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The fact that one apple added to one apple invariably gives two apples helps in the teaching of arithmetic, but has no bearing on the truth of the proposition that 1 + 1 = 2.
-- James R. Newman, The World of Mathematics
Previous meditation 1: If we can only meaningfully talk about parts of the universe that can be pinned down by chains of cause and effect, where do we find the fact that 2 + 2 = 4? Or did I just make a meaningless noise, there? Or if you claim that "2 + 2 = 4"isn't meaningful or true, then what alternate property does the sentence "2 + 2 = 4" have which makes it so much more useful than the sentence "2 + 2 = 3"?
Previous meditation 2: It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently.
Speaking conventional English, we'd say the sentence 2 + 2 = 4 is "true", and anyone who put down "false" instead on a math-test would be marked wrong by the schoolteacher (and not without justice).
But what can make such a belief true, what is the belief about, what is the truth-condition of the belief which can make it true or alternatively false? The sentence '2 + 2 = 4' is true if and only if... what?
In the previous post I asserted that the study of logic is the study of which conclusions follow from which premises; and that although this sort of inevitable implication is sometimes called "true", it could more specifically be called "valid", since checking for inevitability seems quite different from comparing a belief to our own universe. And you could claim, accordingly, that "2 + 2 = 4" is 'valid' because it is an inevitable implication of the axioms of Peano Arithmetic.
And yet thinking about 2 + 2 = 4 doesn't really feel that way. Figuring out facts about the natural numbers doesn't feel like the operation of making up assumptions and then deducing conclusions from them. It feels like the numbers are just out there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them. The Peano axioms might have been convenient for deducing a set of theorems like 2 + 2 = 4, but really all of those theorems were true about numbers to begin with. Just like "The sky is blue" is true about the sky, regardless of whether it follows from any particular assumptions.
So comparison-to-a-standard does seem to be at work, just as with physical truth... and yet this notion of 2 + 2 = 4 seems different from "stuff that makes stuff happen". Numbers don't occupy space or time, they don't arrive in any order of cause and effect, there are no events in numberland.
Meditation: What are we talking about when we talk about numbers? We can't navigate to them by following causal connections - so how do we get there from here?
Followup to: Causal Reference
From a math professor's blog:
One thing I discussed with my students here at HCSSiM yesterday is the question of what is a proof.
They’re smart kids, but completely new to proofs, and they often have questions about whether what they’ve written down constitutes a proof. Here’s what I said to them.
A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.
This is not quite the definition I would give of what constitutes "proof" in mathematics - perhaps because I am so used to isolating arguments that are convincing, but ought not to be.
Or here again, from "An Introduction to Proof Theory" by Samuel R. Buss:
There are two distinct viewpoints of what a mathematical proof is. The first view is that proofs are social conventions by which mathematicians convince one another of the truth of theorems. That is to say, a proof is expressed in natural language plus possibly symbols and figures, and is sufficient to convince an expert of the correctness of a theorem. Examples of social proofs include the kinds of proofs that are presented in conversations or published in articles. Of course, it is impossible to precisely define what constitutes a valid proof in this social sense; and, the standards for valid proofs may vary with the audience and over time. The second view of proofs is more narrow in scope: in this view, a proof consists of a string of symbols which satisfy some precisely stated set of rules and which prove a theorem, which itself must also be expressed as a string of symbols. According to this view, mathematics can be regarded as a 'game' played with strings of symbols according to some precisely defined rules. Proofs of the latter kind are called "formal" proofs to distinguish them from "social" proofs.
In modern mathematics there is a much better answer that could be given to a student who asks, "What exactly is a proof?", which does not match either of the above ideas. So:
Meditation: What distinguishes a correct mathematical proof from an incorrect mathematical proof - what does it mean for a mathematical proof to be good? And why, in the real world, would anyone ever be interested in a mathematical proof of this type, or obeying whatever goodness-rule you just set down? How could you use your notion of 'proof' to improve the real-world efficacy of an Artificial Intelligence?
Followup to: Causality: The Fabric of Real Things
"You say that a universe is a connected fabric of causes and effects. Well, that's a very Western viewpoint - that it's all about mechanistic, deterministic stuff. I agree that anything else is outside the realm of science, but it can still be real, you know. My cousin is psychic - if you draw a card from his deck of cards, he can tell you the name of your card before he looks at it. There's no mechanism for it - it's not a causal thing that scientists could study - he just does it. Same thing when I commune on a deep level with the entire universe in order to realize that my partner truly loves me. I agree that purely spiritual phenomena are outside the realm of causal processes that can be studied by experiments, but I don't agree that they can't be real."
Fundamentally, a causal model is a way of factorizing our uncertainty about the universe. One way of viewing a causal model is as a structure of deterministic functions plus uncorrelated sources of background uncertainty.
Let's use the Obesity-Exercise-Internet model (reminder: which is totally made up) as an example again:
We can also view this as a set of deterministic functions Fi, plus uncorrelated background sources of uncertainty Ui:
This says is that the value x3 - how much someone exercises - is a function of how obese they are (x1), how much time they spend on the Internet (x2), plus some other background factors U3 which don't correlate to anything else in the diagram, all of which collectively determine, when combined by the mechanism F3, how much time someone spends exercising.
I think people who are not made happier by having things either have the wrong things, or have them incorrectly. Here is how I get the most out of my stuff.
Money doesn't buy happiness. If you want to try throwing money at the problem anyway, you should buy experiences like vacations or services, rather than purchasing objects. If you have to buy objects, they should be absolute and not positional goods; positional goods just put you on a treadmill and you're never going to catch up.
I think getting value out of spending money, owning objects, and having positional goods are all three of them skills, that people often don't have naturally but can develop. I'm going to focus mostly on the middle skill: how to have things correctly1.
Suppose a general-population survey shows that people who exercise less, weigh more. You don't have any known direction of time in the data - you don't know which came first, the increased weight or the diminished exercise. And you didn't randomly assign half the population to exercise less; you just surveyed an existing population.
The statisticians who discovered causality were trying to find a way to distinguish, within survey data, the direction of cause and effect - whether, as common sense would have it, more obese people exercise less because they find physical activity less rewarding; or whether, as in the virtue theory of metabolism, lack of exercise actually causes weight gain due to divine punishment for the sin of sloth.
The usual way to resolve this sort of question is by randomized intervention. If you randomly assign half your experimental subjects to exercise more, and afterward the increased-exercise group doesn't lose any weight compared to the control group , you could rule out causality from exercise to weight, and conclude that the correlation between weight and exercise is probably due to physical activity being less fun when you're overweight . The question is whether you can get causal data without interventions.
For a long time, the conventional wisdom in philosophy was that this was impossible unless you knew the direction of time and knew which event had happened first. Among some philosophers of science, there was a belief that the "direction of causality" was a meaningless question, and that in the universe itself there were only correlations - that "cause and effect" was something unobservable and undefinable, that only unsophisticated non-statisticians believed in due to their lack of formal training:
"The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm." -- Bertrand Russell (he later changed his mind)
"Beyond such discarded fundamentals as 'matter' and 'force' lies still another fetish among the inscrutable arcana of modern science, namely, the category of cause and effect." -- Karl Pearson
The famous statistician Fisher, who was also a smoker, testified before Congress that the correlation between smoking and lung cancer couldn't prove that the former caused the latter. We have remnants of this type of reasoning in old-school "Correlation does not imply causation", without the now-standard appendix, "But it sure is a hint".
This skepticism was overturned by a surprisingly simple mathematical observation.
Followup to: Rationality: Appreciating Cognitive Algorithms (minor post)
There's an old anecdote about Ayn Rand, which Michael Shermer recounts in his "The Unlikeliest Cult in History" (note: calling a fact unlikely is an insult to your prior model, not the fact itself), which went as follows:
Branden recalled an evening when a friend of Rand's remarked that he enjoyed the music of Richard Strauss. "When he left at the end of the evening, Ayn said, in a reaction becoming increasingly typical, 'Now I understand why he and I can never be real soulmates. The distance in our sense of life is too great.' Often she did not wait until a friend had left to make such remarks."
Many readers may already have appreciated this point, but one of the Go stones placed to block that failure mode is being careful what we bless with the great community-normative-keyword 'rational'. And one of the ways we do that is by trying to deflate the word 'rational' out of sentences, especially in post titles or critical comments, which can live without the word. As you hopefully recall from the previous post, we're only forced to use the word 'rational' when we talk about the cognitive algorithms which systematically promote goal achievement or map-territory correspondences. Otherwise the word can be deflated out of the sentence; e.g. "It's rational to believe in anthropogenic global warming" goes to "Human activities are causing global temperatures to rise"; or "It's rational to vote for Party X" deflates to "It's optimal to vote for Party X" or just "I think you should vote for Party X".
If you're writing a post comparing the experimental evidence for four different diets, that's not "Rational Dieting", that's "Optimal Dieting". A post about rational dieting is if you're writing about how the sunk cost fallacy causes people to eat food they've already purchased even if they're not hungry, or if you're writing about how the typical mind fallacy or law of small numbers leads people to overestimate how likely it is that a diet which worked for them will work for a friend. And even then, your title is 'Dieting and the Sunk Cost Fallacy', unless it's an overview of four different cognitive biases affecting dieting. In which case a better title would be 'Four Biases Screwing Up Your Diet', since 'Rational Dieting' carries an implication that your post discusses the cognitive algorithm for dieting, as opposed to four contributing things to keep in mind.
Followup to: The Useful Idea of Truth (minor post)
So far as I know, the first piece of rationalist fiction - one of only two explicitly rationalist fictions I know of that didn't descend from HPMOR, the other being "David's Sling" by Marc Stiegler - is the Null-A series by A. E. van Vogt. In Vogt's story, the protagonist, Gilbert Gosseyn, has mostly non-duplicable abilities that you can't pick up and use even if they're supposedly mental - e.g. the ability to use all of his muscular strength in emergencies, thanks to his alleged training. The main explicit-rationalist skill someone could actually pick up from Gosseyn's adventure is embodied in his slogan:
"The map is not the territory."
Sometimes it still amazes me to contemplate that this proverb was invented at some point, and some fellow named Korzybski invented it, and this happened as late as the 20th century. I read Vogt's story and absorbed that lesson when I was rather young, so to me this phrase sounds like a sheer background axiom of existence.
But as the Bayesian Conspiracy enters into its second stage of development, we must all accustom ourselves to translating mere insights into applied techniques. So:
Meditation: Under what circumstances is it helpful to consciously think of the distinction between the map and the territory - to visualize your thought bubble containing a belief, and a reality outside it, rather than just using your map to think about reality directly? How exactly does it help, on what sort of problem?
Here's the new thread for posting quotes, with the usual rules:
- Please post all quotes separately, so that they can be voted up/down separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
- Do not quote yourself
- Do not quote comments/posts on LW/OB
- No more than 5 quotes per person per monthly thread, please.
(This is the first post of a new Sequence, Highly Advanced Epistemology 101 for Beginners, setting up the Sequence Open Problems in Friendly AI. For experienced readers, this first post may seem somewhat elementary; but it serves as a basis for what follows. And though it may be conventional in standard philosophy, the world at large does not know it, and it is useful to know a compact explanation. Kudos to Alex Altair for helping in the production and editing of this post and Sequence!)
I remember this paper I wrote on existentialism. My teacher gave it back with an F. She’d underlined true and truth wherever it appeared in the essay, probably about twenty times, with a question mark beside each. She wanted to know what I meant by truth.
-- Danielle Egan
I understand what it means for a hypothesis to be elegant, or falsifiable, or compatible with the evidence. It sounds to me like calling a belief ‘true’ or ‘real’ or ‘actual’ is merely the difference between saying you believe something, and saying you really really believe something.
-- Dale Carrico
What then is truth? A movable host of metaphors, metonymies, and; anthropomorphisms: in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished, and which, after long usage, seem to a people to be fixed, canonical, and binding.
-- Friedrich Nietzche
The Sally-Anne False-Belief task is an experiment used to tell whether a child understands the difference between belief and reality. It goes as follows:
The child sees Sally hide a marble inside a covered basket, as Anne looks on.
Sally leaves the room, and Anne takes the marble out of the basket and hides it inside a lidded box.
Anne leaves the room, and Sally returns.
The experimenter asks the child where Sally will look for her marble.
Children under the age of four say that Sally will look for her marble inside the box. Children over the age of four say that Sally will look for her marble inside the basket.
This is a chapter by chapter review of Causality (2nd ed.) by Judea Pearl (UCLA, blog). Like my previous review, the intention is not to summarize but to help readers determine whether or not they should read the book (and if they do, what parts to read). Reading the review is in no way a substitute for reading the book.
I'll state my basic impression of the book up front, with detailed comments after the chapter discussions: this book is monumentally important to anyone interested in procuring knowledge (especially causal knowledge) from statistical data, but it is a heavily technical book primarily suitable for experts. The mathematics involved is not particularly difficult, but its presentation requires dedicated reading and clarity of thought. Only the epilogue, this lecture, is suitable for the general audience, and that will be the highest value portion for most readers of LW.