God and Nature Spring 2024
The Mathematics and Philosophy of Georg Cantor
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By Andy Quick
Introduction
In the last months of 1917, a thin, broken man wrote to his wife from the Nervenklinik sanatorium in Halle, Province of Saxony, German Empire, asking if he could return home. His entreaties were not granted, and he died of a heart attack on January 6, 1918, one of countless tragic stories being played out in Europe during those times.
This man was Georg Cantor (1845-1918), the great German mathematician. His best work, done in the years 1874-1884, only became fully appreciated by the wider mathematical community years later. He had some supportive colleagues, notably Richard Dedekind, David Hilbert and Gösta Mittag-Leffler, and he was awarded the Sylvester Medal for contributions to mathematics by the Royal Society in 1904. But Cantor was described as a “scientific charlatan” and a “corrupter of youth” by the prominent mathematician Leopold Kronecker, and he was blocked from attaining a position at the prestigious University in Berlin by Kronecker and others.
Introduction
In the last months of 1917, a thin, broken man wrote to his wife from the Nervenklinik sanatorium in Halle, Province of Saxony, German Empire, asking if he could return home. His entreaties were not granted, and he died of a heart attack on January 6, 1918, one of countless tragic stories being played out in Europe during those times.
This man was Georg Cantor (1845-1918), the great German mathematician. His best work, done in the years 1874-1884, only became fully appreciated by the wider mathematical community years later. He had some supportive colleagues, notably Richard Dedekind, David Hilbert and Gösta Mittag-Leffler, and he was awarded the Sylvester Medal for contributions to mathematics by the Royal Society in 1904. But Cantor was described as a “scientific charlatan” and a “corrupter of youth” by the prominent mathematician Leopold Kronecker, and he was blocked from attaining a position at the prestigious University in Berlin by Kronecker and others.
...he came to believe that his ideas had been directly revealed to him by God. |
Cantor’s Mathematics
Cantor is most known for his contributions to Set Theory, one of the foundational areas of mathematics. He started his professional career by making significant contributions to the mathematical analysis of trigonometric series, which led him to the study of the infinite.
Cantor began with the infinity that all mathematicians were familiar with: that of the “counting”, or natural numbers {0,1,2,...}. Mathematicians say that the natural numbers are “countable”, and that any collection that can be put into a one-to-one correspondence with them is countable. Cantor continued to argue that other, somewhat less intuitive, collections are countable. For example, he showed that the set of rational numbers—that is fractions a/b where a,b ≠ 0 are integers, are countable because a mapping from the natural numbers to every rational number can be constructed. Consider the infinite table in figure 1, which contains all the positive rational numbers.
Figure 1: Positive rational numbers [1]
Cantor is most known for his contributions to Set Theory, one of the foundational areas of mathematics. He started his professional career by making significant contributions to the mathematical analysis of trigonometric series, which led him to the study of the infinite.
Cantor began with the infinity that all mathematicians were familiar with: that of the “counting”, or natural numbers {0,1,2,...}. Mathematicians say that the natural numbers are “countable”, and that any collection that can be put into a one-to-one correspondence with them is countable. Cantor continued to argue that other, somewhat less intuitive, collections are countable. For example, he showed that the set of rational numbers—that is fractions a/b where a,b ≠ 0 are integers, are countable because a mapping from the natural numbers to every rational number can be constructed. Consider the infinite table in figure 1, which contains all the positive rational numbers.
Figure 1: Positive rational numbers [1]
We can define an enumeration containing the positive rational numbers by listing them in a sequence derived from a “winding diagonal path”, starting from the top left corner:
Cantor defined the notion of cardinal numbers, representing the size, or “power” of a set. He called the first infinite cardinal number ℵ0 (Hebrew “aleph” with subscript 0, or aleph-zero) for the cardinality of countably infinite sets such as the natural numbers and rational numbers
Cantor defined the notion of cardinal numbers, representing the size, or “power” of a set. He called the first infinite cardinal number ℵ0 (Hebrew “aleph” with subscript 0, or aleph-zero) for the cardinality of countably infinite sets such as the natural numbers and rational numbers
Another of Cantor’s insights was that the real numbers, or continuum, represent a different level of infinity than the integers or natural numbers. He proved that the continuum is not countable. Cantor’s original proof was complicated, but later he gave a simpler proof. [2] That proof was by a diagonal argument, in which he showed that if the real numbers could be enumerated by their decimal representation in an infinite table, then a real number could be constructed that differed from each number in the digit along the diagonal of the table. That violation of the assumption implies that the real numbers cannot be enumerated in that way. This type of argument has been used to prove foundational results in mathematical logic and in the theory of computation.
The cardinality of the set of real numbers (i.e. the cardinality of the continuum) is 2^ℵ0. The intuitive justification for this is that every real number can be represented by a countably infinite sequence of bits.
Cantor asked a natural question: given that the real numbers are not countable—that is their cardinality, 2^ℵ0 is strictly greater than ℵ0,—are there subsets of the real numbers with cardinality in between these numbers? Cantor believed that the answer was no, but he couldn’t prove it, even after years of trying. This is called the Continuum Hypothesis.
Following Cantor, much work was done by mathematicians on the formalization of the foundations of mathematics. An axiomatic system for Set Theory was named ZF after two of the most important contributors, mathematicians Ernst Zermelo and Abraham Fraenkel. A further axiom, called the Axiom of Choice, was later added, and the standard system of axioms for modern Set Theory is called ZFC.
Cantor’s Philosophy of the Infinite
Cantor made philosophy an equal and intentional partner with mathematics in some of his major publications. He put forward his ideas about “actual infinities”, a concept which most philosophers, mathematicians, and theologians had traditionally opposed. Theologians regarded the idea as a direct challenge to the unique and absolutely infinite nature of God [3] while mathematicians, lead by Leopold Kronecker, were opposed to any idea of the actual infinite, citing the great Carl Friedrich Gauss, who wrote in a letter to Heinrich Schumacher [3]:
But concerning your proof, I protest above all against the use of an infinite quantity (Grosse) as a completed one, which in mathematics is never allowed. The infinite is only a “way of talking”, in which one properly speaks of limits.
Cantor’s Disillusionment
Cantor became discouraged about how his mathematics was viewed by the mathematical establishment in his native land. Further, he became frustrated on his attempt to resolve the Continuum Hypothesis. He knew that Kronecker was calling his work “humbug”, and in late 1883, possibly out of spite, Cantor wrote directly to the Ministry of Education, applying for an open position at Kronecker’s university in Berlin. Cantor felt strongly that he deserved a position at a prestigious university in mathematics, such as Göttingen or Berlin, rather than the provincial university in Halle.
Cantor’s stress seemed to come to a head in the spring of 1884, when he suffered the first of his serious nervous breakdowns.
Thereafter, he devoted an increasing amount of his time to certain historical interests, and to questions of philosophy and theology. [3]
From late 1885 onwards Cantor corresponded with Roman Catholic scholars and theologians on philosophies of the infinite. He even taught philosophy at his university in Halle. Some Catholic theologians were suspicious of any attempt to correlate God’s infinite nature with a concrete, temporal infinity, which they believed suggested pantheism, a heretical idea. But Cantor argued that there was a distinction between the “Absolute Infinity” of God and the “actual infinity” evidenced in nature, in the “actually infinite” number of objects in the universe (3).
Cantor became so convinced of his philosophies that he came to believe that his ideas had been directly revealed to him by God. In 1894 Cantor wrote: [3]
But now I thank God, the all-wise and all-good, that He always denied me the fulfillment of this wish (for a position at university in either Göttingen or Berlin), for He thereby constrained me, through a deeper penetration into theology, to serve Him and His Holy Roman Catholic Church better than I have been able with my exclusive preoccupation with mathematics.
Sadly, beginning in 1899, Cantor was hospitalized several times as a result of “nervous breakdowns”. There are indications from the report of an attending psychiatrist that Cantor may have suffered from bipolar disorder. [4] He retired from his position at the university in 1913. In his final years, Cantor, like many of his countrymen, suffered from the privations of World War I.
Musings
Nowadays, mathematicians rightly separate the mathematics of Cantor’s work from his philosophical ideas. There is really no controversy over the mathematics. Cantor is regarded as a founder of Set Theory, a very important area in the theoretical foundations of mathematics. Similarly, philosophical ideas about mathematics are separated from its applications in the physical sciences. Physicists don’t debate the metaphysics of i = square root of -1 when dealing with quantum physics, they recognize that the mathematics in some way describes physical reality.
Cantor believed in the absolute truth of his theories, and he didn’t separate the mathematics from his philosophy of the infinite. He set himself up for unnecessary tensions in his relationships with other mathematicians, who otherwise supported his work.
It is difficult to say what the root causes of Kronecker’s opposition were. At the very least, there appears to have been a deep divide in personal philosophical views between them. Many people have the impression that the field of mathematics is “black and white” and it is easy to determine whether a mathematical theory has value. But mathematics is done by people, who are part of a social context, and their personal philosophies can clash, sometimes with deeply hurtful consequences.
I cannot help but wonder if Cantor created some unnecessary controversy by mixing philosophy with science. In those times it was culturally acceptable, and even encouraged to mix philosophy with science when writing on a scientific subject. Before the term “scientist” was coined by Whewell in 1834, people who studied the natural world were “Natural Historians” or “Natural Philosophers”. Indeed, as an artifact of that history, many scientists attain a degree of “Doctor of Philosophy” at institutions of higher learning. But in the modern world, no purely scientific journal would accept a paper with philosophy mixed in with the science.
It’s a good thing that we can appreciate Cantor’s genius from the distance of time. We can also be reminded that we live in a social context. We can hope that truth, including scientific truth, will stand the test of time and be appreciated by future generations, even if it isn’t always acceptable in certain social contexts.
References
1.https://commons.wikimedia.org/w/index.php?curid=2203732
2 Cantor, G., 1915. Contributions to the Founding of the Theory of Transfinite Numbers. Trans. P.E.B Jourdain. Chicago: Open Court
3 Dauben, J.W., 1979. Georg Cantor, His Mathematics and Philosophy of the Infinite. Princeton, New Jersey: Princeton University Press
4 Dauben, J.W., 1993. Georg Cantor and the Battle for Transfinite Set Theory. Journal of the ACMS, 2004. https://acmsonline.org/journal/journalarchives/2004-journal/
Andy Quick is a retired software engineer, with a Master's degree in Computer Science. He is a member of the CSCA and lives in Kitchener-Waterloo, Canada.
The cardinality of the set of real numbers (i.e. the cardinality of the continuum) is 2^ℵ0. The intuitive justification for this is that every real number can be represented by a countably infinite sequence of bits.
Cantor asked a natural question: given that the real numbers are not countable—that is their cardinality, 2^ℵ0 is strictly greater than ℵ0,—are there subsets of the real numbers with cardinality in between these numbers? Cantor believed that the answer was no, but he couldn’t prove it, even after years of trying. This is called the Continuum Hypothesis.
Following Cantor, much work was done by mathematicians on the formalization of the foundations of mathematics. An axiomatic system for Set Theory was named ZF after two of the most important contributors, mathematicians Ernst Zermelo and Abraham Fraenkel. A further axiom, called the Axiom of Choice, was later added, and the standard system of axioms for modern Set Theory is called ZFC.
Cantor’s Philosophy of the Infinite
Cantor made philosophy an equal and intentional partner with mathematics in some of his major publications. He put forward his ideas about “actual infinities”, a concept which most philosophers, mathematicians, and theologians had traditionally opposed. Theologians regarded the idea as a direct challenge to the unique and absolutely infinite nature of God [3] while mathematicians, lead by Leopold Kronecker, were opposed to any idea of the actual infinite, citing the great Carl Friedrich Gauss, who wrote in a letter to Heinrich Schumacher [3]:
But concerning your proof, I protest above all against the use of an infinite quantity (Grosse) as a completed one, which in mathematics is never allowed. The infinite is only a “way of talking”, in which one properly speaks of limits.
Cantor’s Disillusionment
Cantor became discouraged about how his mathematics was viewed by the mathematical establishment in his native land. Further, he became frustrated on his attempt to resolve the Continuum Hypothesis. He knew that Kronecker was calling his work “humbug”, and in late 1883, possibly out of spite, Cantor wrote directly to the Ministry of Education, applying for an open position at Kronecker’s university in Berlin. Cantor felt strongly that he deserved a position at a prestigious university in mathematics, such as Göttingen or Berlin, rather than the provincial university in Halle.
Cantor’s stress seemed to come to a head in the spring of 1884, when he suffered the first of his serious nervous breakdowns.
Thereafter, he devoted an increasing amount of his time to certain historical interests, and to questions of philosophy and theology. [3]
From late 1885 onwards Cantor corresponded with Roman Catholic scholars and theologians on philosophies of the infinite. He even taught philosophy at his university in Halle. Some Catholic theologians were suspicious of any attempt to correlate God’s infinite nature with a concrete, temporal infinity, which they believed suggested pantheism, a heretical idea. But Cantor argued that there was a distinction between the “Absolute Infinity” of God and the “actual infinity” evidenced in nature, in the “actually infinite” number of objects in the universe (3).
Cantor became so convinced of his philosophies that he came to believe that his ideas had been directly revealed to him by God. In 1894 Cantor wrote: [3]
But now I thank God, the all-wise and all-good, that He always denied me the fulfillment of this wish (for a position at university in either Göttingen or Berlin), for He thereby constrained me, through a deeper penetration into theology, to serve Him and His Holy Roman Catholic Church better than I have been able with my exclusive preoccupation with mathematics.
Sadly, beginning in 1899, Cantor was hospitalized several times as a result of “nervous breakdowns”. There are indications from the report of an attending psychiatrist that Cantor may have suffered from bipolar disorder. [4] He retired from his position at the university in 1913. In his final years, Cantor, like many of his countrymen, suffered from the privations of World War I.
Musings
Nowadays, mathematicians rightly separate the mathematics of Cantor’s work from his philosophical ideas. There is really no controversy over the mathematics. Cantor is regarded as a founder of Set Theory, a very important area in the theoretical foundations of mathematics. Similarly, philosophical ideas about mathematics are separated from its applications in the physical sciences. Physicists don’t debate the metaphysics of i = square root of -1 when dealing with quantum physics, they recognize that the mathematics in some way describes physical reality.
Cantor believed in the absolute truth of his theories, and he didn’t separate the mathematics from his philosophy of the infinite. He set himself up for unnecessary tensions in his relationships with other mathematicians, who otherwise supported his work.
It is difficult to say what the root causes of Kronecker’s opposition were. At the very least, there appears to have been a deep divide in personal philosophical views between them. Many people have the impression that the field of mathematics is “black and white” and it is easy to determine whether a mathematical theory has value. But mathematics is done by people, who are part of a social context, and their personal philosophies can clash, sometimes with deeply hurtful consequences.
I cannot help but wonder if Cantor created some unnecessary controversy by mixing philosophy with science. In those times it was culturally acceptable, and even encouraged to mix philosophy with science when writing on a scientific subject. Before the term “scientist” was coined by Whewell in 1834, people who studied the natural world were “Natural Historians” or “Natural Philosophers”. Indeed, as an artifact of that history, many scientists attain a degree of “Doctor of Philosophy” at institutions of higher learning. But in the modern world, no purely scientific journal would accept a paper with philosophy mixed in with the science.
It’s a good thing that we can appreciate Cantor’s genius from the distance of time. We can also be reminded that we live in a social context. We can hope that truth, including scientific truth, will stand the test of time and be appreciated by future generations, even if it isn’t always acceptable in certain social contexts.
References
1.https://commons.wikimedia.org/w/index.php?curid=2203732
2 Cantor, G., 1915. Contributions to the Founding of the Theory of Transfinite Numbers. Trans. P.E.B Jourdain. Chicago: Open Court
3 Dauben, J.W., 1979. Georg Cantor, His Mathematics and Philosophy of the Infinite. Princeton, New Jersey: Princeton University Press
4 Dauben, J.W., 1993. Georg Cantor and the Battle for Transfinite Set Theory. Journal of the ACMS, 2004. https://acmsonline.org/journal/journalarchives/2004-journal/
Andy Quick is a retired software engineer, with a Master's degree in Computer Science. He is a member of the CSCA and lives in Kitchener-Waterloo, Canada.