Mathematical truths (which are always proofs because, as mentioned previously, there is no “probably” in pure mathematics) are truths that are certain. We know, because it has been proved, that we will never find a triangle on a plane in which the sum of the three angles is not 180 degrees. (There are, it must be acknowledged, some mathematicians who argue for acceptance of “extraordinarily likely” hypotheses—not merely as clues leading to proofs, but as valid on their own. Most mathematicians hate this.)
This certainty comes at a price. Unlike science, mathematics alone can never tell us anything about the real world. Mathematics can tell us that two apples plus two apples makes four apples, but that is because two anything and two anything makes four anything. Knowledge of the real world, on the other hand, must always include a fact that one could imagine not being a fact. For example, once one knows that the circumference of the earth is about 25,000 miles, then the mathematician can tell us that the earth’s diameter is about 8,000 miles. But mathematics alone cannot tell us the diameter of the earth without a factual starting point. That takes science, with a tremendous amount of help from mathematics.
Two apples and two apples makes four apples, not because observation and experiment have shown this to be true, but because four is, in effect, defined as two plus two. You might well wonder, “Then how come mathematics always works in describing the real world? How come two apples and two apples does make four apples?” Nearly every great thinker has also wondered this, but no one has yet come up with an answer that does much more than restate the question.
Thus, take the truth that “we’ll never find a bachelor with a wife.” The scientist can dismiss this as a trivial truth because it is a “tautology,” a truth that is true by definition. The mathematician can, of course, do the same. But his dismissal is not on the grounds that it is a tautology, but because it is an obvious tautology.
If the mere fact that something is a tautology were to make it trivial, then all mathematics would be trivial because every mathematical proof is a tautology: its conclusion is entailed in the definitions of its premises (e.g., a few axioms and the properties of numbers). Just as we will never find a bachelor who is married, we will never find a triangle on a plane whose angles do not add up to 180 degrees. In each case, the underlying definitions entail the finding. Put another way, a God who is infinitely intelligent but not able to predict the future would know all mathematics instantly but not all that will happen in the future of the real world. (This assumes that reality is random at bottom, which it seems to be.)
Because mathematics is ultimately based on definitions and logic (that is what permits the certainty that is precluded by the real world), it also avoids the vagueness inherent in words. Consider a question such as, “Do chimps have ‘language’?” Superficially, this seems to be a fascinating question. A bit of thought, however, makes it clear that this question is not amenable to a satisfying answer because “language” cannot be sufficiently and rigorously defined to permit an answer. If “language” means “warning a member of your species,” then beavers have language. (They slap their tails or something.) If “language” means “writing a good sonnet,” then most of us do not have language. The interesting scientific question is not “Do chimps have language?” but “What is it that chimps do to communicate?”
Linguistic vagueness is inescapable because words are categories describing a world not built of categories. Even such seemingly ironclad categories as “living” and “nonliving” are ultimately incapable of immunization against the continuity of reality. “Living” and “nonliving” seem easily distinguishable, as indeed is the case if you are comparing a person and a rock. But when you try to give the definitional dividing line, it becomes clear that it is not so easy. Most people will conclude that “living” is defined by “reproducing itself” and will be most annoyed to find that they have defined crystals as “living.” The problem devolves from attempting to capture the virtually continuous nature of nature (at least at the supra-atomic level) with the discrete nature of words.
Here is another example. When I was in school, biologists argued about whether the euglena (one of those little guys you see through a microscope) is a plant or an animal. It has some of the characteristics that define a plant and some that define an animal. This problem of classification did not stop nature from making more euglenas. Today, a new taxonomy has overcome the euglena problem, but no one doubts that we will soon discover a horse with wheels or some other creation that nature whips up to keep us in our taxonomic place.
Thus, all attempts to categorize nature must fail. A new taxonomic system may be without failure momentarily, but nature always eventually creates the uncategorizable. And even sooner, we can always imagine an exception and can specify its properties.
Except in mathematics. Because mathematics defines its universe and is not beholden to nature, only in mathematics can we avoid the taxonomic problem. So, for example, a number is 1 (or 12 or 1,247,351 or pi or whatever) or it is not 1 (or 12 or 1,247,351 or pi or whatever). It cannot be “sort of 1.”
There is, it must be acknowledged, a tiny group of mathematicians who deny “the law of the excluded middle” and claim that we cannot say that a number is 1 or not 1. They are not much fun at parties.
Okay, let’s get this out of the way now and save the mathematicians the awful anticipation of waiting for exposure of this terrible embarrassment. Yes, it is true; they cannot define division by zero. When you or I screw up, we call it a “mistake.” When a scientist screws up, he calls it an “anomaly.” When a Ford executive screws up, he calls it an “Edsel.” When a mathematician screws up, he calls it “undefined.”
Consider what it means to say that “six divided by two is three.” It means that when you multiply two by three you get six. What would it mean to say that “six divided by zero is x”? It would mean that zero times x is six. But there is no number that, when multiplied by zero, equals six. Zero times any number is zero. You can see the problem and the need to call division by zero “undefined.”
When we talk about zero, we are not just talking about any old number (such as 43 or 6,412). We are talking about the beginning of all numbers, the godhead of the mathematical universe. Indeed, once the logician has zero, the name of an “empty set,” he can construct the whole of mathematics.
We in the other academic departments often try to run interference for the mathematicians to save them embarrassment. When we are asked whether “those mathematicians have defined division by zero yet,” we answer that the mathematicians “have had the flu; we’re sure that they will define zero as soon as they recover.” But no one is buying this anymore.
Note that the mathematician’s declaring zero, the birthplace of all numbers, to be functionally undefined resonates with the physicist’s declaring the big bang, the birthplace of the physical universe, to be a singularity. Perhaps at the core of things there can be only mystery. Some would say that this is an expression of the mystery that must remain, even if we were to learn all that can be learned. Others, those with a religious sensibility, would say it is the door to God’s house.
If you have a terrific memory (or were recently in high school), you may remember learning that the different types of numbers we are most familiar with can be graphed on a straight line. First there are the natural numbers (positive integers such as 1, 2, 3, 4 …). Then there are the negative integers (such as the negative $1,200 in your bank account) and 0. Then we can fill in the spaces with fractions that end (such as 1/2, which is 0.5) or do not end (such as 1/3, which is 0.3333333 …, or 2/7, which is .285714285714285714285714 …) but always have repeating decimal patterns. These are called “rational numbers.”
This still leaves some spaces on the number line because the rational numbers (the numbers expressible as a fraction of two integers) have between them an infinity of numbers. Between any two fractions there is an infinity of irrational numbers, numbers not expressible as a fraction composed of two integers. These space fillers are the “irrational numbers.”
The number that is the square root of 2 is an irrational number. You can create fractions that, when squared, get closer and closer to 2, but they are always a tiny bit below or above 2. Thus, the number that, when squared, gives 2 is not a fraction; it is “the square root of two.” “The square root of two” is the name of this number, just like “3” is the name of “three.”
Then came the “imaginary numbers,” multiples of the square root of minus one. At first this scared a lot of mathematicians. There seemed good reason to be scared. The most obvious of these is that a positive number times itself had always given a positive number (1 × 1 = 1 and 4 × 4 = 16). Zero times itself gives zero. And a negative number times itself also gives a positive number (−1 × −1 = 1; −4 × −4 = 16). So what kind of number could it be that, when multiplied by itself, gives a negative number?
Well, the mathematicians finally decided it could be an “imaginary number.” After all, if a new kind of number could not break any of the rules of the old kinds of numbers, then we could not even have had the fractions. Indeed, if a supposed new kind of number did not break any of the rules of the old kinds of numbers, it would not be a new kind of number.
Thus, the unfortunately named “imaginary” numbers are every bit as legitimate as the numbers known long before; the square root of negative one (called “i”) is just as much a number as six or ninety-seven. Such numbers are not any more imaginary than any other numbers. One is tempted to say that they are just as “real,” but, unfortunately, the rational and irrational numbers, taken together, are already named “real.” That is why this new kind of number is called “imaginary.”
As long as a new type of number follows consistent rules, it is legitimate. Consistency in mathematics means that there is no contradiction. If there is a contradiction, a single contradiction, the entire structure collapses. (A system with a contradiction permits one to prove anything.) This is the downside of simplicity and elegance and why mathematics takes such guts. You do not know humiliation until you have published an article in a mathematics journal that has a sophisticated version of 1 + 1 = 7 for all to see.
Incidentally, this “as long as it works and does not introduce inconsistency” is not always easy to swallow. In high school, we learned of Zeno’s paradox, which states, for example, that when you walk toward a wall, you must cover half the distance before reaching the wall, then half that distance, then half that distance, and so on. So you will never reach the wall. But you do reach the wall.
Now, while there are much more sophisticated answers for why you reach the wall, even though it seems you always have some distance to go, the answer we got in high school was that an infinite series (the steps) adds up to a finite amount (the starting distance to the wall). No proof was given for this claim, and it struck me that it had a certain ad hoc quality, that it was whipped up to “explain” why you reached the wall without really explaining things, but only redefining them.
Likewise, we learned that the unending decimal, 0.999 … is the same thing as 1.0. “Says who,” I wondered.
And again, it was claimed that the infinity of even numbers is the same size as the infinity of integers. Now, you would think that the infinity of integers, which includes both the odd and even numbers, is twice as large as the infinity of even numbers. But, no. Size is defined by a matching process (1 matched with 2, 2 matched with 4, etc.). And because you cannot run out of numbers in an infinite series, matching the integers and the even integers is possible, so the infinity of even numbers is as large as, if less dense than, the infinity of all integers. (There are larger infinities—an infinite number of them—that the even numbers, or even all the integers, cannot match. But we will ignore that here.)
This too struck me at first as a bit dubious, an evasive redefinition of things that seemed rather shoddy. But, as indicated, my dubiosity was not justified because all of these practices, of which I had unfairly been skeptical, enlarged mathematical knowledge without introducing any incorrectness or contradiction, just as had been the case with new types of numbers, such as fractions and irrationals.
Now, let us go back to types of numbers. There are many other types of numbers. Next on the ladder is “complex numbers,” which combine real and imaginary numbers. The list goes on until it reaches a level of abstraction that very, very few have observed.
While descriptions of the other types of numbers would take us too far afield even if I knew enough about these numbers to describe them, it is worth making a point about “infinity” because the very properties that make infinities new types of numbers seem so unintuitive.
If you have a hundred hamburgers and a hundred and one people to eat them, the hamburgers and the people will not match up; there will be one person who does not get a hamburger. This is pretty obvious. But the reason the hamburgers cannot be matched up one-to-one with the people is that we are dealing with finite numbers and there are not enough hamburgers. Similarly, if a box holds fifty white balls and fifty red balls and you take out a white ball, the odds (slightly) favor the next ball’s being red. There is an end to the number of white balls. So when the first ball you choose is white, there are more red balls left. This principle would, of course, obtain if the first ball were red.
However, what if we have a box with an infinity of red and white balls? Because infinities never get to an end, taking out a white ball does not imply anything about the color of the next ball. Failure to understand this has led to a lot of people losing their money. The fact that red comes up twelve times in a row on an unbiased roulette wheel does not change the odds from even money on the thirteenth spin of the wheel, which is also the case with the box with an infinity of balls.
It is obvious that the infinity of white balls is the same size as the infinity of red balls. It is obvious in the same sense that twelve hamburgers and twelve hamburger eaters are groups of the same size: they can be matched one-to-one.
Consider, however, two infinities, one of which seems obviously bigger than the other. We might choose A, the infinite set of integers (1, 2, 3, 4 …), and B, the infinite set of even numbers (2, 4, 6, 8 …). Common sense tells us that the first set is bigger—twice as big. But remember that the way we measure the size of a group, finite or infinite, is to match them one-to-one. Can groups A and B be matched?
Yes. In group A, 1 is matched with 2 from group B. Then 2 in group A is matched with 4 from group B—and so on forever. Group A may in some way be more densely packed, but the two infinities are the same size. Infinities of this type are called “countable” because there is an integer to match up to each element. (Note that the infinity of positive and negative integers is the same size as the infinity of just the positive integers.)
Despite the fact that we have seen that infinities can have counterintuitive properties, it seems obvious that all infinities are the same size. In other words, you just match each element in any infinity with an integer and you have shown that all infinities are the same size.
Note, however, as Georg Cantor did a hundred years ago in one of the greatest of all mathematical insights, that it is not self-evident that you can always match every element of an infinity with an integer. Here is Cantor’s wonderful proof that you cannot always do so.
Let us list all of the numbers—including unending decimals—between 0 and 1. We will treat numbers that are not unending by their unending equivalents (i.e., 1 is 1.00000 … . Now, we cannot in reality make an infinite list, but we can list a few numbers and demonstrate that, however long the list of numbers, there will always be a number not included in the list. By doing this, Cantor proved that there is a larger infinity than the ones that are countable. His proof—called Cantor’s Diagonal Proof—gives new meaning to “elegant.”
Cantor made such a list. It does not matter what order the numbers are in, so let’s arbitrarily choose numbers between zero and one. (The same point obtains whatever range one chooses, as all countable numbers are countable with the countable numbers between zero and one.)
0.763498276 …
0.500000000 …
0.267037143 …
0.987654321 …
0.555555555 …
0.273403949 …
etc.
Now, draw a diagonal line, beginning with the first digit of the first number, the second digit of the second number, the third digit of the third number, ad infinitum. The unending diagonal number we get will begin 7074553 … .
Add 1 to each digit of our diagonal number. The new number will begin 8185664 … . Notice that the diagonal number we end up with cannot be the same as the first number because its first digit is an 8, not a 7. It cannot be the same as the second number because its second digit is a 7, not a 6. It cannot be the same as the third number … well, you get the idea. The infinity of listed numbers does not include the diagonal number. Therefore, the diagonal number is a member of an infinity that cannot be counted (i.e., an infinity larger than the countable infinities). This demonstrates that some infinities are larger than others. Moreover, because the same sort of argument can be made against any infinity claiming to be the largest, there is no more a largest infinity than there is a largest number.
The mathematician and science author John L. Casti, has a marvelous way of demonstrating the diagonal proof that is easier for some people to see:
Consider these six names:
Twain
fUrman
beRry
sprIng
lockNer
herzoG
Create a word using the letter after the first letter of the first word, the letter after the second letter of the second word, and so on. You will get “turing.” This must be different from every word on the list because it will differ from the first word by a different first letter, the second word by a different second letter, and so on.
I hope you now have some slight feeling for what elegance is. This book is a compendium of examples of elegance, with some other stuff thrown in just because it is interesting. The point is not to explain in any detail, but to give the reader a taste that will, I hope, lead to exploration of one of the many paths hinted at in these pages.
I cannot stress this final point enough: This book will not teach you mathematics. I am far, far from qualified to do this, and there are many, many mathematicians—some of them, unfortunately, unemployed—who are eminently qualified to do so.
My purpose is simply to provide a sort of tasting menu to whet your appetite. I attempt to serve up fascinating and beautiful findings of mathematics, in the hope of persuading you to follow up on those you find most appealing by searching out the works of mathematicians who have described those findings in detail.
Picture a die, like one used for gambling only much smaller, just one-tenth of an inch on each side. A million of these lined up horizontally will stretch out about one and two-thirds miles. Ignoring the height and width of the dice, this is a one-dimensional arrangement.
Now, let’s arrange the dice flat in a square, a thousand dice long and a thousand dice wide. The square holding the million dice will be a bit over eight feet (a hundred inches—one-tenth inch times one thousand) on each side. Ignoring the third dimension, the height of the die, this is a two-dimensional arrangement.
Finally, let’s make a square of a hundred dice by a hundred dice (i.e., 100 × 100 = 10,000 dice). Now make ninety-nine more such squares and pile each on top of the previous one. You now have a cube, 100 × 100 × 100 = 1,000,000. This cube will be only ten inches by ten inches by ten inches. This is a three-dimensional arrangement, probably much smaller than you would have guessed.
When Carl Gauss, the greatest of all mathematicians in some people’s view, was seven years old, his teacher did not feel like lecturing to his second graders one day. So he gave the boys a math problem that would easily take them the entire class: add the numbers from 1 to 100. Seconds later, young Carl raised his hand and told the teacher that the sum was 5,050. Both the teacher and the other students were dumbfounded.
Carl had realized that he could add pairs from the two ends: that is, (1 + 100) + (2 + 99) + (3 + 98) … . Each pair of numbers adds up to 101. Because there are fifty pairs of numbers, the total is 50 × 101, or 5,050.
Mathematics is full of facts that you would never guess are facts. For example, Joseph Louis Lagrange proved that every natural number (i.e., positive integer) is equal to the sum of four or fewer square numbers: for example, 23 = 9 + 9 + 4 + 1 = 32 + 32 + 22 + 12. Trial and error establishes that not all natural numbers can be represented as the sum of fewer than four squares, though some can: for example, 18 = 32 + 32.
Most people know that the more often interest is compounded, the more money you end up with. Banks know that people know this and compete by offering ever-more-frequent compounding. What banks also know is that the difference between semiannual compounding and daily, or even second-by-second, compounding is insignificant.
Let’s say you have $100 in a savings account and the bank compounds the interest annually (i.e., once a year). Your money will double (i.e., grow to $200) in, for example, ten years. Now, if the bank compounds the interest semiannually, your $100 will grow to about $269, rather than $200, in the same amount of time.
This might lead you to the conclusion that ever-more-frequent compounding will engender similar increases. Banks occasionally exploit this belief by compounding ever-more-frequently.
But that conclusion is not correct. In fact, even if the $100 dollars is compounded second-by-second, it will grow to only about $272 (actually a tad less) in the same time that semiannual compounding would make it grow to $269.
This is because compounding approaches the limit of the “e” (2.7182 …). Why? You might well ask why does e equal 2.7182 … but you will not be satisfied with the answer, which is “because it does.” The e is an irrational number—a number not expressible as a fraction composed of two integers—that is, as we have seen was the case with the square root of 2, just as legitimate a number as 6 or 3/4. The e is one of those numbers, like π, that is a mathematical constant that keeps popping up in divergent mathematical areas for no obvious reason.
Each increase in the number of times your savings is compounded does increase the total; daily compounding is better than weekly compounding. But very soon the increments become so small as to be insignificant.
The reason that we are surprised to find that second-by-second compounding does not make us far richer than monthly compounding is that we tend to think in terms of what is known as linear functions. If you make ten dollars an hour and work for four hours, you get forty dollars. If you work for eight hours, you get eighty dollars, twice as much as for four hours. These are linear functions; the rates of increase are what you intuitively expect.
Many functions, however, are nonlinear and behave in unexpected ways. Compound interest is nonlinear. While there is always some increase when you increase the number of times you compound in a given period, the increases get smaller and smaller very quickly.