Mathematics in 20th century

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MATHEMATICS IN 20TH CENTURY
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• Axiomatic method, strongly influenced by David
  Hilberts example
• The logical formulation of pure mathematics
  suggested by Bertrand Russell
• Pure mathematician became a recognized vocation,
  to be achieved through training.

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Generality and abstraction
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IDEA OF GENERALITY Pure mathematics often
exhibits a trend towards increased
generality. Generality has many different
manifestations.
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Certain benefits of generality
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1. Generalizing theorems or mathematical structures
   can lead to deeper understanding of the original
   theorems or structures by exploring the
   implications of weakening the assumptions, one
   gains a better understanding of the role those
   assumptions play in the original theorems or
   structures.
2. Generality can simplify the presentation of
   material, resulting in shorter proofs or
   arguments that are easier to follow.
3. One can use generality to avoid duplication of
   effort, proving a general results from other
   areas of mathematics.
4. Generality can facilitate connections between
   different branches of mathematics, by emphasizing
   commonality of structure that may not be apparent
   at less general levels. Category theory is one
   area of mathematics dedicated to exploring this
   commonality of structure as it plays out in some
   areas of math.

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20th century Summary
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• The profession of mathematician became much more
  important.
• Jobs are available both in teaching and industry.
• In 1900 David Hilbert presented a list of 23
  unsolved problems
• In the 1910s, Srinivasa Aiyangar Ramanujan
  developed over 3000 theorems.
• In 1931, Kurt Goedel published his two
  incompleteness theorems
• Wolfganag Haken and Kenneth Appel used a computer
  to prove the four color theorem in 1976.
• Andrew Wiles proved Fermats last theorem in 1995
• New areas of mathematics mathematical logic,
  topology, complexity theory, and game theory.
• Mathematics was even findin its way into art, as
  fractal geometry produced beautiful shapes never
  before seen.

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David Hilbert (January 23, 1862 February 14,
1943)
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• German mathematician
• One of the most influential and universal
  mathematicians
• He invented or developed a broad range of
  fundamental ideas, in invariant theory, the
  axiomatization of geometry, and with the notion
  of Hilbert space, one of the foundations of
  functional analysis
• He defended Georg cantors set theory and
  transfinite numbers
• One of the founders of proof theory, mathematical
  logic and
• the distinction between mathematics and
  metamathematics

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Srinivasa Ramanujan Iyengar (22 December, 1887
26 April, 1920)
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• Indian mathematician
• One of the greatest mathematical genius
• He made substantial contributions in the areas of
  mathematical analysis, number theory, infinite
  series and continued fractions
• Independently compiled nearly 3900 results during
  his short lifetime

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Kurt Goedel (April 28, 1906 January 14, 1978)
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• Austrian American mathematician and philosopher
• His work has had immense impact upon scientific
  and philosophical thinking
• Two incompleteness theorems by the age of 25
• for any self-consistent recursive axiomatic
  system powerful enough to describe the aritmetic
  of the natural numbers, there are true
  propositions about the naturals that cannot be
  proved from the axioms.
• He showed that the continuum hypothesis cannot be
  disproved from the accepted axioms of set theory,
  if those axioms are consistent

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Wolfgang Haken (born June 21, 1928)
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• Mathematician who specializes in topology
• In 1976 with Kenneth Appel, solved one of the
  most famous problem in mathematics, the
  four-color theorem.
• One of his key contributions to algorithmic
  topology is an algorithm to detect if a knot is
  unknotted

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Kenneth Appel (born 1932)
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• In 1976 with Wolfgang Haken, solved one of the
  most famous problem in mathematics, the
  four-color theorem.

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The four-color theorem
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• Haken and Appel proved that any two-dimensional
  map, with certain limitations, can be filled in
  with four colors without any adjacent countries
  sharing the same coulor.

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Sir Andrew John Wiles (born April 11, 1953)
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• British research mathematician at Princeton
  University
• Specialised for number theory
• Most famous for proving Fermats Last Theorem

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The End
Ivana Balatinac Irena Brdar Mirna
Brekalo Antonija Chorich Marija Zovko In
Osijek May 19th, 2008
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