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.”
If there’s a foundational skill in the martial art of rationality, a mental stance on which all other technique rests, it might be this one: the ability to spot, inside your own head, psychological signs that you have a mental map of something, and signs that you don’t.
Suppose that, after a tree falls, the two arguers 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; their maps of the world do not diverge in any sensory detail.
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 simplistically model 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 English professor teaches you that the famous writer Wulky Wilkinsen is actually a “retropositional author,” which you can tell because his books exhibit “alienated resublimation.” And perhaps your professor knows all this because their professor told them; but all they're able to say about resublimation is that it's characteristic of retropositional thought, and of retropositionality that it's marked by alienated resublimation. What does this mean you should expect from Wulky Wilkinsen’s 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 assertions that “Wulky Wilkinsen” has the “retropositionality” attribute and also the “alienated resublimation” attribute, so you can regurgitate them on the upcoming quiz. The two 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 retropositional author? Then what do you expect to see because of that? No, not “alienated resublimation”; 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 Élan 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 Wilkinsens writing being retropositional, you can go on arguing 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.
Yes, there are numerous cases where a field in "pure" mathematics proved interesting theorems that mathematicians undertook because of its challenging and elegant nature (like certain theorems possess generality and elegance) which were then to be found to be practically useful, which are called "applied" mathematics. Frankly, this distinction is blurred as pure mathematics are so useful (see Eugene Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences") that the abstract nature of mathematics has huge extensibility and general applications in multiple domains. For instance, Einstein's GR was based on the pure mathematics of Riemannian manifolds, which is an abstract topological structure, not tied to reality in any way initially. Or how algebraic topology is used for data mining, how number theory is used for cryptography, how linear algebra is used for machine learning, group theory is used for particle physics... and even how Bayesian probability theory is used for LW rationality.
Stephen Wolfram has great resources on rulial spaces and the nature of computation for the universe's fundamental ontology (the territory not the map) in which these networks of theorems can correspond to our empirical reality. (psst I am a very new LW user, and I am deciding if I should do a Sequence for this idea of "rulial cover" which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction, would be great if someone thinks this is interesting to explore so I can be motivated)
To link back to Eliezer's post, "floating beliefs" in a Bayesian net can be connected through adjusting the "weights" of the edges that connects that belief using Bayesian inference, and mathematics make these robust inferences from axioms (deductively validity as 100% in weight and 0% in prior). Therefore, anticipation becomes certain under a set of idealized axioms.