Infinity is only the beginning

kw: book reviews, nonfiction, mathematics

Where is Buzz Lightyear when you need him? When he says, "To infinity, and beyond!" it just sounds cute, but it can mean so much, much more. Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math, by Alex Bellos, is a singular pleasure among books that popularize mathematics. It manages to cover mathematical endeavors from one end of mathematical experience to the other in less than 300 pages (291, if you don't count the Bibliography and Index).

Where to begin? The author begins with innumeracy, both cultural and personal. Some peoples speak languages that have no more than three words representing numbers, if they even mean what we do by the concept "Number". "One", "Two", "Many" is kind of an anthropological joke, but it is a reality some folks live with. If you ask one of them how many children he has, he will most likely answer, "Children are a gift from God. Why tempt Him by worrying about that?"

In a way, the book ends where it begins, with numbers for which we, at present, know no more than three words. The first of those words is "Aleph-null", the second is "c", and the third is "d", and you have to pronounce c and d in italics ;> NOTE: I don't find a convenient way to show the Hebrew character Aleph that will work on all browsers, so I will continue to spell it out. We'll get back to what these three terms mean a little later.

In the meantime, think about what the world must have been like when you had to tally the family flock of goats thus: //////////////////. If you want to sell two goats (uh, that's // goats), how many are left? Get out a fresh piece of bone and scratch into it ////////////////. There, wasn't that easy? But can you tell at a glance that the two long tallies are different? This is why herders named their goats. You soon come to recognize them, and it is easy to tell if Hephzibah is missing. The innovation of Roman Numerals made it easier to recognize that XVIII and XVI were different, but it took the invention of positional notation and Indian (AKA Arabic) numerals to render the sale of those goats thus: 18-2=16.

The stories the author tells while recounting such historical concepts insert lots of interesting mathematical concepts as you read along. I found it fascinating that, while the Babylonians and Chinese had a useful concept of Zero a few thousand years ago, somehow the Europeans had to go through more than a century of wrangling about Zero once it was imported from India at the beginning of that amusing period called The Enlightenment. The rest of the world must have been thinking, "It is about time those dullards got with the program."

Mathematics couldn't really progress much beyond basic arithmetic until algebra and symbolic representation came on the scene. I remember my sudden shift of outlook when I realized that the x in 10x=50 represented a quantity all by itself, not just an unknown digit in some number a little bigger than 100, and that I was multiplying it by ten. Thereafter, I have sometimes asked people, "What is the fundamental concept of mathematics?", looking for the answer "Operations". The implied multiplication and equivalence operations in that tiny formula 10x=50 opened up to me a great world of symbolic thought.

Higher and higher mathematical disciplines are actually language lessons, in which new kinds of operations are learned, and a language for describing them. So, for example, we learn very young to use the operators + - × ÷ and = , and soon thereafter the parentheses and √ . Calculus classes are a long series of lessons in the use of the D and the Integral (a skinny "S") operators. In still other classes we learn functional operators like cos and log. And so it goes.

About halfway through the book Bellos introduces logarithms in the clearest presentation I have yet seen. While I can't hope to do better, let me just say that we tend to think in a logarithmic way by nature; it allows us to focus on what is important. The difference from 1,000 to 1,100 "feels" similar to the difference from 10 to 11, even though in absolute terms that difference of 100 is much, much greater. The context matters. Our senses respond to the world logarithmically: take any sound and make it twice as loud; no matter the intensity of the first sound, the difference between the two is felt as the same. But I'll leave it to you to read how the author goes on to introduce the concept of a slide rule (I still have one) and the mathematical power that goes with it.

No book about enjoying mathematics is complete without a chapter on mathematical games, featuring Martin Gardner, who died not long ago. There is enough to enjoy about math that Gardner was able to write a very entertaining column, "Mathematical Games", for 25 years in Scientific American magazine. I remember his columns about different kinds of zero, about "flexagons", about making a machine out of matchboxes that plays perfect Tic-Tac-Toe. The skills needed to keep bowling scores are far from trivial, but because we're enjoying ourselves, we find them easy to learn.

But where does Aleph-null come in? It turns out to be the "smallest" kind of infinity. The term was coined by Georg Cantor, who also invented Cantor Dust. This is not covered in the book, but it ought to be: Take a line of any length. Remove the middle half. For each remaining line segment, remove the middle half. Repeat forever. You are left with an infinite number of points, scattered in a specific way, whose length adds up to zero. Now, start with a triangle, an equilateral triangle. Mark the midpoints of each side, and draw the triangle (also equilateral) that they define. Remove this inner triangle, which has an area of 1/4 the original. Repeat the operation (there's that word) with each of the three remaining triangles, then repeat again and so forth. This results in a triangular doily called the Sierpinski Triangle. It is an extension of the Cantor Dust into two dimensions. If you start with a square, think of it as nine squares and remove the center square, then repeat infinitely, you get the Sierpinski Gasket, a different doily. Both the Triangle and the Gasket are fully-connected (not like the Cantor Dust), having infinite edge lengths but zero area.

OK, Cantor showed that, not only can the numbers on the number line be "counted" all the way to the infinite limit, but so can the fractions. He called any set that can be counted an Aleph-null infinity. The "counting numbers" plus all the fractions (rational numbers) are such a set. Then he showed that one can easily (in concept) produce decimal numbers (or in any base, if you please) that are not parts of any Aleph-null set. These are the irrational numbers. Because of the way they can be constructed, they are not countable. It turns out that between any two rational numbers, there are infinitely many irrational numbers, and this "larger infinity" has come to be called c, from the word continuum. Without the irrational numbers, the rational numbers do not fill the number line, but form a kind of Cantor Dust of zero total length! It takes the irrational numbers to fill the line. Then where does d come from? It is the next letter after c, and no word has been associated with it. It is the set of all functional graphs (or any kind of wiggle whatever) that can be drawn on the two-dimensional plane. It is known to be larger than c. What is not known is if there are other infinities that are larger than Aleph-null but smaller than c, or between c and d.

So we wind up with three kinds of infinity, but we don't know if there are more, nor if these even form a consecutive set. As students of infinity, we are in worse shape than the "one, two, many" people! At least they know that two comes right after one…if they care.