The Universe of Discourse : Mathematical jargon failures

https://blog.plover.com/lang/math-jargon-failures.html

Mathematical jargon failures

A while back I wrote an article about confusing and misleading
technical jargon
, drawing special
attention to botanists’ indefensible misuse of the word “berry” and
then to the word “henge”, which archaeologists use to describe a class
of Stonehenge-like structures of which Stonehenge itself is not a
member.

I included a discussion of mathematical jargon and generally gave it a
good grade, saying:

Nobody hearing the term “cobordism” … will think for an instant that they have any idea what
it means … they will be perfectly correct.

But conversely:

The non-mathematician’s idea of “line”, “ball”,
and “cube” is not in any way inconsistent with what the mathematician
has in mind …

Today I find myself wondering if I gave mathematics too much credit
Some mathematical jargon is pretty bad. Often brought up as an
example are the topological notions of “open” and “closed”
sets
. It sounds as if
they should be exclusive and exhaustive — surely a set that is open is
not closed, and vice versa? — but no, there are sets that are neither
open nor closed and other sets that are both. Really the problem here
is entirely with “open”. The use of “closed” is completely in line
with other mathematical uses of “closed” and “closure”. A “closed”
object is one that is a fixed point of a closure
operator
. Topological closure is an example
of a closure operator, and topologically closed sets are its fixed
points.

(Last month someone asked on Stack Exchange if there was a connection
between topological closure and binary operation
closure
and I was
astounded to see a consensus in the comments that there was no
relation between them. But given a binary operation !!oplus!!, we
can define an associated closure operator !!text{cl}_oplus!! as
follows: !!text{cl}_oplus(S)!! is the smallest set !!bar S!! that
contains !!S!! and for which !!x,yinbar S!! implies !!xoplus yin
bar S!!. Then the binary operation !!oplus!! is said to be “closed
on the set !!S!!” precisely if !!S!! is closed with respect to
!!text{cl}_oplus!!; that is if !!text{cl}_oplus(S) = S!!. But I
digress.)

Another example of poor nomenclature is “even” and “odd” functions.
This is another case where it sounds like the terms ought to form a
partition, as they do in the integers, but that is wrong; most
functions are neither even nor odd, and there is one function that is
both. I think what happened here is that first an “even” polynomial
was defined to be a polynomial whose terms all have even exponents
(such as !!x^4 – 10x^2 + 1!!) and similarly an “odd” polynomial. This
already wasn’t great, because most polynomials are neither even nor
odd. But it was not too terrible. And at least the meaning is simple
and easy to remember. (Also you might like the product of an even and
an odd polynomial to be even, as it is for even and odd integers, but
it isn’t, it’s always odd. As far as even-and-oddness is concerned
the multiplication of the polynomials is analogous to addition of
integers, and to get anything like multiplication you have to compose
the polynomials instead.)

And once that step had been taken it was natural to extend the idea
from polynomials to functions generally: odd polynomials have the
property that !!p(-x) = -p(x)!!, so let’s say that an odd function is
one with that property. If an odd function is analytic, you can
expand it as a Taylor series and the series will have only odd-degree
terms even though it isn’t a polynomial.

There were two parts to that journey, and each one made some sense by
itself, but by the time we got to the end it wasn’t so easy to see
where we started from. Unfortunate.

I tried a web search for bad mathematics terminology and the top hit
was this old blog
article

by my old friend Walt. (Not you, Walt, another Walt.) Walt suggests
that

the worst terminology in all of mathematics may be that of !!G_delta!!
and !!F_sigma!! sets…

I can certainly get behind that nomination. I have always hated those
terms. Not only does it partake of the dubious open-closed
terminology I complained of earlier (you’ll see why in a moment), but
all four letters are abbreviations for words in other languages, and
not the same language. A !!G_delta!! set is one that is a countable
intersection of open sets. The !!G!! is short for Gebiet, which is
German for an open neighborhood, and the !!delta!! is for
durchschnitt, which is German for set intersection. And on the
other side of the Ruhr Valley, an !!F_sigma!! set, which is a countable union
of closed sets, is from French fermé (“closed”) and !!sigma!! for
somme (set union). And the terms themselves are completely opaque
if you don’t keep track of the ingredients of this unwholesome
German-French-Greek stew.

This put me in mind of a similarly obscure pair that I always
mix up, the type I
and type II
errors. One if them is when you fail to ignore something
insignificant, and the other is when you fail to notice something
significant, but I don’t remember which is which and I doubt I ever
will.

But the one I was thinking about today that kicked all this off is, I
think, worse than any of these. It’s really shameful, worthy to rank
with cucumbers being berries and with Stonhenge not being a henge.

These are all examples of elliptic curves:

These are not:

That’s right, ellipses are not elliptic curves, and elliptic curves
are not elliptical. I don’t know who was responsible for this idiocy,
but if I ever meet them I’m going to kick them in the ass.

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