In a formal way to look at reasoning is like the following: you have a set of sentences which are syntactically correct, and describe the properties in say physics say P. You also have a set of sentences which are true call axioms. Like the conservation of energy, F=ma etc. A function f that maps sentences in P to either True or False. This function f is constructed such that for a giving sentence s in P, f(s) is a true, if there is a series of logical operation starting from the a sentence in the Axiom and ends at s. e.g. s = "if mass is increased 10 times, and acceleration is constant, the force is increased 10 times". We have F=ma therefore by increasing m to 10 times, while a is constant, F is increased 10 times, hence f(s) is true, and in the set of the theory of P. It is important that every sentence in the Theory of P should be consistent.

Now, for the simple theory of Integer, it has been proved that such theory is either inconsistent, or incomplete. Meaning either this Theory contains sentences that contradict to others, or there are sentences in P which just don't know whether it is true or now. If you don't even know whether it is true or not, what can you explain?

The explanation is not just description, but also based on this explanation, other properties can be predicted, and inferred through logical operation.

Is that good enough for you?