A new “Mathematician’s Apology”
With the main issues in the OP, I have
struggled for too many years, and I
strongly agree that the main issues are
very important.<p>While the OP makes some solid points,
mostly I disagree with the essay as a
whole.<p>I got into math because (A) I was good at
it and (B) math was presented as useful.
For (A), no way could I please humanities
teachers, but when my math was correct,
easy enough for me, no teacher could
refuse me an A.<p>I got a big shot of enthusiasm about the
usefulness of math as I worked, starting
partly by accident, in applied math and
computing within 100 miles of the
Washington Monument.  There was a LOT of
applied math and computing to do, heavily
for US national security (right, needed to
be a US citizen with a security clearance
of at least Secret, and I had both).<p>Some of the topics were curve fitting,
numerical linear algebra (right, all the
Linpack stuff, the numerical stability
stuff, and the applications), antenna
theory, e.g., for adaptive beam forming
and digital filtering for passive sonar
arrays, multivariate linear statistics
(about a cubic foot of books), statistical
hypothesis testing, the fast Fourier
transform, numerical integration,
optimization (unconstrained non-linear,
constrained linear and non-linear,
combinatorial, deterministic optimal
control, stochastic optimal control,
etc.), time series, power spectra, digital
filtering, numerical solution of
differential equations (ordinary and
partial), integration of functions of
several variables, statistical inference
and estimation, estimation of stochastic
processes, algebraic coding theory, Monte
Carlo simulation of non-linear systems
driven by exogenous stochastic processes,
building good mathematical models of real
systems, etc.<p>For the applied math, I was in water way
over my head, struggling to keep my head
in the air, while drinking from a fire
hose.  I made good money, e.g., quickly
was making in annual salary about six
times what a new, high end Camaro cost.
And I had just such a Camaro and daily
drove it something like road racing all
around within 100 miles of the Washington
Monument, occasionally ate at the best
French restaurants in Georgetown, got a
lot of samples of nearly the best grape
juice from Burgundy (Pommard, Corton,
Nuit-St.  George, Chambertin, Morey-St.
Denis, etc.), occasional samples from the
Haut-Medoc, Barolo from Italy, etc., had
big times at Christmas, enjoyed the
museums on the Mall, etc.  Good times.<p>After some years of that math fire hose
drinking, I got a Ph.D. in applied math
from research in stochastic optimal
control for a problem I&#x27;d identified
before graduate school.<p>For applications to the stock market,
well, for a while the Black-Scholes
formula was popular, but by now that
flurry of interest seems to be over.  For
the more general case, say, of solving the
Dirichlet problem by Brownian motion, that
seems not to be of much interest.<p>Apparently the main success was just the
one by James Simons and his Renaissance
Technologies.  Of course, Simons is a
darned good mathematician.  For just what
his math training contributed to his
investment returns, maybe actually Simons
is an example of the OP&#x27;s remarks about a
math education being good training in how
to think.<p>For the rest of business, my view is that
significant, new applications of math are
dead, walked on like dead insects, and
swept out the door -- very much not wanted
and otherwise bitterly resented and
fought.<p>Or, to work for someone in business who
has money enough to create a good job for
you, they are nearly always rock solidly
practically minded, no nonsense,
conservative, rigid as granite, have for
all their careers rejected thousands of
opportunities to waste money, and never
but never invest even 10 cents in
something THEY do not understand or trust.
So, the first time they see &quot;Theorem&quot;,
they walk away in disgust; never in their
business careers have they ever seen
&quot;Theorem&quot; lead to money made.<p>Such a business person really can make use
of information that is technical,
advanced, obscure, specialized, etc. and
do so frequently from experts they trust
in finance, engineering, medicine, and
law.  Note, math is NOT in that list.<p>Note:  It is true that occasionally some
lawyers want to draw on mathematicians as
expert witnesses to try to win some legal
cases.<p>So, for that context of mainline US
business, math has two huge problems:<p>(A) Math is not a recognized <i>profession</i>
like law, medicine, and much of
engineering.<p>(B) Math has, in business as best as
business leaders can see, from no track
record to dismal, time and money wasting
disasters.  People who have made good
money in US mainline business have seen
many disasters, but relatively few of
their own, and very much want nothing to
do with disasters.<p>In particular, IMHO the OP&#x27;s argument for
math in business based on some version of
intellectual or conceptual <i>diversity</i> or
<i>way of thinking</i> will fly like a lead
balloon or float like a canoe with a
framework of cardboard covered with toilet
paper.<p>For US pure math research, here is my
nutshell view of the situation:<p>As in a famous movie, &quot;The bomb, the
hydrogen bomb, Dimitry&quot;, is one heck of a
big reason.  A little more generally, from
another famous movie, &quot;Mathematics won
WWII&quot; -- not exactly true but darned
close.<p>For a short version, Nimitz, Ike, and
MacArthur slogged and struggled, but the
end was from two bombs in about a week.<p>Those bombs were heavily from some good
applied math and physics, and there were
more really important to just crucial
contributions via code breaking, radar,
sonar, and more.<p>Big lessons tough to miss.<p>Supposedly at the end of WWII Ike said
something like &quot;Never again will US
science be permitted to operate
independent of the US military.&quot;.<p>Since then, Gulf War I showed more of the
overwhelming power of good applied
math&#x2F;physics, e.g., the F-117.<p>Broadly the lesson was:  Basic physics is
super important stuff.  The next country
that discovers something as fundamental,
important, and powerful as nuclear energy
might take over the world in a week.  So,
the US MUST be right at the leading edge
in fundamental research in physics.<p>Much the same for mathematics.<p>To these ends, the US will just ask US
high end research university academics to
be at the world class leading edge,
whatever that is, say, as can be seen in
the internationally competitive aspects of
research and publishing, Nobel prizes,
etc., in basic math and physics.<p>So, what the Harvard, Princeton, MIT,
Chicago, Berkeley, Stanford, Cal Tech,
etc. math and physics departments want for
funding for basic research to be the world
champions, they get.  Period.  For
defending the whole US, it&#x27;s not many
people or much money.<p>The money will come via the NSF, DARPA,
ONR, Air Force Cambridge, Department of
Energy, or wherever, but Congress will
write the checks, no doubts, no delays, no
questions asked.<p>There will be more research funded in
units attached to universities, various
national labs, various companies, etc.
So, there&#x27;s Oak Ridge, Lawrence-Livermore,
Los Alamos, Argonne, Lincoln Lab, Johns
Hopkins University Applied Physics Lab,
Naval Research Lab, Raytheon, Lockheed,
GE, NSA, etc.<p>Still, considering the size of the US, the
size of the US economy and the Federal
budget, and the importance of US national
security, we&#x27;re not talking very many
people or much money.<p>Broadly, research is cheap and a big
bargain.<p>And Congress can lean back, relax, and
easily see that US academic research is
extremely competitive.  Genuinely
brilliant students are awash in
scholarships.  For a new Ph.D., for a good
job at Harvard, Princeton, etc., the
student need only do some really good
research -- one good paper, if really
good, is quite sufficient.  If they keep
the really good papers coming, keep
getting prizes, etc., then the money will
keep coming.  No problemos.  And for the
fundamental research that Congress and the
US DoD want, that competitiveness is
enough.<p>For math in business?  The solution is
easy:  (A) See a good problem, that is,
some nicely big pain in the real world.
(B) Do some applied math research to find
a good solution.  (C) Write software to
implement the solution and deliver it over
the Internet, maybe as just a Web site.
(D) Get a first server, for $1000 or less,
go live, get users&#x2F;customers, revenue, and
earnings.  Slam, bam, thank you mam.
Presto.  Bingo.  Done.<p>Here never have to convince some rock
solid, conservative mainline US business
person that your theorems are valuable.
All such people see is the solution to the
big pain and your happy trips to the bank.<p>Notice that (A)-(D) isn&#x27;t done very often
and don&#x27;t have a lot of examples in the
headlines?  Right.  So, good news; there&#x27;s
not much competition!<p>Accountants can confirm the revenue and
earnings, and that&#x27;s enough for VCs,
private equity types, M&amp;A types,
investment bankers, institutional
investors, stock pickers, stock funds,
etc.<p>Want to improve the situation for math in
business?<p>(i) Okay, need more examples like what I
just outlined in (A)-(D).<p>(ii) Then need to have applied math
graduate schools borrow from law and
medicine and be clinical and professional.<p>Don&#x27;t hold your breath waiting for (ii);
that would mean that good applied
mathematicians would be employees instead
of their own CEOs, and that&#x27;s not so good.
Or, if a good applied mathematician wants
a good job, then they should create it for
themselves by being CEO of their own
successful startup.<p>Back to it!