I've been a bit perplexed about the "problem" of induction.
Hume challenges other philosophers to come up with a deductive reason for the inductive connection. If the justification of induction cannot be deductive, then it would beg the question. To Hume, induction itself, cannot explain the inductive connection. (Wikipedia)
But I ask, why do we need to show that induction is a necessary truth? We can not demonstrate a necessary truth but we can demonstrate that we have a valid reason to believe.
The principle of uniformity refers to the assumption that the same natural laws and processes that operate in the universe now have always operated in the universe in the past and apply everywhere in the universe.
If nature is uniform, crystal gazing may or may not work, but induction works. If nature is not uniform, then induction will fail, but so will any alternative method. Because if the alternative method did not fail, if it consistently yielded true predictions, and the success of that alternative would constitute a uniformity that could be exploited by the inductive method. Because we could inductively infer the future success of the crystal gazer from her past success. Hence, the inductive method will succeed if any alternative method could.
Why do we need to know whether the inductive method is necessarily true if the inductive method will succeed if any alternative method could? We can’t have a reason for believing that induction is necessarily true because we can’t know in advance whether nature is uniform. We can’t have a reason necessarily true, but we can justify the inductive method by saying that it's the best method for making predictions about the future/unobserved, because if nature is uniform, crystal gazing may or may not work, but induction works. If nature is not uniform all methods will fail.
Why we don't need a reason for believing that inductive method is necessarily true.
The problem of induction is a problem of which action to take, or which wager to accept. The injunction is to act so as to maximize expected utility of generate knowledge about the world. A deductive argument can only bring out what is already implicit in its premises, and hence all new knowledge about the world must come from some form of induction.The induction will succeed if any alternative succeed, and if the principle of uniformity doesn't hold, any alternative will not succeed. If an alternative method did not fail, if it consistently yielded true predictions, the success of that alternative would constitute a uniformity that could be exploited we could inductively infer the future success of the alternative method from her past success.
If any rule will lead to positing the correct knowledge, the inductive rule will do this, and it is simplest rule that is successful. The justification of application of Occam's razor is a direct result of generate knowledge about the world. By definition, all assumptions introduce possibilities for error; if an assumption does not improve the accuracy of a knowledge, its only effect is to increase the possibility that the knowledge is wrong.
It is difficult to formulate principle of the uniformity of nature in a coherent and useful manner. For nature is not uniform in all respects and it is uninformative to say simply that nature is uniform in some respects. Inductive inferences are not conclusively verifiable in physics, no crucial experiment is possible. For experiments in physics are observations of phenomena accompanied by interpretations and, therefore, physicists submit whole groups of hypotheses, not a single hypothesis, to experimental tests, and experimental evidence alone cannot conclusively falsify hypotheses. This is known as the Duhem thesis.That all inductive inferences will be justified is false, in any case an inductive inference doesn't need to be necessarily true.
Induction is a matter of following a tentative and self-correcting rule. A key thought is that agents start with their subjective prior hypotheses and then update them by conditionalisation, set within the subjective Bayesian framework. Induction, then, is the process of updating a hypotheses. The problem of justifying conditionalisation on the evidence is again which action to take, or which wager to accept. Again the injunction is to act so as to maximize expected utility of generate knowledge about the world.
Popper wants to say that induction is not justifiable. That a theory has been corroborated in the past "says nothing whatever about future performance." Popper wants to say that it is possible to avoid assuming that the future will, or probably will, be like the past, and this is why he has claimed to have solved the problem of induction. We do not have to make the assumption, he tells us, if we proceed by formulating conjectures and attempting to falsify them. He says that, as a basis for action, we should prefer "the best-tested theory." This can only mean the theory that has survived refutation in the past; but why, since Popper says that past corroboration has nothing to do with future performance, is it rational to prefer this?
Without the inductive assumption, the fact that a theory was refuted yesterday is quite irrelevant to its truth-status today. So demising the inductive assumption makes nonsense of Popper's own theory of the growth of scientific knowledge. The more often a conjecture passes efforts to falsify it, Popper maintained, the greater becomes its "corroboration", although corroboration is also uncertain and can never be quantified by degree of probability. "Corroboration" is a form of induction, and Popper has simply sneaked induction in through a back door by giving it a new name.
Every falsification of a conjecture is simultaneously a confirmation of an opposite conjecture, and every conforming instance of a conjecture is a falsification of an opposite conjecture. If Popper bet on a certain horse to win a race, and the horse won, you would not expect him to shout, "Great! My horse failed to lose!" Astronomers look for signs of water on Mars. They do not think they are making efforts to falsify the conjecture that Mars never had water. For Popper, what Carnap called a "degree of confirmation", a logical relation between a conjecture and all relevant evidence, is a useless concept. Instead, the more tests for falsification a theory passes, the more it gains in "corroboration”. It's not so much that Popper disagreed with inductivists as that he restated their views in a bizarre and cumbersome terminology. Why scratch your left ear with your right hand?