learned it. In J. R. Newman (ed.) The World of Mathematics,
New York: Simon and Schuster, 1956.
North (1861 - 1947)
Through and through the world
is infested with quantity: To talk sense is to talk quantities. It is
not use saying the nation is large .. How large? It is no use saying
the radium is scarce ... How scarce? You cannot evade quantity. You
may fly to poetry and music, and quantity and number will face you in
your rhythms and your octaves. In J. R. Newman (ed.) The World of Mathematics,
New York: Simon and Schuster, 1956.
Mathematics is a logical
method ... Mathematical propositions express no thoughts. In life it
is never a mathematical proposition which we need, but we use mathematical
propositions only in order to infer from propositions which do not belong
to mathematics to others which equally do not belong to mathematics.
Tractatus Logico Philosophicus, New York, 1922, p. 169.
Douglas (1952 - )
Bistromathics itself is
simply a revolutionary new way of understanding the behavior of numbers.
Just as Einstein observed that space was not an absolute but depended
on the observer's movement in space, and that time was not an absolute,
but depended on the observer's movement in time, so it is now realized
that numbers are not absolute, but depend on the observer's movement
in restaurants. Life, the Universe and Everything. New York: Harmony
(1952 - )
The first nonabsolute number
is the number of people for whom the table is reserved. This will vary
during the course of the first three telephone calls to the restaurant,
and then bear no apparent relation to the number of people who actually
turn up, or to the number of people who subsequently join them after
the show/match/party/gig, or to the number of people who leave when
they see who else has turned up. The second nonabsolute number is the
given time of arrival, which is now known to be one of the most bizarre
of mathematical concepts, a recipriversexcluson, a number whose existence
can only be defined as being anything other than itself. In other words,
the given time of arrival is the one moment of time at which it is impossible
that any member of the party will arrive. Recipriversexclusons now play
a vital part in many branches of math, including statistics and accountancy
and also form the basic equations used to engineer the Somebody Else's
Problem field. The third and most mysterious piece of nonabsoluteness
of all lies in the relationship between the number of items on the bill,
the cost of each item, the number of people at the table and what they
are each prepared to pay for. (The number of people who have actually
brought any money is only a subphenomenon of this field.) Life, the
Universe and Everything. New York: Harmony Books, 1982.
(1952 - )
Numbers written on restaurant
bills within the confines of restaurants do not follow the same mathematical
laws as numbers written on any other pieces of paper in any other parts
of the Universe. This single statement took the scientific world by
storm. It completely revolutionized it. So many mathematical conferences
got held in such good restaurants that many of the finest minds of a
generation died of obesity and heart failure and the science of math
was put back by years. Life, the Universe and Everything. New York:
Harmony Books, 1982.
Where there is matter, there
is geometry. (Ubi materia, ibi geometria.) J. Koenderink Solid Shape,
Cambridge Mass.: MIT Press, 1990
The mathematical sciences
particularly exhibit order, symmetry, and limitation; and these are
the greatest forms of the beautiful. Metaphysica, 3-1078b.
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.
Mathematics is not only real,
but it is the only reality. That is that entire universe is made of
matter, obviously. And matter is made of particles. It's made of electrons
and neutrons and protons. So the entire universe is made out of particles.
Now what are the particles made out of? They're not made out of anything.
The only thing you can say about the reality of an electron is to cite
its mathematical properties. So there's a sense in which matter has
completely dissolved and what is left is just a mathematical structure.
Gardner on Gardner: JPBM Communications Award Presentation. Focus-The
Newsletter of the Mathematical Association of America v. 14, no. 6,
The Reader may here observe
the Force of Numbers, which can be successfully applied, even to those
things, which one would imagine are subject to no Rules. There are very
few things which we know, which are not capable of being reduc'd to
a Mathematical Reasoning; and when they cannot it's a sign our knowledge
of them is very small and confus'd; and when a Mathematical Reasoning
can be had it's as great a folly to make use of any other, as to grope
for a thing in the dark, when you have a Candle standing by you. Of
the Laws of Chance. (1692)
"╔ The universe, which stands
continually open to our gaze╔ cannot be understood unless one first
learns to comprehend the language and interpret the characters in which
it is written. It is written in the language of mathematics╔"