On the nature of mathematical proofs
Joel Cohen
Bertrand Russell has defined mathematics as the science in which we never 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 perjorative calculus
Lemma 1: All horses are the same colour.
Proof by induction:
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 been shown to be the same colour. It follows 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. Q.E.D.
Theorem 1: Every horse has an infinite number of legs.
Proof by intimidation:
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 is a horse with a finite number of legs. But that is a horse of another colour, and by the lemma that does not exist. Q.E.D.
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." Q.E.D.
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's The Stolen White Elephant). Therefore all elephants are white. By corollary 1 everything is white. Q.E.D.
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 "forewarned 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. Q.E.D.
It is not to be thought that there are not other types of proofs, which in print shops are recorded on the 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.