``On the Nature of Mathematical Proofs'' by Joel E. Cohen (1961). Bertrand Russel 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 Color (by induction).

Proof: It is obvious that one horse is the same color. Let us assume the proposition P(k) that k horses are the same color and use this to imply that k+1 horses are the same color. Given the set of k+1 horses, we remove one horse; then the remaining k horses are the same color, by hypothesis. We remove another horse and replace the first; the k horses, by hypothesis, are again the same color. We repeat this until by exhaustion the k+1 sets of horses have each been shown to be the same color. It follows then that since every horse is the same color 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 succeding values of k, that is, all horses are the same color.

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 is a horse with a finite number of legs. But that is a horse of a different color, and by the lemma that does not exist.

COROLLARY 1. Everything is the Same Color.

Proof: The proof of lemma 1 does not depend at all on the nature of the object under consideration. The predicate of the anticedent of the universally-quantified conditional ``For all x, if x is a horse, then x is the same color,'' 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 color.''

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 color'' 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 that everything is white; hence Alexander could not have ridden a black horse. Since the consequent of the condition is false, in order for the whole statement to be true the antededent must be false. Hence Alexander the great did not exist. We also have the historically true statement that Alexander was warned by an oracle that he would meet death if he crossed a certian 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 infinte number of limbs.

More humor