# [FOM] R: R: constructivism and physics

Antonino Drago drago at unina.it
Thu Feb 9 18:28:58 EST 2006

```Todd Wilson wrote:
I  would love to include an example of
a result obtained by infinitesimals that has not been confirmed by
"rigorous calculus".  Can you give an example (or, better, a few)?

According to infinitesimals, no essential discontinuity could exists in a
function, i.e. a function was essentially a hand drawn curve; only those
discontinuities were admissible that the hand could perform by its
detachment from the paper.
Just after the introduction of a minimally satisfactory notion of limit
(first by L. Carnot, then by Cauchy) as the basic notion of calculus,
instances of essentially discontinuous functions have been offered. The most
celebrated examples: Cauchy's essentially non-analytical functions, which
destroyed Lagrange's new foundation of calculus (which then disappeared),
based upon the assumption of the existence of a Taylor expansion for
whatsoever function.
The series' convergence was admitted without problems a long time; by
astronomers till up the end of 19th Century, when Poincaré recognised as non
convergent several series often used by astronomers; as a consequence, a lot
of "results" have been destroyed.

For your curiosity, there exist also examples of physical notions which can
be defined by means of infinitesimal, but not by rigorous calculus.
An instance is the notion of reversible process in thermodynamics: a process
composed by states of equilibium; i.e here the state is defined by an exact
number which moreover has to express its belonging to a series (process):
this notion is exactly that of an infintesimal. No definition is possble in
rigorous calculus, because either you have a process of limit, i.e a series
of distinct approximations, or you have the final number only, not both.
One more example is the mathematical formula of the third principle, often
written as limS for T->0 = 0; it is incorrect, because the operative content
of this principle is that exactly S=0 is unachievable; but rigorous calculus
cannot express an unachievability by means of a limit process (or some other
notions); hence a physicist writes the principle as in the above and yet he
thinks it in infintesimal terms.
Best greetings
Antonino Drago
via Benvenuti 5
Castelmaggiore Calci Pisa 56010
tel. 050 937493
fax 06 233242218
-----Messaggio Originale-----
Da: "Todd Wilson" <twilson at csufresno.edu>
A: "Antonino Drago" <drago at unina.it>
Cc: <fom at cs.nyu.edu>
Data invio: giovedì 9 febbraio 2006 4.56
Oggetto: Re: [FOM] R: constructivism and physics

> Antonino Drago wrote:
> > Surely since its beginnings theoretical physicists (Cavalieri,
Torricelli,
> > Newton) made use of infinitesimals, whereas Galilei, Huygens and Leibniz
> > resisted to them. Almost two centuries after, rigorous calculus
confirmed
> > almost all the results obtained by infinitesimals; but not all.
>
> I'm giving an informal talk on infinitesimal analysis at our math
> seminar in a couple of weeks, and I would love to include an example of
> a result obtained by infinitesimals that has not been confirmed by
> "rigorous calculus".  Can you give an example (or, better, a few)?
>
> --
> Todd Wilson
> Department of Computer Science
> California State University, Fresno
>
> --
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```