Knowledge and hope

The traditional understanding of knowledge as “justified true belief” is internalist. That is to say, for a true belief to count as an item of knowledge, it must satisfy a third condition of being “justified”, which is a state of mind. Justification is traditionally understood as being internal to the mind.

More recent “naturalized” epistemology is externalist. That is to say, for a true belief to count as an item of knowledge, it must satisfy a third condition of being connected in a special way to the real world “outside” the mind. There are various ways of characterizing this special connection: it must be reliable, it must “track” truth, it must be sustained by a law-like process, the belief in question must be non-accidentally true. I’ll use the word ‘reliable’. But whichever words we use for it, the connection reaches outside the mind, and so part of it is external to the mind.

I think these two ways of thinking about knowledge correspond to “is” and “ought” in an interesting way.

The internalist is looking for justification — and (in theory at least) he can check whether a belief is justified by examining the way it is linked to his other beliefs through relations of implication. Foundationalists think justified beliefs are implied by “basic” beliefs; coherentists think there is a network of mutual implication. Either way, these other beliefs are in the mind, and so they can potentially be put “before the mind” for inspection. According to this understanding of knowledge, we can have assurance that we know. In fact the main thrust of traditional epistemology is “doctrinal”: it’s aimed at assuring the radical sceptic that we do in fact have knowledge. We know something when a belief is justified, and it is justified or not as a matter of fact — an “is”.

Instead of seeking justification, the externalist wants reliability. And he isn’t “looking” for reliability so much as “hoping” for it. He can’t directly check whether the connection he hopes is reliable actually is reliable, because one end of it lies outside his mind. According to this understanding of knowledge, we can’t have an internal assurance that we know, because some aspects of knowledge are aspirational. We aspire to the goal of having reliably true beliefs. To the potential knower, such aspirations are better expressed by the word ‘ought’ than ‘is’. None of the beliefs he already has — as a matter of fact — can imply that these aspirations are met, because “oughts” cannot follow from “is”s alone.

This aspirational aspect of knowledge might be likened to “the object of the game” for a chess-player. The would-be knower and the chess player have goals: to have reliably true beliefs, and to get the opponent’s king into checkmate, respectively. These goals are the object of “oughts”: the would-be knower’s beliefs ought to be reliably true, and the player’s moves ought to bring the goal of checkmate closer. In both cases, the “ought” guides behavior in a nontrivial way.

Of course neither of these is a moral “ought”. Proper epistemic practice obliges us rationally rather than morally to aim for reliably true beliefs. Chess players’ implicit acceptance of the rules of chess — specifically, the rule that specifies the object of the game — obliges them to aim for checkmate. Someone who gets bored and plays “suicidal” chess to end a game quickly isn’t guilty of a moral failing, he’s just not playing chess properly.

The chess player has to interact with his opponent: he can’t focus on his own moves to the exclusion of his opponent’s moves. Analogously, the potential knower has to interact with the world: he can’t focus on his own beliefs to the exclusion of “answers” the world gives in response to the “questions” he “asks” it. In practice, this “questioning” of the world is the testing of hypotheses. To form new beliefs or new theories by simply building on what one already believes — including “data” — is like playing chess without paying attention to your opponent’s moves. In effect, this is what Bayesians do when they make epistemic decisons on the basis of “degrees of belief”. (I shall have more to say about Bayes’ Theorem in a forthcoming blog post.)

The “object of the game” of empirical knowledge is to transcend internal assurances and aim for reliably true beliefs — an external matter, which usually involves the testing of hypotheses.

I mentioned above that traditionally, epistemology was internalist. The tradition continues to this day, and it affects the way epistemology is taught in university philosophy courses: they tend to begin with Descartes’ Meditations, and typically don’t move very far beyond that. They tend to treat Gettier problems as a mere curiosity. Internalism can also affect the way scientists do science. Some sciences — especially those that appeal to “overwhelming evidence” to counter scepticism — use “internalist” methods of shaping models to fit “data” that may as well have been gathered beforehand. In effect, this is to eschew testing in favor of an internalist sense of assurance.

Proper science and knowledge are aimed at truth, not at assurance. Their aspirational aspects entail that testing is essential. To use another analogy: a miser might get assurance from counting coins he already owns, but he can’t make money unless he runs the risk of losing money by investing it in outside projects. In pursuit of truth rather than profit, science too must “cast its net” beyond “data” already gathered.

Whatever it is, I’m against it

Irish President Michael D Higgins — a sociologist, not a philosopher — is leading a campaign to teach philosophy in secondary schools. Almost everyone seems to welcome this idea. I think it stinks. — Why? What could possibly go wrong with such a laudable enterprise?

It seems to me that whatever eventually gets taught, it will affect students of low, middle and high ability in different ways. It will intimidate those at the bottom, indoctrinate those in the middle, and infuriate those at the top.

Let’s start at the bottom. Philosophy is hard. I don’t just mean that it requires intelligence. More importantly, it requires a “strange” turn of mind with the creativity to juxtapose previously unconnected ideas, the imagination to consider far-fetched scenarios, the ability  to look at things from a “meta-level” perspective, and comfort with abstraction.

Some of the most intelligent, gifted people I’ve ever met didn’t have this required strangeness of mind, and were hopeless at philosophy. They simply didn’t “get it”. They were self-confident adults who knew they had an abundance of talent in other areas, so they weren’t downcast by their lack of ability in this one area. But I shudder to think of what less confident children will make of a topic that is simply incomprehensible to them.

Moving up to students of middling ability: here they will get the central idea, and most will enjoy the classes. But what they are taught won’t be genuine philosophy. It will be what their teachers call “thinking skills” and “ethics”. Proponents of teaching philosophy in schools try to defend the neutrality of “thinking skills” by saying it just means formal logic, “critical thinking” and the like. This is a good time to remember the motto of all good philosophy: “know thyself”. Let’s be honest. What we call “skill” in thinking consists of thinking we approve of. I rate Kant an unskilled thinker, others think he is one of the greatest philosophers who ever lived. Many rate AJ Ayer an unskilled thinker because they disagree with what he thinks. And so on. Even in universities, teachers of philosophy are prone to presenting philosophical ideas to show them up as flawed, and to set up the alternative as more correct. Always, the “more correct” way of thinking is the one that is more highly skilled according to their own lights.

But wait. Formal logic must be neutral, right? Well, logic itself may seem value-free and theoretically uncommitted, but learning logic isn’t. If we treat deductive arguments as the acme of human thought, we will inadvertently promote the most plodding forms of traditional epistemology, which assume that the ideal of reason is to be the conclusion of a valid argument. That’s not a deliberate attempt to brainwash, but it is an insidious form of indoctrination. (Personally, I blame that indoctrination for so much bad science using brainless inductive methods.)

The vast majority of great thinkers — scientists, mathematicians, artists — never took a logic class in their lives. A person doesn’t need to study logic to think in a logical way. I don’t think studying logic improves a person’s ability to think logically. At best it may help describe the patterns that logical thought takes, so those who have a special interest in such things can communicate with one another. Analogously, learning to be a theatre critic doesn’t make one a better playwright. The arrogance of those who assume better thinking results from “thinking like me” is breathtaking.

This urge to shape minds really shifts into high gear with ethics. Just reflect for a moment on how many well-meaning secondary school teachers will be eager to correct the “unskilled thinking” behind such evils as sexism, homophobia, and “climate denial”. On the receiving end, many well-meaning secondary school students will be eager to have their moral prejudices confirmed by the “authority” of a philosophy teacher. When these two kinds of evangelism meet, the result is missionary zeal — for orthodoxy.

That is nothing like real philosophy, which involves awkward questions rather than agreeable-sounding answers. But even if a few teachers and students realize they’re doing nothing like real philosophy, their hands are tied, because strictly speaking these students are children, and children are forbidden to discuss awkward questions. It’s too invasive.  Teachers can’t allow children to discuss such questions as whether suicide is an act of spite, whether homosexuality is a mental disorder, whether sexual or racial differences are innate. Many of the students are bound to be “affected by these issues”, as they warn on BBC. But even those who want to discuss them can’t give their consent to do so because they’re just children.

This brings us to the top level — the small proportion of students who do have some real philosophical acumen. They’ll have a good idea of what philosophy is all about, because they’ll probably have done a bit of it on their own already. They will recognize that the promotion of orthodoxy and curtailment of free discussion falls far short of the real thing. That will anger and frustrate them, and it may even get some of them into trouble. Despite ill-informed rumors to the contrary, anything like real Socratic dialogue is dangerous.

It is this last type of student who might choose to study philosophy at the third level, in university. But how many will be put off by the cardboard sham that passes for philosophy at the second level?

Most philosophy is bad philosophy. Good philosophy mostly consists of un-learning the bad philosophy you learned before, or arrived at on your own before you realized you were doing philosophy.

What is induction?

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”.)

Statistical extrapolation

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.

Proof and science

In many human endeavors, evidence takes the form of proof, or something structurally equivalent to proof. From pure mathematics to everyday discourse, something presently open to question is shown to be derivable via rules of inference from some other things that are presently not open to question. In pure mathematics, theorems are derived from axioms. In everyday discourse, we try to persuade one another of claims by showing that they follow from what we already believe or agree on.

In these endeavors, we start off with stuff we already consider to be “in the bag” — axioms, shared beliefs, whatever — and we go through a rule-guided process that ends with our adding something to our “bag” of claims that we consider “virtually assured or secured, as good as in one’s possession”. (This quotation is taken from from the OED’s entry on ‘in the bag’.)

Our most familiar concepts of knowledge and evidence have been shaped by the idea that some beliefs can be decisively considered beyond question, at least for the time being. Knowledge is supposed to consist of truths that are assured, and evidence is supposed to consist of what can give us the assurance.

There is one human endeavor that isn’t like that at all: science. The object of the game of science is not assurance but truth — usually truths about aspects of reality that cannot be observed directly.

I think scientific knowledge exists, but we have to adjust our concept of knowledge to accommodate it. Scientific knowledge is far from certain, because scientific theories are about unfamiliar and often quite strange things that cannot be seen directly. Our best theories about them might be only approximately true. These theories are representations of reality, but we are often unclear how to interpret them. Our understanding of how they represent reality is not always complete.

More to the point, evidence in science cannot be understood as proof, or as having anything like the structure of proof. It does not consist of deriving something via rules of inference from something else that we are already assured of.

Rather, evidence in science consists of piecemeal peripheral indications that we have got it right. The most important such indications are observational tests. Because most scientific theories describe things that can’t be observed directly, we can’t check directly to see whether they are true. Instead, a theory implies things that can be observed directly, and if direct observation confirms what the theory predicts, the theory is corroborated. This is nothing nothing like following a rule of inference, and it delivers nothing like assurance of the sort that prompts us to say “it’s in the bag”.

Other peripheral indications of a theory’s truth are more “aesthetic”. It explains a lot. It seems simple, modest, general, etc. It has the ring of truth. And so on.

Good science is conducted with a clear awareness that scientific evidence has nothing like the structure of proof. Alas — bad science isn’t conducted with that awareness.