How to explain to the layperson what mathematics is, why it's important, and why it's interesting A mathematician walks into a party.


No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. Somebody asks you: 


  "So, what do you do?"


What do you answer? "I am a mathematician / math student / math enthusiast," whichever applies. Then, two things can happen: 


they are satisfied with this answer, which, almost all of the time, means that they are satisfied with their own misconception of what you do. They perhaps imagine that your main tool is a TI-83 calculator, that you spend all day crunching interest rates or solving quadratic equations using that formula they also memorized and forgot long ago. In the best of cases, they believe that your work consists in doing a bunch of calculations with a bunch of numbers, and that it must be incredibly boring. (The only upshot - they usually believe that "you must be smart".)
if you are lucky, they ask you what you really do, and then you get a shot at explaining.


I'm not sure which of these two situations I find the hardest to deal with, although I have dealt with both countless times. I've come to think that mathematicians are the most misunderstood professionals in the world. Biologists, physicists, writers, sociologists, and maybe even philosophers all have an easier time explaining what they do. Of course, there are popular misconceptions about them also. People might believe that biologists study cells under the microscope, or that physicists study very small things with lasers and very big things with telescopes, or that writers write and drink, or that philosophers study Aristotle. Although none of these beliefs is correct, none seems so far removed from the truth as the belief that mathematicians calculate interest rates with a TI-83 calculator.

Most of the time, I give up before even trying. And those rare times when I do try to give my interlocutor a glimpse of what mathematics is about, I fail horribly. Could it be that the beauty of mathematics is impossible to communicate, except to those people who have already had a taste of it? It might explain why so many mathematicians seem to marry each other - perhaps they feel that no other person could possibly understand them!

So my question is - how do you explain to people what mathematics is? And how do you convey the idea that it is absolutely fascinating and mysterious and that it is one of the greatest human enterprises - all in five minutes? My experience has been that going into any amount of detail - by which I mean uttering a single strange word - is a horrible idea. I'd really love to know how others deal with this.

Thank you all in advance! :-)

Addendum: I am disappointed that this question was closed after less than a day, especially considering that in such a short time it was viewed over 500 times. Obviously this question strikes a chord with some people, and since it is well-posed question which admits real answers, I think it should be reopened. What is the soft-question tag for, if not for a question like this? Why have similar questions not faced the same fate at MO? Anyways, no bitterness - thanks to all of you who shared your insight! (And please vote to reopen if you can. :) )
I am not a mathematician, and I don't care what kind of mathematics someone I just met at a party finds absolutely wonderful - and I don't think anyone else will either.

"I am a mathematician / math student / math enthusiast" doesn't actually answer the question: "So, what do you do?".

In fact, what that question is actually asking is: "How do you make money?".

People already have an idea as to how accountants, programmers, burger-flippers, etc make money. The layman really doesn't have any idea how a mathematician makes money, which is why they may make the assumption that you just sit at a desk doing calculations all day, everyday; probably for an accountant or statistician.

So, to answer their question just tell them how you make money, better yet, let them infer the reason why someone would pay money for them to do what they do.

Examples:


A programmer: "I'm a web-programmer, I make websites similar to facebook."
An engineer for NASA: "I'm an aerospace-engineer, I help NASA [do whatever a NASA engineer does]."
A musician: "I'm a drummer, I play the drums for Lynyrd Skynyrd."
What do mathematicians do ?  Mathematicians do mathematics.  What is mathematics ?  It is an appendage of natural language that is a bit more precise than natural language and any human knowledge that is not based on intuition or experience or custom can often be abstracted into mathematics and surprisingly, in this abstract world, one can see the pure essence of the knowledge captured (or sometimes, but rarely, jumbled).  Mathematics is what offers physics, theoretical chemistry, engineering, medicine, computing, business, logistics, networks, ... thought-itself to have a language beyond natural language with which to express itself and understand itself especially in regions when natural language does not suffice to elucidate.  Mathematics is the language of abstract knowledge.  Mathematicians are the grammarians (and poets) of this language.