BELIEVE   ME   NOT!    - -     A   SKEPTICs   GUIDE  

... Arithmetic.^4.1

No doubt the useful lifetime of this example is only a few more years, since many students now learn to divide by punching the right buttons on a hand calculator, much to the dismay of their aged instructors. I am not so upset by this - one arithmetic manipulation technology is merely supplanting another - except that ``long division'' is in principle completely understood by its user, whereas few people have any idea what actually goes on inside an electronic calculator. This dependence on mysterious and unfamiliar technology may have unpleasant long-term psychological impact, perhaps making us all more willing to accept the judgements of authority figures without question.... But in Mathematics, as long as you have once satisfied yourself completely that some technology is indeed trustworthy and reliable, of course you should make use of it! (Do you know that your calculator always gives the right answers...?)

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... teachers!^4.2

It occurs to me that Grammar school is called Grammar school because it is where we learn grammar - i.e. the conventional representations for things, ideas and the relationships between them, whether in verbal language, written language, mathematics, politics, science or social behaviour. These are usually called ``rules'' or even (when a particularly heavy-handed emphasis is desired) ``laws'' of notation or manipulation or behaviour. We also pick up a little technology, which in this context begins to look pretty innocuous!

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... number^4.3

I do not know the proof that $\pi$ is an irrational number, but I have been told by Mathematicians that it is, and I have never had any cause to question them. In principle, this is reprehensible (shame on me!) but I am not aware of any practical consequences one way or the other; if anyone knows one, please set me straight!

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... scale.^4.4

For instance, it explains easily why the largest animals on Earth have to live in the sea, why insects can lift so many times their own weight, why birds have an easier time flying than airliners, why bubbles form in beer and how the American nuclear power industry got off to a bad start. All in due time....

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... Laws:^4.5

Note that division is not commutative: $a/b \ne b/a$! Neither is subtraction, for that matter: $a-b \ne b-a$. The Commutative Law for multiplication, ab=ba, holds for ordinary numbers (real and imaginary) but it does not necessarily hold for all the mathematical ``things'' for which some form of ``multiplication'' is defined! For instance, the group of rotation operators in 3-dimensional space is not commutative - think about making two successive rotations of a rigid object about perpendicular axes in different order and you will see that the final result is different! This seemingly obscure property turns out to have fundamental significance. We'll talk about such things later.

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... Theorem:^4.6

Surely you aren't going to take my word for this! Convince yourself that this formula is really true!

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... consistent^4.7

In Mathematics we never worry about such things; all our symbols represent pure numbers; but in Physics we usually have to express the value of some physical quantity in units which make sense and are consistent with the units of other physical quantities symbolized in the same equation!

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... generally^4.8

The $\pm$ symbol means that both signs (+ and -) should represent legitimate answers.

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