[FOM] constructivism and physics

[Timothy Y. Chow]

Ben Crowell <fomcrowell06 at lightandmatter.com> wrote:
> Timothy Y. Chow wrote:
> >If I have been following this thread correctly, the claim that there are
> >methods that can be justified by infinitesimals but not by rigorous
> >calculus is about "infinitesimals" in the classical nonrigorous sense, not
> >in the sense of the rigorous infinitesimals of nonstandard analysis.
> 
> No, I think the claim is that even rigorous nonstandard analysis is 
> incapable of proving results that can't be proved using limits. At 
> least, that's my claim.

Certainly that's your claim, and Alasdair Urquhart made the same claim.  
It is uncontroversially true.  But that is not what the discussion was 
initially about.  Antonino Drago wrote:

>Almost two centuries after, rigorous calculus confirmed almost all the 
>results obtained by infinitesimals; but not all.

Clearly, "results obtained by infinitesimals" here does not refer to 
nonstandard analysis, since nonstandard analysis did not precede rigorous 
calculus by two centuries.  Todd Wilson asked for examples, to which 
Antonino Drago responded; I didn't understand the examples, but clearly 
they were *not* examples (or even attempts at providing exmaples) of 
results in nonstandard analysis.

> The problem with the nonrigorous treatment of infinitesimals wasn't that 
> it was insufficiently powerful, it was just that it was logically 
> inconsistent.

History is rife with examples of results that were derived from logically 
inconsistent bases but that were nonetheless correct results in some 
sense.  I'm interested in whether there are examples of such coming from 
the nonrigorous treatment of infinitesimals, that seem to have some 
validity but that currently have no mathematically rigorous 
reconstruction.

Tim Chow