Truth, doubt & Popper #
We know what is false far better than what is true. So the honest posture is Popper’s: propose bold conjectures and try to refute them. These notes circle that idea — through language, through Gödel, and through the difference between agreeing and merely not disagreeing.
Agree, or disagree? #
If you talk with someone and want to express agreement, you can say either “it is not false” or “it is true.” But saying one or the other does not amount to the same thing.
Suppose someone says, “it is good to think before acting,” and you concede the point with “it is true”: you agree without any nuance. Yet if you are walking in nature and see a charging bull, it may be a very good idea to jump onto a nearby tree and think afterwards — not before. You run on survival instinct; there is no time to think, it is too slow, and it would cost you your life. So “it is good to think before acting” is not always true, when the difference between life and death is seconds.
Or, it is not false.
If instead you say “it is not false,” you mean that you detect no falsity in the statement — which does not mean none exists, only that you are not aware of it. You are conscious enough to know you might be wrong, and that ultimately leads to a better decision, since we cannot be sure of many things in life. A conversation in that spirit:
- John — It is good to think before acting.
- Mary — It is not false.
- John — It is always true.
- Mary — You cannot know how much you know, since you cannot possibly know how much you don’t know. You are essentially like an insect with antennae, sensing its environment a centimetre ahead — not even aware that I am observing it, since a motionless watcher barely registers in eyes built to catch movement.
- John — That is a good point. It is not false.
- Mary — Exactly what I mean. If you agree to a theory that is “not false,” it means some part of it does not look false to you at this moment — but that itself may be false. You agree, while remaining aware of the conjectural nature of the statement.
Notes #
Studying to unlearn thinking #
I am studying to unlearn thinking.
We know the false better than the true #
We know better what is false than what is true — for example, what God is not, rather than what he is.
Conjecture and refute #
We know nothing from having merely seen it. The truth is hidden deep, inaccessible to humans — accessible only to God. All we can hope to do is conjecture and refute: propose bold theories and have them tested, or criticized, by others. That is the nature of learning — trying things out, for we know nothing and can know nothing, being human; we are too limited. Just look at a horse: is he aware that the Earth is round? Could you explain statistics to a horse? And does he need statistics to survive? What is the difference between a human and a horse? Both are animals; only, the human has this power of reasoning, which he uses to explain things after the fact. Is that all?
Popper’s truth-seeking #
If the world is chaos and we need a theory that works most of the time, then — I add personally — the assumption is this: the truth is something like an infinite tunnel between dimensions, which cannot be touched, but which we can approach by removing the false, by seeking what is not truth, so that there is less noise. Doing science, testing theories and finding them false, brings us closer to truth. But only theories that can be falsified — that can be tested and, in principle, found not to work. A theory that holds true whether A or B happens is no theory at all, because no test attached to it could make it false. So a theory needs test cases, as in programming, where one program tests another: the test runs the conditions under which the theory (the program) should fail, and expects that failure, proving that under certain conditions the theory breaks.
Are some false theories better than others? #
This is the question the rest of these notes stand or fall on, so let me answer it plainly.
If every theory is a map, and every map distorts, then every theory I hold is false in some degree, this one included. A reader is entitled to ask what is left. If all theories are false, is Ptolemy no worse than Kepler? Is my butcher’s opinion of the economy level with an economist’s?
No — and the reason is that falsity comes in degrees, which is what makes the removing of error worth doing. Newton’s mechanics is strictly false and it put men on the Moon; Aristotle’s physics is also false and would not get them off the ground. A theory is better when it forbids more, when it says more about the world and therefore exposes more of itself to refutation, and when it has stood in front of harsher tests and survived. The one to distrust is the theory that is compatible with every outcome, since it has risked nothing.
I hold this without being able to make it exact, and I should say so. Popper tried to define this closeness to truth formally — his verisimilitude — and in 1974 Tichý and Miller showed the definition fails: on his own terms no false theory comes out closer to the truth than any other false theory. As far as I know no replacement has settled the matter. So I am left with a comparison I use every day and cannot yet justify with the precision I would like. That is an open wound in the position, not a decoration on it, and I would rather leave it visible.
Ignorance breeds confidence #
Darwin put it exactly: ignorance more frequently begets confidence than does knowledge. Why? If you know, then you know the whole world is random and unpredictable. When you don’t know — when you believe dogmatically in ideas and ideologies — that is ignorance. Such a person believes in a simple world with absolute rules, where the believers are the chosen ones. That is one way in which ignorance, with a dose of stupidity, can drive people to commit heinous crimes — as in a religious war, or any war.
The question over the answer #
The question is more important than the answer, because the question is definite, while answers are conjectures that can never be certain to be true.
Change is not progress #
Change in human society does not mean progress. Where there is progress it is because criticism is allowed to do its work — theses put up, attacked, and abandoned when they fail. Science is the clearest case, being built for it. But it is not the only one, and I would be contradicting Popper to say otherwise: an open society is precisely one that has arranged to remove bad rulers and bad rules without bloodshed, which is trial and error carried into politics. Change without that machinery is only movement.
Opinion #
Opinion is ignorance that pretends to be knowledge. (The thought is Plato’s doxa against episteme; the phrasing is mine, and I have not been able to trace it to anyone else.)
The machine that names things #
Popper imagines a machine fitted with a lens and a voice, which names any medium-sized object placed before it — “cat,” “dog” — or, in some cases, says “I don’t know.” It can be made more human still: made to answer only when asked, “Can you tell me what this is?”, or to reply now and then, “I am getting tired, leave me alone for a while.” If such a machine behaved much like a person, we might mistakenly believe it describes and argues — just as someone ignorant of how a radio works might think the receiver describes and argues. Yet an analysis of its mechanism shows nothing of the kind happens: the radio does not argue, though it expresses its physical states and signals. (After Popper, Conjectures and Refutations.)
Language has bugs, like code #
If you take human language and look for paradoxes — “all Cretans are liars,” and other circular or unclassified quirks, or bugs, to use the term from software — you find something worth noticing.
A computer language is a special kind of program, an interface between the human and the chip (a gross simplification, for the sake of the argument). It can contain errors, mistakes and miscalculations that surface over time and are corrected by programmers. Now, if we think of human language in the same terms, it too can have built-in errors and quirks that no one designed — the Cretan paradox being the famous one. (Mathematics I take to be another specialist language, like Latin or COBOL: invented by humans rather than lying ready-made in nature, and developed by degrees, starting perhaps from peoples who count “one,” “two” and “many.” This is a position and not a fact, and the opposite view — that we discover mathematics rather than build it — is held by serious people and gains something from how uncannily well the thing works.)
So you might think the world is irrational — that things sometimes make no sense: an idiot gets voted into office, a paradox turns up in a language we built ourselves. But all these irrationalities exist only in our imagination, or stem from our imagination and knowledge. Nature knows no mathematics, no theories, and needs no humans; it is humans who need nature. So all those irrationalities are failures, in the Popperian sense, of conjectures — either not yet refuted, or refuted and still in use because nothing better is at hand and the rule works well enough.
Consider the atomic clocks flown around the world by Hafele and Keating in 1971. The clocks came back disagreeing with the ones left on the ground, by tens of nanoseconds, in the directions relativity had predicted in advance — speed slowing a clock, altitude speeding it up, the two effects netting out with opposite signs going east and going west. You were, in miniature, in a time machine.
I use this as the strongest case against my own scepticism, because it is one. A theory made a precise, risky, quantitative prediction about an experiment nobody had performed, and the world obliged. That is as good as human knowledge gets, and any account of knowledge that cannot admit it is worthless. What survives the example is only this: passing the test established that the theory held under those conditions, not that it holds everywhere and forever. Newton’s mechanics passed every test put to it for two centuries and was superseded anyway — not shown to be worthless, but shown to be a special case, correct within a boundary nobody had known was there. That is the most likely fate of relativity too. These are failures not of nature but of the theories, for theories — like maps or maquettes — are only approximations and distortions of reality: it will never be possible to compress the same detail into a smaller scale without shedding and distorting something. Human languages are so much talk, sometimes corresponding to reality, sometimes for a long time, sometimes not. As with nations and species, nature eliminates one variant and experiments with another. Men keep making their theories, but it is nature that has the last word.
Gödel, and the drop of poison #
I had this badly wrong, and correcting it in public seems the least I can do, since these notes are supposed to be conjectures offered up for refutation rather than opinions defended.
What I used to think: that mathematics was shown to be an imperfect language, that Gödel proposed to repair it by breaking it into smaller and more manageable parts, and that the repair fails the way a fine carpet thrown over a puddle fails — the mud comes through in the end.
That is the wrong man. The programme of securing mathematics by reducing it to small, safe, mechanically checkable pieces was Hilbert’s, and it was the great hope of the discipline. Gödel is the one who ended it. In 1931 he showed that any consistent formal system strong enough to express ordinary arithmetic contains true statements it cannot prove, and cannot prove its own consistency from within. He did not patch the carpet; he demonstrated that no carpet of that kind can cover the floor.
Two corrections follow, and both cut against what I wrote.
Incompleteness is not inconsistency, and neither is falsity. Gödel did not find a contradiction in arithmetic. He found a limit on what proof can reach. That is a far stranger and more interesting result than a mistake would be, and it leaves mathematics standing. Related: irrational numbers are not an irrationality. That √2 cannot be written as a ratio of whole numbers is a theorem, proved, one of the most beautiful things anyone has ever established — the word “irrational” refers to ratio, not to unreason. I was leaning on a pun and mistaking it for an argument.
And the drop of poison proves too much. My principle was that a single false element spoils the whole theory, as a drop of cyanide spoils the tank. It is a good image for logical inconsistency, where indeed anything at all follows from a contradiction. It is a bad principle about falsity, and if I hold to it I lose everything else I believe. Every theory contains something false, mine as much as anyone’s. If one drop condemns the tank, then Newton and Ptolemy are equally poisoned, no theory is closer to the truth than another, and the whole business of removing error — the reason for these notes — collapses into a shrug. Newton’s mechanics is false and it landed men on the Moon. So the tank must be allowed to be partly drinkable. What I keep is narrower and still worth having: a theory resting on a foundation you know to be broken is a perilous thing to build on, and the deeper the stack of inherited assumptions, the worse the odds — which is the compounding argument from Maps, models & power laws, not a law of contamination.
The Cretan liar, I should add, is a paradox of natural language and self-reference. It is not a bug found inside arithmetic, and I was quietly using it to convict a discipline it never belonged to.
To read, on doubt #
- Sextus Empiricus, Pierre Bayle, Nicolas d’Autrecourt.
- The Name of the Rose (book and film); Charles Sanders Peirce and Victor Brochard (1878), close to Popper; and Hempel’s paradox.