VKS comments on Rationality Quotes April 2012 - Less Wrong

4 Post author: Oscar_Cunningham 03 April 2012 12:42AM

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Comment author: VKS 04 April 2012 10:23:55AM 32 points [-]

Just as there are odors that dogs can smell and we cannot, as well as sounds that dogs can hear and we cannot, so too there are wavelengths of light we cannot see and flavors we cannot taste. Why then, given our brains wired the way they are, does the remark, "Perhaps there are thoughts we cannot think," surprise you?

  • Richard Hamming
Comment author: majus 10 April 2012 11:31:02PM 5 points [-]

In Pinker's book "How the Mind Works" he asks the same question. His observation (as I recall) was that much of our apparently abstract logical abilities are done by mapping abstractions like math onto evolved subsystems with different survival purposes in our ancestors: pattern recognition, 3D spatial visualization, etc. He suggests that some problems seem intractable because they don't map cleanly to any of those subsystems.

Comment author: [deleted] 04 April 2012 08:19:04PM 0 points [-]

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.

Comment author: BillyOblivion 17 April 2012 12:22:57PM 1 point [-]

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.

Comment author: [deleted] 17 April 2012 01:59:34PM 0 points [-]

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.

Comment author: BillyOblivion 21 April 2012 12:20:21AM 2 points [-]

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).

Comment author: Eugine_Nier 05 April 2012 01:07:32AM 3 points [-]

Try visualizing four spacial dimensions.

Comment author: CronoDAS 13 April 2012 08:13:41AM 2 points [-]
Comment author: wedrifid 13 April 2012 07:50:40AM 0 points [-]

Try visualizing four spacial dimensions.

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.

Comment author: Mitchell_Porter 13 April 2012 08:00:18AM *  1 point [-]

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.

Comment author: faul_sname 13 April 2012 06:43:01AM *  0 points [-]

Am I allowed to use time/change dimensions? Because if so, the task is trivial (if computationally expensive).

Comment author: Eugine_Nier 13 April 2012 06:56:12AM *  1 point [-]

Ok, now add a temporal dimension.

Comment author: faul_sname 13 April 2012 08:17:08AM *  0 points [-]

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

Comment author: Multiheaded 07 April 2012 08:26:10PM *  2 points [-]

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.

Comment author: Eugine_Nier 08 April 2012 07:08:54AM 3 points [-]

Now, if only this let you rotate things through the 4th dimension.

Comment author: wnoise 12 April 2012 07:31:23PM 5 points [-]

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.)

Comment author: [deleted] 05 April 2012 02:07:35PM *  0 points [-]

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.

Comment author: Eugine_Nier 06 April 2012 03:03:53AM 1 point [-]

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.

Comment author: [deleted] 06 April 2012 01:57:25PM 0 points [-]

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?

Comment author: Strange7 12 April 2012 08:41:26AM 1 point [-]

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.

Comment author: RichardKennaway 05 April 2012 06:31:10AM *  0 points [-]

Hinton cubes. I haven't tried them though.

ETA: Original source, online.

Comment author: Nominull 05 April 2012 03:03:01AM 5 points [-]

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.

Comment author: tgb 05 April 2012 06:29:37PM 2 points [-]

Is this legit and if so can you elaborate? I bet I'm not the only one here who has tried and failed.

Comment author: Nominull 05 April 2012 07:12:59PM 5 points [-]

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.

Comment author: sixes_and_sevens 05 April 2012 11:18:46PM 5 points [-]

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.

Comment author: wnoise 05 April 2012 02:03:08AM 11 points [-]

Just visualize n dimensions, and then set n = 4.

Comment author: bbleeker 05 April 2012 12:40:14PM *  1 point [-]

You might as well tell me to 'just' grow wings and fly away...

Comment author: NancyLebovitz 05 April 2012 03:14:33PM 3 points [-]

I believe wnoise was making a joke-- one that I thought was moderately funny.

Comment author: bbleeker 06 April 2012 07:59:59AM 4 points [-]

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.)

Comment author: wnoise 12 April 2012 07:40:17PM 4 points [-]

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.

Comment author: NancyLebovitz 06 April 2012 01:20:54PM 0 points [-]

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.

Comment author: Desrtopa 04 April 2012 11:31:58PM 1 point [-]

I'm not sure what kind of evidence could give me good reason to think that there are thoughts that I cannot think.

The existence of other signals your brain simply doesn't process doesn't shift your prior at all?

Comment author: [deleted] 05 April 2012 02:06:40PM 0 points [-]

The existence of other signals your brain simply doesn't process doesn't shift your prior at all?

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.

Comment author: Desrtopa 06 April 2012 01:40:33PM *  1 point [-]

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.

Comment author: [deleted] 06 April 2012 02:02:07PM *  0 points [-]

Can you rotate four dimensional solids in your head?

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.

Comment author: Desrtopa 06 April 2012 02:21:03PM 1 point [-]

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.

Comment author: [deleted] 06 April 2012 03:19:18PM 1 point [-]

If you discount inaccessible qualia, how about accurately representing the behaviors of subatomic particles in a uranium atom?

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.

Comment author: LucasSloan 05 April 2012 08:36:07PM 1 point [-]

What is the difference between a thought you can't think and one you don't think?

Comment author: [deleted] 05 April 2012 09:06:42PM *  0 points [-]

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.

Comment author: TheOtherDave 05 April 2012 10:35:30PM 1 point [-]

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.

Comment author: DanArmak 23 April 2012 12:27:34PM *  0 points [-]

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:

  • They exchanged genetic material, like bacteria, or outright code, like computer programs; which made them behave more similarly.
  • They are programs, one attacked the other, killed it and replaced its computational slot with a copy of itself.
  • A1 gave A2 a copy of its black-box decision maker which both now use to determine their behavior in this situation. However, neither of them understands the black box's decision algorithm on the level of their own conscious thoughts; and the black box itself is not sentient or alive and has no thoughts.
  • One of them observed the other was more efficient and is now emulating its behavior, but they didn't talk about it ("exchange thoughts"), just looked at one another.

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".

Comment author: TheOtherDave 23 April 2012 12:56:08PM 1 point [-]

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.

Comment author: [deleted] 06 April 2012 01:50:23PM 0 points [-]

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.

Comment author: TheOtherDave 06 April 2012 02:08:16PM 3 points [-]

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.

Comment author: [deleted] 06 April 2012 02:59:22PM *  1 point [-]

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.

Comment author: RichardKennaway 05 April 2012 06:37:17AM *  1 point [-]

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.

Comment author: Eliezer_Yudkowsky 04 April 2012 08:10:47PM 25 points [-]

It surprises people like Greg Egan, and they're not entirely stupid, because brains are Turing complete modulo the finite memory - there's no analogue of that for visible wavelengths.

Comment author: Dmytry 05 April 2012 04:58:54PM *  0 points [-]

There's the halting problem, so here you go. There's also the thoughts that you'll never arrive at because your arriver at the thoughts won't reach them, even if you could think them if told of them.

Comment author: AspiringKnitter 05 April 2012 06:05:40AM 23 points [-]

If this weren't Less Wrong, I'd just slink away now and pretend I never saw this, but:

I don't understand this comment, but it sounds important. Where can I go and what can I read that will cause me to understand statements like this in the future?

Comment author: Viliam_Bur 05 April 2012 09:15:23AM *  31 points [-]

When speaking about sensory inputs, it makes sense to say that different species (even different individuals) have different ranges, so one can percieve something and other can't.

With computation it is known that sufficiently strong programming languages are in some sense equal. For example, you could speak about relative advantages of Basic, C/C++, Java, Lisp, Pascal, Python, etc., but in each of these languages you can write a simulator of the remaining ones. This means that if an algorithm can be implemented in one of these languages, it can be implemented in all of them -- in worst case, it would be implemented as a simulation of another language running its native implementation.

There are some technical details, though. Simulating another program is slower and requires more memory than the original program. So it could be argued that on a given hardware you could do a program in language X which uses all the memory and all available time, so it does not necessarily follow that you can do the same program in language Y. But on this level of abstraction we ignore hardware limits. We assume that the computer is fast enough and has enough memory for whatever purpose. (More precisely, we assume that in available time a computer can do any finite number of computation steps; but it cannot do an infinite number of steps. The memory is also unlimited, but in a finite time you can only manage to use a finite amount of memory.)

So on this level of abstraction we only care about whether something can or cannot be implemented by a computer. We ignore time and space (i.e. speed and memory) constraints. Some problems can be solved by algorithms, others can not. (Then, there are other interesting levels of abstraction which care about time and space complexity of algorithms.)

Are all programming languages equal in the above sense? No. For example, although programmers generally want to avoid infinite loops in their programs, if you remove a potential for infinite loops from the programming language (e.g. in Pascal you forbid "while" and "repeat" commands, and a possibility to call functions recursively), you lose ability to simulate programming languages which have this potential, and you lose ability to solve some problems. On the other hand, some universal programming languages seem extremely simple -- a famous example is a Turing machine. This is very useful, because it is easier to do mathematical proofs about a simple language. For example if you invent a new programming language X, all you have to do to prove its universality, is to write a Turing machine simulator, which is usually very simple.

Now back to the original discussion... Eliezer suggests that brain functionality should be likened to computation, not to sensory input. A human brain is computationally universal, because (given enough time, pen and paper) we can simulate a computer program, so all brains should be equal when optimally used (differing only in speed and use of resources). In another comment he adds that ability to compute isn't the same as ability to understand. Therefore (my conclusion) what one human can understand, another human can at least correctly calculate without understanding, given a correct algorithm.

Comment author: AspiringKnitter 05 April 2012 07:51:31PM 6 points [-]

Wow. That's really cool, thank you. Upvoted you, jeremysalwen and Nornagest. :)

Could you also explain why the HPMoR universe isn't Turing computable? The time-travel involved seems simple enough to me.

Comment author: thomblake 05 April 2012 08:57:48PM 7 points [-]

Not a complete answer, but here's commentary from a ffdn review of Chapter 14:

Kevin S. Van Horn
7/24/10 . chapter 14
Harry is jumping to conclusions when he tells McGonagall that the Time-Turner isn't even Turing computable. Time travel simulation is simply a matter of solving fixed-point equation f(x) = x. Here x is the information sent back in time, and f is a function that maps the information received from the future to the information that gets sent back in time. If a solution exists at all, you can find it to any desired degree of accuracy by simply enumerating all possible rational values of x until you find one that satisfies the equation. And if f is known to be both continuous and have a convex compact range, then the Brouwer fixed-point theorem guarantees that there will be a solution.

So the only way I can see that simulating the Time-Turner wouldn't be Turing computable would be if the physical laws of our universe give rise to fixed-point equations that have no solutions. But the existence of the Time-Turner then proves that the conditions leading to no solution can never arise.

Comment author: johnswentworth 09 April 2012 10:52:42PM 3 points [-]

There's also the problem of an infinite number of possible solutions.

Comment author: faul_sname 13 April 2012 05:49:00AM 0 points [-]

The number of solutions is finite but (very, very, mind-bogglingly) large.

Comment author: Nick_Tarleton 06 April 2012 02:04:48AM 7 points [-]

I got the impression that what "not Turing-computable" meant is that there's no way to only compute what 'actually happens'; you have to somehow iteratively solve the fixed-point equation, maybe necessarily generating experiences (waves hands confusedly) corresponding to the 'false' timelines.

Comment author: tgb 10 April 2012 11:29:12PM 2 points [-]

Sounds rather like our own universe, really.

Comment author: AspiringKnitter 05 April 2012 11:14:25PM 2 points [-]

Ah. It's math.

:) Thanks.

Comment author: Nornagest 05 April 2012 06:42:10AM *  3 points [-]

A computational system is Turing complete if certain features of its operation can reproduce those of a Turing machine, which is a sort of bare-bones abstracted model of the low-level process of computation. This is important because you can, in principle, simulate the active parts of any Turing complete system in any other Turing complete system (though doing so will be inefficient in a lot of cases); in other words, if you've got enough time and memory, you can calculate anything calculable with any system meeting a fairly minimal set of requirements. Thanks to this result, we know that there's a deep symmetry between different flavors of computation that might not otherwise be obvious. There are some caveats, though: in particular, the idealized version of a Turing machine assumes infinite memory.

Now, to answer your actual question, the branch of mathematics that this comes from is called computability theory, and it's related to the study of mathematical logic and formal languages. The textbook I got most of my understanding of it from is Hopcroft, Motwani, and Ullman's Introduction to Automata Theory, Languages, and Computation, although it might be worth looking through the "Best Textbooks on Every Subject" thread to see if there's a consensus on another.

Comment author: MarkusRamikin 05 April 2012 07:06:07AM 0 points [-]

infinite memory space

Curious, does "memory space" mean something more than just "memory"?

Comment author: wedrifid 05 April 2012 10:05:50AM *  7 points [-]

infinite memory space

Curious, does "memory space" mean something more than just "memory"?

Just a little more specific. Some people may hear "memory" and associate it with, say, the duration of their memory rather than how many can be physically held. For example when a human is said to have a 'really good memory' we don't tend to be trying to make a claim about the theoretical maximum amount of stuff they could remember.

Comment author: Nornagest 05 April 2012 07:11:42AM *  2 points [-]

No, although either or both might be a little misleading depending on what connotations you attach to it: an idealized Turing machine stores all its state on a rewritable tape (or several tapes, but that's equivalent to the one-tape version) of symbols that's infinite in both directions. You could think of that as analogous to both memory and disk, or to whatever the system you're actually working with uses for storage.

Comment author: MarkusRamikin 05 April 2012 07:13:35AM *  0 points [-]

Right, I know that. Was just curious why the extra verbiage in a post meant to explain something.

Comment author: Nornagest 05 April 2012 07:15:02AM 1 point [-]

Because it's late and I'm long-winded. I'll delete it.

Comment author: jeremysalwen 05 April 2012 06:18:04AM 2 points [-]
Comment author: Vaniver 04 April 2012 08:35:43PM 6 points [-]

brains are Turing complete modulo the finite memory

What does that statement mean in the context of thoughts?

That is, when I think about human thoughts I think about information processing algorithms, which typically rely on hardware set up for that explicit purpose. So even though I might be able to repurpose my "verbal manipulation" module to do formal logic, that doesn't mean I have a formal logic module.

Any defects in my ability to repurpose might be specific to me: I might able to think the thought "A-> B, ~A, therefore ~B" with the flavor of trueness, and another person can only think that thought with the flavor of falseness. If the truth flavor is as much a part of the thought as the textual content, then the second thinker cannot think the thought that the first thinker can.

Aren't there people who can hear sounds but not music? Are their brains not Turing complete? Are musical thoughts ones they cannot think?

Comment author: Eliezer_Yudkowsky 04 April 2012 09:33:35PM 14 points [-]

It means nothing, although Greg Egan is quite impressed by it. Sad but true: Someone with an IQ of, say, 90 can be trained to operate a Turing machine, but will in all probability never understand matrix calculus. The belief that Turing-complete = understanding-complete is false. It just isn't stupid.

Comment author: JulianMorrison 20 April 2012 02:41:23PM -1 points [-]

It means you could, in theory, run an AI on them (slowly).

Comment author: komponisto 05 April 2012 09:57:52PM 3 points [-]

[That human brains are Turing-complete] means nothing, although Greg Egan is quite impressed by it. Sad but true: Someone with an IQ of, say, 90 can be trained to operate a Turing machine, but will in all probability never understand matrix calculus.

It doesn't mean nothing; it means that people (like machines) can be taught to do things without understanding them.

(They can also be taught to understand, provided you reduce understanding to Turing-machine computations, which is harder. "Understanding that 1+1 = 2" is not the same thing as being able to output "2" to the query "1+1=".)

Comment author: Elithrion 05 April 2012 09:39:20PM 1 point [-]

I would imagine that he can be taught matrix calculus, given sufficient desire (on his and the teachers' parts), teaching skill, and time. I'm not sure if in practice it is possible to muster enough desire or time to do it, but I do think that understanding is something that can theoretically be taught to anyone who can perform the mechanical calculations.

Comment author: DanArmak 23 April 2012 01:38:19PM 1 point [-]

I can't imagine how hard it is to learn to program if you don't instinctively know how. Yet I know it is that hard for many people. Some succeed in learning, some don't. Those who do still have big differences in ability, and ability at a young age seems to be a pretty good predictor of lifetime ability.

I realize I must have learned the basics at some point, although I don't remember it. And I remember learning many more advanced concepts during the many years since. But for both the basics and the advanced subjects, I never experienced anything I can compare to what I'd call "learning" in other subjects I studied.

When programming, if I see/read something new, I may need some time (seconds or hours) to understand it, then once I do, I can use it. It is cognitively very similar to seeing a new room for the first time. It's novel, but I understand it intuitively and in most cases quickly.

When I studied e.g. biology or math at university, I had to deliberately memorize, to solve exercises before understanding the "real thing", to accept that some things I could describe I couldn't duplicate by building them from scratch no matter how much time I had and what materials and tools. This never happened to me in programming. I may not fully understand the domain problem that the program is manipulating. But I always understand the program itself.

And yet I've seen people struggle to understand the most elementary concepts of programming, like, say, distinguishing between names and values. I've had to work with some pretty poor programmers, and had the official job of on-the-job mentoring newbies on two occasions. I know it can be very difficult to teach effectively, it can be very difficult to learn.

Given that I encountered a heavily preselected set of people, who were trying to make programming their main profession, it's easy for me to believe that - at the extreme - for many people elementary programming is impossible to learn, period. And the same should apply to math and any other "abstract" subject for which biologically normal people don't have dedicated thinking modules in their brains.

Comment author: David_Gerard 08 April 2012 09:12:38AM *  8 points [-]

I fear you're committing the typical mind fallacy. The dyscalculic could simulate a Turing machine, but all of mathematics, including basic arithmetic, is whaargarbl to them. They're often highly intelligent (though of course the diagnosis is "intelligent elsewhere, unintelligent at maths"), good at words and social things, but literally unable to calculate 17+17 more accurately than "somewhere in the twenties or thirties" or "I have no idea" without machine assistance. I didn't believe it either until I saw it.

Comment author: TheOtherDave 08 April 2012 02:38:45PM 0 points [-]

Do you find this harder to believe than, say, aphasia? I've never seen it, but I have no difficulty believing it.

Comment author: David_Gerard 08 April 2012 03:19:46PM 0 points [-]

Well, I certainly don't disbelieve in it now. I first saw it at eighteen, in first-year psychology, in the bit where they tried to beat basic statistics into our heads.

Comment author: Eliezer_Yudkowsky 05 April 2012 09:43:25PM 12 points [-]

Have you ever tried to teach math to anyone who is not good at math? In my youth I once tutored a woman who was poor, but motivated enough to pay $40/session. A major obstacle was teaching her how to calculate (a^b)^c and getting her to reliably notice that minus times minus equals plus. Despite my attempts at creative physical demonstrations of the notion of a balanced scale, I couldn't get her to really understand the notion of doing the same things to both sides of a mathematical equation. I don't think she would ever understand what was going on in matrix calculus, period, barring "teaching methods" that involve neural reprogramming or gain of additional hardware.

Comment author: NancyLebovitz 24 April 2012 07:45:18AM 1 point [-]

What was your impression of her intelligence otherwise?

Suzette Haden Elgin (a science fiction author and linguist who was quite intelligent with and about words) described herself as intractably bad at math.

Comment author: DanArmak 23 April 2012 01:15:54PM *  0 points [-]

This anecdote gives very little information on its own. Can you describe your experience teaching math to other people - the audience, the investment, the methods, the outcome? Do you have any idea whether that one woman eventually succeeded in learning some of what you couldn't teach her, and if so, how?

(ETA: I do agree with the general argument about people who are not good at math. I'm only saying this particular story doesn't tell us much about that particular woman, because we don't know how good you are at teaching, etc.)

Comment author: matt 13 April 2012 03:50:38AM *  14 points [-]

Your claim is too large for the evidence you present in support of it.

Teaching someone math who is not good at math is hard, but "will in all probability never understand matrix calculus"!? I don't think you're using the Try Harder.

Assume teaching is hard (list of weak evidence: it's a three year undergraduate degree; humanity has hardly allowed itself to run any proper experiments in the field, and those that have been run seem usually to be generally ignored by professional practitioners; it's massively subject to the typical mind fallacy and most practitioners don't know that fallacy exists). That you, "in your youth" (without having studied teaching), "once" tutored a woman who you couldn't teach very well… doesn't support any very strong conclusion.

It seems very likely to me that Omega could teach matrix calculus to someone with IQ 90 given reasonable time and motivation from the student. One of the things I'm willing to devote significant resources to in the coming years is making education into a proper science. Given the tools of that proper science I humbly submit that you could teach your former student a lot. Track the progress of the Khan Academy for some promising developments in the field.

Comment author: DanArmak 23 April 2012 01:13:30PM 1 point [-]

list of weak evidence

Some of it is weak evidence for the hardness claim (3 years degree), some against (all the rest). Does that match what you meant?

Comment author: matt 24 April 2012 07:28:04AM *  1 point [-]

I'd intended a different meaning of "hard". On reflection your interpretation seems a very reasonable inference from what I wrote.

What I meant: Teaching is hard enough that you shouldn't expect to find it easy without having spent any time studying it. Even as a well educated westerner, the bits of teaching you can reasonably expect to pick up won't take you far down the path to mastery.

(Thank you for you comment - it got me thinking.)

Comment author: wedrifid 13 April 2012 05:37:18AM 4 points [-]

humanity has hardly allowed itself to run any proper experiments in the field, and those that have been run seem usually to be generally ignored by professional practitioners

What are the experiments that are generally ignored?

Comment author: Elithrion 05 April 2012 11:52:31PM 5 points [-]

No, I haven't, and reading your explanation I now believe that there is a fair chance you are correct. However, one problem I have with it is that you're describing a few points of frustration, some of which I assume you ended up overcoming. I am not entirely convinced that had she spent, say one hundred hours studying each skill that someone with adequate talent could fully understand in one, she would not eventually fully understand it.

In cases of extreme trouble, I can imagine her spending forty hours working through a thousand examples, until mechanically she can recognise every example reasonably well, and find the solution correctly, then another twenty working through applications, then another forty hours analysing applications in the real world until the process of seeing the application, formulating the correct problem, and solving it becomes internalised. Certainly, just because I can imagine it doesn't make it true, but I'm not sure on what grounds I should prefer the "impossibility" hypothesis to the "very very slow learning" hypothesis.

Comment author: Incorrect 05 April 2012 09:54:53PM *  5 points [-]

I can't imagine how hard it would be to learn math without the concept of referential transparency.

Comment author: MixedNuts 09 April 2012 11:52:01AM 0 points [-]

Not all that hard if that's the only sticking point. I acquired it quite late myself.

Comment author: thomblake 05 April 2012 02:36:48AM 1 point [-]

The belief that Turing-complete = understanding-complete is false. It just isn't stupid.

I'm not sure what you mean by understanding-complete, but remember that the turing-complete system is both the operator and any machinery they are manipulating.

Comment author: Incorrect 05 April 2012 02:17:17AM 1 point [-]

So you are considering a man in a Chinese room to lack understanding?

Comment author: J_Taylor 05 April 2012 02:37:41AM 13 points [-]

Obviously the man in the Chinese room lacks understanding, by most common definitions of understanding. It is the room as a system which understands Chinese. (Assuming lookup tables can understand. By functional definitions, they should be able to.)

Comment author: Incorrect 05 April 2012 12:09:57PM 0 points [-]

But with a person it becomes a bit more complicated because it depends on what we are referring to when we say their name. I was trying to make an allusion to Blindsight.

Comment author: Will_Newsome 04 April 2012 09:13:02PM *  6 points [-]

Aren't there people who can hear sounds but not music?

FWIW I've read a study that says about 50% of people can't tell the difference between a major and a minor chord even when you label them happy/sad. [ETA: Happy/sad isn't the relevant dimension, see the replies to this comment.] I have no idea how probable that is, but if true it would imply that half of the American population basically can't hear music.

Comment author: khafra 06 April 2012 01:21:30PM 0 points [-]

At first, I found it unbelievable. Then, I remembered that I have imperfect perfect pitch: I learned both piano and french horn; the latter of which is transposed up a perfect fourth. Especially when I'm practicing regularly, I can usually name a note or simple chord when I hear it; but I'm often off by a perfect fourth.

Introspecting on the difference between being right about a note and wrong about a note makes me believe people can confuse major and minor, but still enjoy music.

Comment author: Bluehawk 05 April 2012 11:32:31PM 0 points [-]

Might have something to do with the fact that happy/sad is neither an accurate nor an encompassing description of the uses of major/minor chords, unless you place a C major and a C or A minor directly next to each other. I for one find that when I try to tell the difference solely on that basis, I might as well flip a coin and my success rate would go down only slightly. When I come at it from other directions and ignore the emotive impact, my success rate is much higher.

In short: Your conclusion doesn't follow from the evidence.

Comment author: Will_Newsome 06 April 2012 12:15:46AM 1 point [-]

I stated the evidence incorrectly, look at the uncle/aunt of your comment (if you haven't already) for the actual evidence.

Comment author: Bluehawk 07 April 2012 09:54:11PM 0 points [-]

Yeah, I spotted that after making my comment, but after that I wasn't sure whether you were citing the same source material or no. The actual evidence does say a lot more about how humans (don't?) perceive musical sounds. Thanks for clarifying, though.

Comment author: [deleted] 05 April 2012 04:05:43PM 16 points [-]

http://languagelog.ldc.upenn.edu/nll/?p=2074

It shocked the hell out of me, too.

Comment author: arundelo 05 April 2012 09:55:56PM *  0 points [-]

I've had between a dozen and two dozen music students over the years. (Guitar and bass guitar.) Some of them started out having trouble telling the difference between ascending and descending intervals. (In other words, some of them had bad ears.) All of them improved, and all of them, with practice, were able to hear me play something and play it back by ear. I'm sure there are some people who are neurologically unable to do this, but in general, it is a learnable skill.

The cognitive fun! website has a musical interval exercise.

Edit: One disadvantage to that exercise/game for people who aren't already familiar with the intervals is that it doesn't have you differentiate between major and minor intervals. (So if you select e.g. 2 and 8 as your intervals, you'll be hearing three different intervals, because some of the 2nds will be minor rather than major.) Sooner or later I'll write my own interval game!

Comment author: alex_zag_al 06 April 2012 01:26:55AM *  2 points [-]

is this what you're looking for?

http://www.musictheory.net/exercises/ear-interval

Comment author: arundelo 06 April 2012 01:33:25AM 0 points [-]

That's pretty cool. Are there keybindings?

Comment author: alex_zag_al 06 April 2012 04:02:24PM 0 points [-]

I don't know, doesn't look like it.

Comment author: Dmytry 05 April 2012 04:55:31PM *  4 points [-]

This is weird. It is hard for me to hear the difference in the cadence, but crystal clear otherwise. In the cadence, the problem for me is that the notes are dragging on, like when you press pedal on piano a bit, that makes it hard to discern the difference.

Maybe they lost something in retelling here? Made up new stimuli for which it doesn't work because of harmonics or something?

Or maybe its just me and everyone on this thread? I have a lot of trouble hearing speech through noise (like that of flowing water), i always have to tell others, i am not hearing what you're saying i am washing the dishes. Though i've no idea how well other people can hear something when they are washing the dishes; maybe i care too much not to pretend to listen when i don't hear.

This needs proper study.

Comment author: arundelo 22 April 2012 04:56:24PM 0 points [-]

I added another recording. See "Edit 2" in this comment for an explanation.

Comment author: Scottbert 09 April 2012 03:12:11PM 3 points [-]

Ditto for me -- The difference between the two chords is crystal clear, but in the cadence I can barely hear it.

I'm not a professional, but I sang in school chorus for 6 years, was one of the more skilled singers there, I've studied a little musical theory, and I apparently have a lot of natural talent. And the first time I heard the version played in cadence I didn't notice the difference at all. Freaky. I know how that post-doc felt when she couldn't hear the difference in the chords.

Comment author: arundelo 05 April 2012 11:22:03PM *  4 points [-]

The following recordings are played on an acoustic instrument by a human (me), and they have spaces in between the chords. The chord sequences are randomly generated (which means that the major-to-minor ratio is not necessarily 1:1, but all of them do have a mixture of major and minor chords).

Each of the following two recordings is a sequence of eight C major or C minor chords:

Each of the following two recordings is a sequence of eight "cadences" -- groups of four chords that are either

F B♭ C F

or

F B♭ Cminor F

Edit: Here's a listing of the chords in all four sound files.

Edit 2 (2012-Apr-22): I added another recording that contains these chords:

F B♭ C F
F B♭ Cmi F

repeated over and over, while the balance between the voices is varied, from "all voices roughly equal" to "only the second voice from the top audible". The second voice from the top is the only one that is different on the C minor chord. My idea is that hearing the changing voice foregrounded from its context like this might make it easier to pick it out when it's not foregrounded.

Comment author: arundelo 05 April 2012 09:36:22PM 0 points [-]

Maybe they lost something in retelling here? Made up new stimuli for which it doesn't work because of harmonics or something?

Nope, the audio examples are all straightforward realizations of the corresponding music notation. (They are easy for me to tell apart.)

Comment author: Dmytry 05 April 2012 11:20:43PM *  0 points [-]

Still, the notes drag on, the notes have harmonics, etc. It is not pure sine waves that abruptly stop and give time for the ear to 'clear' of afterimage-like sound.

I hear the difference in the cadence, it's just that I totally can't believe it can possibly be clearer than just the one chord then another chord. I can tell apart just the two chords at much lower volume level and/or paying much less attention.

Comment author: tgb 05 April 2012 06:25:54PM 1 point [-]

I am with you on easily telling the two apart in the original chords but being unable to reliably tell the difference in the cadence version.

Comment author: wedrifid 05 April 2012 04:20:43PM 0 points [-]

http://languagelog.ldc.upenn.edu/nll/?p=2074

It shocked the hell out of me, too.

Likewise.

Comment author: TheOtherDave 05 April 2012 04:31:37PM 3 points [-]

I was going to comment about how the individual chords were clearly different to my ear but the "stereotypical I-IV-V-I cadential sequences" were indistinguishable, precisely the reverse of the experience the Bell Labs post doc reportedly reported. Then I read the comments on the article and realized this is fairly common, so I deleted the comment. Then I decided to comment on it anyway. Now I have.

Comment author: wedrifid 05 April 2012 04:42:50PM 2 points [-]

I had to listen to that second part several times before I could pick up the difference too. They sound equivalent unless I concentrate.

Comment author: Dmytry 05 April 2012 05:02:44PM *  1 point [-]

And me. I guess - as most probable explanation - they just lost something crucial in retelling. The notes drag on a fair bit in the second part. I can hear the difference if I really concentrate. But its ilke a typo in the text. If the text was blurred.

Comment author: orthonormal 07 April 2012 08:58:33PM 0 points [-]

The second sequence sounded jarringly wrong to me, FWIW.

Comment author: MixedNuts 04 April 2012 04:06:45PM 3 points [-]

Because thoughts don't behave much like perceptions at all, so that wouldn't occur to us or convince us much once we hear it. Are there any thoughtlike things we don't get but can indirectly manipulate?

Comment author: Vaniver 04 April 2012 05:20:23PM 7 points [-]

Parity transforms as rotations in four-dimensional space.

Comment author: TheOtherDave 04 April 2012 04:33:13PM 2 points [-]

thoughts don't behave much like perceptions at all

Can you expand on what you mean by that? There are many ways in which thoughts behave quite a bit like perceptions, which is unsurprising since they are both examples of operations clusters of neurons can perform, which is a relatively narrow class of operations. Video games behave quite a bit like spreadsheets in a similar way.

Of course, there are also many ways in which video games behave nothing at all like spreadsheets, and thoughts behave nothing like perceptions.

Comment author: MixedNuts 04 April 2012 05:38:31PM 1 point [-]

Naively speaking, if Alice can think a thought, she can just tell Bob, and he will. Dogs can't tell us what ultrasounds sound like, but that's for the same reason they can't tell us what regular sounds sound like.

Comment author: Eugine_Nier 05 April 2012 01:04:13AM 2 points [-]

That's assuming the thought can be expressed in language.

Comment author: TheOtherDave 04 April 2012 05:45:53PM 1 point [-]

Even if we posit that for every pair of humans X,Y if X thinks thought T then Y is capable of thinking T, it doesn't follow that for all possible Ts, X and Y are capable of thinking T.

That is, whether Alice can think the thought in the first place is not clear.

Comment author: MixedNuts 04 April 2012 06:22:22PM 1 point [-]

If you limit yourself to humans, yes. But at least one mind has to be able to think a thought for that thought to exist.

Comment author: TheOtherDave 04 April 2012 06:57:28PM *  0 points [-]

Ah, I thought you were limiting yourself to humans, given your example.

If you're asserting that for every pair of cognitive systems X,Y (including animals, aliens, sufficiently sophisticated software, etc.) if X thinks thought T then Y is capable of thinking T, then we just disagree.

Comment author: MixedNuts 04 April 2012 07:13:07PM 0 points [-]

Yes, transmission of thoughts between sufficiently different minds breaks down, so we recover the possibility of thoughts that can be thought but not by us. But that's a sufficiently different reason from why there are sensations we can't perceive to show that the analogy is very shallow.

Comment author: VKS 04 April 2012 04:17:21PM *  9 points [-]

Extremely large numbers.

(among other things)