
A new “Mathematician’s Apology” - seycombi
https://ldtopology.wordpress.com/2017/03/18/a-new-mathematicians-apology/
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stonesixone
I also became a software developer after getting a PhD in mathematics and
specializing in three-dimensional topology.

One of the things I'm always struck by is how similar the process of writing
code is to writing a math paper. There are similar issues of encapsulation and
organization. Choosing the right abstractions and good names for things are
both important. Definitions correspond to data structures; lemmas correspond
to helper functions; theorems to higher-level functions; and sections to
modules. You can also "refactor" a math paper in the same way you refactor
code (e.g. renaming variables, choosing better names, etc).

What I've found missing in software relative to math is the creative /
research part of math, since the math that comes up in software tends to be
routine, easy stuff.

~~~
amelius
> One of the things I'm always struck by is how similar the process of writing
> code is to writing a math paper.

Except when coding you never have to write down any proofs :)

> the math that comes up in software tends to be routine, easy stuff.

Software is easy until it grows big.

Math is often elegant because the problem can usually be stated in a concise
way. In contrast, software usually has an ever growing list of requirements.
It is balancing those requirements that makes software difficult.

~~~
posterboy
TDD is proof by construction

~~~
merlincorey
TDD is very far from [formal] proofs.

~~~
ben_w
Out of interest, how close are formal methods to the mathematical standard for
proofs? VDM-SL was part of my degree, but the lecturer ended up showing more
limitations than strengths by getting his own example wrong, and sadly I've
had no real-life experience with them because none of my career to date has
involved things that need to be proven correct.

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goldenkey
Here is the levels of absolute truth in our universe in terms of dependency:

Mathematics > Physics > Chemistry > Biology > Physiology > Medicine

Discoveries in mathematics are truths about the universe. They are deeper than
particle physics in some respects. Some parts of mathematics might seem
abstract but every mathematical system uses the naturals in its axioms or
representation. The naturals are directly based on counting, based on the
nature of macroscopic objects in our universe. The universe enforces rules,
and the facts about naturals, and systems built upon them, are truths that
directly point at the nature of information and complexity in our universe.

Why should mathematicians apologize? Hardy was wrong, mathematics can lead to
nukes. But its the base level of truth, there is no other scientific
discipline that discerns the patterns of the most abstract physicality -
objects, and gleans truths, rules for how objects interact.

Solving the Riemann Hypothesis or other conjectures that aren't even known yet
might lead to understandings/models that allow for time machines. It's
impossible to know.

But why not seek to understand the universes' laws at its most generic level.
Its enlightening. Spiritual. Awakening.

~~~
j7ake
Are you talking about the universe as in the physical universe in which we
live ? Because although maths can be used to find out about our physical
universe, its abstractions go beyond what is in our physical universe...

Thinking of the maths involved in certain man made games (eg chess), those
maths aren't necessarily truths about the physical universe.

Maths have been useful for clear thinking to help understand and predict
behavior in the physical world but they remain distinct from the physical.

~~~
inimino
Isn't chess a part of the physical universe?

I think I understand the point you are making, but it's not such a clear
distinction.

~~~
contravariant
Well, maybe, but it's physically impossible (or at least improbable) for all
possible chess games to occur within the physical universe.

~~~
rini17
Wait what, I thought it occurred already in some IBM computer, no?

~~~
contravariant
There are estimated to be over 10^120 possible games, so probably not.

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Kenji
I turned my back on academia because in my eyes, it seems to be very toxic
towards playful exploration of mathematical or other scientific topics. Often,
you are forced into working on one particular issue, whereas exploring maths
is more like jumping from island to island where each one of them contains
secrets, and it definitely makes sense to follow the path wherever it takes
you. The structure is too rigid, every step needs justification. How can you
justify playing around with numbers and formulas, sometimes a bit aimlessly,
when you're in pursuit of a proof? And then you have so much overhead because
you have to document it all. Documentation makes sense, but let it be terse.
And then, of course, there is the pressure to achieve when hard work is only
one part of the equation, the other part being that ideas are essentially 'god
given' and come randomly. Thanks but no thanks.

~~~
rocqua
This seems to be the result of believing the extrinsic value of mathematics
being the proofs and theorems.

If we follow the argument by OP, it says that the extrinsic value comes from
any serious attempt to understand anything in mathematics. I think OP would
agree with you that playful exploration should be possible.

However, that exploration should also be useful to mathematics itself if you
want mathematicians to support it.

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graycat
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.

While the OP makes some solid points, mostly I disagree with the essay as a
whole.

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.

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).

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.

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.

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'd identified
before graduate school.

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.

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's remarks about a math education being
good training in how to think.

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.

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
"Theorem", they walk away in disgust; never in their business careers have
they ever seen "Theorem" lead to money made.

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.

Note: It is true that occasionally some lawyers want to draw on mathematicians
as expert witnesses to try to win some legal cases.

So, for that context of mainline US business, math has two huge problems:

(A) Math is not a recognized _profession_ like law, medicine, and much of
engineering.

(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.

In particular, IMHO the OP's argument for math in business based on some
version of intellectual or conceptual _diversity_ or _way of thinking_ will
fly like a lead balloon or float like a canoe with a framework of cardboard
covered with toilet paper.

For US pure math research, here is my nutshell view of the situation:

As in a famous movie, "The bomb, the hydrogen bomb, Dimitry", is one heck of a
big reason. A little more generally, from another famous movie, "Mathematics
won WWII" \-- not exactly true but darned close.

For a short version, Nimitz, Ike, and MacArthur slogged and struggled, but the
end was from two bombs in about a week.

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.

Big lessons tough to miss.

Supposedly at the end of WWII Ike said something like "Never again will US
science be permitted to operate independent of the US military.".

Since then, Gulf War I showed more of the overwhelming power of good applied
math/physics, e.g., the F-117.

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.

Much the same for mathematics.

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.

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's not
many people or much money.

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.

There will be more research funded in units attached to universities, various
national labs, various companies, etc. So, there's Oak Ridge, Lawrence-
Livermore, Los Alamos, Argonne, Lincoln Lab, Johns Hopkins University Applied
Physics Lab, Naval Research Lab, Raytheon, Lockheed, GE, NSA, etc.

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're not talking
very many people or much money.

Broadly, research is cheap and a big bargain.

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.

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/customers, revenue, and earnings. Slam, bam,
thank you mam. Presto. Bingo. Done.

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.

Notice that (A)-(D) isn't done very often and don't have a lot of examples in
the headlines? Right. So, good news; there's not much competition!

Accountants can confirm the revenue and earnings, and that's enough for VCs,
private equity types, M&A types, investment bankers, institutional investors,
stock pickers, stock funds, etc.

Want to improve the situation for math in business?

(i) Okay, need more examples like what I just outlined in (A)-(D).

(ii) Then need to have applied math graduate schools borrow from law and
medicine and be clinical and professional.

Don't hold your breath waiting for (ii); that would mean that good applied
mathematicians would be employees instead of their own CEOs, and that'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.

Back to it!

