I use the word ‘induction’ a lot. But the word can be a bit slippery. Hume is celebrated for his “problem of induction”, and he was indeed concerned with what we nowadays call “induction”. But Hume himself didn’t use the word ‘induction’ for what he had his problem with.
What do I mean when I use the word ‘induction’?
A classic example of induction is the inference from “the swans I’ve seen so far have been white” to “all swans are white”. This inference assumes that “nature continues uniformly the same” (as Hume put it), so that as-yet unseen swans are similar in the relevant way to the swans I’ve seen already. Nowadays, we would put it in terms of scientific laws, which describe the sort of reliably universal regularities Hume had in mind.
Putting it in terms of scientific laws conveniently illustrates why induction is often unreliable. Swans are not universally white — some are black. Because swans’ color is not regular in the lawlike way required — in other words because nature does not continue uniformly the same from one swan to the next as far as color is concerned — the inference above is unreliable.
Because inferences like that involve generalization, induction is sometimes characterized as inference “from the particular to the general”, but that is actually a rather poor way of characterizing it. Words like ‘all’ can appear in the “premises” of inductive inferences as well as being absent from their “conclusions”. For example, consider the inference from “all of the electrons observed so far have had a charge of minus one” to “the next electron we observe will have a charge of minus one”, or to “any electron has a charge of minus one”, or to “the electron is a subatomic particle with charge minus one”. Superficially, these look like inferences from greater to lesser generality.
The assumption of lawlike regularity
At the risk of belaboring the point, it isn’t always obvious when induction is involved in an inference. For an induction to be reliable, it has to be underwritten by a lawlike regularity in the real world, whether or not we are aware of it — it’s often a matter of sheer luck. But even when an induction is unreliable because there is no real lawlike regularity, it still counts as a case of induction if it assumes that there is.
So, if we’re wondering whether induction is involved in an inference, it’s probably safer to look for an assumption of lawlike regularity than to look for words that typically signal generalization or extrapolation.
Sometimes the required assumption of lawlike regularity isn’t all that obvious. Suppose we take a sample of people and find that 10% of them have red hair. We then use statistical extrapolation — an application of induction — to claim that 10% of the entire population of the world has red hair. For this to be any better than a shot in the dark, the sample must be representative of the world’s population, at least in respect of the proportion who have red hair. It isn’t enough for the sample to reflect this feature of the larger population by accident — it must do so systematically, so that the generalization from sample to entire population is non-accidental. (Laws are sometimes characterized as “non-accidental generalizations”.)
There is more than one way an induction can go wrong. In the current example, if the proportion attributed to the entire population is too precise — for example, if we claim that exactly 10.0001% have red hair because that is the exact proportion in the sample — the detail is overly-fine-grained. Detail of that sort — not underwritten by lawlike regularity — is merely artifactual. That is, it is a misleading by-product of our own methodology rather than a feature of the real world. It is analogous to seeing one of your own eyelashes reflected in the lens of a microscope.
Skillful sampling is vital for reliable statistical extrapolation. A sample should be representative of the population as a whole, and that takes skill. Some rigorous-looking statistical methods are meant to estimate how representative samples are, but too often, these methods themselves rely on induction, by extrapolating the variability of samples to the entire population. To my mind, these statistical methods are the products of a quest for assurance rather than a quest for truth. A better idea is to test sampling techniques. For example, the sampling techniques of voter popularity polls before elections are tested by actual election results. Nate Silver accepted credit for predicting the most recent US election results, but more credit is due to the people who were able to get such representative samples of voters.
Non-numerical examples of induction are tricky enough. Things get worse when numbers are involved. Perhaps worst of all is the completely spurious idea that we can have a numerical measure of “how much the conclusion of an induction deserves to be believed”, usually assumed to be some arithmetical function of “the number of instances an induction is based on”.
Induction versus guessing and testing
I hope it is clear that there are several “problems of induction”. It is a distinctly problematic form of reasoning, mostly because it apes deduction. It’s what people come up with when they try to imagine what an argument would look like if it could deliver “empirical” conclusions that do more than just re-arrange ideas expressed in premises. Behind it lies the malign assumption that evidence consists of being shown to be the conclusion of an argument. (I beseech you, dear reader, to reject this assumption!) When combined with popular ideas about science being “based on observation”, induction can acquire a hallowed status — a status it doesn’t deserve. It’s not the “complementary, alternative form of reasoning to deduction”, and it doesn’t appear much in any of the respectable sciences.
Rather than relying on induction, science is mostly a matter of guessing followed by testing. Rather than starting off with observations and proceeding by extrapolating from them in a mechanical way, science starts off with explanatory hypotheses and proceeds by devising tests for them — feats that call for imagination, creativity, and cunning. Rather than seeking assurances in the form of inductive arguments, science seeks truth by casting a wider net to check for falsity.
Karl Popper recognized the centrality to science of making “bold conjectures” that stand despite the possibility of their refutation. He rejected induction altogether as unscientific. But I think he went way too far here. I also think it was ridiculous to claim as he did that a theory’s passing a test doesn’t give us any reason to think the theory is true.
I would argue that passing tests usually gives us good reason for thinking theories are true. I agree with Quine that induction is a special case of the hypothetico-deductive method (of guessing and testing just mentioned) and that a broader understanding of the latter helps to explain why induction is sometimes reliable.
One of the main attractions of induction is it removes the sense of guesswork from empirical reasoning. Instead of “having a stab” at things, induction “frog-marches” us from observations to conclusion in what might seem a reassuringly “inescapable” way. It has a mechanical feel, like deduction. Let us not be too easily seduced by these attractive features!
Example: modeling a compound pendulum
I’ll try to illustrate why with an example — an artificial one, concocted specially to show how induction can fail to deliver the goods. Image a compound pendulum of two rigid parts that behaves chaotically — that is, its configuration critically depends on initial conditions, so that over time its movements are practically unpredictable. And because they are practically unpredictable, they can’t be modeled in a computer. (Not in practice, anyway.) I can make computer versions of compound pendulums that behave just like real compound pendulums behave, but I can’t make a single computer model that mimics the behavior of a given actual compound pendulum.
But suppose I don’t know that. Suppose I set out to create a computer model of this very apparatus, encouraged by the thought that each of its two moving parts behaves in a well-understood, completely lawlike way. As “inputs” I obviously use my knowledge of the simple, elegant laws that describe their movements. But I also know that more than that’s involved: I need to experiment a bit with different lengths of the rigid bars that make such a pendulum, different masses, different centers of gravity, different moments of inertia, and so on.
Every time I run my model, adjusting one or other of these input variables, I compare the progress of my computer model and the configuration over time of the actual compound pendulum, to see where they begin to diverge. Although I will make progress at first, there will come a point in every single run at which the divergence becomes significant — too large for me to count my model as a model of that actual, given compound pendulum.
Now, if I were asked to defend my computer modeling, I might say that I have been working with numerous models, and that they all have been rigorously “tested”. The models that have failed such a “test” have been diligently thrown out, I might claim, and I have learned from my errors by making new models with better initial values for the relevant variables.
But I would just be kidding myself. None of these models is a “bold conjecture”, nor is any of the “tests” anything like the real test of a hypothesis. A real test involves cunning on the part of the experimenter — and sweating palms on the part of the theorist whose reputation is on the line. There is the possibility of failure rather than the expectation of some further tinkering. What we have here instead is the mechanical adjustment of numerical values to fit some “data” that may as well have been gathered beforehand. Rather than individual models being tested, the entire process of model-generation is being adjusted to fit a series of data-points. In effect, it is the fitting of a curve through them. This fitting is guided by the assumption that future motion of the pendulum will “continue uniformly the same” in matching the progress of the computer model. It assumes there is a lawlike connection between them, or at least that one can be found. (It can’t.)
This is induction. It has many of the problems that attend induction. There is no way around the fatal mismatch between model and reality that I purposely built into the example.
A compound pendulum is a very simple apparatus, whose behavior can’t be captured by induction. I leave it as an exercise how much we can hope that anyone could model more complicated items such as the climate of an entire planet subject to many, many more variables.