It's a great comment, graycat, and I couldn't agree more. Where we might differ is that I think the future of computing is in Computer Science -- but where it will lead to an extension of mathematics (specifically Set Theory). But we're essentially saying the same thing.
For where computing will be based, currently
there are big cultural issues:
With one
paper I published, in a computer science
journal for a problem in computer science
but basically some applied math, I had to discover that, really, it was tough for the computer science
community to review the paper. I sent an
'informal' submission to a nice list of the top
journals appropriate for the paper. From one
Editor in Chief of one of the best journals and
a chaired professor of computer science at one
of the best universities I got back "Neither I
nor anyone on my board of editors has the
mathematical background to review your paper."
Then I got a similar statement from another
such person. For one at MIT, I wrote him
background tutorials for two weeks before he
gave up. Finally I found a journal that
really wanted the paper, but apparently
in the end only the editor in chief reviewed
the paper and did so likely by walking it
around his campus to the math department
to check the math and then to the computer
science department to check the relevance
for computer science.
I had to conclude: For the good future of
computer science via math, the current crop of
CS profs just didn't take the right courses
in college and graduate school.
That's one side of the cultural divide. For
the other side, (1) the math departments
always want to be as pure as possible and
f'get about applied math and (2) in applied
math really don't like doing computer science.
Some obscure math of relativity theory,
maybe, but mostly nothing as practical as
computer science.
Of course, the big 'cross cutting' exception
is the problem P versus NP, apparently first
discovered in operations research (integer
linear programming, yes, in NP-complete),
later in computer science with SAT, and now
at times taken seriously in math, e.g., at
Clay Math.
Here's a 'reason' for the math: When we
write software, we need something prior to
the software as, say, a 'specification', that
is, saying what the software is to do.
Now, where is computer science going to get
that specification? A big source has been
just to program what we know how to do
at least in principle just manually. After
that, computer science starts to lose it
and drift into 'expert systems' (program,
with 'rules' and Forgy's RETE algorithm
what an expert says), intuitive heuristics,
and various kinds of brute-force fitting,
machine learning, neural networks, where
basically where we fit to the 'training'
data, test the fit with the rest of the
data, and stop when get a good fit. So
we throw fitting methods at the data
until something appears to stick.
We need more powerful means of getting that
prior specification. The advantage of
math is that it can start with a real problem,
formulate it as a math problem, solve the
math problem, and then let the math solution
and what is says we needed to do in manipulating
the data be the specification for the core
software. E.g., if want to design an airplane
on a computer, then start with the applied
math of structural engineering, mechanical
engineering, and aeronautical engineering, program
that applied math, and then design the plane.
For software to navigate a spacecraft to the
outer planets, start with Newton's second
law and law of gravity, get the differential
equations of motion, get some numerical
means of solution, and then program what the
numerical analysis says to do.
We need things solid and prior to the software
to know what software to write, and basically
that is we need a math solution first.
For computing itself, as in monitoring, we
can call that a problem in statistics --
more applied math.
A lot in computer load balancing is
some serious applied math. Actually
optimal job scheduling is awash in
NP-complete optimization problems.
Or, we used to have 'metaphysics'. Then
physics became mathematical and made
real progress. Basically the solid logical
chain of correctness given by math theorems
and proofs is just too darned hard to
compete with or, thus, ignore.
