🚀 go-pugleaf

In article <27770.278aef90@kuhub.cc.ukans.edu> kinnersley@kuhub.cc.ukans.edu (Bill Kinnersley) writes:
> In article <27942:Jan902:20:0791@kramden.acf.nyu.edu>, 
>    brnstnd@kramden.acf.nyu.edu (Dan Bernstein) writes:
> > No. I would rather subscribe to the standard definition. One of the
> > flaws of an education in mathematics is that you learn little tolerance
> > for attempts to corrupt the standard definitions of words.
> Maybe that's one of the flaws in *your* education in mathematics.  Apparently
> you were made to memorize and regurgitate a list of definitions at some
> point, and thought that that was mathematics.

Non-mathematicians rarely understand that mathematics is never more than
a step ahead of its definitions. All the great advances in mathematics
were accompanied by advances in terminology and symbolism. Geometry went
nowhere until Euclid made the *definitions* that formalized it. When
Riemann *defined* manifolds he created differential geometry. Now the
Atiyah-Singer theorem is revolutionizing several fields---not because
it's an interesting statement about two different types of index, but
because it *defines* a newly visible structure.

It takes years to introduce a new concept into mathematics. In that
time, several people have probably decided on their own notations and
words for the concept. If the concept becomes popular, each definition
will survive for a while, and papers on the subject have to mention
which definitions they're using. Sometimes competing definitions last
for many decades.

What rarely happens, though, is a *single* term defined in two different
ways in one field. Sure, some people will say ``kernel'' while others
say ``null space''; but neither term is ambiguous. Once somebody had
defined ``null space,'' others using the same concept would either stick
to the *same* term with the *same* definition or introduce a *different*
word with the same definition. There are very few examples of the *same*
word being used with *different* definitions, because it is rare that
two authors will invent the same term independently. (``Graph'' is an
unfortunate couterexample.)

In computer science it's all different. I didn't realize a week ago that
``first-class'' might be meant to imply status compared to other types,
so that first-class objects get all the privileges that any other object
can get. I had seen the standard definition of ``first-class'' in terms
of argument passing and variables, and as a mathematician I assumed that
people wouldn't corrupt a standard definition to suit their purposes.

I was wrong. Did two computer scientists independently invent
``first-class''? Apparently not. So how did the term acquire more than
one meaning? Either somebody doesn't have any respect for standard
definitions, or else people have been using ``first-class'' without
defining it precisely. This isn't immoral; it's just surprising to a
mathematician.

> Never in my life have I seen two mathematicians spend more than five
> seconds worrying about the "official" definition of anything.

Because the *definitions* are so rigid that there's nothing to worry
about. Apparently this isn't true in at least some fields of computer
science.

---Dan