The Age of Rigor

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The Age of Rigor
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A proof tells us where to concentrate our
doubts.
Morris Kline
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The State of Mathematical Analysis in 1800
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The State of Mathematical Analysis in 1800

• The concept of function was unclear.
• Series were used without regard to their
  convergence or divergence.
• The concept of limit, derivative, integral, and
  continuity had no clear definitions.
• It was also universally accepted that continuity
  implied differentiability.

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The Critical Movement

• Several mathematicians resolved to bring order
  out of the chaos by rebuilding analysis solely on
  the basis of arithmetical concepts. The
  beginnings of the movement coincide with the
  creation of Non-Euclidean Geometry although there
  is no direct evidence of any connection between
  the two events. Gauss was the sole member of
  both groups.

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The Critical Movement

• Rigorous analysis begins with the work of
  Bolzano, Cauchy, Abel, and Dirichlet and was
  further developed by Weierstrass and his group.
  Cauchy and Weierstrass are best known in this
  connection.

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Augustin Louis Cauchy(1789 1857)
Cours d'analyse 1821 Sur
un nouveau genre de calcul analogue au calcul
infinitésimal 1826 Leçons sur le
Calcul Différentiel 1829
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Augustin Louis Cauchy(1789 1857)
Cauchy pioneered the study of analysis, both real
and complex, and the theory of permutation
groups. He also researched in convergence and
divergence of infinite series, differential
equations, determinants, probability and
mathematical physics. Stressed defining the
integral as a limit.
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Bernard Bolzano (1781 1848)

• Beyträge zu einer begründeteren Darstellung der
  Mathematik. Erste Lieferung
  1810
• Der binomische Lehrsatz ... 1816
• Rein analytischer Beweis ...
• (Pure Analytical Proof) 1817
• Wissenschaftslehre (Theory of Science)
• 1837
• Paradoxien des Unendlichen 1851

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Bernard Bolzano (1781 1848)

• Gave the proper definition of continuity.
• Bolzano successfully freed calculus from the
  concept of the infinitesimal. He also gave
  examples of 1-1 correspondences between the
  elements of an infinite set and the elements of a
  proper subset.

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Niels Abel(1802 1829)

proved the impossibility of solving the general
equation of the fifth degree in radicals
1824 Recherches sur les
fonctions elliptiques
1827 radically
transformed the theory of elliptic integrals to
the theory of elliptic functions by using their
inverse functions
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Lejeune Dirichlet(1805 1859)

• Vorlesungen über Zahlentheorie 1863
• proposed modern definition of function
• 1837
• convergence of trigonometric series
• the use of the trigonometric series to represent
  arbitrary functions
• founder of the theory of Fourier series
• close friendship with Gauss, Riemann was his
  student

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Carl Friedrich Gauss(1777 1855)

• Prince of Mathematicians
• Disquisitiones Arithmeticae 1801
• Theoria motus . . . 1809
• Disquisitiones generales . . . 1816
• Methodus nova . . . 1816
• Bestimmung . . . 1816
• Theoria attractionis. . . 1816
• Theoria combinationis. . . 1823
• and many more too numerous to list
• almost single handedly made number theory an
  important area of math
• defined curvature for surfaces
• developed method of least squares
• first to prove fundamental theorem of algebra

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Carl Friedrich Gauss(1777 1855)

• Participated in firming up the ideas involved in
  functions and infinite series.
• Gauss worked in a wide variety of fields in both
  mathematics and physics incuding number theory,
  analysis, differential geometry, geodesy,
  magnetism, astronomy and optics. His work has had
  an immense influence in many areas.

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Joseph Fourier(1768 1830)

• On the Propagation of Heat
• in Solid Bodies. 1807
• Théorie analytique de la chaleur
• 1822
• Historical Précis not published
• Fourier studied the mathematical theory of heat
  conduction. He established the partial
  differential equation governing heat diffusion
  and solved it by using infinite series of
  trigonometric functions.

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Karl Weierstrass(1815 1897)
Settled the issue of continuity not implying
differentiability. Developed the idea of uniform
convergence among his students. Weierstrass is
best known for his construction of the theory of
complex functions by means of power series.
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Sofia Kovalevskaya(1850 1891)
Kovalevskaya made valuable contributions to
the theory of differential equations.
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The State of Mathematical Analysis in 1890

• Thanks to work by Cauchy, Bolzano, Weierstrass,
  and others by 1890 the calculus was freed from
  all dependence upon geometrical notions, motion,
  and intuitive understandings. This was not
  without pain.
• For instance, after Laplace heard Cauchys
  presentation on convergence of series he hurried
  home to check all the series in his Traité de
  Mécanique Céleste. Luckily every one was found
  to be convergent

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The State of Mathematical Analysis in 1890

• When Weierstrass work became known through his
  lectures, the effect on rigor was even more
  noticable.
• The improvements in rigor can readily be seen by
  comparing the first edition of Jordans Cours
  d'analyse (1882-87) with the second (1893-96)
  and the third (1909-15)

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• . . . by about 1890, only 6000 years after the
  Egyptians and Babylonians began to work with
  numbers, fractions, and irrational numbers,
  mathematicians could finally prove 2 2 4.
• Morris Kline
• in
• Mathematics,
• The Loss of Certainty

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