86
II - Convergence: Discrete variables
After these examples, and those we shall present later in this chapter, one
will understand perhaps a little better why the mathematicians of the XIXth
century and yet more of the XX th , tired of false proofs, by great mathematicians, of generally correct theorems (there were surely also innumerable false
theorems produced by lesser masters, but they have not passed to posterity),
have finally propounded, at least implicitly, the following basic principles:
(i) every assertion which is not fully proved is potentially false and is only,
at best, an interesting conjecture,
(ii) using an incompletely proved assertion to prove others increases the risk
of error exponentially,
(iii) the duty to prove an assertion falls on the author
even if his colleagues do not refrain, on occasion, from doing so in his place, or
from demolishing it. Naturally there are conjectures which people have tried
to prove for decades, even centuries: Fermat's Last Theorem, the Goldbach
and Riemann Conjectures, etc. But to formulate or prove such hypotheses is
not the lot of everyone ...
The observation of these principles has led to the formidable intellectual
discipline that mathematicians have progressively imposed on themselves for
a century. One finds it nowhere else to the same degree. In physics theoreticians often take great liberties with the mathematics, their inspired intuitions
being sufficient; the experimentalists insist on the reproducibility of their experiences, which, in Big Science, can lead far, even though one sometimes
works on hypotheses which may be revealed to be totally false. True historians may attempt to observe the mathematicians'- rules, but the inevitable
gaps in their information, the need to check sometimes falsified documents,
and to interpret them objectively, makes it difficult. And imagine the career
of a politician who applied these rules.
An example of an assertion falling directly within the scope of principles (i), (ii) and (iii): it is thanks to nuclear arms that the Third
World War has been avoided. Repeated ad nauseam for decades without the least proof being provided (one invokes Munich, or the Soviet
regime, imperialism and arms, or even the precedent of Pearl Harbor;
whereas the Foreign Policy of the Soviets has always been radically
different from that of the Nazis or the Japanese before 1939, etc.),
this assertion ignores all sorts of arguments which do not lead in the
same direction.
(1) Nuclear or not, the horrors of two World Wars and the almost total unpredictability of this kind of enterprise, might have been enough
to dissuade amateurs less mad than Adolf Hitler; look at the enthusiasm of the French, British and Soviet leaders faced with Hitler
already before September 1939; the USSR did not enter the war until attacked by the Nazis; the USA waited for Pearl Harbor and a
II - Convergence: Discrete variables
After these examples, and those we shall present later in this chapter, one
will understand perhaps a little better why the mathematicians of the XIXth
century and yet more of the XX th , tired of false proofs, by great mathematicians, of generally correct theorems (there were surely also innumerable false
theorems produced by lesser masters, but they have not passed to posterity),
have finally propounded, at least implicitly, the following basic principles:
(i) every assertion which is not fully proved is potentially false and is only,
at best, an interesting conjecture,
(ii) using an incompletely proved assertion to prove others increases the risk
of error exponentially,
(iii) the duty to prove an assertion falls on the author
even if his colleagues do not refrain, on occasion, from doing so in his place, or
from demolishing it. Naturally there are conjectures which people have tried
to prove for decades, even centuries: Fermat's Last Theorem, the Goldbach
and Riemann Conjectures, etc. But to formulate or prove such hypotheses is
not the lot of everyone ...
The observation of these principles has led to the formidable intellectual
discipline that mathematicians have progressively imposed on themselves for
a century. One finds it nowhere else to the same degree. In physics theoreticians often take great liberties with the mathematics, their inspired intuitions
being sufficient; the experimentalists insist on the reproducibility of their experiences, which, in Big Science, can lead far, even though one sometimes
works on hypotheses which may be revealed to be totally false. True historians may attempt to observe the mathematicians'- rules, but the inevitable
gaps in their information, the need to check sometimes falsified documents,
and to interpret them objectively, makes it difficult. And imagine the career
of a politician who applied these rules.
An example of an assertion falling directly within the scope of principles (i), (ii) and (iii): it is thanks to nuclear arms that the Third
World War has been avoided. Repeated ad nauseam for decades without the least proof being provided (one invokes Munich, or the Soviet
regime, imperialism and arms, or even the precedent of Pearl Harbor;
whereas the Foreign Policy of the Soviets has always been radically
different from that of the Nazis or the Japanese before 1939, etc.),
this assertion ignores all sorts of arguments which do not lead in the
same direction.
(1) Nuclear or not, the horrors of two World Wars and the almost total unpredictability of this kind of enterprise, might have been enough
to dissuade amateurs less mad than Adolf Hitler; look at the enthusiasm of the French, British and Soviet leaders faced with Hitler
already before September 1939; the USSR did not enter the war until attacked by the Nazis; the USA waited for Pearl Harbor and a