What do grad students in math do all day?

Sounds like what math grad students did 50 years ago, before we solved all the pure math even tenuously related to the 4 dimensional reality we live in.

I prefer the explanation that I developed back when I was a math student:

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.

> Fortunately, math has an incredibly powerful tool that helps bridge the gap. Namely, when we come up with concepts, we also come up with very explicit symbols and notation, along with logical rules for manipulating them. It's a bit like being handed the technical specifications and diagrams for building a vacuum cleaner out of parts.

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.

You are just wrong. For example, algebraic topology is one of the most abstract subjects I hear conversations about week. It has applications in physics, data analysis, and numerical computing.

Also, saying that reality is 4-dimensional is hooding yourself. The machine learning techniques we use everywhere depend on the math of higher-dimensional spaces, and they wouldn't work if reality couldn't be meaningfully viewed as many-dimensional.

I'm a CS PhD, and I agree. I always tell people I'm studying math because it gives them much more of a flavor for what I'm actually doing everyday. When I say computer science, people think I'm just programming something.

Also, when people ask me, "So What do you actually do studying math?" I reply, "Sit and stare at the wall for several hours a day. Occasionally I write something down."