[FOM] Infinitesimals

Hendrik Boom hendrik at topoi.pooq.com
Fri Jul 13 20:12:58 EDT 2012

On Thu, Jul 12, 2012 at 09:32:01AM +0100, Sam Sanders wrote:
> Dear members of the FOM list,
> I would like your opinion on the following statements (Please declare them true/false, with a possible explanation why).
>  ) infinitesimals are used throughout physics and engineering (in some 
> informal way, formalizable in Nonstandard Analysis).  

Yes, they're used informally throughout.  This drove us physics 
studennts mad with indefiniteness until we learned the proper 
epsilon-delta definition of derivatives.

But I learned you can formalize differentials themselves, and 
derivatives become their ratios.  You don't need nonstandard analysis. 
f be a function and we're interested in its differential df at x.
At x, df is a linear function such that for all e > 0 there exists a 
delta > 0 such that for h less than delta,
  |df(h) - f(x) + f(x + h)| < eh
These df functions act like differentials.

The definition also works for functions for Rn to Rm.  The differential 
becomes a linear function from vectors to vectors and the various 
absolute values of difference become the metric on these spaces.

-- hendrik

> 2) when infinitesimals are used in physics and engineering, the choice of infinitesimal does not matter
> (i.e. a calculation involving an infinitesimal \e remains valid if \e is replaced with any other infinitesimal \e' .)  

in the above formulation, switching between e and e' is not much more 
than taking different values of h.  It's not even interesting any more.

> 3) The aforementioned independence (of the choice of infinitesimal) in physics and engineering has been observed before by X (Please fill in X).

No idea who noticed it first. I've had very little use for the 
nonstandard version of differentials.  I've  noticed it, but it seems 
more to ba a matter of how you define the algebra of nonstandard 
infinitesimals than 
a result.

-- hendrik

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