Nineteenth-Century Monsters

The chimeranineteenth century was a time of change, when long-held assumptions about human society and the surrounding world were being questioned and overturned. It shouldn’t come as a surprise that it brought about an upheaval in the foundations of mathematics as well. It was an era of controversy and bitter contention. One of the main points of issue was cardinality.

The theory of sets and the idea of cardinality were first made rigorous by the mathematician Georg Cantor (1845 – 1918). At the time his work provoked widespread hostility, but posterity has vindicated him. As David Hilbert famously remarked:

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

Now his theory of sets forms part of most elementary education preparation programs.

Simply put, two sets are said to have the same cardinality if they can be paired off with one another without using anything more than once or leaving anything out. If a set has the same cardinality as the set

\lbrace 1, 2, 3, \dots \rbrace

of natural numbers, then it is said to be countable. This means that there is a numbering scheme that eventually numbers every object in the set. I discussed this in my post on counting without numbers.

For instance, consider the set of rational numbers (numbers that can be written as fractions) on the line segment from 0 to 1 on the number line. Every number in this set can be written as a/b, where ≤ b. Consider the following list:

\left\{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6},\frac{1}{7},\dots \right\}

It should be clear that every rational number between 0 and 1 will eventually appear on this list. Of course, there are many repeats. For instance, the rational numbers 1/2, 2/4, 3/6, and so on are all equal to each other. So remove any unreduced fraction from the list. This deletes 2/4, and 2/6, and 3/6, and 4/6, and so on.

\left\{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{5}{6},\frac{1}{7},\dots \right\}

This lists every rational number exactly once. Now number off down the list. This shows that the set is countable.

On the other hand, the line segment from 0 to 1 is not countable. No matter how we try to pair the segment off with the natural numbers, there will always be infinitely many points of the segment left out. Intuitively, this is hardly surprising: the line segment is a continuum of points, and we wouldn’t expect that we could list them like we can the natural numbers. This idea of there being different grades of infinity was surprising to Cantor’s contemporaries. Even philosophers and theologians took note of it.

Given that the line segment is “more infinite” that the set of natural numbers, we might ask if there exists an uncountable set of points that isn’t a continuum. It turns out that there is: the Cantor ternary set, introduced by Georg Cantor in 1883. Take the line segment from 0 to 1 and remove the middle third, resulting in the line segments from 0 to 1/3 and from 2/3 to 1. Do the same for each of these segments, and so on, ad infinitum.

Cantor set

The resulting set is uncountable, contains no segments, and in fact has measure 0.

Cantor also showed that the line segment (which has topological dimension 1) has the same cardinality as the unit square (which has dimension 2). The notion that spaces of different dimension could be paired off in this way was so shocking that Cantor himself disbelieved the result for several years. One mathematician remarked:

It appears repugnant to common sense. The fact is that this is simply the conclusion of a type of reasoning which allows the intervention of idealistic fictions, wherein one lets them play the role of genuine quantities even though they are not even limits of representations of quantities. This is where the paradox resides.

Cantor’s ideas were savagely attacked by many of his fellow mathematicians; he was denounced as a “scientific charlatan” and a “corrupter of youth.” This exacerbated the bouts of mental illness he dealt with throughout his life, and he spent much of his time in sanatoriums.

Inspired by Cantor’s discovery, the Italian mathematician Guiseppe Peano set out to find a continuous mapping from the segment to the square that would cover the entire square. He did this in 1890 through an arithmetic description, and the curve was later interpreted geometrically as the limit of an iterative process. The result was the first known plane-filling curve. Another, better-known example was discovered by David Hilbert in 1891:

Hilbert Curve 3

Although none of the iterative steps are self-intersecting, their limit contacts itself at every point. There is no continuous pairing of the line segment with the square. (Cantor’s mapping, while a true pairing, jumps around the square, whereas Hilbert’s mapping, while continuous, revisits each point multiple times.)

Another famous “pathological” object from the end of the nineteenth century is the Koch curve, named for its discoverer, the Swiss mathematician Helge von Koch (1870 – 1924). Beginning with a line segment, replace the middle third with the two legs of the equilateral triangle erected upon it. Repeat this process for each of the four segments that result, and so on, ad infinitum.

Koch curve

Each iterative step is a simple polygonal curve of dimension 1. But the limit of these steps—the Koch curve—is something else entirely. It is a continuous curve with an infinite length. For, if the length of the initial segment is 1 unit, then the length of the second step is 4/3 units, and that of third, 4/3 ⋅ 4/3 = 16/9 units, and so on. This sequence of lengths tends to infinity.

Each portion of the curve winds back and forth infinitely many times; in fact, each portion resembles the whole. More specifically, the Koch curve contains four copies of itself, each scaled down by a factor of one third. It is a continuous curve whose derivative exists at no point; this amounts to saying that at no point is there a single tangent line. The existence of such curves was profoundly disturbing to the mathematicians of the late nineteenth century; in 1893, the mathematician Charles Hermite wrote:

I turn away with fear and horror from this lamentable plague of functions with no derivatives.

If the above process is applied to each side of an equilateral triangle, then the figure known as the Koch snowflake results.

Koch 2

It is possible to compute the area of the snowflake. Taking its initial side length to be 1, its initial area is √3/4. Adding on the three triangles at the second step, the area becomes

\frac{\sqrt{3}}{4}+ 3 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{3}\right)^2

At the third step, we add on 12 triangles, each of side length 1/9, so the area becomes

\frac{\sqrt{3}}{4}+ 3 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{3}\right)^2 + 12 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{9}\right)^2

Continuing this pattern, we find that the area is

\frac{\sqrt{3}}{4}\left(1 + \frac{1}{3} \left(1 + \frac{4}{9} + \frac{4^2}{9^2} + \frac{4^3}{9^3} + \cdots \right)\right) = \frac{2 \sqrt{3}}{5}

On the other hand, we’ve seen that the perimeter tends to infinity. So here we have a figure with a finite area and an infinite perimeter.

Plane-filling curves and the Koch snowflake are brought together in a beautiful way by a family of curves designed by Benoit Mandelbrot in the 1970s.

Snowflake Peano (Fifth Step)

The limiting curve fills the Koch snowflake. At each stage the figure consists of a single open, simply connected polygonal curve. In principle this entire curve can be traced on paper without lifting one’s pencil or crossing a line, beginning at the lower left-hand point of the star and ending at the lower right-hand point. Filling in the polygonal curve produces a closed polygon. As Mandelbrot says:

This advanced teragon, shown as boundary between two fantastically intertwined domains serves better than any number of words to explain what plane-filling means.

The word teragon, coined by Mandelbrot, comes from the Greek roots tera, or “monster,” and gon, or “corner.”

All the animations in this post were created by me using GeoGebra software.


Counting without Numbers

Imagine a person living at the dawn of civilization, a goatherd, let’s say, dwelling somewhere in the Fertile Crescent. Every day the goatherd lets his animals out of their pen into the pasture so they can graze. When evening comes, he opens the gate and calls to his goats, and they return.

One day the goatherd notices that the herd seems to take up less space in the pen. He begins to worry that he may be losing some goats to thieves or wild animals while they’re out in the field grazing. How is he to make certain?

One obvious suggestion might be to count the goats. That’s what you or I would do. But our goatherd is living at a time when there was no systematic way to count.

Think about this. The English language has proper names for the first twelve counting numbers: one, two, three, and so on, up to twelve. Beyond that, we use the base ten numeration system to label the numbers. For instance, twenty-seven is two tens and seven ones. Three hundred and forty-five is three hundreds, four tens, and five ones.

This machinery originated in India in fairly recent times, only one or two thousand years ago. Our goatherd has no such system. If he wants to label the numbers, he just has to make up proper names for them, and there’s only so many proper names you can come up with. For all we know, his culture may not even have a word for two; the aborigines of Australia are said to have words only for one and many. It would be about as reasonable to ask our goatherd to invent a numeration system on the spot as it would be to ask him to build a computer from scratch. So, how is he to keep track?

Here’s an idea. He could gather a big heap of pebbles and get a large basket. As the goats go out in the morning, he puts one pebble in the basket for each animal that passes him. Once the pen is empty, he knows he has exactly as many pebbles in the basket as goats in the pasture. In other words, he knows that he could pair off the goats and the pebbles without leaving anything out.


Then, when the herd returns in the evening, he can remove one pebble for each goat that passes. If he runs out of goats first, he knows he has a problem. If he runs out of pebbles first, well, he knows that nature has taken its course.

This assignment of pebbles to goats is known as a one-to-one correspondence. Various peoples of antiquity actually did use such methods to keep track of amounts. The ancient Sumerians are said to have used baked clay tokens rather than pebbles for their accounting. They would then seal the tokens in a clay pouch, and put as many marks on the pouch as there were tokens inside. Eventually they decided to do away with the tokens and just use the marks. And the first numeration system was born.

cantorYou see, whenever we count, we are establishing a one-to-one correspondence between a list of numbers and a group of objects. The set of counting numbers may thus be viewed as a universal, abstract set of “pebbles.” Instead of pairing goats with pebbles, we pair goats with numbers, and pebbles with numbers. This involves a profound leap in human thought. The same idea forms the foundation of the modern theory of number as formulated by the great German mathematician Georg Cantor (1845 – 1918). It is to Cantor that we owe the knowledge that there are different kinds of infinities, and that the set of real numbers is more “numerous” than, say, the set of counting numbers.

The child psychologist Jean Piaget (1896 – 1980) studied the role of one-to-one correspondence in early childhood development. In The Child’s Conception of Number, he describes several stages. First, the child compares groups of objects by noting their spacial arrangement or extension, much as our goatherd did when he observed the size of his herd in the pen. This frequently leads to incorrect responses. Later, the child may be brought to recognize the equivalence of two sets through observing a pairing. But it is not until the child realizes that anything done respectively to the two groups can be undone, thus restoring them to the paired arrangement, that they arrive at a true grasp of counting. In group-theoretic terms, we would say that the child has to recognize that the operations performed on the sets are invertible.

So here we have a remarkable parallel between the origins of counting at the dawn of civilization, the theoretical foundation of sets and numbers, and the development of the conception of number in the human mind.