TITLE: How to create new mathematics?
QUESTION [44 upvotes]: How do scientists and mathematicians create new mathematics for describing concepts? What is new mathematics? Is it necessarily in format of previous mathematics? Can one person make (invent or discover) a mathematics such that it isn't in format of geometry or algebra or analysis that we know? 
Of course I know we can define a new meter and have a new geometry, or define a new n-ary operation and have a new algebra, etc. But can we create a new mathematics that is not like that?
In fact my question is this: when a scientist (in particular, a physicist), tries to create a mathematical theory for a concept, how does he/she do it? May he/she invent a new mathematics that is not in format of previous available mathematics at all? (i.e it wasn't geometry or algebra or analysis, etc.) Thanks a lot.

REPLY [2 votes]: Creating new types of math is easy*
*for certain values of easy
To create new kinds of math. Just make up some rules. Or take an existing set of mathematical rules, and add a new rule. That's the easy part.
The hard part is creating new math that is consistent, interesting, and useful. 
After you've made up your rules, you probably want to know if your system is consistent. Suppose I wanted to add the rule that there is a biggest natural number n. But then what is n+1? Well, either n+1 = n because it's the biggest number, or I have that n+1 isn't a natural number. In either case I have problems because now. I could decide that the rule is that $ n+1 \neq n $ doesn't hold for all natural numbers anymore or addition on the natural numbers doesn't always result in a natural number. But either of these choices makes my new natural numbers less nice. And there are all sorts of contradictions that can be reached from either of those choices. And if a contradiction can be reached, in some sense, what I'm playing with is non-sense. 
That's what happened to set theory. At one point, they realized that given the current set of rules they were using for sets, you could define a set which both must contain itself and must not contain itself. But instead of giving up on sets, they just added some more rules to their system that prevented such a set from being defined and thereby left them with a set theory that didn't contain such contradictions.
But even if you choose a new rule that is consistent, it might not be very interesting. It might turn out to be something that some one already thought of before but just described in different terms.
And even if it's new and interesting, it might not be useful. But don't let that dissuade you. There are many branches of math that upon their creation were not very useful, but people found uses for them later. Knot theory was once considered completely useless, but it was used to prove the structure of DNA.
Even if it's not new, discovering something on your own that had previously been discovered by others can be very rewarding. It can give you a much deeper understanding of a topic, and it can be gratifying to realize you independently came up with the same thing as some clever person in the past. And it can be just fun.