[FOM] Infinitesimal calculus

[Monroe Eskew]

I worry that without showing that these new rules are sound in the
appropriate way, this would seem like magic.  But showing that
nonstandard analysis works is probably much more advanced than just
doing standard analysis.  For the beginning mathematics student who is
interested in theory and proof, this may be detrimental.

Best,
Monroe

On Sun, May 24, 2009 at 3:35 PM, Louis H Kauffman <kauffman at uic.edu> wrote:
> Dear Charlie,
> I am sure that we can teach calculus with square zero infinitesimals in
> any calculus course where some proofs are possible. We should also teach
> limits in the practical way that limits are usually handled (without
> epsilons and deltas). But consider introducing a new number *
> such that *^2 = 0, and extended real numbers R^ in the form a + b* where
> (a+b*)(c+d*) = ac + (ad + bc)* (a,b,c ordinary reals) and of course
> a+b* = c+d* iff a=c and b=d. It is understood that * > 0 and * < r for all
> positive reals r. Then one defines the derivative via
> f(a + b*) = f(a) + f'(a)b*
> whenever it is clear how to define an extension of a function f(x) over
> the reals to R^. This is where the mathematics in the course
> will become a discussion! How do you extend a given function? Is it
> possible to make a sensible extension. But after all, we are talking about
> a first course (or an honors course) not a complete and axiomatic
> treatment. Then the mathematics develops naturally:
> (x + *)^2 = x^2 + 2x* ----> (x^2)' = 2x.
> (fg)(x+*) = f(x+*)g(x+*) = (f(x) + f'(x)*)(g(x) + g(x)*) = f(x)g(x) +
> (f'(x)g(x) + f(x)g'(x))* ----> (fg)' = f'g + fg'.
> The proofs are so easy that students may actually catch on to the fun
> of making proofs.
> But what about sin(*) and cos(*)? This takes discussion and examination of
> how sin and cos behave for very small numbers. Then a definition has to be
> made and once one makes the definition sin(*) = * and cos(*) = 1 it is
> easy to see that this has good consequences. The process will also spark
> better students to wonder if there is a theory that will encompass such
> excursions in definition. One more definition , e^{*} = 1 + *, suffices
> for most of elementary calculus. No limits for most of the formulas, and
> elegant proofs for all the rules. I think that it can work to add this
> to the first course in calculus. As you say, negative numbers were once
> regarded as strange.
>
> I have to admit that I only tried this seriously in an honors calculus
> course, and there it went just fine. In fact, I motivated differential
> forms by saying that all squares of infinitesimals should vanish. Hence
> (dx + dy)^2 = 0, hence
> dx dy + dy dx = 0.
> Hence dx dy = - dy dx, and off we went to Stokes Theorem and all the rest.
>
> Best,
> Lou Kauffman
>