Induction, as applied in the physical sciences, is always uncertain, because it rests on the belief in a general order in the universe, an order outside of us. On the contrary, mathematical induction, I.e., demonstration by recurrence, imposes itself as a necessity, because it is only a property of the mind itself …
We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different from physical induction but just as fertile, deduction would be powerless to create a science.
Henri Poincare, “The nature of mathematical reasoning”