]]]]]]]]]]]]]]    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.


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

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