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.
I see Mathematics as somewhat like programming but not exactly like you.
Bijections seem to show information equivalence, like two programs outputting the same information but in different encodings.
Mathematical structures like rings, groups, vector spaces, ideals, etc.. are like object classes, if you can cast your current structure to them -- or prove it is that kind of structure via duck typing or whatnot, you then get a whole slew of new methods, transformations, properties, and casts that you can use on the structure to further manipulate it.
Instead of learning APIs, we learn mathematical objects/structures and their properties (necessary to cast other structures or construct them.)
The casting is really key. Once you cast a bunch of disparate things around using their derives properties and you get an equality, you've got a bijection. Connect the islands. Boom, modular form <-> elliptic curve, etc. Langlands program!
I think programming first helped a lot with having the mindset for information equivalence. That's really the key to so many theorems. Things that appear different and cannot be compared until their representations converge through a lot of wormy structure alchemy. Then you either have equivalence or at least, like structures, so a comparison can be made, and thus a decent theorem.
I really want to study representation theory. I hardly know it but it seems to be a computational/information theoretic subset of mathematics.
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.
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.
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.
I think I understand the point you are making, but it's not such a clear distinction.
But the universe isnt just particle configurations. It's the rules of physics too.
Mathematical truths, the ones mathematicians care about, are founded on axioms that have truth values set by our relationship with the physical world, it informs our intuition. This is inescapable.
But maybe you are right, math isn't the epitome of human knowledge. It is the art of learning that knowledge. Still, in some way, individual knowledge constitutes the extent of anyone's own universe.
The old and tired tripe of mathematics being some ethereal detached mental masturbation is so sorely mistaken that I must ask you to respond in as much depth as I did. Because your viewpoint is not only wrong but also the reason so many children grow up to be adults that dislike mathematics. Its a sad disinformation to disconnect the properties of our universe from the science that studies the most abstract type cast of our universe: object.
Imagination is in principle intuitive.
> it recognises no individual things, not even itself,
self-awareness is part of the common definition of intelligence.
> Imagine [something that is patently made up]
> Yet it is true
> it has no intuition of the natural numbers. If this being makes statement on arithmetic, like 1+1=2, ...
You can't just say
> imagine Natural numbers need not be in scope of intuition
> then Natural numbers need not be in scope of intuition
I mean, sure you can, but that is circular reasoning and not at all convincing.
Take a look at this: https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...
The fact that it could mean the universe or the language pretty much lets your argument fall apart.
However, there is another question. Why do we want anyone to do mathematics? Asking, not what math gives the mathematician but instead what math gives society at large. (That is, beyond contented mathematicians).
As long as mathematicians are self-reliant, the second question doesn't matter. However, when mathematicians start making demands of society the second questions is natural.
Note that these questions are orthogonal. Any answer to the mathematics extrinsic value need not inform its intrinsic value. Just like the intrinsic value of mathematics is not related to its extrinsic value.
The point here is that math for math's sake has an awesome byproduct for society: phd's that go commercial.
To be afraid that maths will then get squeezed for such phd's to the detriment of it's intrinsic value massively discounts the will of mathematicians and the long view of society.
Sure, there will be some that try this squeeze. But their effect will be limited. Meanwhile, this is how mathematics can sustain itself at much higher levels.
Do you really think there is so much true extrinsic value in all the proofs of mathematics? I'd say the value of all those who graduated and went on to do other things easily outweighs all those proofs.
It was the creation and later understanding of those proofs that had extrinsic value. This means that any field of mathematics has value no matter how likely it is to ever be applied.
As such, extrinsically it matters little what field you study, all of it is useful.
Apology in this context means "defense of".
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.
While the OP makes some solid points,
mostly I disagree with the essay as a
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
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
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
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
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
(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
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
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
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
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
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,
Want to improve the situation for math in
(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
Back to it!