[FOM] Infinitesimal calculus

Louis H Kauffman kauffman at uic.edu
Sun May 24 18:35:07 EDT 2009

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.

Lou Kauffman

On Sat, 23 May 2009, Charles Silver wrote:

> 	I think the most important reason infinitesimals aren't used in
> calculus books is simply that epsilon-delta was given formal
> justification over a hundred years ago, so we stick to it.   Well,
> sort of.   Since the e-d proofs are too hard for most beginning
> calculus students, they've been dropped from almost all the texts (at
> least in the US).
> 	As far as I know, every proof using e-d is easier using
> infinitesimals.   So, if you want proofs back in intro. calculus
> texts, infinitesimals are the way to go.  But, apparently historical
> momentum trumps sensible thinking.
> 	Whoever said there are lots of infinitesimal calculus books around is
> wrong.   I know of Martin Davis's very fine book, Keisler's carefully
> developed book online, and a nice, simple one you could read (and
> understand) in an hour or so by Jim Henle.  These books are
> essentially all dead: Keisler's not in print, Martin's in Dover, and
> so is Henle's.  (I'm sure I must be missing a couple others.)
> 	The only downside I can see is that infinitesimals are not ordinary
> numbers, but neither were negative numbers once upon a time.
> Charlie Silver
> On May 21, 2009, at 5:54 PM, Monroe Eskew wrote:
> > 2) Aren't the epsilon-delta notions of limits practical?  I know that
> > in experimental science, one wants to approximate and compute all the
> > time, and it is also of interest to do so within a margin of error.
> > This means a FINITE margin of error, since infinitesimal error is not
> > available in the real world.
> >
> > Best,
> > Monroe
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