2Start:Axiomatic method, strongly influenced by David Hilbert’s exampleThe logical formulation of pure mathematics suggested by Bertrand RussellPure mathematician became a recognized vocation, to be achieved through training.
3Generality and abstraction IDEA OF GENERALITYPure mathematics often exhibits a trend towards increased generality.Generality has many different manifestations.
4Certain 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.
520th century: SummaryThe 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 problemsIn the 1910s, Srinivasa Aiyangar Ramanujan developed over theorems.In 1931, Kurt Goedel published his two incompleteness theoremsWolfganag Haken and Kenneth Appel used a computer to prove the four color theorem in 1976.Andrew Wiles proved Fermat’s last theorem in 1995New 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.
6David Hilbert (January 23, 1862 – February 14, 1943) German mathematicianOne of the most influential and universal mathematiciansHe 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 analysisHe defended Georg cantor’s set theory and transfinite numbersOne of the founders of proof theory, mathematical logic andthe distinction between mathematics and metamathematics
7Srinivasa Ramanujan Iyengar (22 December, 1887 – 26 April, 1920) Indian mathematicianOne of the greatest mathematical geniusHe made substantial contributions in the areas of mathematical analysis, number theory, infinite series and continued fractionsIndependently compiled nearly results during his short lifetime
9Kurt Goedel (April 28, 1906 – January 14, 1978) Austrian – American mathematician and philosopherHis work has had immense impact upon scientific and philosophical thinkingTwo incompleteness theorems by the age of 25for 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
10Wolfgang Haken (born June 21, 1928) Mathematician who specializes in topologyIn 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
11Kenneth Appel (born 1932)In 1976 with Wolfgang Haken, solved one of the most famous problem in mathematics, the four- color theorem.
12The 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.
13Sir Andrew John Wiles (born April 11, 1953) British research mathematician at Princeton UniversitySpecialised for number theoryMost famous for proving Fermat’s Last Theorem
14The EndIvana Balatinac Irena Brdar Mirna Brekalo Antonija Chorich Marija Zovko In Osijek May 19th, 2008