> Gordon Fisher wrote:
>
>> 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.
>
My discussion with Schechter, as I saw it, centered on teaching
*elementary* calculus. For more advanced work, I would have other
opinions about the value and usefulness of nonstandard analysis.
> 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 quatifiers (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.
Agreed. And that was one of the most telling selling points concerning
attempts to teach nonstandard analysis to beginning calculus students.
However, in my opinion, the places where NSA is simpler than epsilon-delta
styles was, for purposes of elementary teaching, overcome by places where
it is more difficult. Some of these difficult places are of what I'll call
social origin, but some of them, I think, are intrinsic to the kind of
field extension that one gets in the Robinson manner. These more difficult
places may eventually come to seem simple, too, but I doubt that it will
happen in my time. (Of course, I'm fairly old.)
Also, more quantifiers or not, I think the epsilon-delta styles are better
fitted to approximation techniques and applications, such as appear in
numerical analysis.
> 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.
>
Not at all. I was a little cagey about whether or not Robinson's kind of
enlargement was required because I was allowing for the fact that some
brave soul might try to use the system of Giuseppe Veronese, as refined by
Tullio Levi-Civita and Reinhold Baer, among others, and I believe later
developed by Detlef Laugwitz. I have an article on this called "Veronese's
non-archimedean linear continuum", p 107-145 in a book edited by Paul
Ehrlich called *Real Numbers, Generalizations of the Reals, and Theories of
Continua*, 1997. Compare also my article "The Infinite and Infinitesimal
Quantities of du Bois-Reymond and their Reception:, *Archive for History of
Exact Sciences*, v 24, 1981, p 101-164.
> 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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>
By the way, I remember referring, with profit, to the book by you and Bell
some years ago when I was working with nonstandard analysis. I probably
also read your BJPS article, too.
Best wishes
Gordon Fisher gfisher@shentel.net