Creationists deny transitional forms by asking for evidence of implausible chimeras - "half-fish, half-cow" and the like. That's like denying that coffee cools down because there's no time at which half the cup is scalding and the other half is cold.
[The plausible transitions - that is, actual transitional forms, such as lineages of fossils displaying characteristics of both fish and amphibians, for example - are not asked for much these days - because there are far too many of them, and more all the time.]
(Related post: A creationist looks at Janie's photo album)
Thursday, August 28, 2008
Quote
"Evangelising to people who don't want to hear it [...] is like exposing yourself in public." - Pat Condell
Wednesday, August 27, 2008
Captaining the Titanic
Bad arithmetic can leave us like the captain of the Titanic - convinced we're unsinkable while we confidently steam toward the iceberg.
The inability of the Clinton advisors to perform basic number crunching cost them dearly in their primary campaign.
After the expensive loss in Iowa the Clinton campaign focused on states with a primary, such as Texas, pouring an enormous amount of resources into winning them while Obama racked up win after win in states they weren't even running polling in... only to discover that the time, money and effort had gained them little advantage in several of the contests that they focused on. Clinton won the Texas primary, but it didn't translate into a big advantage in delegates. A little number-crunching (which plenty of people pointed out well before the Texas contest took place) would have shown that a big effort in Texas would have conferred a relatively minor advantage, given the way that Texas' system works. But their campaign apparently didn't understand the issue - in spite of the fact that the issue was well understood by others - until too late; the Clinton camp started whining about it a few days before the primary, but it is not like Texas' circumstances were a secret before then.
Clintons' campaign paid "millions of dollars to consultants who offered up dubious advice".
These "experts" then managed to make further, even simpler elementary mathematical mistakes (by applying a calculation suitable for districts with 6 delegates to districts with different numbers of delegates), which meant that time after time, Clinton must have been mispending money, by allocating resources where they would be certain to be wasted and failing to allocate them where they could make a real difference, in effect multiplying Obama's financial advantage many-fold.
[Mathematics was not the only problem in the Clinton campaign, by any means - but it was a very important one that should never have arisen at all.]
What is it that causes monumental errors on the scale of using a calculation based on six delegates - that the target should be 7/12 of the vote ("the magic number is 59%") - for other districts?
Might it be the Dunning-Kruger effect? Is it just arrogance? Is it getting so focused on things like spin and sound-bites that you can't even remember the rules of the game?
Perhaps the Dunning-Kruger effect might also explain why the California Supreme Court have ruled that courts, not statisticians, will decide which calculations are to be used in cases involving so-called "cold-hit" DNA-matches. Statistical experts are to be reduced to "calculators", performing court approved calculations, whether the circumstances merit the calculations or not.
Innumeracy is what lets a political leader spend a trillion dollars on a largely futile and deadly war, and at the same time veto spending a million dollars on an essential education program, with a stunningly small backlash, partly because many voters don't realize the first is a million times as large as the second. Trillion, billion and million all just become different ways of saying "gazillion", and even the most implausible justifcation can be made to sound iron-clad.
A citizen needs enough mathematics to understand such differences in scale, and a political advisor certainly needs at least enough to be able to formulate a sensible strategy. Without it, we're all in for some very painful and expensive lessons.
The inability of the Clinton advisors to perform basic number crunching cost them dearly in their primary campaign.
After the expensive loss in Iowa the Clinton campaign focused on states with a primary, such as Texas, pouring an enormous amount of resources into winning them while Obama racked up win after win in states they weren't even running polling in... only to discover that the time, money and effort had gained them little advantage in several of the contests that they focused on. Clinton won the Texas primary, but it didn't translate into a big advantage in delegates. A little number-crunching (which plenty of people pointed out well before the Texas contest took place) would have shown that a big effort in Texas would have conferred a relatively minor advantage, given the way that Texas' system works. But their campaign apparently didn't understand the issue - in spite of the fact that the issue was well understood by others - until too late; the Clinton camp started whining about it a few days before the primary, but it is not like Texas' circumstances were a secret before then.
Clintons' campaign paid "millions of dollars to consultants who offered up dubious advice".
These "experts" then managed to make further, even simpler elementary mathematical mistakes (by applying a calculation suitable for districts with 6 delegates to districts with different numbers of delegates), which meant that time after time, Clinton must have been mispending money, by allocating resources where they would be certain to be wasted and failing to allocate them where they could make a real difference, in effect multiplying Obama's financial advantage many-fold.
[Mathematics was not the only problem in the Clinton campaign, by any means - but it was a very important one that should never have arisen at all.]
What is it that causes monumental errors on the scale of using a calculation based on six delegates - that the target should be 7/12 of the vote ("the magic number is 59%") - for other districts?
Might it be the Dunning-Kruger effect? Is it just arrogance? Is it getting so focused on things like spin and sound-bites that you can't even remember the rules of the game?
Perhaps the Dunning-Kruger effect might also explain why the California Supreme Court have ruled that courts, not statisticians, will decide which calculations are to be used in cases involving so-called "cold-hit" DNA-matches. Statistical experts are to be reduced to "calculators", performing court approved calculations, whether the circumstances merit the calculations or not.
Innumeracy is what lets a political leader spend a trillion dollars on a largely futile and deadly war, and at the same time veto spending a million dollars on an essential education program, with a stunningly small backlash, partly because many voters don't realize the first is a million times as large as the second. Trillion, billion and million all just become different ways of saying "gazillion", and even the most implausible justifcation can be made to sound iron-clad.
A citizen needs enough mathematics to understand such differences in scale, and a political advisor certainly needs at least enough to be able to formulate a sensible strategy. Without it, we're all in for some very painful and expensive lessons.
Tuesday, August 26, 2008
Magnetic cows? Well, maybe not.
Science journalists love a kooky headline.
The latest is magnetic cows. It seems rather blown out of proportion.
Google earth photos indicate a tendency for cows (and some other grazers) to line up north-south. But supposedly you can't tell if they are facing north or facing south or any mixture of the two.
So, here's a conjecture*:
In the morning, cold cows stand side on to the sun (which is in the east), thus exposing a greater area to the sun.
In the late afternoon, cold cows stand side on to the sun (which is in the west).
In both cases, they will face either north or south (and probably won't prefer one much over the other).
In the middle of the day, warm cows in a paddock with little shade might even stand backside toward the sun (thus reducing sun exposure when they're hot)... and thus tending to face either north or south, depending on the hemisphere.
Consequently, there would be some tendency for cows to line up north-south, purely as a way of managing sun exposure in order to be more comfortable.
[*Yes, yes, I know it says in the BBC article Their study ruled out the possibility that the Sun position or wind direction were major influences on the orientation of the cattle. Dr Begall said: "In Africa and South America, the cattle (were) shifted slightly to a more north-eastern-south-western direction."
"But it is known that the Earth's magnetic field is much weaker there," she explained. How does that establish anything? If sun and wind hypotheses have been ruled out, why not explain how? The bit about Africa and South America really doesn't help that much, so if that's what they've got, it's damn weak.]
Edit: Well, reading the physorg link more closely, it looks like they used "shadows" to discount the effect of the sun. It's not clear what precisely they did -- there are a variety of possible conjectures that might involve the sun. I'll have to try to get the PNAS paper. I just went looking for it, but in spite of the fact that the library supposedly has electronic access, the article is not showing up, either in the most recently available issue there, nor in the articles that have immediate access.
Indeed, I can think of several other perfectly simple explanations that I'd want to eliminate before I start hypothesizing magnetic cows.
Does that mean that I assert cows don't have a magnetic sense like some birds?
Not at all. Such a thing would be particularly useful for some migrating animals, say, caribou or wildebeest, so it's certainly not completely out of the realm of possibility. But you have to at least show that it's not some rather obvious and simple thing like sun exposure before you take it at all seriously. And you need a lot more than "they looked at shadows" or "the cattle were at a slightly different angle in Africa" before concluding that you've eliminated alternative explanations.
That cows tend to align north-south is interesting. But magnetic cows is a bit overblown on that basis alone. At best its a plausible conjecture.
Well, I guess we wait and see. If anyone does see the paper, I'd be curious to know just how strong the arguments for the elimination of alternative explanations actually was.
The latest is magnetic cows. It seems rather blown out of proportion.
Google earth photos indicate a tendency for cows (and some other grazers) to line up north-south. But supposedly you can't tell if they are facing north or facing south or any mixture of the two.
So, here's a conjecture*:
In the morning, cold cows stand side on to the sun (which is in the east), thus exposing a greater area to the sun.
In the late afternoon, cold cows stand side on to the sun (which is in the west).
In both cases, they will face either north or south (and probably won't prefer one much over the other).
In the middle of the day, warm cows in a paddock with little shade might even stand backside toward the sun (thus reducing sun exposure when they're hot)... and thus tending to face either north or south, depending on the hemisphere.
Consequently, there would be some tendency for cows to line up north-south, purely as a way of managing sun exposure in order to be more comfortable.
[*Yes, yes, I know it says in the BBC article Their study ruled out the possibility that the Sun position or wind direction were major influences on the orientation of the cattle. Dr Begall said: "In Africa and South America, the cattle (were) shifted slightly to a more north-eastern-south-western direction."
"But it is known that the Earth's magnetic field is much weaker there," she explained. How does that establish anything? If sun and wind hypotheses have been ruled out, why not explain how? The bit about Africa and South America really doesn't help that much, so if that's what they've got, it's damn weak.]
Edit: Well, reading the physorg link more closely, it looks like they used "shadows" to discount the effect of the sun. It's not clear what precisely they did -- there are a variety of possible conjectures that might involve the sun. I'll have to try to get the PNAS paper. I just went looking for it, but in spite of the fact that the library supposedly has electronic access, the article is not showing up, either in the most recently available issue there, nor in the articles that have immediate access.
Indeed, I can think of several other perfectly simple explanations that I'd want to eliminate before I start hypothesizing magnetic cows.
Does that mean that I assert cows don't have a magnetic sense like some birds?
Not at all. Such a thing would be particularly useful for some migrating animals, say, caribou or wildebeest, so it's certainly not completely out of the realm of possibility. But you have to at least show that it's not some rather obvious and simple thing like sun exposure before you take it at all seriously. And you need a lot more than "they looked at shadows" or "the cattle were at a slightly different angle in Africa" before concluding that you've eliminated alternative explanations.
That cows tend to align north-south is interesting. But magnetic cows is a bit overblown on that basis alone. At best its a plausible conjecture.
Well, I guess we wait and see. If anyone does see the paper, I'd be curious to know just how strong the arguments for the elimination of alternative explanations actually was.
Wednesday, August 20, 2008
You ain't so special... part whatever
Some birds can tell they're looking at themselves in a mirror.
[Prior&al paper at PLoS Biology]
This is a version of the mirror test, used to gauge self-awareness.
There are now at least nine species where this has been observed; this is the first non-mammal to have unequivocal results - and there are a number of other bird species that I would be utterly unsurprised if they had similar results.
You have to wonder - do they have a theory of mind? From what I've seen from certain corvids, I would have to suppose that they do.
As the wikipedia link (which has already been updated to include the new information) points out, it's not a suitable test for all animals, so there may well be some creatures who don't perform at the mirror test who are nevertheless self-aware.
[Prior&al paper at PLoS Biology]
This is a version of the mirror test, used to gauge self-awareness.
There are now at least nine species where this has been observed; this is the first non-mammal to have unequivocal results - and there are a number of other bird species that I would be utterly unsurprised if they had similar results.
You have to wonder - do they have a theory of mind? From what I've seen from certain corvids, I would have to suppose that they do.
As the wikipedia link (which has already been updated to include the new information) points out, it's not a suitable test for all animals, so there may well be some creatures who don't perform at the mirror test who are nevertheless self-aware.
Too much mathematics homework
Recently I commented over at En Tequila Es Verdad, saying in part that I thought too much mathematics homework was a bad thing, education wise.
The response to headlines about US falling behind in education (say, like this one) is usually to increase homework.
Well, a paper in Econometrics Journal apparently concludes that for average students (about half of them, speaking roughly), lots of mathematics homework is not productive.
[Of course, this is looking at relatively short term effects. What will be the effects of too much homework five years down the line? My guess is that long term it will probably be unproductive for an even larger percentage.]
The linked news article says: According to Henderson, the learning process needs to remain a rich, broad experience.
Which is one of the main points I was getting at in my lengthy comment over at Dana's blog. Nice to see I'm not talking complete bullshit.
Think of it this way:
Imagine art class consisted of having to practice drawing a duck, over and over, until you could produce a good outline of a few very particular kinds of duck, drawn just so. You would do half an hour of ducks every night for homework. Then back to school the next day for more ducks. Then you'd move on to chickens. The generalization to all birds would be sort of handwaved, because the curriculum is kind of packed. It's time to move on to drawing fish! If you didn't learn to draw ducks, you would get even more work on drawing ducks. Some aspects of what you learned in drawing ducks could be used in drawing fish, but the relationships aren't very intuitive, and anyway, there's just so many bits to remember and it's all so confusing and WTF, now I have to go home and do fish for an HOUR?
And then suddenly you're drawing battleships, and while drawing kind of made sense before, suddenly it makes no sense. You never quite got the hang of ducks and now you're trying to catch up that, fish and now battleships? How on earth are you ever going to remember all the parts of a battleship? And god forbid you should draw the parts in the wrong order!
If art was like that, most people would hate it.
Imagine Rembrandt at a party, who desperately wants to convey something of the beauty and importance of chiaroscuro. What would he hear, over and over, as he brought up the topic of art?
"Art? I was never any good at that. I always hated art! My worst subject. All those ducks! You must be very strange."
Most people - if you forced them - would be able to draw a fairly reasonable-looking duck, but there'd be precious little art in their lives. They'd certainly have no sense that it could be moving and beautiful - or indeed that it was about anything other than ducks and fish, and maybe something painful about battleships.
The response to headlines about US falling behind in education (say, like this one) is usually to increase homework.
Well, a paper in Econometrics Journal apparently concludes that for average students (about half of them, speaking roughly), lots of mathematics homework is not productive.
[Of course, this is looking at relatively short term effects. What will be the effects of too much homework five years down the line? My guess is that long term it will probably be unproductive for an even larger percentage.]
The linked news article says: According to Henderson, the learning process needs to remain a rich, broad experience.
Which is one of the main points I was getting at in my lengthy comment over at Dana's blog. Nice to see I'm not talking complete bullshit.
Think of it this way:
Imagine art class consisted of having to practice drawing a duck, over and over, until you could produce a good outline of a few very particular kinds of duck, drawn just so. You would do half an hour of ducks every night for homework. Then back to school the next day for more ducks. Then you'd move on to chickens. The generalization to all birds would be sort of handwaved, because the curriculum is kind of packed. It's time to move on to drawing fish! If you didn't learn to draw ducks, you would get even more work on drawing ducks. Some aspects of what you learned in drawing ducks could be used in drawing fish, but the relationships aren't very intuitive, and anyway, there's just so many bits to remember and it's all so confusing and WTF, now I have to go home and do fish for an HOUR?
And then suddenly you're drawing battleships, and while drawing kind of made sense before, suddenly it makes no sense. You never quite got the hang of ducks and now you're trying to catch up that, fish and now battleships? How on earth are you ever going to remember all the parts of a battleship? And god forbid you should draw the parts in the wrong order!
If art was like that, most people would hate it.
Imagine Rembrandt at a party, who desperately wants to convey something of the beauty and importance of chiaroscuro. What would he hear, over and over, as he brought up the topic of art?
"Art? I was never any good at that. I always hated art! My worst subject. All those ducks! You must be very strange."
Most people - if you forced them - would be able to draw a fairly reasonable-looking duck, but there'd be precious little art in their lives. They'd certainly have no sense that it could be moving and beautiful - or indeed that it was about anything other than ducks and fish, and maybe something painful about battleships.
The delights of Biddleonian nose-wrestling
Every Olympics we get the cries of "that's not a sport!" about ... well, about almost every sport that the speaker is not a rabid follower of.
I have no patience for it.
The delight of the olympics is watching the very best exponents of whichever activity it happens to be duke it out in front of a potential audience probably thousands of times larger than at any time they have competed in their lives.
Some sports seem obscure, but in fact are not particularly - just to our parochial mindset. Not every country likes the same stuff. A sport nobody plays in your country may actually be one of the more popular competitive sports in some other countries.
Some other sports are indeed obscure. That's fine too - we just need to mix em around a bit, so other obscure sports can have their time in the sun. I love them all. Actually, I want more obscure sports. Tennis - meh, I can watch the same people play any old time. Give me Royal Tennis!
And I reckon every olympics should have 5 or 6 completely made up sports, and the best one gets to come back next olympics, along with another 5 or 6 newly-minted modes of competition. If a made up sport wins twice in a row, it should be a regular fixture.
The new sports would be part of the Olympics bid process, so people who like the sound of a new sport can have a nice long training period to get good. Remember Eddie the Eagle from the winter olymics way back? If there were more sports, there'd be more half-mad semi-fit ordinary people having a burl. More power to 'em.
There's a very fertile ground in mixing pre-existing sports. I want to see fencing mixed with trampolining. Now there's a sport.
What about clay pigeon shooting from a hang-glider? That takes skill. None of this "perfect score" shit - you manage to wing a couple, you're probably in line for a medal.
[Edit: I see Greta Christina made some similar points (though she's taking a somewhat different tack). I've had this post in mind since reading a poll a few days ago asking "Which sports don't belong at the olympics?"; any similarity to Greta's post is coincidental]
I have no patience for it.
The delight of the olympics is watching the very best exponents of whichever activity it happens to be duke it out in front of a potential audience probably thousands of times larger than at any time they have competed in their lives.
Some sports seem obscure, but in fact are not particularly - just to our parochial mindset. Not every country likes the same stuff. A sport nobody plays in your country may actually be one of the more popular competitive sports in some other countries.
Some other sports are indeed obscure. That's fine too - we just need to mix em around a bit, so other obscure sports can have their time in the sun. I love them all. Actually, I want more obscure sports. Tennis - meh, I can watch the same people play any old time. Give me Royal Tennis!
And I reckon every olympics should have 5 or 6 completely made up sports, and the best one gets to come back next olympics, along with another 5 or 6 newly-minted modes of competition. If a made up sport wins twice in a row, it should be a regular fixture.
The new sports would be part of the Olympics bid process, so people who like the sound of a new sport can have a nice long training period to get good. Remember Eddie the Eagle from the winter olymics way back? If there were more sports, there'd be more half-mad semi-fit ordinary people having a burl. More power to 'em.
There's a very fertile ground in mixing pre-existing sports. I want to see fencing mixed with trampolining. Now there's a sport.
What about clay pigeon shooting from a hang-glider? That takes skill. None of this "perfect score" shit - you manage to wing a couple, you're probably in line for a medal.
[Edit: I see Greta Christina made some similar points (though she's taking a somewhat different tack). I've had this post in mind since reading a poll a few days ago asking "Which sports don't belong at the olympics?"; any similarity to Greta's post is coincidental]
Monday, August 18, 2008
Sunday, August 17, 2008
Why I look forward to the death of atheism
As should already be obvious, I'm an atheist.
But I'm an atheist who looks forward to a time when there are no people who identify as atheist - to the death, as it were, of atheism.
After all, I don't identify as an a-tooth-fairy-ist; it's just naturally accepted that as an adult, I don't carry such a belief. Being an atoothfairyist is so universally common that it isn't even a term. Atoothfairyism, if it were ever to have existed, is certainly now long dead. It's not necessary to identify as a skeptic of the religious beliefs prevalent in ancient Greece. Such a lack of belief is, essentially, a dead issue.
So it goes with the religions presently at large -- I hesitate to say "modern religion", because there's little about their fundamentals that's modern. I look forward to the day when it's utterly pointless to ever mention a lack of belief in them, because it is essentially universal. When that day comes, atheism as we presently understand it, will be utterly dead. We'll just be people, getting on with life.
Maybe then we can really get to work of fixing the mess we're in.
But I'm an atheist who looks forward to a time when there are no people who identify as atheist - to the death, as it were, of atheism.
After all, I don't identify as an a-tooth-fairy-ist; it's just naturally accepted that as an adult, I don't carry such a belief. Being an atoothfairyist is so universally common that it isn't even a term. Atoothfairyism, if it were ever to have existed, is certainly now long dead. It's not necessary to identify as a skeptic of the religious beliefs prevalent in ancient Greece. Such a lack of belief is, essentially, a dead issue.
So it goes with the religions presently at large -- I hesitate to say "modern religion", because there's little about their fundamentals that's modern. I look forward to the day when it's utterly pointless to ever mention a lack of belief in them, because it is essentially universal. When that day comes, atheism as we presently understand it, will be utterly dead. We'll just be people, getting on with life.
Maybe then we can really get to work of fixing the mess we're in.
Saturday, August 16, 2008
The hunting of the snark
There's a very brief interview with a beautifully snarky Gore Vidal in Esquire that's online here.
(My apologies to whichever blog I saw this at a few days ago - I have completely forgotten).
(My apologies to whichever blog I saw this at a few days ago - I have completely forgotten).
What difference will quantum computing make to me, anyway?
I think, broadly speaking, a lot of what's written about quantum computers is somewhat wrong-headed. It will make a difference to cyptography (in that it will be easier to have "unsnoopable"** communications). It will speed up some algorithms dramatically.
**of course all that means is that the "snooping" shifts to other parts of the process.
It's usually this last thing that gets the attention. What quantum computing essentially does is give you the potential to "halve the exponent". If something was going to take something on the order of 2^60 calculations, then if you're lucky, you may be able to reduce that to on the order of 2^30 quantum calculations. And if the exponents happen to be of that order, that could be enormusly useful. It might be possible, for example, to reduce a computation from centuries to maybe days or weeks (and ordinary parallelism could reduce that further, of course).
But there aren't all that many calculations that matter to us right now that have exponents of that order (well, actually there are a lot, but as a fraction of calculations we want to do, not so much). Most calculations are either much, much smaller or much, much bigger.
There's not much to be gained in taking a calculation from 2^400 to 2^200 - it's still going to take longer than you likely have.
So in the region past the outer limits of what we can practically do with ordinary parallelism, quantum computing will make a big difference - it will revolutionize what we can do with certain kinds of calculation. Most of those kinds of calculation won't impact you directly (indirectly, sure - in things like design of drugs and such) in the sense that you probably won't be doing much in the way of calculation you weren't before. At least not to begin with.
I think the big impacts on our personal lives will come in ways we can't even anticipate right now. The software that will change everything - maybe even decades after we have quantum computers as things we can buy - we don't know what that's going to be like. We won't until we have quantum computers, until we have coding environments and programming paradigms that let us think about things in ways we don't right now. Until we've had a generation of people who grew up with quantum computers.
Because when that exponent-halving is available, we're going to invent new problems that lie in that space above what we can do right now with parallelism, but that quantum computers can do. And we won't know what they are until we're there.
**of course all that means is that the "snooping" shifts to other parts of the process.
It's usually this last thing that gets the attention. What quantum computing essentially does is give you the potential to "halve the exponent". If something was going to take something on the order of 2^60 calculations, then if you're lucky, you may be able to reduce that to on the order of 2^30 quantum calculations. And if the exponents happen to be of that order, that could be enormusly useful. It might be possible, for example, to reduce a computation from centuries to maybe days or weeks (and ordinary parallelism could reduce that further, of course).
But there aren't all that many calculations that matter to us right now that have exponents of that order (well, actually there are a lot, but as a fraction of calculations we want to do, not so much). Most calculations are either much, much smaller or much, much bigger.
There's not much to be gained in taking a calculation from 2^400 to 2^200 - it's still going to take longer than you likely have.
So in the region past the outer limits of what we can practically do with ordinary parallelism, quantum computing will make a big difference - it will revolutionize what we can do with certain kinds of calculation. Most of those kinds of calculation won't impact you directly (indirectly, sure - in things like design of drugs and such) in the sense that you probably won't be doing much in the way of calculation you weren't before. At least not to begin with.
I think the big impacts on our personal lives will come in ways we can't even anticipate right now. The software that will change everything - maybe even decades after we have quantum computers as things we can buy - we don't know what that's going to be like. We won't until we have quantum computers, until we have coding environments and programming paradigms that let us think about things in ways we don't right now. Until we've had a generation of people who grew up with quantum computers.
Because when that exponent-halving is available, we're going to invent new problems that lie in that space above what we can do right now with parallelism, but that quantum computers can do. And we won't know what they are until we're there.
Quantum computers, RSN?
Graphene seems to be the new favourite material - it's cropping up everywhere, the way that fullerenes were a few years ago, and recently SWCNTs in particular. Just lately, it's specifically graphene.
At the same time, quantum computers have a number of serious technical hurdles to deal with.
Well, as highlighted at the physics arXiv blog, in a recent paper at arXiv, "Z"-shaped pieces of graphene nanoribbon are being suggested as a solution to several of those issues. Specifically, you need to have things that don't readily interact with the environment, yet can be manipulated and can interact with each other; electrons are problematic because they interact with the environment (which is why a fair bit of quantum computing focus has been on photons - but they have their own problems). The "corners" in the graphene Z's are where the action happens - the electons are "stored", not interacting with the environment, but where their spins can interact with each other.
The authors say: “Due to recent achievement in production of graphene nanoribbon, this proposal may be implementable within the present techniques.”
You won't see quantum laptops in stores for next Christmas. But instead of "decades" we're maybe now looking at a few years for something on a lab bench, and maybe getting closer to a decade for practical devices, and perhaps another 5 years after that for consumer products. Maybe.
But Vista will still suck on them.
At the same time, quantum computers have a number of serious technical hurdles to deal with.
Well, as highlighted at the physics arXiv blog, in a recent paper at arXiv, "Z"-shaped pieces of graphene nanoribbon are being suggested as a solution to several of those issues. Specifically, you need to have things that don't readily interact with the environment, yet can be manipulated and can interact with each other; electrons are problematic because they interact with the environment (which is why a fair bit of quantum computing focus has been on photons - but they have their own problems). The "corners" in the graphene Z's are where the action happens - the electons are "stored", not interacting with the environment, but where their spins can interact with each other.
The authors say: “Due to recent achievement in production of graphene nanoribbon, this proposal may be implementable within the present techniques.”
You won't see quantum laptops in stores for next Christmas. But instead of "decades" we're maybe now looking at a few years for something on a lab bench, and maybe getting closer to a decade for practical devices, and perhaps another 5 years after that for consumer products. Maybe.
But Vista will still suck on them.
Wednesday, August 13, 2008
Emotion and mathematics
It would be easy for people outside of mathematical areas to assume that the exercise of mathematics is an austere and unemotional activity, and that as a result mathemamaticians are, whether by nature or by habit, cold and disinclined to emotion.
Having observed many people (including myself) doing mathematics and discussing mathematically-related topics, this is far from the case.
I have had many congenially heated arguments with colleagues, and I have even caught myself grinning in delighted anticipation of going another round with a valued fellow-traveller.
(I've been called crazy a lot of times - but more times in mathematically-related discussions than anywhere else - and with unstinting good humour to boot. "You're crazy! You can't do that." "No, really, it's right. You can do it here..." "No, no, it's nuts to do it that way even if it's right." -- and so on back and forth; in fact, I think that's how a lot of mathematical arguments get polished)
Even as a solitary activity, mathematics is for me, intensely emotional, even visceral. Many times, equations I have worked with have various kinds of symmetry, and many of those symmetries will carry through the equations as the argument develops.. this is, I presume, what produces a strong sense of rightness that I often feel as the steps progress. There's also a corresponding sense that there is some mistake - for me a feeling something like that moment on a roller coaster as it begins to descend, though it is sometimes even stronger than that - before being aware of exactly what is wrong, or precisely where it lies.
If you work with particular kinds of expressions a lot, you built up a sense of what they "should" look like, and it becomes easier to recognize that something is wrong before you can say precisely what the problem is; because the intellectual cognition is behind the pattern-recognition, it has an emotional quality.
A really clever manipulation (I can't help but think of them as "tricks") or an inspired substitution that makes a difficult problem easy can produce a tingling sensation up the back of my neck and head. A particularly beautiful piece of mathematics can, on occasion, move me almost to tears.
Then there's joy and delight. On occasion I have had the fortune to look at some neat, if modest, just-derived result and wonder if perhaps I am the first to have ever seen it (it is, obviously, rarely the case that I am - it is not unusual to find that my result has been tucked away in some mathematical corner for many decades ... on one occasion I found I had been beaten by Gauss - but the thrill of discovery is there all the same).
There's also what I call the "stupid feeling". When I'm working on something new or unfamiliar (or even, on occasion on what ought to be familiar), I can spend long periods - days, weeks, or even, shamefully, months - where I feel intensely incompetent, like I'm reaching around in the dark for something that I know is right there, but can't seem to locate it - and then there's a fleetingly brief moment of joy as I see how to do it (often barely long enough to say "Yes!"). Then quickly after, in retropspect (sometimes as I see an even better way to do it), it is all so utterly obvious, so agonizingly plain, that the prior feeling of incompetence seems, if anything, far too mild.
For me, that feeling is occasionally so intense I cannot even bear to write it up properly, or sometimes even to mention it, because the whole thing is so painfully facile. (I doubt that most people feel this quite so keenly; I'd be curious to know.)
Mathematicians don't discuss emotion much; a kindly supervisor might have a few words on dealing with the disappointments that naturally come with trying to get some result to come out, or those that come with trying to get something published. But outside of that, the preference is almost always to talk about the mathematics itself.
But just because we don't talk about our feelings with each other doesn't mean we're not feeling them.
Having observed many people (including myself) doing mathematics and discussing mathematically-related topics, this is far from the case.
I have had many congenially heated arguments with colleagues, and I have even caught myself grinning in delighted anticipation of going another round with a valued fellow-traveller.
(I've been called crazy a lot of times - but more times in mathematically-related discussions than anywhere else - and with unstinting good humour to boot. "You're crazy! You can't do that." "No, really, it's right. You can do it here..." "No, no, it's nuts to do it that way even if it's right." -- and so on back and forth; in fact, I think that's how a lot of mathematical arguments get polished)
Even as a solitary activity, mathematics is for me, intensely emotional, even visceral. Many times, equations I have worked with have various kinds of symmetry, and many of those symmetries will carry through the equations as the argument develops.. this is, I presume, what produces a strong sense of rightness that I often feel as the steps progress. There's also a corresponding sense that there is some mistake - for me a feeling something like that moment on a roller coaster as it begins to descend, though it is sometimes even stronger than that - before being aware of exactly what is wrong, or precisely where it lies.
If you work with particular kinds of expressions a lot, you built up a sense of what they "should" look like, and it becomes easier to recognize that something is wrong before you can say precisely what the problem is; because the intellectual cognition is behind the pattern-recognition, it has an emotional quality.
A really clever manipulation (I can't help but think of them as "tricks") or an inspired substitution that makes a difficult problem easy can produce a tingling sensation up the back of my neck and head. A particularly beautiful piece of mathematics can, on occasion, move me almost to tears.
Then there's joy and delight. On occasion I have had the fortune to look at some neat, if modest, just-derived result and wonder if perhaps I am the first to have ever seen it (it is, obviously, rarely the case that I am - it is not unusual to find that my result has been tucked away in some mathematical corner for many decades ... on one occasion I found I had been beaten by Gauss - but the thrill of discovery is there all the same).
There's also what I call the "stupid feeling". When I'm working on something new or unfamiliar (or even, on occasion on what ought to be familiar), I can spend long periods - days, weeks, or even, shamefully, months - where I feel intensely incompetent, like I'm reaching around in the dark for something that I know is right there, but can't seem to locate it - and then there's a fleetingly brief moment of joy as I see how to do it (often barely long enough to say "Yes!"). Then quickly after, in retropspect (sometimes as I see an even better way to do it), it is all so utterly obvious, so agonizingly plain, that the prior feeling of incompetence seems, if anything, far too mild.
For me, that feeling is occasionally so intense I cannot even bear to write it up properly, or sometimes even to mention it, because the whole thing is so painfully facile. (I doubt that most people feel this quite so keenly; I'd be curious to know.)
Mathematicians don't discuss emotion much; a kindly supervisor might have a few words on dealing with the disappointments that naturally come with trying to get some result to come out, or those that come with trying to get something published. But outside of that, the preference is almost always to talk about the mathematics itself.
But just because we don't talk about our feelings with each other doesn't mean we're not feeling them.
Monday, August 11, 2008
How can we be sure when the bible is being literal?
Over at Friendly Atheist, Hemant quoted Rochelle Weiss of the Freedom from Religion Foundation, who wrote about whether Jesus was righteous, and finds he comes up short.
Needless to say, apologists came in (and as usual, saying different things). This post is based on a comment I made there.
As always, when the bible say uncomfortable things, the apologists come right in with “this doesn’t mean what it says”.
Which is fine, I could accept that - unless, unless they also say about other parts “this means what it says”. You can’t have it both ways. You can’t declare the parts you don’t like to be “figures of speech” unless you accept that the same will be true of parts you would like to be literal; conversely, you can’t say “I think this is literally true” unless you’re prepared to accept that some of the parts you don’t like are also literally true.
It’s astoundingly convenient the way that it’s apparently almost universally only the most inconvenient parts of the bible that are held to be figures of speech (or in some other way should not be taken to mean what they plainly say).
“Okay, here, hate doesn’t mean hate. Actually, it means love, just not quite so much as someone else. But over there, well, it means hate.”
Even George Orwell didn’t imagine wordplay quite as sinister as that.
The problem with the “figure of speech” argument is that people of the time (as with people now) sometimes said similar things literally. There’s nothing in the bible to clearly say that the claim of “figure of speech” is in fact so. And it gets worse, because if some part isn't literal, you have the further problem of guessing exactly what it's supposed to mean.
It’s guesswork. Sometimes it’s educated guesswork, but much of the time it’s just a hopeful guess. Where does this supreme authority come from to know with certainty what is literal and what is not?
If nothing was at stake but academic pride, I wouldn't care.
What if you guess wrong about what’s literal and what’s not? What could loving Jesus have in store for you? Well, he tells us - get the wrong things wrong, and it's infinite torture. For guessing wrong, or believing someone else who claims their guess is right. So before you start casually declaring one bit not literal, and another bit literal, you better be damn sure you’ve got it right. You better have a lot more evidence than is on display in the comment thread there.
Now if Jesus really did mean that bit about hating parents literally, and you don't, or you tell others not to hate theirs, you could be totally screwed, depending on which other bits are literally true (the problem is, if some is literal and some is figurative, there is no solid foundation for any claim). But then again maybe even Jesus’ tender Hell is also just a figure of speech. So maybe nothing's at stake. For the sake of all the apologists, we better hope so, eh?
I see lots of opinion on display from the religious, and precious little fact. Yet, they’ve got the unmitigated arrogance to be happily playing around with their apologetics, apparently putting others in danger of infinite torture.
Fingers crossed, eh? Good luck with that.
Needless to say, apologists came in (and as usual, saying different things). This post is based on a comment I made there.
As always, when the bible say uncomfortable things, the apologists come right in with “this doesn’t mean what it says”.
Which is fine, I could accept that - unless, unless they also say about other parts “this means what it says”. You can’t have it both ways. You can’t declare the parts you don’t like to be “figures of speech” unless you accept that the same will be true of parts you would like to be literal; conversely, you can’t say “I think this is literally true” unless you’re prepared to accept that some of the parts you don’t like are also literally true.
It’s astoundingly convenient the way that it’s apparently almost universally only the most inconvenient parts of the bible that are held to be figures of speech (or in some other way should not be taken to mean what they plainly say).
“Okay, here, hate doesn’t mean hate. Actually, it means love, just not quite so much as someone else. But over there, well, it means hate.”
Even George Orwell didn’t imagine wordplay quite as sinister as that.
The problem with the “figure of speech” argument is that people of the time (as with people now) sometimes said similar things literally. There’s nothing in the bible to clearly say that the claim of “figure of speech” is in fact so. And it gets worse, because if some part isn't literal, you have the further problem of guessing exactly what it's supposed to mean.
It’s guesswork. Sometimes it’s educated guesswork, but much of the time it’s just a hopeful guess. Where does this supreme authority come from to know with certainty what is literal and what is not?
If nothing was at stake but academic pride, I wouldn't care.
What if you guess wrong about what’s literal and what’s not? What could loving Jesus have in store for you? Well, he tells us - get the wrong things wrong, and it's infinite torture. For guessing wrong, or believing someone else who claims their guess is right. So before you start casually declaring one bit not literal, and another bit literal, you better be damn sure you’ve got it right. You better have a lot more evidence than is on display in the comment thread there.
Now if Jesus really did mean that bit about hating parents literally, and you don't, or you tell others not to hate theirs, you could be totally screwed, depending on which other bits are literally true (the problem is, if some is literal and some is figurative, there is no solid foundation for any claim). But then again maybe even Jesus’ tender Hell is also just a figure of speech. So maybe nothing's at stake. For the sake of all the apologists, we better hope so, eh?
I see lots of opinion on display from the religious, and precious little fact. Yet, they’ve got the unmitigated arrogance to be happily playing around with their apologetics, apparently putting others in danger of infinite torture.
Fingers crossed, eh? Good luck with that.
Saturday, August 2, 2008
Pentagonal Tiling...
I was reading Julie Rehmeyer's current column (I like Julie's writing and have been following MathTrek for more than almost a year now) at Science News, which is on quasicrystals.
But this column I think she said something other than what she intended... I'd have commented there but you have to register, which I'm not going to do just to leave one small comment, and anyway, posting it here gives me a change to prattle on and point at pictures and such.
To quote:
"That's why you've never seen a bathroom tiled with pentagons - it'd be impossible to cover the whole surface with no gaps."
Now the problem is, this statement is wrong... in fact, here's a counterexample I knocked up in a few seconds (it's a bit rough, but you can see what's going on easily enough).
This is called the Cairo pentagonal tiling. It's one of fourteen known tilings of pentagons (it's probably not obvious, but "Type 4" on that page is the same type of tiling). Another favourite of mine is the the Floret pentagonal tiling. (Go take a look at those two named ones, they're pretty.)
What Julie meant was "...tiled with regular pentagons". Cos, yeah, that doesn't work.
Anyway, aside from that hiccup, it's a good article; worth a read.
But this column I think she said something other than what she intended... I'd have commented there but you have to register, which I'm not going to do just to leave one small comment, and anyway, posting it here gives me a change to prattle on and point at pictures and such.
To quote:
"That's why you've never seen a bathroom tiled with pentagons - it'd be impossible to cover the whole surface with no gaps."
Now the problem is, this statement is wrong... in fact, here's a counterexample I knocked up in a few seconds (it's a bit rough, but you can see what's going on easily enough).
This is called the Cairo pentagonal tiling. It's one of fourteen known tilings of pentagons (it's probably not obvious, but "Type 4" on that page is the same type of tiling). Another favourite of mine is the the Floret pentagonal tiling. (Go take a look at those two named ones, they're pretty.)
What Julie meant was "...tiled with regular pentagons". Cos, yeah, that doesn't work.
Anyway, aside from that hiccup, it's a good article; worth a read.
Thursday, July 31, 2008
End of mandatory detention
A few days ago the Rudd government in Australia decided to end mandatory detention for asylum seekers while their applications for asylum are processed. This can be a very long process, often taking years.
This is great news; it ends a period of appalling treatment of people who in many cases have already suffered a great deal.
This is great news; it ends a period of appalling treatment of people who in many cases have already suffered a great deal.
Thursday, July 24, 2008
Unasking the question
The problem with asking a theist to give evidence for a God is not so much the fact that they lack evidence.
It's that they don't seem to have a coherent concept of what they mean when they say the word. The God concept slips around in any argument - one moment its one thing, the next moment it's some other, contradictory thing.
(And woe betide anyone should they dare ask for a coherent definition! Your likely fate is to be buried under increasingly thick layers of blather, each one getting further from any sense of definition.)
I have come to think "God" is not a concept in the usual sense. It's a bag for holding and dealing with a large pile of (often conflicting) emotions, with a vaguely concept-like structure imposed on it by hundreds of generations of talking about it.
I wonder if perhaps the debate should shift. We should unask "What evidence is there?", and instead we should perhaps be saying "You keep using that word - what do you think it means?".
If you can get an answer that sounds like it means anything whatever ask "Umm, can we write that definition down, please?"
You still won't get any evidence, of course, but maybe you can get them to talk about the one damn conception of God for more than one sentence at a time.
Well, it depends on what the meaning of "is" is.
It's that they don't seem to have a coherent concept of what they mean when they say the word. The God concept slips around in any argument - one moment its one thing, the next moment it's some other, contradictory thing.
(And woe betide anyone should they dare ask for a coherent definition! Your likely fate is to be buried under increasingly thick layers of blather, each one getting further from any sense of definition.)
I have come to think "God" is not a concept in the usual sense. It's a bag for holding and dealing with a large pile of (often conflicting) emotions, with a vaguely concept-like structure imposed on it by hundreds of generations of talking about it.
I wonder if perhaps the debate should shift. We should unask "What evidence is there?", and instead we should perhaps be saying "You keep using that word - what do you think it means?".
If you can get an answer that sounds like it means anything whatever ask "Umm, can we write that definition down, please?"
You still won't get any evidence, of course, but maybe you can get them to talk about the one damn conception of God for more than one sentence at a time.
Well, it depends on what the meaning of "is" is.
Wednesday, July 23, 2008
The parable of the histogram
I must be some kind of heretic. I'm a statistician, and here I am pointing out the problems in yet another common statistical tool.
We'll see how the histogram, which is a very popular way of displaying the distributional shape of a set of data, must be viewed with a good deal of caution.
Even though histograms are often found in the media, the problems with histograms are almost unknown among the general public. Indeed, most places that teach statistics at university completely fail to mention them.
I'd like to say that the problems are well known among professional statisticians, but that might be too strong. Certainly problems have been pointed out in the literature, and many statisticians are aware of the problems, but it seems many still are not, and the appropriate cautions are not always explained.
I'm going to show you a simple example.
Here's some data (40 observations in this sample), which I'm going to draw a histogram of. I have rounded the numbers off to two decimal places.
I give the numbers so you can (if you are so inclined) confirm for yourself what I will tell you in my little parable.
(Edit added Feb 2012: I noticed that the results didn't quite reproduce in R - three observations in the original data set I gave occurred exactly on bin boundaries for some situations. This was either a problem caused by rounding, or possibly by different conventions of different packages for handling observations at bin boundaries; I have accordingly altered those three observations by tiny amounts to move them off boundaries and avoid the issue, whatever its source. There is R code at the end of the post that works.)
The parable
This data set was given to a student, Annie. She constructs her histogram of the data by counting the number of values between 0 and 1 (but not including exactly 1), between 1 and 2, and so on, and then drawing a series of boxes each of whose base covers the subset of values that the count came from and whose height is the count for that range of values. Annie's histogram is shown in the top-left of the picture below.
She obtains a histogram whose shape corresponds to a distribution that is skewed to the left. See, for example, this description of using histograms to assess distributional shape here (edit: broken link replaced with an alternative) - that's pretty much precisely the way many elementary books on statistics describe the way to assess the shape of a distribution (and usually it's going to give you the right sort of impression).
Note that I could remove the scale and I could still describe the shape - I don't need to know the numbers on the scale in order to arrive at my description.
Three of Annie's friends, Brian, Chris and Zoe (Hah! Psych!) also get data sets with 40 observations, and they all do exactly as Annie did. Their histograms are given below (Annie's data is V1, Brian's is V2 and so on).
(click pic for a larger image)
Correspondingly, Brian describes his distribution as symmetric (and he might add "uniform"). Chris describes his as skewed to the right. Zoe describes hers as symmetric and bimodal (it has two main peaks).
So far so good - this is exactly how the books tell you it all works.
So while they're comparing their histograms, Annie idly starts looking at Brian's actual numbers. She realizes something odd is going on. She quickly places all their data sets side-by-side.
"Look, Chris!" Annie says, "all Brian's values are smaller than mine by 0.25. All yours are a quarter smaller than Brian's, and Zoe's are a quarter smaller than yours!"
They all confirm that she is correct - each set of values is the same, but with its origin merely shifted a little. Their data sets are identical in shape, but the resulting histograms are not.
That is to say, assessment of distributional shape in histograms can be dramatically affected by choice of scale (specifically, by the choice of the origin and width of the histgram bins). Here ends the parable.
It usually isn't this dramatic, of course, but the fact is, if one can generate a seemingly innocuous set of numbers whose histogram will look completely different (and for which many people will assert the distributional shape is completely different) every time we merely add or subtract a quarter, it can happen with real data too. And it does happen. Mostly the difference in impression is more modest... but not always.
So if you see a histogram, just keep in the back of your mind that it's perfectly possible that a different choice of bin boundaries would yield a somewhat different impression of the data.
Imagine I want to show some students that I write "easy" tests (I don't know why this should be such an object of fascination for students since they all do the same test, and marks are generally scaled, but it is). In preparation, I draw a histogram and it turns out to like Chris' - it looks like most students score below the middle of the range of marks. But lo, I discover with a bit of fiddling around that if I make my bin centres where the edges were (and so on), the completely opposite impression is given - just like Annie's histogram. Yay, "easy test" ... and many fewer worried queries from students in the run-up to the test, because they tend to feel there's a good chance of scoring "above the middle".
Did I lie? No. Did I fudge the data? Well, no. I did something, though. Or rather, I didn't do something.
This is a sin of omission. I fail to explain what the data would have looked like given a different choice of bin location.
Clearly, when circumstances are right, the ability to choose the location and width of the bins can give us the opportunity to somewhat alter the impression given by a histogram. Without fudging the numbers themselves, we can sometimes fudge the impression they give.
What do statisticians do? Well, there are other ways to look at distributional shape. Kernel density estimates are popular, and they completely get rid of the "bin-location" issue, though there's still the equivalent of a "bin-width" issue (choice of bandwidth, also called the "window"), which is often dealt with by looking at more than one choice of width (usually a width that gives a nice smooth result and then one that is smaller, giving a "rougher" result, in order that we can see there's nothing unsual hiding away - like the blue and green curves in the graph at topleft right** at the wikipedia link a few lines up). But there are a variety of other tools that might be used (which I don't plan on going into here).
**(did I ever mention that I have trouble with correctly attributing the words "left" and "right"? - well as you see, sometimes I do. But not when describing the shape of a distribution, isn't that odd?)
What can you do? Well, assuming you don't have anything more sophisticated that a basic histogram tool, at the least (with continuous data, anyway), try shifting your bin starts forward or back a fraction of a bin-width (if you're lazy, maybe try something near a half, otherwise maybe try a couple of values). Also try a narrower bin width. If you do a few different histograms that all give the same general impression, it doesn't matter much which one you use. And if they don't give the same impression, you better either say so, show more than one, or find some other way to convey the information.
[Or you can do a kernel density estimate readily enough - many packages (including some free ones) will do them; there are pages online that can draw them if you just paste in some data. Implementing a kernel density estimate of your own is fairly straightforward - you can compute one in a spreadsheet easily enough - if anything, it's probably slightly simpler to compute one than it is to compute counts for a histogram, which is in itself pretty straightforward. ]
Caveat Emptor
___
Added in edit in Feb 2012:
Here is some R code to create the data:
Here is some R code to generate the histograms:
Here is some R code to generate some density estimates:
Here is some R code to generate some other informative displays:
First - the sample cumulative distribution function
Second, a stripchart that shows the positions of the individual observations as they move back.
We'll see how the histogram, which is a very popular way of displaying the distributional shape of a set of data, must be viewed with a good deal of caution.
Even though histograms are often found in the media, the problems with histograms are almost unknown among the general public. Indeed, most places that teach statistics at university completely fail to mention them.
I'd like to say that the problems are well known among professional statisticians, but that might be too strong. Certainly problems have been pointed out in the literature, and many statisticians are aware of the problems, but it seems many still are not, and the appropriate cautions are not always explained.
I'm going to show you a simple example.
Here's some data (40 observations in this sample), which I'm going to draw a histogram of. I have rounded the numbers off to two decimal places.
3.15 2.28 2.06 3.43 4.85 3.22 4.01 4.43 5.46 3.12 5.53 5.51 5.56 5.52 5.31 4.96 3.28 4.10 5.19 2.54 1.89 1.84 2.56 1.90 4.20 3.42 2.39 3.64 4.84 4.31 5.11 5.60 1.98 3.91 1.88 4.33 5.74 2.01 2.58 1.92
I give the numbers so you can (if you are so inclined) confirm for yourself what I will tell you in my little parable.
(Edit added Feb 2012: I noticed that the results didn't quite reproduce in R - three observations in the original data set I gave occurred exactly on bin boundaries for some situations. This was either a problem caused by rounding, or possibly by different conventions of different packages for handling observations at bin boundaries; I have accordingly altered those three observations by tiny amounts to move them off boundaries and avoid the issue, whatever its source. There is R code at the end of the post that works.)
The parable
This data set was given to a student, Annie. She constructs her histogram of the data by counting the number of values between 0 and 1 (but not including exactly 1), between 1 and 2, and so on, and then drawing a series of boxes each of whose base covers the subset of values that the count came from and whose height is the count for that range of values. Annie's histogram is shown in the top-left of the picture below.
She obtains a histogram whose shape corresponds to a distribution that is skewed to the left. See, for example, this description of using histograms to assess distributional shape here (edit: broken link replaced with an alternative) - that's pretty much precisely the way many elementary books on statistics describe the way to assess the shape of a distribution (and usually it's going to give you the right sort of impression).
Note that I could remove the scale and I could still describe the shape - I don't need to know the numbers on the scale in order to arrive at my description.
Three of Annie's friends, Brian, Chris and Zoe (Hah! Psych!) also get data sets with 40 observations, and they all do exactly as Annie did. Their histograms are given below (Annie's data is V1, Brian's is V2 and so on).
(click pic for a larger image)Correspondingly, Brian describes his distribution as symmetric (and he might add "uniform"). Chris describes his as skewed to the right. Zoe describes hers as symmetric and bimodal (it has two main peaks).
So far so good - this is exactly how the books tell you it all works.
So while they're comparing their histograms, Annie idly starts looking at Brian's actual numbers. She realizes something odd is going on. She quickly places all their data sets side-by-side.
"Look, Chris!" Annie says, "all Brian's values are smaller than mine by 0.25. All yours are a quarter smaller than Brian's, and Zoe's are a quarter smaller than yours!"
They all confirm that she is correct - each set of values is the same, but with its origin merely shifted a little. Their data sets are identical in shape, but the resulting histograms are not.
That is to say, assessment of distributional shape in histograms can be dramatically affected by choice of scale (specifically, by the choice of the origin and width of the histgram bins). Here ends the parable.
It usually isn't this dramatic, of course, but the fact is, if one can generate a seemingly innocuous set of numbers whose histogram will look completely different (and for which many people will assert the distributional shape is completely different) every time we merely add or subtract a quarter, it can happen with real data too. And it does happen. Mostly the difference in impression is more modest... but not always.
So if you see a histogram, just keep in the back of your mind that it's perfectly possible that a different choice of bin boundaries would yield a somewhat different impression of the data.
Imagine I want to show some students that I write "easy" tests (I don't know why this should be such an object of fascination for students since they all do the same test, and marks are generally scaled, but it is). In preparation, I draw a histogram and it turns out to like Chris' - it looks like most students score below the middle of the range of marks. But lo, I discover with a bit of fiddling around that if I make my bin centres where the edges were (and so on), the completely opposite impression is given - just like Annie's histogram. Yay, "easy test" ... and many fewer worried queries from students in the run-up to the test, because they tend to feel there's a good chance of scoring "above the middle".
Did I lie? No. Did I fudge the data? Well, no. I did something, though. Or rather, I didn't do something.
This is a sin of omission. I fail to explain what the data would have looked like given a different choice of bin location.
Clearly, when circumstances are right, the ability to choose the location and width of the bins can give us the opportunity to somewhat alter the impression given by a histogram. Without fudging the numbers themselves, we can sometimes fudge the impression they give.
What do statisticians do? Well, there are other ways to look at distributional shape. Kernel density estimates are popular, and they completely get rid of the "bin-location" issue, though there's still the equivalent of a "bin-width" issue (choice of bandwidth, also called the "window"), which is often dealt with by looking at more than one choice of width (usually a width that gives a nice smooth result and then one that is smaller, giving a "rougher" result, in order that we can see there's nothing unsual hiding away - like the blue and green curves in the graph at top
**(did I ever mention that I have trouble with correctly attributing the words "left" and "right"? - well as you see, sometimes I do. But not when describing the shape of a distribution, isn't that odd?)
What can you do? Well, assuming you don't have anything more sophisticated that a basic histogram tool, at the least (with continuous data, anyway), try shifting your bin starts forward or back a fraction of a bin-width (if you're lazy, maybe try something near a half, otherwise maybe try a couple of values). Also try a narrower bin width. If you do a few different histograms that all give the same general impression, it doesn't matter much which one you use. And if they don't give the same impression, you better either say so, show more than one, or find some other way to convey the information.
[Or you can do a kernel density estimate readily enough - many packages (including some free ones) will do them; there are pages online that can draw them if you just paste in some data. Implementing a kernel density estimate of your own is fairly straightforward - you can compute one in a spreadsheet easily enough - if anything, it's probably slightly simpler to compute one than it is to compute counts for a histogram, which is in itself pretty straightforward. ]
Caveat Emptor
___
Added in edit in Feb 2012:
Here is some R code to create the data:
histdata <- c(3.15,5.46,3.28,4.2,1.98,2.28,3.12,4.1,3.42,3.91,2.06,5.53 ,5.19,2.39,1.88,3.43,5.51,2.54,3.64,4.33,4.85,5.56,1.89,4.84,5.74,3.22 ,5.52,1.84,4.31,2.01,4.01,5.31,2.56,5.11,2.58,4.43,4.96,1.9,5.6,1.92)
Here is some R code to generate the histograms:
opar<-par() par(mfrow=c(2,2)) hist(histdata,breaks=1:6,main="Annie",xlab="V1",col="lightblue") hist(histdata-0.25,breaks=1:6,main="Brian",xlab="V2",col="lightblue") hist(histdata-0.5,breaks=1:6,main="Chris",xlab="V3",col="lightblue") hist(histdata-0.75,breaks=1:6,main="Zoe",xlab="V4",col="lightblue") par(opar)
Here is some R code to generate some density estimates:
opar<-par() par(mfrow=c(2,2)) plot(density(histdata,bw=.2),main="Annie") plot(density(histdata-.25,bw=.2),main="Brian") plot(density(histdata-.5,bw=.2),main="Chris") plot(density(histdata-.75,bw=.2),main="Zoe") par(opar)
Here is some R code to generate some other informative displays:
First - the sample cumulative distribution function
plot(ecdf(histdata))
Second, a stripchart that shows the positions of the individual observations as they move back.
x<-c abline="" c="" each="40)" g="" histdata-.25="" histdata-.5="" histdata-.75="" histdata="" pch="|" rep="" stripchart="" v="(2:5),col=6,lty=3)</pre" x=""> end edit
Religion in its purest form... is a vast work of bunkum
Dana at En Tequila Es Verdad discusses an interview in Salon with James Carse.
He (Carse) condemns Richard Dawkins and Sam Harris for not being "historians or scholars of religion" and so "it's too easy for them to pass off a quick notion of what religion is."
This is a common argument; PZ called it the "Courtiers Reply". The fact is, when atheists engage with less sophisticated versions of religion, they're doing it because that's what people believe. Was Dawkins' aim to engage with esoteric, sophisticated versions of religious belief? No. Indeed, it's a pointless effort, because the apologist can simply keep moving their goalposts, and refusing to pin down exactly what it is they would expect the atheist to engage with. Carse's difficulty in defining religion in the interview is a case in point.
What these atheists do is argue with veiws on religion that people actually hold.
My first thought on reading this was "he's got a double standard there - he's not asking Christians to study sophisticated versions of other religions, to be historians or scholars of religion before they reject other religions".
But actually, what made me write this is that I realized it's worse than this. There's an even worse double standard.
Carse says:
"To be an atheist, you have to be very clear about what god you're not believing in. Therefore, if you don't have a deep and well-developed understanding of God and divine reality, you can misfire on atheism very easily."
Apparently, the meaning of atheism is lost on him, since, of course, we lack belief in any gods (unless gods are defined so weakly that the term becomes, essentially, pointless).
That's funny, because Carse is apparently attempting "to find some underlying unity to all religions".
His understanding of atheism is laughable. He also says:
"To be an atheist is not to be stunned by the mystery of things or to walk around in wonder about the universe."
So while he berates atheists for failing to have the sophisticated understanding that would come from being an historian and religious scholar, he can't even do us the courtesy of engaging with an everyday relatively unsophisticated version of atheism. Instead, he makes up his version of atheism out of whole straw.
That's an astounding double standard.
He (Carse) condemns Richard Dawkins and Sam Harris for not being "historians or scholars of religion" and so "it's too easy for them to pass off a quick notion of what religion is."
This is a common argument; PZ called it the "Courtiers Reply". The fact is, when atheists engage with less sophisticated versions of religion, they're doing it because that's what people believe. Was Dawkins' aim to engage with esoteric, sophisticated versions of religious belief? No. Indeed, it's a pointless effort, because the apologist can simply keep moving their goalposts, and refusing to pin down exactly what it is they would expect the atheist to engage with. Carse's difficulty in defining religion in the interview is a case in point.
What these atheists do is argue with veiws on religion that people actually hold.
My first thought on reading this was "he's got a double standard there - he's not asking Christians to study sophisticated versions of other religions, to be historians or scholars of religion before they reject other religions".
But actually, what made me write this is that I realized it's worse than this. There's an even worse double standard.
Carse says:
"To be an atheist, you have to be very clear about what god you're not believing in. Therefore, if you don't have a deep and well-developed understanding of God and divine reality, you can misfire on atheism very easily."
Apparently, the meaning of atheism is lost on him, since, of course, we lack belief in any gods (unless gods are defined so weakly that the term becomes, essentially, pointless).
That's funny, because Carse is apparently attempting "to find some underlying unity to all religions".
His understanding of atheism is laughable. He also says:
"To be an atheist is not to be stunned by the mystery of things or to walk around in wonder about the universe."
So while he berates atheists for failing to have the sophisticated understanding that would come from being an historian and religious scholar, he can't even do us the courtesy of engaging with an everyday relatively unsophisticated version of atheism. Instead, he makes up his version of atheism out of whole straw.
That's an astounding double standard.
Sunday, July 20, 2008
Fighting Mathiness
'I know what you're thinking about,' said Tweedledum: 'but it isn't so, nohow.'
'Contrariwise,' continued Tweedledee, 'if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.'
'Can you do Addition?' the White Queen asked. 'What's one and one and one and one and one and one and one and one and one and one?'
'I don't know,' said Alice. 'I lost count.'
'She can't do Addition,' the Red Queen interrupted. 'Can you do Subtraction? Take nine from eight.'
'Nine from eight I can't, you know,' Alice replied very readily: 'but-'
'She can't do Subtraction,' said the White Queen. 'Can you do Division? Divide a loaf by a knife-what's the answer to that?'
Jordan Ellenberg defined mathiness as "a series of fervent gestures that gives the impression that mathematical ideas are being expressed, but doesn’t actually deliver the goods".
Let us examine some examples of mathiness, and some examples where honest attempts to deal with mathematical situations have foundered, and try to understand how we can be led astray by mathematical arguments.
Skewness
I recently wrote about how in statistics, the measure that is often called skewness doesn't really mean what popular lore holds it to mean, and that it is often misused - for example, when people assert that zero skewness implies symmetry. I later pointed to several sites that made the kinds of errors I was talking about. In the brief time since then, new instances of the same issue have come up on some mathematics-related blogs. It's a case where the verbal "description" of the situation is not in agreement with the mathematical tools being used - mathematical ideas appear to be expressed, but the goods are not being delivered.
Misleading Graphics
In another vein, bad statistical graphics, such as this
lie in graphical form
can mislead us, whether by accident, or as in this case, by design.
(via Andrew Gelman at Statistical Modeling, Causal Inference, and Social Science; there's other good examples to be found there.)
Examples abound in the media. Here's one from the NYT (via the Gallery of Data Visualization’s Missed Opportunities and Graphical Failures - click image for bigger version):
The top plot there is a graph of happiness against GNP-per-capita for a number of countries. The NYT has circled the countries in the top left hand corner, noting that many countries "had higher ... happiness than their economic situation would predict". This is the cardinal sin of treating inherently nonlinear relationships as linear - as they point out at the Gallery, an appropriate transformation - in this case looking at log-GNP, not raw GNP, makes these supposed "outliers" seem much more in keeping with the rest, and the apparent relationship more linear - and indeed, if anything, some entirely different points don't fit the general pattern. We seem to find nearly-linear relationships easier to understand, so transformation is often a useful strategy.
I have discussed the same issues - both the danger of treating nonlinear relationships as linear and the value of transformations in understanding relationships better in another context - relationships involving percentages. It's so easy to fall into the rut of linear thinking that we should consider taking advantage of the tendency to think that way and use transformation to reduce nonlinearity.
A common "nonlinear effect treated as linear" is when people try to average miles per gallon (or miles per hour, or a variety of other rates) - such as "I got 15 mpg going up and 45 mpg coming back, so I averaged 30 mpg overall" (when it's actually 22.5). In terms of transformations - the reciprocal (gallons per mile) - is linear and can be averaged.
Relying on a False Premise
Seemingly mathematical arguments may just be based on bad premises (such as one requiring selecting from the positive integers with equal probability - an impossibility that completely sinks the argument that relies on it).
That "infinity" thingy can be tricky - it seems to cause problems for journalists as well because they tend to underestimate how big it is.
Adding percentiles
Treating percentiles of distributions as if they were additive is unfortunately extremely common. In the case of official estimates of total oil reserves, it means that we probably have a fair bit more oil that we think.
What's the square root of that?
Or, sometimes, it seems, mathiness comes in because someone has no clue what the heck they're talking about, so we can be told that the Maya knew how to take the square root of a rectangle.
Mathematical arguments can feel unsually convincing, even unassailable, and we're awash in them for precisely that reason. It's too easy to forget that just because something seems to be laid out mathematically, it's not necessarily true - or even meaningful at all. Mathiness, like truthiness, is all around us. Even among skeptics, it's possible to put too much store in an argument couched in mathematical terms. We should be at least as skeptical of mathematical arguments - and in basically the same kinds of ways - as any other kinds of arguments, because we're all too often misled by them.
Unfortunately, it seems that we sometimes accept the (often implicit) conclusions of a mathematical argument without even realizing that an argument was being made.
If we fail to treat these arguments with the skepticism they deserve, we're open to being deceived by charlatans.
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