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ABrooks comments on Rationality Quotes April 2012 - Less Wrong
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It would surprise me, since no one could ever give me an example. I'm not sure what kind of evidence could give me good reason to think that there are thoughts that I cannot think.
Try visualizing four spacial dimensions.
Been there, done that. Advice to budding spatial-dimension visualizers: the fourth is the hardest, once you manage the fourth the next few are quite easy.
Is this legit and if so can you elaborate? I bet I'm not the only one here who has tried and failed.
Well, I can elaborate, but I'm not sure how helpful it will be. "No one can be told what the matrix is" and that sort of thing. The basic idea is that it's the equivalent of the line rising out of the paper in two-dimensions, but in three dimensions instead. But that's not telling someone who has tried and failed anything they don't know, I'm sure.
If you really want to be able to visualize higher-order spaces, my advice would be to work with them, do math and computer programming in higher-order spaces, and use that to build up physical intuitions of how things work in higher-order spaces. Once you have the physical intuitions it's easier for your brain to map them to something meaningful. Of course if your reason for wanting to be able to visualize 4D-space is because you want to use the visualization to give you physical intuitions about it that will be useful in math or computer programming, this is an ass-backward way of approaching the problem.
Is it like having a complete n-dimensional construct in your head that you can view in its entirety?
I can visualise 4-dimensional polyhedra, in much the same way I can draw non-planar graphs on a sheet of paper, but it's not what I imagine being able to visualise higher-dimensional objects to be like.
I used to be into Rubik's Cube, and it's quite easy for me to visualise all six faces of a 3D cube at once, but when visualising, say, a 4-octahedron, the graph is easy to visualise, (or draw on a piece of paper, for that matter), but I can only "see" one perspective of the convex hull at a time, with the rest of it abstracted away.
Just visualize n dimensions, and then set n = 4.
You might as well tell me to 'just' grow wings and fly away...
I believe wnoise was making a joke-- one that I thought was moderately funny.
I thought it might be, and if I'd read it elsewhere, I'd have been sure of it - but this is LessWrong, which is chock-full of hyperintelligent people whose abilities to do math, reason and visualize are close to superpowers from where I am. You people seriously intimidate me, you know. (Just because I feel you're so much out of my league, not for any other reason.)
It's a standard joke about mathematicians vs everybody else, and I intended it as such. I can do limited visualization in the 4th dimension (hypercubes and 5-cells (hypertetrahedra), not something as complicated as the 120-cell or even the 24-cell), but it's by extending from a 3-d visualization with math knowledge, rather than specializing n to 4.
For what it's worth, my ability to reason is fairly good in a very specific way-- sometimes I see the relevant thing quickly (and after LWers have been chewing on a problem and haven't seen it (sorry, no examples handy, I just remember the process)), but I'm not good at long chains of reasoning. Math and visualizing aren't my strong points.
Even better - play Snake in four spatial dimensions!
When I was 13 or so, my brains worked significantly better than they currently do, and I figured out an easy trick for that in a math class one day. Just assign a greyscale color value (from black to white) to each point! This is exactly like taking an usual map and coloring the hills a lighter shade and the low places a darker one.
The only problem with that is it's still "3.5D", like the "2.5D" graphics engine of Doom, where there's only one Z-value to any point in the world so things can't be exactly above or below each other.
To overcome this, you could theoretically imagine the 3D structure alternating between "levels" in the 4th dimension every second, so e.g. one second a 3D cube's left half is grey and its right half is white, indicating a surface "rising" in the 4th dimension, but every other second the right half changes to black while the left is still grey, showing a second surface which begins at the same place and "descends" in the 4th dimension. Voila, you have two 3D "surfaces" meeting at a 4D angle!
With RGB color instead of greyscale, one could theoretically visualize 6 dimensions in such a way.
Now, if only this let you rotate things through the 4th dimension.
Doing specific rotations by breaking it into steps is possible. Rotations by 90 degrees through the higher dimensions is doable with some effort -- it's just coordinate swapping after all. You can make checks that you got it right. Once you have this mastered, you can compose it with rotations that don't touch the higher dimensions. Then compose again with one of these 90 degree rotations, and you have an effective rotation through the higher dimensions.
(Understanding the commutation relations for rotation helps in this breakdown, of course. If you can then go on to understanding how the infinitesimal rotations work, you've got the whole thing down.)
I knew a guy who credibly claimed to be able to visualize 5 spacial dimensions. He is a genius math professor with 'autistic savant' tendencies.
I certainly couldn't pull it off and I suspect that at my age it is too late for me to be trained without artificial hardware changes.
The way I would do it for dimensions between d=4 and d=6 is to visualize a (d-3)-dimensional array of cubes. Then you remember that similarly positioned points, in the interior of cubes that are neighbors in the array, are near-neighbors in the extra dimensions (which correspond to the directions of the array). It's not a genuinely six-dimensional visualization, but it's a three-dimensional visualization onto which you can map six-dimensional properties. Then if you make an effort, you could learn how rotations, etc, map onto transformations of objects in the visualization. I would think that all claimed visualizations of four or more dimensions really amount to some comparable combinatorial scheme, backed up with some nonvisual rules of transformation and interpretation.
ETA: I see similar ideas in this subthread.
Am I allowed to use time/change dimensions? Because if so, the task is trivial (if computationally expensive).
Ok, now add a temporal dimension.
Adding multiple temporal dimensions effectively how I do it, so one more shouldn't be a problem*. I visualize a 3 dimensional object in an space with a reference point that can move in n perpendicular directions. As the point of reference moves through the space, the object's shape and size change.
Example: to visualize a 5-dimensional sphere, I first visualize a 3 dimensional sphere that can move along a 1 dimensional line. As the point of reference reaches the three-dimensional sphere, a point appears, and this point grows into a full sized sphere at the middle, then shrinks back down to a point. I then add another degree of freedom perpendicular to the first line, and repeat the procedure.
Rotations are still very hard for me to do, and become increasingly difficult with 5 or more dimensions. I think this is due to a very limited amount of short-term memory. As for my technique, I think it piggybacks on the ability to imagine multiple timelines simultaneously. So, alas, it's a matter of repurposing existing abilities, not constructing entirely new ones.
*up to 7: 3 of space, 3 of observer-space, and 1 of time
Either I can visualize them, and then they're thoughts I can think, or I can't visualize them, in which case the exercise doesn't help me.
If you can, replace 4 with N for sufficiently large N.
If you can't, imagine a creature that evolved in a 4-dimensional universe. I find it unlikely that it would not be able to visualize 4 dimensions.
There's a pretty serious gap between the idea of a person evolved to visualize four dimensions and it being capable of thoughts I cannot think. This might be defensible, but if so only in the context of certain thoughts, something like qualitative ones. But the original quote was inferring from the fact that not everyone can see all the colors to the idea that there are thoughts we cannot think. If 'colors I can't see' are the only kinds of things we can defend as thoughts that I cannot think, then the original quote is trivial.
So even if you can defend 4d visualizations as thoughts I cannot think, you'd have to extend your argument to something else.
But I have a question in return: how would the belief that there are thoughts you cannot think modify your anticipations? What would that look like?
By itself? Not much at all. The fun part is encountering another creature which can think those thoughts, then deducing the ability (and, being human, shortly thereafter finding some way to exploit it for personal gain) without being able to replicate the thoughts themselves.
Hinton cubes. I haven't tried them though.
ETA: Original source, online.
So if Majus's post (on Pinker) is correct, and the underling processing engine(s) (aka "the brain") determine the boundaries of what you can think about, then it is almost tautological that no one can give you an example since to date almost all folks have a very similar underlying architecture.
So what I argued was that thoughts are by nature commensurable: it's just in the nature of thoughts that any thinking system can think any thought from any other thinking system. There are exceptions to this, but these exceptions are always on the basis of limited resources, like limited memory.
So, an application of this view is that there are no incommensurable scientific schemes: we can in principle take any claim from any scientific paradigm and understand or test it in any other.
All I argued was that if their thesis is correct, then unless you've had some very odd experiences, no one can give you an example because everyone you meet is similarly bounded.
That is the limit of what my statement was intended to convey.
I don't know enough neurology, psychology and etc. to have a valid opinion, but I will note that we see at most 3 colors. We perceive many more. But any time we want to perceive, for example, the AM radio band we map it into a spectrum our eyes can handle, and as near as I can tell we "think" about it in the colors we perceive.
It is my understanding that there is some work in this area where certain parts of hte brain handle certain types of work. Folks with certain types of injuries or anomalous structures are unable to process certain types of input, and unable to do certain kinds of work. This seems to indicate that while our brain, as currently constructed, is a fairly decent tool for working out the problems we have in front of us, there is some evidence that it is not a general purpose thinking machine.
(in one of those synchronicity thingies my 5 year old just came up to me and showed me a picture of sound waves coming into an ear and molecules "traveling" into your nose).
The existence of other signals your brain simply doesn't process doesn't shift your prior at all?
For me, it merely brings it to the level of "interesting speculation". What observations would provide strong evidence that there be dragons? Other weak evidence that just leaves it at much the original level is the existence of anosognosia -- people with brain damage who appear to be unable to think certain thoughts about their affliction. But that doesn't prove anything about the healthy brain, any more than blindness proves the existence of invisible light.
Some people seem unable to grok mathematics, but then, some people do. The question is whether, Turing-completeness aside, the best current human thinking is understanding-complete, subject only to resource limitation.
That doesn't seem strictly relevant. Other signals might lead me to believe that there are thoughts I don't think (but I accepted that already), not thoughts I can't think. How could I recognize such a thing as a thought? After all, while every thought is a brain signal, not every brain signal is a thought: animals have lots of brain signals, but no thoughts.
Can you rotate four dimensional solids in your head?
Edit: it looks like I'm not the first to suggest this, but I'll add that since computers are capable not just of representing more than three spacial dimensions, but of tracking objects through them, these are probably "possible thoughts" even if no human can represent them mentally.
Well, suppose I'm colorblind from birth. I can't visualize green. Is this significantly different from the example of 4d rotations?
If so, how? (ETA: after all, we can do all the math associated with 4d rotations, so we're not deficient in conceptualizing them, just in imagining them. Arguably, computers can't visualize them either. They just do the math and move on).
If not, then is this the only kind of thought (i.e. visualizations, etc.) that we can defend as potentially unthinkable by us? If this is the only kind of thought thus defensible, then we've rendered the original quote trivial: it infers from the fact that it's possible to be unable to see a color that it's possible to be unable to think a thought. But if these kinds of visualizations are the only kinds of thoughts we might not be able to think, then the quote isn't saying anything.
If you discount inaccessible qualia, how about accurately representing the behaviors of subatomic particles in a uranium atom?
I'm not a physicist, but I have been taught that beyond the simplest atoms, the calculations become so difficult that we're unable to determine whether our quantum models actually predict the configurations we observe. In this case, we can't simply do the math and move on, because the math is too difficult. With our own mental hardware, it appears that we can neither visualize nor predict the behavior of particles on that scale, above a certain level of complexity, but that doesn't mean that a jupiter brain wouldn't be able to.
I'm not discounting qualia (that's it's own discussion), I'm just saying that if these are the only kinds of thoughts which we can defend as being potentially unthinkable by us, then the original quote is trivial.
So one strategy you might take to defend thoughts we cannot think is this: thinking is or supervenes on a physical process, and thus it necessarily takes time. All human beings have a finite lifespan. Some thought could be formulated such that the act of thinking it with a human brain would take longer than any possible lifespan, or perhaps just an infinite amount of time. Therefore, there are thoughts we cannot think.
I think this suggestion is basically the same as yours: what prevents us from thinking this thought is some limited resources, like memory or lifespan, or something like that. Similarly, I could suggest a language that is in principle untranslatable, just because all well formed sentences and clauses in that language are long enough that we couldn't remember a whole one.
But it would be important to distinguish, in these cases, between two different kinds of unthinkability or untranslatability. Both the infinite (or just super complex) thoughts and the super long sentences are translatable into a language we can understand, in principle. There's nothing about those thoughts or sentences, or our thoughts or sentences, that makes them incompatible. The incompatibility arises from a fact about our biology. So in the same line, we could say that some alien species' language is untranslatable because they speak and write in some medium we don't have the technology to access. The problem there isn't with the language or the act of translation.
In sum, I think that this suggestion (and perhaps the original quote) trades on an equivocation between two different kinds of unthinkability. But if the only defensible kind of unthinkability is one on the basis of some accidental limitation of access or resources, then I can't see what's interesting about the idea. It's no more interesting then than the point that I can't speak Chinese because I haven't learned it.
What is the difference between a thought you can't think and one you don't think?
Well, for example I don't think very much about soccer. There are thoughts about who the best soccer team is that I simply don't ever think. But I can think them.
Another case: In two different senses of 'can', I can and can't understand Spanish. I can't understand it at the moment, but nevertheless Spanish sentences are in principle translatable into sentences I can understand. I also can't read Aztec hieroglyphs, and here the problem is more serious: no one knows how to read them. But nevertheless, insofar as we assume they are a form of language, we assume that we could translate them given the proper resources. To see something as translatable just is to see it as a language, and to see something as a language is to see it as translatable. Anything which was is in principle untranslatable just isn't recognizable as a language.
I think the point is analogous (and that's no accident) with thoughts. Any thought that I couldn't think by any means is something I cannot by any means recognize as a thought in the first place. All this is just a way of saying that the belief that there are thoughts you cannot think is one of those beliefs that could never modify your anticipations. That should be enough to discount it as a serious consideration.
And yet, if I see two nonhuman life forms A1 and A2, both of which are performing something I classify as the same task but doing it differently, and A1 and A2 interact, after which they perform the task the same way, I would likely infer that thoughts had been exchanged between them, but I wouldn't be confident that the thoughts which had been exchanged were thoughts that could be translated to a form that I could understand.
Alternative explanations include:
These are, of course, not exhaustive.
You could call some these cases a kind of thought. Maybe to self-modifying programs, a blackbox executable algorithm counts as a thought; or maybe to beings who use the same information storage for genes and minds, lateral gene transfer counts as a thought.
But this is really just a matter of defining what the word "thought" may refer to. I can define it to include executable undocumented Turing Machines, which I don't think humans like us can "think". Or you could define it as something that, after careful argument, reduces to "whatever humans can think and no more".
Sure. Leaving aside what we properly attach the label "thought" to, the thing I'm talking about in this context is roughly speaking the executed computations that motivate behavior. In that sense I would accept many of these options as examples of the thing I was talking about, although option 2 in particular is primarily something else and thus somewhat misleading to talk about that way.
I think you're accepting and then withdrawing a premise here: you've identified them as interacting, and you've identified their interaction as being about the task at hand, and the ways of doing it, and the relative advantages of these ways. You've already done a lot of translation right there. So the set up of your problem assumes not only that you can translate their language, but that you in some part already have. All that's left, translation wise, is a question of precision.
Sure, to some level of precision, I agree that I can think any thought that any other cognitive system, however alien, can think. There might be a mind so alien that the closest analogue to its thought process while contemplating some event that I can fathom is "Look at that, it's really interesting in some way," but I'll accept that this in some part a translation and "all that's left" is a question of precision.
But if you mean to suggest by that that what's left is somehow negligible, I strenuously disagree. Precision matters. If my dog and I are both contemplating a ball, and I am calculating the ratio between its volume and surface, and my dog is wondering whether I'll throw it, we are on some level thinking the same thought ("Oh, look, a ball, it's interesting in some way") but to say that my dog therefore can understand what I'm thinking is so misleading as to be simply false.
I consider it possible for cognitive systems to exist that have the same relationship to my mind in some event that my mind has to my dog's mind in that example.
Well, I don't think I even implied that the dog could understand what you're thinking. I don't think dogs can think at all. What I'm claiming is that for anything that can think (and thus entertain the idea of thoughts that cannot be thought), there are no thoughts that cannot be thought. The difference between you and your dog isn't just one of raw processing power. It's easy to imagine a vastly more powerful processor than a human brain that is nevertheless incapable of thought (I think Yud.'s suggestion for an FAI is such a being, given that he's explicit that it would not rise to the level of being a mechanical person).
Once we agree that it's a point about precision, I would just say that this ground can always in principle be covered. Suppose the translation has gotten started, such that there is some set of thoughts at some level of precision that is translatable, call it A, and the terra incognito that remains, call it B. Given that the cognitive system you're trying to translate can itself translate between A and B (the aliens understand themselves perfectly), there should be nothing barring you from doing so as well.
You might need extremely complex formulations of the material in A to capture anything in B, but this is allowed: we need some complex sentence to capture what the Germans mean by 'schadenfreude', but it would be wrong to think that because we don't have a single term which corresponds exactly, that we cannot translate or understand the term to just the same precision the Germans do.