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Number Systems

We have seen that units of measurement and indeed the very nature of the dimensions of measurement are arbitrary models of what is significant, constructed for the practical convenience of their users. If this causes you some frustration or disappointment, you are not alone; most students of Physics initially approach the subject in hope of finding, at last, some rigor and reliability in an increasingly insubstantial and malleable reality. Sorry.

What most disillusioned Physics students do next is to seek refuge in mathematics. If physical reality is subject to politics, at least the rarefied abstract world of numbers is intrinsically absolute.

Sorry again. Higher mathematics relies on pure logic, to be sure, but the representation used to describe all the practically useful examples (e.g. ``arithmetic'') is intrinsically arbitrary, based once again on rather simpleminded models of what is significant in a practical sense. The decimal number system, based as it is upon a number whose only virtue is that most people have that number of fingers and thumbs, is a typical example. If we had only thought to distinguish between fingers and thumbs, using thumbs perhaps for ``carrying,'' we would be counting in octal and be able to count up to twenty-four on our hands. Better yet, if we assigned significance to the order of which fingers we raised, as well as the number of fingers, we could count in binary up to 31 on one hand, and up to 1023 using both hands!  However, we have already made use of that information for other communication purposes . . . .

Is mathematics then arbitrary?  Of course not. We can easily understand the distinction between the representation (which is arbitrary) and the content (which is not). Ten is still ten, regardless of which number system we use to write it. Much more sophisticated notions can also be expressed in many ways; in fact it may be that we can only achieve a deep understanding of the concept by learning to express it in many alternate ``languages.''

The same is true of Physics.