FOM: Infinitesmials,Conservation/ReplyToBarwise

Harvey Friedman friedman at
Wed Nov 12 20:38:21 EST 1997

Barwise writes:

>Is there an intuitive concept of infinitesimal?  Conserative exension
>results can't answer THAT kind of question.

What conservative extension results can do is take the focus away from a
potentially fruitless exchange about the status of infinitesimals, and put
it on something fruitful. Aren't they wonderful?

>I think there is.

No there isn't. There is only the fact of some people using them in an
informal way and some people using them in a formal way. In the former
case, there is the project of formalizing this reasoning. In the latter
case, there is the project of trying to see what good it is, and trying to
explain why it is any good. Despite some enthusiastic adherents, do you
agree that most mathematicians do not think it is all that useful, are not
particularly comfortable with it, and would rather ignore it than use it?
One could also work on explaining this.

I don't deny that there is an interesting phenomenon of reasoning about
them - independently of any fully formal use of them. And this can be the
object of interesting study.

There is also interesting work of Piaget and others about childhood
development and how children of a certain age may think that anything that
moves is alive, or that tall thin glasses hold more water than short fat

Also there is interesting work about earth, air, fire, water physics.

>In addition
>to the historical evidence, starting with Archimedes "method," there is
>also pedagogical evidence, formal and informal.

Have you properly taken into account Bill Tait's e-mail of Wed, 12 Nov 97
12:19:24 -0500? The great historical figures he mentions who vehemently
rejected them rejected them for very good reasons. This has nothing to do
with studying the way people who choose to reason with them actually reason
with them. It also has nothing to do with their formal treatment and use as
a technical tool.

>On the informal side, I
>used Keisler infinitesimal calculus book two or three times, with happy
>results.  The students ...

So what happened to this calculus book? It was never considered for use
here, and we have more calculus students here than almost anywhere in the
U.S. Is it regularly used at Indiana? At Princeton? At Harvard? At
Berkeley? At Madison?

> (Perhaps it needs saying, for those who have not looked at
>Keisler's book, that he takes an informal axiomatic approach, he does not
>construct the hyperreals.)

In contrast, a bit more advanced book could construct the real numbers in
the usual way, and be comfortably self contained. Not so with this
approach. What a difference! And Jon, why do you use the word   "the"  ??

>I am not interested in arguing that infinitesimal calculus is "better" than
>epsilon-delta calculus (though I found it a lot easier and more fun to

You found it "a lot easier and more fun to teach" because you are a
radical, who is biased in favor of most things that are new and different
regardless of ...

By the way, I am interested in arguing that epsilon-delta calculus is
better than infinitesimal calculus, and I find the epsilon-delta calculus
so much easier and more fun to honestly teach that I only teach
epilson-delta calculus. E.g., if I was in your class, I would ask you for
an example of an infinitesimal. What would be your answer? Students are
always asking me for examples of anything I talk about in calculus.

> I only want to suggest to Harvey that there is an intuitive
>concept that people can learn to use correctly in doing mathematics
>--without understanding Robinson's construction.  This point is relevant to
>the philosophy of mathematics.

I already said that I was interested in studying the structure of such
reasoning - regardless of its quality. But at some point there will
inevitably be ad hoc restrictions on how this reasoning must be carried out
without leading to contradictions. In fact, these restrictions involve
certain considerations that you (Jon Barwise) would be expected to find
awkward and undesirable - given the way you talk about first order

>A philosophy of mathematics needs to be compatible with the way people do
>correct mathematics. If a student comes across some function and uses
>infinitesimals to correctly calculate its derivative or its integral, then
>surely they are doing mathematics.

It doesn't need to be (and probably can't be) compatible with the way
arbitrary people do correct mathematics, especially if these same people
also do incorrect mathematics. The students you are talking about do more
incorrect mathematics than correct mathematics. That's why they are

Your friend, Harvey.

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