214
III - Convergence: Continuous variables
Retrospectively the most extraordinary thing about these controversies
is that, concerned as they were with the necessity of relieving analysis of
its artistic vagueness, neither Cauchy, nor Abel, nor Dirichlet, i.e. three of
the greatest mathematicians of the XIX th century, apparently had the idea
of using the incomparably simpler example which we have mentioned above,
that one can vary ad libitum, and which has now acquired the status of clicM.
The fact that they never traced the graph of a function is maybe to the point:
these were the first, Abel and Dirichlet much more than Cauchy, who tried to
found analysis on arguments as rigorous as those of the arithmetic and not
on geometric intuition; though not dispensing their users from "arithmetic"
proofs, diagrams are sometimes useful. But the more probable explanation is
that, while only beginning to develop an awareness of the extreme generality
of the concept of function - Dirichlet gave the first good general definition
-, they were still trammelled by their predecessors. They, the functions, are
those that one can represent by algebraic formulae (polynomials and rational
fractions) or by power series. During the XIXth century, following Cauchy,
the theory (of analytic functions on C) was constructed, but, at the other
end of the spectrum which spans from hypernormality to superpathology,
one also introduced "bizarre" functions into analysis, not continuous, not
differentiable, not representable by simple formulae nor by power series, even
resistant to every theory of integration, etc. Habituated as they were to the
virtuosic calculations and beautiful formulae of their predecessors, they did
not see the simple things which stare our contemporaries in the face. Strange?
No. New ideas do not enter the folklore of mathematicians any faster than in
every other intellectual guild.
Finally, one must remark that Cauchy was neither the only nor the first
to have had these ideas; but one lends only to the rich. From 1817, in rather
obscure works, much in advance of their time in their spirit, and rediscovered
later - some were not known before 1930 -, Bernhard Bolzano (1781-1848),
priest and professor of "science of religions" at Prague l l from 1805 to 1819,
published in German (or did not publish) the majority of ideas that one attributes to Cauchy - a precise definition of convergence and the first notion
of the general criterion to be discussed in nO 10, among others - with, here
again, proofs which leave something to be desired; in 1835 he still believed
in Cauchy's false theorem on the limits of continuous functions. Cauchy'S
n Wolfgang Walter, Analysis I (Springer, 1992) tells us that Bolzano's chair was
founded by the Emperor of Austria "to combat the ever more powerful influence
of the free thinkers of the Enlightenment". Bolzano, having become the principal spokesman of the Enlightenment in Bohemia, was dismissed in 1819. See
Jan Sebestik, Logique et mathematiques chez Bernard Bolzano (Paris, Vrin, 1992)
where one will see particularly that Bolzano already had ideas on set theory, but,
too much the philosopher, clearly lacked the proper language to express them
mathematically, B. Bolzano, Paradoxien des Unendlichen, 1851 (trans. Paradoxes of the Infinite, Routledge and Kegan Paul, London, 1950), and Franc;ois
Rivenc and Philippe de Rouilhan, Logique et fondements de mathematiques. Anthologie (1850-1914) (Paris, Payot, 1992).
III - Convergence: Continuous variables
Retrospectively the most extraordinary thing about these controversies
is that, concerned as they were with the necessity of relieving analysis of
its artistic vagueness, neither Cauchy, nor Abel, nor Dirichlet, i.e. three of
the greatest mathematicians of the XIX th century, apparently had the idea
of using the incomparably simpler example which we have mentioned above,
that one can vary ad libitum, and which has now acquired the status of clicM.
The fact that they never traced the graph of a function is maybe to the point:
these were the first, Abel and Dirichlet much more than Cauchy, who tried to
found analysis on arguments as rigorous as those of the arithmetic and not
on geometric intuition; though not dispensing their users from "arithmetic"
proofs, diagrams are sometimes useful. But the more probable explanation is
that, while only beginning to develop an awareness of the extreme generality
of the concept of function - Dirichlet gave the first good general definition
-, they were still trammelled by their predecessors. They, the functions, are
those that one can represent by algebraic formulae (polynomials and rational
fractions) or by power series. During the XIXth century, following Cauchy,
the theory (of analytic functions on C) was constructed, but, at the other
end of the spectrum which spans from hypernormality to superpathology,
one also introduced "bizarre" functions into analysis, not continuous, not
differentiable, not representable by simple formulae nor by power series, even
resistant to every theory of integration, etc. Habituated as they were to the
virtuosic calculations and beautiful formulae of their predecessors, they did
not see the simple things which stare our contemporaries in the face. Strange?
No. New ideas do not enter the folklore of mathematicians any faster than in
every other intellectual guild.
Finally, one must remark that Cauchy was neither the only nor the first
to have had these ideas; but one lends only to the rich. From 1817, in rather
obscure works, much in advance of their time in their spirit, and rediscovered
later - some were not known before 1930 -, Bernhard Bolzano (1781-1848),
priest and professor of "science of religions" at Prague l l from 1805 to 1819,
published in German (or did not publish) the majority of ideas that one attributes to Cauchy - a precise definition of convergence and the first notion
of the general criterion to be discussed in nO 10, among others - with, here
again, proofs which leave something to be desired; in 1835 he still believed
in Cauchy's false theorem on the limits of continuous functions. Cauchy'S
n Wolfgang Walter, Analysis I (Springer, 1992) tells us that Bolzano's chair was
founded by the Emperor of Austria "to combat the ever more powerful influence
of the free thinkers of the Enlightenment". Bolzano, having become the principal spokesman of the Enlightenment in Bohemia, was dismissed in 1819. See
Jan Sebestik, Logique et mathematiques chez Bernard Bolzano (Paris, Vrin, 1992)
where one will see particularly that Bolzano already had ideas on set theory, but,
too much the philosopher, clearly lacked the proper language to express them
mathematically, B. Bolzano, Paradoxien des Unendlichen, 1851 (trans. Paradoxes of the Infinite, Routledge and Kegan Paul, London, 1950), and Franc;ois
Rivenc and Philippe de Rouilhan, Logique et fondements de mathematiques. Anthologie (1850-1914) (Paris, Payot, 1992).