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A new “Mathematician’s Apology” (ldtopology.wordpress.com)
65 points by seycombi 311 days ago | hide | past | web | favorite | 33 comments

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.

I am getting a PhD in mathematics after having programmed since I was 10, basically PhD level CS degree equivalent.

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.

> 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.

TDD is proof by construction

TDD is very far from [formal] proofs.

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.

That's shifting the goal post. But sure, jesuslop's pointer to the Curry-Howard-correspondance is closer to the meat.

You know, propositions are types, proofs are programs :-), thanks to Howard & Haskell Curry, somewhat culprit of the functional frenzy.

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.

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.

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.

Yes it's true chess is part of the physical universe and I can see why it's a difficult point to articulate. Perhaps one way to say it is that the rules of chess and strategy should not depend on the parameters that define your universe, only should depend on the rules of chess you decide to play.

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

maybe if you constrain the game to take place on a wooden board, pieces moved by players. Otherwise, e.g. the ions inside some exotic star may be moving in chess like patterns (for a while). who's to say. Where does the disagreement with the Pythagorean ideal stem from, anyway?

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

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

You seem to be confusing particle configurations for conceptual truth. Truths about triangles are truths about physical triangles to an extent. Same with circles although you are probably thinking "but theres no perfect circle in Nature!!." True.

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.

I would not disagree that many axioms that we use have direct relationships with the physical world, this is one of many ways math is useful in describing nature. I was trying to say that axioms in general do not have to be set to accommodate the physical world.

Those would be strangely wrong axioms though and probably not what OP had in mind. Just because a theory with axiom 1==0 contains numbers doesn't mean it's math, at least if it is not consistent.

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.

Mathematical truths have nothing to do with the universe.

Extremely naive thing to say. Mathematical truths are merely derived truths about complex structures. These structures borrow intuitive concepts like "sets" or "natural numbers" in their foundations. Mathematics has everything to do with the format of this universe. A statement is only meaningfully true if it is based on axioms that are intuitively true. The direct relationship between what mathematicians agree on as meaningful systems to derive truth from, ie. ZFC , has everything to do with the relstionship between the axioms and intuition that can only be provided by living in this world as a human being.

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.

Natural numbers need not be in scope of intuition. Imagine a diffuse intelligence formed in a fluid (such a thing is at least imaginable), it has no fingers, it recognises no individual things, not even itself, it has no intuition of the natural numbers. If this being makes statement on arithmetic, like 1+1=2, according to you this statement is not "meaningfully true". Yet it is true, and this being has derived it from (abstract) thought. So what is "meaningfully true"?

> need not be in scope of intuition.

> Imagine

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

utter nonsense

> 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.

You prove his point with your fluid example. Integers have _everything_ to do with our world. If we lived in a fluid world, then the mathematics of that world will not have integers.

Take a look at this: https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...

Perhaps my point is misunderstood here: it is that if we think of mathematical truth as being based on intuition, then since what can be intuited depends on the species, on the society, even on the individual (99% of what was intuitive to Grothendieck is completely beyond me), then mathematical truth becomes relative. I say no, 1+1=2 universally, whether it is intuitive to you or not.

Language too has nothing to do with the Universe - that doesn't prevent sentences in it from "mean"-ing something.

> that doesn't prevent sentences in it

The fact that it could mean the universe or the language pretty much lets your argument fall apart.

Mathmen are more Van Eyk than Pollok.

There is great beauty in this approach to mathematics. It is why I am currently studying mathematics. It is a great defense of why anyone would want to do mathematics.

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.

"Why should mathematicians apologize?"

Apology in this context means "defense of".

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.

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.

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!

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