What I Do

Math is neither an art nor a science, it's mathematics.

Tonight is one of those nights where I drank some caffein pretty late. This causes me to not be able to sleep as I usually don't drink much caffein, but I am not really able to concentrate at full capacity either. So, in the state I'm in, I've decided it's a good time to make my profile a little less empty.

I've got math stuff all over my profile already, but it is what I have spent most of my time doing this last year. I am a first year graduate school student and I have to pass these exams at the end of the year or I am kicked out. So, starting Monday, I am studying 6 hours a day, 6 days a week, for the next two months to learn all of this material backwards and fowards. I hope you all can understand if I seem a little obsessive. Also, that being said, I'm actually a pretty mediocre student here. That's not what I want to talk about though...

It's been my experience that most people have horrible misconceptions of what mathematics is, and though I don't pretend to understand it I would like to just get some thoughts out about it. Also, I would like to learn more about how art and mathematics are related as I have seen they have some things in common, but I am not really involved in the art community and I don't really know what artists 'do'.

A few (all too) common misconceptions:
1) Math is linear - Many people think you learn algebra, then geometry, then calculus. So if I am in algebra, I don't know as much math as someone taking calculus. In fact, I have never met somebody who does research in calculus, but algebra is still a big field of research. I plan on taking a year long algebra sequence next year which will cover groups, rings, fields, modules, vector spaces, categories, etc at the graduate level. These are all algebraic structures.
2) Math is about computation - Most mathematicians rarely work with numbers. People concerned with computation usually work in 'computer' science. In the topology class I am taking this year, I don't think I've seen a number bigger than 10 ever come up.
3) Math is easy - Just because you could follow the algorithms presented to you in high school or low level college math courses doesn't mean you are good at math. In fact, the real test is when you begin doing research for the first time, so I don't really know if I am good at math yet. On the other hand, doing bad in courses doesn't mean you are necessarily bad at math either as math isn't really the same thing as coursework. Good or bad though, math is hard, and that is why most questions in mathematics are currently unsolved.
4) The purpose of math is to apply it to the real world - Most mathematicians have no use in mind for math they are working on, and usually they don't really care if it has a use (although it is nice to think one day it might).

Most people think of math as following some steps or rules laid out for you to find a solution to an equation or simplify a formula. Somewhere throughout the many years of learning 2000 year old algorithms most people are introduced to 'proofs'. Once you have gone far enough in college courses, everything becomes proof based. About when the engineers are done taking math classes, a major shift is made and you are expected to view math in a whole new way. Rather than following some steps someone gave you to solve a problem, you are given a type of problem and you have to come up with the steps needed to solve such a problem (kind of).

In higher level math courses, you usually start with definitions and axioms. These are your building blocks. In lecture, the professor will prove results with these building blocks and in homework a student will usually get results that have much less difficult arguements. The arguements are based on logic, linking ideas, using contradiction, contrapositive, induction as different techniques. Some homeworks can be 10 pages of writing, they look like essays. Every arguement must be solid and rigorous or points are deducted and some of the arguements get really tricky and creative. The thinking becomes very abstract and sometimes concepts are hard to grasp, such as in topology where one must get used to working with geometric objects that live in an arbitrary number of dimensions.

To do real math, to get a Ph.D. and official be called a mathematician, one must prove a theorem that has never been proven before. Believe it or not, but most questions in mathematics have never been answered, although they take a lot of background to understand and one needs to have practice with the techniques necessary to solve a problem that no one has solved before.

Note where these axioms come from, however. We make them up. There is no basis for them, we just assume some things and then play the game "if those things are true, what can we deduce?" What is the point of all this? Why do we care? Most math isn't actually applicable to anything anyway. But then, what is the point of anything? Why do we really care about anything?

Math is as old as humanity and seems to be a very innate part of us. It is a totally fictitious system of abstract objects that live only in our minds, but are defined with perfect understanding and clarity within our thoughts. This is why our society tries to quantize everything, this is why business and science want to model and approximate with mathematics. It is our most definitive tool for communication, and as we expand mathematics we expand our abilities to clearly communicate with one another. Let me give a simple example. Say two people with baskets full of oranges are having a conversation:
Person 1: "I have several oranges in my basket"
Person 2: "I have a whole lot of oranges in my basket, I can hardly carry them all"
Person 1: "Let's count our oranges and compare"
Person 2: "It seems I have 50 oranges"
Person 1: "I have 80"
This is a silly example off the top of my head, but it conveys the idea that our definitions tend to be vague until we quantize them.

Math is human understanding. I think to understand humans, one needs to understand mathematics. There is no underlying force in nature that I can see which says "If A implies B and B implies C then A implies C" or "If A then B is the same as If not B then not A". These understandings are built into humans and are more concrete than anything else. Why is it that all of our technology is based off of some kind of mathematics? Because math is something we can trust, more than anything people have faith in math. How can 2+2 not be 4 unless we just define it that way. If I say "Here is a new system of math and in it 2+2=0" this is perfectly valid (and happens sometimes). It always works because I defined it that way or I assumed the axioms that made it true. Once we can relate physical objects to mathematics, now we have something that is really tangible to us on a deeper level, a mental level, we know exactly the rules of math. It is truly coincidence that they work beautifully with nature (although nature probably influenced math).

How can one then not see the aesthetic value of mathematics? It connects people universally in a way nothing in the physical world can do, perfect understanding shared at a mental level. Anyway, this crazy math guy has rambled on long enough, and it's really late, and as I said in the beginning I'm not working at full mental capacity due to not being able to fall asleep although I'm tired.

If anyone actually read this I would enjoy comments.