(Swans - July 12, 2010) Mathematics is part of most scientific measurements and experiments. Those whose activities involve mathematics often think of numbers and other mathematical ideas as Plato did. They are forms, objects in another world that we only apprehend with the mind's eye. They connect only approximately with the dirty world of data sometimes called the real world or the day-to-day world. It is only through mathematics that we can hope to apprehend the real real world, the world of pure forms, where everything obeys laws. But we also believe that the everyday world also obeys mathematics. It is the queen of the sciences and the language the sciences speak.
Let's turn to mathematics using our standard method. What is mathematics? Clearly, we must start with counting, and what is counting? Well, we make a series of sounds, words, that must be produced in a definite order. One, two, three, and so on. What do these words mean? "Two" for example. What does "two" mean? A number of images come to mind. Two logs, two inches. What do they have in common? Is there a "twoness" there? If so, it is funny that the word "twoness" is not a word. For how could we use it? We seem almost to be able to imagine it -- twoness. But our image is always of one or another of the images, the pile of logs, the two inches lined up, or maybe several images we go through like the flipping of pages in a book. We are sure we can see the "twoness" in the two logs, but what about "elevenness?" Well, we can't see it but we can count the objects.
Of one thing we are sure: when counting, these words (numbers) must be said in a definite order. One, two, three. Count differently and you are wrong, and everyone will treat you with scorn. When it comes to numbers, order is everything, and we insist on it absolutely. Anyone who refuses to say these words in the proper order is doomed, reviled, cast out, stupid, probably insane. And you will not be allowed to enter second grade. Later we associate marks on paper or on a computer screen with the sounds, and when counting write these down in a certain order. We can then just use the marks on the screen and skip the sounds. So the sounds themselves are not the numbers, nor the marks on the page, nor the pixels on the computer screen. So what are the numbers? Are they indeed transcendent entities? No, the numbers are a collection of the things we do with them. The number is not the written sign, nor the spoken one, but a tool used in many ways.
Before we consider what we do with numbers let's look at the order a little more. To help us make or say or write ever larger numbers we make the later out of the earlier sounds. Thus twenty-one, twenty-two, and so on. Because of the building block arrangement of the ordered sounds we are convinced that we could "go on forever" if not for mere problems such as fatigue or death arising from our imperfect world. The numbers go far far beyond the amount of time the universe provides to say them. We only believe that children know how to count when, if asked how far they can count, they reply, "forever." It seems that the numbers are in another world that extends beyond what anyone has ever counted. We seem to apprehend numbers with the mind's eye, for surely they are nowhere all written down. We could not have invented them, for they go beyond what we have ever experienced. We forget that it all happens out here, where we meet and do things others can understand. We have devised an operation that has no end, that is all.
This series of sounds (or marks on the page) comes in handy. Suppose we want to trade for firewood. If we are trading with Joe, we move a log from Joe's pile to ours with each number spoken. At the same time Joe moves a shekel from our pile to his. When we run out of shekels or Joe runs out of logs the deal is done and everybody is happy. One might wonder how trading was done prior to counting, and of course we can only guess. My guess is that they simply eyeballed the two piles and argued. Eventually they agreed. Or maybe there was no trading before counting, but then why did they bother to learn to count? Since logs might vary considerably in thickness, simply eyeballing the piles might have been a more accurate way to measure their value.
I have purposely chosen an example in which the objects piled up could vary in size and quality to show that counting, used in this way, necessarily turns the objects counted into tokens of a particular type, in this case "logs." Counting, by its nature, forces us to accept the idea of a form that the objects counted "imitate." By counting them we assert that they are all the same, at least for the purpose served by our counting. So counting is useful for measuring the number of things in a pile that, necessarily for this operation, are all the same, and the operation as a whole consists of uttering these sounds in order while moving one of these tokens with each sound.
Let's take this one step further. What is addition? Suppose we want to add two numbers. What do we do? Of course we now have very quick ways to do this, but let us build addition out of counting. It will now involve two counters. The first counter counts to, let's say, ten. Then a second counter starts to count to five. The first counter continues to count beyond ten, counting in unison with the one who is counting to five. Of course when the second guy reaches five the first guy will reach fifteen. And fifteen is called the sum of ten and five. The utility of such a practice is, I think, obvious.
How about subtraction? This involves an entirely new idea, counting backwards. To subtract, say, five from fifteen the first guy counts to fifteen, then, while the second guy counts to five he counts backwards starting with (notice!) fourteen in unison with the second guy. When the second guy says "five" he says a number and stops. That number is fifteen minus five.
Subtraction leads us to the mysterious "number" zero. For if you subtract fifteen from fifteen what do you get? Zero! What the hell is that? Zero, as mathematicians readily admit, is not one of the "natural numbers." It doesn't show up when we count (The natural numbers start with one.) When we pair numbers with objects in a pile we always start with "one." To utter "zero" in counting would be to count nothing. Absurd! It is simply not one of the sounds in our original string of sounds. So what is it? Well, in this example it is a sound one says when the backward counting erases all the work of counting. We have done a lot of work and ended up back where we started. It's as if two operations balanced out, so to speak. At that point we might think that it is a sound one makes to indicate not doing anything at all or, having done things, had them all undo each other. For subtraction is an operation where we undo what we have just done. It all came to naught! Zero is the result when everything cancels out.
So how do we use "zero"? It hardly ever comes up in carpentry, and I am going to guess that it is rare in any making of actual objects. Of course it is useful in counting money. In recognition that you have paid your debt, your balance is ZERO. What a satisfying sound, "zero," when it is a question of debt, and what a terrifying one when it is a question of savings. Where else do we use zero? Scientific laws, but that will have to wait.
It is sometimes said that zero is a "placeholder." What does this mean? Clearly, it involves the writing down of numbers, the notation. We operate with a "tens" system. We really only need ten symbols. Moving to the left of a decimal point the first place is the "ones" column, the next the "tens," and so on. With this new arrangement our ability to handle numbers becomes a lot easier. With written numbers we can do manipulations that are impossible with spoken ones. A one then a zero means we have one ten and no ones. The order of the symbols has meaning.
Is zero the placeholder the same as zero the answer to a subtraction problem? It certainly has a different role in a different operation. After all, the place holder only makes sense when we are writing numbers, but zero as the answer to a subtraction problem comes up when the numbers are spoken too. Clearly zero can have two entirely different roles. A number of mathematical entities play more than one role.
For example, let us look at the minus sign, the written symbol that tells us to subtract. We can add two numbers of any size without a problem, but what happens if we subtract a larger from a smaller number? We get a "negative" number. What are these? We count backwards, get to zero, and just keep going as if we were counting forwards except that we say "minus" before every number. The minus sign in front of the negative number does not tell us to do anything, unlike the minus sign that tells us to subtract.
What are these negative numbers? If we were using subtraction with our pile of logs we would run out of logs at zero. What would we do then? We wouldn't have anything to move from pile to pile. No doubt the clever reader will have anticipated the answer. We would be in debt. In our basic connection between counting and the day-to-day world negative numbers mean debt. What does it mean to say that I am "in debt" for two logs of firewood? It seems to mean that we keep counting after one of the piles, the logs, is exhausted, but the other one isn't. We don't move logs, having none, but write down the conclusion of our counting. Shekels keep moving from the unexhausted pile. Sometime in the future I have to put two logs on his pile without him putting two shekels on mine. What does it mean to "have to" do something? It is a duty, a new and rather complex idea. If I don't do it I am a bad person. My goodness, how much is involved with these negative numbers! Not only will I be a bad person, but other people will feel justified to come after me and hurt me if I don't do it? So there is a slightly veiled threat. Somehow, negative numbers connected to this operation force me to promise something, put me at risk for being a bad person forevermore, put me in danger, in some cases, of bodily harm, and require me to have a concept of time (the future) and a way of measuring it. There is also the notion of a "person" who can become bad because of something he didn't do. Accepting the concept of negative numbers is tantamount to accepting the concept of debt, duty, linear time, and personhood. No wonder young students with the best imaginations find negative numbers daunting.
Of course negative numbers in and of themselves don't have to be associated with debt. Where else might they be useful? The only thing I can see is something rather artificial, like numbering subbasements. What else? In science, arbitrary designations of spin for an electron.
To explicate negative numbers mathematicians introduce a new technique, the number line, a horizontal line with numbers evenly spaced along it. The numbers are all lined up in a nice picture. Zero is in the middle and the positive numbers march off to the right, the negative to the left. This does give us a graphic way to understand addition and subtraction, counting in the one case forward and in the other backward. But what does this picture tell us about applications of negative numbers to day-to-day life? What does it represent other than a way of picturing negative numbers to ourselves? But how does this picture help us to see other useful applications? To be sure, the number line tends to persuade us that the numbers all fit together in a nice continuum, that negative and positive numbers are not that different from one another. But does it help us in any other way? For surely the use of negative numbers in debt involves very much that is not involved with the simple exchanges positive numbers help.
Let's leave negative numbers to one side and ask another question. Is addition part of the nature of things? Do we add because, well, that's just the way things are? As we have seen, distance involves counting. We travel along and count ten miles, stop for lunch, then travel along again for five miles. Can we add ten and five and say we are fifteen miles from our starting point? Not so fast, monsieur! Were we going in the same direction after lunch? Perhaps we turned back after lunch so in the end we have only gone five miles. Perhaps, instead of adding we should have subtracted. Or perhaps after lunch we went off at an angle so that calculation of the distance traveled involves a more complicated operation. To add distances together we need to pay attention to "direction." But how do we pay attention to direction, and how did we know that we needed to do so? Clearly, sometimes addition is useful; sometimes we need something else. The good student knows that direction of travel can be incorporated into a whole process of addition of "vectors," which can be "added" using a far more complex technique that will allow us to figure out just how far from home we are. So it is clear that addition is not always applicable where it might seem to be. Sometimes it is, sometimes it isn't.
Then, of course, there is time. Scientific time is measured by a rhythmic counting. If we wait for Jill for five minutes, then wait for five minutes more, we have waited for ten minutes. All well and good. But what about subtraction? How can that be employed with time? Nothing prevents us from making a clock that goes backwards, but would this allow us to somehow "go into the past"? What would this mean? Well, it is not hard to imagine the real world looking like a movie run in reverse. Brown leaves would swirl around, attach themselves to trees, grow green, and then shrink into buds. People would emerge from graves, grow younger, and disappear into wombs. And so on. It is somewhat titillating, and, if you look at it too closely, disgusting.
But if time really went in reverse it wouldn't be a clock made to count backwards that would measure it. No, our ordinary clocks, made to count forward, would count backwards of themselves. Everything we tried to do would be an undoing. Our elaborate production lines for cars would become elaborate dismantling plants. All learning would be unlearning. War would be constructive, peace destructive. Murder, birth; birth, burial.
But all this is just fantasy, a phantasmagoria sprung from the idea of applying counting backwards, that is subtraction, to time. Could it be possible? Could we devise a technique to do it? No, of course not. For if we devised such a technique and put it into operation the first thing that would happen is that we would reach a time prior to our having this ability. The technique, in application, would obliterate itself. If it were a mechanism it would self-dismantle. If a human skill, be forgotten.
Our concern is not with such fantasies except as a caution. Techniques, such as subtraction, are not always applicable where it seems they ought to be. Addition with periods of time has meaning; subtraction leads us to this fantasy. It is tempting to think that wherever we can add we can subtract. We are easily fooled by analogy. Mathematics is sometimes applicable, sometimes not, and we have to know where to use it.
I have outlined only the simplest of mathematical operations. To be sure it is an exercise of looking at what, to most people, is obvious. But if we can learn to look at the obvious with fresh eyes we can see that even the idea of "doing the same thing again" requires rules that are part of a whole culture. My purpose is only to dispel the idea that mathematics is an excursion into a world of pure ideas somehow grasped with the mind's eye. It is a collection of operations done with words and symbols according to rules, operations that can often be used in our daily lives. When these symbolic operations can be used in the performance of operations in our daily lives, doing the symbolic operations allows us to predict the result of what might, imprecisely, be called real operations. But what is certain is that it all remains within the realm of things we do, and involves no mysterious Platonic objects.
(Next time: Scientific Theories And Experiments.)
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About the Author
Michael Doliner studied with Hannah Arendt at the University of Chicago (1964-1970) and has taught at Valparaiso University and Ithaca College. He lives with his family in Ithaca, N.Y. (back)