MATHEMATICS IN 20TH MATHEMATICS IN 20TH CENTURYCENTURY

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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

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Generality and abstractionGenerality and abstraction

IDEA OF GENERALITYIDEA OF GENERALITYPure mathematics often exhibits a

trend towards increased generality.

Generality has many different manifestations.

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Certain benefits of Certain benefits of generalitygenerality

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: 20th century: SummarySummary

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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 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.

David Hilbert David Hilbert (January 23, 1862 – February 14, 1943)(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 cantor’s set theory and transfinite numbers

One of the founders of proof theory, mathematical logic and

the distinction between mathematics and metamathematics

Srinivasa Ramanujan Iyengar 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 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

Wolfgang Haken Wolfgang Haken (born June 21, 1928)(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

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.

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.

Sir Andrew John Wiles 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 Fermat’s Last Theorem

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The End

Ivana BalatinacIrena Brdar

Mirna BrekaloAntonija Chorich

Marija Zovko

In OsijekMay 19th, 2008