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
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
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.