NOTE: This transcription was contributed by Martin P.M. van der Burgt, who has devised a process for producing transcripts automatically. Although its markup is incomplete, we believe it serves a useful purpose by virtue of its searchability and its accessibility to text-reading software. It will be replaced by a fully marked-up version when time permits. —HR
More mathematical folklore.
"Fermat's general method, that of "infinite descent", is profoundly
ingenious, but has the disadvantage that it is often extremely difficult
to apply."
E.T.Bell in "The Development of Mathematics" (1945) |
The purpose of this debunking note is to expose a few time-honoured mathematical concepts as mere artefacts of an unfortunate notation. The unfortunate notation writes non A ≥ C or non C ⇒ A instead of the symmetric disjunction A or C .
The indirect proof.
Mathematicians call a proof of the theorem A ⇒ B "indirect" when, instead of proving A ≥ B "directly", one proves non B ⇒ non A . But you cannot prove the one without proving the other: both are equivalent to non A or B . The distinction is void.
The reductio ad absurdum.
With C and non C as the prototype of a contradiction —an "absurdity"— the reductio ad absurdum proceeds as follows. In order to prove A
0) | one proves | non A ⇒ C |
1) | one proves | non A ⇒ non C |
2) | one concludes | non A ⇒ (C and non C) , hence A . |
But, apart from notation, the above proof must be equivalent to
0) | proving | non C ⇒ A |
1) | proving | C ⇒ A |
2) | concluding | (C or non C) ⇒ A , hence A . |
0) | proving | A or C |
1) | proving | A or non C |
2) | concluding | A |
Mathematical induction.
I was taught two patterns for mathematical induction.
Pattern 1:
1.0) | prove P(0) | ("the base") |
1.1) | prove (A n: n ≥ 0: P(n) ⇒ P(n+1)) | ("the induction step) |
1.2) | conclude (A n: n ≥ 0 P(n)) |
2.1) | prove (A n: n ≥ 0 (A i: 0 ≤ i < n: P(i)) ⇒ P(n)) |
2.2) | conclude (A n: n ≥ 0: P(n)) |
The following are comments on the way in which I was introduced to them— a way that, I am afraid, is rather standard.
Pattern 1 , which regretfully is the better known one, was postulated and by renaming, substitution, and simplification pattern 2 was derived from it.
My first complaint is that no one pointed out to me that the two are equivalent, i.e. that by suitable renaming, substitution, and simplification the validity of pattern 1 can again be derived from the postulated validity of pattern 2.
My second complaint is that no one ever pointed out to me that the transition from pattern 1 to pattern 2 represents an obviously permissible strengthening of the induction hypothesis, thereby offering the opportunity for a simpler proof obligation.
My third complaint is that no one pointed out to me that for given P pattern 2 is in a sense canonical because the transition from 1 to 2 is idempotent. The moral of the story is that pattern 2 should always be preferred.
Fermat's "infinite descent" consists of proving (A n: n ≥ 0: P(n)) by showing that the assumption non P(n) leads to the existence of a natural value i < n , such that non P(i) , which then leads to a contradiction.
I quote from E.T.Bell again —but this time from his "Men of Mathematics"— "His own account is both concise and clear, so we shall give a free translation from his letter of August, 1659, to Carcavi.
"For a long time I was unable to apply my method to affirmative propositions, because the twist and the trick for getting at them is much more troublesome than that which I use for negative propositions. Thus, when I had to prove that every prime which exceeds a multiple of 4 by 1 is composed of two squares, I found myself in a fine torment. But at last a meditation repeated many times gave me the light I lacked, and now affirmative propositions submit to my method. The course of my reasoning in affirmative propositions is such: if an arbitrarily chosen prime of the form 4n+1 is not a sum of two squares, [I prove that] there will be another of the same nature, less than the one chosen, and [therefore] next a third still less, and so on. Making an infinite descent of all the numbers of this kind I arrive at the number 5, the least of all the numbers of this kind [4n+1]. [By the proof mentioned and the preceding argument from it], it follows that 5 is not a sum of two squares. But it is. Therefore we must infer by a reductio ad absurdum that all primes of the form 4n+1 are sums of two squares." "
But what is Fermat's proof obligation? With
P(n): the n-th prime of the form 4k+1 is the sum of two squares |
(A n: n ≥ 0 : non P(n) ⇒ (E i: 0 ≤ i < n: non P(i))), |
(A n: n ≥ 0: (E i: 0 ≤ i < n: non P(i)) or P(n)). |
Plataanstraat 5 | 1 May 1980 |
5671 AL NUENEN | prof.dr.Edsger W.Dijkstra |
The Netherlands | Burroughs Research Fellow |
Transcribed by Martin P.M. van der Burgt
Last revision