[FOM] Notation. was: Responses re Certainty, Simplified Foundations

[W.Taylor@math.canterbury.ac.nz]

Harvey Friedman wrote:

> As far as I know, the automated deduction/proof checking people have
> paid very little attention to the crucial issue of language construction.

And as Harvey notes soon after, this is very close to the topic of
good NOTATION in math.  It has long been recognized that good notation
generally means good math, or at least, vice versa.     Harvey extends
this to good languages mean good math computing.  It is hard to disagree.

The two matters are not *quite* identical, OC, as notation tends to be
a lot more 2-dimensional than programs can be (as of now).

Given that good notation is such an important part to practical math,
it is odd that this FoM list devotes so little time to it.  As do FoM
studies in general.  This might be a good topic for us to pursue?

> Certain notation built up over centuries in mathematics
> is fatally bad for this purpose, yet mathematicians cling to it.

Indeed!  Harvey picks at a sore point here!  Mathies are notoriously
conservative in putting up with bad notation long after its sell-by date.
Partly this is pressured onto us by text-book requirements, but I suspect
it is mostly due to personal inertia.

Certainly, well-established notations *can* disappear.  An example was
the old habit, now virtually unknown, thankfully, of writing integrals
with the integral-sign dx *in front of* the integrand, rather than
surrounding it as now.  One can see some point to this, but it was
terrible for (e.g.) change of variable and the like.  And though dy/dx
is a good notation in general, especially for working with differentials
and increments; its extension to d^2y/dx^2 is rather hideous.  (For 2nd
and higher derivatives maybe Newton's old  "y-dot-dot"  was better?)
And one can't help feeling that partial derivatives could also have
been handled better - they are rather badly behaved regarding change
of variables, at least compared to plain derivatives.  Maybe that can't
be replaced by anything fully suitable, though.


One thing that *could* be changed, but I suspect never will be,
(in spite of efforts by some computer scientists and categorists),
because of the VAST re-writes it would entail, is function notation.
Like so many things (about 50% of them!) where some beginning researcher
had to make a 50-50 decision, this one, (like getting the positive and
negative electric chrages the wrong way round) was guessed wrong.
AFAICT from Cajori it was Laplace we have to blame for this.
He seems to have been the first to write the general function as f(x).
And, as we almost all know now, it would have been far better in almost
every respect, to write it as  (x)f.  At about the same time, the notation
for factorials, n! , was invented, almost the only standard function
to be written standardly in this direction.  It is a real shame that
the math world didn't take heed and agree on that, round about 1810.

Still, at least we no longer have (or do we?) books using the awful
forms "consider a potential V = V(x)", with its ghastly conflation
of function and value thereof.  My student days were full of this
monstrosity, and I loathed it even then (though I could see advantages).
In passing, on this sub-topic, I note that mathematicians tend to write
f for the function and f(x) for the particular value, whereas physicists
tend to do it the other way round!  Amusing, except when we get V = V(x)!


It would also be nice, for students at least, if the upper and lower
bounds on definite integrals and sums and products were written
the same way.  At present we have the slightly misleading mixture
of "n = 1" at the bottom of a sum, and just "10" at the top.  It would
be better to have "n=10" at the top as well.  Similarly, though equally
cumbersome, it would be nice to have integral limits written as "x = 0"
and "x = 1" at the top and bottom of integral signs, rather than just
"0" and "1".    With the dx already at the other end, this is not so
vital, but could be very handy for (at least) change of variables again.


Finally, while I'm in rant mode, let me extend my griping to unfortunate
terminology, rather than just bad notation.

Topology is particularly bad at this.    For example, I think the word
"connected" had been far better used for what we call "arcwise connected",
which is what I'm sure almost all mathematicians think of it as. Then what
we call "connected" could be called "inseparable" or whatever.

But my *huge* disgruntlement in terminology is for terms beginning
with the "non-" particle!     AFAIAC, these are non-words!  :)
We already have terms like "strictly positive" and "strictly increasing",
so why couldn't we have done the obvious and used "positive" and "negative"
to include zero.  It is almost universal for mathies to include border
cases in definitions, (e.g. squares are rectangles etc), and use "strict"
when we mean strict.  I gather the French *do* do the sensible thing and
use "positif" and "negatif" to include zero; I wish we did the same!
"Non-decreasing" is another such monstrosity.   Grrr.  The "non" in
the word is largely bypassed by the brain and we then find ourselves
concentrating on the wrong aspect of things.  We could take a leaf out
of the constructivist manual and speak of "inhabited sets" and so forth.

However, perhaps the terminology thing is merely pedagogical; rather than
truly logical (as the notation matters are), and thus off-topic here.

------------------------------------------------------------------------------
   Bill Taylor                              W.Taylor at math.canterbury.ac.nz
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