Trying to make sense of it doesn’t make sense.

A spa town in the Czech Republic, Marienbad was a favored vacation spot of European royalty and celebrities during the 19th and early 20th centuries. Some mathematicians came, too: Karl Weierstrass, Gösta Mittag-Leffler, and Sofia Kovalevskaya were all drawn by the combination of the restful atmosphere and sparkling social life that could be found at the spa.

Though its golden era ended in the early 20th century, Marienbad remained popular between the world wars. Among the visitors during that time was a young Kurt Gödel. According to some accounts, Gödel’s interest in the sciences was kindled by a teenage visit to Marienbad, during which he and his brother studied Goethe’s philosophical theory of color and found it lacking in comparison to Newton’s more strictly scientific account.

After the war we were in Marienbad quite often with my brother, and I remember that we once read Chamberlain’s biography of Goethe together. At several points, he took a special interest in Goethe’s theory of color, which also served as a source of his interest in the natural sciences. In any case, he preferred Newton’s analysis of the color spectrum to Goethe’s.

-Rudolf Gödel

Goethe himself was a frequent visitor to Marienbad. During an 1823 trip, the 73-year-old Goethe became infatuated with the 18-year-old Baroness Ulrike von Levetzow. The pain caused by her rejection of his marriage proposal led him to write the famous Marienbad Elegy (updated in 1999 by the great W.G. Sebald).

…who dance, stroll up and down, and swim in the pool, as if this were a summer resort like Los Teques or Marienbad.

-Adolfo Bioy Casares, The Invention of Morel

In 1961, Last Year at Marienbad was released. Directed by Alain Resnais and written by Alain Robbe-Grillet, the film is as beautiful as it is inexplicable. On its face, the film is set at a resort hotel; an unnamed man (‘X’) becomes infatuated with an unnamed woman (‘A’) and attempts to convince her that they had an affair the previous year. The film unfolds in combinatorial play, with narration and scenes repeated in ever-evolving and bewildering variation.

A popular theory is that Last Year at Marienbad is actually an adaptation of Adolfo Bioy Casares’ The Invention of Morel, a novel of which our friend Jorge Luis Borges wrote, “To classify it as perfect is neither an imprecision nor a hyperbole.” I will not say much about this, so as not to spoil the book (you should go read it right now), but will only mention that Morel was, in a way, an homage to Louise Brooks, a Hollywood actress with whom Casares was somewhat obsessed and whose performance in Pandora’s Box provided a model for Delphine Seyrig’s performance as ‘A’ in Marienbad. The idea that there is a direct line from Casares and Brooks to the main characters in Morel to ‘X’ and ‘A’ in Marienbad, and that the obscurities of both novel and film are at heart simply odes to the power of cinema is an appealing one.

This theory about the connection between Marienbad and Morel was never acknowledged by the filmmakers (some say that the source text for the film is not Casares’ novel, but rather Wittgenstein’s Philosophical Investigations). Perhaps its fullest explication is given by this article in Senses of Cinema, in which the only sources given by the author are the dust jacket of a different Casares work and an Encyclopedia Britannica article which has since been removed from the online archives. Regardless of the theory’s truth, though, when one views the film through the lens of Morel, it comes tantalizingly close to making sense; the characters of the film lose their agency, consigned to repeating their roles ad infinitum.

At first sight, it seemed impossible to lose your way. At first sight…

The third main character in Marienbad is ‘M’, a man who may or may not be the husband of ‘A’. Throughout the film, we see ‘M’ playing a version of the mathematical game of Nim with ‘X’.

This version of Nim became known by the name “Marienbad” and was a brief craze in certain circles. It even got written about in Time:

Last week the Marienbad game was popping up at cocktail parties (with colored toothpicks), on commuter trains (with paper matches), in offices (paper clips) and in bars (with swizzle sticks). Only two can play, but any number can kibitz — and everyone, it seems, has a system for duplicating “X’s” talent for winning.

-“Games: Two on a Match,” Time, Mar. 23, 1962

The game is a theoretical win for the second player, although, as it is unlikely that a player will stumble upon the winning strategy by accident, ‘X’ is able to win even as the first player. We will return to general winning strategies for Nim and other games in later posts.

-The one who starts, wins.

-You must take an even number.

-You must take the smallest odd number.

-It’s a logarithmic series.

-You must switch rows as you go.

-And divide by three.

-Seven times seven is forty-nine.

-kibitzers in Last Year at Marienbad

I leave you now with Nick Cave’s exquisite “Girl in Amber.” Another secret adaptation of The Invention of Morel? Possible…

# Cantor and the Absolute (Universal Structures IV)

As long-time readers of this blog will know, the mathematical discipline of set theory, and with it the modern mathematical investigation of infinity, was almost single-handedly initiated by Georg Cantor, a mathematical giant of the late 19th century. Among the factors contributing to Cantor’s revolutionary achievements was his willingness to work with actual infinities, to consider infinite sets as completed entities that can be manipulated as mathematical objects in their own right. Two of the foundational ideas made possible by Cantor’s acceptance of actual infinity are:

• the existence of multiple infinite cardinalities;
• the concept of transfinite ordinal numbers.

It was realized quite early in the development of set theory, though, that a naïve conception of infinity and sets would lead to paradoxes. (Indeed, the existence of these paradoxes is one of the reasons for the reluctance of mathematicians to accept actual infinities and provided fodder for Cantor’s mathematical enemies.) Two of the most consequential of these paradoxes are Russell’s Paradox and the Burali-Forti Paradox.

Russell’s Paradox: Essentially, Russell’s Paradox asserts that there can be no set of all sets. For suppose that such a set exists (call it $S$). Then, by any reasonable formulation of set theory, we could form the set of all sets that do not contain themselves. Formally, this set, which we will call $R$, would be $\{x \in S \mid x \not\in x\}$. Now comes the crucial question: can $R$ be a member of itself? Either answer leads to contradiction: if $R \in R$, then $R \not\in R$, and if $R \not\in R$, then $R \in R$. Thus, there can be no set of all sets.

Burali-Forti Paradox: The Burali-Forti Paradox goes even further than Russell’s Paradox: not only can the set of all sets not exist, but the set of all ordinal numbers cannot exist either. For suppose that $T$ is the set of all ordinal numbers. Recall that an ordinal number is, essentially, the order type of a well-ordered set. Recall also that the class of ordinal numbers is well-ordered: any set of ordinals has a least element. But this implies that $T$ itself is a well-ordered set, but, since $T$ contains all of the ordinals, its order type must be larger than all the ordinals numbers. This is a contradiction.

The collection of all sets and the collection of all ordinal numbers are therefore not sets. But it seems that we are nonetheless able to conceive of them, to reason about them. They seem to have some form of reality. So what are they?

The standard response in modern set theory is simply to say that these collections are what is known as proper classes. They are collections that are “too large” to be sets, that are “too large” to be manipulated in ways that sets are manipulated. This has served us well, to this point banishing paradox from the subject. Cantor’s response was similar but was imbued with something more. Consider this excerpt from a letter sent by Cantor to the English mathematician Grace Chisholm Young:

I have never assumed a “Genus Supremum” of the actual infinite. Quite on the contrary I have proved that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”, which is by many called “God”.

For Cantor, infinity comes in not two but three varieties: the potential infinity of limits, sequences, and series; the transfinite infinity of the infinite cardinal and ordinal numbers; and the absolute infinity, which transcends mathematics and human comprehension, and which is identified with God.

Much has been made of the possibility that Georg Cantor’s family had Jewish roots and that Cantor’s conceptions of infinity and mathematical philosophy were influenced by Jewish mysticism and Kabbalah. On the former issue we will just say that there is evidence, from correspondence and genealogical records, pointing in both directions. On the latter, we will simply note that Cantor’s choice to denote infinite cardinals was ‘aleph’ (ℵ), the first letter of the Hebrew alphabet and a letter of importance in Kabbalah as the opening of the words Ein Sof (infinity, roughly) and Elohim (a name for God).

Also, here’s a little-known fact: the symbol that Cantor used to denote the class of all cardinal numbers (too big to be a set, of course) is ‘tav’ (ת), the last letter in the Hebrew alphabet and the representative, in Kabbalah, of perfection, of the synthesis of all that exists.

Another fascinating intersection between Cantor’s set theory and theology involves the Catholic Church. Long-time readers may recall that, the last time the Church appeared in our story, the mathematician involved was Galileo, and the relationship between the two could be described, at the risk of understatement, as antagonistic.

250 years later, the relationship between the Catholic Church and the day’s leading provocateur of the infinite would turn out decidedly differently. This was due in large part to Pope Leo XIII, who assumed the position in 1878 and, the following year, issued the encyclical Aeterni Patris. The encyclical called for the revival of the Scholastic philosophy of Thomas Aquinas. It sought to modernize the Church and to increase its interest and participation in scientific inquiry.

For, the investigation of facts and the contemplation of nature is not alone sufficient for their profitable exercise and advance; but, when facts have been established, it is necessary to rise and apply ourselves to the study of the nature of corporeal things, to inquire into the laws which govern them and the principles whence their order and varied unity and mutual attraction in diversity arise. To such investigations it is wonderful what force and light and aid the Scholastic philosophy, if judiciously taught, would bring.

-Leo XIII, Aeterni Patris

Neo-Scholastic scholars thus gained prominence in the Church, and the Church began engaging more with the cutting-edge scientific work of the day. Cantor’s brand new ideas about infinity were naturally of interest. Cantor, with religious tendencies of his own and upset by the skeptical or hostile reactions of most of his fellow mathematicians to his work, was himself eager to explain his ideas to the Church and ensure that they were properly understood. And thus began the remarkable correspondence between Cantor and the leading Catholic scholars and clergy of the day.

In any case it is necessary to undertake a serious examination of the latter question concerning the truth of the Transfinitum, for were I correct in asserting its truth in terms of the possibility of the Transfinitum, then there would be (without doubt) a certain danger of religious error for those of the opposite opinion, since: error circa creaturas redundat in falsam de Deo scientiam.

-letter from Cantor to Jeiler von Pfingsten, 1888

One of the issues that arose in Cantor’s correspondence with the Church involved the fundamental existence of Cantor’s transfinite numbers. (Note that I am in no way a theological scholar, so this account is probably grossly oversimplified.) A number of Catholic intellectuals were hesitant to accept the objective existence of Cantor’s transfinite numbers on the grounds that, as God is often theologically identified with the infinite, the acceptance of the existence of transfinite numbers would lead inevitably to Pantheism, a doctrine which was not only implicity but explicitly (by Pius IX in 1861) condemned by the Church. Cantor responded by emphasizing his distinction between the absolute infinity and the transfinite infinity, between an “Infinitum aeternum increatum sive Absolutum” and an “Infinitum creatum sive Transfinitum.” The former is reserved for God; the latter manifests itself in mathematics and in the universe.

The neo-Scholastic thinkers were largely convinced by Cantor’s distinction and came to accept many of his ideas about infinity. Cantor himself took great pride in this achievement, even going so far as, in a moment of mathematical self-doubt engendered by his inability to solve the Continuum Problem, to express the following remarkable sentiment in an 1894 letter to the French mathematician Charles Hermite:

Now I only 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 would have been able to with my probably weak mathematical powers through an exclusive occupation with mathematics.

P.S.: Cantorian set theory recently made a surprise appearance in the New York Times crossword puzzle:

Rex Parker was not happy!

Acknowledgement: The material about Cantor’s correspondence with the Catholic Church came largely from Joseph W. Dauben’s excellent paper, “Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite.”

Cover Image: Harald Sohlberg, “Winter Night in the Mountains,” 1901. Nasjonalgalleriet, Oslo.

# Surreal Numbers (Universal Structures III)

“You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”

-John Horton Conway