[FOM] Infinitesimal calculus

Harvey Friedman friedman at math.ohio-state.edu
Mon May 25 01:09:50 EDT 2009

> On May 23, 2009, at 5:31 PM, 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

This issue was discussed many years back on the FOM in great detail.  
My line was that if we use an extended form of the real number system,  
then we lose categoricity - at least in any familiar sense.

The usual systems - the system of reals, with the field operations,  
and a predicate for the integers - are second order categorical. This  
is the right kind of categoricity to use for such a discussion.

Specifically, I raised the point that there is no definition in the  
language of set theory which, in ZFC, can be proved to form a system  
having the required properties.

I then considered whether there is a definition in the language of set  
theory which, in ZFC, can be proved to form a set (or even class) of  
systems having the required properties, all of which are isomorphic. I  
think there were similar negative results.

You can point to "the negative numbers", or "the complex numbers",  
etc. But note the use of the word *the*!

Perhaps this topic can be profitably revisited.

Harvey Friedman

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