How are creative math ideas come up with to solve problems?

[trees and plants]

Hello.Well, this is a general question.when solving a math problem what are the ways the mind finds to prove the problem?How about giving new definitions for abstract objects or posing new problems on your own?What are the motivations to do these things?Should they be of very high value to be accepted?Should they be very difficult or be interesting to be accepted?I know that Aristotle in ancient Greece tried to solve problems other posed and he also posed his own problems and then tried to solve them.Is this done today in math or physics?Are mathematicians of two kinds, problem solvers and theory builders?What are theory builders?Thank you.

 

[jedishrfu]

What is the context for your questions? Is this for a school project? Is it personal interest? Do you have any other questions? Should I answer these? or should I wait until others decide to answer first? ...

The first person to prove something or the first person to come up with a new idea in math and other fields often uses out of the box thinking. Others may have tried before but didn't think in the same way. There is no method to this style of thinking but you must be open to breaking the rules that you learned in order to see beyond then.

Some historical examples:
- Archimedes and his mathematical solutions to the surface area and volume of a sphere.
- Gallileo's observation about falling objects
- DaVinci's art, many inventions...

Sometimes ideas are extensions of existing ideas that are powerful in themselves:
- having counting numbers and then debit counting to negative numbers
- adding zero to the counting numbers
- fractional numbers to rational numbers to real numbers

Sometimes problems like the quadratic polynomial have no solution in real numbers:
- imaginary numbers then to complex numbers
- then to quaternions
- and octonians

These are just my personal feelings on these ideas.