Georg Cantor was an exceptional mathematician who is probably best known for making a vital breakthrough in our understanding of the concept of infinity. Until Cantor, infinity had been a subject largely reserved to religion, but Cantor showed that it was possible to apply it to the world of numbers. He is also remembered for his long struggle with depression, which affected not only his research but also his family and colleagues.
Cantor was born in St. Petersburg, Russia in 1845 to German parents. His father, Georg Waldemar Cantor was a very prosperous merchant in the city. While his father had originally come from Denmark, Georg’s mother, Maria Anna Böhm, was Russian and came from a very musical family of violinists. Both his parents were very refined people who were passionate about the arts. Indeed, under his mother’s influence, Georg learned to play the violin to a high level. Overall George was pushed hard by his parents to excel and this pressure to succeed was a constant feature of his early years. Religion was also a very important aspect of Georg’s life. His mother was catholic while his father held strong protestant beliefs, and it was Protestantism that would dominate his own belief system.
In his early years, Cantor received specialised one-to-one private tuition at home and then went on to primary school in St Petersburg. Towards the end of his early education, his parents decided to move to Germany, partly due to his father’s bad health and his struggle to cope with the exceptionally cold Russian winters. They settled in the city of Wiesbaden where Cantor studied at the Gymnasium. Subsequently, he attended a specialist secondary boarding school (Realschule) in Darmstadt, intended to encourage pupils to have a particular career focus. Cantor was an exceptional student and it was during his time at this school that he developed his interest in and excelled at mathematics. His father wanted to push the young Georg toward a more practical career choice in engineering, but he eventually gave in to his son’s plea to enter university to study mathematics. After an initial period of study in Zurich, Cantor had to cope with the death of his father in 1863. Cantor then moved to the University of Berlin where he learned from some of the leading German mathematicians of the day. After a brief period at the University of Göttingen, Cantor came back to Berlin in 1867 where he wrote his dissertation on number theory.
Cantor’s social life tended to revolve around his intellectual interests. In Berlin, he was an active member of the Mathematical Society and he became president of that organisation in 1864. He frequently met with other student mathematicians at a local wine house. Cantor became interested in teaching as a way of making a living and worked in a school for girls for a short time. In 1868 he got involved in the Schellbach Seminar for the teaching of mathematics. He later met Richard Dedekind with whom he shared a strong interest in number theory. Over a long period, they would correspond and exchange ideas that had an influence on their respective research outputs. Cantor was later responsible for the establishment of the Association of German Mathematicians in 1890. This was of great importance as it played an international role in bringing together leading mathematicians from all over the world at congresses.
Two years after receiving his doctorate, Cantor was given an academic position at the University of Halle. Cantor immediately began to flourish in his new post and was rapidly promoted to an Assistant Professor role at the university in 1872. In addition to his rising academic rank, Cantor also began to publish regularly and focussed on analysis. Some of his papers examined aspects of trigonometry.In work carried out between 1873 and 1874, Cantor established the fundamental outlines of what would become inﬁnite set theory. Cantor’s major achievement was to offer a logical understanding of infinity in a mathematical form.
Aristotle argued a principle that lasted for around two thousand years: the idea of infinity was viewed with almost fearful reverence. He believed that, if the infinite existed at all, it was nothing more than “potential existence”. In maths, such a potential existence manifested itself in a never-ending numerical sequence: 1, 2, 3, 4, 5 and so on. Philosophers and mathematicians thought that an in-depth understanding or measurement of infinity was impossible. Before Cantor’s breakthrough in the 1870s, the overwhelming majority of philosophers were in agreement that the application of the idea of infinity was very much a religious matter beyond the understanding of human beings and in the hands of God. Over time there were a number of developments in mathematics, such as the discovery of rational numbers in ancient Greece or the debate on infinitesimals (quantities so small that we cannot measure them) in the seventeenth century. However, none of these developments were as important as Cantor’s work. This is because he was able to provide a mathematical explanation to suggest that there were different levels of numerical infinity.
Cantor began with a straightforward list of whole numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and so on. He then decided to compare these numbers with a smaller set of numbers along the lines of: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Cantor showed that these two sets of numbers have the same size because it is possible to express them as pairs: 1 and 10, 2 and 20, 3 and 30, 4 and 40, 5 and 50, 6 and 60, 7 and 70, 8 and 80, 9 and 90, 10 and 100 and so on. On first glance, one might question what Cantor means when he says that they are the same size. What he means is that they are infinitely the same. He is not separately counting each set he is matching them, 1 with 10, 2 with 20 and so on. In this way, we have a correspondence between the two numbers.
He then looked at fractions. There are infinitely many fractions between the whole numbers themselves, suggesting that perhaps the infinity of fractions might be greater than that of whole numbers. Cantor devised an ingenious way of pairing the fractions and the whole numbers. He organised the positive fractions into an infinite grid. By doing this Cantor was again able to show that there is a correspondence between the fractions and the whole numbers and so there are infinitely the same. However, while these sets of numbers have the same infinity, Cantor makes the remarkable discovery that decimal numbers have a greater infinity. In the above lists, we are always able to achieve a correspondence of numbers. Cantor devised a procedure to produce a new decimal number that is missing from any given list of decimal numbers. In this way, he was able to establish that no list of decimal numbers could be seen as complete. The other conclusion here is that there are, in fact, different sorts of infinities as some sets of numbers are larger than others.
In the late nineteenth century, Cantor’s idea was very much ahead of its time. Conservative mathematicians, especially Leopold Kronecker, showed strong resistance to recognizing the idea. He stated: “I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there”.Kronecker was an influential figure who rejected Cantor as a potential candidate for an academic position at the University of Berlin. Another leading mathematician at the time, the Swede Gösta Mittag-Leffler, found Cantor’s work to be interesting. He had established a new mathematical journal called Acta Mathematica in 1882 and the two men had exchanged a series of letters towards the latter part of 1884 in which Cantor outlined elements of his new research. Cantor sought to get his work published and Mittag-Leffler understood that Cantor was an innovator ahead of his time. Nonetheless, after Cantor had sent Mittag-Leffler a more complete and amended manuscript in the winter of 1885, the latter responded and informed Cantor that he would need to offer more concrete proof of his “continuum hypothesis” before the work could be published. This was a huge blow for Cantor at a time when his mental health was beginning to fail.
Cantor remained stoic in the face of this opposition and sought to challenge himself further. Earlier, in 1874, he had turned his attention to another complex problem: he wanted to prove that there could be no correspondence between a unit square and a unit line segment. However, in 1877 he learned to his own great surprise that the points of a unit line segment could, in fact, correspond with the points of a unit square. Again Cantor had reached a result that went against the intuition of conventional mathematics. The results of his research were important as he was effectively questioning the traditional understanding of dimension.
In the latter part of his life, Cantor became obsessed with the nature of the infinite set of decimal numbers. He had shown that they had a greater infinity than whole numbers. Then he wondered if there might be a set in between the whole numbers and the decimal numbers. Cantor had had his work rejected by Gösta Mittag-Leffler on the basis that he had not managed to prove this theory. Yet Cantor became increasingly fascinated with his continuum hypothesis and the possibility of proving that the order of infinity of decimal numbers followed on from that of the whole numbers. At one point Cantor thought that he had proved the hypothesis to be false, but, much to his frustration, he discovered that he was mistaken the next day. Cantor subjected himself to years of intellectual torture in trying to resolve this complex problem which proved in later years to be unresolvable.
Cantor was very fortunate in that his father had left him half a million Marks to ensure that his family could live in security. A significant portion of this sum was used by Cantor to build an impressive house for his family in a wealthy area of Halle. The two-storey building was luxuriously appointed with the best interior features, such as high ceilings and magnificent furniture. He had married Vally Guttman in 1874 and they had five children together. By 1879, Cantor had become a fully fledged professor and everything seemed to point to an ideal family situation. Unfortunately, this idyll did not last for long. In 1884 Cantor started to experience symptoms of depression. During the summer of that year, his depression was such that he was unable to work or function at all. His family was very worried about the sudden onset of this condition, as were his colleagues. Cantor recovered and returned back to work, but over time, his depression kept returning. His university was very supportive during these years and made sure that he retained his academic position. When he fell into depression he was sent to the Halle University Nervenklinik where he was given a large room with his own facilities.
Historians usually associate Cantor’s depression with his mathematics because each onset of the illness coincided with intense work on his continuum hypothesis. Recent research suggests that his work or even conflicts with others were unlikely to be the actual cause of the condition, although they may have exacerbated it. Cantor’s struggle with the illness lasted until his death on January 6, 1918. He died of heart failure in the Nervenklinik and he was buried in a small, local cemetery. There were not many people at his funeral only his wife and children, as well as a few others.
Georg Cantor was one of the world’s most remarkable mathematicians. He was feared by his critics for the simple reason that he questioned certain assumptions of conventional mathematics in the nineteenth century. The mathematicians of the day saw themselves as practitioners of a very powerful discipline with laws that allowed us to have a sense of certainty about numbers and their practical application to the world around us. Kurt Gödel,working in the 1930s, argued that there were many statements based on axioms that cannot be proved and that such systems of mathematical reasoning were deeply flawed. One of the important and sober conclusions that have been drawn from Cantor’s work is that some aspects of mathematics do not offer the certainty that had been asserted in the past. However, this should not necessarily be seen in a negative light. Cantor’s work also points to the beauty and imagination of mathematics and he inspired many new mathematicians who emerged in the twentieth century.