Why be moral? Because doing otherwise doesn't make sense. Doing otherwise doesn't work well in the long run. Doing otherwise will result in shooting yourself (and others) in the metaphorical foot. In a nutshell, not being moral is shortsightedly stupid.
I'm not talking about why just [insert group here] should be moral, here, I'm talking about why anyone should be moral, and that's because the reasons that immorality doesn't work have nothing to do with supernatural entities. Whether the universe is blamed on Brahma, Yahweh, Allah, The Force, or nothing at all, there are certain aspects that couldn't have been put together any differently (at least, not without having a meaningless, completely self-contradictory undifferentiated mass in which everything that exists does not exist, and vice versa). There are some constraints on how a natural world could be which can't be violated by the most super of supernatural forces, and morality arises from those constraints.
Concisely: Morality, coming from a source beyond any supernatural entity, and being a property of every natural system, is relevant to everyone, whether they add supernatural entities to their worldview or not.
One concern many theists have seen is that many atheists seem to think that the morality of theists is somehow inferior. At this point, perhaps it's clear why. If someone said to you (I'm temporarily assuming the reader is a theist) that morality didn't come from your god, but rather came from, say... following laws... you'd be rightly concerned. There's a misattribution going on, and worse yet there's the risk that if people run around thinking "Morality equals lawfulness" and some jerk puts a bad law on the books, people will think that following the bad law is moral. Ack! Worse yet, since the law is open to various degrees of interpretation, someone with the authority to decide on how the law is interpreted might attribute the wrong spirit to the letter, and so now everyone is compelled to follow a law ("be moral") even if it is what you know to be immoral! Double ack! You could even have people saying "Well, without the law, there'd be no reason to be moral... might as well drink and be merry, for tomorrow you die!" when you know full well there are more important reasons than the law to be moral. Triple ack! And on top of it all, you're painted as immoral for worrying about higher levels of morality than that dictated by the law. Quad overkill ack!
That's the kind of misattribution that many atheists go "Quad overkill ack!" over when they hear people claiming that morality comes from supernatural sources. There's a misattribution, there's the ability to subject morality to various biased interpretations, the bad interpretations can be handed down in some cases from a priviledged few and thus spread alarmingly quickly, people start saying things like "Sure you could be moral, but why?", and brights in general are often suspected of having an inferior class of morality simply because they're trying to get their morality from the proper source.
When atheists are saying that morality without gods is superior to a morality dictated by gods, it often comes from a similar concern to a theist saying that morality dictated by gods is superior to a morality dictated by human laws. Sure, you might get lucky and manage to keep in line with the true source, but there's far too much room for human bungling and misattribution of why things are important.
Then again, sometimes it's coming from a common combination of bitterness and pompousness. Every group has people like that, too.
There's also something similar to misattribution that's a concern, though it's not quite the same. I'm thinking of an experiment that was done at a daycare center. In a nutshell, there were often a few parents every week who picked their kids up late, which wasn't so great for the daycare workers that had to stay to watch the not-gone-yet kids. The center started charging a 'late pick-up fee', and... late pickups went up! What was a matter of being inconsiderate before was now merely a matter of a very small fee for a very convenient service. Worse yet, when the center dropped the late fee, the late pickups didn't go down. Hey, they're doing the extra service for free now! No conscience pangs AND no hit on the wallet, how great is that?
By misattributing an issue from the realm of morality to the realm of externally enforced consequences (in this case, by adding a penalty fee) people were led to ignore the moral aspect and think only of the inflicted consequences aspect and, worse yet, when the consequences were taken away, they didn't revert to moral behavior, but acted as if the action were now just fine!
Whenever I hear someone say "Sure, you could be a moral atheist, but why?" I worry that here is someone who wouldn't be moral if they weren't essentially being told to, and it's scary and sad. Perhaps there are people who couldn't be moral if they weren't told to, in which case it's a good thing someone's telling them to... but I suspect there are a lot of people who could once have been moral simply because it doesn't make sense not to, but who've been led to misattribute the source of morality and, in so doing, have lost their capacity to actually be moral, and only retain the ability to act morally.
It's also possible that there are people who believe in supernatural forces but don't mistakenly label those things as sources of morality... people who are fundamentally moral, and simply add a layer of religious decoration on top of it without mistaking the decoration for the real thing. I have not, to the best of my knowledge, ever met anyone or heard of anyone who fit this description, so it certainly seems unlikely to me, but perhaps that is more an upshot of how pervasive modern religious piekories have to be in order to survive, and less a property of religious piekories in general.
3.27.2007
Welcome to Daveworld
I'd like to take you to an imaginary reality called Daveworld, named after the first person to suffer (imaginary) death therein.
Upon arriving in Daveworld, I first bring a visitor to a large board. On the board is a notice, "You are type..." below which are listed some letters, with a lot of gaps in between. It's as if someone put up a whole alphabet, and then removed most of the letters. Let us imagine that you are type __C____H_JK_____, according to the sign.
Now, for purposes of efficiency, I've brought some other readers along as well, and you can see some of them looking around, having just completed reading the board as well. The conversation that you hear goes as follows:
Such are the confusions that can arise in Daveworld.
If you'd like to puzzle out what's going on, feel free. For everyone else, this may be useful: The big board has, despite appearances, all the letters of the alphabet on it. The board itself is an object of type ABCDEFGHIJKLMNOPetcetera, but each letter is only the type shown on the letter: The letter A is a type A object, the letter Q is a type Q object. In Daveworld, two things can only interact directly if they share a type. The letter Q can stay on the board because both are type Q (among possibly other types). You can see the letters CHJ and K because you are type CHJK, and since you're type C (among other things) you can see the letter C, since you're type J (among other things) you can see the letter J.
In a similar fashion, you share type C with readers 2 and 6, so can hear and see them, type H with reader 3 and 5, but you don't share any types with reader 4 (who's DEFG), so in many regards reader 4 does not exist to you. The same goes for readers 5 and 6, who don't share any types, so don't exist to each other. They can't hear, see, or even bump into the other reader.
For those who are morbidly curious, Dave was a type AB who was a passenger in a type A car driven by a type A person. There was a type BC car driven by a type C driver who couldn't see the type A car or the type AB Dave. Alas, though the BC car would have passed through the A car without damage, it couldn't pass through poor type AB Dave, who in turn could not pass through his type A seat.
The car Dave was riding in spun out of control, what with the passenger seat being forcibly ripped backward due to Dave suddenly exerting a lot of force on it (due to another car exerting a lot of force on him), and the oncoming vehicle got a crumpled front end. While the type A driver and the type C driver survived with substantial injuries and repair bills, Dave was little more than a thin film of liquefied organic matter by the time the cars ground to a halt.
This tale is told to everyone visiting Daveworld in order that they will have an immediate appreciation for the fact that it is not safe to ignore something or someone just because you have non-overlapping types.
It is now time to depart Daveworld, but perhaps we will return another time, if something would be clearer here than in the world we've grown up in.
Incidentally, all the readers you share a type with have assured you that reader 4 wants you to know that, as soon as you're back in the real world, you can borrow that book.
Upon arriving in Daveworld, I first bring a visitor to a large board. On the board is a notice, "You are type..." below which are listed some letters, with a lot of gaps in between. It's as if someone put up a whole alphabet, and then removed most of the letters. Let us imagine that you are type __C____H_JK_____, according to the sign.
Now, for purposes of efficiency, I've brought some other readers along as well, and you can see some of them looking around, having just completed reading the board as well. The conversation that you hear goes as follows:
- Reader 2: "I'm type BCDE? I wonder what that means?"
- Reader 3: "I don't know, but it says I'm a type DHI. How about the rest of you?"
- Reader 4 seems to not be there, which is too bad, because they were going to lend you an interesting book. Oh well.
- Reader 5 says "You're a DEFG? Cool, that's like mine, I'm a EFGH." at the same time as Reader 6 says "DEFG, eh? Well I've got you beat, I'm an ABCD."
- Oddly, neither reader 5 nor reader 6 (who are notoriously bad about being gracefully interrupted) seem in the least bit bothered by both talking at once. Readers 2 and 3 also look a bit surprised by this.
- There's a bit of a pause, then 5 and 6 both object at the same time, asking what reader 4 means by "please speak one at a time."
- 5 also mentions that you, reader 1, are right over where you are, while 6 claims that everyone is there except 5, and doesn't know what 4 is talking about.
- 2 and 3 try to reassure everyone that everyone is present, and try to herd 5 and 6 together because obviously something is wrong with their eyes...
- And suddenly it seems something is wrong with your eyes, because 5 and 6 end up standing in exactly the same spot... they didn't bump into each other, they just... overlapped. And not only does this seem to not bother them in the least, but they still seem unaware of each other.
Such are the confusions that can arise in Daveworld.
If you'd like to puzzle out what's going on, feel free. For everyone else, this may be useful: The big board has, despite appearances, all the letters of the alphabet on it. The board itself is an object of type ABCDEFGHIJKLMNOPetcetera, but each letter is only the type shown on the letter: The letter A is a type A object, the letter Q is a type Q object. In Daveworld, two things can only interact directly if they share a type. The letter Q can stay on the board because both are type Q (among possibly other types). You can see the letters CHJ and K because you are type CHJK, and since you're type C (among other things) you can see the letter C, since you're type J (among other things) you can see the letter J.
In a similar fashion, you share type C with readers 2 and 6, so can hear and see them, type H with reader 3 and 5, but you don't share any types with reader 4 (who's DEFG), so in many regards reader 4 does not exist to you. The same goes for readers 5 and 6, who don't share any types, so don't exist to each other. They can't hear, see, or even bump into the other reader.
For those who are morbidly curious, Dave was a type AB who was a passenger in a type A car driven by a type A person. There was a type BC car driven by a type C driver who couldn't see the type A car or the type AB Dave. Alas, though the BC car would have passed through the A car without damage, it couldn't pass through poor type AB Dave, who in turn could not pass through his type A seat.
The car Dave was riding in spun out of control, what with the passenger seat being forcibly ripped backward due to Dave suddenly exerting a lot of force on it (due to another car exerting a lot of force on him), and the oncoming vehicle got a crumpled front end. While the type A driver and the type C driver survived with substantial injuries and repair bills, Dave was little more than a thin film of liquefied organic matter by the time the cars ground to a halt.
This tale is told to everyone visiting Daveworld in order that they will have an immediate appreciation for the fact that it is not safe to ignore something or someone just because you have non-overlapping types.
It is now time to depart Daveworld, but perhaps we will return another time, if something would be clearer here than in the world we've grown up in.
Incidentally, all the readers you share a type with have assured you that reader 4 wants you to know that, as soon as you're back in the real world, you can borrow that book.
3.24.2007
Are you a bright?
As immediately evidenced by its being an already existing word, 'bright' is not one of my own new words. Nonetheless, the efforts of the Brights Movement are worth noting, and 'a bright' is a useful noun to add to the vocabulary.
From the site:
From the site:
What is a bright?Considering that, if this blog were a person, it would be a bright (noun, noun, gotta remember it's a noun. This is not site which is bright, it's a site written by a bright!) it seemed appropriate to give The Bright's Net some linkspace over in the sidebar.
- A bright is a person who has a naturalistic worldview
- A bright's worldview is free of supernatural and mystical elements
- The ethics and actions of a bright are based on a naturalistic worldview
3.22.2007
Symmetry breaking
Meet Allen Marshall and Marshall Allen.
Allen's a good person. Not perfect, but well above average. He's got a good life. Again, not perfect, but well above average. Allen knows this comes from holding A above all else. The same holds, for the most part, for Marshall. The main differences are that Marshall's good fruits come from holding M above all else, and he knows that the standard of A that allows Allen to be a good person with a good life will, eventually, run out of steam, whereas Marshall's own standard of M won't. Allen feels similarly, except he knows that M will eventually fail.
This is sad on two counts, the first being that Allen and Marshall are friends, so Allen would much prefer that Marshall realize he's headed down a dead end before running into it face-first (and vice-versa) and the second being that by holding himself to the higher standard of M, Allen would still end up meeting his original, lower standard of A as well. At least, so Marshall claims. Allen, in turn, knows that Marshall would still get M by putting in the extra work to meet the higher standard of A.
Alas, nothing Allen says or does seems to make this point. Allen's got a good life, but it's not good enough to persuade Marshall that A is more important than M. Allen's a good person, but Marshall says that's despite Allen holding A above M, not because of it, and so isn't an indicator that Allen's right. Allen may hope that Marshall will come to accept the importance of A, but as time goes on, there's less and less evidence that it'll happen. Nor does Marshall have it any better, as all of his evidence is just as unconvincing to Allen.
With such a fundamental disparity, are there any worthwhile grounds on which Allen and Marshall can be friends in a meaningful sense, or is "friendly acquaintance" the best that can be hoped for? What range of things might they be able to both improve upon together, from their separate starting points? How broad is the range of things that are between A and M which can only be reached from the 'higher' one, and not the 'lower'?
If there's a yet-'higher' standard Q which, if it were followed, would subsume both A and M, what are the chances of either Allen or Marshall being able to become aware of it? Are their chances at all improved if they have access to each other?
Allen's a good person. Not perfect, but well above average. He's got a good life. Again, not perfect, but well above average. Allen knows this comes from holding A above all else. The same holds, for the most part, for Marshall. The main differences are that Marshall's good fruits come from holding M above all else, and he knows that the standard of A that allows Allen to be a good person with a good life will, eventually, run out of steam, whereas Marshall's own standard of M won't. Allen feels similarly, except he knows that M will eventually fail.
This is sad on two counts, the first being that Allen and Marshall are friends, so Allen would much prefer that Marshall realize he's headed down a dead end before running into it face-first (and vice-versa) and the second being that by holding himself to the higher standard of M, Allen would still end up meeting his original, lower standard of A as well. At least, so Marshall claims. Allen, in turn, knows that Marshall would still get M by putting in the extra work to meet the higher standard of A.
Alas, nothing Allen says or does seems to make this point. Allen's got a good life, but it's not good enough to persuade Marshall that A is more important than M. Allen's a good person, but Marshall says that's despite Allen holding A above M, not because of it, and so isn't an indicator that Allen's right. Allen may hope that Marshall will come to accept the importance of A, but as time goes on, there's less and less evidence that it'll happen. Nor does Marshall have it any better, as all of his evidence is just as unconvincing to Allen.
With such a fundamental disparity, are there any worthwhile grounds on which Allen and Marshall can be friends in a meaningful sense, or is "friendly acquaintance" the best that can be hoped for? What range of things might they be able to both improve upon together, from their separate starting points? How broad is the range of things that are between A and M which can only be reached from the 'higher' one, and not the 'lower'?
If there's a yet-'higher' standard Q which, if it were followed, would subsume both A and M, what are the chances of either Allen or Marshall being able to become aware of it? Are their chances at all improved if they have access to each other?
3.20.2007
A Fair Share
(A) Are you dong your fair share around the house? (B) Do you think you're doing your fair share? These are, for better or worse, not the same question. To many folks, it's obvious that (A) is a different question from (C) Do other people in the house think you're doing your fair share, and that (B) and (C) are likely to be mismatched, but realizing that (A) and (B) are different often takes some practice to really get used to.
That's not to say that people don't try. I know many folks who use the rule of thumb
Since a post here wouldn't be a post here without a good digression or tangent, I'd like to take a minute to talk about cake. Cake is a classic object for "how do you fairly divide something" problem, and for a homogenous cake, the "you cut, I choose" or "I cut, you choose" strategies work just fine for two people. But what about when you've got more people? For homogenous cakes, there are similar strategies that ensure that 'fairness' is satisfied, at least in the form of "If I cut an unfairly sized piece, I'm the one who'll end up with the unfairly sized piece."
But what about a non-homogenous cake? What about a chocolate/vanilla marble cake with frosting (thick in some places, thin in other), walnuts in most of it, some pieces with strawberries on top, some pieces with candlewax drippings and a candlehole, and let's toss a lovely edible candy effigy on the top. Suddenly, the scene changes, because different pieces of the cake can be more desirable to different people. Perhaps you're allergic to walnuts: you'd probably rather have a small portion without walnuts than a large portion that's stuffed with nuts. Perhaps I reaaaaally like strawberries. I could happily pass up a large piece without fruit if it meant I could get a small piece with fruit. Some third person might have a moderate preference for chocolate to vanilla, so would be just as happy with a medium-small chocolate piece as a medium-large vanilla piece.
With a heterogeneous cake and two people, "You cut, I choose" isn't symmetric with "I cut, you choose" any more, because where that cut falls can be important! You might cut things in a way that presents me with two equally bad options from your point of view, but I might like one option far more than the other. In the sense of "You suffer if you cut unfairly", you still have that issue, but in the sense of "If you cut fairly, I don't care which piece I get either", the situation has changed! On the other hand, you might be able to cut the cake in a way that you felt one piece was better than the other, and I thought the other piece was better than the one, so unless I'm a masochist, we both end up getting more than our 'fair share' of goodness.
Reeling the tangent back in closer to reality, consider the 'cake' of housework. Pretend, for the sake of argument, that while you may not exactly enjoy mopping floors, you find it far less onerous than balancing checkbooks. For the same argument, I'll also pretend that I don't like balancing checkbooks. I don't have to pretend that I prefer that activity to mopping, because it's true. We could split up the chores by having you do the books and me do the floors. According the the rule of thumb, that would be a good solution, since I'd feel like I was doing the harder/less pleasant part, and you'd feel the same way. And, indeed, if there really were some objective way of measuring bookkeeping against cleaning, we might indeed each be doing our fair share. But, if that were the case, if we switched roles, we'd also be doing our fair share.
The rule of thumb, however, would throw a red flag. "You're doing the easier stuff!" it would say, "You're not pulling your weight!" Even if both of you were pulling your weight (with reversed roles or not), the rule of thumb would give a false negative. Lacking enough communication to get a broader picture of what's going on, this could lead to switching roles (not so great, because while still fair, everyone's stuck with more than their share instead of less), or it could lead to rule of thumb indoctrinees building up a resevoir of guilt over getting away with doing less than their fair share, or it could lead to anti-guilt defensive anticipation due to expecting the other person to eventually complain about doing more than their fair share.
So, as a second approximation... another rule of thumb that still isn't exactly on the mark but gets closer... I'd suggest the following:
That's not to say that people don't try. I know many folks who use the rule of thumb
"If it doesn't feel like you're doing more than your share, you're doing less than your share".I've used it. I think it's often appropriate. It's kept me from accidentally complaining about doing more than was fair when, in actuality, I wasn't even pulling my own weight. As a first approximation goes, I think it's a good one, but I don't think it's spot-on.
Since a post here wouldn't be a post here without a good digression or tangent, I'd like to take a minute to talk about cake. Cake is a classic object for "how do you fairly divide something" problem, and for a homogenous cake, the "you cut, I choose" or "I cut, you choose" strategies work just fine for two people. But what about when you've got more people? For homogenous cakes, there are similar strategies that ensure that 'fairness' is satisfied, at least in the form of "If I cut an unfairly sized piece, I'm the one who'll end up with the unfairly sized piece."
But what about a non-homogenous cake? What about a chocolate/vanilla marble cake with frosting (thick in some places, thin in other), walnuts in most of it, some pieces with strawberries on top, some pieces with candlewax drippings and a candlehole, and let's toss a lovely edible candy effigy on the top. Suddenly, the scene changes, because different pieces of the cake can be more desirable to different people. Perhaps you're allergic to walnuts: you'd probably rather have a small portion without walnuts than a large portion that's stuffed with nuts. Perhaps I reaaaaally like strawberries. I could happily pass up a large piece without fruit if it meant I could get a small piece with fruit. Some third person might have a moderate preference for chocolate to vanilla, so would be just as happy with a medium-small chocolate piece as a medium-large vanilla piece.
With a heterogeneous cake and two people, "You cut, I choose" isn't symmetric with "I cut, you choose" any more, because where that cut falls can be important! You might cut things in a way that presents me with two equally bad options from your point of view, but I might like one option far more than the other. In the sense of "You suffer if you cut unfairly", you still have that issue, but in the sense of "If you cut fairly, I don't care which piece I get either", the situation has changed! On the other hand, you might be able to cut the cake in a way that you felt one piece was better than the other, and I thought the other piece was better than the one, so unless I'm a masochist, we both end up getting more than our 'fair share' of goodness.
Reeling the tangent back in closer to reality, consider the 'cake' of housework. Pretend, for the sake of argument, that while you may not exactly enjoy mopping floors, you find it far less onerous than balancing checkbooks. For the same argument, I'll also pretend that I don't like balancing checkbooks. I don't have to pretend that I prefer that activity to mopping, because it's true. We could split up the chores by having you do the books and me do the floors. According the the rule of thumb, that would be a good solution, since I'd feel like I was doing the harder/less pleasant part, and you'd feel the same way. And, indeed, if there really were some objective way of measuring bookkeeping against cleaning, we might indeed each be doing our fair share. But, if that were the case, if we switched roles, we'd also be doing our fair share.
The rule of thumb, however, would throw a red flag. "You're doing the easier stuff!" it would say, "You're not pulling your weight!" Even if both of you were pulling your weight (with reversed roles or not), the rule of thumb would give a false negative. Lacking enough communication to get a broader picture of what's going on, this could lead to switching roles (not so great, because while still fair, everyone's stuck with more than their share instead of less), or it could lead to rule of thumb indoctrinees building up a resevoir of guilt over getting away with doing less than their fair share, or it could lead to anti-guilt defensive anticipation due to expecting the other person to eventually complain about doing more than their fair share.
So, as a second approximation... another rule of thumb that still isn't exactly on the mark but gets closer... I'd suggest the following:
"If nobody wants to swap workloads with you, you're not doing less than your share."Of course, this requires communicating with other people, so automatically involves more work than the first rule of thumb, but weighed against alternatives like "everyone doing one of the fair shares they dislike most" and "feeling unnecessarily guilty" and "becoming unnecessarily pre-emptively defensive", perhaps it'd be worth the extra effort sometimes.
3.15.2007
Evidence vs The Dayalektik
Intuitionistic proof
Hands up if you're aware that logicians talk about all sorts of different logics. Emphasis on the plural. If you are currently afflicted with an elevated hand, you probably have heard about intuitionistic logic, and can safely skip down a ways. And put your hand down. Unless it's enjoying the view, of course.
For all the rest of you, I'll begin by apologizing (mildly) for the name. I didn't come up with the name, so I can't apologize much for it, but given how nearly opposite its original meaning (and usage in this term) is from any of the current meanings that you may already be puzzling through... well, someone ought to apologize, and I'm not afraid to throw that first stone.
This isn't feminine intuition, it's not mathematical intuition, it doesn't have to do with intuitive people... heck, I'm not even going to explain what kind of intuition it is, because it's not really all that necessary to understanding the concept. I'll just recommend this: whenever I say "intuitionistic", pretend I instead said "evidential" or "monotonic".
If 'monotonic' makes you think of medicine taken by mononucleosis sufferers, just go with 'evidential' instead.
So, how does intuitionistic logic differ from run-of-the-mill everyday logic? In one sense, it all boils down to rejecting the law of the excluded middle. That is to say, it rejects Sherlock Holmes' maxim that "when you have excluded the impossible, whatever remains, however improbable, must be the truth." On the face of it, this might seem a disastrous thing to do (or, at least, I thought it was) but consider this not-quite-accurate-but-illuminating example:
A man is told there may be a pot of gold buried under his farmland. He goes out and digs a hole. If he finds a pot in the hole, then he has established that, yes, there is a pot of gold buried under his farmland. If, on the other hand, the hole has nothing but dirt, he hasn't proven that there's no gold under his land! He could try again, in a different spot. If he finds gold, he's proven that the statement was true, but if he fails, he has not proved it false, but again he could dig another hole. In this manner, any 'true' hole can prove the statement true, but no 'false' hole proves the statement false.
Perhaps farmlands are small enough, in your mind, to be able to dig up the whole place looking for gold, in which case I simply ask that you change 'farmland' to 'nation' and 'gold' to 'WMDs', which should be enough to convince you that there are relevant issues here.
Now, proving something doesn't exist is notoriously more difficult than proving something does exist, but even when it comes to proving that something does exist, intuitionistic logic holds more stringent standards. Here's another example which has recently entertained me:
Someone wants to know if there are any two numbers, x and y, such that both of them are irrational numbers (numbers like Pi, which can't be exactly expressed as the ratio of two whole numbers) but that x^y (x to the power y) is a rational number. Here's a non-intuitionistic proof that this is true: The square root of 2 is an irrational number, so perhaps x=sqrt(2), y=sqrt(2) satisfies the property that x^y is rational, and maybe it doesn't. If it does, well then, the claim is true. If it doesn't, that means that sqrt(2)^sqrt(2) is an irrational number... which isn't so useful, except that this in turn means that perhaps x=sqrt(2), y=sqrt(2)^sqrt(2) satisfies the property. Indeed, the value of x^y turns out to be 2, which is a rational number!
So, either x=sqrt(2), y=sqrt(2) satisfies the property, or (if that's not the case) x=sqrt(2), y=sqrt(2)^sqrt(2) satisfies the property, so the property is true. Any way you cut it, there are two numbers that satisfy the property.
There's a proof that there are such numbers, but... uh... what ARE those numbers? Is it the first pair? Is it the second pair? We've just proved that some pair of numbers exists without nailing down what the numbers actually are, and while that's allowed in everyday logic, it's not kosher by intuitionistic rules. If we want an intuitionistic proof that something exists, it's necessary to actually go out and find an actual specimen, not just show that it's impossible for one to fail to exist.
It's provable (intuitionistically and otherwise) that 'normal' (i.e. classical) logic and intuitionistic logic are equally powerful. What you can do with one, you can do with the other, so it wouldn't make sense to say that one is better than the other in terms of being able to prove more, or failing to prove less, or the suchlike. There are, however, ways in which using a different (and equally capable) system of logic can be handy.
For an example, I turn to C.S. Lewis. Mere Christianity is a fun read, and some parts of it appear tremendously persuasive. Alas, it's an excellent example of what some folks I know call "The Dayalektik" (any resemblance to 'dialectic' that your ears may detect should be ignored, as The Dayalektik is a method for obfuscating truth and making people agree with things by pushing them through a series of subtle false dichotomies, rather the opposite of the meaning of Socratic dialectic, and I'm not sure how it relates to Marxist dialectic.) Lewis sets up a series of arguments where each major step essentially ran "Either A or B or C must be true, A and B are clearly false, so C is true." Then, of course, going on with C assumed to be true, another step was taken, and so forth.
While the method of "List all the cases, eliminate the impossible ones, and say the remaining one is true" is logically impeccable, the method of "List some of the cases, eliminate all but one, and say the remaining one is true" isn't. If you haven't listed all the cases, it might be that you haven't even listed the 'true' case, so you can end up 'proving' that something false is true by elimination. It's a powerful method for clouding minds, evading the truth, and coming to unsound conclusions, and the worst part is that it's very, very, very easy to use this method completely by accident and with the best of intentions. Yes, people who want to cloud the truth have good reason to use this method, but without discipline and training, people who want to reveal the truth can blunder into it even while trying to avoid it.
I raise this example because it's a place where intuitionistic reasoning nicely fails to present a big danger. To show, intuitionistically, that C is true, it's necessary to give evidence that it is true, not simply accumulate evidence that everything else isn't true. False dichotomies aren't a big problem, because knocking out a strawman collection of opposition isn't intuitionistically enough to support a pet view... there actually has to be something substantial supporting it before you can claim victory.
Now, to those who raised their hand way back at the beginning, I am aware that there's a certain amount of proof-by-eliminating-alternatives that's still intuitionistically acceptable, but I still assert that it requires enough work to prevent the false dichotomy from being a pitfall that's excessively easy for the untrained to stumble into. And if your hand is still up at this point, you might consider giving it a rest.
So, the take-home of today is that if you're worried you're dealing with a false dichotomy, consider taking a hiatus from the law of the excluded middle and demand positive proof of an assertion, not just negative evidence against its negation.
Hands up if you're aware that logicians talk about all sorts of different logics. Emphasis on the plural. If you are currently afflicted with an elevated hand, you probably have heard about intuitionistic logic, and can safely skip down a ways. And put your hand down. Unless it's enjoying the view, of course.
For all the rest of you, I'll begin by apologizing (mildly) for the name. I didn't come up with the name, so I can't apologize much for it, but given how nearly opposite its original meaning (and usage in this term) is from any of the current meanings that you may already be puzzling through... well, someone ought to apologize, and I'm not afraid to throw that first stone.
This isn't feminine intuition, it's not mathematical intuition, it doesn't have to do with intuitive people... heck, I'm not even going to explain what kind of intuition it is, because it's not really all that necessary to understanding the concept. I'll just recommend this: whenever I say "intuitionistic", pretend I instead said "evidential" or "monotonic".
If 'monotonic' makes you think of medicine taken by mononucleosis sufferers, just go with 'evidential' instead.
So, how does intuitionistic logic differ from run-of-the-mill everyday logic? In one sense, it all boils down to rejecting the law of the excluded middle. That is to say, it rejects Sherlock Holmes' maxim that "when you have excluded the impossible, whatever remains, however improbable, must be the truth." On the face of it, this might seem a disastrous thing to do (or, at least, I thought it was) but consider this not-quite-accurate-but-illuminating example:
A man is told there may be a pot of gold buried under his farmland. He goes out and digs a hole. If he finds a pot in the hole, then he has established that, yes, there is a pot of gold buried under his farmland. If, on the other hand, the hole has nothing but dirt, he hasn't proven that there's no gold under his land! He could try again, in a different spot. If he finds gold, he's proven that the statement was true, but if he fails, he has not proved it false, but again he could dig another hole. In this manner, any 'true' hole can prove the statement true, but no 'false' hole proves the statement false.
Perhaps farmlands are small enough, in your mind, to be able to dig up the whole place looking for gold, in which case I simply ask that you change 'farmland' to 'nation' and 'gold' to 'WMDs', which should be enough to convince you that there are relevant issues here.
Now, proving something doesn't exist is notoriously more difficult than proving something does exist, but even when it comes to proving that something does exist, intuitionistic logic holds more stringent standards. Here's another example which has recently entertained me:
Someone wants to know if there are any two numbers, x and y, such that both of them are irrational numbers (numbers like Pi, which can't be exactly expressed as the ratio of two whole numbers) but that x^y (x to the power y) is a rational number. Here's a non-intuitionistic proof that this is true: The square root of 2 is an irrational number, so perhaps x=sqrt(2), y=sqrt(2) satisfies the property that x^y is rational, and maybe it doesn't. If it does, well then, the claim is true. If it doesn't, that means that sqrt(2)^sqrt(2) is an irrational number... which isn't so useful, except that this in turn means that perhaps x=sqrt(2), y=sqrt(2)^sqrt(2) satisfies the property. Indeed, the value of x^y turns out to be 2, which is a rational number!
So, either x=sqrt(2), y=sqrt(2) satisfies the property, or (if that's not the case) x=sqrt(2), y=sqrt(2)^sqrt(2) satisfies the property, so the property is true. Any way you cut it, there are two numbers that satisfy the property.
There's a proof that there are such numbers, but... uh... what ARE those numbers? Is it the first pair? Is it the second pair? We've just proved that some pair of numbers exists without nailing down what the numbers actually are, and while that's allowed in everyday logic, it's not kosher by intuitionistic rules. If we want an intuitionistic proof that something exists, it's necessary to actually go out and find an actual specimen, not just show that it's impossible for one to fail to exist.
It's provable (intuitionistically and otherwise) that 'normal' (i.e. classical) logic and intuitionistic logic are equally powerful. What you can do with one, you can do with the other, so it wouldn't make sense to say that one is better than the other in terms of being able to prove more, or failing to prove less, or the suchlike. There are, however, ways in which using a different (and equally capable) system of logic can be handy.
For an example, I turn to C.S. Lewis. Mere Christianity is a fun read, and some parts of it appear tremendously persuasive. Alas, it's an excellent example of what some folks I know call "The Dayalektik" (any resemblance to 'dialectic' that your ears may detect should be ignored, as The Dayalektik is a method for obfuscating truth and making people agree with things by pushing them through a series of subtle false dichotomies, rather the opposite of the meaning of Socratic dialectic, and I'm not sure how it relates to Marxist dialectic.) Lewis sets up a series of arguments where each major step essentially ran "Either A or B or C must be true, A and B are clearly false, so C is true." Then, of course, going on with C assumed to be true, another step was taken, and so forth.
While the method of "List all the cases, eliminate the impossible ones, and say the remaining one is true" is logically impeccable, the method of "List some of the cases, eliminate all but one, and say the remaining one is true" isn't. If you haven't listed all the cases, it might be that you haven't even listed the 'true' case, so you can end up 'proving' that something false is true by elimination. It's a powerful method for clouding minds, evading the truth, and coming to unsound conclusions, and the worst part is that it's very, very, very easy to use this method completely by accident and with the best of intentions. Yes, people who want to cloud the truth have good reason to use this method, but without discipline and training, people who want to reveal the truth can blunder into it even while trying to avoid it.
I raise this example because it's a place where intuitionistic reasoning nicely fails to present a big danger. To show, intuitionistically, that C is true, it's necessary to give evidence that it is true, not simply accumulate evidence that everything else isn't true. False dichotomies aren't a big problem, because knocking out a strawman collection of opposition isn't intuitionistically enough to support a pet view... there actually has to be something substantial supporting it before you can claim victory.
Now, to those who raised their hand way back at the beginning, I am aware that there's a certain amount of proof-by-eliminating-alternatives that's still intuitionistically acceptable, but I still assert that it requires enough work to prevent the false dichotomy from being a pitfall that's excessively easy for the untrained to stumble into. And if your hand is still up at this point, you might consider giving it a rest.
So, the take-home of today is that if you're worried you're dealing with a false dichotomy, consider taking a hiatus from the law of the excluded middle and demand positive proof of an assertion, not just negative evidence against its negation.
3.09.2007
Are you a wise man, or a fool?
It is said that the wise man listens, and sees his folly, but the fool hears, and sees only his wisdom.
The fool thinks he sees his folly. The wise man fears he sees a false reflection of himself as wise.
It is said that a man broke his leg, and took up crutches, and then the whole village took up crutches, and then everyone forgot how to walk without crutches. It is said that one day a boy threw down his crutches and cried "One does not need crutches to walk!" and promptly fell on his face.
The fool thinks he is the boy, brave champion of a truth he cannot yet prove. The wise man fears he may be a villager who scoffs at the boy he should heed.
It is said that a king dreamed of a statue with a head of gold, a chest of silver, a belly of bronze, legs of iron, and feet of iron mixed in clay, and that a stone shaped not by human hands broke the statue's feet and scattered the edifice to dust.
The fool thinks that he is the stone, shaped by greater forces to knock down the increasingly valueless structures of the world. The wise man fears he may be the statue that would be a precious thing if it were not solidly standing on a foundation that cannot withstand the truth.
It is said there was a farmer who planted an olive tree, but it would not bear fruit, so he grafted on some wild olive nearby that, while sparse, bore fruit, but his tree only bore fruit for a brief time, and so the gardener cut away the tree to a stump. Time passed, and he returned to the tree, and coaxed a small shoot out of the old, dry root, and he wondered if history would repeat itself.
The fool thinks he is the shoot, bright recreation of a patient gardener destined to bear good fruit. The wise man fears he may be the farmer, trying to grow tame trees in his own fatal image of order rather than learn from the greater order that has always known how to bear fruit.
It is said there was a landowner who hired men in the morning, and hired men in the evening, and when the work was done, he gave them each the same wage.
The fool thinks himself the worker, who will receive a full wage as long as he is hired before the end of the day. The wise man fears that he may be the landowner, who shall seldom find anyone to work for him all day when they could only work for the evening.
It is said that a man had a fig planted, but for three years it bore no fruit, so he told the gardener to cut it down and stop wasting the soil, but the gardener replied that it should be let alone for one more year, dug around and manured, and that it might then bear fruit, and if not it could be cut down.
The fool thinks himself the fig, ready with potential and soon to bear great fruit if given good soil. The wise man fears he may be the man without patience to see the fig bear fruit, who cuts down good trees before he knows their worth.
The fool thinks he sees his folly. The wise man fears he sees a false reflection of himself as wise.
It is said that a man broke his leg, and took up crutches, and then the whole village took up crutches, and then everyone forgot how to walk without crutches. It is said that one day a boy threw down his crutches and cried "One does not need crutches to walk!" and promptly fell on his face.
The fool thinks he is the boy, brave champion of a truth he cannot yet prove. The wise man fears he may be a villager who scoffs at the boy he should heed.
It is said that a king dreamed of a statue with a head of gold, a chest of silver, a belly of bronze, legs of iron, and feet of iron mixed in clay, and that a stone shaped not by human hands broke the statue's feet and scattered the edifice to dust.
The fool thinks that he is the stone, shaped by greater forces to knock down the increasingly valueless structures of the world. The wise man fears he may be the statue that would be a precious thing if it were not solidly standing on a foundation that cannot withstand the truth.
It is said there was a farmer who planted an olive tree, but it would not bear fruit, so he grafted on some wild olive nearby that, while sparse, bore fruit, but his tree only bore fruit for a brief time, and so the gardener cut away the tree to a stump. Time passed, and he returned to the tree, and coaxed a small shoot out of the old, dry root, and he wondered if history would repeat itself.
The fool thinks he is the shoot, bright recreation of a patient gardener destined to bear good fruit. The wise man fears he may be the farmer, trying to grow tame trees in his own fatal image of order rather than learn from the greater order that has always known how to bear fruit.
It is said there was a landowner who hired men in the morning, and hired men in the evening, and when the work was done, he gave them each the same wage.
The fool thinks himself the worker, who will receive a full wage as long as he is hired before the end of the day. The wise man fears that he may be the landowner, who shall seldom find anyone to work for him all day when they could only work for the evening.
It is said that a man had a fig planted, but for three years it bore no fruit, so he told the gardener to cut it down and stop wasting the soil, but the gardener replied that it should be let alone for one more year, dug around and manured, and that it might then bear fruit, and if not it could be cut down.
The fool thinks himself the fig, ready with potential and soon to bear great fruit if given good soil. The wise man fears he may be the man without patience to see the fig bear fruit, who cuts down good trees before he knows their worth.
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