Every observed emerald is green because the laws of nature make it necessary that every emerald be green. Of these two attempted explanations, the second seems best. Indeed, the first is not so much an explanation of the fact that every observed emerald was green as the claim that this fact is just an inexplicable coincidence. So we can infer that the best explanation of the fact that every observed emerald was green is that, as a matter of natural law, every emerald is green. But clearly, if the laws of nature make it necessary that every emerald be green, we can also infer that the next emerald to be dug out of the ground will be green, and that every emerald, observed or unobserved, is green.
Induction is rational, because it is a form of inference to the best explanation.
In mathematics , induction is one of the major methods used to prove results. For the natural numbers, induction works as follows. So induction is used to go from a large set of particular statements to a single general statement. This use of the principle of induction as a means of mathematical proof is based on the idea of induction in the philosophy of science . This idea is that the universe is basically uniform, so that, for example, it is a scientific 'fact' that the Sun will rise tomorrow, because it has done so every day in your experience so far.