> I can sum up my final view by saying that I came to
> believe that a complete linearly ordered (therefore Archimedean) field,
> augmented with some limit theory in the spirit of Cauchy and Weierstrass
> was simpler (that word "simple" again!) than using a non-Archimedean
> field, and also easier to relate to applications, especially in connection
> with the sort of approximations one works with when using computers.
And Eric Schechter agreed:
> Oh, I quite agree. When I mentioned the non-Archimedean fields,
> I didn't mean to suggest that I prefer them. Personally, I would
> be reluctant to try using a book like Kiesler's. I don't know
> nonstandard analysis very well, but for the little bit of it that
> I do know, epsilons and deltas are just as good as infinitesimals
> for doing calculus, and they don't require quite as much explanation.
> I actually see them as being nearly the same thing, anyway. In the
> epsilon-delta argument, you have a variable delta which varies through
> smaller and smaller, "ordinary" (not infinitesimal) positive numbers;
> delta eventually passes and gets smaller than any particular fixed
> positive ordinary number that you might compare it with. In the
> infinitesimal argument, you have a constant (not variable) infinitesimal
> number delta, which is positive but is *already* smaller than any
> particular positive ordinary number that you might compare it with.
> In my mind, the real difference is how "time" enters our intuition.
> But perhaps there are more substantial differences that I am not
> aware of.
I disagree with both Gordon and Eric. I do have reservations about teaching
a first course in calculus via nonstandard analysis (see refs [1] and [2]).
However, when it comes to understanding and doing maths NSA can offer
far-reaching, even spectacular advantages. I shall give just two examples.
1. The delta-epsilon definition of continuity of a (real) function f at a
(real) point x is of the form Sigma^0_3. In other words, it has the form
forall epsilon thereexists delta such that forall x',
if |x' - x| < delta then ....
The equivalent NS charaterization has the form Pi^0_1:
forall delta ~ 0: f(x+delta) ~ fx.
Instead of three quantifiers (of alternating kinds) there is just one. This
is a considerable logical and psychological simplification. Note also
another thing: the function f 'goes' from its domain to its range (or
codomain), and we instictively tend to think of it as moving in that
direction. But the delta-epsilon definition starts at the 'end', at the
range of f, where the epsilon is given; then it moves to the domain, where
the delta is found, and then moves back to the range, where we have
|fx' - fx| < epsilon. By contrast, the NS characterization starts at the
beginning, in the domain, and goes to the range.
So using infinitesimals is not at all 'nearly the same thing' as using
delta-epsilon.
2. The standard definition of topological compactness of a set A is of the
form Pi^2_2:
for every cover of A there exist a finite subcover of A.
Here there are two alternating quantifiers, but of higher type: not over
points or over sets-of-points but over sets-of-sets-of-points. The NS
characterization has the for Pi^0_1:
for every point x \in A and every *point x':
if x' ~ x then x' \in A.
Just one quantifier (two successive quantifiers of the same kind count as
one) and they are of lowest type! This is a massive simplification.
As a result, most situations involving continuity, and even more so
compactness, are *much* simpler to visualize and reason about using NSA.
Of course, these two examples are very far from illustrating all the
various advantages of NSA.
One point on which I think both Gordon and Eric are mistaken is that they
seem to believe that a more or less arbitrary non-Archimedean or perhaps
some other kind of ordered field will do the job. The whole point about
using infinitesimals is to study, say, the good old field R of reals,
functions defined on R and so on. For infinitesimals to behave correctly,
the extended field *R (of all standard and nonstandard *reals) must satisfy
the 'same' (ie, *formally* the same) laws as the original R. For this, *R
needs to be an enlargement of R in the sense of A Robinson. Any other kind
of extension (even if it is non-Archimedean) will not behave correctly.
References
[1] Bell and Machover, A course in mathematical logic, North-Holland 1977,
1986 see section 7 of ch. 11.
[2] Machover, The place of nonstandard analysis in mathematics and in
mathematics teaching, Brit. J. Phil. Sci. 44 (1993) 205-212.
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