03.30.08
Posted in Idependence Friendly logic, game theory, logic, measurement, philosophy at 8:44 pm by nogre
The Monty Hall Problem illustrates an unusual phenomenon of changing probabilities based upon someone else’s knowledge. On the game-show Let’s Make a Deal the host, Monty Hall, asks the contestant to choose one of three possibilities - Door One, Two or Three - with one door leading to a prize and the other two leading to goats. After the contestant selects a door, another door is opened, one with a goat behind it. At this point the contestant is allowed to switch the previously selected door with the remaining (unopened) door.
Common intuition is that this choice does not present any advantage because the probability of selecting the correct door is set at 1/3 at the beginning. Each door has this 1 out of 3 chance of having a prize behind it, so changing which door you select has no effect on the outcome.
In hindsight, this intuition is wrong. If you initially selected the first goat and then switch when you get a chance, you win. If you selected the second goat and switch, you win. If you selected the prize and switch, you lose. Therefore if you switch, you win 2 out of 3, whereas if you do not switch you win only 1/3 of the time.
So what has gone horribly wrong here:
- Why is most everyone’s intuition faulty in this situation?
- How does switching doors make any difference?
- When did the 1/3 probability turn into a 2/3 probability?
At the beginning of the game you have a 2 out of 3 chance of losing. Likewise the game show has a 2 out of 3 chance of winning (not giving you a prize) at the beginning of the game. Both of these probabilities do not depend upon which door the prize is behind, but only upon the set-up of a prize behind only one of three doors. For instance, an outside service (not the game show) could have set everything up such that both you and the game show would be kept in the dark: there would still be 2 goats and a prize, but neither you nor the game show would know which door led to the prize.
Now imagine that it is the game show that is playing the game. The game show is trying to win by selecting a goat. From this perspective, whichever door that was chosen is good: this door has a 2 out of 3 probability of being a winner (being a goat). Therefore when given the opportunity to change (after the outside service opens a door and shows a goat), there is no reason to do so.
Of course you, the contestant, are the one making the selection, and you do not want a goat. However, if you imagined yourself in the position of the game show at the beginning, as trying to select a goat, you would reasonably assume that, just as the game show did, you were successful in choosing a goat. When given the choice to switch, now that the other goat has been removed, it seemingly makes sense to change your selection.
In this case the easiest way to view the situation is in terms of how to lose, or by considering all the possible outcomes (as mentioned above). Though this is a guess, it seems that our first blush reaction to this problem is always to view it in terms of winning and this is the reason we do not immediately recognize the benefit in switching. We start out with a 1/3 chance of winning and switching doors doesn’t immediately seem to increase this percentage.
To answer how switching doors makes a difference we need to look more closely at the doors. The door that was initially selected has a 1 out of 3 chance of being a prize, and this does not change. If you were to play many times and ignore changing doors, then you would win 33.3% of the time. At the outset the other two doors each have the exact same chance of being a winner, 1 out of 3. So the other two doors combined have a 2 out of 3 chance of containing a winning door.
Now the game show changes the number of doors available from 3 to 2, with one door guaranteed to contain a prize. If you were presented this situation without knowledge of the previous process, then you would rightly put the chance of selecting the prize at 1 out of 2, 50%.
However, you know something about the setup: The door that was initially selected had a probability of having a prize behind it set at 1 out of 3. The thing behind the other door, though, has been selected from a stacked deck: Whatever is behind the door was selected from a group of objects with a 2 out of 3 chance of containing a prize (1/3 + 1/3). You know that the odds on this door are stacked in your favor because the game show knowingly reveals the goat: In the 2/3 case in which you have previously selected a goat, the prize is behind one of the other two doors. When the game-show reveals (and removes) a goat, it guarantees that the prize is behind the last door. Therefore switching doors at the end is equivalent to combining and selecting the probability associated with the two doors not initially selected.
If the game show did not knowingly reveal the goat, you would not be able to take advantage of the stacked deck. Imagine that you select the first door and then another door is opened randomly, revealing a goat. By randomly eliminating this door (and not looking behind the unselected doors) the door that was initially selected becomes unrelated to the present choice: Only by looking behind the unselected doors does the initial selection become fixed in reference to the other doors. Since no one looked behind the doors, some bored, but not malicious, demon could have come and switched whatever was behind the selected and remaining door and neither you nor the game-show would be able to tell. Therefore switching doors when a goat is randomly revealed provides no advantage because the initial selection cannot be related to the probable location of the prize.
Only when the contestant can fix the probable locations of the prize because the location of the prize is known by the game-show, is it possible to assign interdependent probabilities on the location of the prize and the previous selection made. The odds are then tilted in the contestant’s favor by switching away from the low probability initial selection to the door that has the combination of remaining probabilities.
The logic of this needs to be represented game-theoretically with the different quantifiers representing different players of a game of incomplete information. The game would run
* like this:
Domain={prize, goat, goat}
| |
Contestant |
Game Show |
| 1. |
- |
∃x∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=x & b=y & c=z) |
| 2. |
- |
∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=y & c=z) |
| 3. |
- |
∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=z) |
| 4. |
∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=p) |
- |
| 5. |
- |
∃b∀c/x,y,z(p=g & b=g & c=p) |
| 6. |
∀c/x,y,z(p=g & g=g & c=p) |
- |
| 7. |
∀d∀c/x,y,z(d=g & g=g & c=p) |
- |
| 8. |
∀c/x,y,z(g=g & g=g & c=p) |
- |
| 9. |
(g=g & g=g & p=p) |
- |
Line 1 is the initial setup of the prize game: the goal is for the contestant to make his or her placement of the prize and goats match the game show’s placement. Whatever is on the left side of an = will be what the contestant thinks is behind a door and what is on the right of an = will be what the game show puts behind the door, such that each = represents a door. If the formula is satisfied then the contestant will have successfully guessed the location of the prize.
Lines 2, 3 and 4 represent the results of the Game Show placing the prize and goats. Line 5 is the result of the first move of the contestant choosing where he or she thinks the prize is: the ‘a/x,y,z’ means that whatever placed in spot a has to be done independently, i.e. without knowledge, of what x or y or z is. Then the game show reveals a goat behind one of the doors not selected by the contestant. Line 7 represents the choice that is given to the contestant to switch his or her initial placement of where the prize is. Line 8 is the important step: since the contestant does not know what is behind the doors (c/x,y,z) it looks as if there is no advantage to switching. However, the contestant does know that when making a choice to reveal a goat in line 6 that at this point the game show had to know what was behind every door. This means that c is dependent upon b which was depended upon x, y, and z. With this knowledge the contestant can figure out that there is an advantage to switching because the selection of b in line 6 fixed the locations of the prize & goats and in doing so fixed the odds. Since the odds were intially stacked against the contestant, switching to the only remaining door flips the odds in the contestant’s favor, and is done so in this example. Line 9 shows that all the contestant’s choices match up with what the game show has placed behind the doors and hence she or he has won the prize.
* To do a better representation would require keeping the gameshow from not placing a prize anywhere by using a line like ‘x≠y or x≠z’. For graphical brevity I left it out.
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03.21.08
Posted in fun, internet at 11:12 am by nogre
Not that everyone hasn’t already seen everything on the ‘net.
 |
- YouTube - Boston Dynamics Big Dog (3:29)
- The quote from Core77 is priceless: You have to see this video–this %#&*@ robot simply won’t go down. Watch as the guy tries to kick it over at the 00:36 mark. Watch as it goes uphill and downhill in snow, traverses slippery ice, and leaps accurately over demarcated distances. The most disturbing thing is that once it loses its balance, like at the 1:29 mark, it frantically scrabbles to regain its footing with the grotesque urgency of a locomoting cockroach. This machine looks like it will carry out its mission at any cost.
- Ping Mag - Geospatial Technology: Mapping For Human Rights
Last, but certainly not least
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03.18.08
Posted in biology, ethics, evolution, fitness, mind, philosophy, technology at 9:47 pm by nogre
The only ways to enhance the mind is to learn or evolve. Since evolution is out of our hands, all that is left is to learn.
Drugs that purport psychopharmacological enhancement do not do what their name states: they may change certain psychological factors but there is no drug that will make you smarter. This would be to eat from the tree of knowledge.
However drugs may be able to let you do things that you were unable previously, but this is nothing mysterious. If you do not breath enough oxygen, you will not be able to run. You get enough oxygen, you will be able to do more things. Now is oxygen a performance enhancing drug? It depends: the World Anti-Doping Agency recently ruled on oxygen tents (tents that vary the amount of oxygen inside) because using these tents can affect red blood cell counts. This example illustrates two things: that there is nothing inherently special about any particular chemical, be it oxygen or a newfangled drug, and secondly, that drugs only affect intermediary situations, not the final outcome.
The first point is that there is no moral dimension associated with the chemicals themselves. If it is possible to use the most fundamental of chemicals required for our survival in a way that could be seen as inappropriate, then any other chemical could be equally accused. If any chemical can be equally accused, then there is nothing unique about any individual chemical that makes its use morally wrong.
The second point is that drugs only have a specific range of effects. In the above example, the oxygen tents affect red blood cell count. An increased red blood cell count can be used to boost endurance, but this benefit will only appear under certain situations. The tents themselves do not increase endurance: they merely increase red blood cells. If a different drug was consumed to weaken the muscles, then the two ‘drugs’ would counteract each other and there would be no change in ability. Therefore it is not a drug that gives people an ability, such as endurance, but a drug may change how an ability is expressed.
The question is (and always was), “What do you want?” Since drugs have no moral dimension nor imbue the user with unknown (super-human) ability, the only issue is of fair play. Fair play in terms of other people and with your own goals. If you want to be able to lift heavy things, then you can use a machine, you can use drugs or you can work hard. Using a machine or drugs is to use someone else’s technology to assist whatever ability you have. If you use discipline to achieve the same results, then the technology that is being used is your own. Therefore if you are trying to play fair with others, then you have to ensure everyone has access to the same technology, be it machine or drug. If you are trying to achieve something yourself, then only you know whether or not using drugs makes a difference.
As we learn what is safe(r), we are going to have a fun future. Nothing changes our natural born ability or the hard work we have put in, but that has never stopped us from trying. Better drugs are on the way and this means options will be open to us that weren’t possible in the past. Good luck, be safe, have fun.
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03.14.08
Posted in motivation, philosophy at 9:42 pm by nogre
A Philosophy Job Market Blog is a great blog. This is a quote from the last sentence of my previous post:
If anyone asks me about solved philosophy, I’ll tell them about the life and world-changing ideas that make philosophy amazing…
I felt that I ought give some sense of what makes philosophy amazing after writing this last sentence of the last post, but most of what I find amazing takes a while to explain and requires you to know something about me, how I thought and what I now think. Not that this would make for bad blog material - I try very hard to explain things I now believe - but I figured that many philosophers have had a similar experience and subjecting y’all to a biography isn’t my goal.
Then today PJMB busts out with ‘I Mean the Game of Going Insane‘:
In comments, Anon. 5:14’s talking about the howling fantods:
I’m (happily but stressfully) in the middle of my tenure evaluation, and I’ve been remembering lately just how much worse–worlds worse–l I felt during my two years of being on the market and not getting a job. (Sometimes three times really is the charm.) One day just after finding out, for the second year in a row, that I wasn’t going to get a job, I was sitting at the front of the classroom looking at my watch to see if it was time to begin. I looked down and saw the socks I was wearing. They were perfectly ordinary socks. No different from any of the other socks I usually wear. But I was gripped with a deep, undeniable feeling that they were The Wrong Socks. I was wearing The Wrong Socks. I couldn’t do anything right. Not even socks.
Howling.
I studied philosophy like my sanity depended upon it. It may very well have– I try not to have illusions about my failings and I can see how things could have gone very wrong for me. No one speaks of the dangers of philosophy. It is all too easy to lose reality; you lock yourself away working for long enough and there may be no telling how you will emerge. You may start to fear your socks.
One of the more scary facts about insanity is that you won’t be able to tell when you are going insane. It takes an experience such as Anon. described above before you realize something is wrong. And I’d have to qualify that experience as lucky and minor because no one got hurt: imagine it wasn’t Anon.’s socks but Anon.’s significant other’s socks that suddenly caused a breakdown.
Now it might be nice to round out this post by claiming that for all the dangers that philosophy may pose to yourself and others there is an equally great upside of triumph and understanding that may be reached. As I said, “the life and world-changing ideas that make philosophy amazing.” You probably weren’t thinking that I meant something like not having to worry about fearing your socks. But imagine Anon 5:14. I won’t say that Anon is better for that experience of The Wrong Socks, but that the difference in Anon from that day to now is ‘worlds‘ of difference.
It does all boil down to who you are and what you want. If you’re willing to gamble your sanity on the chance that you may one day be happy (if stressed) - Anon may have, and I did so knowingly - then you need not worry about any of this. After you have made your bet, it does no good to worry about it while waiting for the cards to show. If this is news to you, then yes, amazing and world-changing things are possible, but just know that you may have to change equally and change doesn’t come cheap.
When I think about solved philosophy, I remember the confusion, the disturbed contorted thoughts and pain that could have been averted if only the right arguments were available at the right time. This sounds personal, but all the good arguments I have come up with have remained so and have life beyond me with those I’ve told. If I have helped someone understand this world a bit more, then I have solved a philosophical problem.
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03.08.08
Posted in philosophy, science at 12:09 am by nogre
I was reading the philo-blogs and today (7 March) Richard Brown has taken issue with Richard Chappell’s Examples of Solved Philosophy. Brown holds that there is no such thing as solved philosophy (or problems are “only solved from a theoretical standpoint” and hence “involve substantial begging the question”), whereas Chappell happily provides examples that “are at least as well-established as most scientific results.”
Now there is something to be said for both sides: Brown is right when he says that all solutions are theory dependent and Chappell is right when says that we used to argue about certain things and now we don’t (don’t take this as my endorsement of his examples). However, this disagreement is just the two sides of one issue within philosophy of science: Thomas Kuhn’s scientific paradigms.
Thomas Kuhn stated that science evolves by scientific paradigm, that science works under one major governing theory until it is overthrown by another. For instance everyone worked within Newton’s version of the universe until Einstein came along, and now we work under Einstein’s relativity theory. Eventually it is possible that there will be a further paradigm shift away from relativity theory.
Now Brown, I think, makes the claim that philosophical problems (of the sort Chappell indicates) are not solved without question begging. Well, if Chappell is going for the sort of consensus that happens in science - which he looks like he is - then this is not a problem: All problems and solutions are determined within and by the overarching theoretical framework, the paradigm. This is to specifically say that there is no such thing as an answer to a question outside of some theoretical framework: some meta-theory always determines what sort of thing counts as a solution. Therefore Brown has conflated being part of a paradigm with begging the question. Begging the question involves assuming what you set out to prove, whereas being part of a paradigm merely assumes the general rules about what determines a solution to a problem when answering.
However, philosophy is not science, and Brown has a point when he says, “all we can mean by ’solved’ is ‘generally agreed to be true by philosophers/philosopher X’”. Now the paradigm cuts the other way: philosophy does not work by paradigms and hence there is no background framework on which Chappell can base his solved philosophy. Even if all the top philosophers of the day agree to an extent about a good number of issues, it only takes a Kant or a Wittgenstein to turn philosophy on its head. Even simple issues, what might be seen as obvious mistakes made only by laymen, can take on new significance. For example, many people believe that everyone’s perceptions of color are their own, that each of us can’t know what other peoples’ perception of color are like. Perhaps this is true, but personally I believe that it makes no sense to say that you have something if you logically exclude other people from having it (Philosophical Investigations #398) and therefore if you have color perceptions then I can have the same color perception. By no means should my view be taken as correct, but it should illustrate that there is nothing so simple as to be considered solved if all it has is a consensus.
So what of solved philosophy? Is it all just us shifting our assumptions around? The logician De Morgan recognized that his logic (the logic of antiquity until the mid 1800s) was too weak to derive the statement “All heads of horses are heads of animals.” With the advent of modern logic, the statement was derivable. This is an example of solved philosophy: At a certain point we had a problem, were unable to do something, and then later we were able to do it. If we want to solve philosophical problems we have to first find problems, phenomena that no theory can explain, and then find a way to explain it using the unique tools available to philosophers. Taking down bad theories and clarifying issues is a worthwhile endeavor, progress is made, but nothing is solved.
If anyone asks me about solved philosophy, I’ll tell them about the life and world-changing ideas that make philosophy amazing, not about all the bunk theories we had to go through to get there.
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03.02.08
Posted in metaphysics, mind, philosophy, science, technology at 4:24 pm by nogre
This interview with Hubert Dreyfus (just the parts about computers: part 1, part 2. via Continental Philosophy) briefly outlines one of the major criticisms leveled against artificial intelligence: computers will never be intelligent because our intelligence is based upon our physical interactions in and with the world. Very briefly, our intelligence is fundamentally tied to our bodies because it is only through our bodies do we have any interaction with the world. If we separate our intelligence from the body, as in the case with computers, then whatever it is that the computer has, it is not intelligence because intelligence only refers to how to bodily interact with the world.
As Dreyfus says this problem is attributed to a Merleau-Ponty extension of Heidegger and the only proposed solution is to embody computers by providing them with a full representation of world and body. I don’t think there is generally much faith in this solution; I certainly don’t have much faith in it.
However, this bodily criticism is a straw man. Computers have ‘bodies,’ they are definitely physical things in the world. But what of the physical interactions required for intelligence? Computers interact with the world: computers are affected by heat, moisture, dirt, vibration, etcetera. The only differences are the actual interactions that computers have as compared to humans: we experience humidity one way and they experience it differently. So yes, computers will have different interactions and hence they will never have the same intelligence that we have, but that does not imply that computers cannot have an embodied intelligence. It only means that computer embodied intelligence will be significantly different than our own intelligence. Therefore the above argument against computer intelligence only applies to those people who are trying to replicate perfect human intelligence and does nothing against people trying to create intelligence in computers.
For example, light-skinned and dark-skinned people have very slightly different physiologies. Now I see the above argument as saying that someone of different skin color cannot have the same sort of intelligence that you have because their interactions with the world are inherently different. Sure, everyone experiences things slightly differently due to having different bodies, but to claim that this creates incompatible intelligences is obviously wrong: No one on the face of the earth would be able to communicate with each other due to everyone being physically unique. Computers may be physically different to a greater extent, but this does not impact intelligence.
The criticism of computer intelligence based upon the need for a body is no more than subtle techno-racism.
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