Re: Calculus XOR Probability

Dave Rusin said:
In article <b4dd4$4417ec3d$82a1e228$16583@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:
Tony Orlow wrote:

Calculus is fine. It acknowledges the notion of the infintiesimal.

That's what I asked to David Ullrich: is it indeed that infinitesimals
are recognized in standard mathematics? Or is that a forbidden area as
well? I mean, as a physicist, I have no problems with infinitesimals ..

"Forbidden"?

Mathematicians are a libertine, almost anarchistic group.

That's a laugh, coming from the guy who patiently tried to explain how I was
wrong about countably infinite sets, and then demanded that I either accept
that there are an infinite number of finite naturals or be banished to his
killfile, ala the Spanish Inquisition. Very open minded. If someone responds to
this post, maybe he'll see my response.

Nothing is really forbidden -- you're welcome to pursue whatever
you like. The only requirement is that you define your terms and
clarify your assumptions, and then proceed logically.

That didn't seem to be good enough a year ago.


There are indeed perfectly logical and useful notions of "infinitesimals".
Emphasis on the plural here -- e.g. one definition of "infinitesimal" is
the set of nilpotent elements in the ring F[x]/(x^2) where F is
a field. I'll bet that's not the one you use in physics, but what
you DO have in mind I don't quite know.

One can build a theory of the calculus on a logical foundation which
involves another kind of infinitesimal; one can even prove a
transfer theorem which allows us to decide which results from
that theory apply to ordinary calculus over the real field.
Some folks prefer this approach, others start in measure theory, etc.
Different strokes for different folks. We're a "big tent" kind of group,
so that's OK.

Socks with sandals? Not a problem. Unkempt hair? Par for the course.
Unintentionally insults the host's spouse? Happens all the time.
None of these things is forbidden in mathematics. The only thing
that will get you thown out of the club is an insistence on vague
language and a refusal to accept logical deduction from clear
starting principles.

Or, perhaps, starting from different clear principles than the ones that a
particular mathematician has invested their life in, or not accepting blindly
their decrees as to truths which they base on unfounded axioms. Just those
extras things, that's all.


dave




--
Smiles,

Tony
.

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