# 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.

# Hippasus and the Infinite Descent

Today, dear Reader, we bring you a story. A story of Mathematics and Music, of Reason and Passion, of Drama and Irony. It is the story of Hippasus of Metapontum, of his remarkable life and his equally remarkable death. Before we begin, a note of warning. Not everything presented here is true, but all of it is meaningful.

A Greek philosopher and mathematician from the 5th century BCE, Hippasus was a follower of Pythagoras. The Pythagoreans believed in the transmigration of souls, subscribed to the belief that All is Number (where “Number” is, of course, “Whole Number”), made great strides in the study of musical harmony, and eschewed the eating of beans. Our hero was a particularly illustrious Pythagorean. He performed experiments linking the sizes of metal discs to the tones they emit upon being struck, developed a theory of the musical scale and a theory of proportions, and showed how to inscribe a regular dodecahedron in a sphere. The regular dodecahedron is a twelve-sided solid whose faces are regular pentagons, shapes which were dear to the Pythagoreans and central to our story. The pentagram, the five-sided star formed by extending the sides of a regular pentagon, and whose tips themselves form a regular pentagon, was a religious symbol of the Pythagoreans and a mark of recognition amongst themselves.

A corollary to the Pythagorean doctrine that All is Number was a belief, at the time, that any two lengths are commensurable, i.e., that given any two lengths, they are both whole number multiples of some fixed smaller length. In modern language, this amounts to the assertion that, given any two lengths, their ratio is a rational number, i.e., can be expressed as the ratio of whole numbers. It is only fitting that the first evidence to the contrary would come from the pentagram.

It happened when Hippasus was stargazing. He saw five stars forming a perfect regular pentagon, and inside this regular pentagon he formed a pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram. An infinite web of similar triangles was woven through his mind and, in a flash of insight, he realized something terrible: the lengths of one side of the regular pentagon and one side of the pentagram found inside it are incommensurable.

The next day, he told his fellow Pythagoreans of his discovery, and they were horrified. They could not have this knowledge, which struck at the core of their belief system, getting out into the wider world. So they took Hippasus far out to sea, and they threw him overboard.

Infinite descent is a proof technique that morally dates back to the ancient Greeks but really came into its own in the work of Pierre de Fermat in the 17th century. The idea behind it is simple and immediately appealing. Suppose we want to prove that there are no positive integers satisfying a particular property. One way to prove this would be to show that, given any positive integer $n$ satisfying this property, we could always find a smaller positive integer $n'$ satisfying the same property. Repeating the argument with $n'$ in place of $n$, we could find a still smaller positive integer $n''$ satisfying the same property. Continuing, we would construct a decreasing sequence of positive integers, $n > n' > n'' > n''' > \ldots$, the aforementioned “infinite descent”. But of course there can be no infinite decreasing sequences of positive integers (if a decreasing sequence starts with $n$, it can of course have at most $n$ elements). One thus reaches a contradiction and concludes that there are no positive integers with the given property.

Fermat made great use of the method of infinite descent in his work on number theory. One particularly striking application came in the proof of a special case of his famous Last Theorem: there are no positive integers $a,b,c$ such that $a^4 + b^4 = c^4$. Fermat showed that, if $a_0,b_0,c_0$ are positive integers such that $a_0^4 + b_0^4 = c_0^4$, then we can construct positive integers $a_1, b_1, c_1$ such that $a_1^4 + b_1^4 = c_1^4$ and $c_1 < c_0$. One can then continue, with $a_1, b_1, c_1$ in place of $a_0, b_0, c_0$, and obtain $a_2, b_2, c_2$ with $a_2^4 + b_2^4 = c_2^4$ and $c_2 < c_1$. This of course leads to the infinite descent $c_0 > c_1 > c_2 > \ldots$ and a contradiction.

We will use the method of infinite descent to prove the result of Hippasus mentioned above. This will not be exactly the way the ancient Greeks would have presented the proof, but it is very similar in spirit, and we make no apologies for the anachronism. Let’s get started.

Suppose, for the sake of an eventual contradiction, that we have a regular pentagon whose side length is commensurable with the side length of the inscribed pentagram. Let $s$ denote the side length of the pentagon and $t$ denote the side length of the pentagram (i.e., the diagonal of the pentagon). In modern language, our assumption is that $\frac{t}{s}$ is rational, i.e., that there are positive integers $p$ and $q$ such that $\frac{t}{s} = \frac{p}{q}$. By scaling the pentagon, we can in fact assume that $t = p$ and $s = q$. So let this be our starting assumption: there is a regular pentagon whose side length $s$ and diagonal length $t$ are both positive integers.

Let us label the vertices of the pentagon by the letters A,B,C,D,E. The center of the pentagram forms another regular pentagon, whose vertices we shall call a,b,c,d,e. This is shown in the diagram below. Note that $s$ is the length of the line segment connecting A and B (we will denote this length by |AB|), and $t$ is |AC|. Let $s'$ denote the side length of the inner pentagon, i.e., |ab|, and let $t'$ denote the length of the diagonal of the inner pentagon, i.e., |ac|.

We now make use of the wealth of congruent triangles in the diagram. We first observe that, whenever a pentagram is inscribed into a regular pentagon, the diagonals that form the pentagram exactly trisect the angles of the pentagon. (Exercise: Prove this!) Therefore, the triangle formed by A, E and D is congruent to that formed by A, e, and D, on account of their sharing a side and having the same angles on either end of that side. In particular, we have |Ae| = |AE| = $s$.

Next, consider the triangle formed by A, b, and d. By our observation at the start of the previous paragraph, the angle at b in this triangle has the same measure as the angle at A. But then this triangle is isosceles, so we have |Ad| = |db| = $t'$. Of course, we clearly have |Ad| = |Bd| = |Be| = |Ce| =…, so all of these lengths are equal to $t'$.

Let’s now see what we have. Consider first |AC|. By definition, we have |AC| = $t$. But also |AC| = |Ae| + |Ce|, and we saw previously that |Ae| = $s$ and |Ce| = $t'$. All together, we have $t$ = |AC| = |Ae| + |Ce| = $s + t'$, or $t' = t-s$.

Next, consider |Ae|. We have already seen that |Ae| = $s$. But we also have |Ae| = |Ad| + |de|. We saw previously that |Ad| = $t'$, and by definition we have |de| = $s'$. All together, we have $s$ = |Ae| = |Ad| + |de| = $t' + s'$, or $s' = s - t'$. Since we already know that $t' = t-s$, this yields $s' = 2s - t$.

This might not seem like much, but we’re actually almost done now! The key observation is that, since $s$ and $t$ are integers, and since $s' = 2s - t$ and $t' = t-s$, it follows that $s'$ and $t'$ are also (positive) integers. We started with a regular pentagon whose side length $s$ and diagonal length $t$ were integers and produced a smaller pentagon whose side length $s'$ and diagonal length $t'$ are also integers. But now we can continue this process, producing smaller and smaller regular pentagons and producing infinite descending sequences of positive integers, $s > s' > s'' > \ldots$ and $t > t' > t'' > \ldots$.

This is a contradiction, and we have thus shown that $\frac{t}{s}$ is irrational. But what is this ratio exactly? Well, it turns out to be a fascinating number, easily #2 on the list of Most Famous Ratios of All Time: $\phi$, a.k.a. the golden ratio! But that is a story for another time…

As a short addendum to today’s story, here’s a little-known fact about the end of Hippasus’ life. It turns out that, after being thrown overboard by his fellow Pythagoreans, he has not and will never in fact reach the seabed! For to get there, he would first have to travel halfway down, and then he would have to travel half of the remaining distance, and then half of the still remaining distance, and so on and so forth, completing an endless sequence of tasks, and thus he remains to this date in the midst of an infinite descent of his own.

Notes:

(1) $\sqrt{2}$ is more commonly cited as being the first irrational number discovered by the Pythagoreans, and it is almost always the first number proven to be irrational in classrooms today. However, the proof of the irrationality of $\sqrt{2}$ possessed by the Greeks is not the simple number-theoretic proof used today and is in fact a rather complex elaboration of the proof of the incommensurability of the side and diagonal lengths of a regular pentagon. Given this fact and the centrality of the pentagram in Pythagorean intellectual life, some scholars have suggested that perhaps this was in fact the first proof of the existence of incommensurability and that the proof of the irrationality of $\sqrt{2}$ came later. We have adopted this hypothesis for the purpose of our story today.

(2) This story is derived from legends that significantly post-date the death of Hippasus. It seems unlikely actually to have happened as presented here.

(3) This post was in part inspired by the episode “Drowned at Sea”, from the excellent podcast, Hi-Phi Nation, by philosopher Barry Lam. Check it out!

Cover image: “Rainstorm Over the Sea” by John Constable

# Circle Inversion and the Pappus Chain

There is a pledge of the big and of the small in the infinite.

-Dejan Stojanović

In the next two posts, we are going to look at two interesting geometric ideas of the 19th century involving circles. Next time, we will consider Poincaré’s disk model for hyperbolic geometry. Today, though, we immerse ourselves in the universe of inversive geometry.

Consider a circle in the infinite 2-dimensional plane:

This circle divides the plane into two regions: the bounded region inside the circle and the unbounded region outside the circle (let’s say that the points on the circle belong to both regions). A natural thing to want to do, now, especially in the context of this blog, would be to try to exchange these two regions, to map the infinite space outside the circle into the bounded space of the circle, and vice versa, in a “natural” way.

I could be bounded in a nutshell, and count myself a king of infinite space.

-William Shakespeare, Hamlet

Upon first reflection, one might be tempted to say that we want to “reflect” points across the circle. And this is sort of right, but reflection already carries a meaning in geometry. Truly reflecting points across the circle would preserve their distance from the circle, so the inside of the circle could only be mapped onto a finite ring whose outer radius is twice that of the circle two. Moreover, it would not be clear how to reflect points from outside this ring into the circle.

Instead, we want to consider a process known as “inversion.” Briefly speaking, we want to arrange so that points arbitrarily close to the center of the circle get sent to points arbitrarily far away from the center of the circle, and vice versa. For simplicity, let us suppose that the circle is centered at the origin of the plane and has a radius of 1. The most natural way to achieve our aim is to send a point $P$ to a point $P'$ that lies in the same direction from the origin as $P$ and whose distance from the origin is the reciprocal of the distance from $P$ to the origin. Here’s an example:

One can check that, algebraically, this inversion sends a point $P$ with coordinates $(x,y)$ to a point $P'$ with coordinates $(\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2})$. Points inside the circle are sent to points outside the circle, points outside the circle are sent to points inside the circle, and points on the circle are sent to themselves. Moreover, as one might expect from the name, the inversion map is its own inverse: applying it twice, we end up where we started. Perfect!

Wait a second, though. We’re being a little too hasty. What about the origin? Where is it sent? Our procedure doesn’t seem to tell us, and if we try to use our algebraic expression, we end up dividing by zero. Since the origin is inside the circle, it should certainly be sent to a point outside the circle, but all of those points are already taken. Also, since points arbitrarily close to the origin get mapped to points arbitrarily far from the origin, we want to send the origin to a point as far away from itself as possible. At first glance, we might seem to be in a quandary here, but longtime readers of this blog will see an obvious solution: the origin gets mapped to a point at infinity! (And the point at infinity, in turn, gets mapped to the origin.)

(Technical note: Since we’ve added a point at infinity, the inversion map should be seen not as a map on the plane $\mathbb{R}^2$, but on its one-point compactification (or Alexandroff compactification), $\hat{\mathbb{R}}^2$. In fact, the inversion map is a topological homeomorphism of $\hat{\mathbb{R}}^2$ with itself.)

Let’s examine what the inversion map does to simple geometric objects. We have already seen what happens to points. It should also be obvious that straight lines through the origin get mapped to themselves. For example, in the image above, the line connecting $P$ and $P'$ gets mapped to itself. (Here we are specifying, of course, that every line contains the point at infinity.)

A bit of thought and calculation will convince you that lines not passing through the origin get sent to circles that do pass through the origin.

Since the inversion map is its own inverse, circles passing through the origin get mapped to lines that don’t pass through the origin. Circles that don’t pass through the origin, on the other hand, get mapped to other circles that don’t pass through the origin.

There’s an important special case of this phenomenon: a circle that is met perpendicularly by the circle through which we are inverting gets mapped to itself.

We thus have a sort of duality between lines and circles that has been revealed through the process of circle inversion. Lines, when seen in the right light, are simply circles with an infinite radius. We’re going to move on to some applications of circle inversion in just a sec, but, first, a pretty picture of an inverted checkerboard.

The introduction of the method of circle inversion is widely attributed to the Swiss mathematician Jakob Steiner, who wrote a treatise on the matter in 1824. When combined with the more familiar rigid transformations of rotation, translation, and reflection, the decidedly non-rigid transformation of inversion gives rise to inversive geometry, which became a major topic of study in nineteenth geometry. It was perhaps most notably applied by William Thomson (later to become 1st Baron Kelvin, immortalized in the name of a certain temperature scale), at the age of 21, to solve problems in electrostatics. Circle inversion also allows for extremely elegant proofs of classical geometric facts. We end today’s post with an example.

Consider three half-circles, all tangent to one another and centered on the same horizontal line, with two placed inside the third, as follows:

This figure (or, more precisely, the grey region enclosed by the semicircles) is known as an arbelos, and its first known appearance dates back to The Book of Lemmas by Archimedes. A remarkable fact about the arbelos is that, starting with the smallest of the semicircles in the figure, one can nestle into it an infinite sequence of increasingly small circles, each tangent to the two larger semicircles and the circle appearing before it, thus creating the striking Pappus chain, named for Pappus of Alexandria, who investigated the figure in the 3rd century AD:

Let us label the circles in the Pappus chain (starting with the smallest semicircle in the arbelos) $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2$, etc. (So, in the picture above, $P_1$ is the center of $\mathcal{C}_1$, $P_2$ is the center of $\mathcal{C}_2$, and so on.) Clearly, the size of $\mathcal{C}_n$ decreases as $n$ increases, but it is natural to ask how quickly it decreases. It is also natural to ask how the position of the point $P_n$ changes as $n$ increases. In particular, what is the height of $P_n$ above the base of the figure? It turns out that the answers to these two questions are closely related, a fact discovered by Pappus through a long and elaborate derivation in Euclidean geometry, and which we will derive quickly and elegantly through circle inversion.

Let $d_n$ denote the diameter of the circle $\mathcal{C}_n$, and let $h_n$ denote the height of the point $P_n$ above the base of the Pappus chain (i.e., the line segment $AB$). We will prove the remarkable formula:

For all $n \in \mathbb{N}$$h_n = n \cdot d_n$.

For concreteness, let us demonstrate the formula for $\mathcal{C}_3$. The same argument will work for each of the circles in the Pappus chain. As promised, we are going to use circle inversion. Our first task is to find a suitable circle across which to invert our figure. And that circle, it turns out, will be the circle centered at $A$ and perpendicular to $\mathcal{C}_3$:

Now, what happens when we invert our figure? First, consider the two larger semicircles in the arbelos, with diameters $AC$ and $AB$. The circles of which these form the upper half pass through the center of our circle of inversion and thus, as discussed above, are mapped to straight lines by our inversion. Moreover, since the centers of these circles lie directly to the right of $A$, a moment’s thought should convince you that they are mapped to vertical lines.

Now, what happens to the circles in the Pappus chain? Well, none of them pass through $A$, so they will all get mapped to circles. $\mathcal{C}_3$ is perpendicular to the circle of inversion, so it gets mapped to itself. But, in the original diagram, $\mathcal{C}_3$ is tangent to the larger semicircles in the arbelos. Since circle inversion preserves tangency, in the inverted diagram, $\mathcal{C}_3$ is tangent to the two vertical lines that these semicircles are mapped to. And, of course, the same is true of all of the other circles in the Pappus chain. Finally, note that, since the center of $\mathcal{C}_0$ lies on the base of the figure, which passes through the center of our inversion circle, it also gets mapped to a point on the base of the figure. Putting this all together, we end up with the following striking figure:

The circle with diameter $AB$ gets mapped to the vertical line through $B'$, and the circle with diameter $AC$ gets mapped to the vertical line through $C'$. Our Pappus chain, meanwhile, is transformed by inversion into an infinite tower of circles, all of the same size, bounded by these vertical lines. Moreover, the circle $\mathcal{C}_3$ and the point $P_3$ are left in place by the inversion. It is now straightforward to use this tower to calculate the height $h_3$ of $P_3$ in terms of the diameter $d_3$ of $\mathcal{C}_3$. To get from $P_3$ down to the base, we must first pass through half of $\mathcal{C}_3$, which has a height of $\frac{d_3}{2}$. We then must pass through the image of $\mathcal{C}_2$ under the inversion, which has a height of $d_3$. Then the image of $\mathcal{C}_1$, which also has a height of $d_3$. And, finally, the image of the smallest semicircle of the arbelos, which has a height of $\frac{d_3}{2}$. All together, we get:

$h_3 = \frac{d_3}{2} + d_3 + d_3 + \frac{d_3}{2} = 3d_3$.

Pretty nice!

For further reading on circle inversion, see Harold P. Boas’ excellent article, “Reflections on the Arbelos.”

Cover image: René Magritte, The false mirror

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…

# 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.

# An Infinitude of Proofs, Part 1

In our previous post, we gave three elementary number-theoretic proofs of the infinitude of the prime numbers. Today, in an unforgivably delayed second installment, we provide two proofs using machinery from more distant fields of mathematics: analysis and topology.

A small word of warning: today’s proofs, though not difficult, require a bit more mathematical sophistication than those of the previous post. The first proof will involve some manipulation of infinite sums, and the second proof uses rudiments of topology, which we introduced here.

Without further ado, let us begin. Our first proof today dates to the 18th century and is due to the prolific Leonhard Euler, of whom Pierre-Simon Laplace, an exceptional mathematician in his own right, once said, “Read Euler, read Euler, he is the master of us all.”

Proof Four (Euler): Suppose there are only finitely many prime numbers, $\{p_1, p_2, \ldots, p_n\}$. For each prime number $p$, consider the infinite series $\sum_{i = 0}^\infty \frac{1}{p^i} = 1 + \frac{1}{p} + \frac{1}{p^2} + \ldots$. As you probably learned in a precalculus class, the value of this infinite sum is precisely $\frac{1}{1-\frac{1}{p}}$.

Now recall that every natural number greater than 1 can be written in a unique way as a product of prime numbers. Since $\{p_1, \ldots, p_n\}$ are all of the primes, this means that, for each natural number $m \geq 1$, we can express $\frac{1}{m}$ as $\frac{1}{p_1^{s_1}} \cdot \frac{1}{p_2^{s_2}} \cdot \ldots \cdot \frac{1}{p_n^{s_n}}$ for some natural numbers $s_1, s_2, \ldots, s_n$. Moreover, it is clear that this gives a one-to-one correspondence between natural numbers $m \geq 1$ and the corresponding $n$-tuples of natural numbers, $\langle s_1, s_2, \ldots, s_n \rangle$. Putting this all together, we obtain the following remarkable formula:

$\sum_{m=1}^\infty \frac{1}{m} = \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}}$.

The sum on the left hand of this equation is the famous harmonic series, and it is well-known and easily verified that this sum diverges to infinity. On the other hand, the product on the right hand is a finite product of finite numbers and is therefore finite! This gives a contradiction and concludes the proof.

Our next proof was published in 1955 by a young Hillel Furstenburg, who is still active at the Einstein Institute of Mathematics here in Jerusalem.

Proof Five (Furstenberg): For all integers $a$ and $b$, let $U_{a,b}$ be the set $\{a + bk \mid k \in \mathbb{Z} \}$, i.e., $U_{a,b}$ contains $a$ and all integers that differ from $a$ by a multiple of $b$.

Claim: For all pairs of integers $(a_0, b_0)$ and $(a_1, b_1)$, either $U_{a_0,b_0} \cap U_{a_1, b_1} = \emptyset$ or there are $a_2, b_2$ such that $U_{a_0,b_0} \cap U_{a_1, b_1} = U_{a_2, b_2}$.

Proof of Claim: If $U_{a_0, b_0} \cap U_{a_1, b_1} \neq \emptyset$, then let $a_2$ be any element of the intersection, and let $b_2$ be the least common multiple of $b_0$ and $b_1$. It is easily verified that $U_{a_0,b_0} \cap U_{a_1, b_1} = U_{a_2, b_2}$, thus finishing the proof of the claim.

It now follows that we can define a topology on $\mathbb{Z}$ by declaring that a set $U \subseteq \mathbb{Z}$ is open if and only if $U = \emptyset$ or $U$ is a union of sets of the form $U_{a,b}$. In particular, for each prime number $p$, the set $U_{0,p}$, which is the set of all multiples of $p$, is open. However, $U_{0,p}$ is also closed, since it is the complement of the set $\bigcup_{0 < a < p} U_{a,p}$, which is an open set.

Now suppose that there are only finitely many prime numbers, $\{p_1, \ldots, p_n\}$. Since each set $U_{0,p_k}$ is closed and a finite union of closed sets is closed, we have that $\bigcup_{k = 1}^n U_{0,p_k}$ is a closed set. Notice that every integer that is not $1$ or $-1$ is a multiple of some prime number, so $\bigcup_{k = 1}^n U_{0,p_k}$ is precisely the set of all integers except $1$ and $-1$. Since it is closed, its complement, which is $\{-1,1\}$, is open. But every non-empty open set is a union of sets of the form $U_{a,b}$, each of which is clearly infinite, so there can be no finite non-empty open sets. This is a contradiction and concludes the proof.

Cover Image: Passio Musicae by Eila Hiltunen, a monument to Jean Sibelius in Helsinki, Finland. Photograph by the author.

# An Infinitude of Proofs, Part 0

Today, we take a break from our recent philosophical musings to return to some good old-fashioned mathematics (indeed, the mathematics today could be seen as all being “old-fashioned,” as the theorem we will be considering dates back to the ancient Greeks).

I’m sure my readers have all been introduced to the prime numbers, but, to refresh any memories, let me remind you that a prime number is a natural number, at least 2, that has no divisors other than 1 and itself. The first few prime numbers are, therefore: 2, 3, 5, 7, 11, 13…

The prime numbers can be thought of as the building blocks of number theory, as the backbone of the natural numbers. They have occupied a central place in mathematics and offered endless fascination for millennia. They are a source of great mystery even today.

In this post, we look at a classic theorem, commonly attributed to Euclid. Euclid’s proof is one of the gems of mathematics, a proof that is taught to every mathematician at the beginning of their true mathematical career. The theorem, as we will state it, is simply this:

Euclid’s Theorem: There are infinitely many prime numbers.

Euclid himself did not state the theorem in exactly this form, perhaps because of the ancient Greeks’ general antipathy towards the existence of infinite sets. He instead put forward the equivalent statement, “For every finite set of prime numbers, there is a prime number not in that set.”

Euclid’s Theorem has collected a vast and varied array of delightful proofs throughout the years, and, in a proposed infinite series of blog posts, I plan to cover all of them. Today, we will look at three proofs, from three very different centuries, all using only elementary number-theoretic techniques. First, of course, is Euclid’s proof itself.

Proof One (Euclid): Suppose that $P = \{p_0, p_1, \ldots, p_n\}$ is a finite set of prime numbers. We will show that there is a prime number that is not in $P$. To do this, let $q = p_0p_2\ldots p_n$, and let $r = q+1$. For all $i \leq n$, $p_i$ divides $q$ with no remainder (as $q = p_i(p_0\ldots p_{i-1}p_{i+1}\ldots p_n)$), so, as $r = q+1$ and $p_i \geq 2$, $p_i$ does not divide $r$. Since $r > 1$ and every integer greater than 1 has prime divisors, there is at least one prime, $p^*$, that divides $r$. But we just saw that no element of $P$ divides $r$, so $p^*$ is a prime that is not in $P$.

The next proof we will consider is due to the eighteenth-century German mathematician Christian Goldbach, largely known today for the statement of the still unproven Goldbach’s Conjecture. This proof (like Goldbach’s Conjecture itself) appears in a letter from Goldbach to Leonhard Euler (whose analytic proof of Euclid’s Theorem we will visit in a future post).

Proof Two (Goldbach): For each natural number $n$, let $F_n = 2^{2^n} + 1$, so $F_0 = 3, F_1 = 5, F_2 = 17$, etc. ($F_n$ is known as the $n^{\mathrm{th}}$ Fermat number.)

Claim 1: For every natural number $n$, $F_{n+1} = F_0F_1\ldots F_n + 2$.

Proof of Claim 1: Suppose that the Claim is false, and let $n$ be the smallest natural number such that $F_{n+1} \neq F_0F_1 \ldots F_n + 2$. Since the Claim can easily be verified by inspection for $n=0$, we may assume $n > 0$. But now we have the following sequence of calculations where, because we are assuming that $n$ is the smallest counterexample to the Claim, we can use the fact that $F_n = F_0F_1\ldots F_{n-1} + 2$.

$\begin{array}{rcl}F_{n+1} & = & 2^{2^{n+1}}+1 \\ & = & 2^{2^n}\cdot 2^{2^n} + 1 \\ & = & (F_n-1)(F_n-1)+1 \\ & = & (F_n)(F_n) - 2F_n + 2 \\ & = & (F_n)(F_0F_1\ldots F_{n-1} + 2) -2F_n + 2 \\ & = & F_0F_1\ldots F_n + 2F_n - 2F_n + 2 \\ & = & F_0F_1\ldots F_n + 2 \end{array}$

But this calculation shows that $n$ is not a counterexample to our Claim, contradicting our assumption and finishing the proof of the Claim.

We now need a very simple number-theoretic Claim.

Claim 2: If $a$ and $b$ are natural numbers, $p$ is a prime, and $p$ divides both $a$ and $a+b$, then $p$ divides $b$.

Proof of Claim 2: Suppose $a = mp$ and  $a+b = np$, where $m$ and $n$ are natural numbers. Then $b = a+b-a = np - mp = (n-m)p$, so $p$ divides $b$.

Claim 3: If $m < n$ are natural numbers, then there is no prime number $p$ such that $p$ divides both $F_m$ and $F_n$.

Proof of Claim 3: Suppose that the Claim is false and that there are natural numbers $m < n$ and a prime number $p$ such that $p$ divides both $F_m$ and $F_n$. Note that $F_m$ is one of the factors in the product $F_0F_1\ldots F_{n-1}$, so $p$ divides $F_0F_1\ldots F_{n-1}$. Since $F_n = F_0F_1\ldots F_{n-1} + 2$, we also conclude that $p$ divides $F_0F_1\ldots F_{n+1} + 2$, so, by Claim 2, we know that $p$ divides $2$. As $2$ is prime, this means that $p = 2$. But $F_m$ and $F_n$ are both odd, so this is impossible.

We are now ready to finish the proof. For each natural number $n$, choose a prime number $p_n$ that divides $F_n$. By Claim 3, for all natural numbers $m < n$, we have $p_m \neq p_n$, so $\{p_n \mid n \in \mathbb{N}\}$ is an infinite set of prime numbers.

Remark: The Fermat numbers were introduced, unsurprisingly, by the seventeenth-century French mathematician Pierre de Fermat (of Fermat’s Last Theorem fame). Fermat conjectured that every Fermat number is in fact prime (this would obviously imply Claim 3 from the previous proof). The first four Fermat numbers (3, 5, 17, 257) are easily verified to be prime, and the next Fermat number, 65537, can be seen to be prime with a bit more work. However, Euler proved in 1732 that $F_5$ is not prime. Indeed, $F_5 = 2^{32}+1 = 4294967297 = 641 \times 6700417$. Many mysteries remain regarding the Fermat numbers. For example, even the following two very basic questions remain unsolved to this day:

• Are there infinitely many prime Fermat numbers?
• Are there infinitely many non-prime Fermat numbers?

The Fermat numbers grow so quickly that, even with computers, it can be hard to analyze even “relatively small” Fermat numbers. For example, it is unknown whether $F_{20}$ or $F_{24}$ are prime. On the other hand, the truly gigantic number $F_{3329780}$ is known to be non-prime: one of its prime factors is $193 \times 2^{3329780} + 1$.

Finally, a proof that combines ideas from Euclid’s and Goldbach’s proofs was given recently by Filip Saidak.

Proof Three: (Saidak) Note first that, for every natural number $n$, we have that $n$ and $n+1$ do not have any shared prime divisors. Now define an infinite sequence of natural numbers, $\langle n_0, n_1, \ldots \rangle$ as follows. Let $n_0$ be any natural number that is at least 2. Let $n_1 = n_0(n_0+1)$. Since $n_0$ and $n_0 + 1$ do not share any prime divisors, and since both $n_0$ and $n_0+1$ have at least one prime divisor, it follows that $n_1$ must have at least 2 distinct prime divisors.

Now let $n_2 = n_1(n_1+1)$. Again, $n_1$ and $(n_1 + 1)$ do not have any shared prime divisors. We have shown that $n_1$ has at least 2 distinct prime divisors, and we know that $n_1 + 1$ has at least one prime divisor, so $n_2$ must have at least 3 distinct prime divisors. Continuing in this way, defining $n_{k+1} = n_k(n_k+1)$ for every natural number $k$, one proves that, for each natural number $k$, the number $n_k$ has at least $k+1$ distinct prime divisors. In particular, there are infinitely many prime numbers.

Stay tuned for a future post, in which we will provide proofs of Euclid’s Theorem from further flung areas of mathematics, including real analysis and topology.