Stories from the

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Stories from the History of Mathematics
David M. Bressoud Macalester College, St. Paul,
MN MathFest, Knoxville, August 12, 2006
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The task of the educator is to make the childs
spirit pass again where its forefathers have
gone, moving rapidly through certain stages but
suppressing none of them. In this regard, the
history of science must be our guide. Henri
Poincaré
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• Cauchy and uniform convergence
• The Fundamental Theorem of Calculus
• The HeineBorel Theorem

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• Cauchy and uniform convergence
• The Fundamental Theorem of Calculus
• The HeineBorel Theorem

A Radical Approach to Real Analysis, 2nd edition
due January, 2007
A Radical Approach to Lebesgues Theory of
Integration, due December, 2007
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What Weierstrass Cantor did was very good.
That's the way it had to be done. But whether
this corresponds to what is in the depths of our
consciousness is a very different question
Nikolai Luzin
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I cannot but see a stark contradiction between
the intuitively clear fundamental formulas of the
integral calculus and the incomparably artificial
and complex work of the justification and their
proofs.
Nikolai Luzin
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Cauchy, Cours danalyse, 1821
explanations drawn from algebraic technique
cannot be considered, in my opinion, except as
heuristics that will sometimes suggest the truth,
but which accord little with the accuracy that is
so praised in the mathematical sciences.
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Niels Abel (1826)
Cauchy is crazy, and there is no way of getting
along with him, even though right now he is the
only one who knows how mathematics should be
done. What he is doing is excellent, but very
confusing.
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Cauchy, Cours danalyse, 1821, p. 120
Theorem 1. When the terms of a series are
functions of a single variable x and are
continuous with respect to this variable in the
neighborhood of a particular value where the
series converges, the sum S(x) of the series is
also, in the neighborhood of this particular
value, a continuous function of x.
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Abel, 1826
It appears to me that this theorem suffers
exceptions.
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n depends on x
x depends on n
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If even Cauchy can make a mistake like this, how
am I supposed to know what is correct?
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What is the Fundamental Theorem of Calculus? Why
is it fundamental?
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Differentiate then Integrate original fcn (up
to constant)
I.e., integration and differentiation are inverse
processes, but isnt this the definition of
integration?
Integrate then Differentiate original fcn
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1820, The fundamental proposition of the theory
of definite integrals
Siméon Poisson
If then
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1820, The fundamental proposition of the theory
of definite integrals
Siméon Poisson
If then
Definite integral, defined as difference of
antiderivatives at endpoints, is sum of products,
f(x) times infinitesimal dx.
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Cauchy, 1823, first explicit definition of
definite integral as limit of sum of
products mentions the fact that
en route to his definition of the indefinite
integral.
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Earliest reference (known to me) to Fundamental
Theorem of the Integral Calculus The Theory of
Functions of a Real Variable, E. W. Hobson, 1907
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Granville (w/ Smith) Differential and Integral
Calculus (starting with 1911 ed.), FTC definite
integral can be used to evaluate a limit of a sum
of products.
William A. Granville
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• The real FTC
• There are two distinct ways of viewing
  integration
• As a limit of a sum of products (Riemann sum),
• As the inverse process of differentiation.
• The power of calculus comes precisely from their
  equivalence.

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Riemanns habilitation of 1854 Über die
Darstellbarkeit einer Function durch eine
trigonometrische Reihe

• Purpose of Riemann integral
• To investigate how discontinuous a function can
  be and still be integrable. Can be discontinuous
  on a dense set of points.
• To investigate when an unbounded function can
  still be integrable. Introduce improper integral.

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Riemanns function
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This part of the FTC does not hold at points
where f is not continuous.
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Volterra, 1881, constructed function with bounded
derivative that is not Riemann integrable.
Vito Volterra
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Perfect set equals its set of limit
points Nowhere dense every interval contains
subinterval with no points of the set
Non-empty, nowhere dense, perfect set described
by H.J.S. Smith, 1875
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Perfect set equals its set of limit
points Nowhere dense every interval contains
subinterval with no points of the set
Non-empty, nowhere dense, perfect set described
by H.J.S. Smith, 1875
Then by Vito Volterra, 1881
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Perfect set equals its set of limit
points Nowhere dense every interval contains
subinterval with no points of the set
Non-empty, nowhere dense, perfect set described
by H.J.S. Smith, 1875
Then by Vito Volterra, 1881
Finally by Georg Cantor, 1883
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Perfect set equals its set of limit
points Nowhere dense every interval contains
subinterval with no points of the set
Non-empty, nowhere dense, perfect set described
by H.J.S. Smith, 1875
SVC Sets
Then by Vito Volterra, 1881
Finally by Georg Cantor, 1883
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• Volterras function, V satisfies
• V is differentiable at every x, V' is bounded.
• For a in SVC set, V'(a) 0, but there are points
  arbitrarily close to a where the derivative is
  1, 1.
• ? V' is not Riemann integrable on 0,1

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Volterra, 1881, constructed function with bounded
derivative that is not Riemann integrable. FTC
does hold if we restrict f to be continuous or if
we use the Lebesgue integral and F is absolutely
continuous.
Vito Volterra
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• Lessons
• Riemanns definition is not intuitively natural.
  Students think of integration as inverse of
  differentiation. Cauchy definition is easier to
  comprehend.

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• Lessons
• Riemanns definition is not intuitively natural.
  Students think of integration as inverse of
  differentiation. Cauchy definition is easier to
  comprehend.
• Emphasize FTC as connecting two very different
  ways of interpreting integration. Go back to
  calling it the Fundamental Theorem of Integral
  Calculus.

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• Lessons
• Riemanns definition is not intuitively natural.
  Students think of integration as inverse of
  differentiation. Cauchy definition is easier to
  comprehend.
• Emphasize FTC as connecting two very different
  ways of interpreting integration. Go back to
  calling it the Fundamental Theorem of Integral
  Calculus.
• Need to let students know that these
  interpretations of integration really are
  different.

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HeineBorel Theorem
Any open cover of a closed and bounded set of
real numbers has a finite subcover.
Émile Borel 18711956
Eduard Heine 18211881
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HeineBorel Theorem
Any open cover of a closed and bounded set of
real numbers has a finite subcover.
Émile Borel 18711956
Eduard Heine 18211881
Due to Lebesgue, 1904 stated and proven by Borel
for countable covers, 1895 Heine had very little
to do with it.
P. Dugac, Sur la correspondance de Borel
Arch. Int. Hist. Sci.,1989.
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1852, Dirichlet proves that a continuous function
on a closed, bounded interval is uniformly
continuous.
The proof is very similar to Borel and Lebesgues
proof of HeineBorel.
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1872, Heine reproduces this proof without
attribution to Dirichlet in Die Elemente der
Functionenlehre
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1872, Heine reproduces this proof without
attribution to Dirichlet in Die Elemente der
Functionenlehre
Weierstrass,1880, if a series converges uniformly
in some open neighborhood of every point in
a,b, then it converges uniformly over a,b.
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1872, Heine reproduces this proof without
attribution to Dirichlet in Die Elemente der
Functionenlehre
Weierstrass,1880, if a series converges uniformly
in some open neighborhood of every point in
a,b, then it converges uniformly over a,b.
Pincherle,1882, if a function is bounded in some
open neighborhood of every point in a,b, then
it is bounded over a,b.
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Harnack, 1885, considered the question of the
measure of an arbitrary set. Considered and
rejected the possibility of using countable
collection of open intervals.
Axel Harnack 18511888
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Harnack, 1885, considered the question of the
measure of an arbitrary set. Considered and
rejected the possibility of using countable
collection of open intervals.
Axel Harnack 18511888
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Harnack, 1885, considered the question of the
measure of an arbitrary set. Considered and
rejected the possibility of using countable
collection of open intervals.
Axel Harnack 18511888
Harnack assumed that the complement of a
countable union of intervals is a countable union
of intervals, in which case the answer is YES.
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Harnack, 1885, considered the question of the
measure of an arbitrary set. Considered and
rejected the possibility of using countable
collection of open intervals.
Axel Harnack 18511888
Harnack assumed that the complement of a
countable union of intervals is a countable union
of intervals, in which case the answer is YES.
Cantors set 1883
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Borel, 1895 (doctoral thesis, 1894), problem of
analytic continuation across a boundary on which
lie a countable dense set of poles
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Arthur Schönflies, 1900, claimed Borels result
also holds for uncountable covers, pointed out
connection to Heines proof of uniform
continuity. First to call this the HeineBorel
theorem.
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Arthur Schönflies, 1900, claimed Borels result
also holds for uncountable covers, pointed out
connection to Heines proof of uniform
continuity. First to call this the HeineBorel
theorem.
1904, Henri Lebesgue to Borel, Heine says
nothing, nothing at all, not even remotely, about
your theorem. Suggests calling it the
BorelSchönflies theorem. Proves the Schönflies
claim that it is valid for uncountable covers.
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Arthur Schönflies, 1900, claimed Borels result
also holds for uncountable covers, pointed out
connection to Heines proof of uniform
continuity. First to call this the HeineBorel
theorem.
1904, Henri Lebesgue to Borel, Heine says
nothing, nothing at all, not even remotely, about
your theorem. Suggests calling it the
BorelSchönflies theorem. Proves the Schönflies
claim that it is valid for uncountable covers.
Paul Montel and Giuseppe Vitali try to change
designation to BorelLebesgue. Borel in Leçons
sur la Théorie des Fonctions calls it the first
fundamental theorem of measure theory.
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• Lessons
• HeineBorel is far less intuitive than other
  equivalent definitions of completeness.

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• Lessons
• HeineBorel is far less intuitive than other
  equivalent definitions of completeness.
• In fact, HeineBorel can be counter-intuitive.

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• Lessons
• HeineBorel is far less intuitive than other
  equivalent definitions of completeness.
• In fact, HeineBorel can be counter-intuitive.
• HeineBorel lies at the root of Borel (and thus,
  Lebesgue) measure. This is the moment at which it
  is needed. Much prefer Borels name First
  Fundamental Theorem of Measure Theory.

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This PowerPoint presentation is available at
www.macalester.edu/bressoud/talks A draft of A
Radical Approach to Lebesgues Theory of
Integration is available at www.macalester.edu/br
essoud/books