Circles 0

Life is a full circle, widening until it joins the circle motions of the infinite.

-Anaïs Nin

No shape has captured the human imagination quite like the circle has. Its perfect symmetry and constant radius stand in contrast to the messy variability of our everyday lives. We have inner circles, vicious circles, fairy circles, crop circles, family circles, circles of influence, the circle of life. Circles permeate our conception of time, as they provide the shape of our clocks, and when things return to their original configuration, we say that they have come full circle.

We have seen intimate connections between circles and infinity in many previous posts. The circle is the one-point compactification of the infinite straight line, which itself can be thought of as a circle with infinite radius. The process of circle inversion provides a useful duality between the finite region inside a circle and the infinite region outside. The Poincaré disk model provides an elegant finite setting for a decidedly infinite instantiation of hyperbolic geometry.

In the next few posts, we will be exploring circles more deliberately, through lenses of mathematics, philosophy, linguistics, art, and literature. We will think about circular definitions and closed timelike curves, about causality and the Liar Paradox.

These are for future weeks, though. For today, to lay the foundations and whet the appetite, please enjoy these three essential pieces of circle-related culture:

Although popularly every one called a Circle is deemed a Circle, yet among the better educated Classes it is known that no Circle is really a Circle, but only a Polygon with a very large number of very small sides. As the number of the sides increases, a Polygon approximates to a Circle; and, when the number is very great indeed, say for example three or four hundred, it is extremely difficult for the most delicate touch to feel any polygonal angles. Let me say rather, it would be difficult: for, as I have shown above, Recognition by Feeling is unknown among the highest society, and to feel a circle would be considered a most audacious insult. This habit of abstention from Feeling in the best society enables a Circle the more easily to sustain the veil of mystery in which, from his earliest years, he is wont to enwrap the exact nature of his Perimeter or Circumference. Three feet being the average Perimeter, it follows that, in a Polygon of three hundred sides, each side will be no more than the tenth part of an inch; and in a Polygon of six or seven hundred sides the sides are little larger than the diameter of a Spaceland pin-head. It is always assumed, by courtesy, that the Chief Circle for the time being has ten thousand sides.

Edwin A. Abbott, Flatland: A Romance of Many Dimensions

The Subtle Art of Go, or Finite Simulations of Infinite Games

Tatta hito-ban to
uchihajimeta wa
sakujitsu nari

Saying `just one game’
they began to play . . .
That was yesterday.

-Senryū (trans. William Pinckard)

A few weeks ago, I found myself in Virginia’s finest book store and made a delightful discovery: a newly published translation of A Short Treatise Inviting the Reader to Discover the Subtle Art of Go, by Pierre Lusson, Georges Perec, and Jacques Roubaud (two mathematicians and two poets, all associates of the French literary workshop Oulipo), originally published in France in 1969.

Go, of course, is a notoriously difficult and abstract board game, invented in China over 2500 years ago and further developed in Korea and Japan. After millennia of being largely unknown outside of East Asia, it has in the last century become popular throughout the world (the publication of this book played a significant role in introducing it to France) and has even been in the news recently, as computer programs using neural networks have defeated some of the best professional go players in highly publicized matches.

The stated goal of Lusson, Perec, and Roubaud’s book is to “[provide], in a clear, complete, and precise manner, the rules of the game of GO” and to “heighten interest in this game.” As a practical manual on the rules and basic tactics and strategy of the game, the modern reader can do much better with other books. As a literary meditation on play, on art, and on infinity, it is dazzling. It is this latter aspect of the book that I want to touch on here today.


A theme running throughout the book is the idea that the practice of go is akin to a journey into infinity, and this theme is expressed both with respect to one’s relationship with other players and with one’s relationship to the game itself.

A joy of learning any game is developing relationships and rivalries with other players, and this is especially true with go, for two main reasons. First, an individual match is not simply won or lost but rather is won or lost by a certain number of points. Second, there is a robust handicapping system whereby a substantially weaker player can legitimately compete with a stronger player, in a match of interest to both players, by placing a specific number of pieces on specific points of the board before the first move. Through these two mechanisms, a rich and rewarding go relationship can thus develop, even between players of unequal skill, over not just one match but a progression of matches, during which handicaps can be finely calibrated and can, indeed, change over time, as the players learn more about each other’s play and about the game in general.

As such, GO, at its limits, constitutes the best finite simulation of an infinite game. The two adversaries battle on the Goban the way cyclists pursue each other in a velodrome.

This is not as much the case with, say, chess, in which the facts that the result of a game is purely binary and that handicap systems are clumsier and more substantially alter the character of the game mean that matches between players of unequal skill will frequently be frustrating for the weaker and boring for the stronger.


It is a cliché to say that one can never truly master a subject, that there is always more to learn. But the richness of go makes it especially true here, and, in a sense, quantifiably so. The number of legal positions in go is more than 2 \times 10^{170}. This is a truly astounding number, dwarfing the estimated number of atoms in the observable universe (10^{80}) or the estimated number of legal positions in chess (a piddling 10^{43}). The possibilities in go are, for all intents and purposes, infinite. No matter how much one learns, one knows essentially nothing.

Crucially, though, there is a well-developed hierarchy through which one progresses and by which one can measure one’s skill, even if remains practically zero when taking a wider view. Lusson, Perec, and Roubaud write about this better than I could in the following two excerpts, so let us simply listen to them:

The genius of GO stems precisely from what it hides as well as what it reveals, at any moment, at any level, in its different, hierarchized mysteries whose progressive mastery transforms the game every time:

A garden of bifurcating pathways, a labyrinth, the game of Babel, each step forward is decisive and each step forward is useless: we will never have finished learning..

(Note the surely intentional nod to Borges in the last sentence above).

From a beginner to a classified amateur in the bottom ranks of kyu, a player can rise to the top kyus and then, one by one, climb the six amateur echelons, approaching (but only approaching) the inaccessible regions where the true players reign, the professionals…

In this last excerpt, we hear echoes of a mathematical concept from set theory, my personal area of expertise. The authors temper the reader’s (and their own) dreams of go mastery by noting that, no matter how high an amateur go player may ascend in their study of go, they will still never reach the “inaccessible” realm of the true masters of the game. These professionals also inhabit a hierarchy, but it is a separate hierarchy, visible but unreachable from below.

This calls to mind the concept of an inaccessible cardinal, which is an uncountable cardinal number \kappa that cannot be reached from below through the standard procedures of climbing to the next cardinal, applying cardinal exponentiation, or taking unions of a small number of small sets. (More formally, \kappa is (strongly) inaccessible if it is regular, uncountable, and, for all \lambda < \kappa, we have 2^\lambda < \kappa.)

It cannot be proven that inaccessible cardinals exist or are even consistent, and the assumption of the consistency of such cardinals has significant implications for what one can prove (see a previous post for more information about inaccessible and other “large cardinals”). On the simplest descriptive level, an inaccessible cardinal divides the hierarchy of infinite cardinals into two realms that cannot communicate via the standard operations of arithmetic: those above and those below.

(A modern version of the book would surely posit a third separate region of hierarchies in go: that of the neural networks that with stunning swiftness have become stronger than the strongest professionals.)


So why bother? If one can spend one’s whole life studying go and still hardly understand the game, if one can develop to one’s full potential and still be nowhere close to the level of professional players, let alone the newly ascendant artificial intelligences, then why start at all?

The authors consider this question, but ultimately they reject its premises. The study of go is not worthwhile in spite of the fact that it is an “infinite pathway.” It is worthwhile because of it.

And this clearly has implications outside of go. Why devote much of my life to mathematical research if I can never know more than a miniscule portion of what remains undiscovered? Why write if it is physically impossible to write more than about 10,000,000 words in a life, and if everything I may write is already contained in Borges’ Library of Babel anyway? Perhaps because the best life is a finite simulation of an infinite life.

Only one activity exists to which GO may be reasonably compared.

We will have understood it is writing.


PS: We have had occasion to mention chess and its relation to the infinite in a previous post. One of the joys of A Short Treatise… is the exaggerated contempt expressed by its authors for the game of chess. We end by offering you just a taste:

Good news!

One of the best European players, Zoran Mutabzija, abandoned chess, to which he had devoted himself since the age of four, as soon as someone taught him GO!

In related news.

We just received a detailed report concerning a certain Louis A. caught in the act of robbing a gas station attendant on the Nationale 6. According to the report, whose source and information cannot be called into question, Louis A. is a notorious chess player.

Non-Euclidean Geometry and a Goldfish

We’ll be back, probably next week, with a new post about common knowledge. Today, though, a couple of links.

First, coming off of our recent posts about non-Euclidean geometry, a delightful 1970s BBC program on the subject:

Second, a poem, by Sara Baume and published at Granta, about a goldfish.

Enjoy!

 

Independence

I am no bird; and no net ensnares me: I am a free human being with an independent will.

-Charlotte Brontë, Jane Eyre

An object is independent from others if it is, in some meaningful way, outside of their area of influence. If it has some meaningful measure of self-determination. Independence is important. Nations have gone to war to obtain independence from other nations or empires. Adolescents go through rebellious periods, yearning for independence from parents or other authority figures. Though perhaps less immediately exciting, notions of independence permeate mathematics, as well. Viewed in the right light, they can even be seen as direct analogues of the more familiar notions considered above: in various mathematical structures, there is often a natural way of defining an area of influence of an element or subset of the structure. A different element is then independent of this element or subset if it is outside its area of influence. Such notions have proven to be of central importance in a wide variety of mathematical contexts. Today, in anticipation of some deeper dives in future posts, we take a brief look at a few prominent examples.

Graph Independence: Recall that a graph is a pair G = (V,E), where V is a set of vertices and E is a set of edges between these vertices. If u \in V is a vertex, then its neighborhood is the set of all vertices that are connected to u in the graph, i.e., \{v \in V \mid \{u,v\} \in E\}. One could naturally consider a vertex’s area of influence in the graph G to consist of the vertex itself together with all of its neighbors. With this viewpoint, we can say that a vertex u \in V is independent from a subset A \subseteq V if, for all v \in A, u is not in the neighborhood of v, i.e., u is not in the area of influence of any element of A. Similarly, we may say that a set A \subseteq V of vertices is independent if each element of A is independent from the rest of the elements of A, i.e., if each v \in A is independent from A \setminus \{v\}.

Independent_set_graph
The blue vertices form a (maximum) independent set in this graph. By Life of Riley – Own work, GFDL

 

Fun Fact: In computer science, there are a number of interesting computational problems involving independent sets in graphs. These problems are often quite difficult; for example, the maximum independent set problem, in which one is given a graph and must produce an independent set of maximum size, is known to be NP-hard.

Linear Independence: Let n be a natural number, and consider the real n-dimensional Euclidean space \mathbb{R}^n, which consists of all n-tuples of real numbers. Given \vec{u} = (u_1,\ldots,u_n) and \vec{v} = (v_1, \ldots, v_n) in \mathbb{R}^n and a real number r \in \mathbb{R}, we can define the elements \vec{u} + \vec{v} and r\vec{u} in \mathbb{R}^n as follows:

\vec{u} + \vec{v} = (u_1 + v_1, \ldots, u_n + v_n)

r\vec{u} = (ru_1, \ldots, ru_n).

(In this way, \mathbb{R}^n becomes what is known as a vector space over \mathbb{R}). Given a subset A \subseteq \mathbb{R}^n, the natural way to think about its linear area of influence is as \mathrm{span}(A), which is equal to all n-tuples which are of the form

r_1\vec{v}_1 + \ldots + r_k\vec{v}_k,

where k is a natural number, r_1, \ldots, r_k are real numbers, and \vec{v}_1, \ldots, \vec{v}_k are elements of A.

In this way, we say that an n-tuple \vec{u} \in \mathbb{R}^n is linearly independent from a set A \subseteq \mathbb{R}^n if \vec{u} is not in \mathrm{span}(A). A set A is linearly independent if each element \vec{u} of A is not in \mathrm{span}(A \setminus \{\vec{u}\}), i.e., if each element of A is linearly independent from the set formed by removing that element from A. It is a nice exercise to show that every linearly independent subset of \mathbb{R}^n has size at most n and is maximal if and only if it has size equal to n.

Fun Fact: Stay tuned until the end of the post!

Thou of an independent mind,
With soul resolv’d, with soul resign’d;
Prepar’d Power’s proudest frown to brave,
Who wilt not be, nor have a slave;
Virtue alone who dost revere,
Thy own reproach alone dost fear—
Approach this shrine, and worship here.

-Robert Burns, “Inscription for an Altar of Independence”

Algebraic Independence: If A is a set of real numbers, then one can say that its algebraic area of influence (over \mathbb{Q}, the set of rational numbers), is the set of all real roots of polynomial equations with coefficients in A \cup \mathbb{Q}, i.e., the set of all real numbers that are solutions to equations of the form:

r_kx^k + \ldots + r_1x + r_0 = 0,

where k is a natural number and r_0, \ldots, r_k are elements of A \cup \mathbb{Q}. With this definition, a real number s is algebraically independent (over \mathbb{Q}) from a set A \subseteq \mathbb{R} if s is not the root of a polynomial equation with coefficients in A \cup \mathbb{Q}. A set A \subseteq \mathbb{R} is algebraically independent (over \mathbb{Q}) if each element s \in A is algebraically independent from A \setminus \{s\}.

Fun Fact: Note that a 1-element set \{s\} is algebraically independent over \mathbb{Q} if and only if it is transcendental, i.e., is not the root of a polynomial with rational coefficients. \pi and e are famously both transcendental numbers, yet it is still unknown whether the 2-element set \{\pi, e\} is algebraically independent over \mathbb{Q}. It is not even known if \pi + e is irrational!

Logical Independence: Let T be a consistent set of axioms, i.e., a set of sentences from which one cannot derive a contradiction. We can say that the logical area of influence of T is the set of sentences that can be proven from T, together with their negations. In other words, it is the set of sentences which, if one takes the sentences in T as axioms, can be proven either true or false. A sentence \phi is then logically independent from T if neither \phi nor its negation can be proven from the sentences in T.

Logical independence is naturally of great importance in the study of the foundations of mathematics. Much of modern set theory, and much of my personal mathematical research, involves statements that are independent from the Zermelo-Fraenkel Axioms with Choice (ZFC), which is a prominent set of axioms for set theory and indeed for all of mathematics. These are statements, then, that in our predominant mathematical framework can neither be proven true nor proven false. The most well-known of these is the Continuum Hypothesis (CH), which, in one of its formulations, is the statement that there are no infinite cardinalities strictly between the cardinality of the set of natural numbers and the cardinality of the set of real numbers. To prove that CH is independent from ZFC, one both produces a mathematical structure that satisfies ZFC and in which CH is true (which Kurt Gödel did in 1938) and produces a mathematical structure that satisfies ZFC and in which CH is false (which Paul Cohen did in 1963). Since Cohen’s result in 1963, a great number of natural mathematical statements have been proven to be independent from ZFC.

In our next post, we will consider a logical independence phenomenon of a somewhat simpler nature: the independence of Euclid’s parallel postulate from Euclid’s four other axioms for plane geometry, which will lead us to considerations of exotic non-Euclidean geometries.

Fun Fact: In the setting of general vector spaces, which generalize the vector spaces \mathbb{R}^n from the above discussion of linear independence, a basis is a linearly independent set whose span (what we referred to as its linear area of influence) is the entire vector space. A basis for \mathbb{R}^n is thus any linearly independent set of size n. Using the Axiom of Choice, one can prove that every vector space has a basis. However, there are models of ZF (i.e., the Zermelo-Fraenkel Axioms without Choice) in which there are (infinite-dimensional) vector spaces without a basis. Thus, the statement, “Every vector space has a basis,” is logically independent from ZF.

Solitude is independence.

-Hermann Hesse, Steppenwolf

Infinite Cities: Calvino and Chess

…trying to master chess is like trying to master the infinite, and the psychological consequences can be transcendent or terrifying.

I’ve been busy traveling lately, so, in lieu of a new post, I’m just giving you a couple of literary links today.

First, a piece at The Millions about depictions of chess in literature. Chess, like the infinite, is often depicted in the popular imagination as an object of obsession, a pursuit that can lead either to transcendence or madness. This is often a little overwrought, but certainly entertaining.

“Plunging Into the Infinite: How Literature Captures the Essence of Chess” by Matthew James Seidel

Next, at LitHub, a selection of artwork inspired by Italo Calvino’s wonderful Invisible Cities, a novel exploring the infinite permutability of the urban environment.

Art Inpired by Italo Calvino’s Invisible Cities” by Emily Temple


Cover image: “Zenobia” by Maria Monsonet

Marienbad

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

-Last Year at Marienbad

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…

Last Year at Marienbad


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…

The Simurgh (Universal Structures I)

To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour

-William Blake, “Auguries of Innocence”

In Persian mythology, the Simurgh is a bird that lives in the mountains of Alborz. Sometimes she has the head or body of a dog, sometimes of a human. She has witnessed the destruction of the world three times. The wind of her beating wings is responsible for scattering seeds from the Tree of Life, creating all plants in the world.

The Simurgh is, in some tellings, the archetype of all birds. Her name resembles the Persian phrase si murg, meaning “thirty birds.”

In The Conference of the Birds, Farid ud-Din Attar’s 12th-century masterpiece, the birds of the world undertake a journey to find the Simurgh. And they succeed.

Their life came from that close, insistent sun
And in its vivid rays they shone as one.
There in the Simorgh’s radiant face they saw
Themselves, the Simorgh of the world – with awe
They gazed, and dared at last to comprehend
They were the Simorgh and the journey’s end.
They see the Simorgh – at themselves they stare,
And see a second Simorgh standing there;
They look at both and see the two are one,
That this is that, that this, the goal is won.

-Farid ud-Din Attar, The Conference of the Birds

The Simurgh is a bird that contains all birds. She is a universal bird.

Unsurprisingly, the Simurgh shows up a number of times in the works of Jorge Luis Borges, in both his short stories and his essays. One reference appears in the masterful story, “The Aleph,” a particularly rich and dense work which you should certainly read for yourself.

“The Aleph” is partly about how we create our own worlds, how we approximate the unknowable universe within our lives and our art. The narrator of the story, also named Borges, grieving the loss of his beloved Beatriz, pays repeated visits to the home of her father and her cousin, the poet Carlos Argentino Daneri. On one of these visits, Carlos Argentino takes Borges to his basement to show him the source of his poetry, the titular Aleph, a single point that contains the universe.

On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a backyard of Soler Street the same tiles that thirty years before I’d seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand; I saw a woman in Inverness whom I shall never forget; I saw her tangled hair, her tall figure, I saw the cancer in her breast; I saw a ring of baked mud in a sidewalk, where before there had been a tree; I saw a summer house in Adrogué and a copy of the first English translation of Pliny — Philemon Holland’s — and all at the same time saw each letter on each page (as a boy, I used to marvel that the letters in a closed book did not get scrambled and lost overnight); I saw a sunset in Querétaro that seemed to reflect the colour of a rose in Bengal; I saw my empty bedroom; I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters, which Beatriz had written to Carlos Argentino; I saw a monument I worshipped in the Chacarita cemetery; I saw the rotted dust and bones that had once deliciously been Beatriz Viterbo; I saw the circulation of my own dark blood; I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe.

-Jorge Luis Borges, “The Aleph”

Aleph (\aleph) is of course the letter chosen by Georg Cantor to represent transfinite cardinals and the first letter of the Hebrew alphabet. It plays a special role in Kabbalah as the first letter in “Ein Sof,” roughly translated as “infinity,” and in “Elohim,” one of the names of the Hebrew god. We will surely return to these matters.

This is the first installment in a mini-series on what we will call “universal structures,” objects that contain all other objects of their type. We will continue to look at examples from literature and religion, and will delve into the existence of universal structures in mathematics, a topic which continues to drive cutting-edge research to this day. Next week, we will look at a particular universal structure in mathematics, the wonderfully named “random graph.” I hope you will join us.