Origin: Modeling

Modeling natural phenomena is the ultimate goal of natural sciences. Mathematics too pays the tribute to this noble activity. Actually, practically all mathematical abstractions are rooted in practical problems. On the other hand, novel areas of applications call for developing of new mathematical concepts. In the last (XX-th) century, mathematics penetrate practically all natural and social sciences, from physics to linguistics and correspondingly new branches of mathematics were originated.

How different the modern mathematics would be if Newton paid attention to biology or sociology instead of physics?

However, when the intellectual ``math engine'' starts to develop a theory in connection with the need of a natural or engineering problem, this development puts forward its own goals and develops its own means.

Is the situation out of control? Sure it is, and this is wonderful! Actually, mathematical theory may return the favor and become a useful tool for science in the future, or it may not. No one can judge on this matter.

The celebrated example of development of Non-Euclidean geometry illustrates this principle. Here how the story went:

• Euclid suggested system of ``obvious'' axioms of geometry that reflects our feeling about space.
• Lobachevsky rejected one of these axioms; was ridiculed then become famous. Non-Euclidian geometry was born as an intellectual exercise that deals with ``imaginary'' curved spaces.
• Modern physics accepted the idea that the Universe is curved which brought Non-Euclidean geometry back to the physical world.
• Geometry of curved spaces developed, generalized geometrical concepts, and originated many new applications, including the computer vision.

Next sections demonstrate the methodology of expansion of mathematical knowledge.