# Cantor v. Crank

No one shall expel us from the paradise that Cantor has created for us.

-David Hilbert

To the extent that a mathematical theorem can be considered controversial, Cantor’s Theorem has historically been quite a controversial statement. The theorem, which, we remind you, states that, for any set $A$, the power set $\mathcal{P}(A)$, which consists of all subsets of $A$, is strictly larger than $A$ (or, in a commonly cited special case, the set of all real numbers, $\mathbb{R}$, is strictly larger than the set of all natural numbers, $\mathbb{N}$), was attacked by a number of Cantor’s illustrious contemporaries, among them Kronecker and Poincaré, who objected to Cantor’s manipulation of infinite sets as mathematical objects in their own right.

Cantor’s work also had prominent defenders, though, most notably David Hilbert, whose quote in the header of this post has become among the most iconic proclamations in modern mathematics. As mathematicians settled into the twentieth century, Cantor’s Theorem and the field of set theory that it helped establish became widely accepted in the community. Today, it is safe to say that Cantor’s Theorem is uncontroversial among the vast majority of mathematicians. Resistance to the theorem or to its extrapolations has not died out, though; it remains present in certain pockets of mathematics, such as ultrafinitism, in some philosophy departments, and among amateur mathematicians. Today, we will look at an example from this last category.

Let us turn now to Dilworth’s paper. The impetus for the work appears to be Dilworth’s displeasure with the Banach-Tarski Paradox, which is a theorem stating roughly that one may take a three-dimensional ball, decompose it into finitely many pieces, and move these pieces around in space so that they form two complete balls, each of the same size as the original ball. On its face, this seems absurd. We have seemingly doubled the amount of stuff we have! Of course, closer inspection dispels the aura of paradox here. The pieces needed to perform the construction are infinitely complicated and non-definable and certainly could not be used in some brilliant gold-proliferating get-rich-quick scheme in the real world. And, when we think about it, there are just as many points in two balls as there are in one (continuum-many, in both cases), so maybe this theorem isn’t so surprising after all?

Dilworth was apparently not convinced, though, He found the Banach-Tarski Paradox to be clearly wrong, and since the result depended on the techniques of the then-young field of set theory, and since set theory was born out of Cantor’s Theorem, there must be something wrong with Cantor’s Theorem.

The paper makes for alternately entertaining and maddening reading. His reasoning is often muddled, but Dilworth does sometimes have a way with words and, though he perhaps goes a bit too far in this direction, the writing has character and flair in a way that I sometimes wish more technical mathematical writing did. For example, when describing Cantor’s diagonal argument and what he sees as its undeserved acceptance by the mathematical community, Dilworth writes,

Historically and up to this date, he has won. The horrendous “alephs” of his endless infinities thunder through the evening skies of academe “with hooves of steel”, as the songwriter put it.

There is a decided lack of Johnny Cash references in today’s mathematical writing. (And yes, I know that Cash’s version of this song was released in 1979 and therefore can’t be the version referenced in this 1974 paper…)

The paper also ends memorably, with an assertion of a conspiracy among mathematicians to cover up the fallacies in Cantor’s proof, followed by a final, simple, cryptic directive:

Remember the spheres.

In substance, though, Dilworth’s writing is at first incomprehensible, and when its meaning has been at least partially uncovered, his error is immediately evident. In attempting to show that the size of the set of all real numbers is exactly the same as the size of the set of all natural numbers, he is not in fact considering all real numbers, but just those whose decimal representations only have finitely many digits (or, in a different reading of his argument, only those whose decimal representations end in endlessly repeating patterns; or, in a still different reading, only those which are definable by a finite sentence). And he is correct in the sense that this subset of the real numbers does have the same size as the set of natural numbers. But by restricting his vision to this subset, he is excluding the vast majority of real numbers.

This is a mistake that could have been pointed out to Dilworth by any mathematician or any sufficiently advanced student of mathematics. And it apparently was pointed out to him on multiple occasions, for Dilworth includes accounts of his interactions with mathematics professors in the paper. For example, in one passage, Dilworth acknowledges the obvious objection to his argument, namely that he is not considering all of the real numbers, but dismisses it out of hand without argument. In another passage, when discussing Cantor’s claim that, if one tries to pair the real numbers off in a one-to-one fashion with the integers, the integers will necessarily “become exhausted” before the process is complete, he relates the following incident:

“Yes sir,” the head of the mathematics department of a Univ. of Illinois section said matter-of-factly to my face, “The integers will become exhausted.” Believe it or not, Georg Cantor made these remarkable claims stick with the world’s mathematicians of his time, and they stick unto this day. The effects of the Cantorian grip on the professional mind have to be experienced to be believed.

It is difficult to judge how much of this disconnect between Dilworth and the professional mathematicians is due to Dilworth’s own stubbornness and unwillingness to admit his error and how much of it is due to the mathematicians’ inability to clearly articulate somewhat complex mathematical ideas to a non-mathematician (something that, as I have discovered through writing this blog, is often quite difficult to do). In any event, something went wrong in this case, which is unfortunate. It is encouraging, and all too rare, to see non-mathematicians become enthusiastic about and engaged in mathematics, so it is always a shame to see them go astray.

Let us take a slight detour here to make an observation about education and training. I don’t think it will be particularly controversial to say that the primary aim of mathematics education, particularly at the graduate level, is not the learning of calculation techniques or the statements and proofs of theorems (though this will of course come), but rather a training in the actual practice of doing mathematics, of thinking mathematically, and of being able to communicate mathematical ideas with other mathematicians. In my experience, competence in this practice of mathematics is much harder to come by than mathematical knowledge, perhaps almost impossible to come by without a dedicated mentor, a role typically filled by one’s PhD advisor. It is precisely this conversance with mathematical practice that, at least in modern times, seems to be almost a prerequisite to making any sort of substantial contribution to mathematical knowledge, and it seems to be precisely what Dilworth was lacking.

There is a common and mildly insulting label that often gets applied to people such as William Dilworth: crank. In 1992, Underwood Dudley, then a professor at DePauw University in Indiana, published a comprehensive and encyclopedic compendium of crankery, Mathematical Cranks. In the introduction to the book, Dudley clarifies what he means by the word.

They’re not nuts. Well, a few are, but most aren’t. A lot of them are amateurs — mathematical amateurs who don’t know much mathematics but like to work on mathematical problems. Sometimes, when you can’t convince them that they haven’t done what they thought they’ve done, they turn into cranks, but cranks aren’t nuts, they’re just people who have a blind spot in one direction.

One chapter of Dudley’s book is devoted to cranks attacking Cantor’s Theorem. We meet a person who found five separate mistakes in Cantor’s proof, a person who held that Cantor’s conception of “number” was incorrect, and people who dispute the existence of mathematical infinity. At the end of the chapter, we meet William Dilworth. After running through the problems with Dilworth’s paper, Dudley concludes the chapter with the following sentences (Dudley does not identify the state responsible for publishing Dilworth’s paper, and also refers to Dilworth as W.D.):

His article reads as if it is by someone convinced, whose mind is not going to be changed by anything. It is by, in two words, a crank, and it is no credit to the state of X.

Dilworth was not happy about his inclusion in Mathematical Cranks. As an engineer, an academic outsider, it was hard enough for him to publish his ideas and to be taken seriously by the mathematical establishment. After being publicly labeled a crank, it would become nearly impossible. And so, in 1995, he sued Underwood Dudley for defamation.

The suit was dismissed by a district judge for “failure to state a claim.” More precisely, the judge held that the word “crank” is incapable of being defamatory; it is mere “rhetorical hyperbole.” Dilworth appealed this ruling, and the case ended up in the Seventh Circuit Court of Appeals, before none other than Judge Richard Posner, the “most cited legal scholar of the 20th century” and one of the most prominent modern American judges not to have been appointed to the Supreme Court. And if this whole story gave us nothing else, it would have been worth it solely for Posner’s remarkable decision.

Posner begins by noting that it is crucial for Dilworth’s claim to establish that Dudley acted with “actual malice” in calling Dilworth a crank:

The allegation of actual malice is necessary because the plaintiff is a public figure. Not, it is true, a “public figure” in the lay sense of the term. Dilworth is an obscure engineer. But anyone who publishes becomes a public figure in the world bounded by the readership of the literature to which he has contributed.

(It is good to know that I am, at least legally speaking, a public figure!)

Posner then goes on to consider the district judge’s ruling that “crank” cannot be defamatory because it is mere “rhetorical hyperbole.” He begins by considering past cases that deal with precisely this issue.

Among the terms or epithets that have been held (all in the cases we’ve cited) to be incapable of defaming because they are mere hyperbole rather than falsifiable assertions of discreditable fact are “scab,” “traitor,” “amoral,” “scam,” “fake,” “phony,” “a snake-oil job,” “he’s dealing with half a deck,” and “lazy, stupid, crap-shooting, chicken-stealing idiot.”

These terms, Posner asserts, have both literal and figurative usages. Figurative usages cannot be defamatory; they are mere “rhetorical hyperbole.” Literal usages, on the other hand, that make real factual claims, can be defamatory if false. Decisions on the defamatory nature of these terms, therefore, hinge on a judgment as to whether they are intended literally or figuratively. Posner then goes on to consider whether “crank” falls into this category.

“Crank” might seem the same type of word, but we think not. A crank is a person inexplicably obsessed by an obviously unsound idea — a person with a bee in his bonnet. To call a person a crank is to say that because of some quirk of temperament he is wasting his time pursuing a line of thought that is plainly without merit or promise. An example of a math crank would be someone who spent his time trying to square the circle. To call a person a crank is basically just a colorful and insulting way of expressing disagreement with his master idea, and it therefore belongs to the language of controversy rather than to the language of defamation.

And, before affirming the district judge’s opinion, Posner includes this sentence designed to flatter set theorists everywhere.

As we emphasized in the Underwager case, judges are not well equipped to resolve academic controversies, of which a controversy over Cantor’s diagonal process is a daunting illustration…

Acknowledgments: Our thanks to Peter Smith and his blog, Logic Matters, where we first learned of this story. Cover image from the Bad Postcards Tumblr page.

# L’escalier du Diable

Welcome one, welcome all to the Point at Infinity sideshow, where today we present a tantalizing and diabolical selection of musical and mathematical curiosities. Just watch your step; these stairs can be a bit tricky.

A few months ago, you may recall, we published two posts about the Shepard tone and the Risset rhythm, aural illusions in which a tone or rhythm seems to perpetually rise or fall in pitch or in tempo but is actually repeating the same pattern over and over again, the musical equivalents of Penrose stairs.

To accompany the posts we created some sound samples so the readers could hear the illusions themselves. A couple of weeks ago, one of these samples was used in an internet radio program on audio paradoxes released by Eat This Radio, paired with some work of Jean-Claude Risset. The entire program is really excellent, ranging from a piece by J.S. Bach to mid-twentieth century audio experiments to modern electronic music, and I encourage all of you to listen to it.

One of the pieces in the radio program is a piano étude written by György Ligeti in the late twentieth century. The étude is named L’escalier du diable, or The Devil’s Staircase, and its repeated ascents of the keyboard have a striking resonance with the never-ending ascent of the Shepard tone.

The Devil’s Staircase is also the colloquial name given to a particular mathematical function introduced by Georg Cantor in the 1880s. It is a function defined on the set of real numbers between 0 and 1 and taking values in the same interval, and it has some quite curious properties. Before we discuss it, let’s take a look at (an approximation to) the graph of the function.

To appreciate the strangeness of this function, let us recall some definitions regarding functions of real numbers. Very roughly speaking, a function is called continuous if it has no sudden jumps, or if its graph can be drawn without lifting the pencil from the page. Continuous functions satisfy a number of nice properties, such as the intermediate value theorem.

The derivative of a function at a given point of its domain, if it exists, measures the rate of change of the function at that point. If the x-axis measures time and the y-axis measures the position of an object along some one-dimensional track, then the derivative can be thought of as the velocity of that object. If a function is differentiable at a point (i.e., if its derivative exists there) then it must be continuous at that point, but the converse is not necessarily true. (For example, if the graph of a function has a sharp corner at a point, then the function cannot be differentiable there.)

Let’s think about what it means for a function to have a derivative of 0 at a point. It means that, at that point, the rate of change of the function has vanished. It means that, if we zoom in sufficiently close to that point, the function should look like a constant function. Its graph should look like a horizontal line. What would it mean for a function to have a derivative of 0 almost everywhere? (Here “almost everywhere” is a technical term (which I’m not going to define) and not just me being vague.) One might think that this must imply that the function is a constant function. At almost every point in its domain, the rate of change of the function is 0, so how can the value of the function change?

One will quickly discover that this is not quite right. Consider the function defined on the real numbers whose value is 0 at all negative numbers and 1 at all non-negative numbers.

This function has derivative 0 everywhere except at 0 itself, and yet it increases from 0 to 1. It does this quite easily by being discontinuous at 0, which, in hindsight, seems sort of like cheating. So what if we also require our function to be continuous? Now we need more exotic examples, and this is where the Devil’s Staircase comes in, for the Devil’s Staircase is a continuous function, it is differentiable almost everywhere, it has a  derivative of 0 wherever its derivative is defined, and yet it still manages to increase from 0 to 1. Wild!

What is the Devil’s Staircase exactly? I’ll give two different definitions. The first proceeds via an iterative construction. Start with the function $f_0(x) = x$. Its graph, between 0 and 1, is simply a straight line segment increasing from (0,0) to (1,1). Now, look at the midpoint of this increasing line segment, and draw a horizontal line segment centered there whose length is 1/3 of the horizontal line of the original increasing segment. Now connect the ends of this line segment via straight lines to (0,0) and (1,1). This new curve is the graph of a function that we call $f_1$. It consists of two increasing line segments with one horizontal line segment between them. Now repeat the process that took us from $f_0$ to $f_1$ on each of these increasing line segments, and let $f_2$ be the function whose graph is the result. Continue in this manner, constructing $f_n$ for every natural number $n$.

It turns out that, as $n$ goes to infinity, the sequence of functions $\langle f_n \mid n \in \mathbb{N} \rangle$ converges (uniformly) to a single function. This function is the Devil’s Staircase.

A more direct but also more opaque definition is as follows: Given a real number $x$ between 0 and 1, first express $x$ in base 3 (i.e., using only 0s, 1s, and 2s). If this base 3 representation contains a 1, then replace every digit after the first 1 with a 0. Next, replace all 2s with 1s. The result has only 0s and 1s, so we can interpret it as a binary (i.e., base 2) number, and we let $f(x)$ be this value. Then the function $f$ defined in this manner is the Devil’s Staircase. Play around with this definition, and you might get a feel for what it’s doing.

And now, on our way out, some musical addenda. An encore, if you will. First, after making the Risset rhythms for the aforementioned post, I did some further coding and wrote a little program that can take any short audio snippet and make a Risset rhythm out of it. Here’s an example, first accelerating and then decelerating, using a bit from a Schubert piano trio.

You may recognize the sample from the soundtrack to Barry Lyndon.

Finally, I can’t help but include here one of my favorite pieces by Ligeti, Poema sinfónico para 100 Metrónomos.

Cover image: Devil’s Staircase Wilderness, Oregon, USA

# 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.

# 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.

# Measuring III: Paris

The Setting:

Fin de siècle Paris. The 1900 International Congress of Mathematicians takes place in the city. David Hilbert gives the opening address, laying out his vision for the future of mathematics in a seminal list of 23 unsolved problems. First on the list is Cantor’s Continuum Hypothesis, the most prominent problem in the fledgling field of set theory. Motivated by the Congress, Hilbert’s speech, and a venerable professor receptive to Cantor’s radical ideas, three brilliant young French mathematicians make sudden, bold advances in set theory and analysis, changing mathematics forever in just a few short years.

The characters:

Camille Jordan. A respected Parisian mathematician and professor. Introduces Cantorian set theory to Paris through his lectures. Makes substantial progress in measure theory by extensively studying a finitely-additive measure on certain sets of real numbers that comes to be known as the Jordan measure. Known for his unorthodox notational decisions.

Émile Borel. Doctoral thesis in 1898. Mathematician, socialite, politician. Member of the French National Assembly (1924-1936), Minister of Marine in the cabinet of Paul Painlevé (1925) (also a mathematician), member of the French Resistance during World War II. Introduces the class of Borel sets of real numbers and shows that a measure can be defined on them. Provides refuge to Marie Curie during her scandal of 1911. Becomes frustrated by set theory. Stops doing mathematics in 1914.

René-Louis Baire. Doctoral thesis in 1899. Early student of Borel. Grows up in a poor Parisian family, suffers his entire life from delicate health and agoraphobia. Feels the world treats him unfairly, does poorly on oral exams, is given teaching posts beneath his ability. Defines a new hierarchy of functions that goes beyond the continuous functions that had predominated in mathematics. Proves what becomes known as the Baire Category Theorem, one of the seminal theorems in general topology. Breakdowns in health lead him to leave mathematics around 1914.

Henri Lebesgue. Doctoral thesis in 1902: “Integral, Length, Area” contains many breakthroughs. Also an early student of Borel. Develops the theory of Lebesgue measure and Lebesgue integration. Lebesgue measure is a countably additive measure, extending the Jordan measure and forming the foundation for modern analysis. Strained relations develop between Lebesgue and Borel, his doctoral supervisor, over a number of issues, including the Dreyfus Affair and competition over priority for ideas in measure theory.

The Axiom of Choice. Introduced in the early twentieth century and instantly controversial for its assertion of the existence of sets that cannot be explicitly constructed. The young French mathematicians, who so readily adopted Cantor’s ideas on infinity, feel that the Axiom of Choice goes too far and retreat from their earlier explorations, and, in some cases, even from mathematics. Set theory, seemingly going against the strict rationalist tradition of the Enlightenment that prevails in France at the time, moves east, to Poland and Russia.

Selected Quotes:

Higher infinities … have a whiff of form without matter, which is repugnant to the French spirit.

-Henri Poincaré, rival of David Hilbert and opponent of set theory

Like many of the young mathematicians, I had been immediately captivated by the Cantorian theory; I don’t regret it in the least, for that is one mental exercise that truly opens up the mind.

-Borel

Such reasoning does not belong to mathematics.

-Borel, on the Axiom of Choice

I will try never to speak of a function without defining it effectively; I take in this way a very similar point of view to Borel … An object is defined or given when one has said a finite number of words applying to this object and only to this one; that is when one has named a characteristic property of the object.

-Lebesgue

Preview of Next Act

Cracow, Poland, 1916: Hugo Steinhaus, fresh from his graduation in Göttingen, walks through a park and overhears a stranger say the words, “Lebesgue integral.” He introduces himself to said stranger, who happens to be Stefan Banach. The two go on to found the great Lwów school of mathematics, which revolutionizes set theory, topology, and analysis.

Credits

Quotes sourced from the fascinating book, Naming Infinity, by Loren Graham and Jean-Michel Kantor.

Cover image: Paris Street; Rainy Day by Gustave Caillebotte

# The (Ultra)finitists

Infinity is a fascinating and seductive topic, but it is also a contentious one and, throughout intellectual history, has been attacked on religious, philosophical, and practical grounds. Today, we turn away from the vastness of infinity and toward those who deny its existence.

Consider the Pythagoreans, followers of the famous 6th century BC Greek philosopher. According to the Pythagoreans, numbers are the first of all beings the “dominant and self-produced bond of the eternal continuance of things.” Pythagoreans saw numbers in everything, from the motions of the planets to musical harmony. But they saw only rational numbers, i.e. ratios of finite integers; according to legend, the Pythagorean Hippasus was drowned at sea after proving that $\sqrt{2}$ is irrational.

As we have mentioned in earlier posts, actual infinity did not establish a real foothold in mathematics until the late 19th century and the work of Dedekind and Cantor, which naturally engendered fierce opposition. Perhaps the most prominent opponent of Cantorian set theory was Leopold Kronecker, who is widely known for having said, “God made the integers, all else is the work of man” and thought that irrational numbers do not exist. I have given this exact quote before, but I will do so again now:

I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.

Finitism, the philosophy of mathematics that denies the existence of infinite objects, is today a minority view but is nonetheless very much alive, and a number of mathematicians are doing work to remove infinities (and, hence, irrationalities) from mathematics. Among these is N.J. Wildberger, a professor at the University of New South Wales who is developing what he calls Rational Trigonometry, a reworking of trigonometry replacing the notions of ‘distance’ and ‘angle,’ which readily produce irrational numbers even when applied only to points with rational coordinates, with ‘quadrance’ and ‘spread’ to measure the amount of separation between two points and two lines, respectively.

(A related but separate movement, which we will return to in a later post, is the attempt by prominent scientists such as Max Tegmark and Raphael Bousso, to remove infinities from physics. The assertion that our physical universe is entirely finite is a weaker (and, in my opinion, much more plausible) assertion than the assertion that infinity does not exist mathematically, and we will devote attention to it in the future.)

There are some, though, for whom the finitists do not go far enough. These people, known as ‘ultra-finitists,’ not only deny the existence of mathematical infinity but even refrain from accepting the existence of very large finite integers. For example, let $N$ be the largest integer less than $e^{e^{e^{79}}}$, a number known as Skewes’ number and that has appeared in proofs in number theory. An ultra-finitist will likely refrain from accepting the existence of $N$ on the basis that this natural number has not actually been calculated and may in fact be too large to be physically calculated at all.

Ultra-finitism in its modern guise was initiated by Alexander Yessenin-Volpin, a mathematician who was a prominent human-rights activist in the Soviet Union, for which he was imprisoned in 1968. There is a wonderful anecdote about him from Harvey Friedman, who had occasion to confront him about his extreme ultra-finitist views.

I have seen some ultrafinitists go so far as to challenge the existence of 2100 as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 21, 22, 23, … , 2100 do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 21 and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 22, and he again said yes, but with a perceptible delay. Then 23, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2100 times as long to answer yes to 2100 then he would to answering 21. There is no way that I could get very far with this.

Harvey M. Friedman “Philosophical Problems in Logic”

Perhaps the most prominent ultra-finitist working today is Doron Zeilberger, a mathematician at Rutgers. Zeilberger, like many ultra-finitists, believe there is a largest natural number. When asked the inevitable question about what happens when you add 1 to it, he replies that, in a very elegant circularity, you go back to 0. Zeilberger also has a fascinating web page of his Opinions. I present to you in full Opinion 146: Why the “fact” that 0.99999999…(ad infinitium)=1 is NOT EVEN WRONG.

The statement of the title, is, in fact, meaningless, because it tacitly assumes that we can add-up “infinitely” many numbers, and good old Zenon already told us that this is absurd.

The true statement is that the sequence, a(n), defined by the recurrence

a(n)=a(n-1)+9/10n   a(0)=0   ,

has the finitistic property that there exists an algorithm that inputs a (symbolic!) positive rational number ε and outputs a (symbolic!) positive integer N=N(ε) such that

|a(n)-1|<ε for (symbolic!) n>N   .

Note that nowhere did I use the quantifier “for every”, that is completely meaningless if it is applied to an “infinite” set. There are no infinite sets! Everything can be reduced to manipulations with a (finite!) set of symbols.

We end with the classic instructional video, “Look Around You (Maths),” which addresses ultra-finitism in its first segment. Enjoy!

# Infinity Dots

When I was a kid and my parents were out of the house, I would sometimes slip into their bathroom. I would do this not to snoop around in the drawers or to try to ascertain the secrets of the lives of adults, but rather to find a piece of infinity.

An explanation of the topography of the bathroom, in particular of the bathroom sink, and especially of the medicine cabinet above the bathroom sink, should elucidate the situation. You see, the medicine cabinet has three partitions, each with a mirrored door, the outer two of which both swing open towards the center. Therefore, if, say, a ten-year-old boy were to open these two doors just the right amount and stick his head between them in just the right place, with his eyes pointing in just the right direction, all of these details of course meticulously fine-tuned during multiple sessions, he would seemingly find himself placed right in the middle of an infinite greenish-silver corridor, alternating images of the front and back of his head lined up as far as he could see. It was intoxicating to find such an expansive space in a room that, at first glance, appears so limited in dimensions.

(This desire to find infinity in unexpectedly small places is perhaps in part responsible for my later infatuation with the short fictions of Jorge Luis Borges, whose techniques, as Lois Parkinson Zamora aptly states in an essay on Borges and trompe l’œil, create “the illusion of infinity in a tightly contained narrative space.”)

In the Mexican War Streets neighborhood of Pittsburgh, located in a former mattress warehouse, there is a remarkable contemporary art museum called, appropriately, Mattress Factory. Among the highlights of the museum are three light installations by James Turrell and two mirrored rooms by Yayoi Kusama, where, as a graduate student, I found more fully realized versions of the infinite corridor in my parents’ bathroom.

The first, Infinity Dots Mirrored Room, is an empty room with a polka-dotted floor and mirrored walls and ceiling, illuminated by black light. The effect is eerie; one feels entirely alone and lost in a formless landscape which grows increasingly indistinct as it recedes to the horizon.

The second, Repetitive Vision, is slightly less disorienting and slightly more surreal. The polka dots are all bright red, the light is white rather than black, and the viewer is joined by mannequins, covered in red polka dots themselves.

During the 1960s, Kusama, who was born and raised in Japan, was a prominent figure in the New York art scene, working alongside Donald Judd, Claes Oldenburg, and Andy Warhol and romantically involved with Joseph Cornell. She worked obsessively, creating huge installations and staging bizarre happenings in public places, including many in which, anticipating Repetitive Vision, Kusama would paint polka dots on the bodies of nude models.

Her time in New York, though, was also characterized by severe health problems, and she was hospitalized on many occasions. In 1973, at the advice of a doctor, Kusama moved back to Japan, and since 1975 she has resided at the Seiwa Hospital for the Mentally Ill, where she continues to create art. She suffers from depersonalization disorder, in which an individual has recurring feelings of disconnection from their body or thoughts. They may feel as if they are watching their life in a movie. They may have hallucinations. According to Kusama, her artwork is largely an expression of her experiences with mental disease; this seems particularly the case with her mirrored rooms.

Depersonalization disorder is typically thought to be caused by severe traumatic events. In a 1999 interview in Bomb Magazine, which due to Kusama’s residence at a mental hospital was conducted via fax, she describes being abused by her mother as a child:

My mother was a shrewd businesswoman, always horrendously busy at her work. I believe she contributed a great deal to the success of the family business. But she was extremely violent. She hated to see me painting, so she destroyed the canvases I was working on. I have been painting pictures since I was about ten years old when I first started seeing hallucinations.

In her artist’s statement accompanying Repetitive Vision, created and installed in the Mattress Factory in 1996, Kusama describes a hallucination that inspired the work:

One day, I was looking at a tablecloth covered in red flowers, which was spread out on the table. Then I looked up towards the ceiling. There, on the windows and even on the pillars, I would see the same red flowers. They were all over the place in the room, my body, and entire universe. I finally came to a self-obliteration and returned to be restored to the infinity of eternal time and the absoluteness of space. I was not having a vision. It was a true reality. I was astounded. Unless I got out of here, the curse of those flowers will seize my life! I ran frantically up the stairs. As I looked down, the sight of each step falling apart made me stumble. I fell all the way down the stairs and sprained my leg.

There is in the popular imagination a strong link between infinity and madness, in particular a causal link posited from the first to the second, a persistent image of a person staring into the void and never again being quite the same. For example, it is commonly asserted that Georg Cantor went insane from thinking about infinity too much. And while it is true that he spent the end of his life in sanatoria, this likely had little to do with his contemplations of infinity. Rather, experts believe that he suffered from bipolar disorder, which was possibly exacerbated by strong criticism of his work from other mathematicians.

In actuality, I suspect (and I write here with no real authority on the matter) that some amount of immersive experience with infinity is beneficial, that it can help us gain a healthy perspective on our relationship with the world. Much of Kusama’s work, and, I think, much other contemporary art as well, seeks to engender this experience. Standing in one of her mirrored rooms, you temporarily lose yourself in its vastness. You become more aware of the true scale of the world and your size within it. And in my experience this turns out, when taken in small doses, to be a surprisingly comforting sensation.

Kusama herself, in the Bomb interview, puts things a bit more forcefully:

By obliterating one’s individual self, one returns to the infinite universe.