Many of the greatest discoveries of the twentieth century are discoveries of the inherent limitations that will forever withhold knowledge from even our smartest descendants.
Gödel’s Theorem: For millennia mathematicians dreamt of discovering an all-encompassing mathematical system in which it was possible to prove every mathematical truth. Half a century ago, however, Kurt Gödel proved that every nontrivial logical and mathematical system will possess truths not provable in that system. (Trust me, you do not want to read here how he proved this.)
We can demonstrate the truths using a more inclusive system, but we cannot prove that a larger system will have truths without going to an even more robust system. (Ad infinitum.)
The dream is dead.
Heisenberg’s Uncertainty Principle: There is a limit beyond which full knowledge is impossible. In other words, there are things we can never, even in principle, know. For example, we can never learn both the position and velocity of a subatomic particle. We can learn either with as much precision as we wish, but the more precisely we know one, the less precisely we can know the other.
Here’s why. When we measure something, we use particles to see the thing we measure and to make the measurement. For example, we use photons, particles of light, to see and measure the tabletop we wish to measure. The fact that we use these particles makes no practical difference when we are viewing and measuring tabletops, elephants, or even bacteria; the effect of the tiny photons on the thing we measure is essentially nonexistent and has no more effect than a ping-pong ball thrown at a mountain range. However, when we attempt to measure particles on the tiny scale of the particles we use to measure them, the effect of the latter on the former is tremendous and sets limits on what we can ever know about the measured particles.
Unpredictability: Even when a system is determinative, it is often the case that complexity and sensitive dependence on initial conditions renders the system forever unknowable in practice. This is why we will never be any good at predicting weather conditions more than a couple of weeks ahead.
Unsolvable in Practice (Probably): There are a large number of problems that are probably inherently unsolvable in a finite amount of time. Some of these are problems that you would think would be easy pickings for the mathematician armed with a computer. The traveling salesman problem is a good example.
Say you are a traveling salesman who must visit a certain number of cities, and you want to take the shortest route possible. You might think that there is no practical application of mathematics more easily accomplished than this. And, in fact, the problem is easy to solve by trial and error if the number of cities is relatively small.
For any specific number of cities, the number of routes is the number of cities “factorial”; take my word for this. The meaning of “factorial—which is indicated by an exclamation point—is most easily seen by example: 5! means 5 × 4 × 3 × 2 × 1, or 120. Thus, there are 120 different routes for visiting five cities one time each.
If you must visit ten cities, the number of routes your trial-and-error methods must compare is over three and a half million. To be sure, you could obviously immediately disregard inefficient routes such as New York to Los Angeles to Boston to Seattle … , but there would remain a daunting number of possible routes that would require actual comparison. With the help of a computer you could do this in a reasonable amount of time, but this is for only ten cities.
Perhaps you must visit thirty thousand cities. A mathematician cannot give you the shortest route that reaches all of the cities; he cannot even tell you with certainty whether there is any way other than trial and error for finding the shortest route. There are shortcuts that guarantee a route not more than about 20 percent longer than the shortest route, but no known method other than trial and error that guarantees the shortest route.
Now, you may well say, “Who has to visit thirty thousand cities?” And, of course, you would be right. But there are a great many analogous problems (circuit design, storing and transporting millions of packages, and the like) that have the equivalent of many thousands of “cities.” A saving of a few percent in the length of trucking routes, for example, can save the industry billions of dollars a year.
If a method better than trial and error for guaranteeing the most efficient (shortest, quickest, etc.) answer for even one of these problems is found, it will work for all of them. If, on the other hand, it can be proved that there can be no such method for even one of these problems, then there can be no such method for any of them. Most mathematicians believe that there can be no such method.
You no doubt remember the Pythagorean theorem:
a2 + b2 = c2
Pythagoras was considering right triangles. His theorem states that the lengths of the two shorter sides determine the length of the longer side (or the other way around, which is the same thing). Specifically, the square of one short side plus the square of the other short side equals the square of the long side (the hypotenuse). In other words, a2 + b2 = c2. If the two short sides are 3 and 4 inches, the hypotenuse is 5 (i.e., 9 + 16 = 25; the square root of 25 is 5). The lengths need not be integers (nor need they be consecutive numbers).
Now, what about a solution to the equation ax + bx = cx when x (the exponent—the power) is an integer greater than two? Is there a nontrivial solution? (A trivial solution would be that a, b, and c are all 0 and x is anything you want.)
The question of whether there is a nontrivial solution to the equation ax + bx = cx (x is an integer > 2; a and b and c are positive integers) was the most famous unsolved problem in mathematics. It is referred to as Fermat’s Last Theorem, after Pierre de Fermat (1601–1665), who wrote in the margin of a book that he had “discovered a truly marvelous” proof that there could be no solution.
Fermat did not provide the proof. His margin note added that “demonstration of this proposition (is one) that this margin is too narrow to contain.” Now, who would actually bother to write those words if he did not plan on someone’s reading them? It is not inconceivable that Fermat was a practical joker of a particularly insidious type—after all, the man wrote in Latin—and that he knew he could flummox three and a half centuries of great mathematicians by claiming he had a proof.
In any case, there are many reasons for believing that Fermat found his “proof,” if he had one, to be faulty. If someone had found just one solution for ax + bx = cx, where x is greater than 2, that exception to Fermat’s claim would, of course, have been sufficient to disprove Fermat’s conjecture and to demonstrate that there could be no valid proof. No one ever found such an exception, but failure to find an exception can never prove that there is none. The exception might be the next number after the one you where you stopped checking. Here we see an asymmetry: failure to find any exception does not prove the conjecture is true, but one exception does prove the claim is untrue.
However, it was proved that there is no solution for exponents less than one million. Thus, do not bother checking to see whether, say, 163 + 233 = 473, or any other combination. It does not. If there were a solution, it would have had to include numbers unimaginably much larger than those necessary to count all the particles in the universe.
Unlike science, which deals in probability-like realities, mathematics recognizes only proofs and solutions. The Fermat question can be answered only by finding a solution to the equation, which would show the theorem to be false, or proving that there could not be one. Incidentally, as Ian Stewart (Game, Set, and Math: Enigmas and Conundrums) has pointed out, there are many “close calls”:
63 + 83 = 93 − 1
The philosopher W. V. Quine has shown that Fermat’s Last Theorem can be stated in terms purely of power, rather than addition and power. This is the first of a few entries in this book that are included simply because they are nifty. An explanation would take us too far afield—out to that area where I would have no idea what I was talking about.
Fermat’s theorem has finally, after three centuries, been proved, by the British mathematician Sir Andrew John Wiles. There is no solution to ax + bx = cx (x is an integer > 2; a and b and c are positive integers). Do not even ask about what the proof was. It took Professor Wiles over seven years and depended on techniques discovered in areas of mathematics unknown for centuries after Fermat. Of course, it is always possible that Fermat had a different, simple proof. But do not bet on it.
Until recently, most mathematicians merely felt, rather than knew, that there is no solution to the equation. This was certainly plausible (and, as we have seen, turned out to be true). One would think that, if there were a solution, it would show up long before the unimaginably high numbers that are the lowest that could possibly provide a solution.
Some “Fermat-like” equations do have simple solutions that are easy to find. For example:
a3 + b3 + c3 = d3 is solved by 33 + 43 + 53 = 63
However, there are many equations that have very high numbers as first solutions, and this renders inductive thinking in mathematics extremely dangerous.
Logic and mathematics work by deduction. Each step is logically entailed in the previous step(s). Thus, if (A) Andy is taller than Bob, and (B) Bob is taller than Charles, then (C) Andy is taller than Charles. You do not have to measure Andy and Charles to know that Andy is taller than Charles.
Logic and mathematics do not even care whether A and B are true; their interest is only in the relationship between the premises. That is one of the things that makes math so great: you do not have to know anything. Science, while nearly always making use of logic and mathematics, works differently. It considers empirical truths (“facts”) and generalizes about them. Where mathematics is certain, science is always tentative.
We have excellent reasons for believing that no cow can fly, but we always leave open the possibility that tomorrow we will spot a flying cow and all start carrying umbrellas. A science that is not tentative in this way would subordinate nature to science. And the first rule of science is that nature is never wrong, because the explanation of nature is the very purpose of science. That is why mathematics has (pretty much) only proof, while science can never have proof. The dangers of induction in mathematics is nicely demonstrated by Albert H. Beiler in his book Recreations in the Theory of Numbers.
Suppose you multiplied 2 by itself 7 times and subtracted 2 (multiplying 2 by itself 7 times is 2 to the 7th power, or 27):
27 − 2 = 126
This is evenly divisible by the exponent (i.e., 7):
126/7 = 18
Now let’s try multiplying 2 by itself 5 times and subtracting 2:
25 − 2 = 30
This is also evenly divisible by the exponent (i.e., 5):
30/5 = 6
You might be tempted to conclude that 2x − 2 is always divisible by x (x stands for any integer you choose, in this last example 5). But you then try
24 − 2 = 14 (i.e., 16 − 2 = 14), which is not evenly divisible by 4,
and
26 − 2 = 62, which is not evenly divisible by 6,
and
28 − 2 = 254, which is not evenly divisible by 8.
Aha, you say, 2x − 2 is always divisible by 2 only when x is an odd number (as in our first example, 27 − 2 = 126). But you then try
29 − 2 = 510 and find that 510 is not evenly divisible by 9.
So where does that leave you? Well, x = 2 and x = 5 worked, and this might suggest to you that when x is a prime number 2x − 2 will be divisible by x. But when x is not a prime number, 2x − 2 will not be divisible by x. (Reminder: a prime number is an integer divisible without a remainder by only 1 and itself.)
At this point you are probably not sanguine about your “discovery,” having been cowed by dashed hopes so many times. But now you learn that Fermat proved that 2x − 2 is always divisible by p when p is prime.
You will feel so proud that it will not even bother you to learn that when x is not prime, 2x − 2 is sometimes divisible by x and sometimes not. And you will be very glad that you did not continue using trial and error to try to find an exception to the rule that 2x − 2 is not divisible by p when p is not prime. You will be glad not because the rule is true (so that you could have never found an exception), but because you would not have continued trial and error long enough to find the exception and would have incorrectly concluded that 2p − 2 is never evenly divisible by p when p is not prime. For it would have taken you a long time to reach the smallest exception: 2341 − 2 is evenly divisible by 341, and 341 is not a prime number. (341 equals 11 times 31.)
Incidentally, we saw that 2p − 2 is always divisible by p when p is prime. The 2s need not be 2s; they can be any integer. For example, 53 − 5 = 120 (i.e., 125 − 5 = 120), and 120 is evenly divisible by 3. Likewise, 92 − 9 = 72, and 72 leaves no remainder when divided by 2.
Here is another example of the dangers of induction. Consider Euler’s incorrect conjecture that there is no solution to w4 + x4 + y4 = z4. There are solutions to this equation. The first (i.e., smallest) was discovered by American mathematician Roger Frye:
95,8004 + 217,5194 + 414,5604 = 422,4814
(Damn, I stopped three short.)
A mathematical system that always permits addition, multiplication, and subtraction without resulting in a new kind of number is called a “ring.” For example, the mathematical system that includes only the positive and negative integers (and zero) is a ring because when you add, subtract, or multiply any numbers in the system, you get a number in the system (for example, 7 + 9 = 16; 6 − 6 = 0; 12–16 = −4). However, this “ring” does not (always) permit division: 10 divided by 6 gives a fraction, and there are no fractions in the mathematical system that includes only the positive and negative integers (and zero).
A system that permits addition, subtraction, multiplication, and division is a called a “field.” The integers alone do not constitute a field, as we have just seen.
You will remember that the rational numbers (numbers expressible as a fraction, such as 3/4, 12/12, or 4/3) together with the irrational numbers (real numbers that cannot be so expressed, such as the square root of 2) constitute the line we call the x-axis. The numbers on this line constitute a field because not just addition, subtraction, and multiplication always result in a number in the system; division also does (12 divided by 5 = 2 2/5, which is a number in this system).
The purpose of these terms is to enable mathematicians to communicate that a set of numbers has certain attributes without having to redefine the attributes relevant to the specific case. There are types of rings and fields that permit other operations without needing to go outside the system, but discussion of them would take us too far afield.
Say you have an infinitely long line. This is a one-dimensional entity that is described as having one degree of freedom because the only direction you can move on the line is forward or backward (i.e., forward toward plus infinity or back toward minus infinity). Degrees of freedom are important in social statistics because each degree of freedom can represent one variable, such as like height, country of birth, and so on.
Now, say that you put a marker anywhere on the line you like. You then flip a coin. If the coin comes up heads, you move the marker one integer toward plus infinity; if tails, you move it one integer toward minus infinity. You keep tossing the coin and following the same rule. What are the chances that you will someday return to your starting point?
You may, of course, return to your starting point in just two tosses of the coin (head-tail or tail-head). But it could also be that your first million tosses are heads (unlikely as this is) and then your coin begins to act normal (coming up heads about half the time).
Even if this were to happen, sooner or later you will return to your starting place. You probably guessed this. After all, you have an infinite number of tosses, and sooner or later (probably an unimaginably long time later) you will get back to your starting place). Such is the nature of infinity.
Let’s now take an infinite checkerboard instead of a line. Now we are working in two dimensions (forward/back/left/right; diagonal moves do not change anything, so we will ignore them here). Instead of a coin, we will use a spinner marked “forward,” “back,” “left,” and “right.”
Will you ever return to the square on the checkerboard where you started? Yes, indeedy. It may be in two moves or it may take a quadrillion years. But, given an infinite number of spins, you will definitely return home.
Now, let’s take a three-dimensional entity (say, an infinite pile of infinite checkerboards) and a spinner with sixteen possibilities. There are sixteen because, assuming you must move to another level and not stay on the same level, you must move to one of the eight adjacent squares on the level above or one of the eight adjacent squares on the level below.
You have forever, so, once again, there is a 100 percent probability that you will return to your starting point, right? Well, actually, no. There is a mathematical proof that you have only a one-third chance of ever returning to your starting square.
As Ian Stewart, the British mathematician and science-fiction author, points out in Scientific American, this means that if you are lost in the desert, you will definitely get back to your starting point at the oasis if you walk long enough (though you may be walking for a million or more years, so take lots of water). But if you are lost in space, you have only about a one-third chance of ever getting home.
Just between us, I do not understand this either. After all, you have forever. But when a proof conflicts with intuition and common sense, proof wins.
How many people have to be in a room for the odds to be greater than fifty-fifty that two of them have the same month and day birthday? One hundred? Two hundred? Surprisingly, the answer is twenty-three. (We assume that equal numbers of people are born each day, and leap year does not change the outcome.)
If there is only one person, then, obviously, the probability is 0, (There is no one to match.) If there are two people, the probability is 365/365 times 364/365 that the two do not match, and so on, until the twenty-third person arrives. Then the probability that no two have the same birthday is 365/365 times 364/365 times 363/365 and so on to 342/365. Multiply those numbers and you will find that they equal less than one-half. If the probability that no two people of the twenty-three have the same birthday, then the probability that two do have the same birthday is greater than one-half.
Johnny Carson once asked whether any member of the audience shared his birthday. Finding that no one did, he lightly mocked the guest who had given him the problem. Johnny misunderstood the problem; the problem did not claim that there is a fifty-fifty likelihood that one of twenty-three people will have a birthday on a specific date (in this case, Johnny’s birthday), but that two of the twenty-three will have the same birthday, a much greater likelihood. In other words, picture twenty-three people being asked whether any of them has a birthday on, say, August 12. Now picture twenty-three people being asked whether any two of them have the same birthday. The latter is clearly much more likely to be the case. Indeed, the probability of one of the twenty-three having a specific birthday is one in two hundred and fifty-three, according to mathematician Warren Weaver in Lady Luck: The Theory of Probability.
The gist of the reasoning is this: Think of the problem as asking the probability that no two people share the same birthday. Now take two people. The odds of the second person having the same birthday as the first are 364 to 1. Now take three people. The odds of any two of the three having the same birthday is 364 × 363/365 × 365. When you get to twenty-three people, you find that the probability of two not sharing a birthday is less than 0.50, so the probability that two do share a birthday is greater than 0.50 (i.e., more likely than not).
In general, the probability of n people not sharing a birthday is 364 × 363 × … . (364 - n)/365n.
P-KB4 P-K3
P-KN4 Q-R5
Black mates White.
(Martin Gardner, Wheels, Life, and Other Mathematical Amusements)
Mathematically, a “degree of freedom,” a “dimension,” and a “variable” are all the same thing. To say that we can move in three dimensions is equivalent to saying that our movement is a function of three variables (up-down, left-right, forward-back).
A human being is the representation of so many variables (sex, race, income, status, position in family, etc.) that it is amazing that social scientists ever find out anything. They work with what is effectively an “infinite dimensional manifold.”
You may have noticed that scientific articles very rarely use the words “billion,” “trillion,” “quadrillion,” and so forth. The primary reason for this is the clear superiority of scientific notation. Scientific notation puts numbers in terms of exponents. For example, a billion (1,000,000,000) is 109 (the 9 means 10 multiplied by itself nine times, which equals 1 followed by nine zeros); 1/1,000,000,000 is 10−9; and 1, 230,000,000 is 1.23 × 10−9. Scientific notation is much better when you are dealing with huge numbers.
But another reason why you never see “billion,” “trillion,” and so on, in scientific writing is because when the British use the words “billion,” “trillion,” or “quadrillion,” they refer to different numbers than when Americans use the same words, and this inevitably causes confusion. If you do run into these terms, however, you need not be confused (as long as you know whether British or American usage is being employed). There is a simple way to remember the various meanings.
In British usage, the prefix indicates the number of sets of six zeros. Thus, a British “billion” is a 1 followed by two sets of six zeros (1,000,000,000,000), and a British “trillion” is a 1 followed by three sets of six zeros (1,000,000,000,000,000,000).
In American usage, the prefix is one less than the number of sets of three zeros. Thus, an American “billion” is a 1 followed by three sets of three zeros (1,000,000,000), and an American “trillion” is a 1 followed by four sets of three zeros 1,000,000,000,000).
Note that this all works for a “million” (considering “mi” means “one”). Both British and American usage call 1,000,000 a “million.” This number has one set of six zeros (British usage) and two sets of three zeros (American usage).
| American | British | |
|---|---|---|
| Million | 1,000,000 | Million |
| Billion | 1,000,000,000 | Thousand million |
| Trillion | 1,000,000,000,000 | Billion |
| Quadrillion | 1,000,000,000,000,000 | Thousand billion |
| Quintillion | 1,000,000,000,000,000,000 | Trillion |
In other words, an American “trillion” (four sets of three zeros) is a British “billion” (two sets of six zeros). An American “quintillion” (six sets of three zeros) is a British “trillion” (three sets of six zeros).
The difference between the systems arises because Americans consider a thousandfold increase as requiring a new level, while the British see the need for a word change only with a millionfold increase (for numbers over one million). Thus, the Americans introduce “billion” at one thousand million, while the British call this number “one thousand million” and introduce the term “billion” at the million million level.
One can imagine a more elegant system in which a thousand thousand is a “million,” a million million is a “billion,” a billion billion is a “trillion,” a trillion trillion is a “quadrillion,” and so on. But no nation uses this system, which is probably just as well.
Say you wanted the world to have many more girls than boys. It doesn’t matter why; this is a math problem. You might then think that the following is a great idea.
If every woman kept having babies until she had a boy, but stopped having babies as soon as she had a boy, then you would think there would be a lot more girls. Things seem like this: Only women whose first (and therefore only) child is a boy will have more boys than girls (i.e., one boy, no girls). Some women, those who have a girl and then a boy, will have the same number of boys and girls (one of each). All other women will have more girls; for example, the woman who has six girls and then a boy will have five more girls than boys (whereas no woman can have sons outnumber daughters by more than one and this is the case only with the woman who has a boy and then, as she must, stops at that point).
Well, with all those women who have a number of girls before having a boy, and with no women having more than one boy, it sure seems like there would be many more girls than boys in the world. It makes sense, but is completely wrong.
This is why. For every first child who is a girl, there will be one who is a boy. The women who have boys stop having babies. The women who have girls have a second baby. But half of these second babies are boys, so there are still as many boys as girls. The other half, the women who now have two girls, have a third baby. Half of these third babies are boys—and so forth.
Let’s use numbers now. If 60 women have boys and 60 women have girls, the ratio is 1 boy to 1 girl. The 60 women who had boys stop there, while the 60 women who had girls give birth a second time. Now there are 30 new girls and 30 new boys for a total of 90 boys and 90 girls. The ratio is still 1 girl to 1 boy. The 30 women with only girls give birth a third time. Now there are 105 boys and 105 girls. And so it goes.1
This may make things more clear. Half of the women have a boy, but no girl. No women have a girl, but no boy. The number of the boys of women whose firstborn is a boy plus the number of boys who follow one or more girls equals the number of girls.
On the other hand, this may be the rare situation where mathematics makes things less clear than common sense. After all, half of all births are girls and half are boys. Presenting the problem in the way it has been presented serves only to introduce an element that is irrelevant.
1.From Marilyn vos Savant’s column, Ask Marilyn, PARADE, October 19, 1997.