# Term Paper: Georg Cantor: A Genius Out

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Cantor's most criticized concept presented in this paper is that the power of the continuum is independent from its number of dimensions. It was commonly believed that points in two dimensional space cannot be traced back to one dimensional space, and Cantor had thought that he could get higher transfinite powers by going from the one-dimensional to the multi-dimensional. Here also is the famous conjecture by Cantor that the two powers of the rational numbers and the real numbers exhaust all possibilities for infinite subsets of the continuum.

Cantor traveled to Switzerland often and it was here that he met Richard Dedekind who was to have a great influence on both his personal and professional life. They corresponded for many years and Dedekind's logic had an influence on Cantor's own work. Cantor respected Dedekind's opinion and it was Dedekind, who encouraged him to continue even when all others told him to give up on his arguments. Had ti not been for Dedekind, Cantor may have given up and the theory of sets would not exist as we know it today.

Kronecker was skeptical of the new and strange methods being used by Cantor, and apparently used his influence in his position on the editorial staff of Crelle's Journal to hold up publication of the paper. Cantor was tempted to withdraw the manuscript and publish it as a special article, but Dedekind persuaded him against this. Cantor never again published in Crelle's Journal. Cantor believed in his work and continued to produce works containing some concepts that are now familiar such as well-ordering, closed, dense, and connected.

Cantor has been quoted as saying,

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things" (Georg Cantor in Dunham, 1990).

Cantor clearly was not afraid to stick by his beliefs, even under great pressure from others to give up his ideas.

Another work of Cantor's, Grundlagen einer aligemeinen Mannichfaltigkeitslehre (Grundlagen for short), gives an account of the way transfinite numbers developed in Cantor's mind. It contains a discussion of what he would later call ordinal numbers. He discusses the rationale for the introduction of a new concept, the first ordinal number. Toward the end of the nineteenth century, Cantor published a double treatise which clarified and systematized much of what he had done before.

Cantor suffered a nervous breakdown in the spring of 1884, due to the heavy criticism and lack of acceptance of his works. Mental illness plagued him throughout his life, often causing him to act irrationally in public, but the crisis was essentially over early in 1885.

In 1897, with his work barely accepted by the Mathematical community, mostly due to several influential people, paradoxes began to appear. The first paradox was discovered in 1897 by Cesare Burali-Forti in Cantor's theory of ordinal numbers and soon other paradoxes of an even more basic nature began to appear in set theory. After much discussion by many schools, the most accepted resolution of the paradoxes has been to axiomize set theory. The first formal axiomization was by Ernst Zermelo in 1908. Other versions of the axioms soon appeared.

In 1904, Georg Cantor was awarded a place in the London Mathematical Society and the Society of Sciences in Gottingen. He was awarded a medal for his theories. However, this did not help Cantor to recover and on January 6, 1918, he died in the mental institution of a heart attack (Breen, 2000).

Conclusions

IT has been said that true genius is a form of madness. If that is the case, then Georg Cantor would certain fall into this category. Many great minds have held to beliefs that others of their time thought were absurd. Christopher Columbus was one of these people. Thomas Edison and Albert Einstein were some others to name a few. These men knew they were right and believed in themselves, even when everyone else did not. Had it not been for the spirit of these men, our world would be a very different place today.

Georg Cantor did not show the signs of genius early in his life and it was only through discipline and hard struggle that these theories came into being. Many contemporaries of his own time, including some, such as Kronecker whom he had earlier held in high esteem, considered his work to be a complete failure. His career at Halle was also considered to be a complete failure by the standards of others who were there. Now Cantor is considered to be one of the greatest mathematicians of all time and our ideas about set theory, finite numbers, infinity and many other topics would not exist had it not been for this man.

By what means is a man to be judged? Can we judge a man by his wealth while he is alive, or by the gifts that he leaves behind? This is the question that arises when we examine the life of Georg Cantor. In his time he was the brunt of many cruel jokes by those whom he had one admired. However, it is difficult to imagine the field of mathematics today without his contributions. Articles have been written attacking his logic from an Aristotelean point-of-view. However, despite this criticism, his work is still considered a standard part of the teaching of mathematics.

The evolution and publication of the paradoxes associated with his theory and the need to axiomize the theories shows that at one time his work was considered highly questionable at best. When then did we finally accept his work? Do we consider set theory to be standard? How can it be considered to be standard, if the theory is so unsound in its basis? These are the questions that the study of Georg Cantor raises. Cantor's life was one of questioning strongly held standard beliefs. This leads us too, to question the validity of set theory and ask ourselves if it is sound math, or if we too have accepted standardized ideas which have been thrust upon us.

Many of these questions have no standard answer, and each must explore them for themselves. In the tradition of Cantor, it would seem that the act of questioning is sometimes more important than the answers. The educational system is filled with standardized ideas, and sometimes when we examine the source of the information, we find it to be not as credible as we believe. This is the purpose behind studying the history of mathematical theory. It is so that we can track the development of the theories, in order that we do not fall into the trap of accepting what everyone else holds to be true.

One thing is certain, if no on ever questions the source and validity of knowledge, then no new discoveries will ever be made from this time forward. Those who question, lead the way for tomorrow's discoveries. We also must remember to have faith in ourselves and forgive those who cannot conceive of true genius. This is the lesson of Georg Cantor and one which we all should strive to live, in our lives, whereever we may go in our careers.

Works Cited

Breen, Craig. Georg Cantor (1845-1918) History of Mathematics. July 2000. Retrieved at http://www.geocities.com/CollegePark/Union/3461/cantor.htm. July12, 2002.

Johnson, Phillip E. The Late Nineteenth Century Origins Of Set Theory. Department of Mathematics. UNC Charlotte, NC.Volume V, 1997. Retrieved at http://www.aug.edu/dvskel/JohnsonSU97.htm. July11, 2002.

Rucker, Rudy. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press, Princeton University. 1995

Dunham, William. Journey through Genius: Great Theorems of Mathematics. New York: Wiley.

Bibliography

Breen, Craig. Gerog Cantor (1845-1918) History of Mathematics. July 2000. Retrieved at http://www.geocities.com/CollegePark/Union/3461/cantor.htm. July12, 2002.

Fraenkel, Abraham. "Georg Cantor," Jahresbericht der Deutschen mathematiker Vereinigung 39

1930, 189-266.

Johnson, Phillip E. The Late Nineteenth Century Origins Of Set Theory. Department of Mathematics. UNC Charlotte, NC.Volume V, 1997. Retrived at http://www.aug.edu/dvskel/JohnsonSU97.htm. July11, 2002.

Rucker, Rudy. Infinity and the Mind:…
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