Georg Cantor: A Genius Out of Time

Georg Cantor: A Genius Out of Time

If you open a textbook, in high school or college, in the first chapter you will be introduced to set theory and the theories of finite numbers, infinite numbers, and irrational numbers. The development of many theories of math took years upon years and the input of many mathematicians, as in the example of non-Euclidean geometry. This was the case with most math theories, however set theory was primarily the result of the work of one man, Georg Cantor. In his time, these hypotheses were considered greatly controversial by other mathematicians. However, now they are an integral part of the study of mathematics. Georg Cantor received more criticism than complement in his time and it eventually led him to mental illness. However, one must remember that many other things, once thought to be controversial are now considered to be fact. Georg Cantor should be considered one of the pioneers of modern mathematical theory.

Mathematics can be considered a language in its own right. It is the language that we use to describe our world. Math tells us vital information such as how big, how fast, and describes the relationship between two things. A set is a group of things that can be treated as a single unit. There are two ways to describe a set. The first method is to simply list the elements of a set. The second way is to describe the members of a set and define what characteristics determine which elements will be included or not be included in a particular set.

Family and Early Life

Georg Ferdinand Ludwig Phillip Cantor was born on March 3, 1845 in St. Petersburg, Russia. The family lived in Russia for eleven years until his father's failing health forced the family to move to the milder climate of Frankfort, Germany in 1856. It was here that Cantor would spend the rest of his life. Georg was the eldest of the three children. His father was a wealthy merchant, Georg Walsematr Cantor and his mother was a famous artist, Maria Boehm. The other children had exceptional artistic talents like their mother. Georg's brother, Constantine, was an army officer and also a fine pianist. His sister, Sophie Nobiling, was an accomplished designer. However, Georg excelled in Math (Johnson, 1997). Georg Cantor came from a family with a wealth of talent in math, physics, and philosophy. His brothers and sisters also displayed talent in math.

Cantor had a strict religious upbringing, and he carried a strong religious sense all through his life. His father was Jewish, but later converted to Protestantism around the time of Georg's birth. His mother was a devout Catholic. This difference of religious opinions did not sway Georg's own beliefs and he became a knowledgeable theologian as well as mathematician (Johnson, 1997). It was no doubt that this diverse religious background made him the type to question his surroundings and stand by his ideas, even when everyone else said he was wrong.

Education

Georg attended several private schools in Frankfurt, and in 1859, entered the distinguished Grossherzoglich Hessiche Provinzialrealschule in Darmstadt. He left this institution in 1860 with high recommendations in mathematics. His father discouraged the study of math due to the fact that he wished him to become an engineer, a job that paid considerably more than mathematics. He originally attended Grossherzogliche Hoehere Gewerbeschule (Grand-Ducal Higher Polytechnic, later changed to Technische Hochschule) at Darmstadt following his father's wishes and studying Engineering. Later, when Georg convinced his father that his heart was truly in math, his father relented and he began the study of Mathematics in 1862 (Johnson, 1997).

Cantor began his higher studies in Zurich, the fall of 1862. He left in Spring of 1963 due to the death of his father. In Fall of 1863, he entered the University of Berlin to study mathematics, physics, and philosophy. The University of Berlin was home to three famous mathematicians Ernst Eduard Kummer, Karl W.T. Weierstrass and Leopold Kronecker. These three men made the University of Berlin one of the top schools for the study of mathematics in the entire world. The student population was small and therefore the students were in close contact with these three great minds. Cantor was heavily influenced by the works of Weierstrass. Kronecker was also a great influence, but would later become one of his greatest critics. It was customary in Germany, at the time, to study at another University for a period time. He studied at the Cantor attended the University of Gottingen during the summer term of 1866 (Johnson, 1997). Cantor received the degree of doctor on December 14, 1867. His dissertation was based on a study of the Disquisitiones Arithmeticae of Carl Friedrich Gauss, another contemporary mathematician of his time, and on the number theory of Adrien-Marie Legendre (Johnson, 1997).

Cantor's thesis centered around one of the ideas that Gauss had left aside concerning the solutions in integers x, y and z of the indeterminate equation ax2 + by2 + cz2 = 0, where a, b and c are any given integers. The full title of the thesis was "De aequationibus secundi gradus indeterminatis" ("On indeterminate equations of the second degree") and as was customary, dedicated to his guardians, Eduard Flersheim and Bernhard Horkheimer. As was also the custom for Doctoral candidates, Cantor also defended three theses against opposing doctors. All fo which were translated from the original Latin, the theses were "In arithmetic merely arithmetic methods far surpass analytic methods," "Since it is disputed, the question of the absoluteness of space and time is more important than its solution" and "In mathematics the art of proposing a question must be held of higher value than solving it." (Johnson, 1997). Cantor's early works were considered to be excellent among his peers, but no on ever suspected the genius that would emerge in his later writings.

In Spring of 1869, Cantor began his career as a Privatdozent at the University of Halle on the basis of his paper "De transformatione formarum ternariarum quadraticarum" ("On the transformation of ternary quadratic forms") (Johnson,1997). Cantor specialized in Number theory. Cantor's became Extraordinarius at Halle in 1872 and Ordinarius in 1879. He was released from his official duties in 1905 and resigned his post altogether in 1913.

In 1874,Cantor published his first paper on the Theory of Sets. In that same year, he married Vally Guttman in the summer. They had two sons and four daughters, none of whom, it might be noted were gifted in mathematics. One of Cantor's daughters, Frau Gertrud Vahlen, was an important source of information for the biography of Cantor by A.A. Fraenkel, which was written in the latter part of his life. When Cantor received his professorship on the Theory of Sets in 1879 (Breen, 2000), not everyone agreed with and readily accepted Cantor's ideas. His lectures on the theory were not well attended. They criticized his set theory and believed that only numbers were integers and that negatives, fractions, and imaginary numbers did not belong in the field of mathematics proper, but rather as a type of metaphysics. Cantor's career at Halle would not be considered to be a success by many standards. He produced few researchers and doctorate candidates, unlike those who had influenced him. Cantor stood by his theories and now is considered to be one of the greatest mathematicians in history (Breen, 2000).

In 1874, Cantor wrote a paper which appeared in Crelle's Journal. The paper proposed that there are two different orders of infinity. Cantor showed that the set of real numbers can be put in one-to-one correspondence with the set of natural numbers. However the real numbers cannot be put into one-to-one correspondence with the set of natural numbers. This theory in itself raised no eyebrows and stirred no controversy. The fact was established in this paper that the set of real numbers is larger than the set of natural numbers. Cantor was the first to use nested intervals to proved that the set of real numbers is not countable rather than use his diagonal processes produced in his later work.

Cantor commented on his theory as such,

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds (George Cantor in Rucker, 1995).

Cantor's next paper appeared in 1878 with the central idea of one-to-one correspondence and a number of theorems concerning such correspondences given along with suggestions for classifying sets based on these assumptions. This paper contains the proof that the set of rational numbers is countable. Cantor used the word "power" ("Machtigkeit") for the first time to establish that two sets, which can be put in one-to-one correspondence with each other, have the same power. Cantor discusses in some length sets as having the smallest infinite power and demonstrates that any infinite subset has this same power. 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.