The Problem of Induction
The problem of induction is usually associated with the name of David Hume. Induction is the method of inference, by which we go form a number of specific observations [enumeration] to a generalization about a whole class or prediction about the future. For instance, every emerald we have observed - all of us, ever - has been green. Therefore [this therefore is crucial - it shows that this is a method of inference], all emeralds are green [generalization]; and every emerald we will find in the future will also be green [prediction or forecast.] Is induction a good [reliable] method of inference?
You might be tempted to say that a lot depends on the 'sample,' so to speak. What if we have observed millions of emeralds? Isn't this different from my enumeration of, say, swans? Suppose I see one, two, three, …, a hundred swans. It is probable that all the swans I have observed are white. Why shouldn't I apply induction and infer that 'all swans are white?' Why shouldn't I infer, by induction, that any swan I am going to encounter tomorrow, or in one year from tomorrow, will also be white? And yet, there are black swans. Of course, I might have been luckier in my enumeration - I might have had the opportunity to observe a black swan. But, if all the swans I have seen have been white, then my induction will produce the statement 'all swans are white.' The next black swan I see will prove me wrong. Is induction itself in trouble? Are we realizing that the method itself is not reliable? Even if I have observed thousands of an entity, event, etc., I am still dealing with what happened before. Do I have perfect assurance that the same thing will happen always? How do I know this if all I know is that I just happened to observe something being the same every time so far. Unless I understand what forces are operating to cause this thing, event, etc., to be what it is, then I have no reliable knowledge based on induction.
But what if I know what causes what - what if I know that x causes y? In that case, it is not simply that I have seen x cause y and I am applying induction; my knowledge of the causal connection between x and y makes my knowledge strong - never mind that I have not yet observed if x will cause y tomorrow in the same way that I have not yet seen a black swan. Hume challenged this view of causality. All you know is that [x,y]: every time x, then y followed. This is a sequence; why are we calling it 'causation' and investing it with such a glorious claim to knowledge? The problem of induction, which we began to see above with swans, re-appears here: unless you know what forces make x bring about y, you only have enumeration [specific observed events, x-y]; even if you have observed x,y a thousand times, you have no guarantee that x will be followed by y tomorrow. You do not have the reliable kind of knowledge that allows you to generalize. This is not like the Pythagorean theorem - it applies always to every triangle of a certain kind and will, no doubt, be true tomorrow and in a thousand years from tomorrow even if no one is present to think of the theorem. But, induction is not like what we do in geometry - that is called deduction. We do not say that the Pythagorean theorem is true because we observed it being the case in a thousand, a million, and so on, times. Either knowledge IS like that, or it is not! With swans and emeralds and everything in your experienced life, it is NOT like it is in geometry: you infer by observing; the more you observe the more certain you feel - you even form a habit to the effect - that what you observed holds regardless of the observation - that it IS the case and will be the case in the future; but how do you know? If I press you on this, you will have to say that you know because it happened before: in this way, you are begging the question - you are promising to show me that induction is reliable and you 'show' this to me by telling me that induction has been reliable before [in other words, you are telling me that induction is reliable because … induction is reliable.] You see the problem? Does this change if we keep adding hundreds and even thousands of more and fresh observed cases of something? It does not change - think about for a moment and you will see that this is, unfortunately, the case. Hume figured this out; and he applied it even to causation itself. Causation too is enumeration of observed sequences - when x, then, right afterwards, y. We still do not know what makes emeralds green and what causes lightning. Do we? You might think that we do - science tells us - but think again: Chemistry [explaining why emeralds are green] reaches a point when it speaks of electrons and particles interacting and so on: aren't these things mysterious? Do you really know what is going on? Chemistry does not give you a metaphysics - chemistry has observed and classified, for a very long period of time; chemistry, in other words, has been always, and happily, performing induction. Chemists cannot pose to ask the questions Hume asked. They justify what they do by pointing to the results but they cannot show that their knowledge is reliable in the way Hume is pressing us to show how induction can be reliable. Well, it seems that induction cannot be reliable - period. And here is a more formal presentation of the problem of induction.
When we perform induction, we draw an inference from x1, x2, x3, …, xn to all xs', and to the characteristics of any x in the future. We are assuming one more thing, which we do not spell out: We are assuming that there is uniformity in what we observe - that whatever it is that made x be green, square, etc., will always operate in the same fashion - so that x will always have those features. In that case, we don't need to know more about the deeper stuff of which xs' are made. It is sufficient that we have observed xs' and no x ever appeared to be different. [This is not an analytical truth, of course: see that you can negate it, by saying that tomorrow a red x will be found, and what you have just said sounds odd and impossible but it is NOT logically impossible. It is NOT like saying that a triangle will not have three angles tomorrow. The point, though, is: can we still rely on induction? Well, maybe, if nature is UNIFORM - if x will come about always in the same way - even if I don't really know exactly how x comes about.] So, if we can prove that nature is uniform, then we can legitimately rely on induction. But, can we prove that nature is uniform? The answer is no - and see how this is the case. A strong proof would be deductive - like the proofs we have in geometry. But, since this principle is about how nature really operates, the principle is not a necessary truth like the ones we find in geometry. Notice that there is no definition involved here either: We do NOT define nature as uniform operation, or as the source or underlying support of uniform operations. We can think of nature as non-uniform without violating logic. [The only way you could perhaps have a deductive proof of nature's uniformity would be by claiming that God is running the operations of nature and does the same thing always because God always maximizes performance or does the best possible thing, and so on. But even this would not work, given the way God is defined in monotheistic religions. Can you see why?] So, it looks that the only possible proof for the principle of the uniformity of nature is an inductive proof. Can we prove inductively that nature is uniform? Well, what would this be like? Nature has been uniform so far. Therefore, nature is uniform - period; and nature will always be uniform. Notice the problem here: This inductive argument PRESUPPOSES that induction is reliable. But this is exactly what we are trying to prove - that induction is reliable. So, we are begging the question, as the logicians say. To conclude, the principle of the uniformity of nature cannot be proven. This makes induction unreliable.
Perhaps, we are led to this sorry state - even unto skepticism, radical doubt about the possibility of knowing everything reliably - because we started with foundationalist views of knowledge - looking for indubitable knowledge at the foundation. See my comments under 'epistemology' on this.
Now, it IS rational to depend on induction. Since emeralds have been green in, say, 100% of observed instances, it is rational to assume, and act under the assumption, that emeralds will still be green in the future. It is sobering, however, to recall the problems with induction. Good examples are to be found in meteorology - weather forecast - and in the lives of … viruses. Because viruses multiply at an exponentially high rate, they scan evolutionary millennia within a few hours or days. Induction - past observations about how a virus behaves - do not have forecasting power. The problem of induction is something like this, but, notice that the problem of induction goes even more deeply than that.
Here is an example of what follows when you take into account the problem of induction. [This example is in part based on a puzzle introduced by philosopher Nelson Goodman.] Suppose that we come in contact with an alien species. We manage to understand their language and we are able to interact and communicate with them, and they can do so with us too. Their language, we found out, has a time component built into it. For unknown reasons, they signify things in such a way that the word takes into account shifts in features over time. So, in their language, the aliens call things GRUE if they are green until the year 2020 and blue afterwards. Aren't our planet's emeralds GRUE indeed? Does that mean that the emeralds will be blue after 2020? Maybe the aliens insist that our emeralds are GRINK - green until 2030 and pink afterwards. How do you argue about - or discuss - this issue with the aliens? What is the problem here?