Knowledge and the Gettier Problem

It has long been accepted in epistemology that the conditions for something to be considered "knowledge" are that it must be a justified, true belief. We would not say that one knows that P (P being a proposition where, say, P is "the sky is blue") if that person does not hold the belief that P, and it should be fairly clear that beliefs are fundamentally necessary for any knowledge. That belief must also be true, for we wouldn't allow false beliefs to pass for knowledge lest we must allow the delusions of a madman to count as knowledge. Lastly, we must enforce that one must be justified in having a true belief, for we would not say that someone who dreamt the winning lottery numbers actually knew the winning lottery numbers.

Let me just say that I don't intend to dwell (just yet) on the metaphysical implications of belief, the nature and degree of justification, referential ambiguity, etc., as interesting or relevant they may be. I think several of these might be mentioned later, but for now and for the sake of explaining the Gettier problem, let's adopt the common-sense notion of knowledge.

The philosopher Edmund Gettier posed some serious questions against the justified, true belief conditions for knowledge (henceforth JTB). In his short paper "Is Justified True Belief Knowledge?" published almost half a century ago, Gettier exposed some flaws in the epistemological definition of knowledge that (as far as I know) have yet to be reconciled while remaining consistent with the common-sense notion of knowledge. There are two original cases that Gettier gives, and are as follows:

[original found here]

Case 1

Smith and Jones are both applying for a job. Smith was told by the hiring manager that Jones would get the job. Smith also knows that Jones has 10 cents in his pocket because Jones told Smith and Jones has no reason to lie (and yes, Jones has 10 cents in his pocket). Now, we can now say that Smith knows the following:

P1-Jones will get the job.
P2-Jones is a man with 10 cents in his pocket.
P3-A man with 10 cents in his pocket will get the job.

We can see here that given the P1 and P2, we can, by the transitive property, derive P3. We know that P1 and P2 satisfy JTB conditions, and likewise, the P3 satisfies the JTB conditions and thus, Smith knows that "a man with 10 cents in his pocket will get the job."

But it just so happens that the hiring manager had a last minute change of heart and decided to hire Smith instead. And as it turns out, Smith, not knowing so himself, also has 10 cents in his pocket. Would we say that Smith knew that P3? P3 turned out to be both justified and true (and of course, it's a belief), but I (and most other people, I bet) would hesitate to designate such to be knowledge.

Case 2

Smith knows that Jones owns a Ford: Jones works for Ford, is absolutely in love with Ford cars, and has vowed to never own a vehicle of another brand. So we have:

P1-Jones owns a Ford.

Let's now say that Smith has a friend named Brown whose location in unknown to Smith. Given P1, we can make:

P2-Jones owns a Ford or Brown is in Barcelona.

We are able to do this by creating a disjunctive proposition in which JTB conditions are preserved via P1. By creating [P1 v anything], and assuming that P1 is true and justified, [P1 v anything] is also true and justified (by virtue of the logical fact that the truth of at least one component in a disjunctive proposition makes the whole proposition true). We, then, are to accept that Smith knows that P2.

However, unknown to Smith, Jones had just sold his Ford to pay for his mortgage after being laid off from the Ford factory. It also happens to be the case that Brown, with whom Smith hasn't spoken to in years, is in Barcelona. P2 turns out to be true, but does Smith know that "Jones owns a Ford or Brown is in Barcelona"? Again, I'm hesitant to categorize this as knowledge.

So...

How do we explain away these problems? While there are many possible routes to take, many that were proposed by Gettier's contemporaries were unable to remedy the necessary and sufficient conditions of knowledge without destroying the common-sense notion of knowledge. Put your suggestions in the comment, but try to keep your theory consistent with the common-sense notion of knowledge. Also, while many related topics are certainly relevant to the discussion, I would like to focus on the logical intricacies of the problem.

Moral Frameworks: Ends and Means

Imagine a terrorist came up to you while you were riding a bus and told you to kill the stranger sitting next to you or he would set off a bomb killing everyone on the bus--including you, the stranger beside you, the bus driver, all the other passengers and the terrorist himself. What would you do? This isn't a riddle--there's no trick. The terrorist really has a bomb, he'll actually let you go if you kill the stranger and you can't make a run for it or kill the terrorist. The options are kill the stranger or do nothing and risk the terrorist blowing up the bus.

Mean Morality - do nothing
This standard is a philosophy grounded in the principle of not harming others and keeping one's hands clean of blood. Killing the stranger may be permissible but it is not right. Intuitively, killing innocents is never right. Means morality reminds us that the lesser of two evils is still an evil.

Means morality is the mark of saints, who are said to have done no wrong in their lifetimes. It was the philosophy of Rev. Martin Luther King who once said, "it is more important that we do justice then get justice." Mathematically, it is expressed as the Pareto efficency.

Legally, you are almost always immune from liability for non-action. America is famous for it's 'no duty to rescue' rule, meaning you could watch someone slowly drown in a three inch pool of water when all you would have to do is flip them over. However, many European countries impose such a duty when the risks of injury are slight. If you let the terrorist blow up the bus and miraculously survived no court could convict you.

End Morality - kill the stranger
By contrast, 'end morality' looks not to what is right but what is justified. Because the stranger will die either way, it is pointless not to kill him to save the lives of others. End morality is grounded in the principles of utilitarianism and the 'greater good.'

End morality is the mark of knights, avenging angels and Jack Bauer of 24. Practically, it is the philosophy of governments and provides the philosophical basis for a 'just war' a la WWII (end the holocaust) and the Civil War (end slavery). Mathematically, it finds its analog in the Kaldor-Hicks efficency.

Legally, people are immune from liability when acting under 'duress' or 'necessity.' In the terrorist on a bus hypothetical, you would have an airtight duress defense if you killed the stranger. Even if there was no terrorist 'putting a gun to your head', you could avoid liability under 'necessity' if you could show you were forced to choose between one life and a busload. Curiously, you may still be liable in civil suits when you invoke the 'necessity' doctrine for personal benefit (under the theory that people should internalize all the costs and benefits of their actions when making their decisions) but the government pays the costs when 'public necessity' is invoked (like the bus hypothetical).

These two theories provide a logical and simple framework to determine what is moral, free from the coercive influence of history, government or religion. In practice, most people practice a combination of both ends and means morality but an individual's balance may explain their choice in a moral dilemma.

The Monty Hall Problem

The Setup:
You're on a game show and are given the choice of three doors: Behind one door is a car; behind the other two, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?"

Before you go off trying to solve this, there are a few things to remember:

• the player may initially choose any of the three doors (not just Door 1) and the host may open any different door revealing a goat (not necessarily Door 3).
• the host is constrained to always open a door revealing a goat.
• the host is constrained to always to make the offer to switch.
• the host opens one of the remaining two doors randomly if the player initially picked the car.

The Problem:
Is it to your advantage to switch your choice?

The Solution:
Yes, you should switch—doing so doubles the probability of winning the car from one in three to two in three.

The Explanation:
In contrast to Blue Eyes, the Monty Hall problem is explained by raising the numbers rather than reducing them. Imagine instead of three doors, there were three hundred doors but still only one contained a car. After you selected your door, let's say No. 1 again, the host opened two hundred and ninety-eight doors, all with goats behind them, leaving only the door you chose and one other, let's say No. 164. Now intuition tells us to switch because there's a 299/300 chance our initial guess was wrong and the other door holds the car.
The lower numbers in the Monty Hall problem conceal the fact that the host is not choosing one door to open but choosing one door to keep closed (the car). The only time you lose by switching is the 1 in 300 time when you actually guessed correctly to begin with and the host has selected a random door. Likewise, the only time switching loses in the Monty Hall problem is the 1 in 3 time when you guessed the correct door initially.

Savants: Seeing with numbers?

A continuation of Follow up to Secrets of Success.

I stumbled across an interview with autistic savant and author of the book Born on a Blue Day, Daniel Tammet, in Scientific American. If you read and agree with my last article relating to autistic savants, you'll realize that the title of the article "Learn to Think Better: Tips from a Savant" is a bit deceptive. The only tip he gives is one promoted by most psychologists nowadays. Anyway, go read the interview. It's pretty interesting and I hope to get a chance to read the book soon.

There are several things that Tammet says that made me wonder...

    The first thing he says about numbers is that numbers, to him, are multidimensional. He claims that each number takes on complex forms that he can visualize, later claiming that he perceives them as (pseudo)physical objects; that is, they have "form, color, texture and so on." This makes me wonder in what sense he means this, barring that he's a synesthete (though he might be, or even might need to be in order to have his talents*). In the few hours I've had to reflect on this, I can't seem to push my imagination beyond the point where the numbers merely relate to visual images. The furthest I have gotten so far is that the number 111 strikes an image in my mind's eye that is like round lumpy oatmeal. Of course, this is only because I just adopted Tammet's description of the number and tried to employ it with my own understanding, and as such, the roundness of the number 111 has no inherent connection with the number 3 (or any other number) and likewise for lumpiness. I'd bet that most of you who try this mental exercise will find yourself at a similar realization: one in which the relationship between the conceptual number and its visual characteristics is limited to mere association.

    Since Tammet's understanding of numbers is far beyond mine and I am utterly incapable of even partially internalizing his description of the numerical domain with the powers of my imagination (maybe I can draw upon the powers of LSD), I've not but my speculation to depend on. It leads me to assume that his understanding of numbers is not just associations with visual characteristics but rather that the visual characteristics are fundamental to the numeral; that is, the number 111 and round, lumpy oatmeal are one in the same (to a similar degree in which "bachelor" and "unmarried man" is the same for us). Tammet has already given us a description of his thought process when cognizing the number 111, but I would not find it surprising if to the description "round, lumpy oatmeal" of a numeral triggered the number "111". Essentially, his mind somehow bi-directionally maps appropriate numbers (e.g., multiples of 3) to the corresponding "visual characteristic" (e.g., roundness).

    I think that several people might find that act of associating numbers to images undermines my point of not being able to imagine another's thought processes, so I feel I need to try to make an important distinction here (if such a distinction is possible). The very act of emulating Tammet's described thought process is fundamentally restricted to certain cognitive faculties; namely, of numbers, visual imagery, and language. I would guess that when we entertain the idea of Tammet's understanding of numbers, the mental connection between the concept of a number and imagery is the product of internalizing an idea through language. That is, we are merely taking a "sentence" (i.e., the idea) and entertaining that, not the actual conscious or mental experience of the process itself. And here lies the great distinction: one involves internalizing an idea and the other involves internalizing an understanding. I dare someone who previously did not have such prowess to perform this mental experiment and honestly claim "Aha! It all makes sense now! The number 3 is no longer abstract but shares a perceptual trait with all other multiples of three. Oh look, it's number 3's friend number 111! I can tell because they look alike. Number 111 also looks like number 37! And the number 37 looks similar to number 116.28571..." (of course, I can say all of this, but I cannot actually do it or even properly think about doing it).

    Another thing I find intriguing is this visual map that Tammet indirectly claims he has. He says that the number 111 is round, like the number 3, and also lumpy like the number 37. I don't think it would terribly wrong here to assume that other visual features are identified with other numbers; shininess or smoothness with numbers 5 and numbers not wholly divisible by the number 19, and so on. So I'm left to wonder: how exactly did these visual features come to represent what they represent? Why are the particular features chosen to represent particular numerical relationships? Are they chosen by virtue of some connection between the nature of a numerical relationship and something inherent in the visual feature, or are they simply arbitrary? Are there any numbers that strike two contradicting visual features, or have all of the intricacies of mathematical relationships been taken into account? All of these questions lead my speculation down a wide, tricky, and confusing path which I will spare you (and myself) the pain of working through it. But there is one obvious thing that could account for this: since Tammet is autistic, it would not be unlikely for him to have simply focused his compulsions and obsessions on numerical calculations and then associating numerical relationships with objects around him. It wouldn't be impossible for years and years of such behavior to lead to the kind of understanding that Tammet has.

    I am left to conclude, after exhausting my faculties, that Tammet just has an abnormal mental domain or platform which lies at the foundation of his comprehension. This relates back to my other post in that it shows how a savant's understanding is incompatible with conventional minds. It is not accessible or cognizable to anyone but him, and he even proclaims that he finds it "surprising that other people don't think in the same way," indicating that his imagination is similar to ours in that it cannot escape its own understanding. Further, he states "I find it hard to imagine a world where numbers and words are not how I experience them!" Maybe, Daniel, you should read my post on why that is. =)

    That is not to say that these abilities are forever elusive to empirical inquiries. Tammet offers some potential biologically grounded suspects for his abilities. Hyperconnectivity, he says, could account for his rich understanding of numbers and words. For example, cross-modular connections between his calculating/numbers/whatever part of the brain and somewhere in his visual cortex could help explain why Tammet associates form, color, texture and etc. with numbers, though the mystery of its structural organization remains (for example, the semantic organization of his visual understanding of features such that it corresponds to certain numerical functions--and I think this is the really cool part). I can only hope that neuroscience will one day explain Tammet's vast mental network of numbers and visual characteristics. However, this empirical faith does nothing to help my understanding of Tammet's conscious experience nor would it be wise of me to hope that it ever will.

    The point I'm trying to make is the same as the one in my last post regarding this topic. This is just presented in another context.

*Another interview I came across a few days ago with a synesthesia specialist. Not that good of a read but at least the topic is interesting.

A Theory of God: the Great Animator

"In the beginning was the word and the word was with God and the word was God"

- John 1:1

What is man but God's fiction? Let us consider the relationship between an author and his writing. If he wills a character to die, they die. If he wants to revive them, they may rise from the grave. If he wishes to rewrite the laws of physics so that gravity were to cease functioning and all creation were to fly off the earth, he could do so with a single sentence.

Novels leave many undefined grey areas to the reader's imagination so let us instead consider the comparatively despotic world of animated cartoons. The artist is master of his fiction, both omniscient and omnipresent. Mickey Mouse doesn't blinks unless the artist takes the time to make him blink. The sun does not shine unless the animator places it in the sky and decides where it will cast its light. Nothing happens without him willing it. He knows everything that happens in every inch of his world because he is behind it. Everything is deliberate, no matter how small the detail.

The laws of physics do not operate automatically in an animated world. If an apple were to drop from a tree, it does not land unless the artist draws it doing so. Presumably, the artist will try to mimic the physics of the real world in his fictional one but there is nothing to make sure he gets them right or even keeps them consistent. If the artist says it takes an apple two seconds to fall ten feet, it does--even if in the real world it would take one second or three. The operation of physics is illusory, and persists only on the continued goodwill of the artist.

Yet the artist faithfully emulate rules of physics. In fact, his job is maintaining consistency and believability. Better artists make fewer mistakes, their worlds are internally cohesive. The larger the scale, the more work they must do to make sure everything operates predictably. Every tree that falls must make a sound. There can be no shortcuts. This is the great duty the artist owes his world, to keep sun rising every morning.

Now consider where free will fit into this picture. Characters take on personalities, they have quirks and mannerisms and opinions. Like physics, there are patterns behind their behavior. If Micky Mouse were too cease smiling, if Buggs Bunny ceased being a trickster or Goofy stopped being Goofy, people would stand up and take notice. They would say they have acted out of character. The artist much yield to the personality just as he yields to the laws of physics.

So what does this analogy tell us about God? First of all that he is a great rule maker and vigilant enforcer of them. It is skeptical of miracles and divine intervention because God's role is enforcing the laws of physics not breaking them. Enforcing laws of physics is 99.99% of what he does. Of course, the theory doesn't say he can't suspend or sidestep the rules of the universe (in fact the theory demands that he retain this power) but suggests that it would be "out of character" for him to do so.

Secondly, it challenges his perfection. Just because Walt Disney possess omniscience, omnipotence and omnipresence in Snow White's world does not mean he lacks character traits, and even flaws, in the real world. While God must be the sovereign master of our world, there is nothing to suggest he is anything more than a humble painter in his. Perhaps he has a name, and a wife, and a scratchy beard. Maybe he IS Jesus. Or maybe he is the only being in his universe and created us for company or out of boredom. There's simply no way to know unless he tells us. This theory only roughly sketches out God's power and tells us about our reality, not his.

Since this post has been pretty heavy, let's end by applying the theory to a 'lighter' situation:

comic provided courtesy of Bob the Angry Flower

Is this possible under the great animator theory? yes and no, God could make one of his characters behave like him and he could behave like one of his characters (he authors their thought process after all so knows them perfectly) but he couldn't actually switch realities with them anymore than L. Frank Baum could physically enter the land of Oz (or pull the Tinman into our world). Hence, the 'swap' would just be an exercise in method acting that God could stop at any moment. Even when his hand isn't on the wheel, God still has a foot on the brake. But would God use his veto power or would that conflict with his commitment to the rules? Likely answer: he wouldn't get in this predicament to begin with.

Three Headed Dragon

This is a follow up to Two Headed Dragon: a Quick Logic Riddle

Three Headed Dragon - difficulty: medium
The Setup:
You walk down a long tunnel and come to the end with a door on either side. One door leads to paradise, and the other door leads to suffering, but they are both identical. A three headed dragon sits in between the doors. Each head must either lie or tell the truth. You are allowed one question to ask to help determine the correct road to paradise.

• All three heads may lie, all three may tell the truth, two heads may lie and one may tell the truth or two heads may lie and one may tell the truth.

The Problem:
What question do you ask to discover the correct door?

The Solution:

Ask, "what would your answer be if I asked you whether the first door will lead me to paradise?" and follow the dragon's advice.

This answer is also an alternate solution to the two dragon riddle.

Random Three Headed Dragon - difficulty: hard
The Setup:
You walk down a long tunnel and come to the end with a door on either side. One door leads to paradise, and the other door leads to suffering, but they are both identical. A three headed dragon sits in between the doors. One of the heads always lies, the other always tells the truth, the third answers randomly. There is no way to tell which one is which but you are allowed two question to ask to help determine the correct road to paradise.

• What the second question is, and to which head it is put, may depend on the answer to the first question.
• The random head may answer truthfully, dishonestly or totally randomly. If you asked him if he was a dragon, he may respond 'yes', 'no' or 'pizza.'
  

The Solution:

Ask dragon B, "If I asked you 'Does head A answer randomly?', would you say 'no'?" If he says no, your next question goes to head C. If he says yes, your next question goes to head A. Either way you ask, "what would your answer be if I asked you whether that door will lead me to paradise" and go through that door.

The Explanation:
The first move is to find a head that you can be certain is not Random, and hence is either True or False. If you ask dragon B, "If I asked you 'Is A Random?', would you say 'no'?" you get six possible results:

A B C Answer t=truth dragon, f=false dragon, r=random dragon
F T R yes
T F R yes
T R F yes, no or pizza
F R T yes, no or pizza
R T F no
R F T no

yes - A isn't random
no - C isn't random

Having isolated a non-random dragon, the question, "what would your answer be if I asked you whether that door will lead me to paradise?" will always produce the true answer.

Two Headed Dragon: a Quick Logic Riddle

The Setup:

You walk down a long tunnel and come to the end with a door on either side. One door leads to paradise, and the other door leads to suffering, but they are both identical.  A two headed dragon sits in between the doors. One of the heads always lies, the other always tells the truth. There is no way to tell which one is which but you are allowed one question to ask one of the heads to help determine the correct road to paradise.

The Problem:

What question do you ask to discover the correct door?

The Solution:

Ask, "what door would the other head say leads to paradise?"  then go through the opposite door.

The Analysis:

The two dragon riddle poses an excellent example of extraneous v. necessary information. At first blush, you think to solve the riddle you must deduce which dragon lies and which tells the truth. Of course, even after the riddle is solved we still never know which is which. Because we never identify the honest dragon, we instead rely on the dragons to cross check eachother's answers--one will lie about the truth, the other will faithfully report a lie. The answer we get is always wrong but we know how its wrong and can deduce the correct answer from it. If we had more questions, we could figure out both which dragon lies and which door lead to paradise--but when our question economy is limited we can only choose one.

This type of intellectual shortcut is used all the time when the opportunity costs of inquiry outweigh the utility. For instance, most people have no idea how their car's engines operates or heart pumps. In complex software coding, different programmers code different functions and may have no idea how the final software works. Like the Two Dragons riddle suggests, some information may be sacrificed in a search for answers.

Follow up to Secrets of Success: Futile to Covet

This is a follow up to The Secret of Success.

I always fantasize about what it would be like to have extraordinary mental powers. Who wouldn't want to have a level of comprehension beyond all comprehension? However, I quickly realized that this is nothing like fantasizing about being a professional athlete or a famous celebrity. I begin by considering the underlying mental processes that these prodigies, geniuses, and the like, have. There have been studies already that suggest that people's understanding of certain things are completely different from the norm (assuming, of course, that there are at least some fundamental things that are uniform within the norms). For example, it's now known that chess masters don't think serially; that is, they don't "play out" possible moves as the early chess playing programs did. Rather, they recognize scenarios and patterns, which are assigned values, and work to achieve the higher valued scenario. Now this kind of thing seems common in our everyday experiences of familiar and frequently practiced activities, but if you consider the sheer volume of all possible board setups, it's quite astounding. (Anyone who plays poker even semi-extensively using ICM should be able to relate).

Consider other so-called geniuses: idiot savants (warning: incoming tangent--my professor Jimmy M. made it clear that "autistics aren't like Dustin Hoffman from Rainman... if they were, they would be fucking geniuses, not retarded"). Their extraordinary abilities range from being able to calculate large (but relatively simple) mathematical calculations in an extremely short period of time. For example, some could give you an answer to, say, 1,495,420 x 235 in under 5 seconds without writing anything down. These people, given that they are autistic, can do few other things well, but they have, literally, a different understanding of numbers. There are others that can calculate dates, have perfect spatial recognition (i.e., when drawing), and memorize enormous amounts of information through what seems to be nothing more than rote memory and, in the case of musical savants, replicate a piece of music to the exact note and tempo after a single hearing. We have to assume that all of these people, too, have a different understanding than everyone else not like them.

So you have to wonder: how exactly do they understand things? The philosopher Nagel posed the question "what's it like to be a bat?" By this, he meant for his audience to imagine the character of a bat's experience. I don't think he meant the content of a bat's experience (like living in a cave hanging upside down and eating insects) but the nature of it; what is it like to be able to discern the size, shape, and motion of objects through auditory processing such as we do through sight? Is this even possible to imagine, or speculate? After all, we necessarily know only the character of our own experience.

It would be a waste to speculate on the sensory experience of idiot savants, but we can speculate as to their understanding of things in which they excel. It would be the opposite of imagining what it would be like to have the mind of someone with a lower IQ than ourselves. "How can they not see this emerging pattern?" "How could they not deduce this?" Likewise, we must ask of the savants "how can they find that pattern so quickly? or "how could they memorize all of that in a matter of seconds?" But any attempts to imagine what it is like to have their understanding would be stopped dead in its tracks by an inherent contradiction in the nature of the the thought experiment itself. That is, we can't understand something in a different way when the only and every thing we understand is limited by our own understanding. In other words, we can't transcend the character of our own experiences as everything we experience is bound to the character of that experience. And, (not trying to beat a dead horse here... or maybe I am) we can't speculate what it is like to have speculative powers beyond our own speculative powers (and, of course, I don't mean fantasizing about your ingenuity winning you fame and admiration of your peers and that of the girl you've been eye-humping). I can go on and on here, but hopefully, you get the point that you can't ask of something a performance beyond its own capability.

There is the biological and neurological element of idiot savants. Sure, they must have some neurological anomalies that account for the differences in processing. Many of you might be tempted to claim that this is all there is to their abilities. Quite frankly, I agree completely. However, that is not what I was getting at, but rather that we can't, by imagination and speculation alone, come even remotely close to an accurate mental picture of what it would be like to have that understanding. But how about this: imagine that neurologists were able to isolate the precise neurological anomaly that accounts for quick mathematical calculations and were able to artificially replicate that structure in already living humans such that all other mental functions were left in-tact except for being able to do mathematical calculations. Now tell me what the mental steps you think you would be taking to do 1,495,420 x 235 in under 5 seconds after you undergo this procedure. What do you think? I would bet the first thing you would say is, if I twisted your arm, that you are doing what you learned in 4th grade math class but do it much faster. Without getting into how this is not actually an adequate answer (there is no adequate answer), I would ask you to imagine doing the calculation at the speed at which it is required to compute the equation in under 5 seconds. Ha! Got'ya. You can't. It's simply outside of your mental capacity to do so.

That is not to say that one's mental capacity is forever static. With enough training, practice, drugs, surgery or what have you, you can ultimately rewire the neural connections in your brain such as to achieve that level of processing. We all, as beings that age and become more wise with every experience, have fluctuating mental capacities. This, however, is gradual and it's difficult, if not outright impossible, to pin-point moments of improving mental processing. If anyone was able to achieve an understanding beyond their understanding merely by the exercises proposed above, it would require an almost instantaneous and dramatic neurological transformation. Yet, there is not a day that goes by where I don't wish I had such epiphanies.

EDIT 4/19 - I was catching up on some reading and it seems that The Economist had an article on The link between autism and extraordinary ability | Genius locus. It touches up on similar elements that was in my post... but the date of it makes it seems like I copied it from them (I didn't!). First off, don't get confused about the "theory of mind" which, in the article, is in reference to the ability to empathize, or 'to put yourself in someone else's shoes." While this is possible for us, I wasn't referring to such, dare I say, superficial notion of thinking what it would be like to be someone else. The article refers to theory of mind as the ability to recognize others as conscious agents, to put it into shallow terms. My idea of "thinking what it would be like to be someone else" refers to the thoughts of what it would be like to have their consciousness.

The Secret of Success

What separates the successful from the unsuccessful? A lot of my friends play online poker—smart kids, kids with more than enough talent to go to the top. The question is: why don’t they?
Poker, despite the large element of luck, is a game of skill. Unlike basketball or football you don’t have to be born seven feet tall or anything else. Other than, perhaps, a basic intelligence requirement, pretty much anyone could be the best if they really wanted to …Or could they?
That’s a loaded phrase: “if they really wanted to.” I have friends that play 12+ hours a day and still can’t put together a bankroll. Do they just “not want to” enough?
They don’t have the skills—at least not yet. And frankly, fifteen years of experience is gonna tweak their game not overhaul it. They’ll never make it to the World Series of Poker—at least not like this. Hence, the elusive “X Factor” – what separates the pros from the amateurs? Experience? Luck? Determination? Or do they just think differently?
I tend to lean towards the latter of the options. The existence of prodigies indicates that it’s the correct answer. If someone with virtually no experience can sit down and play a game like a master it’s because the way they think about the game is entirely different from how most people do. The question becomes how do regular people learn to think like this?
First of all, there can be no debate that it’s possible to learn their mode of thought. Literate men can be taught everything from the theory of relativity to how to play baseball. However the real challenge would be engineering the theory yourself if nobody has invented it yet. So how do you do this?
I tend to think it’s a matter of thinking about the subject at hand as much as possible: every angle—every perspective—every detail. The subject should never leave your mind. To quote Forrest Gump (the ping-pong prodigy), “I played ping pong so much I even played it in my sleep.”
If you‘re always thinking about your subject, soon you start to see everything in terms of it. Pedestrians crossing the street became bouncing ping pong balls, dollars bills become poker hands. By the time you actually sit down to pursue your sport, you know it inside-out, upside-down and backwards. You see everything and then it becomes easy. Great ones always make their actions look effortless because to them, it is.
I end this post with a story about famed violinist Fritz Kreisler that I think shows the difference between the people that do succeed and the people that just want to. After a concert, a woman rushed up to Kreisler and said, “I would give my life to play as beautifully as you did.” Kreisler responded, “I did.”

The Two Envelopes Paradox

The Setup:
You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other (i.e., ten dollars and twenty dollars). The player may select one envelope and keep whatever amount it contains, but upon selection, is offered the possibility to take the other envelope instead.

The Problem:
I open up enevelope #1 and it contains ten dollars. There is a 50% chance the other envelope contains twenty dollars, 50% it contains five. The average value of the other envelope is twelve and a half dollars ((20 + 5) / 2) so it is always worth it to swap--I don't even have to open my envelope to know this. In fact, if I don't open my envelopes its always +ev (expected value) to keep swapping envelopes back and forth forever.

The logician Raymond Smullyan expressed the problem in a way which doesn't involve probabilities. The following plainly logical arguments lead to conflicting conclusions:
1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the potential gain is strictly greater than the potential loss.
2. Let the amounts in the envelopes be Y and 2Y. Now by swapping, the player may gain Y or lose Y. So the potential gain is equal to the potential loss.

Status: unsolved.

Game - Blue Eyes

This was brought to my attention by G, and can be found at XKCD's site. My intention is to attempt at giving a clearer setup and clearer explanation of how to derive the answer. 

This game can be solved using a bit logic, math, and a thought experiment. I'm sure there are a variety of ways to reach the solution, but I feel that I've stumbled across a relatively easy solution, though it may be less elegant than the others. 

The Setup:

200 people--who are perfectly logical beings and completely aware of everyone else at all times--are trapped on an island. These people can see everyone else's eyes but NEVER their own (no reflection, etc.) and they have no means of communication whatsoever (no getting other people to tell you the color of your own eyes). Because they can never know the color of their own eyes, no one person knows the complete distribution of eye colors. They can only deduce the color of their own eyes from only what is given in the rest of this setup.

Every night at midnight, a ferry comes to pick up those who know, with absolute certainty, the color of their own eyes, and bring them back to the mainland. 

One morning, a guru (who happens to have green eyes) descends on this colony of logical and aware beings and makes the statement (a true one): "I see a person with blue eyes." She then disappears to never return again.

The Problem:

If there are 100 people with brown eyes and 100 people with blue eyes, how many days does it take before people can start leaving? Who can leave and how many? The answer must include 1) the color of the eyes of the people leaving, 2) number of people leaving, and 3) the number of days it takes them to leave.

Before you go off trying to solve this, there are a few things to remember.

• There IS an answer to this puzzle that fits the criteria above and does not transcend the scope of the given setup. Id est, someone leaves on a certain day. 
  
• No person initially knows the precise distribution of eye colors; that is, a person might see 100 brown eyed people and 99 blue eyed people, but does not know if he himself has brown eyes, blue eyes, or purple eyes (could be 101 brown eyes and 99 blue, or 100 brown and 99 blue and 1 purple, etc.) 
  
• Remember that in order to leave, a person must know with absolute certainty! 
  
• These people are capable of perfect logic, and know all and only what is given in the setup above.

More than arriving at the correct answer, it's more important (I think) on how you arrived at the answer. Please leave your thoughts (candidates, algorithms, and especially QUESTIONS REGARDING THE SETUP) in the comments section. I'll return periodically to leave hints and, eventually, the answer.

A Solution:
Luke used the bottom-up approach to this problem. By beginning at the simplest possible scenario, you'll find an emerging pattern. I'll begin by entertaining the thought process that a blue eyed would need to go through to achieve absolute certainty (after all, that's all that matters--you'll see later).

br = brown, bl = blue

199br, 1bl on island
The blue eyed person will realized that of the 200 people on the island, he is the only one who could have blue eyes. He leaves.

198br, 2bl on island
We'll assign letters to the two blue eyed people: A and B. A will see that B has blue eyes, and deduce that B will do the following: if B does not sees another blue eyed person, B will leave on day 1, but if B sees another blue eyed person--which must be A--B will wait to see if A leaves on day 1. Since A and B both see each other, they will both wait until day 2, realize that there must have been another blue eyed person, deduce that it is themselves, and leave that night.

197br, 3bl on island
A, B, and C have blue eyes. A will see that there are two blue eyed people and they will, at the very least, adopt the strategy above. Furthermore, if A happens to find that B and C do NOT leave on day 2, A can deduce that he has blue eyes.

There's really no point in going any further as you should see the pattern by now; many elements of which I find to be quite interesting. By the way, the answer is that all 100 blue eyed people leave on day 100.

If the number of days that blue eyed people remain on the island is greater than the number blue eyed people you see, you are the blue eyed person.

A Moral Dilemma

Scenario 1:
A trolley is running out of control down a track. In its path are 5 people who have been tied to the track. Fortunately, you can flip a switch, which will lead the trolley down a different track to safety. Unfortunately, there is a single person tied to that track. Should you flip the switch?

Everybody to whom I have put this hypothetical case says, Yes, you may. Some people say something stronger than that it is morally permissible for you to turn the trolley: They say that you must turn it -- that morality requires you to do so. Others do not agree that morality requires you to turn the trolley, and even feel a certain discomfort at the idea of turning it. But everybody says that it is true, at a minimum, that you may turn it -- that it would not be morally wrong in you to do so.

Scenario 2:
As before, a trolley is hurtling down a track towards five people. You are on a bridge under which it will pass, and you can stop it by dropping a heavy weight in front of it. As it happens, there is a very fat man next to you - your only way to stop the trolley is to push him over the bridge and onto the track, killing him to save five. Should you proceed?

Why is it that you may switch the trolley tracks though may not push the fat man? In both cases, one will die if the agent acts, but five will live who would otherwise die -- a net saving of four lives. Is our society becoming too soft on fatties? If anyone can come up with a clean way to distinguish the scenarios please share.

This is a test post.

This is the third update of this post. New blog about whatever thoughts come to mind. Whatever we feel like writing about. Hopefully, we'll stir up some good arguments and civilized discussions. God knows that's not possible when this is done in person.