David Hume is well known for having argued that it is impossible to find reasonable grounds for induction:
Our foregoing method of reasoning will easily convince us, that there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience. We can at least conceive a change in the course of nature; which sufficiently proves, that such a change is not absolutely impossible. To form a clear idea of any thing, is an undeniable argument for its possibility, and is alone a refutation of any pretended demonstration against it.
Probability, as it discovers not the relations of ideas, considered as such, but only those of objects, must in some respects be founded on the impressions of our memory and senses, and in some respects on our ideas. Were there no mixture of any impression in our probable reasonings, the conclusion would be entirely chimerical: And were there no mixture of ideas, the action of the mind, in observing the relation, would, properly speaking, be sensation, not reasoning. ‘Tis therefore necessary, that in all probable reasonings there be something present to the mind, either seen or remembered; and that from this we infer something connected with it, which is not seen nor remembered.
The only connection or relation of objects, which can lead us beyond the immediate impressions of our memory and senses, is that of cause and effect; and that because ’tis the only one, on which we can found a just inference from one object to another. The idea of cause and effect is derived from experience, which informs us, that such particular objects, in all past instances, have been constantly conjoined with each other: And as an object similar to one of these is supposed to be immediately present in its impression, we thence presume on the existence of one similar to its usual attendant. According to this account of things, which is, I think, in every point unquestionable, probability is founded on the presumption of a resemblance betwixt those objects, of which we have had experience, and those, of which we have had none; and therefore ’tis impossible this presumption can arise from probability. The same principle cannot be both the cause and effect of another; and this is, perhaps, the only proposition concerning that relation, which is either intuitively or demonstratively certain.
Should any one think to elude this argument; and without determining whether our reasoning on this subject be derived from demonstration or probability, pretend that all conclusions from causes and effects are built on solid reasoning: I can only desire, that this reasoning may be produced, in order to be exposed to our examination.
You cannot prove that the sun will rise tomorrow, Hume says; nor can you prove that it is probable. Either way, you cannot prove it without assuming that the future will necessarily be like the past, or that the future will probably be like the past, and since you have not yet experienced the future, you have no reason to believe these things.
Hume is mistaken, and this can be demonstrated mathematically with the theory of probability, unless Hume asserts that he is absolutely certain that future will definitely not be like the past; that he is absolutely certain that the world is about to explode into static, or something of the kind.
Suppose we consider the statement S, “The sun will rise every day for at least the next 10,000 days,” assigning it a probability p of 1%. Then suppose we are given evidence E, namely that the sun rises tomorrow. Let us suppose the prior probability of E is 50% — we did not know if the future was going to be like the past, so in order not to be biased we assigned each possibility a 50% chance. It might rise or it might not. Now let’s suppose that it rises the next morning. We now have some new evidence for S. What is our updated probability? According to Bayes’ theorem, our new probability will be:
P(S|E) = P(E|S)P(S)/P(E) = p/P(E) = 2%, because given that the sun will rise every day for the next 10,000 days, it will certainly rise tomorrow. So our new probability is greater than the original p. It is easy enough to show that if the sun continues to rise for many more days, the probability of S will soon rise to 99% and higher. This is left as an exercise to the reader. Note that none of this process depends upon assuming that the future will be like the past, or that the future will probably be like the past. The only way out for Hume is to say that the probability of S is either 0 or infinitesimal; in order to reject this argument, he must assert that he is absolutely certain that the sun will not continue to rise for a long time, and in general that he is absolutely certain that the future will resemble the past in no way.