Re: Epistemology 201: The Science of Science

From: Dave Rusin (rusin_at_vesuvius.math.niu.edu)
Date: 03/22/05 

Date: 22 Mar 2005 20:41:54 GMT

In article <R6_%d.14990$Fy.2002@okepread04>,
Albert Wagner <albertwagner@cox.net> wrote:

> I cannot get a mathematiker to define 'infinity' in
> such a way that mathematical operations can be performed upon it.

I suppose you think this is a failing of the mathematicians.
Has it occured to you that they demur for exactly the same reason
that they cannot define "two" in such a way that it's half of five,
or for the same reason that they cannot define "God"? I don't
think mathematicians are under any obligation to define terms that
that they don't use, and mathematicians don't use terms that are
patently contradictory or extraneous to their subject of discourse.

Oh, we try to appeal to students intuition and ask them to compute
lim_{x->infinity} x/(2x-1) ; but we right away say exactly what
this means in terms that don't include "infinity". (Personally I
prefer to say, "... as x increases without bound".) And we do
try to help popular audiences think about other things besides
ordinary finite sets of numbers -- "infinity" is a term that they
like and it's got a sort of poetry appropriate for such an occasion.

But you don't actually hear mathematicians speak professionally of "infinity"
itself very often, and when they do it's a (carefully defined) shorthand
(applied in specific situations). That is, it's the name of a specific point
on the Riemann sphere, or a line in some projective varieties, or a
summary of the behaviour of certain types of functions near a single point.
We will be the first to tell you that you can't perform (all) mathematical
operations on the things called "infinity", e.g. "infinity - infinity"
cannot be assigned a value in any useful way, as I have repeatedly
told my calculus students (to little avail).

We DO talk about _infinite sets_. They're sets. They're not finite.
That's really all there is to say there -- it's not some sort of
black hole/chaos/post-modern mystical thing. "All possible moments in
time" --- that's an infinite set; "All possible real numbers" -- that's
an infinite set. "All the stupidity on Usenet" -- a very large infinite set.
These are all infinite sets; none of them is called "infinity".

We do lots of "mathematical operations" with infinite sets: we form
their unions, their intersections, and their powersets (set of all subsets);
we consider functions between them; we introduce equivalence relations.
Lots of fun mathematics. Possibly of no interest to non-mathematicians,
but hey, I don't like bowling or bridge; chacun a son gout. Did you mean
the "arithmetic operations"? We don't perform those with infinite sets
in general -- but see the next paragraph.

We DO talk about cardinals, which happen to be infinite sets, and on
which the mathematical operations + * and ^ are defined. (Subtraction
is not.) We don't call any of these cardinals "infinity".

But I'm repeating myself. Sci.math had a long thread on this just a
year ago so you can see for yourself that I'm not just making up
these claims to suit this present discussion.
  http://groups-beta.google.com/group/sci.math/browse_thread/thread/a84f13f44128de81

dave

PS -- I suppose some of the participants in this thread intend the
term "mathematiker" to be somehow demeaning, or at least signifying
something different from "mathematician". I can't imagine what
difference it makes to use the German word in place of the English one.
Don't be such a mensch.

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