Argumentation or way of reasoning by analogy, induction, deduction, and proportional reasoning, as well as algebraic, geometric, arithmetic, probabilistic, or statistical reasoning.
Mathematical reasoning refers to rules of inference and deduction that uses definitions, accepted statements as premises, laws, or properties, results previously obtained also using reasoning, in order to demonstrate hypotheses or conjectures.
By its process, mathematical reason is distinguished from intuition, prediction, and revelation by the fact that it only proceeds through the application of precise rules to abstract concepts with clearly stated properties (accepted as axioms or demonstrations).
Example
Saying that there is an infinite number of prime numbers, because we have found a lot of them, is based on intuition. Making a demonstration (proof) is based on mathematical reasoning. This is what Euclid showed in proposition 20 of Volume IX of Elements. Other mathematicians, like Leonhard Euler, have also demonstrated the infinity of the set of prime numbers.