Ann Varela : Who was Kurt Gödel and what did he have to do with Einstein?

Feb 13, 2018 by

An Interview with Ann Varela : Who was Kurt Gödel and what did he have to do with Einstein?

Michael F. Shaughnessy –

1)  What do we know about the early childhood of this mathematician who was also involved in religion and spirituality?

Kurt Gödel was born in 1906 in Austria-Hungary (now Czech Republic). He was brought up in a German-speaking household. Gödel’s father, Rudolph, was a businessman, more specifically, the manager of a textile factory. His mother, Marianne, was a well educated and sophisticated woman.

Gödel survived rheumatic fever at the age of six, but convinced himself that his heart was somehow impaired. Throughout his life, Gödel experienced anxiety and hypochondria regarding his health.

Gödel was an excellent student during his gymnasium years and had no difficulty matriculating into university studies at the University of Vienna. Apparently, Gödel was undecided about his focus of study at first, but later decided to declare mathematics (number theory) as his major. This decision may have had something to do with the fact that one of his professors had a major impact on him, partly due to a health-related issue. After all, his mathematics professor, Philipp Furtwängler, lectured from a wheel chair with an assistant to write on the chalkboard. Given Gödel’s obsession with his own personal health and subject to bouts of paranoia, I suppose he could relate to this particular professor and deemed him a role mode, a symbol of hope.

Later, Gödel studied mathematical philosophy, focused on mathematical logic, and completed his doctoral dissertation in 1929 under the tutelage of Hans Hahn. Gödel’s dissertation concentrated on the completeness theorem for first-order logic.

2)  Gödel escaped Germany and Hitler just before World War II- where did he go and what were some of his contributions there?

Seemingly oblivious to the events unfolding around him in Europe, Gödel kept busy with work until it became unsafe to remain any longer. His Jewish friends, colleagues, and professors were experiencing hardships, which were soon likely to include him, so Gödel and his wife, Adele, left Austria in 1940 and moved to the United States. After completing an interview and having a few people confirm the nature of his character, he was granted citizenship.

Since he had previously collaborated with the Institute of Advanced Study in Princeton, New Jersey, the institute assisted him with securing an exit visa for both his wife and himself. By 1946, he became a permanent faculty member of the institute. Seven years later, he advanced to professor status and became a member of the National Academy of Sciences.

Paranoia haunted him so much so, that he thought there were poison gases escaping from his refrigerator. At this time, Albert Einstein began to spend a lot of time with Gödel, and they took walks together almost every day.

3) The generalized continuum hypothesis-what is this stating?

First, it is relevant to understand what the continuum hypothesis states: there is no set whose cardinality lies between that of the natural numbers and that of the reals. Cardinality refers to the number of elements in the set or grouping. It has been proven that integers and the natural numbers have the same cardinality. To have the same cardinality, there must be a bijection (one-to-one correspondence) between the elements of two sets. Each element from one set is paired with exactly one unique element from the other set as shown in Figure 1. Gödel’s results showed that the accuracy of the continuum hypothesis depends on the version of set theory being used.

Figure 1. A Bijective Function.

https://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/768px-Bijection.svg.png

https://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/768px-Bijection.svg.png

In set theory, the Generalized Continuum hypothesis is about the possible sizes of infinite sets. Infinite sets may be countable or uncountable (contains too many elements to count). The hypothesis states that if A is any set, there is no set whose cardinality lies between the cardinality of A and the cardinality of the set of all subsets of A. In other words, the cardinality of set A is the same size as the cardinality of its subsets.

For example, if we choose the set of integers to represent A, then whole numbers and natural numbers are the subsets of integers. The cardinality of integers is equal to the cardinality of whole numbers and natural numbers. See Figure 2.

Figure 2. Real Number System.

The real number system is somewhat like an onion. If one starts with the center of the onion, that would be the natural numbers. Natural numbers are the counting numbers starting with 1, 2, 3, etc.

Contained in the next layer out are the whole numbers. Whole numbers include ALL of the natural numbers and zero: 0, 1, 2, etc. The natural numbers are a subset of the whole numbers.

Integers are the next layer out. Integers include ALL of the whole numbers; therefore, they also include ALL of the natural numbers. Natural numbers and whole numbers are subsets of integers.

Rational numbers are the next layer out and can be written in the form a/b, where the fraction either terminates or is a repeated pattern and b cannot equal zero. Integers, whole numbers, and natural numbers are all subsets of rational numbers.

Irrational numbers are numbers that are in the form a/b and result in non-terminating decimals (numbers like pi or square root of 2). Irrational numbers do not share any numbers with the other real numbers. They are a subset of real numbers, but not any of the other subsets because they have nothing in common with those other subsets.

Real numbers include all of the underlying subsets: irrational, rational, integers, whole, and natural numbers.

4) Mathematical logic—why did this interest Gödel? And what were his contributions?

Mathematical logic became a fascination of Gödel’s after reading and studying Bertrand Russell’s book Introduction of Mathematical Philosophy in a seminar run by Moritz Schlick. Gödel felt that mathematical logic was the foundation of all sciences.

Gödel was also influenced by the lectures of David Hilbert because they focused on the topics of completeness and consistency of mathematical systems. Gödel was so intrigued with these topics that he wrote his doctoral thesis on completeness entitled, On the Completeness of the Calculus of Logic.

In his thesis, Gödel proved the completeness theorem, which uses mathematical reasoning and logic to analyze quantified statements, such as “all” and “some”, and break them down into simpler and logical mathematical concepts.

5) What is Gödel’s ontological proof?

Gödel wrote a proof discussing the plausibility of God’s existence. Apparently, Gödel kept his manuscript private until 1970, when he thought he was dying, because he did not want to give people an impression that he believed or disbelieved in God, he was merely engaged in a logical investigation, based on his completeness theory. According to some sources, including Gödel’s wife, he was religious, believed in an afterlife, and read the Bible regularly, but was not a member of any organized religious organization.

Gödel’s ontological proof reminds me of “if-then” loops in computer programming languages because he analyzes a series of attributes, starting with whether the notion is a “positive property” or “not a positive property”. If not, then its opposite must be true, but not both notions can be true simultaneously.

Next, Gödel suggests that all of the positive properties are possibly demonstrated in some object in some world. If an object has all positive properties, it is considered Godlike; therefore, the object is also deemed a positive property. Gödel goes on to prove that a Godlike object is present in every possible world. Moreover, what is quite interesting, is that Gödel uses Leibniz’s law of indiscernibles (objects are identical if they are composed of only identical properties) to prove that there is only one God.

6) It is said that at the end of his life- Einstein said to a friend, that his own work did not mean much but that he came to Princeton- to the Institute of Advanced Studies-to have the privilege of walking home with Gödel” How much did Einstein and Gödel share over the course of their lifetimes?

Einstein began his tenure at the Institute of Advanced Studies about twelve years prior to Gödel’s arrival. Apparently, Einstein and Gödel struck up a friendship despite their seemingly opposite personalities. Einstein was recognized for his jovial and sociable demeanor, while Gödel was more reserved and cynical. Their differing tastes in music and food were also apparent.

Despite their obvious differences, Einstein and Gödel frequently spent time walking together and conversing with each other about politics, physics, and other topics deemed of “genius importance”. It was reported that Gödel did not feel anxious or tentative when speaking with Einstein; for that matter, Gödel even challenged some of Einstein’s ideas. What these men did share was the belief that the world was autonomous from the human mind. Both men were socially awkward as well. Perhaps this is what enticed them to be close friends and confidants.

7)  There is a Kurt Gödel Society. What do they foster and support?

The Kurt Gödel Society is a worldwide organization, founded in 1987 in Vienna, Austria. The purpose of the society is to promote the advancement of research in the areas of logic, philosophy, and the history of mathematics. A periodically recurring lecture series was also formed in Gödel’s honor.

8) In his later life, Gödel apparently developed a fear of being poisoned- and when his wife was hospitalized for a while, Gödel apparently did not eat. Like the movie, A Beautiful Mind, it appears that some mathematicians have mental issues or problems. Your thoughts?

First of all, I do not think that mental illness is limited to intelligent mathematicians. Current events around the world may attest to this opinion. According to Medical Daily, over thirty studies have shown an association between mental illness and intelligence, while some studies show no such association. When it comes to mental illness, I believe both genes and the environment may play a major role in a person’s mental health.

In Gödel’s case, he was extremely ill with rheumatic fever as a young child, was privy to numerous death reports and atrocities in Austria before World War II, and suffered the loss of his professor at the hands of a former student. These tremendously tragic life events may have compromised his mental fitness.

9) Douglas Hofstadter is well known for his book, published in 1979 entitled Gödel, Escher, Bach: An Eternal Golden Braid, to review the work and thoughts of these three giants in their fields.  Could you just briefly try to summarize the book?

Douglas Hofstadter’s book, Gödel, Escher, Bach: An Eternal Golden Braid is written in a unique style. Each chapter is organized in such a way that he begins with a narrative relating to one of Gödel’s theorems, axioms, proofs, formal systems, or philosophies. From there he relates the topic to artistic and musical qualities such as geometric structure, musical structure, insight, patterns, etc.

Hofstadter intertwines poetry, music, illustrations, puzzles, and analysis to guide the reader’s mind through various cognitive science topics such as meaning, organization, planning, and logic.

The author himself states that his purpose for writing the book was to explain how conscious beings can come out of inorganic matter. Hofstadter attempts to answer the questions, “What is self?” and “What is I?” The author uses analogy and “strange loops” throughout the book to assist with connecting animate and inanimate elements. Hofstadter believes these “strange loops” reveal “consciousness.”

10) What have I neglected to ask?

Kurt Gödel was privileged to earn the first Albert Einstein Award in 1951. The receipt of this particular award must have had special meaning to Gödel, as he and Einstein knew each other quite well. In 1974, Gödel received the Nation Medal of Science. The Gödel Prize is awarded in the area of theoretical computer science each year in his honor.

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