Arithmetization of Analysis

ARITHMETIZATION OF ANALYSIS
19TH Century Mathematics of Germany
Prepared by: Ma. Irene G. Gonzales

When the theory of a
mathematical operation is
poorly understood, what is
likely to happen?

o Attracted by the powerful applicability of
analysis, and lacking a real understanding of
the foundations upon which the subject must
rest, mathematicians manipulated analytical
processes in an almost blind manner.
o A gradual accumulation of absurdities was
bound to result until some conscientious
mathematicians felt bound to attempt the
difficult task of establishing a rigorous
foundation under the subject.

Jean-le-Rond d’Alembert (1717 – 1783)
 He gave the first
suggestion of a
real remedy for the
unsatisfactory
state of the
foundations of
analysis.

Joseph Louis Lagrange (1736 – 1813)

Joseph Louis Lagrange (1736 – 1813)
 An Italian Mathematician
 Earliest mathematician of the first rank to
attempt a rigorization of the calculus.
 His attempt was based upon representing a
function by aTaylor’s series expansion.
 His attempt was published in 1797 in
Lagrange’s monumental work, Theorie des
fonctions analytiques.

Carl Freidrich Gauss
 He broke from intuitive ideas and set new
high standards of mathematical rigor.
 The first adequate consideration of the
convergence of an infinite series was
encountered in Gauss’ treatment of the
hypergeometric series.

Augustin-Louis Cauchy (1789 – 1857)

Augustin-Louis Cauchy (1789 – 1857)
 A French Mathematician
 He successfully executed d’Alembert’s
suggestion, by developing an acceptable
theory of limits.
 He defined continuity, differentiability, and
the definite integral in terms of the limit
concept.
 Cauchy’s rigor inspired other mathematicians
to join the effort to rid analysis of formalism
and intuitionism.

Karl Theodor Wilhelm Weierstrass

 a German Mathematician
 He discovered the theoretical existence of a continuous
function having no derivative (in other words, a continuous
curve possessing no tangent at any of its points).
 He saw the need for a rigorous “arithmetization” of calculus.
 He advocated a program wherein the real number system itself
should be first rigorized, then all the basic concepts of analysis
should be derived from this number system.

 And today all of analysis can be logically
derived from a postulate set characterizing
the real number system.
 Along with Riemann andAugustin-Louis
Cauchy,Weierstrass completely reformulated
calculus in an even more rigorous fashion,
leading to the development of mathematical
analysis.

Bernhard Bolzano
 a Bohemian priest
 one of the earliest mathematicians to
begin instilling rigor into mathematical
analysis
 He gave the first purely analytic proof of
both the fundamental theorem of
algebra and the intermediate value
theorem, and early consideration of sets
(collections of objects defined by a
common property).

Joseph Fourier
 son of a tailor in
Auxere
 He received his
education through the
Benedictine Order
 Best known for his
book Théorie
analytique de la
chaleur

Contributions of Fourier
 Fourier Series
 An important advance in mathematical
analysis
 Periodic functions that can be expressed as the
sum of an infinite series of sines and cosines
 It is a powerful tool in pure and applied
mathematics.
 Fourier series are used in solving differential
equations that arise in the study of heat flow
and vibration.

Johann Peter Gustav Lejeune Dirichlet

 In 1837, he suggested a very broad definition of
function,
 Dirichlet Function
 When x is rational, let y = c and when x is
irrational, let y = d ≠ c
 He gave the first rigorous proof of the
convergence of Fourier Series for a function
subject to certain restrictions.
 In pure mathematics, he is well – known for his
application of analysis to the theory of numbers.

 In the field of mathematical analysis, a
general Dirichlet series is an infinite series
that takes this form

Bernhard Riemann
 He arrived at deep theorems relating number
theory and classical analysis.
 Euler had noted connections between prime-
number theory and the series

Bernhard Riemann
 The Riemann zeta function ζ(s) is a function of
a complex variable s = σ + it. The following
infinite series converges for all complex
numbers s with real part greater than 1, and
defines ζ(s) in this case:

Bernhard Riemann
 In 1859, Riemann conjectured that all the
imaginary zeros of the zeta function have their
real part

Contributions
 refinement of the definition of the integral by
the definition of the Riemann integral
 emphasis on the Cauchy – Riemann differential
equations
 Riemann Surfaces – are ingenious scheme for
uniformizing a function
 Riemann surfaces are nowadays considered the
natural setting for studying the global behavior
of these functions, especially multi-valued
functions such as the square root and other
algebraic functions, or the logarithm.

Arithmetization of Analysis

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