Proof based on geometric intuition

Question

When a proof or concept can be represented by some kind of picture, it becomes quite intuitive, but usually it is not formal. My question is that are there any tools in mathematics, which can formalize this kind of intuition. Any tools that can translate idea such as in analysis into a geometric one, so proof can be done using the techniques of geometry.

The idea comes from usually geometric or topological argument can simplify technical part of the proofs in analysis. I have seen it quite a lot of time. So I wonder is there any tools that are specialized in this task.

Answer 1

Yes, indeed we use geometry to find answers for some problems in calculus ,real analysis,topology and even in statistics.

For example in calculus if you want to find an integral from $-a$ to $a$ and the integrand is an odd function we use symmetry to conclude that the integral is $0$

If you have an integral which represent a known volume then we know the answer from geometry.

In general topology when you want to prove that an open ball is open, you get ideas from graphs to find the radius the smaller open ball centered at the given arbitrary point in the larger open ball.

In statistics we find probabilities by looking at the density function and find the area or the volume which represent the probability.

In summary we use figures to get ideas and sometimes answers but if a formal proof is required a picture is not a proof.

Answer 2

I would say that a lot of the intuition is gained from experience. I know when presented with a geometry problem and have to prove some angle equal to another or some length equal to another there is often a systematic search for congruent or similar triangles, symmetry, parallel lines and alternate or corresponding angles, and properties of triangles, quadrilaterals and circles etc often resulting in additional construction to create useful shapes with known properties. The more this is practiced the better one becomes at seeing these problem solving characteristics.