Usually, if a die lands on edge we say it was a spoiled throw and do it over. Similarly if a Dark Lord writes 37 on the face that lands on top, we complain that the Dark Lord is spoiling our game and we don't count it.
We count 6 possibilities for a 6-sided die, 5 possibilities for a 5-sided die, 2 possibilities for a 2-sided die, and if you have a die with just one face -- a spherical die -- what's the chance that face will come up?
I think it would be interesting to develop probability theory with no boundaries, with no 0 and 1. It works fine to do it the way it's done now, and the alternative might turn up something interesting too.
"To say that human beings "invented numbers" - or invented the structure implicit in numbers - seems like claiming that Neil Armstrong hand-crafted the Moon. The universe existed before there were any sentient beings to observe it, which implies that physics preceded physicists."
No, there's a conflation of two things here.
Have you ever really looked at a penny? I'm looking at a 1990 penny now. I know that if you look at the front and you see the bas-relief of Lincoln, and the date 1990, and it's a penny, then you can be sure that the back side will have a picture of the Lincoln memorial. It works! And you can find all sorts of connections. Like, there's a single "O" on the front, in the name GOD in the phrase IN GOD WE TRUST. And there's a single "O" on the back, in the phrase ONE CENT. One O on the front, one O on the back. A connection! You could make lots and lots of these interconnections between the front and the back of the penny, and draw conclusions about what it means. You could invent a discipline of Pennyology if only somebody would fund it.
Is it true that Pennyology is implicit in pennies? In a way. Certainly the pennies should exist before the Pennyology. But the pennies are only whatever they are. The existence of pennies doesn't tell us much about what the practitioners of the discipline of Pennyology will actually notice. They might never pay attention to the pair of O's. There could be a fold in Lincoln's coat that after the proper analysis provides a solution to the whole world crisis, and they may never pick up on it. While it's predictable that different independent attempts at Pennyology would have a whole lot in common since after all they all need to be compatible with the same pennies, still they might be very different in some respects. You can't necessarily predict the Pennyology from looking at the penny. And you can't predict what mathematics people will invent from observing reality.
You can predict some things. A mathematics that invents the same 2D plane we use and that proves a 3-color theorem has something wrong with it. But you can't predict which things will be found first or, to some extent, which things will be found at all.
If there's a reality that mathematics must conform to, still each individual version of human mathematics is invented by humans.
Similarly with physics. Our physics is invented. The reality the physics describes is real. We can imagine a platonic-ideal physics that fit the reality completely, but we don't have an example of that to point at. So for example before Townsend invented the laser, a number of great physicists claimed it was impossible. Townsend got the idea because lasers could be described using Maxwell's equations. But people thought that quantum mechanics provided no way to get that result. it turned out they were wrong.
Actual physics is invented. Certainly incorrect physics must be invented. There's nothing in reality that shows you how to do physics wrong.
Robs, religions tend to thrive among people who work hard and behave well and such. The two tend to go together. But just as priests do well in that environment so do politicians. How can we tell whether these people are helping to maintain the prosperity, or merely parasitising it? In general parasites do better with healthy hosts.
I do not claim that religion is useless. I claim that you are assuming your conclusion that it is not. Ideally we might find some sort of data. We might for example look at examples where people who previously had no religion get proselytised into a religion and see how much better they get at working hard and behaving well and all that. To actually find out we should test our ideas against reality rather than just assuming that our beliefs are objectively true.
Razib, I see you argue that different religions can compete, and what they compete for is converts who perhaps are comparing the benefits the competing religions provide them. Whether individuals make rational choices or whether they irrationally gravitate to the religions that appear to bring prosperity, either way the religions compete.
But I'm talking about how religions could have gotten their start. If people who are predisposed to religion are better at living in larger-than-kinship groups, and if people who live in larger groups survive better, then the spread of predisposition-for-religion can be explained by individual selection without requiring group selection arguments.
Lots of people think that the main thing religions do is to bind people together. The etymology of the word works that way, right? re-ligio.
If they think that's what religion does, then it's only natural they'd think that's what it gets selected for. No mystery there.
Does it take group selection? People can stay in family groups with kin selection. No mystery. Suppose that people in groups all tend to survive better than loners. That's plausible. Then anything that helps people work together in larger groups (without too many side effects) could be individual-selected. Each individual is selected for the ability to join groups and the ability to maintain them, because the more time he spends alone or in a broken group, the worse his survival. Each individual is selected to correctly choose which others to throw out -- throw out good members sand the group is weakened. Fail to throw out members that will make the group collapse and each individual member is threatened.
It's only group selection when it's religious groups competing against nonreligious groups. It's individual selection when it's individuals in groups competing against loners.
I didn't think it was tremendously funny. But I thought it was funny enough to recite the whole thing to my wife while she sat at her own keyboard, instead of just send her a link. She didn't think it was tremendously funny. But she politely stopped typing to listen, and she laughed some.
It seems to me like at least a B effort. The humor was in everybody wanting to believe.
In reality, wasn't there a claim that the midwife confirmed Mary was a virgin? If I lived in the village I'd probably accept that as sufficient evidence, though in my namesake's tradition I'd rather stick my own fingers in to confirm it.
For myself it's pretty much irrelevant. There was an ecumenical joke when I was a kid -- a preacher compared the different protestant faiths as being like different roads that could be taken to get your cotton to a cotton gin. And when you get there, the engineer isn't going to ask you "Which road did you take?" He's going to ask you, "How good is your cotton?".
By the same reasoning, when Jesus comes to me with a morality for me to follow, I don't ask him "What miracles did you have about your birth?". I ask him, "How good is your morality?".
"If you want to see an example of a measured response, take a look at the UK's after the London Underground bombings of 7th July 2005. Admittedly the bombings weren't of the same league as the September 11th attacks, but virtually nobody in the UK was saying "let's bomb the f*ers" And a month or two later (at the most) it was as if nothing had ever happened."
Mike K, I tend to agree with you, but....
The fact is, the british empire is gone and the british are ex-colonialists. As a nation they're old and tired and wimpy. It's different for us -- you can't be the world's only superpower and let anybody get away with anything. If we let one terrorist group have the WTC they'll all want one.
So as the winners of the cold war we have to respond to any provocation -- we have no choice. Any little group of terrorists can tell us who to invade and we have to do it, or we let the terrorists win. Either we kill off every terrorist group that isn't under our direction, or we lose our special status and have to admit we aren't in control of the world.
Think about it.
"If you believe invading Afghanistan was a correct choice then I'm not sure how you could say Iraq was a complete mistake. The invasion of Afghanistan was aimed at eliminating a state that offered aid and support to an enemy who would use that aid and support to project power to the US and harm her citizens or the citizens of other western states. Denying that aid and support would hope to achieve the purpose of reducing or eliminating the ability of the enemy to project power.
"Any other state that might offer aid and support to the enemy would enable the enemy to rebuild their ability to project power. Iraq was one possible source of aid and support."
Brandon, your reasoning is compelling. However, it has a subtle flaw that I think will be easier to see when I rephrase the argument as follow:
We will be safer after we conquer every potential enemy.
The claim is obviously true, and yet....
"I would have preferred, for example, that the U.S., Russia, China, UK, Israel and perhaps France announced that in one year they will declare war an any other nation that either has weapons of mass destruction or doesn't allow highly intrusive inspections to make sure they don't have weapons of mass destruction."
James D. Miller, I think your idea has possibilities. However, it would be very hard for it to succeed with israel on the list of nations that has nukes but denies them to others. Israel would have to be one of the nations that would be destroyed if it keeps nuclear weapons or refuses highly intrusive inspections.
What about india? Shouldn't they be on the list? We don't want war with india, they haven't threatened anybody except, well, pakistan.
And what about pakistan? If we let india keep nukes it would be hard to invade pakistan over their nukes. Should pakistan be one of the nations of the alliance that will destroy anybody else who has nukes?
Now it looks like a hard problem. No, your idea does not look workable. Allow russia to have nukes but not china? No. Allow china to have nukes but not india? Hardly. Allow india to have nukes but not pakistan? Tempting, but no. Allow pakistan to have nukes but not israel? It would be a good idea but it won't fly. Allow israel to have nukes but not syria? A pleasant thought but not practical. Allow israel and arab nations both to have nukes? Not practical either.
There's a logic here that hasn't played out yet. It goes:
Therefore:
The world as a whole hasn't recognised this logic yet because there haven't been any graphic examples. Probably after the second nuclear war, when the world sees what happens to the "winner", people will have a much clearer idea about it. But two nuclear wars will be hard on the world. Ideally these wars would involve small countries so they can be small nuclear wars.
So most of us will be better off if lebanon gets nukes. Then a nuclear war between lebanon and israel could be one of the smallest possible nuclear wars.
The next obvious choice is a war between libya and chad.
After 2 nuclear wars the world as a whole will be much more ready for disarmament then they are now, with nuclear war a threat that has not materialised for 62 years.
Ir'a much much easier to stop people from doing something they didn't want to do in the first place, than stop them from something they think can keep you from dominating them.
Chris and Ben, we create axiom systems and we discover parts of "mathematics". There are probably only a finite number of theorems that can be stated with only 10 characters, or 20 characters, or 30 characters, provided we don't add new definitions. But the number of possible theorems quickly gets very very large. Will each independent group of mathematicians come up with the same theorems? Probably not. So we get different mathematics.
How different could alien mathematics be? I don't know. We could look at a variety of alien mathematics and see. Except, we don't have much of that. We presumably had different mathematical traditions in china, india, and europe, and we got some minor differences. But they were solving similar real-life problems and they could have been in communication. If you want trade in silk then you need a lot of it for it to make much difference. A very few mathematicians traveling could spread ideas easily.
It's easy to see alternate physics, and alternate technology is kind of arbitrary. I think alien math might be pretty different depending on which theorems they proved first. But I don't have good examples to demonstrate it.
The Pennyologist who notices the O's is not that different from the Pennyologist who notices there's one L on each side. A particular Pennyology might notice one of those, or the other one, or both. Out of the many relationships you could pick out from the penny, which ones will people pay attention to?