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**Mathematics in 20th century**

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Start: Axiomatic method, strongly influenced by David Hilbert’s 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**

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**

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. Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow. One can use generality to avoid duplication of effort, proving a general results from other areas of mathematics. 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 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 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 Fermat’s 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)**

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 cantor’s 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)**

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 results during his short lifetime

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**Kurt Goedel (April 28, 1906 – January 14, 1978)**

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)**

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) 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**

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)**

British research mathematician at Princeton University Specialised for number theory Most famous for proving Fermat’s 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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