Empirically it works fine; you seem to have only philosophical

objections to it. But in any case, your problem is that you have too

restrictive a view of "infinity". The concept crops up all over the

place in mathematics, and therefore in science. Some infinities aren't

even "big"; eg, the "line at infinity" in geometry, which we all quite

happily draw as the horizon in a picture. Whether it "really is" at

infinity is irrelevant.

[...]

*Post by Pancho**Post by abelard**Post by Roger*But there is a lot of stuff in maths which cannot equate to the

real world, and yet is very useful in practical calculations. The

square root of minus one springs to mind.

It's a misconception that sqrt(-1) "cannot equate to the real

world", arising from the misuse of "real", "imaginary" and "complex"

to describe certain numbers. Sqrt(-1) is a perfectly normal point in

an Argand diagram; it just requires a different view of the notion

of "multiplication" from the one you learn as a small child, but one

which maps onto that notion in simple cases.

*Post by Pancho**Post by abelard*no argument

but i'd rather that instead of 'infinity' larners were taught to

use some arbitrary large number

You can't sensibly teach "larners" that way. A smart-Alick

4yo is bound to ask you what happens to this number [which *my*

smart-Alick 4yo invented for herself and called, for no reason we

ever discovered, "kwess"] if you add one to it, or whether it's odd

or even or prime or ...; you finish up with more questions than

answers, and you can't answer them in a satisfactory way. There

are reasons why we discovered the different sorts of number in the

order we did, and you can't buck that process without causing much

worse problems.

OTOH, it would have been perfectly possible to discover

mathematics in a slightly different, more operational, way. For

example, in game theory, moves form a natural primitive currency;

and "infinity" corresponds to a blank cheque. So if I have a

blank cheque, then all I need to do to win is to fill it in with

more moves than you possess and cash it; in real life, quite a

modest sum often suffices, perhaps less than 10, very likely less

than 1000000, but larger numbers are available if needed. Such

an "infinity" has some, but not all, of the properties of more

conventional numbers; it's a number, but not as we know it, Jim.

Life then gets more interesting if you also have a blank cheque;

then whoever is forced to fill in his cheque first loses. After

that, there is a whole arithmetic of blank cheques, in case we

each have several, or even an arbitrary number, and of "entailed"

cheques [worth "infinity/2", or "sqrt(infinity)", ...], and another

arithmetic of "infinitesimal" cheques.

*Post by Pancho*So how many natural numbers should we tell learners there are.

Is this number also the biggest natural number?

What makes you think that the answer to "how many?"

questions has to be a natural number? But that's a question of

philosophy rather than of mathematics. We have only one Real

World, but many ways of describing it, and many different ways

to extend mathematics beyond "1, 2, 3, ..." to a powerful tool

to use in those descriptions. Or just to enjoy.

--

Andy Walker,

Nottingham.