For a mathematician, the path he takes to solve an equation can be summed up in the following diagram:

• Doubt : Discovery of possible paths.
• Belief : Knowledge, use of rules and theorems, instruments in his possession.
• Proof : paths by which he interconnects these elements to find the best choice or solution.

Logic built in the mathematician head is the result of Doubt-Belief-Proof loop.
This is the very meaning of the Reflection process. Which is in fact an idea or solution that sprouted in the mind after a Doubt-Liver-Proof process.