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David Gale* sible not only to professional mathematicians but also to rea- sonably knowledgeable and interested nonmathematicians.

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Avoiding Numbers One of the more frustrating things about being a math- ematician is that almost no one other than professional scientists has the faintest idea what the subject is about or what it is we do. Ask a person chosen at random what the central concept of mathematics is and the answer will probably be numbers. Now I don't know myself what mathematics is about (more on this later), but I'm quite sure the central concept is not numbers. Still there is a central concept. It is proof. The examples to follow are intended to illustrate how sometimes the reflex ac- tion of trying to use numbers can make proofs more dif- ficult.

Counting Beans

At the most naive level, consider the problem of decid- ing which of two jars contains more beans. Easy, you say, just count the number of beans in each jar. True, but a person who didn't know how to count might find the quicker solution. Remove pairs of beans simultane- ously from each jar, and stop when one of them is empty. This fable illustrates (1) the advantage of "par- allel computation" and (2) at a very primitive level, the notion of cardinality of sets via one-to-one correspon- dences.

Mixing Wine

A bit more sophisticated, but the same idea is illustrated in this well-known problem. You are given a glass of red and a glass of white wine and are told to take a spoonful of the red, add it to the white, and mix com- pletely. Then take a spoonful of the mixture and return it to the red. Question: Is there more red in the white or white in the red? (Warning: this problem may have the effect of alienating people, who may argue vehemently for wrong answers like, "You added pure red to the

white but a mixture to the red so there must be more red in the white.")

Now of course one can work it out quantitatively. Let R and W be the given volumes of red and white wine (measured in some units) and let S be the volume of the spoon. Then it is not hard to get the formula for the amounts in question in terms of R, W, and S. The point is that this is all unnecessary, and, further, the condition of complete mixing is a red herring. Here is an alterna- tive scenario. You have a full glass of wine (of either color). Throw some of it in the ocean, wait 5 minutes, and then refill the glass from the now slightly polluted ocean. Question: Is there more wine in the ocean or (pure) ocean in the wine? Answer: The two amounts are equal. Reason: At the end of the experiment the ocean in the wine glass has re- placed the wine that was originally there. Exercise: Where is the missing wine?

The significance of this example is that it not only avoids numbers but, one can argue, it avoids mathe- matics of any sort, using nothing but common sense. I will return to this matter in the next section.

Qualitative Geometry

The preceding questions and those to follow are quali- tative as opposed to quantitative problems. A qualita- tive question deals with concepts like more, fewer, greater, equal, less, bigger, smaller, whereas a quantitative prob- lem involves attaching numbers to objects. The exam- ple par excellence of a qualitative theory, and I was sur- prised to realize this, is the geometry of Euclid. This is not to be confused with what we now refer to as Euclidean geometry with the "Euclidean metric." There is no metric, no concept of distance, no notion of the size of an angle in Euclid's Elements. It is true that Euclid adds, subtracts, and compares objects, but these objects are the segments and angles themselves, not their lengths or sizes. Here, for example, is the triangle in- equality, the celebrated Proposition XX.

*Column editor's address: Department of Mathematics, University of California, Berkeley, CA 94720 USA.

In any triangle two sides taken together are greater than the remaining side.

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 3 �9 1996 Springer-Verlag New York 23

Here is Euclid's proof: Given triangle ABC, extend side BC to D so that DC

equals AC (here and elsewhere for "equals" read "is congruent to").

D

C

B

Then, since triangle ADC is isoceles, angles a and ]3 are equal (Proposition V). Further, a is less than (read, "con- tained in") T. But in triangle ABD, the side opposite ~, is AC+CB and the side opposite/8 is AB, so by (previ- ously proved) Proposition XIX, DC+CB = AC+CB is greater than AB.

Qualitative all the way! But let us continue to back- track. Proposition XIX is the immediate converse of

To show ]3 is greater than % connect A to the midpoint M of BC and extend the line AM to D so that MD equals AM. Then triangles AMC and DMB are congruent (side-angle-side), so T equals 3, which is contained in, hence less than,/3.

Remark. We know, and so does Euclid, the much stronger fact that the exterior angle is the sum of the op- posite interior angles (Proposition XXXII). Why then does Euclid settle for this much weaker result? The rea- son is surely that he wants to go as far as possible with- out having to make use of the parallel postulate. In to- day's language we would say that the three propositions proved above are theorems of absolute geometry, hold- ing in elliptic and hyperbolic as well as Euclidean geom- etry. Once again we see an example of the astonishing sophistication of the mathematics of 2000 years ago.

Stacking Rectangles

Two congruent rectangles are stacked one on top of the other as shown in the figure below on the left. Question: Does the upper rectangle cover more or less than half of the lower one?

PROPOSITION XVIII. In any triangle the greater side subtends the greater angle.

Referring to the figure below, we suppose AB is longer than BC.

Choose D on AB so that BD equals BC. Then CBD is isoceles, so angles E and 3 are equal (Prop V); but e is contained in, hence less than % and 3, an exterior angle of triangle ADC, is greater than a.

This in turn used PROPOSITION XVI. The exterior angle of a triangle is greater than either of the interior opposite angles.

Proof.

C

~ D

M

A B

The first impulse is probably to bring on the numbers, i.e., calculate the areas of the two uncovered triangles. One can do this, of course, but even after arriving at the somewhat messy expression, it is not at all obvious which way the inequality should go, whereas a little thought suggests drawing the two lines shown in the figure on the right and observing the two pairs of con- gruent triangles. Indeed, one does not even need to have the concept of area to present the problem. We need only agree that one region is bigger than another if the second is "piecewise congruent" to a subset of the first.

A Look Back

In winding up my 5-year tenure as editor of the "Mathematical Entertainments" column it seems appro- priate to look back, try to take stock, and report any cos- mic illuminations I may have gained from the experience.

The 22 columns starting in the winter of 1991 have treated a sort of hodgepodge of topics. Here is a rough inventory.

24 THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 3, 1996

Topic No. of columns

1. Sequences 3 2. Games and paradoxes 3 3. Automata (the Ant) 3 4. Tilings by rectangles 2 5. Triangles 3 6. Privacy-preserving protocols 2 7. Optimization 2

It's pretty hard to detect any unifying theme in all this. Computers played a major role in the first five topics, sometimes in solving problems but more often in posing them. In contemplating these "computer-generated mys- teries" it seems to me that the problems they posed were of two very different sorts, solvable and unsolvable.

Examples Our very first column featured the sequences discovered by Michael Somos and their generalizations. These sequences are given by rational recursions, but the terms ttLrn out for no apparent reason to be integers. Ingenious arguments have proved integrality for some cases, but most cases remain unexplained. Here is an ex- ample which as far as I know is still mysterious. We con- sider the sequence (ak) given by the recursion,

ak+s = (ak+lak+7 + (ak+4)2)/ak,

where the first eight terms are 1. This is just the simplest case of a 4-parameter family

of algorithms which grind out integers as far as the eye can see. How come? Now I believe this is a solvable prob- lem, by which I mean there exists a proof that these gen- eralized Somos sequences always yield integers. This is not to say that anyone will necessarily find the proof; but one feels strongly that such a proof exists.

The above is in contrast to other sequence problems which I speculate are unsolvable. A good example is the shuffle problem which appeared back in the winter is- sue of 1992. In case you have forgotten, we considered a countably infinite deck of cards numbered 1, 2 . . . . . An n-shuffle consists of taking the first n cards of the deck and interlacing them with the next n. Thus, after a 5-shuffle the deck would look like: 6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, 12, 13 . . . . . We are told to start with a 1-shuf- fie, giving 2, 1, 3, 4 , . . . , then a 2-shuffle, giving 3, 2, 4, 1, 5, 6 . . . . . and so on; and the conjecture, essentially due to Richard Guy, is that if you keep going long enough, every card will eventually rise to the top of the deck. What about "experimental evidence"? As was reported in the column, all numbers up to 5000 have been tested, and they all do manage to rise to the top, although some of them take a long time to get there. It takes more than 250 million shuffles for card 54 to make it, and more than 21 billion for card 3464.

Now I have no idea whether Guy's conjecture is true or false, but I do have a strong feeling that there simply is no proof either way, meaning no proof from any set

of sensible axioms. This is based on the hunch that when phenomena exhibit this sort of seemingly random, chaotic behavior, it is an indication of unprovability.

A similar and simpler question was raised in the 1992 summer issue. Consider the sequence consisting of the powers of 2. Do all but a finite number of these have, say, a seven in their decimal expansion? The largest known sevenless power of 2 (which, amazingly, is also fiveless and nineless) is

271 = 2 3 6 1 1 8 3 2 4 1 4 3 4 8 2 2 6 0 6 8 4 8 .

It seems to me unlikely that there is a proof that all pow- ers of 2 from here on have at least one 7 in their decimal expansion (assuming it's true), but Hillel Furstenberg be- lieves that one may be able at least to show that there are only finitely many sevenless powers of 2.

All of which leads to the following meta-meta-ques- tion. Consider again statements like, "every card even- tually reaches the top of the deck," or "there are only finitely many sevenless powers of 2." Suppose it is true that there are no proofs of these assertions and, there- fore, we can never know the answer. Are they never- theless either true or false?

We are back to the question of the law of the excluded middle. Must every mathematically well-formed state- ment be either true or false? Is it a fact (there's another word to conjure with) that Guy's conjecture is true? If so, was it a fact before he conjectured it? During the dinosaur era? At the time of the Big Bang? My feeling is that words like "true," "false," and "fact" are not appropriate for dealing with these infinite questions, questions whose verification would depend on performing impossible ex- periments, like calculating all the powers of 2.

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 3, 1996 25

So much for cosmic speculation. Coming back down to earth, I'm still offering a $200 prize to anyone who can tell me who wins Chomp on the non-negative lat- tice points of R 3. And I'm still tormented by the asser- tion by Scott Huddleston that he can show that Chomp on a 3 x ~ board is a second-player win. Huddleston seems to have vanished, if not from the face of the earth, at least from the face of the Internet. If you read this, Scott, please phone home.

And Finally, What Is Mathematics Anyway? After much rumination I've reached the conclusion that there's no such thing. Let me elucidate. Consider again the wine-water problem. If one "deconstructs" it prop- erly, all the mathematics disappears.

Here is another well-known question which doesn't seem to be mathematics at a l l . . , or is it?

Doin g it w i t h Mirrors

Why is it, people have asked, that mirrors reverse right and left but not top and bottom?

This hardly seems like a question in mathematics, but if not, what kind of a question is it? Physics? Optics? There is a fairly substantial literature on this question, but I have yet to see a completely satisfactory discus- sion. Here's how I look at it.

1. The mirror is irrelevant. The phenomenon occurs in other quite different contexts. Write some words on a thin sheet of paper and then turn it away from you and hold it up to the light. The words are still there but the writing runs backward but not upside down. Is this a strange property of paper? Of course not. The phenomenon was produced by you, the experimenter, when in turning the paper toward the light you chose to turn it about a vertical rather than a horizontal axis.

2. My local barbershop has a sign painted on its win- dow which viewed from the street looks like this,

Hair Cut

Connection

but as I look at it from inside while the barber is at work it looks like this,

JU3 7toll

not.toenno3

26 THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 3, 1996

but not like this.

COUUSC,r!ou

Hg4L cnf

.

.

How come? Again it was I, the "experimenter," who caused this to happen, but in this case I moved myself rather than the window. In looking at the win- dow from the inside I was facing in the opposite di- rection from when I was looking at it from the street, and I achieved this by turning myself about a verti- cal axis. If I had wanted to, and was sufficiently well coordinated, I could have faced the window from in- side the barber shop by standing on my head, in which case the writing would have appeared to be like the last figure above. The mirror phenomenon is felt to be paradoxical be- cause the mirror seems to reverse sides but not top and bottom. But this is not true. My mirror is on the north wall of my bedroom and my window is on the west wall, looking out at San Francisco Bay. When I look in the mirror, the reflected window is also on the west wall of the reflected room. Clearly, the mir- ror does not reverse east and west. Why then are we confused? Nevertheless, you may say, why is it that the guy I am looking at in the mirror seems to be wearing his watch on his right wrist (rather than his left ankle).

The point is that these paradoxes are of our own mak- ing. Right and left, unlike east and west or top and bot- tom, are in the eye of the beholder. So are backward and upside down. All of the italicized words are defined rel- ative to the observer, whereas east, west, north, south, up, down are objective. The sun always rises in the east, no matter where I happen to be, but it may rise on my right or left or in front or in back depending on my position. The fallacy is in mixing relative with absolute concepts, the "subjective" left-right with "objective" east-west. When I say my reflection wears a watch on his right hand rather than his left ankle, it is because if I were to replace "him" in the mirror, I would do so by making an about-face, keeping my feet on the ground, rather than going into a handstand. The fact that we humans, along with most other animals, are bilaterally symmet- ric makes us feel that the only natural way to "turn around" is about a vertical rather than a horizontal axis, but it doesn't have to be that way.

But let us get back to the issue at hand. In what way can the arguments above be thought of as mathemati- cal? I claim there is some similarity both in form and content. As to form, we have insisted, as in mathemat- ics, on having precise definitions of the words we use. As to content, the key words, left and right, are the

everyday expression of the notion of orientation, which is pervasive in much of mathematics.

Here is a last example which is even more remote from what one would normally consider mathematics. I recall reading in a child's science book the fairly ob- vious optical fact that the images of objects on the retina of the eye are upside down. The book went on to say that the brain by some remarkable unexplained trick manages to turn them right side up. This is another ex- ample of the subjective-objective fallacy. To whom does the image on the retina appear upside down? To some- one peering into my eye from the outside. To say that it appears upside down to me implies the absurd idea that I am looking at my own retina.

The point of these examples is to illustrate the gray area between what is and is not mathematics, but, more importantly, to show how we get tricked and trapped by words if we fail to analyze them rigorously enough. It is my belief that we are victims of a similar wordtrap when we try to define mathematics.

Looking for a definition of mathematics is chasing a will-o'-the-wisp. We don't even have a definition of, for example, the word "green." My wife insists that this green tie I'm wearing is actually blue, and there is no way either of can prove we are right. On a more high- brow level, I recall that many years ago there was an ongoing exchange of letters in the New York Times Book Review section on the question "What is History."

Scholar A said history is simply everything that hap- pens, but scholar B insisted that, no, history is only those things that happen that make other things happen, or words to that effect--and the debate went on. I think these things illustrate what I call the linguistic fallacy. We invent words like "history," "mathematics," "green," in order to be able to communicate with each other. But then we seem to think that because the words exist they must correspond to some actual outside en- tity, that the concept of History or Mathematics has some sort of existence of its own, perhaps in the mind of God, and so we seek to discover what these things really are, forgetting the fact that it was we, not He/She, who invented them in the first place.

What is mathematics? My answer is that there is no such thing. For some things, e.g., up, down, left, right, it is crucial to have definitions. For others, like mathe- matics, searching for a definition becomes mere word- play. Let's not waste time on it.

And a Final Perspective

This from a granddaughter now in second grade who is somewhat ambivalent on the subject of mathematics. As she puts it, she likes carrying, but she can't stand borrowing--which probably says it all. Over and out.

DG

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