## Sunday, November 3, 2013

### Fun with (Negative) Numbers

Disclaimer: In high school I took math up through algebra and geometry, thereby missing trigonometry, analytical geometry, and calculus. I’m an English major by training and predilection, not a mathematician. I was, however, trained in logic and remain fiercely loyal to clear and open reasoning. Over the years I have worked as a technical writer alongside engineers and scientists, as well as written novels of “hard” science fiction—all of which required me to go back and learn mathematical principles and gain some facility with manipulating numbers. I just don’t always like them very much.1

Mathematics is an intellectual exercise, carried out in the human brain. It is a species of logical reasoning. While we all subscribe to the belief—at least since the Greeks of ancient times, and more recently with the Enlightenment of the 17th century and beyond—that the universe and the physical world around us are defined and best described by mathematics, this remains a matter of conviction and belief. It is based on the fact that mathematics is the dominant way in which we have come to perceive and study this part of reality. Because math has worked in describing many of the phenomena that we can see, we now believe it’s the best way. Still, that does not preclude some other means of achieving a deeper understanding.

Just as it’s possible to write nonsense in perfectly grammatical English sentences, it is possible to write equations with perfectly valid mathematical operations that do not represent actual relationships. For example, I can represent the speed of an automobile in miles traveled per hour, and the rate of my metabolism in calories burned per hour. I can then equate the hours and compare calories burned per miles traveled. That doesn’t mean the one has anything to do with the other.

That’s a ludicrous example, of course. It’s logical inconsistency would be obvious to a child. But many scientific calculations—especially those in modern physics—extend to dozens if not hundreds of computations involving abstractions that are supposed to reference real situations. The logic is not always easy to follow and check. Great minds may work over these calculations and agree about what’s going on. But logical fallacies and incautious comparisons are sometimes difficult to trace and discover at this level of theorizing. Many great minds may take the same wrong turn together and follow each other out the window.2

One obvious rip in the fabric of mathematical logic that makes me uneasy has to do with the extension of the number system into both positive and negative territory. There it becomes difficult to keep track of the difference between what’s mathematically possible and what’s going on in the real world.

By the laws of mathematics, you can add or subtract positive and negative numbers to get a positive or negative result. So, if I start with five apples and subtract two apples (i.e., add -2 apples), I have three apples. Or, if I start with five apples and subtract seven apples (i.e., add -7 apples), I end up with two apples less than none. Weird but logical. In this case, I owe somebody—probably the person who took all seven apples—two more apples than I started with.

Of course, without positing some such obligation as “I owe you two apples,” the negatives of a real object are difficult to identify. One cannot point to the space where two apples might exist and say “there are -2 apples.” You can’t point to holes in the air to find missing quantities. Of course, you can imagine an apple crate or some other framework with two empty slots and say “those empty holes are where two more apples would probably fit.” That’s somewhat like working with a negative number. But without such a framework or crate, the whole remains abstract. “See the two apples that I just don’t have!”

But when it comes to multiplication and division, the result is not even so intuitive. The rule says that when multiplying or dividing numbers of the same sign (i.e., +2 times +2, or -2 times -2), the result is always positive. But when multiplying or dividing numbers of different signs (i.e., +2 times -2), the result is always negative. This may make a tidy rule and easy for a mathematician to remember, but what does it have to do with representations in the real world? Multiplying or dividing with negatives does not seem to have any corollary with what you can do with physical objects.3

This leads to problems within the intellectual structure of mathematics itself.

Take, for example, “imaginary numbers.” These are numbers whose square root (i.e., the number multiplied by itself) is a negative number. So, by the above rules, the square of -5 is easy to understand: -5 times -5 equals +25, just as +5 times +5 equals +25. But while we can imagine that a negative number like -25 might have a square root (i.e., the number which, multiplied by itself, equals -25) we can’t validly calculate it, because the only way to get -25 as a square is to multiply +5 times -5—which is not multiplying the same number by itself. The square root of +25 is may be either +5 or -5, but the square root of -25 is, according to the laws of mathematics, gibberish.

But let’s close our eyes to the mystery of imaginary numbers. Back to just dealing with positives and negatives …

What becomes nonsense with apples makes perfect sense in the relationship of apples or money to the activities that we call commerce and finance. If you have contracted to sell me five apples and yet have only produced three apples, I will take your three apples and agree that you owe me, sometime in the future, a further two apples. You have acquired an obligation to provide me with more two apples. These are not apple-holes-in-the-air but apples-yet-to-be-produced.

The notion that two apples exist to be provided is an intellectual construct. You may obtain the apples which you owe me through purchase from another apple supplier, or we may agree to discharge your debt through payment of money. But the transaction is carried in our heads and communicated through our speech and perhaps in written form. Negative apples do not exist in the real world. If you obtain them from a supplier, he still has to grow and make them real before they can be traded.

In similar fashion, we can make a difficult fit of negative numbers to notions of time. Suppose that at one o’clock you promise to deliver something to me by a deadline of two o’clock—or one hour’s time into the future. I wait. You do not arrive on time. And then you show up at 2:15—or 15 minutes after the deadline. I can say, “You should have been here 15 minutes ago.” That’s a perfectly valid grammatical construct. It’s also valid mathematically: 2 hours 15 minutes plus -15 minutes equals 2 hours. But the concept of “15 minutes ago” is physically impossible. You cannot take back those minutes and arrive on time.

Our experience of time, like our experience of physical apples, only works with positive numbers. We can play with negative time as an intellectual exercise and as a form of reproach,4 just as we can contract for a debt of apples-to-be-delivered. But we cannot directly experience either.

This is why I am leery about much of today’s physics, which is so heavily dependent upon mathematics. In particular, string theory seems to be based entirely on mathematical reasoning absent concrete observations.

Quantum mechanics may find a mathematical relationship between a proposed particle like the Higgs boson—which is too heavy to exist in our spacetime and may only ever have existed at some micro-instant following the Big Bang—and the proposed field conditions which this boson would generate, were it to exist. Without the Higgs, all the other particles in the Standard Model of quarks, leptons, gluons, and the rest would not have mass, and so the measurable quantity—although indefinable quality—of gravity would not exist in the universe. I’m still trying to wrap my head around a field that exists in nature without an actual particle being present to create it. It sounds like experiencing electric or magnetic fields without the actuality of a moving photon. Would just the possibility of the photon create a measurable field? I think not.5

String theorists have an elegant proposition all worked out, where every particle we can see and measure in the Standard Model is actually a tiny bit of energy, drawn out in a tiny loop like a bit of … string vibrating at a particular wavelength. This doesn’t actually work in the three dimensions which we can actually see and experience: x, y, and z, or “side-to-side,” “up-and-down,” and “in-and-out.” But if you propose a universe infested with a kind of spacetime that adds eight more dimensions at microscopic levels which we can’t see or detect, it all works out fine. Uh-huh.

These theories are all valid mathematically. For the rest of us, to whom the proposed situation makes no sense, or who suspect a possibly missing step that might hide a logical trap, we are told that’s just because we don’t understand the mathematics involved. But, for my money, it’s also possible that the tightly knit community of particle physicists is spinning a shared fantasy of elegant, complex equations that just might be equating calories burned to miles traveled.

I’m not anti-science. I love science. I believe that the scientific method and scientific reasoning have given us a quality of life and an outlook on reality that have never before been shared by human beings. But that doesn’t mean I automatically accept whatever a scientist says. And when what a group of scientists says seems to defy logic and common sense, or violate observed reality somewhere else, I become cautious.

Science and mathematics are still human endeavors. Even when pursued with the most fine-grained instruments and powerful computers, I remember also that these machines are still products of the human mind. While two heads may be better than one, people in pairs and groups are still susceptible to hopes, dreams, illusions, misinterpretations, and stubborn folies à deux—and sometimes folies à plus.

I want to see test results that can be shown concretely, without invoking negative apples and holes in the air.

1. For two similar entries on this theme, see: Fun With Numbers (I) and Fun With Numbers (II) from 2010.

2. In this context, I’m drawn back to Stephen Hawking’s explanation about why the universe does not abound in micro black holes. See If You Can Believe … from February 17, 2013.

3. For that matter, multiplication and division appear to be simple shorthand techniques applied to addition and subtraction. If I want to give each of three people five apples apiece, I can add five apples three times, or I can simply multiply five times three. Similarly, if I want to determine how many apples each of three people gets when I have fifteen apples and want to distribute them fairly, I can divide fifteen by three.
Signs and negative numbers don’t seem to come into it, except in the abstract sense that 15 actual apples divided among three people who aren’t there would be -5 apples, while giving five apples that don’t exist to each of three living people is likewise -15 apples. Somewhere apples are owed that do not exist, and so the debt is canceled.
That’s a case of multiplying or dividing a negative by a positive. But the rule doesn’t make any more sense when multiplying or dividing a negative by a negative. If two people less than nobody each take three apples less than none, the mathematical result becomes six real, positive apples. Huh? Is that like the double negative in English, where “I won’t never go to the store again” grammatically comes out meaning “I will always go to the store”? But, of course, the ungrammatical person using the double negative still means “no, not ever.”

4. Of course, we can’t even deal with positive expressions of time except in our imagination. I might imagine that I will be somewhere else in the next two hours, but until those hours pass and that instant of time becomes my reality, it’s all just make-believe. This is why I classify science fiction novels involving time travel, including my own recent The Children of Possibility, as a kind of science fantasy. Fascinating to think about, impossible to do.

5. Oh, yes. The people at CERN’s Large Hadron Collider just “discovered” the Higgs boson. From what I understand, they did no such thing. They smashed together two protons moving near the speed of light and got a flash of energy and detected a shower of particles. They traced back the decay lines of at least two of those particles to a common origin—which means they started out as something much bigger that briefly appeared and immediately disintegrated—and came up with a mass of ~125 GeV (billion electron volts), which correlates to the theoretically proposed mass of the Higgs. Clearly, they did this more than once and came up with answers each time that satisfied everyone all around. But that’s not like they put the thing in a bottle.