thinking

there is no probability theory - only zuul!

2009-04-30T22:39:00.000-07:00

The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance.

Probability is relative, in part to this ignorance, in part to our knowledge.

-Pierre Simon de Laplace, in A Philosophical Essay on Probabilities

I've always been fascinated by the question of determinism. Is everything about the future already contained in the state of the world today?

Addressing this question, one has to come to terms with what it could mean for the world to be non-deterministic. Intuitively, it means that certain events are non-predictable - we can think of them as random. And here is where we have to confront the fact that any useful model of the world, even a probabilistic one, has to be formally deterministic.

Why? Because, in math, there is no random. When a mathematician or statistician studies random behavior, they consider functions whose input is thought of as random. And by thinking of an input as random, the intuition and practical applications follow.

For example, suppose you ask what the standard deviation is for choosing the number 5 a third of the time, and the number 17 the rest of the time. You can model this with a function f:[0,1]->{5,17} which maps [0,1/3) to 5 and [1/3,1] to 17. From here you can study the random behavior of this experiment to your heart's content. The point is that the function f is completely deterministic, and there is nothing here to give us a single example of an actually random number.

So when it comes to a probabilistic model of the universe, we would still have to use a deterministic function. To include the idea of randomness, we could add an extra input, which we think of as unknown or external to the world. For a moment, pretend that the world moves forward in discrete time units. Let x be the state of the world, y an unknown input, and the formula

x' = f(x,y)

gives the state of the world a moment later. The presence of y is the non-determinism.

The thing that's easy to forget is that probability theory does not dictate the way the world will behave. Rather, it is nothing more than a way to do something with a balance of partial knowledge, and an awareness of our own ignorance.

Here's a quick thought experiment to help illustrate the difference:

Choose a number between 1 and 1000. I'll also choose a number in the same range. What's the probability that they're the same?

It's a trick question. They either are the same, or they're not. Of course, if you assume that we're both equally likely to choose any of the 1000 numbers, then you can say the odds are 1 in 1000. But the point is that this model is entirely in your mind. Since it's a one-off experiment, and we don't really even know that these numbers are chosen in any manner we could call random, there's no real basis for that probability model. Maybe I always choose 7 and you always choose 23, and the numbers were bound to be mismatched from the start. The point is that, while probability is incredibly useful, it is wise to keep in mind - as Laplace did - that it is a mitigation of the unknown.

One last thought experiment toward comprehending what a probabilistic model might mean:

Some physicists play with the idea of parallel universes, so let's do something like that. Suppose I toss a coin (heads/tails = H/T) several times in a row, and write down the outcomes in order. If each outcome is random, then we can model that in at least two different ways. One way is to assume that there is an external source of randomness, and that every result pulls in information from that random source - this would be analogous to the extra input to the function f(x,y) as above. Another way is to assume that every time a choice is to be made, the universe forks into two parallel versions: one in which the coin lands H, and another where it lands T. There is never a decision to be made, so no extra information is needed. Which model is better? Is there a way to choose between these two models, if we were in that world?

There is basically no difference. Why not? Try this out: write down every H/T possibility for any number of coin tosses. For 3 tosses, you would get HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Most of those lists look random. In fact, you wouldn't be amazed at any of them. Do the same thing for 50 tosses in a row. 50 heads in a row would be surprising, but the number of lists besides that one is orders of magnitudes larger than the total number of humans who have ever lived so far.

In other words, no matter what, most of the outcomes will look random. In fact, they have to, by any reasonable definition of looking random. This is simply because a probability is nothing more than the number of outcomes including a certain event divided by the total number of outcomes (implicitly assuming they are equally likely). So any very likely event, such as there being about as many tails as heads, will by definition include most of the H/T sequences you write down.

A random model looks the same as a parallel model.

So when I say there is no probability theory (only zuul), what I really mean is: probability theory is just the study of deterministic functions where you really don't know the input, but you pretend to know it's distribution.

experimental integrity and the search for causality

2009-03-24T18:43:00.000-07:00

The phrase the scientific method implies that there is some universal, automated process that investigators blindly follow in order to do science. In truth, there is a great deal of improvisation and creativity required for the doing of good science. Great leaps forward, such as general relativity or the complex (as in complex numbers) proof of the prime number theorem, often rely on bold, inspired insights into the nature of an unsolved problem.

However, there are a few common principles that unite the rational attitudes of modern research. I want to highlight a few that I feel are somewhat neglected. They are:

experimental candor,
easily reproducible experiments, and
induced correlation.

Experimental candor

Here's a nice way to get great results: suppose you think that drug A will help people lose weight. Conduct a thousand studies on small groups of test subjects. Suppose one of those studies shows good results - publish those good results, and throw away the rest of the results.

This may sound a bit unrealistic, but something like this can happen much more easily in computer science. In this case, there is a growing field of algorithms which are both probabilistic and approximate - very similar to experimental drugs in medicine. If they do pretty well most of the time, that's good enough. Yet with an algorithm, it's incredibly easy to run a million trials of your code, and only publish the best subset of that. Even if the quality of your results are completely random, it's just a matter of time before one small subset of the test results look good.

Hence the need for experimental candor. It's important to reveal all the relevant experiments performed, including the negative or inconclusive ones. The web is the perfect platform for this kind of data disclosure - you can pre-publish your intended experiments and hypotheses before you actually run the experiments. This way, good results look better, and other researchers won't waste time on previously failed experiments. Of course, it's always possible that an experiment failed for unaccounted-for parameters (including human error), which is why experimental reproducibility is also crucial to good research.

Easily reproducible experiments

This scientific tenet is well agreed upon, but poorly executed. In practice, I know of very few experiments which can be very easily reproduced at the research level. In some cases, one may wish to build upon the work of another, such as by augmenting a biochemical procedure with a new step. Articles involving experimental lab work do indeed contain careful procedural explanations meant just for this purpose, which is great. But in many cases, even this is not enough for other researchers - in my days as a grad student, I would see other grad students emailing or calling other investigators (often ones who were considered serious competitors) to ask for critical clarifications in procedure.

We can do better than that.

I'm going to pick on computer scientists for a moment, because they're the worst offenders. An algorithmic experiment has the most potential to be easily reproducible. Ironically, it seems typical to leave out necessary parameters to perform the experiments used in many papers. In order to reproduce a certain graph of time complexity versus input size on a certain real-world dataset, for example, a reader will often have to code up the algorithm based on very vague pseudocode and hand-wavy explanations, guess at parameter values, and separately download the dataset. I've even seen code used which was nowhere available in either pseudocode or executable code - the reference given was by personal communication with another researcher (who won't answer my emails).

There is no excuse for this. Any good algorithmic experiment can be reproducible at the click of a button. The experimenters have already written the code - it is simply a matter of adding a link to this code to a website. It would be friendly to add a little documentation; or better yet, to follow a pattern of operation for the field, in much the same way that some software installation procedures have become standardized.

Induced correlation

This point is a call for the conscious recognition of an idea that's been implicitly used for some time.

Certain experiments have the goal of looking for something like a causal relationship. If a drug company is testing a weight-loss drug, they want to know that their drug causes the weight loss, as opposed to it causing something else, or something else causing the weight loss.

Unfortunately, there's no fool-proof way to experimentally test causality. This is a well-known problem. It's also interesting to note that, philosophically, causality itself is subjective in nature, although that is the matter of another post.

Here's the trouble: Let's hypothesize that chemical X causes weight gain. As an experiment, get a large group of people together. We randomly select some folks as the control - they won't change their diets, and we randomly select some others to change their diet to no longer consume chemical X. We see the desired results: the control group gains a little weight on average, but the experimental group (no chemical X) actually loses some.

Does that mean anyone can prevent weight gain by avoiding chemical X? Absolutely not. Here is one possible explanation: Suppose that the vast majority of foods contain both chemicals X and Y together, or not at all. So when the experimental group avoided X, they were also avoiding Y without knowing it. Now you unleash your study on the world, and everyone starts avoiding X. But there are some foods with chemical Y in it, without X. It could happen that those foods become more popular, or that certain people subconsciously crave Y. In either case, we have people consuming Y, not X, and gaining weight.

Is there anything we can do to experimentally show something stronger than mere correlation? A little bit, yes - we can show induced correlation. This is a correlation between parameters which was observed specifically by either turning on or off the cause in each trial, and purposefully leaving all other known parameters the same. Let's use the term natural correlation to indicate experiments where the cause was either present or absent without any control by the experimenters. Induced correlation gives more evidence of causality than natural correlation since there is more evidence that we can control the effect by controlling the cause.

I think this general idea has been understood already, but I'm not sure that it has been explicitly recognized. My goal throughout this post has been to encourage the codification and emulation of a few good core principles of scientific investigation. There are definitely more key principles, although I've been reminded many times that at least these three could use a little more awareness and observation.

thoughts on junk DNA

2009-03-06T20:33:00.000-08:00

It's interesting to think of DNA as the source code for life. A lot of ideas fall into place nicely with this analogy.

You need some sort of compiler or interpreter; this role is given to RNA. You need a basic set of atomic instructions, and something like labels to certain parts of the code base - pointers into memory. Codons are the instruction set, with start codons helping to act as labels. A central processing unit executes the commands - ribosomes turn the codon sequences into proteins, and the proteins interact to achieve various goals. Chemistry itself is the ultimate processor, but it takes more focused form in the complex interaction of the enzymes produced by the DNA. Some of the proteins act as inhibitors, decreasing the activity of enzymes; others are activators, doing the opposite. These constructed molecules are capable of effecting or halting the production of still other amino acid complexes. The end result is a logically sophisticated dance worthy of the millennia of evolution which produced it.

As I write code on my own, in an experimental fashion, I sometimes don't worry about the readability of the code. It is in this scenario that the evolution of source code best matches that of DNA. There is a small cost to having extra/old code, yes, but it is far outweighed by the raw functionality created.

Looking at some source which has grown up just a little bit, mostly unsupervised, offers a few suggestions about bits of information that may, at first glance, appear non-functional (aka junk DNA):

Old functions which are never or rarely ever called
As code evolves, some functions become less useful, or replaced by newer ones. It would make sense that some codon sequences would become obsolete, and the encoding would remain in the DNA.
Literal strings and other initialization data
There might be a bit of initialization data in DNA - information not obviously functional, yet still used. For example, some DNA may only be active for a very short time when an embryo is first developing, or triggered temporarily at certain key development stages. An even more interesting hypothesis is the possibility that some instincts, or primal knowledge, are somehow encoded in DNA, in a manner somewhat different than traditional protein transcription.
Debug code
Debug code is useful for figuring out what part of a process has failed. Although there may not be a conscious debugger to check the output, we could still hypothesize that a little extra information about each step in a procedure could give enough information to locate and react to a failure or attack in the system. In this case, the usually non-functional code would be rarely and temporarily activated as a defensive mechanism.

top movies of 2008

2009-02-24T18:19:00.000-08:00

There are two things that make a movie worth seeing: because it moves you or makes you think by reflecting on reality, or because it entertains you by helping to escape reality.

If we are pained by a sympathetic situation that was experienced by someone in history, or alive today; or if we feel vicarious joy for a simple act of triumph (say, winning a spelling bee), the reality of the situation, symbolic or literal, is a key factor in our empathy. We are moved because this is the way life really is. On some level, we can relate to the plights and victories of these characters.

On the other hand, it's nice to tickle your imagination from time to time with an escape. We don't really believe Indiana Jones could easily be real, or that Spider Man might one day exist. Nor do we expect a monster like Godzilla to ever attack a nearby city (after all, Tokyo is far away for most people). The entertainment here lies with a contrast to reality. Everyday lives are kind of boring. Monsters don't attack, nobody wins the lottery, most days you don't fall in true love for the first time, or find an alternate dimension, or save the world. But it can be cool to daydream.

The best movies work with these principles - they choose a side. Sometimes you can mix these two aspects, but you have to be careful about it. If a piece of a film is just-for-fun, there's no harm in bending - or even reversing - reality. But if you're trying to move your audience, trying to comment on a state of the human condition, you have to be more careful. Symbolism and admitted exaggeration can work, because we understand the reality being represented. But to toy with reality to suite the message of the film is to defeat your own purpose. For example, Syriana presents a terribly bleak and pessimistic view of political and industrial intrigue. We are lead to believe that this situation could be real, but it felt like they were stretching a little too far; as a viewer I felt bereft of both entertainment and reality.

This is some background for my top movie picks of 2008. I think in ten years, these movies will still be worth watching, while a lot of other highly anticipated films from this year will be forgotten.

Gran Torino
Clint Eastwood knows film. He's been involved in about a half century of movie evolution, and I think he's kept the good parts of more traditional film alive here. The film says a lot without being overly symbolic, and the characters are visceral and quotable without feeling cliched. This is a contemporary, somewhat realistic (even if parabolically so) film about redemption and opportunity. It's good because we leave the theatre feeling for the story, not worrying about its plausibility. There are no u-turns or magic revelations. Everyone is flawed or troubled, and nobody wins everything. Yet there are pieces of fun, of power, of thought, of sacrifice, and of compassion.
Wall•e
Only Pixar could seriously attempt this: Let's make a dialogue-free, post-apocalyptic love story between two robots caught in a conspiracy that might crush the threadbare hopes of the space-stranded remnants of humanity. For kids. And somehow it works. Wall•e is visually rewarding, touching, whimsical, nostalgic, and engrossing. It's sci-fi speculation is escapist entertainment first, and social commentary far second. It gets away with allegorical statements on the irresponsibility of humanity because the reality in it is not presented as the truth, but rather as a kind of cautionary fable.
Iron Man
There are two common superhero movie mistakes: they don't know how seriously to take themselves, and the heroes are often portrayed as everyday people who happen to have a heart of pure gold. This film tackles that second mistake - Tony Stark is neither your everyday guy, nor endowed with such heart. We like him because, unlike our super/spider/batmen, when we become iron man, we don't have to shoulder the great responsibility of great power, and we don't have to cower under a shroud of modesty. We can just do our thing and enjoy the moment. Somehow I find Tony Stark more realistic and more entertaining at the same time. Of course, this film is not about the human condition - to spell out the obvious, this is just for fun. And it succeeds.
Be Kind Rewind
See The Science of Sleep before you see Be Kind Rewind. Michel Gondry is a child with the ability to turn his daydreams into movies, and to really appreciate the world you've entered, it helps to speak the language. This one got a number of poor reviews because it's outside the realm of normal moviedom for casual viewers. It's unusual Gondryan style is cubism in crayon. And this is the subtle genius of it. When a critic is confused, they have to decide if it's because the movie is above them or below them to avoid looking dumb. With Be Kind Rewind, the confusion is simply a different narrative medium -- the film is just for fun, but seriously so. If you try to use Duchamp's fountain the way you're used to, you'll be missing the point.
Enough defense. Be Kind Rewind is good because it's fun. The characters and the plight - the foundations - are tangible. Beyond this - the devices and exposition - there is not much pretension of reality. The key components are in place - what's real is what moves us, what's art is what makes us laugh.
Wanted
Like Iron Man, Wanted breaks the chains of the stereotypical hero movie. In this case, it really doesn't take itself too seriously. On top of this, the dramatic tension is very personal - Wesley (our protagonist) desperately wants to avenge the death of his father. Saving the world takes the backseat. It works because it doesn't really bother with the less entertaining aspects of the world - things like the rules of physics applied to bullet trajectories, or oracles more traditional than giant looms.

Honorable mentions: Cloverfield, Kung Fu Panda, Pineapple Express

Enter The Tangent Space (dot com)

2009-01-26T14:34:00.000-08:00

Here's a new site for exploring cool math ideas: thetangentspace.com

My blog posts here have often alternated between technically detailed mathy or algorithmic thoughts, and more informal musings on life, the world, and interesting things in it. In my mind, there are really two audiences - one for the mathy stuff, and one for everything else. So it probably makes sense to have two places to put these different thoughts. From now on, my posts here will be less mathy, and I'll feel free to go math-crazy (or algorithm-crazy, as the case may be) on thetangentspace.

I'm trying a new type of blog with thetangentspace. It's about math research, and research is about communicating and collaborating. Even if it's a slow channel, it's an interactive process. So thetangentspace is both a blog and a wiki. The blog is meant as an easy stream of intuitive ideas - something you can keep up with, without investing too much thought. The wiki is where the details go - the full proofs and formal definitions. It's also a place for other mathematicians to make significant additions - beyond what you can leave in the comments of a blog - using the same software as wikipedia. My hope is that some of the ideas and questions I post will inspire others to build on these initial offerings.

Check it out!
thetangentspace.com

mathskool.com

2009-01-01T18:23:00.001-08:00

I just launched the alpha version of mathskool.com.

This is a website I've been working on for the past month, meant to help connect great math teachers with motivated middle and high school students. The idea is to provide a centralized library that many math teachers can contribute to, and which gives students free access to short, focused videos. I imagine teachers recommending them as supplementary material to classes, or students searching for a single particular topic while stuck on their homework or studying for a test, or even curious people learning new things on their own.

I plan to continue adding features and videos to this site gradually over time. YouTube and other math-oriented sites already offer videos, but I think mathskool is unique in focusing on math education, being free, and encouraging a more interactive community with a nice question/answer system. For now I've included a few videos of my own, and several from other people.

Let me know if you know any math teachers who might be interesting in using or contributing to the site. The next step is to start building a community of users - teachers and students.

Check it out! mathskool.com

the scaled interest principle

2008-12-30T23:52:00.000-08:00

Here's an idea that I've seen in action throughout my life, although I've never seen it explicitly put into words:

Events of interest tend to happen more quickly at smaller scales, and slower on large scales.

Interpreting relativity as putting a speed limit on the flow of information, gives a natural justification of the principle in the physical world. The idea jumps out when you consider the (admittedly imperfect) analogy between atoms and solar systems.

We can also see it in other ways. Small companies usually react more quickly than big ones. Flies move more quickly, and die more quickly, than elephants or whales. Smaller computer programs often run faster than large ones. Things happen faster in dense cities than in a sparse countryside. An idea of little interest fades faster than a popular meme. A simple system is easier to work with than a complex one.

driving is kind of crazy

2008-11-30T22:27:00.000-08:00

It can be fun to ask yourself questions that help re-think our world. An example:

Suppose an advanced alien civilization discovered our human life on earth, and studied it carefully. There would certainly be some things that stand out to them as particularly strange, ironic, or just stupid. Which things would stand out?

I'm willing to bet that our current modes of popular transportation would stand out -- in particular, driving a car somewhere (as opposed to taking a trian, boat, or plane).

Why? Let's evaluate the danger of an action, very approximately, as the frequency you take this action multiplied by the probability at any point during the action that you will become seriously injured, mauled, and/or deadified. On average, this is basically proportional to the number of "man-made" fatalities caused per year in the course of performing an action (I say man-made as opposed to pre-existing medical conditions -- otherwise, we would have to say "having a beating heart" is a dangerous action, since many people have heart attacks).

Along this line of reasoning, it stands out immediately that driving (or being a passenger in a car) is the single most dangerous activity you're likely to perform on a regular basis. And, if you think about it just a little bit, it's not so surprising.

Let's add another test to help discover perilous conducts: does being drunk make the activity stupidly dangerous? Clearly, this is not true for 99% of the actions you take every day. You can read your email while drunk, you can walk around your house, you can listen to music, eat food, watch movies, try to do paper work, chat with friends, play with your dog or cat, read a book, sing karaoke, dance, or play games while drunk without any danger to your person. Yet, clearly, drunk driving is so dangerous that it is a serious legal and societal offense.

I don't seriously expect anyone's behavior to change in light of these thoughts -- I still drive around all the time. But it continues to surprise me to apparently be alone in considering driving, while pragmatically necessary, a highly precarious practice. Certainly we can imagine worlds in which people move about in some way where a strong twitch at the wrong moment can't kill anyone. Packets of information fly around the world following routing protocols and get safely where they're going. Snail mail networks provide another huge and relatively safe means of transportation. Passenger trains, boats, and planes are operated by professionals under careful coordination with much lower risk of collision and higher standards of maintenance. Why not cars?

fairness is contageous

2008-10-31T14:22:00.000-07:00

Here is a trick to simulate a coin toss between two people if both of you are caught coinless: each person secretly chooses a number, either 0 or 1, and both announce their choice simultaneously. If both numbers match, this is heads; otherwise tails.

This idea can give surprisingly fair (i.e. close to 50/50) results.

Suppose you choose fairly, but your so-called friend is nefarious, and chooses their number in any non-50/50 manner. For example, they could always choose 1, or always choose 0, or always try to predict your answer and choose the same number as you.

As long as you choose fairly, though, the ultimate answer will also be exactly fair. Why? No matter what your friend chooses, you'll choose either the same or the different number with 50/50 probability. If you're really choosing your number fairly, then there's no way for your friend to make any kind of prediction (by the way, this unpredictable property is one way to think about probabilistic independence).

Of course, the situation is symmetric: if your friend plays fair, then the ultimate outcome is fair, too, no matter how nefarious you are. Fairness trumps nefarness.

Even cooler: even if you both play unfairly, though still independently, then the ultimate outcome will still be more fair than either of you would have been acting on your own. The math on this is pretty simple.

Suppose you choose 0 with probability 1/2 + b₁, and your friend chooses 0 with probability 1/2 + b₂. I'm writing the probabilities this way because it makes the calculations easier. We can think of the b's as the "bias" of each person's randomness. A bias closer to 0 means a more fair result -- closer to 50/50.

Using b₁ and b₂, what is the probability that the outcome will be a match? It's Prob(both heads) + Prob(both tails) =

(1/2 + b₁)(1/2 + b₂) + (1/2 - b₁)(1/2 - b₂) =
1/2 + 2b₁b₂.

In other words, the combined bias is 2b₁b₂. Notice that each individual bias is in the range [0, 1/2], so the combined bias is also in that range. Also notice that, if both biases are < 1/2, then the combined bias is less than either individual bias. This is what I meant by saying that the combined outcome is more fair than either player alone.

In fact, things are a lot more fair since this is a multiplicative effect. Suppose you're sitting around with the unshakable urge to produce fair random binary digits. Alas, you empirically discover that you seem to choose 0 with probability 60%, and 1 the other 40% of the time. What are you to do??

Just write down a few random 0/1's in a row, and take the XOR of this list of numbers. This is just a slight generalization from the above 2-player version. (By the way, this is the same as giving an ultimate outcome of 1 if there are an odd number of 1's in your sequence; 0 otherwise.) If you started with bias b, then taking the XOR of n bits in a row will give you an ultimate answer with bias (2b)ⁿ/2.

Why? We can confirm this formula by repeated application of the above derivation that 2 players end up with combined bias 2b₁b₂. The sequence of biases for a single player looks like this:

b → 2b² → 4b³ → ... → (2b)ⁿ/2.

To get an idea of how incredibly useful this convergence is, suppose that your personal bias is b=60%, and that you want to be within 1 millionth of perfect fairness. How many times n must you perform a single (60%-biased) choice in order to arrive at an XOR which is this close to perfect fairness? Only nine times! This works because (2*.1)⁹/2 < 1/1,000,000. If you ask me, this is a pretty small price to pay to go from a 10% bias down to 0.0001%.

exposé: poorly Lego'd Eggos

2008-09-21T17:19:00.000-07:00

It's always interesting when a new mix of two different worlds pops up. Remember when Drew Barrymore married Tom Green? Sometimes these things work out - like the glory of rock stardom and video games. Other times the result is disturbing and creepy - like anime and ronald mcdonald.

I try to be optimistic, so I was pretty excited when I found out Eggo had released a Lego-themed waffle. At long last, food I can legitimately play with! This was awesome - two things I really enjoyed as a kid (and still do, though less often) - creativity and edibility, all in one.

Imagine the profound depths of my chagrin when I discovered the horrible truth. Gentle reader, let my folly be your tale of caution:

You can't actually build anything with Lego Eggos.

Someone, somewhere along the way made an unspeakably heinous design decision and decided that three holes on the bottom of the wafflebricks would fit just fine with the 8 pegs on top. Please refer to the photographic evidence.

Why?!?!? As a mathematician, I can attempt to quantify the magnitude of this engineering catastrophe with a simple formula:

8 pegs + 3 holes = WTF

As you can see, these wafflebricks do not stack any better than standard waffles.

You have been warned.

mistaking your wife for a hat

2008-08-30T15:51:00.000-07:00

In 1985, neurologist Oliver Sacks published the book The Man Who Mistook His Wife for a Hat - a collection of 24 essays exploring fascinating case studies in neuropsychology. The title essay describes a man suffering from a form of visual agnosia - a result of brain damage in which vision is intact, but comprehension of vision is impaired.

For some reason, the man in question is able to see and understand the general shape of things, but has immense trouble "seeing" people and certain other things. The most curious example is the case when, as the man was leaving an office, he searched for his hat, and apparently attempted to remove his "hat" from the top of his wife's head. Since he could understand that the top of her head was hat-shaped, he surmised that it must be the hat he was looking for.

In another case, he was easily able to recognize a wristwatch. Yet, when he was shown a glove, he had a great deal of trouble calling it a "glove", or recalling its function. Instead, he was able to very accurately describe its general shape - a flexible series of tube-like structures ending in half-spherical tops, all joined at one end with a hole. In general, he had a great deal of trouble recognizing people and familiar faces, including his own wife.

I would like to add a hypothesis to the essay: that this form of agnosia suggests a neurological dichotomy between recognition of organic vs inorganic objects. Basically, this story fits the idea that there's one part of the brain designed to recognize many natural objects - faces, hands (& gloves), dogs, cats, etc - and another part designed to recognize artificial or learned objects - wristwatches, tubes, cubes, etc. Such a dichotomy also makes sense evolutionarily, since brains that had built-in recognition of things that it could "evolutionarily know" were around, such as familiar human faces, would have a great survival advantage. The human race probably hasn't had wristwatches long enough to evolve a glob of neurons just for them, so we may store the visual recall center for watches in a more learning-oriented part of the brain.

By the way, if you have any interest whatsoever in curious neuropsychological phenomena (who doesn't??), I highly recommend Sack's book. It also includes a case of a "Jimmy G." who suffers from Korsakoff's syndrome (to simplify - severe amnesia) which I'm guessing was an inspiration for the (very cool) movie Memento; and another case about an elderly woman who's happy to be suffering the tertiary stages of a usually fatal disease, which I'm guessing was an inspiration for a certain subplot in an episode of House, MD :)

PS Tangent: why does it "sound wrong" to say "an usually fatal disease" when I know that's grammatically correct?

should considered harmful

2008-07-16T21:24:00.000-07:00

When I was young, I learned a few words to never say. The entire concept of "bad words" always seemed funny and weird. It really doesn't make much sense to disallow the utterance of a few key phonetic components, as if these somehow could really be inexcusably more offensive than other more coherent and meaningful strings of sound.

As I've played with the character of a personal dialect, I came to my own decision to swear less because, often, swearing simply means you couldn't think of a more articulate way to express yourself. To stretch an analogy, swearing is to communicating as drinking is to having fun. It might seem cool when you first try it, but in the end it's more a pain than a panacea.

Finally getting to the point: this post is not all about the classically crude curses, but rather about the existence of a few choice words - well, one in particular so far - that might actually deserve all the hullabaloo of a terminological taboo: the word "should". It is extraordinarily easy to invoke this word as a means of detached and elitist censure.

So what makes a truly bad word? I proffer that a word which strongly implies, or depends upon, the acceptance of questionable ideas as a basis for its meaning, is a word which allows for easy abuse. In the case of should, the questionable idea is that of either a universally-agreed moral code, or a moral expert. A phrase like "you should never burp at dinner" carries with it the weight of unjustified authority. In fact, such advice might leave you a rude dinner guest in some cultures.

To be constructive, let's consider what can be said instead. If you want to avoid the embarrassment of someone new to the dining customs of the U.S., you might say that many American hosts consider it rude to burp loudly during meals. If you're teaching your child about your moral position on theft (let's assume you're against it), you could explain the legal, material, and emotional consequences of stealing, along with your own personal disapproval. If you have your own moral decisions that you've reached rationally, then it's always possible to express yourself without resorting to vague implied universal standards.

Am I so vulgar as to say you should never say should? Absolutely not -- I'm really trying to emphasize personal choice and rationality here, after all. So, with respect to any readers who chuckle at such lexical eccentricities, this post aims at nothing more than a reminder to speak responsibly :)

Turning a convex partition into a Voronoi diagram

2008-06-29T00:31:00.000-07:00

[New wiki page on Voronoi diagrams]

A Voronoi diagram (named after Russian mathematician Georgy Voronoi, 1868-1908) is a partition of space based on a set of points. This is the main idea: we are given a set - usually small and sparse - of points which we'll call Voronoi centers. Let's suppose that each of these center points has its own color. Then we can color all of space by coloring any point according to the center it is closest to.

In the figure here (from wikipedia), each black dot is a Voronoi center, and the cells are colored accordingly.

It's easy to think of some places where we might see similar patterns in nature. For example, the shapes on a turtle's shell, or the hexagons of a honeycomb resemble Voronoi diagrams. Close-packed living cells sometimes do as well, maybe because as they became further squished together, their boundaries approached a shape optimized for packing.

You might notice that every cell in a Voronoi diagram is convex - it is a shape without "dents" - every line between two points in a Voronoi cell is contained within that cell (contrast this with, say, pac man). So every Voronoi diagram is a convex partition. Is every convex partition also a Voronoi diagram? No! In fact, there are many ways to draw a convex partition which is impossible to represent as a Voronoi diagram (see the wiki for some details).

But for those convex partitions which allow a Voronoi diagram, how hard is it to find the corresponding center points? With the right algorithm, it's not so difficult. Here's the main idea: draw a circle around some intersection of three boundaries in the convex partition, pick a random point, and reflect it around the boundaries. After reflecting three times, take the average of this point, and the starting point, along the circle. What you get is a working Voronoi center for that intersection. Reflect this point around to find the other two centers (if you reflect it three times, you'll just get back to your original point). Check out the figure.

Once we have candidate Voronoi centers for any two adjacent 3-intersections, you can expand the circles on which these centers exist until the centers line up. These centers are guaranteed to be the correct unique center for their cells (assuming that the convex partition is a Voronoi diagram). From here, just reflect the centers over the neighboring boundaries of the partition to find all other centers.

This is just a quick summary of a few of the thoughts on the wiki page.

Company ToDo: Something New

2008-06-02T13:20:00.000-07:00

Let’s imagine: tomorrow your employer dissolves. All corporate assets are split evenly among all employees, and everyone is told to work for themself, or in a group of former co-workers. Would the innovation in your sector get better or worse?

Let’s call this hypothetical post-employer innovation your company’s creative baseline.

A lot of companies follow goals that look like this: make money by solving problems we can get paid to solve. Often, they think this way because it’s just a job to most employees -- including executives.

Without any serious experience or business degrees to justify my claims (well, beyond some raw rationalization below), I’m going to propose the following business paradigm:
1. Do one thing very well
2. Keep your creative baseline high
3. Exceed your baseline

Of course, these ideas go together. You could always claim 1 as your ultimate goal, and argue that money and innovation are just subgoals to keep doing that one thing really well, for more customers, over a longer timespan.

But it seems as if a lot of companies aren’t thinking about 2 and 3.

Keeping your baseline high means working with people who think for themselves - and are good at it. If you’re hiring people you couldn’t imagine as worthy competitors, why are you expecting anything more from them at your company?

Exceeding your baseline is even trickier. But this is the fundamental idea of growing a company in the first place - you can do better together than you could apart. The problem is that an employee, as opposed to a founder, inherently has less freedom and less accountability for their contributions. At the same time, you can offer them access to invaluable resources they’d probably miss on their own: capital, an existing corporate reputation, and each other.

Why should these goals, among the plethora of proffered principles, be singled out?

Because stagnant and errant businesses die, but sectors keep on going. Company Old doesn’t disappear because customers stop caring about a general problem they need solved -- it gets replaced by Company New that offers a better approach. The golden ticket for Company Old to avoid replacement is to employ the creative minds that can create the new - and to enable them to build Company New from within.

wiki time

2008-05-15T22:46:00.000-07:00

Have you ever wanted to put something very silly, esoteric, zany, or just very poorly edited on wikipedia, but never bothered because you knew it would be removed immediately anyway? I know the feeling. But now at long last is your chance to edit a very new, wide-open, and very-poorly-edited wiki!

Introducing the thinking blog's wiki.

To get things started, I've added a few brainteasers. Add your own! Or anything you'd like! Enjoy.

a measure of civilization

2008-05-06T23:11:00.000-07:00

civilization = opportunity available to every individual

It might be good to have a way to measure civilization, since it's certainly something we want to improve throughout the world, and (a fact which I think is often overlooked), things are not always improving. If you compare the enlightened advances made by Greek polymaths and efficient Roman administration to the backwards thinking and atrocious living conditions of the dark ages, you might be able to imagine how such a relative change might still occur from our current seemingly-secure quality of life. It's not an Orwellian dystopia or world war III we should fear so much as it is a gradual and subtle shift of cultural and moral attitudes toward dogmatic, authority-based, or mystically-inspired modes of "thinking".

The point is that we should keep our eyes open to how well the world is doing.

Of course, there are too many factors and subjective terms to immediately quantify "civilization" as if it were some kind of test score. Nonetheless, numbers, as objectively defined as is reasonably possible, are less likely to fall prey to misinterpretation.

So how can we quantify opportunity? As any engineer is likely to suggest, when you're trying to improve the overall performance of a complex system, it's a good idea to start with the bottlenecks - the pieces that are holding everything else back.

With that in mind, I propose that we could currently use the number of years of wasted life as a number for how much the world could be improved. It will be a great day when we can pragmatically measure opportunity and civilization in some more optimistic terms, such as average years of education per individual, or amount of nutrition practically attainable to anyone. For now it seems that preventable (by education or otherwise) diseases, conditions, or circumstances (such as war or human-created accidents) is certainly the most significant reason typical for an individual to be deprived of a great deal of living opportunities.

To list just a few major preventable causes of death at large today in the world:

cause	deaths per year (approx)	source(s)
HIV/AIDS	3 million	1
vehicular accidents	1 million	1
malaria	1 million	1
measles	1 million	1
malnutrition/bad drinking water	1 million	2
war	1 million (est.)	3, 4, 5, 6

Current (very rough approximation) civilization score (in #deaths / year): -8 million

proofs as explanations

2008-03-31T22:08:00.000-07:00

Most proofs that I've seen feel like defensive, motley soups of disparate thoughts whose composition does indeed imply the theorem in question; but the reader is left without a strong intuition as to why the fact is true.

To illustrate: Consider one of the most fundamental theorems of mathematics: the Pythagorean. Given a right triangle with sides of lengths a, b, and hypotenuse c, we must have a² + b² = c². Euclid, in The Elements, presented an argument for a proof involving many steps (14, by wikipedia's current count) which involves showing the similarity of several triangles and the equivalence in area of various different shapes. Check out wikipedia for the full proof.

Euclid's proof is mathematically sound. It achieves its aim of demonstrating, beyond a doubt, that in fact a² + b² = c² for all (Euclidean) right triangles. What it leaves wanting is intuition. Why? How can a human, without resorting to memorization, understand that the sides must relate in this manner, upon seeing any right triangle? [The word grok comes to mind.]

I think that some proofs do achieve this higher standard of explaining the fact that they defend. Here, in a single picture, is a complete proof of the Pythagorean:

If I were a teacher, trying to convey as best I could the intuition behind this theorem, I would animate this figure so that the ratio a/b transitioned from 0 to infinity and back repeatedly -- the point being to further the deep comprehension that the equality in area between the squares in question is independent of the acute angles of the triangle.

To give another example: one way to view De Moivre's theorem is

cos (nx) = Real[ (cos x + i sin x)ⁿ ].

De Moivre proved this before Euler's formula, e^ix = cos x + i sin x, was known. One proof follows by induction on n -- assume it holds for n, then show it still holds for n+1. (The base case, n=1, is trivial.)

Suppose you saw this formula and the proof by induction. Would you really say that you knew why the formula was true? I am guessing that most people, including mathematicians, would gain profound new insight into this theorem when seen in terms of Euler's formula. In that case, we should restate De Moivre's theorem as

(cos x + i sin x)ⁿ = cos(nx) + i sin(nx),

and it becomes a very straightforward corollary of Euler's formula:

(e^ix)ⁿ = e^i(nx).

With Euler's formula, De Moivre is just one perspective of a special case; it's a shadow on our allegorical wall, while Euler shows us the true form of the fact.

Most math work today is presented as if intuition were a burden to be borne rather than the enabler of our creativity. Gauss in particular is infamous for hiding the how-d'you-think-of-that of his work. Perhaps the math community as a whole would move along more constructively if we embraced the major role intuition and deep understanding play in our creative efforts.

One does not discover a complex path of reasoning without a guess at the lay of the land. Why ignore the curve of the lines while we strive to connect the dots?

Music and emotion

2008-02-16T13:18:00.000-08:00

I've always been curious about why certain musical entities "feel" a certain way to most people -- for example, why does a major chord feed nicer than a minor, or just a bunch of other random notes?

For elements of feeling which seem to be built in for most people, it makes sense to look for an evolutionary cause. Here are two possibilities that might relate our backgrounds with our current musical interpretations:

(1) Rising vs. dropping pitch. Usually when you hear an increasing pitch, it's associated with increasing anticipation, while decreasing notes often reflect a release or departure. Simple examples include the half-note rising motive of the Jaws theme, or the ultimate farewell denoted by Chopin's falling funeral march. Perhaps these connotations could have arisen from the natural Doppler effect -- when a predator is approaching, or we are approaching prey, any noises from the nearing object will be increasing in pitch, while if our prey is escaping or we from a predator, the pitch is decreasing.

(2) Major and similar "nice" chords vs. dissonance. It seems to me that many "nice-sounding" chords are composed of pitch spectrums which could occur naturally as the overtones of a single sound-producer, while more dissonant progressions have less natural frequency ratios. In nature, we could associate something like a major chord with a single producer -- be it friend or foe, it is easy to understand and work with. Dissonance could only arise from a large group (who are not inclined to speak in pitch). So our sense of harmony among chords might have some foundation in the mere plurality vs unity of a sound-producing unit.

I'd be interested to see more work along these lines. Perhaps if we can physiologically or evolutionarily understand why some things sound good to us, and others less so, we could be more conscious of how to make good music :) with more awareness than the current traditional teachings -- which I think are more historically based than scientifically discovered -- would otherwise encourage.

puzzle answer: avg(sums(a set))

2008-02-10T09:59:00.000-08:00

This is the answer to the last math puzzle. You should check out the puzzle before you read the answer!

It turns out that

avg(sums(S)) = ½ ∑S.

Why?

The main observation is that, for any T ⊂ S, avg({∑ T, ∑(S-T)}) = ½ ∑ S. We should also note that if ∑ T₁ = ∑ T₂, then ∑ S-T₁ = ∑ S-T₂; this means that whenever the sums of T₁ and T₂ overlap in sums(S), so do the sums of their complements. If we think of adding each pair {∑ T, ∑(S-T)} one at a time to build sums(S), then either both elements of the pair are already in the set, or both are not -- either way, the average remains the same.

math puzzle: average of the set of sums

2008-02-03T05:31:00.000-08:00

My friend Chris Altomare once asked me a math question which inspired this one:

Given a set S of real numbers, define the set

sums(S) := {t: t = ∑ T for some T ⊂ S}

and the set function

avg(S) := ∑ S / #S,

where #S denotes the size of the set S (we can leave avg(empty set) undefined).

As an example, let S = {2, 5, 7}. Then

sums(S) = {0, 2, 5, 7, 9, 12, 14};

the 0 is always included in sums(S) as the sum of the empty set. In this example, we also have avg(sums(S)) = 7.

Can you find a general formula for avg(sums(S)), for any set S?

This is actually not super-hard, but the trick is that the "obvious proof" is wrong -- in other words, if we thought of the sums function as giving a multiset, then it becomes a linear function, which would make everything straightforward (since avg is based on linear functions, so avg(sums) is still linearly analyzable and thus easy to find and prove). But this is not the case!

Good luck! I'll post the answer soon.

how to pour soda on ice

2008-01-14T00:29:00.000-08:00

Have you ever noticed that you can pour soda directly into a cup of ice at most restaurants, but when you try it at home, your cup fizzes out the wazoo and all your carbonation is gone, like so many childhood daydreams dashed by the cruel, merciless fist of fate?

As a dedicated soda fan, this problem used to present no small helping of consternation at mealtime. Alas, by experimentation and some web browsing, I found that one small factor makes all the difference -- the surface quality of the ice itself. If you simply make sure your ice is wet (no, ice is not always wet) then you will have copious carbonation upon pouring. Check out the video for some (informal) experimental evidence.

The inquisitive reader may inquire at this point: (a) how does that explain the way restaurant pouring works? and (b) why does this work at all?

(a) I believe many restaurants store ice not in a freezer, but rather in a bucket or other receptacle from which it may be easily dispensed. By allowing the ice to start thawing, the surface begins to melt, so that the ice is covered by a thin layer of water (i.e. it's wet).

(b) Rob Landolfi describes the action of the surface of the ice as a collecting point for carbon dioxide molecules. CO₂ is non-polar, as opposed to H₂O, so that when a bubble of the gas starts to form within the liquid, it leads to a cascading effect (more CO₂ is likely to run into a large gas bubble) which results in a larger bubble that rises to the surface and escapes. In the absence of a rough surface to catch a few CO₂ molecules, they are more likely to never hit each other, and the bubbles are less likely to form.

Notice in the experiment that many factors remain the same -- the temperature of the soda, the number of ice cubes, the surface area of the ice and the glass -- even the temperature of the ice itself remains very similar. This should debunk some other possible hypotheses about how ice removes carbonation.

Enjoy your drink!

gnomes puzzle : hint + answer

2007-12-29T15:43:00.000-08:00

I just wrote up a quick hint and a solution to the gnomes-in-a-circle puzzle (see the last post for the puzzle).

The gnomes in a circle brainteaser

2007-11-10T18:24:00.000-08:00

A friend of mine (Neil S) recently told me about the following puzzle:

A group of n gnomes have been captured by an evil wizard. The wizard, in his twisted ways, has granted the gnomes a tricky opportunity to save themselves from a grisly demise.

The wizard has chosen n colors and he will individually paint each gnome's foreheard one of these colors. Then all the gnomes will stand in a circle, so they can see everyone else's colors, but not their own. They then must simultaneously write down a guess at their own forehead's color. If any gnome is correct (even just one), they all go free. Otherwise they will be forced into a lifetime career of unsolicited telemarketing.

Beforehand, the gnomes know what all n colors are, and they have one chance to meet to agree on a strategy. Can you think of a strategy which is guaranteed to save all the gnomes?

Just to be clear, the wizard is allowed to do strange things such as paint everyone's forehead a single color, or use only 3 colors even if there are 100 gnomes (and therefore 100 possible colors).

I should admit ahead of time that the only answer I know is pretty mathy :)

I'll post the answer in a little while.

a better way to buy and sell airline tickets?

2007-10-23T23:56:00.000-07:00

If you've ever used farecast.com, then you've probably experienced a little wonder at how major airlines choose to price their tickets. Sometimes, as you walk down a jetway built in the 60's to board a plane that feels like it was built at least 20 years ago, you kind of have to wonder how airlines seemed to have missed the capitalism boat. I wonder if price fixing could actually harm an industry?

As an analogy, we could think of an ecosystem in which a single predator eats more than his/her share of the prey to spite competing predators. But in the ensuing system, the entire species of predator without competition thrives -- there is little basis for natural selection, and the usual advantages of evolution are lost. By eliminating the competition, the predator becomes lazy and less competent without the filtering of survival of the fittest.

What can we prey do to improve our lazy incompetent predators? (Actually, this idea also hopes to improve things for both parties.)

How about: airlines could sell tickets ahead of time for half their desired price, allow tickets to be resellable through a centralized system, and then collect half of the final customer's price when that customer boards the plane.

Here is a quick example: Awesome Airlines sells Barbara a ticket from NYC to LA on a certain date and time for $200. Barbara is now free to re-sell the ticket as she sees fit. Demand for tickets goes up, and Barbara sells to Carl for $300. Demand goes way up, and Carl, who did want to fly to LA, decides he's better off flying later, and agrees to sell for $400 to Darlita. Darlita is the final customer, and before she boards, she pays Awesome Airlines another $400, so her final effective price is $800. In this case, Awesome Air got $600 for the flight, while Barbara and Carl each made $100. You could argue that Darlita got hosed, but in the current system it is more likely that either she would have had no way of getting on that flight at all (if the tickets were sold out), or that the airline would then be asking for $1000/ticket since it had anticipated high demand -- a price too dear for Darlita.

We could also imagine that demand goes down after Barbara buys her ticket for $200, in which case she can cut her losses by selling for $100 to Ebert. In this case, Barbara's misplaced hope that the demand would go up cost her $100. The airline still collected $300 for that ticket despite their misjudgment of hoping to get $400 for it, and Ebert was able to fly despite his unwillingness to pay $400.

Exactly why is this system any better? Primarily, it avoids the hassle (for both the airline and customer) of having to guess where the best price is, while at the same time hedging the risk for the airline by encouraging that they usually earn at least 50% of their desired revenue for every flight.

It is good for customers because customers won't get screwed by high prices when there is really not much demand. If there is high demand, then prices may temporarily go up, but ultimately, airlines will naturally move to fill in the flights of higher demand, so that supply and demand stabilize and prices will come back down.

It is good for airlines because (1) they are more likely to always sell at least half of their desired ticket prices, and (2) they can get much more money where there is high demand. In addition, they get much more direct feedback about where the financial demand lies for certain flights.

I can imagine one objection being that purchasing flights now is somewhat convenient (I really think it could be a lot better, though!) but it may seem to be much less convenient if you need to go and find a random person from whom to negotiate your price.

Aha! But I think, if the system is well-designed, purchasing tickets would actually become easier! Why? Think of a large stock exchange system -- trades are effectively made between a plethora of owners at lightning speed. Because the trading is centralized and well-organized, there need be no concern about finding a seller or face-to-face negotiations. In addition, it becomes much more reasonable to book flights which include a layover, since we can now very reasonably work with completely unassociated airlines in choosing a multi-flight trip.

Unfortunately, I doubt this would happen. Some people would be concerned about the security risks of knowing who is buying tickets in the resell system, although this could certainly be handled as well as the current system. Airlines would be concerned about the increased financial risk, although in the long run it is actually reduced risk for any half-decent airline. Finally, it would probably require some legislative support, which means some popular support, and that seems extremely unlikely as long as any single powerful airline or politician is against it.

In any case, it's still a cool system, and we could apply the same idea to many goods - not just airline tickets. Basically, this could work for any redeemable good, and makes sense whenever there is often varying or difficult-to-predict demand. You could try the same system with broadway shows, tickets to sporting events, or major concerts. I can also imagine using this system on pre-release sales for anticipated products, such as a new model of computer, video game system, or even a car.

zen + google = zengle

2007-10-06T11:15:00.000-07:00

I really dig simple pages. Simple and to-the-point. That's the motivation behind what I'm calling "zengle", a simple front-page for google. Just type your query and hit enter - no need for any buttons :)

zengle