Georg Cantor: The Man who went Beyond Infinity

Georg Cantor (1845-1918)

Georg Cantor is a German mathematician whose work in set theory and transfinite numbers remains as one of the most important conributions to mathmatics. His theorem implies the existence of the "infinity of infinitied".

His work was not well received initially as it was counter-intuitive and question the authority of much of modern mathematics. His history-making paper was at first denied publication and when it was finally published, reception was by no means cordial.

His ideas encountered resistance from other leading mathematicians especially Leopold Kronecker who was once his mentor.

By 1900, his work had gained some respect in the mathematical community, however, it was still not universally accepted.

Sources:
Evolution of mathematical concepts, R.L. Wilder
The Story of Mathematics, Motz & Weaver
http://dissertations.bc.edu/cgi/viewcontent.cgi?article=1092&context=ashonors

Point.

The traditional views of the mathematicians prevented them from acknowledging and accepting Cantor’s work. For a theorem to be considered as part of mathematical knowledge, it has to be first accepted. Is there then a social aspect in mathematics?

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Abel's Mistake

Niels Hendrik Abel (1802-1829)

1823: At the age of 21, Abel published a paper where he gave the first general solution to fifth degree algebraic equations .

(Equations of type Ax⁵ + Bx⁴ + Cx³ + Dx² + Ex + F = 0)

A year later, Abel retracted the claim in a famous paper in which he proved that the general 5th degree equation cannot be solved in terms of algebraic radicals.

Source: The Story of Mathematics, Lloyd Motz & Jefferson Hane Weaver

Point.

Even in the realm of mathematics, certainty is not guaranteed. As hard as it is to believe, mathematicians are human. Thus, it is not unreasonable to believe that years from now, some of today's mathematics will be proven wrong or invalid.

More about Abel:

http://www.math.wichita.edu/history/men/abel.html

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html

http://www.everything2.com/index.pl?node_id=700029

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The Four Color Theorem

Theorem: A plane separated into regions can be colored by no more than four colors such that regions which share a common boundary do not have the same color.

Conjecture first made by Francis Guthrie

1879: Alfred Bray Kempe provided a proof for the conjecture and received great acclaim

1890: Percy John Heawood showed that Kempe's proof was wrong.

1976: Four color conjecture becomes four color theorem for a second time. Four Color theorem proven by Appel and Haken, using a computer.

Hence, the first computer assisted proof was born. The four color theorem cannot be verified by other mathematicians without the use of a computer.

Point.

Is it ok for mathematicians to put their faith in computers? Can such proofs be accepted? (By the way, Mathematicians have already accepted it.)

Small point.

Mathematicians can make mistakes. Maths can't be true all the time. (see what happened to Kempe)

Sources:
http://www.mathpages.com/home/kmath266/kmath266.htm
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html.
http://mathworld.wolfram.com/Four-ColorTheorem.html

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maths in reality or reality in maths?

"mathematical reality exists on an abstract plane, and its objects are as real as those in everyday life."

- G. H. Hardy, mathematician.

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