# Bizarre Illusions

11 27 January 2010 06:25PM

Illusions are cool. They make me think something is happening when it isn't. When offered the classic illusion pictured to the right, I wonder at the color of A and B. How weird, bizarre, and incredible.

Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, something is wrong with how I am thinking about colors." I am being stupid when my I look at this illusion and I interpret the data in such a way to determine distinct colors. My expectations of reality and the information being transmitted and received are not lining up. If they were, the illusion wouldn't be an illusion.

The number 2 is prime; the number 6 is not. What about the number 1? Prime is defined as a natural number with exactly two divisors. 1 is an illusionary prime if you use a poor definition such as, "Prime is a number that is only divisible by itself and 1." Building on these bad assumptions could result in all sorts of weird results much like dividing by 0 can make it look like 2 = 1. What a tricky illusion!

An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the visual system is reporting RGB style colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. In this case, my visual system is distinguishing between something else entirely. Now that I have the right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It should stop being weird, bizarre, and incredible. It merely highlights an obvious reality.

Addendum: This post was edited to fix a few problems and errors. If you are at all interested in more details behind the illusion presented here, there are a handful of excellent comments below.

Sort By: Best
Comment author: 28 January 2010 01:26:42AM 18 points [-]

Why do I keep thinking A and B are different colors?

Meanwhile I am thinking 'Wow! My brain can automatically reconstruct a 3D image from limited 2D input and even compensate for shadows and lighting. That is orders of magnitude more complex than the reverse, generating such images from a model such as those we add 3D cards to computers for'.

I don't particularly consider this an 'illusion', especially when it is not simultaneously acknowledged that it is an 'illusion' that A and B are squares on a 3D 'square X a bit' thing that also has a cylinder on top of it.

Comment author: 28 January 2010 04:11:25PM 8 points [-]

Wow, good point! I never thought about it like that. It raises the question: Why are people amazed when you say, "Tiles A and B are actually the same color -- check for yourself!" but they roll their eyes when you say, "There are no squares in this image -- check for yourself!"? In both cases, you can respond with, "Well, yeah -- if you don't interpret it like the scene it's trying to represent!"

I'm not a very good artist, so learning about how to create these illusions sounds like a good reason to take an art class, and help me appreciate what artists are doing. (Why didn't the first major breakthrough in cognitive science come from painters and sketchers?)

Of course, it probably wouldn't do much to help me understand why they can count random smears on a canvas as "art"...

Comment author: 28 January 2010 04:41:10PM 0 points [-]

That last line is coming from a decidedly unrational state of mind!

Comment author: 28 January 2010 07:22:56PM *  2 points [-]

How so? I wasn't spouting the usual greedy/fake reductionist cliches; I was talking about the paintings that look like a 3-year-old made a mess, yet get classified as art, and noting that an art class probably wouldn't convince me this is appropriate.

What specific criticism of that claim do you have?

Comment author: 28 January 2010 07:48:56PM *  4 points [-]

I was talking about the paintings that look like a 3-year-old made a mess, yet get classified as art, and noting that an art class probably wouldn't convince me this is appropriate.

What specific criticism of that claim do you have?

Short version:

High art is about a lot of things, not least of which is impact on the viewer.

In the case of Pollock, for instance, a lot of the interesting thing that was going on there, was that he depicted a process - not by painting a representation of himself doing it, but by actually doing it. You can look at a Pollock and see how he constructed it without being distracted by exactly what he was trying to construct. And being able to see that aspect of art and be aware of it, will in turn give you a greater appreciation of medieval cathedrals and Greek sculpture.

Possibly related: no one knows what science doesn't know

Comment author: 28 January 2010 08:34:00PM 2 points [-]

To add to this with a similar example, consider that some people prefer listening to foreign language vocalists because it allows one to appreciate the sound of the vocal instrument without focusing on the words.

Comment author: 29 January 2010 11:42:46PM 0 points [-]

To me, most music sounds like a foreign language (though one that sounds exactly like English), unless I'm familiar with the lyrics beforehand, in which case I can "hear" them just fine.

Comment author: 29 January 2010 11:49:08PM 2 points [-]

Like this?

Comment author: 31 January 2010 05:41:34AM 0 points [-]

Probably.

Comment author: 29 January 2010 11:54:02PM -1 points [-]

Yes, most people don't care much about the actual lyrics. Which explains the phenomenon thomblake was trying to use for tenuous support of another hypothesis, yet remains modded to 3 for some reason (7 if you include his parent comment).

Comment author: 28 January 2010 09:54:01PM 0 points [-]

Right, because the words (i.e. the lyrical semantics, as differentiated from the qualities of the sounds the words make) are a small, perhaps negligible component of what people like about many of these songs.

If you were trying to draw some other inference from this fact, you're going to have to be more specific about why that inference follows.

Comment author: 28 January 2010 08:06:12PM *  1 point [-]

Definitely related: Truly Part of You

If I erased your knowledge (and everyone else's) of what the kewl kids had classified as "good art", would it grow back? Would you eventually re-recognize the same works as being good, with the same relative merit, for the same reasons?

If your answer is no, that's a big red flag that you're dealing in bullshit.

(The correct reaction to the parable of The Emporer's New Clothes is not, "Well of course a kid isn't going to see the clothes! What's your point?")

Comment author: 28 January 2010 09:48:14PM 2 points [-]

If I erased your knowledge (and everyone else's) of what the kewl kids had classified as "good art", would it grow back? Would you eventually re-recognize the same works as being good, with the same relative merit, for the same reasons?

I imagine that the answer to this is yes for a great deal of art. I don't know much about it myself, but when I think about art that I like I can find reasons aside from cultural significance or peer pressure.

This assumes the question is ignoring the lens created by my limited expose to art. I highly doubt that any of the artists I like would have been experienced by me if others hadn't considered them worthwhile.

Music is an easier analogy for me to make. I can more accurately describe what I like in music because I know a few more terms. But also, when I listen to a song, I find that my opinions are more distinct. I assume this is because my tastes are becoming refined; I am open to other interpretations.

Comment author: 28 January 2010 08:20:10PM 2 points [-]

That's not related.

If you took away everyone's knowledge of English, and someone laid King Lear at your feet, what would you do with that? The fact that art is rooted in culture and context, some of which is the result of stochastic processes, does not mean you're dealing in bullshit.

Comment author: 28 January 2010 08:37:27PM 3 points [-]

I would be very surprised to discover that a King Lear in an unfamiliar language had been produced by an ape. I am not surprised by hoaxes like this. I think that is indicative of a meaningful difference.

Comment author: 28 January 2010 09:51:46PM *  1 point [-]

After Peter had created a number of paintings, Axelsson chose what he considered to be the four best and arranged to have them exhibited in an art show at the Christina Gallery.

Emphasis added to indicate flaw in experimental protocol.

Edit: This point is much weaker than it appears at first glance. See responses.

Comment author: 28 January 2010 10:03:00PM 1 point [-]

I would still be surprised if the monkey King Lear was chosen as the very best of the monkey's literary oeuvre.

Comment author: 28 January 2010 09:57:37PM 1 point [-]

Yeah, I noticed that too. I felt that it was still a valid test of critics' ability to interpret art considering that most artists will do the same thing with their collection before entering an exhibition.

Comment author: 28 January 2010 08:45:51PM 0 points [-]

That's a far cry away from "eventually re-recogniz[ing] the same works as being good, with the same relative merit, for the same reasons."

Comment author: 28 January 2010 08:49:43PM 0 points [-]

Erasing everyone's knowledge of English is a far cry from erasing their knowledge of "what the kewl kids had classified as 'good art'".

Comment author: 28 January 2010 08:44:28PM 1 point [-]

But it does mean that the writing of King Lear is less of an epistemic achievement than, say, the laws of physics, which are not dependent on a particular species' form of communication.

If King Lear is (claimed to be) a good work, given a certain language (humanity? evolutionary history? political history?), does the recognition of its supposed greatness survive deletion of the knowledge about what the kewl kids think is great?

If people continued to speak English, but King Lear fell out of fashion and later was found, but disconnected from anyone's recommendation, would people still decide it was better than most other works? Would they decide it for the same reason?

Do children spontaneously flock to King Lear at a certain age, even when it's not recommended to them by a True Literary Authority?

Comment author: 28 January 2010 08:49:32PM 0 points [-]

If King Lear is (claimed to be) a good work, given a certain language (humanity? evolutionary history? political history?), does the recognition of its supposed greatness survive deletion of the knowledge about what the kewl kids think is great?

Of course not. It doesn't even come with 3D special effects!

Comment author: 28 January 2010 08:16:28PM 0 points [-]

If your answer is no, that's a big red flag that you're dealing in bullshit.

I seriously doubt that the correct answer is "no".

Obviously, there would be a little bit of wobble - I might not care who Pollock is, but I expect there would be something else I'd find that would illuminate the same aspects of the aesthetic experience. But I think being the first to do it that way counts for something.

Comment author: 28 January 2010 08:36:23PM *  2 points [-]

Sorry, all I got out of that was a name-drop and (what seemed like) a dodge.

Could you answer again, and this time maybe explain it a little differently? Specifically:

-Are you claiming that Pollock discovered a way of satisfying the aesthetic senses that allowed generalization of the method in other forms?

-Let's say I knew a wacko who believed that "By historical accident, Pollock became a focal point for people of high-status to identify each other, despite there being nothing special about his work." What evidence would you point me to that has a low Bayes factor against that hypothesis?

Comment author: 28 January 2010 09:07:47PM 1 point [-]

I'm claiming that Pollock's work demonstrated easily-neglected and valuable parts of the aesthetic experience.

As for evidence against that hypothesis, I think that depends largely upon how seriously you take some of the relevant premises in your wacko's model. According to some, there is virtually nothing to all of culture other than status games (though in this case the clause "despite there being nothing special about his work" would make little sense). According to others, there really is quite a bit to aesthetics, and perhaps it's worth listening to the folks who've spent their lives studying it.

There are a lot of different kinds of things in the world, and many of them are valuable in unexpected ways.

Comment author: 28 January 2010 09:28:01PM *  1 point [-]

So you really can't think of anything that is less likely to be observed if "it's all bullshit" than if it's not? There isn't any kind of aesthetic feeling you could feed to the wacko that he couldn't help but burst out in appreciation for?

As for evidence against that hypothesis, I think that depends largely upon how seriously you take some of the relevant premises in your wacko's model.

Not when I'm asking for evidence with a low Bayes factor, rather than a guaranteed low posterior.

Maybe an example would be in order. Let's say Bob is the wacko, but about quantum physics. Bob believe that the claims of quantum physics are just a big status game, and so are the results of the particle accelerators and everything. I could point to evidence like the atom bomb. If they were just arguing over meaningless crap the whole time, and assigning truth purely based on who has the most status, how did they ever get the understanding necessary to build an atom bomb, Bob?

There are a lot of different kinds of things in the world, and many of them are valuable in unexpected ways.

Right, like Halo. Except that millions of people like Halo even in the absence of a well-funded indoctrination campaign, and the fact that expressing appreciation for Halo won't endear them to the kewl kids of art.

It's not very impressive if people start to enjoy something after they've

spent their lives studying it.

You have to adjust for stuff like that.

Comment author: 28 January 2010 09:09:51PM 0 points [-]

Well said!

Comment author: 28 January 2010 09:25:04PM 1 point [-]

I just realized we seem to be arguing over wine again. I fold.

Comment author: 28 January 2010 09:34:15PM *  1 point [-]

Yes, we are.

If you spend ten years associating wine with a good time, and are expected to have a refined palette for wine to be part of the kewl kids club, then guess what -- you can make yourself like wine! The fact that you like wine in such a scenario does not, to me, count as a genuine liking, in the sense in which I judge beverages. Any substance, even bat urine, will find connosieurs under those conditions!

What I want to do is, find out what's good about something, that isn't simply an artifact of practices that can make anything look good.

That's why I'm not impressed by "enjoy this because people are telling you to enjoy it", which the support for much high art and alcohol amounts to.

Instead of refusing to engage the issue, maybe you should start to think about the recursivity of your criteria for quality?

Comment author: 28 January 2010 07:53:01PM 0 points [-]
Comment author: 28 January 2010 08:05:11PM 1 point [-]

Meh.

Never trust a site with that many banners.

I, for one, have never heard seriously disparaging things about 19th-century or Academic art in general, and the essay sounds a bit... rabid. It seems like the writer was operating under the common heuristic "I don't understand it, so it must be stupid."

Comment author: 28 January 2010 10:53:51PM 1 point [-]

And I never trust anyone that says, "No, no, this stuff is good, it really is! Just pay for 10 years worth of education in this specific area, and then you'll see the light!"

(Note the similarity to Scientology practices...)

Comment author: 28 January 2010 11:04:32PM 2 points [-]

But then what do you do when something really does take that long to explain? People say category theory is beautiful; is the nonmathematician supposed to call them liars?

Comment author: 28 January 2010 11:09:41PM 2 points [-]

Category theory doesn't take 10 years to explain. You should be able to explain to a willing, intelligent friend in two full days, and get them to a point where they see the beauty.

I've done similar things, like explaining the elegant beauty of aircraft component structural analysis -- got a decent appreciation across in 10 minutes. ("You know how a chain is as strong as its weakest link? A component is as strong as its weakest failure mode...")

The point is, you can explain it. That's a lot more than you can do for (much of) art, it seems.

Comment author: 28 January 2010 11:12:27PM 0 points [-]

At least where category theory is concerned, you don't have to pay.

Comment author: 28 January 2010 07:35:38PM *  0 points [-]

Disliking Pollock is irrational. As is disliking Cage. Or Joyce. Or PEZ.

Comment author: 28 January 2010 07:37:07PM 4 points [-]

I love 4'33". It helps me get to sleep.

Comment author: 28 January 2010 07:47:42PM 1 point [-]

Edit: OMG! Is my humor too subtle people? PEZ!

It was, yes - maybe we need to make emoticons more normal here, since this is a recurring problem. :P

(Downvote removed)

Comment author: 28 January 2010 08:45:57PM *  2 points [-]

maybe we need to make emoticons more normal here, since this is a recurring problem.

But then how will we signal our sophistication and mutually affirm our status as an intellectual subculture? ;)

Comment author: 29 January 2010 02:32:51AM 2 points [-]

Anime references. Duh.

Comment author: 28 January 2010 08:47:55PM 2 points [-]

:P~~~~~

;D

Comment author: 28 January 2010 08:43:40PM 0 points [-]

Neutral vote. I like the PEZ juxtaposition but 'arational' would fit better. A simply false assertion doesn't fit well with the irony.

Comment author: 29 January 2010 12:01:43AM 0 points [-]

As it was mocking bgrah's assertion, and bgrah used "unrational", and in my estimation his meaning was closer to "irrational" than "arational", I used the former. Perhaps using "unrational" would have been better, though.

Comment author: 28 January 2010 07:44:52PM 0 points [-]

Just consider it evidence of the level of culture you'll find hereabouts. Savages.

Comment author: 29 January 2010 01:53:09AM 1 point [-]

Why are people amazed when you say, "Tiles A and B are actually the same color -- check for yourself!" but they roll their eyes heir eyes when you say, "There are no squares in this image -- check for yourself!"?

It's about expectations. People expect to be able to take (physical) objects that appear to be different colors, examine them under a variety of contexts, and always perceive them as different. People incorrectly extrapolate that expectation to images, and thus find the fact that removing the context reveals these images to be the same colorRGB surprising. They also expect to be presented with representations, so pointing out the fact that they're looking at a representation seems silly to them.

Comment author: 28 January 2010 08:39:50PM 1 point [-]

Of course, it probably wouldn't do much to help me understand why they can count random smears on a canvas as "art"...

By way of reversing the ADBOC concept, I disagree denotationally but confirm your connotation. As you explain in the cousin several times removed post, many kinds of art are bullshit. Cultural preferences that would not be particularly likely to be rediscovered if all trace was removed. This differs from other forms of art which are more specifically directed at aesthetic preferences intrinsic to humans.

Of course, immersing yourself in a culture and experiencing the flow of status first hand is the perhaps the best way to get an intuitive anthropological understanding. I found, for example, that having done a research degree in a subfield of AI helps me understand how peer affiliation by persisting with researching silly ideas can be counted as 'science'.

http://lesswrong.com/lw/1om/bizarre_illusions/1iub

Comment author: 27 January 2010 10:41:38PM *  11 points [-]

I identified a useful and cogent point in your post and it was this: Whenever you receive data from any source (your brain, your eyes, a drug study, Less Wrong) you've got to be aware of how that data has already been packaged. Taking the data at face value -- for example imagining your brain is actually making a claim about the RGB values of the pixels -- can lead to problems, misconceptions, mistakes.

Comment author: 27 January 2010 07:12:28PM 8 points [-]

Your eye didn't evolve to report trivia like, "These two colors are actually the same." Your eye is reporting the most useful information - from which direction the light is coming, the shaded region under it, and the fact that the floor is tiled.

Which is more amazing - this picture, or a picture that somehow tricked the average person into noticing two colors were the same, but didn't notice the picture also had floor tiling, light directionality, and shading? I'd say this picture is pretty tame in comparison to the picture that could do that.

Comment author: 27 January 2010 07:55:46PM 13 points [-]

I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors.

Not at all. In the context of the scene that this picture represents, A and B are absolutely different shades. On the contrary, I think your perceptual system would be poor indeed if it did not reconstruct context, and under-interpreted the picture as a meaningless 2D array of pixels.

(BTW, as with the necker cube, I find that I can consciously exert to experience the interpretation that I choose, without too much difficulty.)

Comment author: 27 January 2010 10:03:13PM 3 points [-]

Hmm. I can with the necker cube, but not at all with this one.

Comment author: 27 January 2010 10:13:39PM 6 points [-]

I was never able to do it with this one before, either. What I'm doing now is concentrating hard on the two tiles of interest, until the rest of the picture fades into the background. The two tiles then seem to be floating on a separate top layer, and appear to be the same shade.

Comment author: 28 January 2010 02:51:29AM 2 points [-]

That worked! Cool!

Comment author: 28 January 2010 08:12:03AM *  1 point [-]

If you go to an Art or Design school. Seeing and producing illusions like this is one of the assignments that they usually will give you in a 2D design class.

As it has been described above, if you can concentrate (if school, we learn how to look at them by squinting as we would when discerning simple shape or color - or, if you have ever learned how to look at one of those weird 3D images made out of what looks to be paint splatter) on the two squares, then you will be able to see that they are indeed the same shade (not color, color is used to describe something else)

Comment author: 28 January 2010 11:36:44AM *  2 points [-]

Ah, that worked for me. For people wondering how to do the technique to see "Magic Eye " images, you focus your eyes so that the image doubles and and overlaps the image. That causes a stereoscopic illusion when done on any things that overlap. You could practice it here. Focus your eyes so that the the first abc overlaps the second abc -- now you have three abc's in your vision, the 1st and 3rd abc are being seen out of one eye and the abc in the middle appears to be almost 3d.

a...b...c......................a...b...c

In this case, I could see that A and B are the same color by tilting my head and then focusing so I saw a double image of A overlapping B.

Comment author: 29 January 2010 08:02:25AM 1 point [-]

Exactly... We spent a total of 6 weeks in Art School design class learning how to do this specific trick with a variety of images. From color, to line length (you know those "which line is longer" tricks that make you think one line is longer when they are usually the same length), to line thickness, to shading and tinting aliasing.

We spent those weeks consuming a lot of aspirin and Tylenol.

Comment author: 28 January 2010 08:18:10AM 0 points [-]

Interesting, when I try this technique the shades seem even more distinct.

Comment author: 28 January 2010 11:21:12AM 1 point [-]

It takes some practice.

We were taught that if you put your nose right in the center of the image, and then let your focus go, and pull back from the image, that at a certain distance from the image (as your focus is still at ∞) various structures of the image will begin to resolve. So contrasts, similarities, and shades will all resolve at different focal lengths from the image.

It was rare that any one person would be able to pick up immediately upon all the effects perceptible in an image. I was able to pick up on certain shades of the color green that are used in contrast to red, but it took me a long time to get the shading of black-white (as in this optical illusion - and it is but one of many).

When we were tested on this, we would not be told what was similar, or where optical tricks were used, and we would have to pick them out of an image (and this was long before the internet, so we couldn't just go online to do a search for optical illusions to find images to study that had their illusions spelled out for us). So, it is a skill that can be learned. For me, eventually I had to learn how to focus upon each square with a different eye, while squinting, and letting the focus go back and forth between my right and left eye. eventually, I get the images resolved as a single shade as I go back and forth between my eyes.

Comment author: 27 January 2010 10:39:06PM *  -1 points [-]

The point of the illusion is that they seem different in context. Ignoring context to make them appear similar isn't a proper resolution.

Comment author: 28 January 2010 03:56:49AM 0 points [-]

I found two ways to do it myself:

(A) Cover up areas of the image to see what causes what to change color in your perception. Slowly reveal the full image again and sometimes A and B look alike

(B) Let your focus drift until the lines of the image get fuzzy. Look at the two squares without actually looking at them. I find that the colors look alike here. If I "snap" focus back they still look alike but nothing is fuzzy anymore.

B works better.

Comment author: 27 January 2010 08:48:25PM 1 point [-]

I don't understand this. Are you saying that A and B are not the same color?

Comment author: 27 January 2010 09:22:07PM *  12 points [-]

AndyWood gave a good explanation, but let me elaborate. If you saw the scene depicted, but in real life -- rather than on a flat paper or 2D screen -- you would be correct to infer that the actual, invariant colors of the tiles are different. But, since they are just pixels on paper or a screen, their invariant colors are the same, and yet your eyes tell you otherwise.

So are the eyes "wrong" in any serious sense? Well, let me put it this way: do you want

a) a visual system that gives the right interpretation of scenes that you are actually going to encounter often, but is tripped up by carefully designed optical illusions?

or do you want:

b) a visual system that gives the right interpretation for carefully designed optical illusions, but fails to catch many attributes of common scenes?

(Yes, there is a tradeoff. Your visual system encounters an "inverse optics" problem: given the retina images, what is the scene you're looking at made of? This is ill-posed: many scenes can generate the same retinal images. E.g. a given square could be far away and big, or close and small. To constrain the solution set, you need assumptions, and any set of assumptions will get some scenes wrong.)

Yes, you are wrong to think that the tiles have different colors. You are not wrong to prefer a visual system that gets most scenes right at the cost of getting a few scenes (like this one) wrong.

(Incidentally, I really like this optical illusion, and have it by my desk at work. What's so great about it is that once you see it, you can actually strip away everything that you think is causing the illusion, and yet they still look different!)

Comment author: 27 January 2010 08:58:35PM 5 points [-]

Your understanding of the word 'colour' does not match what properties of the world your brain is trying to identify and categorize when it interprets 'colour'. The interesting constant property of objects in the world that makes 'colour' useful to your visual system for purposes of object identification and categorization is really the surface properties that interact with incident lighting. Your brain attempts to ignore effects due to lighting variation and assign a 'colour' label to objects that is more or less an invariant property of the surface under a variety of different lighting conditions. This is in general not a solvable problem since the same incident photons can be produced by a number of different lighting and material combinations. Optical illusions like this merely reveal the heuristics your visual system uses to identify the relevant constant aspects of the scene and ignore the irrelevant lighting variation. They generally work quite well.

Comment author: 27 January 2010 09:09:26PM 4 points [-]

When we covered this phenomenon in my psychology degree it was referred to as colour constancy. I now work as a 3D graphics programmer and so know a lot about the physics of light transport. The illusion does not surprise me any more, in fact it seems a little surprising that I ever could have thought that the RGB colour value of an onscreen pixel was directly related to the property of objects in the real world that we call 'colour'.

Comment author: 27 January 2010 09:47:50PM 0 points [-]

Well said (including your later comment about color constancy). Along the same lines, this is why cameras often show objects in shadows as blacked out -- because that's the actual image it's getting, and the image your own retinas get! It's just that your brain has cleverly subtracted out the impact of the shadow before presenting it to you, so you can still see significant contrast and colors in the shadowed objects.

Comment author: 27 January 2010 10:14:25PM 4 points [-]

Along the same lines, this is why cameras often show objects in shadows as blacked out -- because that's the actual image it's getting, and the image your own retinas get! It's just that your brain has cleverly subtracted out the impact of the shadow before presenting it to you

That doesn't explain why faithful reproductions of images with shadows don't prompt the same reinterpretation by your brain.

Comment author: 27 January 2010 11:32:35PM 4 points [-]

Blacked out shadows are generally an indication of a failure to generate a 'faithful' reproduction due to dynamic range limitations of the camera and/or display medium. There is a fair amount of research into how to work around these limitations through tone mapping. High Dynamic Range cameras and displays are also an area of active research. There's not really anything to explain here beyond the fact that we currently lack the capture or display capability to faithfully reproduce such scenes.

Comment author: 27 January 2010 11:00:48PM 0 points [-]

Sure it does -- Faithful reproductions give the shadowed portion the appropriate colors for matching how your brain would perceive a real-life shadowed portion of a scene.

Comment author: 27 January 2010 11:26:08PM *  0 points [-]

Umm, that's not what I meant by "faithful reproductions", and I have a hard time understanding how you could have misunderstood me. Say you took a photograph using the exact visual input over some 70 square degrees of your visual field, and then compared the photograph to that same view, trying to control for all the relevant variables*. You seem to be saying that the photograph would show the shadows as darker, but I don't see how that's possible. I am familiar with the phenomenon, but I'm not sure where I go wrong in my thought experiment.

* photo correctly lit, held so that it subtends 70 square degrees of your visual field, with your head in the same place as the camera was, etc.

Comment author: 27 January 2010 11:31:17PM 0 points [-]

I thought you meant "faithful" in the sense of "seeing this is like seeing the real thing", not "seeing this is learning what your retinas actually get". If you show a photograph that shows exactly what hit the film (no filters or processing), then dark portions stay dark.

When you see the scene in real life, you subtract off the average coloring that can be deceiving. When you see the photo, you see it as a photo, and you use your current real-life-background and lighting to determine the average color of your visual field. The darkness on the photo deviates significantly from this, while it does not so deviate when you're immersed in the actual scene, and have enough information about the shadow for your brain to subtract off the excessive blackness.

Been a long day, hope I'm making sense.

Comment author: 27 January 2010 10:08:24PM 4 points [-]

As others have pointed out, the difficulty here is more in the semantics of "color" than in the optics.

As a simplification, we can consider the projected color.P of a tile to be a product of its surface properties (color.S) and the intensity of the incident light. The illusion straightforwardly contrives one of these terms - the light intensity - so that the color.P of tile A equals the color.P of tile B. But the brain, interpreting the image as a 3D scene with light and shadow, reports the color.S-es of the tiles, which are different under that very reasonable and useful interpretation.

I'm sorry if this is a big distraction from the point of your post. I'm still interested in the point, so perhaps you can find another way of getting it across.

Comment author: 27 January 2010 10:45:56PM 2 points [-]

As others have pointed out, the difficulty here is more in the semantics of "color" than in the optics.

Yeah. I missed the semantic shift. All it took was someone pointing out that there were two uses of Color drifting around and almost all the comments snapped back into making sense.

I'm sorry if this is a big distraction from the point of your post. I'm still interested in the point, so perhaps you can find another way of getting it across.

The point is that an illusion generally gives off a sense of bizarreness because we are expecting X but the illusion gives us Y. In the case of the color example, I once expected boxes A and B to appear to be the same color (perceived) if and only if they were the same color (RGB). The illusion shows this is not the case. Being curious, I sought to understand the underlying principles behind why we perceive two different colors. Once this is understood, the illusion should no longer seem bizarre but a trivial example of the underlying principles.

In trying to find where I went wrong with the post, I come up with this:

• "Color" is an extremely ambiguous term. I should have seen this one coming.
• I think some people thought I was trying to give an explanation of the illusion in the post. I was not.
• I think some people thought I was saying that the visual system itself was stupid or broken and we needed to "fix" our brain to adjust for its shortcomings. I was not. I was trying to say that our feeling of "bizarre" was stupid because we are expecting something different from our visual system than what the visual system provides.
• I deliberately wrote this post more aggressively and concisely than I generally write. Perhaps this degraded its clarity even further.

I am half tempted to take this post down, rewrite it, and put it back up, but I don't know how much that would help.

Comment author: 27 January 2010 11:08:03PM 4 points [-]

Well, don't do anything that takes down the comment section. Many of the comments are insightful and, um, say things that should have been in your original post.

Demystifying optical illusions, and visual cognition in general, is a very good exercise in rationalist reduction.

Comment author: 27 January 2010 11:22:22PM 0 points [-]

Okay. Do you think it would be valuable to just edit the post in place?

As best as I can tell, these are the trouble paragraphs:

Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, that is not what my visual system is trying to tell me." I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors. That information is not being transmitted and received. If it were, the illusion wouldn't be an illusion.

An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the system is reporting True Colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. Once I have found the right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It should stop being weird, bizarre, and incredible. It should highlight an obvious reality.

Is this better?:

Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, something is wrong with how I am thinking about colors." I am being stupid when I look at this illusion and interpret the data in such a way to determine distinct colors. My expectations of Reality and the information being transmitted and received are not lining up. If it were, the illusion wouldn't be an illusion.

An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the system is reporting True Colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. In this case, the visual system is distinguishing between [what term goes here?], not individual RGB style colors. Now that I have right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It stops being weird, bizarre, and incredible. It merely highlights an obvious reality.

Comment author: 28 January 2010 12:07:48AM *  1 point [-]

It seems to me that you are still using the word colour in a way that suggests you haven't really grasped the insight that makes this illusion seem not-bizarre. That insight is fundamentally that the statement "this ball is blue" is not equivalent to the statement "a digital photo of a scene containing this ball would have pixel values of 0, 0, 255 at pixel locations where light from the ball reached the sensor". It is a much more complex (and more useful) statement than that. The bad assumption is that 'colour' when used to refer to a property of objects in the world determined through visual perception has any simple relationship with RGB values recorded by a digital camera. You still seem to be talking as if RGB values are somehow 'true' colours.

Comment author: 28 January 2010 01:37:54AM *  1 point [-]

The bad assumption is that 'colour' when used to refer to a property of objects in the world determined through visual perception has any simple relationship with RGB values recorded by a digital camera.

Especially in the case of human tetrachromats.

Comment author: 28 January 2010 12:22:30AM *  0 points [-]

I am trying to find a way to say what you said with one phrase or word. I feel like I am struggling to find a term.

Comment author: 28 January 2010 12:35:30AM 4 points [-]

I think the key for me in understanding this type of illusion (and the general phenomenon of colour constancy) was to realize that 'colour' in common usage ("this ball is blue") is perceived as a property of objects and we infer it indirectly based on light that reaches our retinas. That light also has a 'colour' (subtly different meaning) but it is not something we perceive directly because it is not very useful in itself.

This makes perfect sense when you think about it from an evolutionary perspective - we evolved to recognize invariant properties of objects in the world (possibly fruit in trees for primates) under widely varying lighting conditions. Directly perceiving the 'colour' (RGB) of light would not tell us anything very useful about invariant object properties. There is enough overlap between the two meanings of colour for them to be easily confused however and that is really the root of this particular illusion.

In computer graphics we commonly use the term 'material' to describe the set of properties of a surface that govern how it responds to incident light. This encompasses properties beyond simple colour ("shiny blue ball", "matte blue ball", "metallic blue ball"). I don't know if that usage is well understood outside of the computer graphics field however.

Comment author: 28 January 2010 01:12:39AM 0 points [-]

I completely agree with you. At this point, I am just trying to clean up the article to help clarify the answer behind the illusion. Does the phrase, "I should stop thinking that the visual system is reporting RGB style colors" mesh okay? That is the only location of RGB as of this edit.

Comment author: 28 January 2010 12:32:49AM 0 points [-]

How is "Color gross of lighting conditions"?

Comment author: 27 January 2010 11:26:23PM 1 point [-]

It sounds like you're trying to come up with a sentence or two that captures all of the insight on color that the commenters have given. While I'm a big fan of summarizing, and a big critic of those who can't, I don't think you can get it to work here. Instead of your final bolded change (the others are good), just point to or quote a few good comments that show what the visual system is doing, and how the optical illusions trick it.

Comment author: 27 January 2010 11:33:58PM *  0 points [-]

In this case, the visual system is not trying to distinguish between individual RGB style colors (more details in the comments).

EDIT: I updated it with something similar. Hopefully it was an improvement. :) Thanks again for your help (which isn't to say that I wouldn't mind more help...)

Comment author: 27 January 2010 11:16:56PM *  0 points [-]

I am half tempted to take this post down, rewrite it, and put it back up, but I don't know how much that would help.

Taking this and SilasBarta's thoughts together: can you apply this same meta-principle to something substantially different in a new post, written with a recognition of these confusions? That post could cite this post with a "Followup to:" line, and elaborate on your discovery in some way.

Comment author: 27 January 2010 11:23:42PM 0 points [-]

Would it be better to just replace the content of this post? I can archive the original in a comment here for future context.

Comment author: 28 January 2010 03:06:52AM 4 points [-]

I would be disinclined to that course, but hard-pressed to justify it more effectively than by my idiosyncratic generalization of one of a number of principles I have heard - I quote from the post:

Don't try to rewrite history. Look, we make mistakes. We all do. Sometimes we post an essay and we get stuff wrong in it. Sometimes that stuff makes the whole essay wrong. Sometimes, we put up an essay innocently and it turns into a firestorm of controversy we never meant. Sometimes, we find ourselves in a crucible on all sides.

The temptation is to go back. Revise. Reword what we said. Take the essay down entirely.

It is never a good idea. Ever.

I don't think you have anything to be ashamed of in this post. It's not deep, it's not extraordinary in its conclusions, but it is correct and brief. The complaints seem to me best addressed by elaboration and discussion - things which require far more than a brief edit placed at the end of the post.

As SilasBarta mentioned, there's a lot of commentary on this post that is worth preserving, and should be preserved with the original post. It would be unfair to the commenters to render their comments incomprehensible - even briefly - by distortion of that to which they responded.

And, if I may be frank, if the idea which inspired this post is interesting, it is probably capable of generalization. The idea of my own which I promoted to a post I did so because I saw that it was applicable beyond the scope of its origination, and in a manner which was natural, elegant, and interesting. It proved of interest to a number of people here, despite its unabashedly algebraic treatment. If you can find a profitable extension of your concept, it will be likely to be worth reporting in a followup post (and if you are concerned about the appropriateness of it, I - as one remaining upvoter of the OP - will have sent my email to you in a PM, and be willing to comment on any draft you wish to send).

If you cannot find a profitable extension of your concept, it is probably not worth the time to revise. Consider your post dubiously successful (it is still in positive territory, is it not?) and leave it be.

Comment author: 28 January 2010 03:50:47AM 1 point [-]

I don't think you have anything to be ashamed of in this post. It's not deep, it's not extraordinary in its conclusions, but it is correct and brief. The complaints seem to me best addressed by elaboration and discussion - things which require far more than a brief edit placed at the end of the post.

It's not so much that I am ashamed; I am just frustrated. The behavior of this post caught me completely off-guard. It was upvoted to +5 within a few hours and people started asking questions. After my responses, the post dropped to +1. The karma itself doesn't mean much to me, but the feedback here was evidence of something greater than a non-interesting or incorrect post.

People were willing to talk about it, so I stuck it out for as much feedback as I could. The investment was completely worth it. I got several comments worth of extremely valuable insights to my writing style and how to better post here at LessWrong.

I think the post itself failed, but the whole experience has been a net gain.

As SilasBarta mentioned, there's a lot of commentary on this post that is worth preserving, and should be preserved with the original post. It would be unfair to the commenters to render their comments incomprehensible - even briefly - by distortion of that to which they responded.

I agree. My intent in the revisions has been to keep people from being distracted by my quirks and leading them into a wonderful discussion in the comments. This particular illusion has a lot more history behind it than I originally thought; I learned a lot.

And, if I may be frank, if the idea which inspired this post is interesting, it is probably capable of generalization. The idea of my own which I promoted to a post I did so because I saw that it was applicable beyond the scope of its origination, and in a manner which was natural, elegant, and interesting. It proved of interest to a number of people here, despite its unabashedly algebraic treatment. If you can find a profitable extension of your concept, it will be likely to be worth reporting in a followup post (and if you are concerned about the appropriateness of it, I - as one remaining upvoter of the OP - will have sent my email to you in a PM, and be willing to comment on any draft you wish to send).

Thank you very much. I have to sit on the events of today and ponder if there is a next step to take. If a followup is coming I will certainly take you up on your offer.

Comment author: 28 January 2010 03:19:53AM 0 points [-]

An addendum - as far as my recollection of the original goes, your edits appear reasonable, although I would not have risked them on my own post. I congratulate you on a successful revision, but my offer stands.

Comment author: 27 January 2010 08:51:28PM *  3 points [-]

In the spirit of not disputing definitons, may I suggest: A and B are the same colorRGB, but, interpreting the image as a picture (as the eye does), not the same colorALBEDO.

Edit: Correction - "as the visual system does".

Comment author: 27 January 2010 09:00:11PM 1 point [-]

This is perhaps beating a dead horse, but "albedo" is supposed to be a ratio between reflected and incident lights, and I would bet that the albedo of these two patches of screen is also identical, just as their RGB values are identical.

Comment author: 27 January 2010 09:05:58PM 2 points [-]

Not in the actual 3D scene that your brain interprets the picture to be of, only in the context of a 2D printout of the image (albedo is not really a relevant property for emissive display devices like LCDs or CRTs).

Comment author: 27 January 2010 09:04:59PM *  2 points [-]

When I said "interpreting the image as a picture", I meant, "interpreting the image as a picture of a checkerboard with a cylinder casting a shadow on it" - the albedos in question are of the squares A and B on the depicted board.

Comment author: 27 January 2010 09:10:58PM 3 points [-]

Ah, "inferred albedo". In that case we agree.

Comment author: 27 January 2010 08:55:08PM 0 points [-]

Thank you.

Comment author: 27 January 2010 09:11:56PM 2 points [-]

They are the same screen-color, but different inferred-colors.

Comment author: 27 January 2010 11:42:21PM 4 points [-]

I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors.

Tell that to your ancestors who escaped from the saber-tooth cat hiding in the shadows at dusk.

Comment author: 28 January 2010 12:29:44AM 2 points [-]

A side note: The only reason that prime numbers are defined in such a way as to exclude 1 and negative numbers is because mathematicians found this way of defining them a bit more useful than the alternative possibilities. Mathematicians generally desire for important theorems to be stated in a manner that is as simple as possible, and the theorems about primes are generally simpler if we exclude 1. There is a more detailed analysis of this question here:

Comment author: 27 January 2010 10:31:23PM *  2 points [-]

if you use a poor definition such as, "Prime is a number that is only divisible by itself and 1."

I have a fondness for this particular definition, and like to think of 1 as a "very special" prime number. To the extent that I usually give a little speech whenever an opportunity arises that (ahem) the only reason I know of that '1' is excluded from the primes (more often than not) is because almost every theorem about prime numbers would have to be modified with an "except 1" clause. But a natural definition (anything along the lines of "already completely factored") would include it. If you disagree, which definition --- or the satisfaction of which theorem -- do you think is more compelling?

(Just in case you perceived you were getting too much heat about "colour"...)

Comment author: 27 January 2010 10:40:03PM 11 points [-]

I'd have to pretty strongly disagree. To me, the "essence" of primes is that you can factor any number into primes in a unique way. That's the most natural definition. They're the multiplicative building blocks of the natural numbers; everything can be reduced to them. If 1 were prime, you could no longer factor uniquely.

Comment author: 27 January 2010 10:45:32PM *  4 points [-]

Your comment and this comment were adjacent in my message folder which I found amusing.

Thomblake wrote:

Caring about what's right might be as arbitrary (in some objective sense) as caring about what's prime, but we do actually happen to care about what's right.

It's funny how we do care.

Comment author: 27 January 2010 11:52:09PM 5 points [-]

Your comment and this comment were adjacent in my message folder which I found amusing.

That's awesome. Thanks for sharing.

Comment author: 27 January 2010 10:50:06PM 1 point [-]

Possibly related: my comment about paperclips.

Comment author: 28 January 2010 12:41:00AM *  3 points [-]

Hmm... I agree this is compelling. However, since I'm resistant to updating my world view about 1-the-discriminated-prime-number, I'll continue to proffer counter-arguments:

• the Fundamental Theorem of Arithmetic is pretty important, but may still not be the "essence" of what prime is

• the FTA itself requires the "except 1" clause: "all natural numbers can be uniquely factored into primes except 1" -- which would make someone thing 1 ought to be prime

• the FTA already assumes 'modulo permutations', we could easily throw in 'modulo 1'

• Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is. (My own emphasis added.)

The best argument I came up with for not including 1 as prime, because I tend to worry about how things are constructed, was with the seive of Eratosthenes.

The seive of Eratosthenes says that you can find the primes by starting with all the natural numbers > 1; let 2 be the first prime number, and then begin eliminating all multiples of 2 and the multiples of subsequent primes as you find them. If you included '1' in the first step, then you would eliminate all the numbers in the first step.

Comment author: 28 January 2010 01:43:25AM *  14 points [-]

I think you're really failing to grasp the content of the unique factorization theorem here. Firstly we don't think about factored numbers as products of primes up to permutation, we think of them as products of distinct prime powers (up to permutation, I suppose - but it's probably better here to just take a commutative viewpoint and not regard "up to permutation" as worth specifying). But more importantly, you need to take a multiary view of multiplication here, not a binary one. 1 is the empty product, so in particular, it is the product of no primes, or the product of each prime to the 0th power. That is its unique prime factorization. To take 1 as a prime would be like having bases for vector spaces include 0. Almost exactly like it - if we take the Z-module of positive rationals under multiplication, the set of primes forms a free basis; 1 is the zero element.

Comment author: 28 January 2010 04:52:13PM *  2 points [-]

Information and expertise like this is why hanging out at Less Wrong is worth the time. I estimate that I value the information in your comment at about $35, meaning my present self would advise my former self to pay up to$35 to read it.

So, I get it. My brain is more wired for analysis than algebra; so this isn't the first time that linear algebra has been a useful bridge for me. I see that we could have a 'vector space' of infinite-dimensional vectors where each vector (a1, a2, ..., an, ...) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)... and Pi are the ordered primes. Clearly 1 is the zero element and would never be a basis element.

I should admit here that my background in algebra is weak and I have no idea how you would need to modify the notion of 'vector space' to make certain things line up. But I can already speculate on how the choice of the "scalar field" for specifying the a_i would have interesting consequences:

• non-negative integer 'scalar field' --> the positive integers,
• all integers 'scalar field' --> positive rational numbers,
• complex integers --> finally include the negative rationals.

I'd like to read more. What sub-field of mathematics is this?

Comment author: 08 February 2010 10:05:35AM 0 points [-]

I see that we could have a 'vector space' of infinite-dimensional vectors where each vector (a1, a2, ..., an, ...) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)... and Pi are the ordered primes.

Oh! And orthogonal vectors are relatively prime!

Comment author: 08 February 2010 11:15:17AM 1 point [-]

I'm not sure that the idea of orthogonality is defined for modules, is it? Is there a standard definition of an inner product for a Z-module?

Comment author: 08 February 2010 06:49:48PM 1 point [-]

I'm not sure that the idea of orthogonality is defined for modules, is it? Is there a standard definition of an inner product for a Z-module?

Yes; the same definition works. See here.

Comment author: 09 February 2010 12:05:14AM 0 points [-]

Yay! I actually got something right!

Comment author: 28 January 2010 05:07:58PM 0 points [-]

What sub-field of mathematics is this?

Number theory, ne? Or is that too general?

Comment author: 28 January 2010 05:57:34PM 2 points [-]

It looks like it's more abstract algebra (possibly applied to number theory) that byrnema is interested in. Check out Wikipedia on module.

Comment author: 28 January 2010 06:32:53PM 1 point [-]

Precisely! Thanks also.

Comment author: 28 January 2010 05:33:24PM *  1 point [-]

A second comment...

You've certainly convinced me that '1' should not be included in the set of things that are used to uniquely factor numbers. However, how I can I know if this set is the set of "primes"?

I guess I was thinking that the essence of primes was about their irreducibility/atomic-ness. The number 5 would be considered prime because you can't describe it multiplicatively in any way except by using the number 5. Using my preferred notion, the number 0 and the number -1 would also be "prime" (as Mr Hen guessed). Is there a different word for this concept?

Comment author: 28 January 2010 05:47:59PM *  1 point [-]

See wikipedia on natural generalizations of prime numbers. In particular note that most of the definitions say "units" instead of "1", like "Irreducible elements are ones which cannot be written as a product of two ring elements that are not units." which rules out 0 for the integers, +, x and includes the possibility of multiple units (-1 and 1).

I don't know offhand of any nice, commonly referenced property P(S,O) that is: A,x,y in a structure S with operation O: A is P just when if x O y = A then either x = A or y = A. Which I believe is the general property you're thinking about?

Edit: with O commutative I do believe

Comment author: 28 January 2010 05:56:55PM 0 points [-]

Thank you. And yes, that is the property.

Comment author: 28 January 2010 04:58:05AM *  0 points [-]

For some reason, I never imagined factors this way.

18 = 3^2 * 2^1
97,020 = 2^2 * 3^2 * 5 * 7^2 * 11

I suppose I have seen them printed out that way, but the deeper structure there never clicked. Cool.

Comment author: 28 January 2010 08:27:20AM *  1 point [-]

As it happens I'm partway through "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery at the moment. Lots of problems are incredibly easy to solve given this structure. The example that springs to mind is the very straightforward proof why the combinatorial formula n! / (r! (n-r)!) always gives you an integer.

Update: Well having been scored up I feel like I should give a hint on the actual proof: for any prime p and any n, the greatest power of p that divides n is

\sigma_{i=1}^{\infty} floor( \over{n}{p^i} )

and for any real numbers a, b, floor(a + b) >= floor(a) + floor(b).

Oh for real TeX markup!

Comment author: 28 January 2010 05:12:39PM 0 points [-]

Do you recommend the book? If I were interested in the subject, is this good to pick up or can you think of a better option?

Comment author: 28 January 2010 06:10:59PM 0 points [-]

I'm enjoying it, but it touches on abstract algebra as an alternative approach rather than leaning on it for everything; I'd kind of prefer the latter.

Comment author: 28 January 2010 06:40:17PM *  1 point [-]

You may be a good person to ask this question:

I was wondering if there was a function f(x, y, z) so that x and z represent the left and right sides of common mathematic operators and y represents the level of operation. So f(1, 2, 4) would be 1 + 4 and f(2, 2, 4) would be 2 * 4. Better versions of f(x, y, z) would have fewer end cases hardcoded into it.

The reason behind this is to handle operator levels greater than addition, multiplication, and exponents. The casual analysis from my grade school and undergrad level math shows the pattern that multiplication is repeated addition and exponents are repeated multiplication.

My quick attempts at coming up with such a function are spiraling into greater and greater complexities. I figured someone else has to have thought about this. Do you know of a place I can start reading up on ideas similar to this? Is what I am doing even plausible?

Quick thoughts based on me playing around:

• Addition may be level 0, not level 1
• The sequences never really look exactly like multiplication tables, but the patterns are similar enough to appease me
• Ideally, everything can be reduced to the simple concept of X + 1 so as to walk along the number line
• In practical terms, I have no idea how to express "negative" levels. Division and roots are unapproachable at this point in my playing around.
Comment author: 28 January 2010 06:49:26PM *  4 points [-]
Comment author: 28 January 2010 06:55:50PM *  1 point [-]

Hyper operators. You can represent even bigger numbers with Conway chained arrow notation. Eliezer's 3^^^^3 is a form of hyper operator notation, where ^ is exponentiation, ^^ is tetration, ^^^ is pentation, etc.

If you've ever looked into really big numbers, you'll find info about Ackermann's function, which is trivially convertable to hyper notation. There's also Busy Beaver numbers, which grow faster than any computable function.

Comment author: 28 January 2010 05:52:15AM 3 points [-]

Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is.

A measure of the arbitrariness is the history, which is that 1 was considered prime up to the 19th century and was a matter of fashion during the 19th century. That suggests that unique factorization is not, in itself, enough to motivate the definition. Perhaps its extension to the gaussian integers or the more radical version for general number rings prompted the definiton.

Comment author: 28 January 2010 08:01:47AM 0 points [-]

This reminds me. Pre-19th century it was thought that part of what it was to be a mammal was to give live birth, in addition to having mammary glands. 1 is the platypus of numbers.

Comment author: 27 January 2010 10:56:51PM 4 points [-]

natural definition: "A prime is a natural number with exactly two factors"

I'm not sure I quite understand your suggestion: we should define 1 as prime, but then write "except for 1" every time we use the word prime? Wouldn't it be quicker just to exclude 1 in the first place (even if there were some sense in which 1 was prime)?

Comment author: 27 January 2010 11:28:39PM 1 point [-]

But a natural definition (anything along the lines of "already completely factored") would include it.

How do you see 0 or -1, using this definition?

Comment author: 28 January 2010 01:07:01AM *  0 points [-]

A factor of a number M is a number that evenly divides M with no remainder. Zero has infinitely many factors, definitely not prime.

...regarding -1, I can't think of anything relevant that I know about the relationship between negative numbers and prime numbers.

Later edit: Then I completely changed my mind... and decided 0 and -1 should be prime relative to how I would define it's essence. I note that you intuited what I really meant by prime better than I did!

Comment author: 28 January 2010 01:13:38AM 0 points [-]

Yeah, I was just curious. I like toying around with the fundamentals behind the maths and seeing what happens. :)

Comment author: 27 January 2010 06:49:02PM 2 points [-]

Uh? What do you mean by "obvious" in that last sentence?

(Post otherwise interesting, and I for one like them short.)

Comment author: 27 January 2010 07:21:56PM *  0 points [-]

"Obvious" as in not "weird, bizarre, or incredible." Would "simple" be a better word there?

Comment author: 27 January 2010 07:33:23PM 2 points [-]

I'm just not seeing what obvious reality it highlights, so either I'm particularly dense or it's not in fact obvious.

So, rephrasing: what reality is being highlighted by the "illusion" ?

Your prime number analogy suggests that it's in fact the "both colors are the same" assertion which is an illusion. The perceptual reality is that the pixels in these areas are discriminated as different colors. The illusion consists of looking at pixel with identical RGB values and thinking "Oh, these have the same position in colorspace, I expect my brain to perceive them as identical."

The reality suggested by the "illusion" is that this expectation doesn't hold in general, it's a stupid model. A smarter model would take more things into account before it predicted what our brain will perceive as identical colors.

But this is very much non-obvious...

Comment author: 27 January 2010 07:52:22PM 1 point [-]

I'm just not seeing what obvious reality it highlights, so either I'm particularly dense or it's not in fact obvious. [...] But this is very much non-obvious...

The post is keying off of Think Like Reality.

Reality has been around since long before you showed up. Don't go calling it nasty names like "bizarre" or "incredible". The universe was propagating complex amplitudes through configuration space for ten billion years before life ever emerged on Earth. Quantum physics is not "weird". You are weird. You have the absolutely bizarre idea that reality ought to consist of little billiard balls bopping around, when in fact reality is a perfectly normal cloud of complex amplitude in configuration space. This is your problem, not reality's, and you are the one who needs to change.

Looking at the image it should be obvious that the colors do not look the same. This is reality. We think they should look the same even though it is obvious they don't. Once we find the right answer to why they don't look the same, the illusion should stop being bizarre.

If you find an explanation, return to the illusion, and still think the illusion is bizarre, than something is wrong. You fall into the category that EY is discussing in Think Like Reality.

I am convinced that most of what we consider to be fancy illusions will be considered obvious to future generations. They will look at this image and wonder why we thought it was so fascinating. When our optics system is solved it would completely ridiculous to assume that we would look at that image and think that the two squares should look like the same color.

Comment author: 27 January 2010 08:19:46PM *  4 points [-]

But your post hasn't offered an explanation. And I don't, in fact, look at that image and think that the two squares should look like the same color.

A and B are in fact different colors, for a value of "in fact" which takes into account that the picture is a picture of something - a checkerboard. My visual system makes the correct inference, conditioned on the assumption that I'm looking at a checkerboard.

EDIT: what I should say is that I'm still surprised, knowing what I know about my visual system and how it works, when you tell me that the pixels have the same RGB values. But that's not a "reality is weird" surprise, it's more like the surprise of learning some interesting bit of trivia.

To really be totally unsurprised, I'd have to enhance not just my knowledge of the visual system, but my visual system - to include an RGB calibration system.

Comment author: 27 January 2010 08:33:50PM *  1 point [-]

EDIT: Oh, okay, I read your edit and that makes much more sense. I agree that it may be difficult to get to the point of being unsurprised. Getting there isn't obvious. You know you are there when you are unsurprised by the illusion. Once Reality is unsurprising and obvious, you are there.

I feel like I have lost the point of this conversation. What, in the following, do you disagree with?

• The image is an optical illusion
• The squares marked A and B appear to be different colors
• In reality, the squares marked A and B are the same color
• It is more correct to say that A and B are the same color than to say they are different colors
• The reason behind the optical illusion explains why A and B appear to be different colors
• This reason is contained somewhere inside of the "visual system"

• It is better to not be surprised by Reality

• The squares A and B should appear to be different colors
• We should not be surprised when A and B look like different colors

• It is incorrect to call Reality bizarre as per Think Like Reality

• "Obvious" is not a good word for the opposite of "bizarre"
Comment author: 27 January 2010 08:37:45PM 2 points [-]

I disagree with #3 and #4. Also #6, mildly - it's not just our visual system that's at issue here, it's our color vocabulary and our "meta" thinking about our color system that explains the "illusion" - that explains why we might think it bizarre.

Comment author: 27 January 2010 08:53:16PM 1 point [-]

Oh, okay. Well... I guess I don't feel like debating the definition of color since that is completely irrelevant to the point of the post. I wish this was made clear earlier.

Uh? What do you mean by "obvious" in that last sentence?

It has nothing to do with the actual answer to the example illusion. What I mean is that once we have the answer to an illusion, the illusion should stop surprising us.

Comment author: 27 January 2010 09:09:08PM *  3 points [-]

Neither do I feel like debating the definition of color.

What I am is disappointed. You brought up the "color constancy" phenomenon as an instance where "think like reality" is applicable, and then failed to follow through with an analysis of what is actually going on. You sound as if you are content to know that the phenomenon is in principle explicable; as if the post has done its job by demonstrating your commitment to "thinking like reality". I would prefer you had gone deeper into the object-level analysis and offered your own explanation of what is going on in this particular case.

This is a little bit like parents who lecture their children about the importance of being truthful, vs. parents who demonstrate being truthful - and being OK with confronting unpleasant truths.

EDIT: I didn't mean to sound sanctimonious (I realize I do sound sanctimonious). My main intent is to express a wish regarding what I'd like to see in future posts of this type.

Comment author: 27 January 2010 09:38:53PM 2 points [-]

What I am is disappointed. You brought up the "color constancy" phenomenon as an instance where "think like reality" is applicable, and then failed to follow through with an analysis of what is actually going on.

The point of this post is not to debate, discuss, or analyze color constancy. The point of the post is to talk about illusions and how we think of them as bizarre when we shouldn't.

I have not once debated color or the theories behind the illusion. All I did was use a word one way when other people use it another way. I am not trying to offer some strange, new truth about a picture. I used it as an example because people recognize it, not because that particular example matters.

EDIT: I didn't mean to sound sanctimonious (I realize I do sound sanctimonious). My main intent is to express a wish regarding what I'd like to see in future posts of this type.

I apparently have completely missed with this post. I have watched its karma swing all over the place in just a few hours and all of the 36 comments so far are talking about something I consider completely irrelevant to the intended point. The same thing happened with my last post, too, so something is very off in my expectations regarding people's responses to the post. Something I did caused you to expect something that I had absolutely no intention of providing. It sucks for you and it sucks for me.

I don't really mind you sounding sanctimonious because I don't care about our relative moral status. I find it frustrating that we had to go back and forth so long to end up where we did. I am not fully convinced my point ever did get across, but at least now I know how you perceived the post. Hopefully by the next one I will have figured out what went wrong.

Comment author: 27 January 2010 07:46:04PM 0 points [-]

In other words, this is the visual version of "if a tree falls in the forest...", except that we already defined 'color' as qualia rather than wavelengths, right?

Comment author: 27 January 2010 09:55:01PM *  3 points [-]

Since you mention it, that's something I should have brought up in one of the Mitchell_Porter consciousness threads: the colors you see are not actually matched up one-to-one with the wavelengths hitting your retina. Rather, the visual system does something like subtracting away the average color.

Meaning, the color that you experience seeing depends on all the colors in the scene, not just the wavelength of the light coming off each specific object.

Some people were talking as if you were getting direct knowledge of (something equivalently expressible as) wavelengths, which is unfortunate, since part of the path to demystifying qualia is understanding this kind of processing.

Comment author: 27 January 2010 07:59:45PM 0 points [-]

Um, "we already defined" - the referent(s) of that phrase are very ambiguous, I'm afraid. Who's "we" and where was that definition ?

I definitely agree that color discriminations in the brain (the processes that eventually end up with color words coming out of our mouths) are about way more than wavelengths. I'd prefer the term "discrimination" to "qualia", the latter carries philosophical baggage that I'd rather do without.

Comment author: 27 January 2010 08:06:26PM 0 points [-]

Um, "we already defined" - the referent(s) of that phrase are very ambiguous, I'm afraid. Who's "we" and where was that definition ?

It's a royal 'we' in this case: Some subset of the group of commenters here at LW, and that subset doesn't include me. It was discussed at some length in the recent discussion of consciousness. I wasn't paying much direct attention to the conversation, though, so I can't be more specific than that. (I'm not even sure that the relevant bits are all in one post's comments.)

Comment author: 27 January 2010 07:38:44PM 1 point [-]

I think what you're trying to say is that the phenomenon becomes highly expected, not aberrant with respect to your world model.

Comment author: 27 January 2010 07:55:24PM 2 points [-]

Yes.

Comment author: 18 February 2010 08:07:51PM *  1 point [-]

There's a rather awesome colour constancy optical illusion in this American Scientist article - click on the enlarge image link on the rubik's cube image. I've mirrored the image here in case the link goes dead. The blue tiles in the left image are the same shade of grey in RGB terms as the yellow tiles in the right image. H/T to this article.

Comment author: 28 January 2010 05:16:08AM 1 point [-]

If I look at the picture without focusing (i.e. thereby seeing it as 2D), A and B look the same color.

If I look at the picture and focus but cover the green cylinder with my hand, they can look either the same or different (depending on whether or not I notice the shadow in that case.)

I agree with Andy Woods: this is no illusion at all, except in the sense that it is an "illusion" that the board is three dimensional.

Comment author: 27 January 2010 07:35:46PM *  0 points [-]

Since the topics are related (but I'll admit I'm biased toward seeing that) - maybe a "Related" link to the "Adaptive bias" post would make sense.

Comment author: 27 January 2010 07:13:41PM *  0 points [-]

"Why do I keep thinking A and B are different colors? Obviously, that is not what my eyes are trying to tell me." I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors. That information is not being transmitted by my eye. If it were, the illusion wouldn't be an illusion.

I think this is a bit misleading. To the extent that your eyes can be thought to be trying to tell you anything, "these are different colors" is exactly what they are trying to say. It's fallacious to assume that there is some sort of "true" input that one's eyes are reporting, and which the post-processing stages corrupt. (I'm not saying you committed this mistake, but people might get that impression.) Instead the meaning emerges purely from the post-processing stages. In fact, they're doing a pretty darn good job, as one might note from the fact that optical illusions nearly never noticeably distort our judgment in natural conditions.

Comment author: 27 January 2010 07:21:07PM 1 point [-]

Yes, I completely glossed over the finer points of eye-brain interaction. I did not think it was needed to get the point across. I suppose I also used a bit of linguistic sleight-of-hand by implying the eye "tells" me things. I sort of just called everything from the occurrence of light leaving an object to me perceiving the light leaving the object as "The Eye."

If you can think of a better way to say it without adding a heck of a lot of complexity I am more than willing to edit the post.

Comment author: 27 January 2010 08:04:59PM 1 point [-]

I'm not sure if the eye-brain interaction is the relevant part. Even if you change "eyes" to "the visual system", the point of "these are of a different color is the very thing your visual system is trying to tell you" remains true.

Comment author: 27 January 2010 08:16:33PM 0 points [-]

What would you call it? The point remains that the colors are not the same. If we think the colors are the same, we are incorrect.

Comment author: 27 January 2010 10:10:28PM 0 points [-]

They are not the same, but our visual system is trying to tell us they are the same - and you can't really say it's wrong to make that judgment, as doing so leads to correct results the overwhelmingly vast majority of the time. (Basically, I'm saying the same thing as AndyWood's comment and the responses to it.)

Comment author: 27 January 2010 10:48:10PM 1 point [-]

They are not the same, but our visual system is trying to tell us they are the same.

Yeah, this makes sense. The trick is asking, "The same what?" The answer, "Color" is not descriptive enough.

I never meant to say it is right or wrong for the visual system to do whatever it is doing. I mean to say it is wrong to expect something different from the visual system than what it is doing.

Comment author: 27 January 2010 07:23:19PM 1 point [-]

"my optic system"? "my visual system"?

Comment author: 27 January 2010 07:27:41PM 0 points [-]

Optics system works for me. I changed a few of the sentences to use this instead of "eye."

Is it "optics system" or "optic system"?

Comment author: 27 January 2010 07:41:02PM 0 points [-]

*checks wikipedia*

Apparently it's either 'optical system' or 'visual system', and the latter seems to be preferred for organisms.

Comment author: 27 January 2010 07:54:25PM 0 points [-]

Updated to visual system.