234
III - Convergence: Continuous variables
Some criticised his Precis for demanding of school-Ieavers a capacity for
abstraction that they do not possess and forcing a brutal rupture from their
old habits, to which Meray replied that "it was not [his] fault if they are bad
to the point that makes a rupture necessary21" (in 1872 ... ).
One sees that all these ideas - cuts a la Dedekind, increasing sequences,
least upper bounds, decimal expansions, Cauchy sequences, nested intervals,
BW, etc. - overlap each other, and consist of always repeating the same
thing in various disguises, leaving one to choose, in practice, the most directly
usable form. When once one has understood these ideas, one has made an
enormous leap forwards in the comprehension of analysis: one has overtaken
Newton and Euler, 1665-1750, Dedekind, Weierstrass, Cantor, etc. Do not
worry if you do not assimilate them instantly and completely: it needed all
of the XIX th century and the reflections of a good dozen of mathematicians
of the highest order to elaborate it and even more in order to give it, in the
XX th century, the almost perfect form and the enormous generality that it
has acquired today.
11 - Cauchy's criterion for series: examples
Cauchy's criterion applies to series I: Un; one applies it to the sequence of
partial sums Sn of the series. Since
Sq - sp = U p+l + ... + uq for p::; q,
we obtain the following result:
where, analysing the reward system for scientists, he comes to the same conclusion as Saint Matthew: "For whosoever hath, to him sl).all be given, and he shall
have more abundance: but whosoever hath not, from him shall be taken away
even that he hath". Merton bases himself on the distribution of Nobel Prize,
on the order in which the authors of articles written in cooperation are cited,
on interviews with scientists who stress that new ideas are much more easily
accepted and recognised as such when they emanate from established scientists,
etc. See Science, Vol. 159, 5/1/1968, or the collection of articles from Robert K.
Merton, The Sociology of Science (U. of Chicago Press, 1973), a new discipline
which has developed greatly recently and not always congenial to the scientists,
who are deeply allergic to not uniformly admiring commentary on their activities, and even more, if possible, to see themselves subjected to the same methods
of enquiry as the amazonian tribes or the "white collars" of the General Motors,
their general theory being that scientists alone are able to understand their own
activities. .. See for example Bernard Barber, Science and the Social Order
(Macmillan/Free Press, 1952), W. O. Hagstrom, The Scientific Community (Basic Books, 1965), D. Crane, Invisible Colleges (Chicago UP, 1972), J. R. Cole &
S. Cole, Social Stratification in Science (Chicago UP, 1973), Harriet A. Zuckerman, Scientific Elite: Studies of Nobel Laureates in the United States (Chicago
UP, 1974), Bruno Latour & Steve Woolgar, Laboratory Life (Sage Publications,
1979), etc. There is even at least one journal, Social Studies of Science, entirely
devoted to the subject.
21 See Dugac's thesis, pp. 81-88.
III - Convergence: Continuous variables
Some criticised his Precis for demanding of school-Ieavers a capacity for
abstraction that they do not possess and forcing a brutal rupture from their
old habits, to which Meray replied that "it was not [his] fault if they are bad
to the point that makes a rupture necessary21" (in 1872 ... ).
One sees that all these ideas - cuts a la Dedekind, increasing sequences,
least upper bounds, decimal expansions, Cauchy sequences, nested intervals,
BW, etc. - overlap each other, and consist of always repeating the same
thing in various disguises, leaving one to choose, in practice, the most directly
usable form. When once one has understood these ideas, one has made an
enormous leap forwards in the comprehension of analysis: one has overtaken
Newton and Euler, 1665-1750, Dedekind, Weierstrass, Cantor, etc. Do not
worry if you do not assimilate them instantly and completely: it needed all
of the XIX th century and the reflections of a good dozen of mathematicians
of the highest order to elaborate it and even more in order to give it, in the
XX th century, the almost perfect form and the enormous generality that it
has acquired today.
11 - Cauchy's criterion for series: examples
Cauchy's criterion applies to series I: Un; one applies it to the sequence of
partial sums Sn of the series. Since
Sq - sp = U p+l + ... + uq for p::; q,
we obtain the following result:
where, analysing the reward system for scientists, he comes to the same conclusion as Saint Matthew: "For whosoever hath, to him sl).all be given, and he shall
have more abundance: but whosoever hath not, from him shall be taken away
even that he hath". Merton bases himself on the distribution of Nobel Prize,
on the order in which the authors of articles written in cooperation are cited,
on interviews with scientists who stress that new ideas are much more easily
accepted and recognised as such when they emanate from established scientists,
etc. See Science, Vol. 159, 5/1/1968, or the collection of articles from Robert K.
Merton, The Sociology of Science (U. of Chicago Press, 1973), a new discipline
which has developed greatly recently and not always congenial to the scientists,
who are deeply allergic to not uniformly admiring commentary on their activities, and even more, if possible, to see themselves subjected to the same methods
of enquiry as the amazonian tribes or the "white collars" of the General Motors,
their general theory being that scientists alone are able to understand their own
activities. .. See for example Bernard Barber, Science and the Social Order
(Macmillan/Free Press, 1952), W. O. Hagstrom, The Scientific Community (Basic Books, 1965), D. Crane, Invisible Colleges (Chicago UP, 1972), J. R. Cole &
S. Cole, Social Stratification in Science (Chicago UP, 1973), Harriet A. Zuckerman, Scientific Elite: Studies of Nobel Laureates in the United States (Chicago
UP, 1974), Bruno Latour & Steve Woolgar, Laboratory Life (Sage Publications,
1979), etc. There is even at least one journal, Social Studies of Science, entirely
devoted to the subject.
21 See Dugac's thesis, pp. 81-88.