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A while back I wrote an article about confusing and misleading

technical jargon, drawing special

attention to botanistsâ€™ indefensible misuse of the word â€śberryâ€ť and

then to the word â€śhengeâ€ť, 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 â€ścobordismâ€ť â€¦ will think for an instant that they have any idea what

it means â€¦ they will be perfectly correct.

But conversely:

The non-mathematician’s idea of â€ślineâ€ť, â€śballâ€ť,

and â€ścubeâ€ť is not in any way inconsistent with what the mathematician

has in mind â€¦

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 â€śopenâ€ť and â€śclosedâ€ť

sets. It sounds as if

they should be exclusive and exhaustiveÂ â€” surely a set that is open is

not closed, and vice versa?Â â€” but no, there are sets that are neither

open nor closed and other sets that are both. Really the problem here

is entirely with â€śopenâ€ť. The use of â€śclosedâ€ť is completely in line

with other mathematical uses of â€śclosedâ€ť and â€śclosureâ€ť. A â€śclosedâ€ť

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 â€śclosed

on the set !!S!!â€ť 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 â€śevenâ€ť and â€śoddâ€ť 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 â€śevenâ€ť polynomial

was defined to be a polynomial whose terms all have even exponents

(such as !!x^4 – 10x^2 + 1!!) and similarly an â€śoddâ€ť 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!! setsâ€¦

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 *fermĂ©* (â€śclosedâ€ť) 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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