You missed his point. Conjectures may be correct. That doesn't make them proofs. We don't call it a proof until we're certain that it's correct (even then, we're sometimes wrong). It's not the correctness, it's the proofness that's the issue here.
I agree that some axioms seem more intuitive and obvious than others, but you can't really decide which of the two:
"There exists a higher power."
"No higher power exists."
is correct based on logic. You have to make an assumption. Given the number of prominent scientists and mathematicians who have believed #1 (who I would assume have/had at least as much abstract reasoning capabilities as an arbitrary person), I wouldn't be so quick to dismiss it for the reasons you state (I personally agree with #2, but realize that I have as much proof as proponents of #1).
The whole point is that you can't derive the axioms using reasoning, whether your assumption is that triangles have 180 degrees, or that god exists.
Except that for any system of logic, you eventually get to the axioms that are really just assumed to be true. You have to start with something from nothing. And from Godel, we know that there will be things that are true, but unprovable.
The integers are really a ring, in algebraic terms. They have addition (subtraction just being addition of a negative integer), multiplication, a zero and unity. Division is not well defined, since you can get results outside of the set of integers.
My point is just that integers are just symbols, and you can make them mean what you want them to mean. Sometimes we make them mean letters, sometimes they mean rational numbers. You're correct, in that an integer, and all valid operations on the ring of integers, will always be an integer, and you shouldn't try to do non-integer things to integers. But if you add a layer of abstraction, you can work in that layer, and do non-integer things with integers, according to the rules of your abstraction.
Oh, hang on. What happens if we have a language with an infinite choice of characters that can be made up of any number of those characters?
I'm not sure, but I think it might be possible. If you have an uncountably infinite number of characters to choose from, then the answer is definitely no. If the characters are countable (meaning there's a 1-to-1 mapping between them and integers), then I'm not sure if it is possible.
Reason to think that it might be is that the rationals can be thought of as a program of length 2. Now you have two characters picked from an infinite set. There is a 1-to-1 mapping between your programs and the rationals, so since the rationals are countable, so is your set of programs with 2 characters. I'm just not sure if this would hold for more than 2 characters.
Not necessarily. Instead of storing things as integers, you could store them as rational numbers, which would mean storing pairs of integers. We do this anyway, since our computers can only store 1 and 0. Then we interpret those 1's and 0's to make floating point numbers.
Yes, I understand that. It's the enumeration problem. So it depends on how smart the BrainDead compiler is. But it *is* possible. Consider the set of all programs. If his method of enumeration were done in a manner such as how you suggest:
x => printf('furbee');
x + 1 => printf('furbee'); printf('furbee');
x + 2 => printf('furbee'); printf('furbee'); printf('furbee'); ...etc...
Then yes, because the only thing it will ever be able to do is print lots of furbees. Instead, imagine that the enumeration were done based on the enumerated equivalent program's length, so first you enumerate all the 1-character programs, then all the 2, 3, etc. Now you have a way to enumerate *all* possible programs, using a single 'character'. Give me an equivalent program, and I can tell you what it's character is. Give me a character, and I can tell you what it's equivalent program is. Of course, you've got a lot of buggy program's in there, but why should BrainDead be any different from all other languages?
...there are an infinite number of even numbers, but you can't use them to express odd numbers...
How so? You could always assume that you have to divide by 2, so
2 => 1
4 => 2
6 => 3
etc...
Re:Just because you CAN...
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What they never mention is that when you calculate 1000! recursively, you push a thousand function calls onto the stack, and basically waste a whole lot of the computers time. "for(i=1;i=1000;++i) result*=i;" is a much more efficient and practical solution.
Ok, ignoring your actual point, but I couldn't resist...What's practical about 1000!? And what were the specs on the machine that ran the code?
So, what "fundamental economics principles" are you citing?
How about the market setting the price? If people are willing to pay $200 for some piece of software, then that piece of software is worth $200. Of course, if no one is willing to even pay the marginal cost, then it's likely that the product won't be made and sold.
But that doesn't mean that you can simply ignore development costs because they're sunk at the time of sale. What about before the development is done? Do you think a company would pay for development if they couldn't get their dev costs back? Yes, they could charge for something other than the software itself, like support, but it's the same thing. They're not going to pay for something unless they expect that it will generate a return that covers more than the costs to develop.
(Note that I'm not saying that SW patents are a good thing here, just saying that your 'economic arguments' are wrong.)
What about the hyperlink? It's just a few characters worth of harmless text, until you feed it through the right program, and the program connects to another site where the file may be downloaded. If one level of indirection is wrong, why wouldn't two? What's the real difference?
You have to list where you'll be staying (address and phone number) when you visit South Korea. Of course, when traveling on an American passport, they seem to pretty much wave you through...
Why don't we just take the FICA payroll deductions and buy T-Bills with them? It doesn't really change anything, except that we'd be able to do things like pass the money on to our heirs. The benefit of SS is supposed to be for the people, not so the govt can use the funds to spend however they want. I guess the problem is that people will figure out that they've been hoodwinked into a giant Ponzi scheme, and will want to be able to invest in things other than T-Bills, and, hey, what do you know, we've got privatized accounts.
Sure, the health insurance companies would balk and probably start up all kinds of FUD and lobbying against it. Why? They've become more concerned with making money than doing what they're supposed to do... cover your health care expenses.
No, making money is exactly what they're supposed to do. The way they do that is by spreading risk among their policy holders, so that some people will pay more in premiums than the insurance companies will have to pay out in claims (just like all other forms of insurance).
But I don't think it would be the insurance companies throwing up FUD if you could cover your retired parents (though your proposal is interesting). Why would they, when it would open up a huge new customer base--one that's supposed to be really really huge as the baby boomers all retire in the next decade or two? No, it would be people with vested interests in the status quo, like Sen. Kennedy, who would throw up a bunch of FUD about killing Medicare.
Actually, decimal arithmetic is more accurate than binary. Any binary fraction can also be represented in decimal, but not all decimal fractions can be represented in binary, since 2 divides 10 evenly, and 5 doesn't divide 2 evenly.
To make a real Nuke, you need 85-95% enrichment. And a pretty sophisticated bomb design - you can't just pack it with TNT and hope.
Actually, you can. If you really have that 85-95% HEU, you can make a gun assembly type weapon pretty easily. The hard part is getting the HEU in the first place.
Yeah, that's possible, too, but it seems more likely that a minority of the people (and not all of them Iraqis) are interested in attacking US (and coalition, and Iraqi) troops. The rest of them are just trying to live there lives. What about all the Iraqis signing up for the police and armed forces? Are they all 5th columnists, too?
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No, they really are students. South Korean students have demonstrations like American students have keg parties.
You missed his point. Conjectures may be correct. That doesn't make them proofs. We don't call it a proof until we're certain that it's correct (even then, we're sometimes wrong). It's not the correctness, it's the proofness that's the issue here.
- "There exists a higher power."
- "No higher power exists."
is correct based on logic. You have to make an assumption. Given the number of prominent scientists and mathematicians who have believed #1 (who I would assume have/had at least as much abstract reasoning capabilities as an arbitrary person), I wouldn't be so quick to dismiss it for the reasons you state (I personally agree with #2, but realize that I have as much proof as proponents of #1).The whole point is that you can't derive the axioms using reasoning, whether your assumption is that triangles have 180 degrees, or that god exists.
Except that for any system of logic, you eventually get to the axioms that are really just assumed to be true. You have to start with something from nothing. And from Godel, we know that there will be things that are true, but unprovable.
I first heard about him when I read Hardy's A Mathematician's Apology. A great read. Also heard about him when I took number theory.
It's not that Phillip K Dick until there's been a nuclear holocaust.
It's intuitive in the sense that it's following natural language conventions. We often speak this way. People who code perl often code this way.
The integers are really a ring, in algebraic terms. They have addition (subtraction just being addition of a negative integer), multiplication, a zero and unity. Division is not well defined, since you can get results outside of the set of integers.
My point is just that integers are just symbols, and you can make them mean what you want them to mean. Sometimes we make them mean letters, sometimes they mean rational numbers. You're correct, in that an integer, and all valid operations on the ring of integers, will always be an integer, and you shouldn't try to do non-integer things to integers. But if you add a layer of abstraction, you can work in that layer, and do non-integer things with integers, according to the rules of your abstraction.
Reason to think that it might be is that the rationals can be thought of as a program of length 2. Now you have two characters picked from an infinite set. There is a 1-to-1 mapping between your programs and the rationals, so since the rationals are countable, so is your set of programs with 2 characters. I'm just not sure if this would hold for more than 2 characters.
Not necessarily. Instead of storing things as integers, you could store them as rational numbers, which would mean storing pairs of integers. We do this anyway, since our computers can only store 1 and 0. Then we interpret those 1's and 0's to make floating point numbers.
Yes, I understand that. It's the enumeration problem. So it depends on how smart the BrainDead compiler is. But it *is* possible. Consider the set of all programs. If his method of enumeration were done in a manner such as how you suggest:
...etc...
x => printf('furbee');
x + 1 => printf('furbee'); printf('furbee');
x + 2 => printf('furbee'); printf('furbee'); printf('furbee');
Then yes, because the only thing it will ever be able to do is print lots of furbees. Instead, imagine that the enumeration were done based on the enumerated equivalent program's length, so first you enumerate all the 1-character programs, then all the 2, 3, etc. Now you have a way to enumerate *all* possible programs, using a single 'character'. Give me an equivalent program, and I can tell you what it's character is. Give me a character, and I can tell you what it's equivalent program is. Of course, you've got a lot of buggy program's in there, but why should BrainDead be any different from all other languages?
2 => 1
4 => 2
6 => 3
etc...
How about the market setting the price? If people are willing to pay $200 for some piece of software, then that piece of software is worth $200. Of course, if no one is willing to even pay the marginal cost, then it's likely that the product won't be made and sold.
But that doesn't mean that you can simply ignore development costs because they're sunk at the time of sale. What about before the development is done? Do you think a company would pay for development if they couldn't get their dev costs back? Yes, they could charge for something other than the software itself, like support, but it's the same thing. They're not going to pay for something unless they expect that it will generate a return that covers more than the costs to develop.
(Note that I'm not saying that SW patents are a good thing here, just saying that your 'economic arguments' are wrong.)What about the hyperlink? It's just a few characters worth of harmless text, until you feed it through the right program, and the program connects to another site where the file may be downloaded. If one level of indirection is wrong, why wouldn't two? What's the real difference?
Perhaps. But then you could just move to California, where they just voted to borrow $3Billion to pay for stem stell research.
You have to list where you'll be staying (address and phone number) when you visit South Korea. Of course, when traveling on an American passport, they seem to pretty much wave you through...
Why don't we just take the FICA payroll deductions and buy T-Bills with them? It doesn't really change anything, except that we'd be able to do things like pass the money on to our heirs. The benefit of SS is supposed to be for the people, not so the govt can use the funds to spend however they want. I guess the problem is that people will figure out that they've been hoodwinked into a giant Ponzi scheme, and will want to be able to invest in things other than T-Bills, and, hey, what do you know, we've got privatized accounts.
But I don't think it would be the insurance companies throwing up FUD if you could cover your retired parents (though your proposal is interesting). Why would they, when it would open up a huge new customer base--one that's supposed to be really really huge as the baby boomers all retire in the next decade or two? No, it would be people with vested interests in the status quo, like Sen. Kennedy, who would throw up a bunch of FUD about killing Medicare.
Maybe, but a lot of that stuff is starting to migrate to Windows and Linux, because x86 is so cheap and powerful for the $$.
Actually, decimal arithmetic is more accurate than binary. Any binary fraction can also be represented in decimal, but not all decimal fractions can be represented in binary, since 2 divides 10 evenly, and 5 doesn't divide 2 evenly.
Yeah, that's possible, too, but it seems more likely that a minority of the people (and not all of them Iraqis) are interested in attacking US (and coalition, and Iraqi) troops. The rest of them are just trying to live there lives. What about all the Iraqis signing up for the police and armed forces? Are they all 5th columnists, too?