The Mysterious Importance of Numbers

In fact each of these examples is counterintuitive. Imagine yourself five hundred years ago, before science. What did people back then use numbers for? Counting things of commercial import: money, seed, goods, lengths, people. Numbers were also used in symbolism and religion: the twelve tribes of Israel, the four winds. They had nothing to do with images, communication, weather, understanding your body, or the like. In daily life there were no calorie counts, no 98.6 degrees for the normal human temperature, no speed limits, no wind speed measurements, no measuring your pulse with a stopwatch. Numbers had their niche in daily life and they stayed there. Certain mystics and accountants might devote their lives to numbers, but that was their specialty.

No sensible person would have suggested that numbers are at the center of everything from understanding your own health and sickness to facilitating dialogue between cultures to agriculture. To get an idea of how incongruous that statement would have seemed, pick some other important part of daily life (fire, postage stamps, clouds, etc.) and substitute it for "numbers" in the preceding sentence. Sure, you can find connections between fire and practically everything else, but saying that fire is at the center of communication would be more than a little far fetched. Someone who is not simply holistic or poetic about fire, but really believes that it is the answer to everything, would be considered at best a dreamer or enthusiast or mystic, and at worst just crazy. That's the same reception that any visionary of today's numbers would have received before the success of numbers.

Even our own language betrays the unusual importance we give to numbers. We now speak of "quantum leaps", and the word "quantum" refers to counting by ones; a quantum simply means a single indivisible unit, the difference between one and two. Imagine encountering a culture where people spoke of "fire leaps" as the apotheosis of progress. What would you think? And what were the chances that someone before the victory of numbers would have thought our present attitudes as anything but crazy?

The worst of it was that there were no hints that numbers might be so useful for so many things. Throughout the ages, our visual experiences never gave us any hint of the billions of numbers that could be used to describe them. No doctor putting his hand on a fevered patient's forehead ever had the temperature suddenly revealed as a number. Even at the height of concentration no musician began hearing numbers emanating from her instrument. Our universe appeared then as it ever does, without numbers.

I don't think that science does anything to take away the mystery of the link between numbers and nature; nor does it give warning of the limits beyond which numbers provide no insight. Instead natural philosophers simply began with the intuition that numbers are the most important way of describing their experimental observations, and then began inventing and refining machines which attach numbers to the world around us. Every number we use to describe our actual physical world was mined from the world using special machinery: thermometers, speedometers, cameras, etc. (Footnote: There are two important numbers, pi and e, whose values can be derived independently of machinery, but neither are observed directly in the natural world - this is related to the issue of physical units, which we may discuss later in this blog.)

Physics and Numbers

We began this introduction by talking about how physics began as philosophy, a sort of thinking that tries to understand things more deeply, and doesn't have a results-oriented focus. But soon the first natural philosophers crossed a divide which continues to separate physics and all the sciences from philosophy. This divide has three hallmarks: experimental observation of nature's actual behavior, complete control over experiments including the ability to repeat them at will, and always using numbers to describe nature. We started by talking about experimental observation, and more recently discussed the focus on experimental control, how it results in a corresponding narrowing of "scientific truth" to those things that can be controlled and repeated at will, and how it unfortunately can be part of a vicious cycle which focuses on power and abuses both man and nature. Now we will turn to the third hallmark of natural philosophy: its focus on numbers.

Numbers and the things we can do with them will be the daily bread of the rest of this blog. (That's why I discussed this hallmark last.) But before we get to know numbers I want to point out that there is a great mystery here: why are numbers useful for describing nature? Certainly this doesn't seem like a mystery today, when the success of numbers is a fact of life. The movie reproductions we watch via DVD or high definition television are composed of billions of numbers; indeed any image can be represented by numbers, as are the voice signals that we listen to on our cell phones and the songs we listen to on CDs. Mapquest and GPS have quantified our moving around: 5.3 miles north, then turn right and 1.4 miles east. And time also: we watches to tell the exact date and time. Clearly numbers have been very very successful, but their success alone does not explain why you're successful. If we think that the utility of numbers is obvious, it is only because we are prejudiced by our daily experience.

Natural Philosophy

Of course natural philosophers - physicists - have the same philosophical bent. As a case in point, consider the efforts to come up with a Theory Of Everything, or a Grand Unified Theory. In those efforts physicists are trying to understand something that is far beyond human experience, with almost no connection to the real world, so that we have to build billion dollar machines if we want to hear even the slightest whisper from nature about these matters. Not practical at all.

But going back to the natural philosophers, you can imagine how it looked at the time when a person asked whether heavy things really fall faster than light things, or debated the commonplace knowledge that vacua can not exist in nature, or claimed that everything is made out of very small invisible indivisible pieces. This last was proposed by the Indians and Greeks but had no connection with real world evidence until two millenia later. These people clearly had an unworldly bent, the sort of attitude that is epitomized in so many pictures of Einstein.

But there was something a bit different about these natural philosophers, which made them far different from other philosophers, and still lies between the two like a canyon. The natural philosophers emphasized numbers, experimental verification of their ideas against nature's actual behavior, and complete control over experiments. I'll expand on all three.