What do grad students in math do all day?

When it's finally your turn to try your hand at actual research, it turns out that your contributions are barely a couple of side notes on a restricted subset of a problem in the hope that someone will use that information to find out something that is actually relevant in practice.

Rather than turning me off from academia, it makes me marvel at the tower of minuscule pebbles upon which our modern civilization rests. One day, I might get to place a few more of them on top.

It does give you a bit of a cynical outlook on new tech in general.

The Man Who Almost Invented The Vacuum Cleaner

The man officially credited with inventing the vacuum cleaner is Hubert Cecil Booth. However, he got the idea from a man who almost invented it.

In 1901 Booth visited a London music-hall. On the bill was an American inventor with his wonder machine for removing dust from carpets. The machine comprised a box about one foot square with a bag on top.

After watching the act -- which made everyone in the front six rows sneeze -- Booth went round to the inventor's dressing room.

"It should suck not blow," said Booth, coming straight to the point. "Suck?", exclaimed the enraged inventor. "Your machine just moves the dust around the room," Booth informed him. "Suck? Suck? Sucking is not possible," was the inventor's reply and he stormed out. Booth proved that it was by the simple expedient of kneeling down, pursing his lips and sucking the back of an armchair. "I almost choked," he said afterwards.

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There's a story here somewhere that aught to relate to mathematics, if only I could find it...

Imagine that all of mathematics is represented as a solid sphere (ball). At the core (origin) of this sphere are the most basic concepts in math, that we all learn in elementary and high school. On top of the core are many layers of knowledge that are all interconnected, but lead in different directions from the origin. These layers have names like algebra, calculus, and geometry. On top of those are other layers with names like topology, set theory, number theory, analysis, etc. These layers continue, like an onion, all the way to the surface of the sphere.

All mathematics students begin at the core and climb outward, towards the surface of the sphere. They choose different directions and set out to learn and practice everything that they encounter on their paths through the sphere. Eventually, after many years of study, the students who survive finally reach some point on the outside surface of the sphere. Which exact point on the surface each student reaches depends on the direction that he chose at the start. Once the students reach the surface of the sphere, they have understood everything that is known to mankind about some specific series of subjects within their speciality. Standing on the surface of the sphere, they must then must work to add a new layer to the sphere, to add something new and original to human knowledge.

As time goes by, new theories are developed and added as new layers onto the sphere. In ancient times, a great mathematician could learn everything in the whole of the sphere within a single lifetime. The sphere has grown exponentially, however, and in modern times no one person could ever visit every place in the sphere within a single lifetime. The sphere has become so vast as to defy comprehension, as generation after generation has expanded it with new layers.

When the mathematics students were starting out in the core of the sphere, they were all in the same place. They could easily see and speak to one another. However, when they reach the surface of the sphere, it is as if they are scattered across different points of the surface of a huge planet. A student might be lucky to find himself at a popular spot on the surface, where there are perhaps a handful or even a few dozen other students who he can talk to. Another student, less fortunate, may find himself stranded in a deserted place where there is no one that can hear him and there is no one for him to speak to. The surface is a lonely place, where few souls are encountered and if you do encounter some wandering traveller then you are unlikely to speak to the same language and must communicate by crude gestures like waving of hands.

As more and more matter is added to sphere, the dwellers on its surface drift further and further apart, as the surface area expands. Furthermore, the surface of the sphere grows farther and farther way from the core of the sphere. It takes longer and longer for the students to reach the surface of the sphere, as there is more and more volume to traverse.

Just to play the devil's advocate ... math has wonderful symbols and methods, but they don't really assist in comprehension unless everyone agrees on their meaning, and then only if everyone already understands the underlying concepts, the axioms. For example, starting in 1910, Bertrand Russell and Alfred North Whitehead published "Principia Mathematica" (a borrowed title):

http://en.wikipedia.org/wiki/Principia_Mathematica

But, notwithstanding their high intellectual level and the ambitions behind the project, and notwithstanding the system of symbols used, Russell and Whitehead missed a crucial, central point -- their plan to systematize mathematics, place it on a solid logical foundation, make it immune from uncertainty and doubt, was doomed from the start. Kurt Gödel demonstrated this a few years later:

http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo...

So much for the power of symbols. The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds.

Somewhere I read: "There is a famous recipe for rabbit stew that starts out, 'First catch a rabbit'.", and I changed that to, "There is a recipe for how to do applied mathematics, first get an application.".

For more, commonly the main criteria for 'research' is that it be "new, correct, and significant". And quite broadly in some powerful places, e.g., a famous David report, there were complaints that a result in math that met the first two but had no visible applications, inside or outside math, was likely short on "significant". So, eventually it dawned on me that if start with an application (something significant in the real world, although inside math would do also but tends to be more difficult and less highly valued outside math) and get a good solution for that application, then have "significant" handled. Yes, these thoughts did occur to me, but they were only secondary: My real interest was 'significant' outside of math and, in particular, in my bank account.

So, on to "new, correct": In math, "correct" is comparatively easy -- just work in the style of definitions, theorems, and proofs where it is fairly easy to check math correctness.

That leaves the part "new": Surprise! If start with a significant problem from the real world, then likely there is no solid solution for that problem on the shelves of the research libraries. Why? Because it's a really complicated real world out there! So, find in your real problem where current math doesn't really provide a solution and then do some more math to get some math for a better solution for the real problem. Now maybe the math just did that was "new" is not as earth shaking for pure math as, say, resolving the Riemann hypothesis, or, now, P versus NP, but still have covered "new, correct, and significant" and, besides, may have something powerful and valuable for the real problem outside math.

And that new math result got for that one real problem has a nice property: Given a new result in 'pure' math, the probability of an application in the next 12 months is small. Given a new result in math that has an application, the probability of another application in the next 12 months is nicely higher. Moreover, that probability appears to be monotone increasing with the number of known applications. Indeed, one skeptical way to evaluate such a result is to look for two significant applications instead of just one!

There is more going for this approach: Are taking math directions and 'values' based heavily on what solves some problems outside math. Well, where'd we get calculus? Sure, trying to make sense out of elliptical orbits of planets. And calculus is the main well spring of the part of math called 'analysis' that is so far by a wide margin the most applicable part of math (I know, number theory can do good things for computer security; maybe some people studying string theory in physics will want some topology; and people in logic may value work in foundations). But, tough not to notice that calculus led to the study of heat flow and Fourier theory which did great things for signal processing.

But in part the OP is correct: When I went through measure theory, it seemed fantastic stuff, especially since finally I had a better theory of integration for applications. But in Rudin's 'Real and Complex Analysis' he discussed regular Borel measures, and I never saw just why he cared about the 'regular'. Maybe if I'd go back and think about those few pages for a few days I'd see it. Yet, if I did see it, then I'd write it down so I wouldn't have to work to see it again, and I wish that Rudin had done that in his book. The precise definitions, theorems, and proofs are crucial, but too often pure math is written with too little explanation of the view from 50,000 feet.

My view is that the key to much more value from computing over the next few decades will be some novel uses of math and its techniques of definitions, theorems, and proofs. Why? Because for what to do in building our hardware and system software, applications, and larger systems, we need more powerful tools than intuitive heuristics or just programming what in principle we see how to do manually.

So, right, in the short term, my approach to math is to do 'applied' math where essentially we start with an application. Then in the longer term my hope is that such math, as calculus did, will lead to new, grand, powerful structures in pure math.

Yes, if the pure mathematicians can make good progress as isolated from applications as in the OP, then good for them, but I concluded that, in effect, good math needs some good applications from outside math.

Much of this 'philosophy' has come to pass whether deliberately or not: Quite broadly it is accepted that the best research 'mathematizes' its field. So, yes, the leading example is mathematical physics, but math is now just crucial in mechanical, electrical, electronic, and civil engineering, statistics, and operations research. Other fields that try to be more mathematical include finance, economics, psychology, sociology, and, now, genetics. And of course computer science is becoming increasingly mathematical.

As powerful as math has been for these other fields, pure math has essentially been left suffering as the applications, grants, and students based on applications of math go to fields outside math. So, if I were a chair or a dean over a math department, then I would welcome serious attention on important problems from outside math. I would keep fully high standards of definitions, theorems, and proofs. But, for more, first cut it would seem that the criteria "new, correct, and significant" would be easier than those criteria with also "applicable, powerful, and valuable" outside of math, but my view is that this is false, that being applicable is easier just to publishable papers but more importantly to real significance both inside and outside math.

This is essentially the idea behind Natural Semantic Metalanguage[1], Andrzej Bogusławski's theory of cross-cultural semantics from the 70s.

[1] http://en.wikipedia.org/wiki/Natural_semantic_metalanguage

The whole thing took about a year, after which I stared my post-doc in public policy. After that, I became a stock analyst. After that, I tried some startups (not successfully). Then I became a self-employed industry analyst.

What could be more straightforward than that? :)

In college a I was told "quantum mechanics = linear algebra" then I hear "electrical engineering = linear algebra".

They were right.

The PhD is a little different. It becomes so specialized... your papers are geared towards only a handful of experts. Literally you and a few other people are the only in the world who understand it. I thought mathematicians had a lot more common ground, but they don't.

"Tool to get rid of dust."