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]]]]]]]]]]]]]] On the nature of mathematical proofs [[[[[[
By Joel E. Cohen (2/24/1989)
[From R.L. Weber, compiler, and E. Mendoza, editor, A Random Walk
In Science (New York: Crane, Russak & Co. Inc., 1973), pp. 34-36.
This piece is condensed from Opus, May 1961]
Kindly uploaded by Freeman 10602PANC
Bertrand Russell has defined mathematics as the science in
which we never know what we are talking about or whether what we
are saying is true. Mathematics has been shown to apply widely
in many other scientific fields. Hence most other scientists do
not know what they are talking about or whether what they are
saying is true. Thus providing a rigorous basis for
philosophical insights is one of the main functions of
mathematical proofs.
To illustrate the various methods of proof we give an example
of a logical system.
THE PEJORATIVE CALCULUS
Lemma 1. All horses are the same colour (by induction).
Proof. It is obvious that one horse is the same colour. Let us
assume the proposition P(k) that k horses are the same colour and
use this to imply that k + 1 horses are the same colour. Given
the set of k + 1 horses, we remove one horse; then the remaining
k horses are the same colour, by hypothesis. We remove another
horse and replace the first; the k horses, by hypothesis, are
again the same colour. We repeat this until by exhaustion the
k + 1 sets of k horses have each been shown to be the same
colour. It follows then that since every horse is the same
colour as every other horse, P(k) entails P(k + 1). But since we
have shown P(1) to be true, P is true for all succeeding values
of k, that is, all horses are the same colour.
Theorem 1. Every horse has an infinite number of legs. (Proof
by intimidation).
Proof. Horses have an even number of legs. Behind they have two
legs and in front they have fore legs. This makes six legs,
which is certainly an odd number of legs for a horse. But the
only number that is both odd and even is infinity. Therefore
horses have an infinite number of legs. Now to show that this is
general, suppose that somewhere there in a horse with a finite
number of legs. But that is a horse of a another colour, and by
the lemma that does not exist.
Corollary 1. Everything is the same colour.
Proof. The proof of lemma 1 does not depend at all on the nature
of the object under consideration. The predicate of the
antecedent of the universally-quantified conditional `For all x,
if x is a horse, then x is the same colour,' namely `is a horse'
may be generalized to `is anything' without affecting the
validity of the proof; hence, `for all x, if x is anything, x is
the same colour.
Corollary 2. Everything is white.
Proof. If a sentential formula in x is logically true, then any
particular substitution instance of it is a true sentence. In
particular then: `for all x, if x is an elephant, then x is the
same colour' is true. Now it is manifestly axiomatic that white
elephants exist (for proof by blatant assertion consult Mark
Twain `The Stolen White Elephant'). Therefore all elephants are
white. By corollary 1 everything is white.
Theorem 2. Alexander the Great did not exist and he had an
infinite number of limbs.
Proof. We prove this theorem in two parts. First we note the
obvious fact that historians always tell the truth (for
historians always take a stand, and therefore they cannot lie).
Hence we have the historically true sentence, `If Alexander the
Great existed, then he rode a black horse Bucephalus.' But we
know by corollary 2 everything is white; hence Alexander could
not have ridden a black horse. Since the consequent of the
conditional is false, in order for the whole statement to be
true, the antecedent must be false. Hence Alexander the Great
did not exist.
We have also the historically true statement that Alexander
was warned by an oracle that he would meet death if he crossed a
certain river. He had two legs; and `fore-warned is four-armed.'
This gives him six limbs, an even number, which is certainly an
odd number of limbs for a man. Now the only number which is even
and odd is infinity; hence Alexander had an infinite number of
limbs. We have thus proved that Alexander the Great did not
exist and that he had an infinite number of limbs.
It is not thought that there are not other types of proofs,
which in print shops are recorded on proof sheets. There is the
bullet proof and the proof of the pudding. Finally there is 200
proof, a most potent spirit among mathematicians and people
alike.
* * *
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