David Hume claimed that it is:
[…] impossible for us to satisfy ourselves by our reason, why we should extend that experience beyond those particular instances, which have fallen under our observation. We suppose, but are never able to prove, that there must be a resemblance betwixt those objects, of which we have had experience, and those which lie beyond the reach of our discovery.
And this then gives rise to the problem of induction: how can we defend assuming the so-called uniformity of nature that we take to exist when we generalize our limited experience to that which lies “beyond the reach of our discovery”? For instance, how can we justify our belief that the world of tomorrow will, at least in many ways, resemble the world of yesterday? Indeed, how can we justify believing that there will be a tomorrow at all?
A thing worth highlighting in response to this problem is that, even if we were to assume that we have no justification for believing in such uniformity of nature, this would not imply, as may perhaps seem natural to suppose, that we thereby have justification for believing the opposite: that there is no uniformity of nature. After all, to say that the patterns we have observed so far do not predict anything about states and events elsewhere would also amount to a claim about that which lies “beyond the reach of our discovery”, and so this claim seems to face the same problem.
The claims 1) “there is a certain uniformity of nature” and 2) “there is no uniformity of nature” are both hypotheses about the world. And if we look at the limited part of the world about which we do have some knowledge, it is clear that the former hypothesis is true about it: patterns at one point in (known parts of) time and space do indeed predict a lot about patterns observed elsewhere.
Does this then mean that the same will hold true of the part of the world that lies beyond the reach of our discovery? One can reasonably argue that we do not have complete certainty that it will (indeed, one can reasonably argue that we should not have complete certainty about any claim our fallible mind happens to entertain; not even when it comes to claims about idealized formal systems, as there is always the possibility that we have failed to instantiate these formal systems properly). Yet if we reason as scientists — probabilistically, endeavoring to build the picture of the world that seems most plausible in light of all the available evidence — then it does indeed seem justifiable to say that hypothesis 1 seems much more likely to be true of that which lies “beyond the reach of our discovery” than does hypothesis 2; not least because to say that hypothesis 2 holds true would amount to assuming an extraordinary uniqueness of the observed compared to the unobserved, whereas believing hypothesis 1 merely amounts to not assuming such an extraordinary uniqueness.
And if we think in this way — in terms of competing hypotheses — then Hume’s problem of induction suddenly seems rather vacuous. “You cannot prove that any given hypothesis of this kind is correct.” This seems true (although the fact that we have not found such a proof yet does not imply that one cannot be found), but also quite irrelevant, since a deductive proof is not required in order for us to draw reasonable inferences. To say that we have no purely deductive argument for a given conclusion is not the same as saying that we have no justification for believing it (and if one thinks that it is, then one is also committed to the belief that we have no justification for believing, based on previous experience, that the problem of induction also exists in this very moment; more on this below).
Applying Hume’s Claim to Itself
According to Hume’s quote above, the belief that we can make generalizations based on particular instances can never be “satisfied by our reason”. The problem, however, is that, according to our modern understanding of the world in physical terms, all we ever can generalize from, including when we make deductive inferences, is particular instances — particular spatiotemporally located states and processes found in our brains (equivalently, one could also say that all we can ever generalize from, as knowing subjects, are particular states of our own minds).
Thus, Hume’s statement that we can never prove such generalizations must also apply to itself, as it is itself a general claim based on a particular instance of reasoning taking place in Hume’s head in a particular place and time (indeed, Hume’s claim would appear to pertain to all generalizations).
So what justification could Hume possibly provide for this general claim of his? According to the claim itself, no proof can be given for it. Indeed, if Hume could provide a proof for his claim that it is impossible to find a proof for the validity of generalizations based on particular instances, then he would have falsified his own claim, as such a proof is the very thing that the claim holds not to exist. And such an alleged proof would thereby also undermine itself, as what it supposedly shows is its own non-existence.
This demonstrates that Hume’s claim is unprovable. That is, based on this particular instance of reasoning, we can draw the general conclusion that we will never be able to provide a proof for Hume’s claim. And thereby we have in fact proven Hume’s claim wrong, as we have thus provided a proof for a general claim that also pertains to that which lies beyond the reach of our discovery. Nowhere, neither in the realm of the discovered nor the undiscovered, can a proof for Hume’s claim be found.
So we clearly can prove some general claims about that which lies beyond the reach of our experience based on particular instances (of processes in our brains, say), and hence the claim that we cannot is simply wrong.
Yet one may object that this conclusion does not contradict what Hume in fact meant when he claimed that we cannot prove the validity of generalizations based on particular instances, since what he meant was rather that we cannot prove the validity of inductive generalizations such as “we have observed X so far, hence X will also be the case in the next instance/in general” — i.e. generalizations whose generality seems impossible to prove.
The problem, however, is that we can also turn this claim on itself, and indeed turn the problem of induction altogether on itself, as we did in a parenthetical statement above: the mere fact that we have not been able to prove the validity of any inductive claims of this sort so far does not imply that such a proof can never be found. In particular, the claim that we cannot prove the validity of any such inductive claim that seems impossible to prove is itself an inductive claim whose generality seems impossible to prove (i.e. it seems to rest on the argument: “we have not been able to prove the validity of any inductive claim of this nature so far, and hence we cannot[/we will never be able to] prove the validity of such a claim”).
And if we accept that this claim, the very claim that gives rise to the problem of induction, is itself a plausible claim that we have good reason to accept in general (or at least just good reason to believe that it will apply in the next moment), then we indeed do believe that we can have good reason to draw (at least some plausible) non-deductive generalizations based on particular instances, which is the very thing Hume’s argument is often believed to cast doubt upon. In other words, in order to even believe that there is a problem of induction in the first place, one must already assume that which this problem is supposed to question and be a problem for.
Indeed, one can make an argument along these lines that it is in fact impossible to give a coherent argument against (the overwhelming plausibility of at least some degree of) the uniformity of nature. For in order to even state an argument or doubt against it, one is bound to rely thoroughly on the very thing one is trying to question. For instance, that words will still mean the same in the next moment as they did in the previous one; that the argument one thought of in the previous moment still applies in the next one; that the problem one was trying to address in the previous moment still exists in the next; etc.
Thus, it actually seems impossible to reasonably, indeed even coherently, doubt that the world has at least some degree of uniformity, which itself seems to constitute a good argument and reason for believing in such uniformity. After all, that something cannot reasonably be doubted, or indeed doubted at all, usually seems a more than satisfying standard for believing it.
So to reiterate: If one thinks we have good reason to take the problem of induction seriously, or indeed just to believe that this problem still exists in this moment (since it has in previous ones), then one also thinks that we do have good reason to make (at least some plausible) non-deductive generalizations about that which lies “beyond the reach of our discovery” based on particular instances. In other words, if one takes the problem of induction seriously, then one does not take the problem of induction seriously at all.
How to then draw the most plausible inferences about that which “lies beyond the reach of our discovery” is, of course, far from trivial. Yet we should be clear that this is a separate matter entirely from whether we can draw such plausible inferences at all. And as I have attempted to argue here, we have absolutely no reason to think that we cannot, and good reason to think that we can.