All math pays rent.
For all mathematical theorems can be restated in the form:
If the axioms A, B, and C and the conditions X, Y and Z are satisfied, then the statement Q is also true.
Therefore, in any situations where the statements A,B,C and X,Y,Z are true, you will expect Q to also be verified.
In other words, mathematical statements automatically pay rent in terms of changing what you expect. (Which is) the very thing it was required to show. ■
In practice:
If you demonstrate Pythagoras's Theorem, and you calculate that 3^2+4^2=5^2, you will expect a certain method of getting right angles to work.
If you exhibit the aperiodic Penrose Tiling, you will expect Quasicrystals to exist.
If you demonstrate the impossibility of solving to the Halting Problem, you will not expect even a hypothetical hyperintelligence to be able to solve it.
If you understand why you can't trisect an angle with an unmarked ruler and a compass (not both used at the same time), you will know immediately that certain proofs are going to be wrong.
and so on and so forth.
Yes, we might not immediately know where a given mathematical fact will come in handy when observing the world, but by their nature, mathematical facts tell us exactly when to expect them.
In practice, most of the time people figure out what to connect it to later. More precisely, most of it probably doesn't connect to anything, but what does connect to stuff usually isn't found to do so until much later than it is invented/discovered.
Some ungrounded concepts can produce your own behavior which in itself can be experienced, so it's difficult to draw the line just by requiring concepts to be grounded. You believe that you believe in something, because you experience yourself acting in a way consistent with you believing in it. It can define intrinsic goal system, point in mind design space as you call it. So one can't abolish all such concepts, only resist acquiring them.
For any instrumental activity, done to achieve some other end, it makes sense to check that specific examples are in fact achieving the intended end.
Most beliefs may have as their end the refinement of personal decisions. For such beliefs it makes sense not only to check whether they effect your personal experience, but also whether they effect any decisions you might make; beliefs could effect experience without mattering for decisions.
On the other hand, some beliefs may have as their end effecting the experiences or decisions of other creatures, such as in the far future. And you may care about effects that are not experienced by any creatures.
Elizer, your post above strikes me, at least, as a restatement of verificationism: roughly, the view that the truth of a claim is the set of observations that it predicts. While this view enjoyed considerable popularity in the first part of the last century (and has notable antecedents going back into the early 18th century), it faces considerable conceptual hurdles, all of which have been extensively discussed in philosophical circles. One of the most prominent (and noteworthy in light of some of your other views) is the conflict between verificationism and scientific realism: that is, the presumption that science is more than mere data-predictive modeling, but the discovery of how the world really is. See also here and here.
Rooney, as discussed in The Simple Truth I follow a correspondence theory of truth. I am also a Bayesian and a believer in Occam's Razor. If a belief has no empirical consequences then it could receive no Bayesian confirmation and could not rise to my subjective attention. In principle there are many true beliefs for which I have no evidence, but in practice I can never know what these true beliefs are, or even focus on them enough to think them explicitly, because they are so vastly outnumbered by false beliefs for which I can find no evidence.
It's amazing how many forms of irrationality failure to see the map-territory distinction, and the resulting reification of categories (like 'sound') that exist in the mind, causes: stupid arguments, phlogiston, the Mind Projection Fallacy, correspondence bias, and probably also monotheism, substance dualism, the illusion of the self, the use of the correspondence theory of truth in moral questions... how many more?
I think you're being too hard on the English professor, though. I suspect literary labels do have something to do with the contents of a book, no matter how much nonsense might be attached to them. But I've never experienced a college English class; perhaps my innocent fantasies will be shaken then.
Michael V, you could say that mathematical propositions are really predictions about the behavior of physical systems like adding machines and mathematicians. I don't find that view very satisfying, because math seems to so fundamentally underly everything else - mathematical truths can't be changed by changing anything physical, for instance - but it's one way to make math compatible with anticipation.
It's amazing how many forms of irrationality failure to see the map-territory distinction
Should have been "how many forms of irrationality result from failure...". Sorry.
I agree with those who say it's okay to figure things out later. If my music professor says a certain composer favors the Aeolian mode, I may not be able to visualize that on the spot but who cares? I can remember that statement and think about it later. Likewise with phlogiston, I have a vague concept of what it is and someday the alchemists will discover more precisely what's going on there.
Too much cognitive effort would be spent if, every time I thought about linear algebra, I had to visualize the myriad concrete instances in which it will be applied. I bet thinking in abstractions results in way more economical use of thinking time and thinking-matter.
Mark: Believing that beliefs should be grounded anticipates that there is absolutely no change in anticipation if one were to change these free floating ideas. Of course this doesn't really answer your question because it just restates the definition of 'free floating beliefs' in different words. This belief actually follows from Eliezers belief in Occam's Razor, which predicts that when faced with unexplained events, if one creates a set of theories explaining these events, any predictions made by the simple theories are more likely to actually happen tha...
Jan: Occam's razor is not so much a rule of science but an operating guideline for doing science. It could be reduced to "test simple theories first". In the past this has been very useful in keeping scientific effort productive, the 'belief' is that it will continue to be useful in this way.
This led to a fun read of "occam's razor" wikipedia entry. Hickum's dictum in particular was a great find (generalized beyond medicine, it could be that explanations for unexplained events can be as complex as they damn well please). As a practical corrective, it seems to me that probability theory suggests that the best accessible explanation to us for unexplained events is in the set of simpler theories, but is probably not one of the absolute simplest.
Eliezer once wrote that "We can build up whole networks of beliefs that are connected only to each other - call these "floating" beliefs. It is a uniquely human flaw among animal species, a perversion of Homo sapiens's ability to build more general and flexible belief networks.
The rationalist virtue of empiricism consists of constantly asking which experiences our beliefs predict - or better yet, prohibit."
I can't see how nearly all of the beliefs expressed in this post predict or prohibit any experience.
"Alchemists believed that phlogiston caused fire"
How is that different than our current belief that oxygen causes fire?
You're giving phlogiston qualities no one who held that theory gave it. If you want to call the absence of oxygen phlogiston, okay, but you aren't talking about the same phlogiston everyone else is talking about. Moreover, thinking about fire this way is clumsy and incompatible with the rest of our knowledge about physics and chemistry.
We already had a conception of matter when phlogiston was invented... and phlogiston was understood as a kind of matter. To say the phlogiston is really this other kind of thing, which isn't matter but a particular kind of absence of matter is both unhelpful and a distortion of phlogiston theory. The whole point of the phlogiston theory was that they thought there was a kind of matter responsible for fire! But there isn't matter like that.
Now by defining phlogiston as the absence of oxygen you might be able to model combustion in a narrow set of circumstances-- but you couldn't fit that model with any of your other knowledge about physics and chemistry.
In short neither the original kind nor your kind of phlogiston exist.
Because one of these allows you to make predictions, and the other doesn't. Saying "fire has a cause, and I'm going to call it 'phlogiston'!" doesn't tell you anything about fire, it's just a relabeling.
The hypothesis went a little deeper than that. "Flammable things contain a substance, and its release is fire" lets you make many predictions — e.g., that things will burn in vacuum, or that things burned in open air will always lose mass (this is how it was falsified).
I loved this post, but I have to be a worthless pedant.
If you drop a ball off a 120-m tall building, you expect impact in t=sqrt(2H/g)=~5 s. But that would be when the second-hand is on the 1 numeral.
What about knowledge for the sake of knowledge? For instance I don't anticipate that my belief that The Crusades took place will ever directly affect my sensory experiences in any way. Does that then mean that this belief is completely worthless and on the same level as the belief in ghosts, psychics, phlogiston, etc.?
Wouldn't taking your chain of reasoning to its logical conclusion require one to "evict" all beliefs in everything that one has not, and does not anticipate to, personally see, hear, smell, taste, or touch? After all, how much personal sensory experience do you have that confirms the existence of atoms, for example?
DP
You write, “suppose your postmodern English professor teaches you that the famous writer Wulky Wilkinsen is actually a ‘post-utopian’. What does this mean you should expect from his books? Nothing.”
I’m sympathetic to your general argument in this article, but this particular jibe is overstating your case.
There may be nothing particularly profound in the idea of ‘post-utopianism’, but it’s not meaningless. Let me see if I can persuade you.
Utopianism is the belief that an ideal society (or at least one that's much better than ours) can be constructed, for example by the application of a particular political ideology. It’s an idea that has been considered and criticized here on LessWrong. Utopian fiction explores this belief, often by portraying such an ideal society, or the process that leads to one. In utopian fiction one expects to see characters who are perfectible, conflicts resolved successfully or peacefully, and some kind of argument in favour of utopianism. Post-utopian fiction is written in reaction to this, from a skeptical or critical viewpoint about the perfectibility of people and the possibility of improving society. One expects to see irretrievably flawed characters, i...
Would you consider Le Guin's The Dispossessed to be post-utopian? I think she intends her Anarres to be a good place on the whole, and a decent partial attempt at achieving a utopia, but still to have plausible problems.
I think it's both. "Brave New World" portrays a dystopia (Huxley called it a "negative utopia") but it's also post-utopian because it displays skepticism towards utopian ideals (Huxley wrote it in reaction to H. G. Wells' "Men Like Gods").
I don't claim any expertise on this subject: in fact, I hadn't heard of post-utopianism at all until I read the word in this article. It just seemed to me to be overstating the case to claim that a term like this is meaningless. Vague, certainly. Not very profound, yes. But meaningless, no.
The meaning is easily deducible: in the history of ideas "post-" is often used to mean "after; in consequence of; in reaction to" (and "utopian" is straightforward). I checked my understanding by searching Google Scholar and Books: there seems to be only one book on the subject (The post-utopian imagination: American culture in the long 1950s by M. Keith Booker) but from reading the preview it seems to be using the word in the way that I described above.
The fact that the literature on the subject is small makes post-utopianism an easier target for this kind of attack: few people are likely to be familiar with the idea, or motivated to defend it, and it's harder to establish what the consensus on the subject is. By contrast, imagine trying to claim that "hard science fiction" was a meaningless term.
Indeed. Some rationalists have a fondness for using straw postmodernists to illustrate irrationality. (Note that Alan Sokal deliberately chose a very poor journal, not even peer-reviewed, to send his fake paper to.) It's really not all incomprehensible Frenchmen. While there may be a small number of postmodernists who literally do not believe objective reality exists, and some more who try to deconstruct actual science and not just the scientists doing it, it remains the case that the human cultural realm is inherently squishy and much more relative than people commonly assume, and postmodernism is a useful critical technique to get through the layers of obfuscation motivating many human cultural activities. Any writer of fiction who is any good, for instance, needs to know postmodernist techniques, whether they call them that or not.
How is this not just a simple arguement on semantics (on which I believe a vast majority of arguements are based)?
They both accept that the tree causes vibrations in the air as it falls, and they both accept that no human ear will ever hear it. The arguement appears to be based solely on the definition, and surrounding implications, of the word "sound" (or "noise" as it becomes in the article) - and is therefore no arguement at all.
"Or suppose your postmodern English professor teaches you that the famous Wulky Wilkinsen is actually a "post-utopian". What does this mean you should expect from his book? Nothing."
When I first read this I thought, "Huh? Surely it tells you something, because I already have beliefs about what 'utopian' probably means, and what the 'post' part of it probably means, and what context these types of terms are usually used in... That sounds like a whole bag of reasons to expect certain things/themes/ideas in his book!"
But I think ...
Thus begins the ancient parable:
If a tree falls in a forest and no one hears it, does it make a sound? One says, "Yes it does, for it makes vibrations in the air." Another says, "No it does not, for there is no auditory processing in any brain."
Suppose that, after the tree falls, the two walk into the forest together. Will one expect to see the tree fallen to the right, and the other expect to see the tree fallen to the left? Suppose that before the tree falls, the two leave a sound recorder next to the tree. Would one, playing back the recorder, expect to hear something different from the other? Suppose they attach an electroencephalograph to any brain in the world; would one expect to see a different trace than the other? Though the two argue, one saying "No," and the other saying "Yes," they do not anticipate any different experiences. The two think they have different models of the world, but they have no difference with respect to what they expect will happen to them.
It's tempting to try to eliminate this mistake class by insisting that the only legitimate kind of belief is an anticipation of sensory experience. But the world does, in fact, contain much that is not sensed directly. We don't see the atoms underlying the brick, but the atoms are in fact there. There is a floor beneath your feet, but you don't experience the floor directly; you see the light reflected from the floor, or rather, you see what your retina and visual cortex have processed of that light. To infer the floor from seeing the floor is to step back into the unseen causes of experience. It may seem like a very short and direct step, but it is still a step.
You stand on top of a tall building, next to a grandfather clock with an hour, minute, and ticking second hand. In your hand is a bowling ball, and you drop it off the roof. On which tick of the clock will you hear the crash of the bowling ball hitting the ground?
To answer precisely, you must use beliefs like Earth's gravity is 9.8 meters per second per second, and This building is around 120 meters tall. These beliefs are not wordless anticipations of a sensory experience; they are verbal-ish, propositional. It probably does not exaggerate much to describe these two beliefs as sentences made out of words. But these two beliefs have an inferential consequence that is a direct sensory anticipation—if the clock's second hand is on the 12 numeral when you drop the ball, you anticipate seeing it on the 1 numeral when you hear the crash five seconds later. To anticipate sensory experiences as precisely as possible, we must process beliefs that are not anticipations of sensory experience.
It is a great strength of Homo sapiens that we can, better than any other species in the world, learn to model the unseen. It is also one of our great weak points. Humans often believe in things that are not only unseen but unreal.
The same brain that builds a network of inferred causes behind sensory experience, can also build a network of causes that is not connected to sensory experience, or poorly connected. Alchemists believed that phlogiston caused fire—we could oversimply their minds by drawing a little node labeled "Phlogiston", and an arrow from this node to their sensory experience of a crackling campfire—but this belief yielded no advance predictions; the link from phlogiston to experience was always configured after the experience, rather than constraining the experience in advance. Or suppose your postmodern English professor teaches you that the famous writer Wulky Wilkinsen is actually a "post-utopian". What does this mean you should expect from his books? Nothing. The belief, if you can call it that, doesn't connect to sensory experience at all. But you had better remember the propositional assertion that "Wulky Wilkinsen" has the "post-utopian" attribute, so you can regurgitate it on the upcoming quiz. Likewise if "post-utopians" show "colonial alienation"; if the quiz asks whether Wulky Wilkinsen shows colonial alienation, you'd better answer yes. The beliefs are connected to each other, though still not connected to any anticipated experience.
We can build up whole networks of beliefs that are connected only to each other—call these "floating" beliefs. It is a uniquely human flaw among animal species, a perversion of Homo sapiens's ability to build more general and flexible belief networks.
The rationalist virtue of empiricism consists of constantly asking which experiences our beliefs predict—or better yet, prohibit. Do you believe that phlogiston is the cause of fire? Then what do you expect to see happen, because of that? Do you believe that Wulky Wilkinsen is a post-utopian? Then what do you expect to see because of that? No, not "colonial alienation"; what experience will happen to you? Do you believe that if a tree falls in the forest, and no one hears it, it still makes a sound? Then what experience must therefore befall you?
It is even better to ask: what experience must not happen to you? Do you believe that elan vital explains the mysterious aliveness of living beings? Then what does this belief not allow to happen—what would definitely falsify this belief? A null answer means that your belief does not constrain experience; it permits anything to happen to you. It floats.
When you argue a seemingly factual question, always keep in mind which difference of anticipation you are arguing about. If you can't find the difference of anticipation, you're probably arguing about labels in your belief network—or even worse, floating beliefs, barnacles on your network. If you don't know what experiences are implied by Wulky Wilkinsen being a post-utopian, you can go on arguing forever. (You can also publish papers forever.)
Above all, don't ask what to believe—ask what to anticipate. Every question of belief should flow from a question of anticipation, and that question of anticipation should be the center of the inquiry. Every guess of belief should begin by flowing to a specific guess of anticipation, and should continue to pay rent in future anticipations. If a belief turns deadbeat, evict it.