This is such bullshit. Historically (and lets talk about history since the Industrial revolution, for fucks sake) policework and addressing the causes of terrorism is the best way of ending terrorosim. Thing Northern Ireland, for instance. For islamic terrorism, the police actions in Germany after 9/11 did a very good job of busting up al quaida in Europe.
Not participating in it (by sending weapons to Israel so that they may kill more Palestinians, or to Saddam's Iraq in the 80's and congratulating him on slaughtering thousands of Kurds) is another very good way.
Re:Free Speech in Denmark??
on
Press freedom
·
· Score: 1
So congrats to Denmark. Danish journalists clearly like Denmark and consider themselves free there.
I think that you've hit the nail on the head. As a non-Dane who's lived in Denmark, the one thing you do notice is that the Danes really love their country. The bad thing is that you are not really a Dane unless your name ends with "sen" and it is one of the more openly racist societies I've lived in.
For example, the media regularly refers to Danes with an ethnic background (parents born outside of Denmark) as "andengenerationsinvandrer" which translates to "second generation immigrant". So, if a crime is committed by such a Dane, it is reported as a "second generation immigrant" committing the crime -- implying that it is not *really* a Danish person. This used to regularly piss me off -- these are people born in Denmark and speak Danish as their first language!
Recently the Danish government introduced a law which forbids a Danish person from marrying a non-EU person if the Danish person is under 24. Although this particular law was a codified way of "keeping the muslims out", it seems to me that overall the Danish society is on a slippery-slope away from freedom. I can't imagine too many other free societies accepting being told who they are or they are not allowed to marry.
But as I said, the Danes love Denmark very much, so clearly these sort of surveys will always look very good for Denmark. The reality is usually quite different since other Western societies have a tradition of constructive criticism of the society by the society -- a tradition which doesn't really seem to exist in Denmark.
Just about any first-year university mathematics introductory textbook on calculus will define the real numbers using dedekind cuts or cauchy completions. These definitions are second-order logic statements and they provably define the set of real numbers up to isomorphism. Once you have the real numbers, you use one of two equivalent methods of obtaining the set complex numbers, which is again defined up to isomorphism.
Summing up, the complex numbers, like the natural numbers, are a well-defined entity, up-to isomorphism (renaming of the individual components). Thus, just as there is only one set of of natural numbers, there is only one set of complex numbers.
We've been talking about programs without inputs. Or if you like, a program P with a fixed input x, and the question is "does P halt on x?". This is trivially decidable, because either it does or it doesn't. But of course you are right, it is impossible to write a program which takes as its input a description of a program P and a description of an input x and decides whether P halts on x.
This is where we disagree. You see, there is only one set of complex numbers. It is defined up to isomorphism, let's say by Dedekind cuts. There is only one zeta function. The hypothesis is true or false.
Of course, there could be many axiom systems for reasoning about the complex numbers. And you could, conceivably, change the definition of complex numbers to something more general; but then what you have is not the Riemann hypothesis anymore and I grant you that it may turn out that whatever that is, it may be independent of some set of axioms.
But the real Riemann hypothesis is either true or false, there is not other way around it.
Look, surely you have to agree with me that any particular program halts or it doesn't. It certainly can't change it's mind halfway through. We are talking about deterministic programs and it is determined from the start whether a particular program will halt or not.
So let's get our definitions straight. The question: "does program x halt or not?" is always trivially decidable - meaning that either program A or program B decides it. Of course, figuring out which of these programs decides it may be hard, but that is a different question which has nothing to do with decidability.
Godel's theorem is about formal systems of axioms. Axiom systems are for reasoning about particular models. It may well be that your particular axiom system is not powerful enough to prove the Riemann hypothesis in a model which includes the set of complex numbers. But the hypothesis is true or false, independent of the axioms -- the model is defined up to isomorphism and exists independently of axiom systems.
I never claimed that the P=NP problem is easy or trivial. In fact it is proving to be one of the hardest problems of mathematics. I just said that it is definitely not undecidable.
This means that either P=NP or not.
Do you have a good mathematical reason to believe that RH is decidable? Or are you just high on a Logical Positivist emotion?
Igonring the second question, I'll answer the first;) Yes, it is definitely decidable. This is because the zeta function is a function on the complex numbers. Questions about the complex numbers are either true or false. There is either a zero which disproves the Riemann hypothesis, meaning that it is false, or there isn't, meaning that it is true. Of course, I'm not claiming that it is an easy problem, just decidable.
Bluntly, yes. The continuum hypothesis is independent of the axioms of standard set theory. This means that there exist set theories where the continuum hypothesis is true and there exist set theories where it is false.
On the other hand, the Riemann hypothesis is a particular question about a precisely stated function on the precisely defined set of complex numbers. There is only one such function and only one such set of complex numbers, no matter which set theory you are talking about. The hypothesis is either true or false: there either exists a zero not on the strip or there doesn't. There is no third option.
Decidability means: there exists a program which decides this problem.
Consider program A: "ignore input, write YES" and program B: "ignore input, write NO". One of these programs decides whether your program halts. I don't have to tell you which one, and indeed it may be difficult to say which one, but the problem is *trivially* decidable.
if P == NP, the world will have been turned on its head
Not necessarily, what if P=NP but the fastest P algorithm which computes an NP complete problem is n^100000? The complexity of this is still much larger than anything feasible.
Which is why compexity theory is a bit funny, since for a complexity theorist, if a problem is in P, it's easy.
P=NP is really a purely mathematical question. It's solution will have absolutely no impact on software engineering, although of course it will have a huge impact on theoretical computer science.
It's really sad how many people keep bringing up this "well, it could be undecidable" line.
Firstly, the halting problem is trivially decidable for any particular program, the program either halts or it doesn't. There is no mystery here, it is just impossible to write a program that will *check* whether an *arbitrary* program will halt or not. There is no magical program that will do both or neither, this is because programs halt or they do not. The undecidability of the halting problem demonstrates the limitations of computers, not the limitations of mathematicians.
Similarly, it's trivially decidable whether a particular program is in P or NP. It's either true or false.
Returning back to the topic P=NP is either true or false. It's a matter of set equality:
1) Let P denote the set of all programs in P 2) Len NP denote the set of all programs in NP 3) compare the two sets, there are only two possible outcomes, since P is in NP
a) there is a program in NP which is not in P, then P is not equal to NP
b) there isn't, then P=NP
It's as simple as that. And when I say program, I mean program: that means you can choose whichever formalism you want, as long as it's turing complete: this includes C, Java, Perl and whatever else you want.
Unfortunately, this fallacy persists because people just do not understand Godel. I even read a book about the Riemann Hypothesis recently where some serious mathematicians were quoted as saying that the Hypothesis could be undecidable... bullshit.
This brings up an interesting issue. Is Duke University actually encouraging/allowing their students to record lectures? I know that this is a reasonably big deal in the UK where, as far as I know, it is illegal; meaning that before recording, consent has to be asked of each individual lecturer. Many of the lecturers I know do not like the practice of students recording lectures for various reasons including:
1) They own the IP of their own lectures 2) Students tend to be easier distracted when they know (or at least think) that they will listen to the lecture again 3) The audio is only a small part of a whole presentation which includes writing on the blackboard, overheads etc.
Anyway, it seems a little strange to me that American universities are encouraging this so openly.
Actually, this is not so unbelievable. Heisenberg apparently did believe that Germany would win and was working on developing the atomic bomb for Hitler.
There has been a lot of attention devoted to a meeting, in 1941, between Heisenberg and Niels Bohr, a Danish physicist, in occupied Copenhagen. There has even been a play about it, called "Copenhagen".
You can read some documents about the meeting here .
Yeah, a meaningless job at Walmart or Mc Donald's!
That is certainly more rewarding than taking advantage of the opportunity to have your studies funded by a third party. You would actually learn some things you enjoy learning and be able to repay society with a rewarding job in which you do things that you are good at. But no...
Clearly flipping burgers at McDonald's is better and prepares you for the *real life*, whatever the fuck that is. You may even get promoted to store assistant manager...
What you say is true. However it refers to proofs given a particular set of axioms. Godels sentence is not provable in the system but it is true. He proved it. Outside the system. In the unlikely case that ZF+AoC is not enough to prove Riemanns Hypothesis, mathematicians will step outside that formal system. In fact, if this is the case, we should get some more set theorists real soon.:)
I personally like the Turing proof of Godels theorem. The set of all theorems is recursively enumerable, but the set of all true statements about the natural numbers is not. The latter is a reduction from the complement of the halting problem, using computation histories of turing machines. It's really cool, the details can be found in Kozen's "Automata and Computability" which by the way is an excellent textbook on introductury computability theory.
Although I know that Platonism (art for art's sake) is regarded as naive, I suspect that a lot of pure mathematicians are Platonists at heart. I don't know if it is such a bad thing.
Most mathematical advances at the theoretical level (starting at the concept of zero and then the negative integers) have been considered "way too abstract" at some time or another. Time and time again, the rest of science catches up and finds use for these things.
Just to continue the timeline, consider complex numbers (electrical engineering) and group theory (chemistry). Even things that were considered abstract nonsense even a few years ago are now finding application (for example 2-category theory in physics). So I'm more of the opinion that we should let mathematicians do whatever they find interesting, it's worked up to now and I think that it will continue to do so.
As for the AoC, I think that most mathematicians don't consider it right or wrong, natural or unnatural. As Hilbert famously once said, "It's not mathematics, it's theology". But they are happy to use it if it will help them prove theorems in their mathematical world which consists of real things like the natural or the complex numbers. To most mathematicians, questions like the validitity of the continuum hypothesis are simply not interesting. I'm a computer scientist, but I do have some knowledge of maths departments. How many set theorists do you know of? I have never met one.
The Riemann Hypothesis (RH) is either true or false. If it is false if and only if there is a counterexample. It is the mathematician's job
to show if it true or false.
RH is not on the same plane as the Axiom of Choice, which is independent of the standard (ZF) axioms of set theory. This means that there are set theories (mathematical objects) with the Axiom of Choice true and others with it false.
However, there is only one set of complex numbers
(or the natural numbers).
The complex numbers or the natural numbers are mathematical objects. Godel's (second?) incompleteness theorem says only that we will never be able to come up with finitely many (actually, recursively enumerable) axioms that will have the natural numbers or the complex numbers as their only model. This is a failure of first-order logic, not of complex number theory or number theory.
Doesn't anyone remember the article by Jon Katz in which he was emailing a ``geek'' just outside of Kabul who was downloading and playing movies on a C64?
I have absolutely no idea of what are you talking about..:)
Still, I'm worried that when it comes to science people have a generally closed minded approach about "accepted" theories (see most of the posts above). After all, all the really large breakthroughs in science came from people who disagreed in what was "accepted" knowledge at the time. This can only be bad for science...
This is not as much a problem in Maths of CS as a proof is either correct or it isn't; there isn't much ground to discussion.
I'm still not convinced that evolution answers the question of where we come from and I am of course really curious. I just hope that there are some biologists who share that curiosity.
Come on, this is bullshit. You argument has degenerated into using another definition of the work "evolving". Surely you're not saying that the kernel hackers are applying random changes! The changes are (usually) well thought out and they try to fix a problem, surely not a random mutation...
I agree with everything you say, except you don't refute my main point. You agree with me that the linux kernel thing wouldn't work. So where exactly do we come from then?
Humans have developed intelligence because it gives a better chance of survival. Why is it then so inconceivable that a full linux kernel wouldn't magically appear after a few million generations of semi random code-tweaking?
Why? Because such a system gets incredibly complicated and small mutations just don't cut it. Evolution (this way defined) simply doesn't scale.
I am not a biologist. I am a computer scientist. However, I can explain to a 10 year old why P=NP would mean the death of RSA.
I would like to have a laymans explanation of how complex systems can be brought about by small mutations? Especially since importing the idea into something a a coder would understand simply would not work. Maybe my idea is too simple, if so why don't we get a biologist to tell us how macro evolution works, get a Beowulf cluster and make some cool complex software by starting with a "Hello World" program and introducing small random changes?
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This is such bullshit. Historically (and lets talk about history since the Industrial revolution, for fucks sake) policework and addressing the causes of terrorism is the best way of ending terrorosim. Thing Northern Ireland, for instance. For islamic terrorism, the police actions in Germany after 9/11 did a very good job of busting up al quaida in Europe.
Not participating in it (by sending weapons to Israel so that they may kill more Palestinians, or to Saddam's Iraq in the 80's and congratulating him on slaughtering thousands of Kurds) is another very good way.
I think that you've hit the nail on the head. As a non-Dane who's lived in Denmark, the one thing you do notice is that the Danes really love their country. The bad thing is that you are not really a Dane unless your name ends with "sen" and it is one of the more openly racist societies I've lived in.
For example, the media regularly refers to Danes with an ethnic background (parents born outside of Denmark) as "andengenerationsinvandrer" which translates to "second generation immigrant". So, if a crime is committed by such a Dane, it is reported as a "second generation immigrant" committing the crime -- implying that it is not *really* a Danish person. This used to regularly piss me off -- these are people born in Denmark and speak Danish as their first language!
Recently the Danish government introduced a law which forbids a Danish person from marrying a non-EU person if the Danish person is under 24. Although this particular law was a codified way of "keeping the muslims out", it seems to me that overall the Danish society is on a slippery-slope away from freedom. I can't imagine too many other free societies accepting being told who they are or they are not allowed to marry.
But as I said, the Danes love Denmark very much, so clearly these sort of surveys will always look very good for Denmark. The reality is usually quite different since other Western societies have a tradition of constructive criticism of the society by the society -- a tradition which doesn't really seem to exist in Denmark.
Just about any first-year university mathematics introductory textbook on calculus will define the real numbers using dedekind cuts or cauchy completions. These definitions are second-order logic statements and they provably define the set of real numbers up to isomorphism. Once you have the real numbers, you use one of two equivalent methods of obtaining the set complex numbers, which is again defined up to isomorphism.
Summing up, the complex numbers, like the natural numbers, are a well-defined entity, up-to isomorphism (renaming of the individual components). Thus, just as there is only one set of of natural numbers, there is only one set of complex numbers.
We've been talking about programs without inputs. Or if you like, a program P with a fixed input x, and the question is "does P halt on x?". This is trivially decidable, because either it does or it doesn't. But of course you are right, it is impossible to write a program which takes as its input a description of a program P and a description of an input x and decides whether P halts on x.
This is where we disagree. You see, there is only one set of complex numbers. It is defined up to isomorphism, let's say by Dedekind cuts. There is only one zeta function. The hypothesis is true or false.
Of course, there could be many axiom systems for reasoning about the complex numbers. And you could, conceivably, change the definition of complex numbers to something more general; but then what you have is not the Riemann hypothesis anymore and I grant you that it may turn out that whatever that is, it may be independent of some set of axioms.
But the real Riemann hypothesis is either true or false, there is not other way around it.
Look, surely you have to agree with me that any particular program halts or it doesn't. It certainly can't change it's mind halfway through. We are talking about deterministic programs and it is determined from the start whether a particular program will halt or not.
So let's get our definitions straight. The question: "does program x halt or not?" is always trivially decidable - meaning that either program A or program B decides it. Of course, figuring out which of these programs decides it may be hard, but that is a different question which has nothing to do with decidability.
Godel's theorem is about formal systems of axioms. Axiom systems are for reasoning about particular models. It may well be that your particular axiom system is not powerful enough to prove the Riemann hypothesis in a model which includes the set of complex numbers. But the hypothesis is true or false, independent of the axioms -- the model is defined up to isomorphism and exists independently of axiom systems.
This means that either P=NP or not.
Igonring the second question, I'll answer the first
Bluntly, yes. The continuum hypothesis is independent of the axioms of standard set theory. This means that there exist set theories where the continuum hypothesis is true and there exist set theories where it is false.
On the other hand, the Riemann hypothesis is a particular question about a precisely stated function on the precisely defined set of complex numbers. There is only one such function and only one such set of complex numbers, no matter which set theory you are talking about. The hypothesis is either true or false: there either exists a zero not on the strip or there doesn't. There is no third option.
Decidability means: there exists a program which decides this problem.
Consider program A: "ignore input, write YES" and program B: "ignore input, write NO". One of these programs decides whether your program halts. I don't have to tell you which one, and indeed it may be difficult to say which one, but the problem is *trivially* decidable.
Not necessarily, what if P=NP but the fastest P algorithm which computes an NP complete problem is n^100000? The complexity of this is still much larger than anything feasible.
Which is why compexity theory is a bit funny, since for a complexity theorist, if a problem is in P, it's easy.
P=NP is really a purely mathematical question. It's solution will have absolutely no impact on software engineering, although of course it will have a huge impact on theoretical computer science.
It's really sad how many people keep bringing up this "well, it could be undecidable" line.
Firstly, the halting problem is trivially decidable for any particular program, the program either halts or it doesn't. There is no mystery here, it is just impossible to write a program that will *check* whether an *arbitrary* program will halt or not. There is no magical program that will do both or neither, this is because programs halt or they do not. The undecidability of the halting problem demonstrates the limitations of computers, not the limitations of mathematicians.
Similarly, it's trivially decidable whether a particular program is in P or NP. It's either true or false.
Returning back to the topic P=NP is either true or false. It's a matter of set equality:
1) Let P denote the set of all programs in P
2) Len NP denote the set of all programs in NP
3) compare the two sets, there are only two possible outcomes, since P is in NP
a) there is a program in NP which is not in P, then P is not equal to NP
b) there isn't, then P=NP
It's as simple as that. And when I say program, I mean program: that means you can choose whichever formalism you want, as long as it's turing complete: this includes C, Java, Perl and whatever else you want.
Unfortunately, this fallacy persists because people just do not understand Godel. I even read a book about the Riemann Hypothesis recently where some serious mathematicians were quoted as saying that the Hypothesis could be undecidable... bullshit.
You mean scientologically accurate.
This brings up an interesting issue. Is Duke University actually encouraging/allowing their students to record lectures? I know that this is a reasonably big deal in the UK where, as far as I know, it is illegal; meaning that before recording, consent has to be asked of each individual lecturer. Many of the lecturers I know do not like the practice of students recording lectures for various reasons including:
1) They own the IP of their own lectures
2) Students tend to be easier distracted when they know (or at least think) that they will listen to the lecture again
3) The audio is only a small part of a whole presentation which includes writing on the blackboard, overheads etc.
Anyway, it seems a little strange to me that American universities are encouraging this so openly.
Actually, this is not so unbelievable. Heisenberg apparently did believe that Germany would win and was working on developing the atomic bomb for Hitler.
There has been a lot of attention devoted to a meeting, in 1941, between Heisenberg and Niels Bohr, a Danish physicist, in occupied Copenhagen. There has even been a play about it, called "Copenhagen".
You can read some documents about the meeting here .
Yeah, a meaningless job at Walmart or Mc Donald's!
That is certainly more rewarding than taking advantage of the opportunity to have your studies funded by a third party. You would actually learn some things you enjoy learning and be able to repay society with a rewarding job in which you do things that you are good at.
But no...
Clearly flipping burgers at McDonald's is better and prepares you for the *real life*, whatever the fuck that is. You may even get promoted to store assistant manager...
What you say is true. However it refers to proofs given a particular set of axioms. Godels sentence :)
is not provable in the system but it is true. He proved it. Outside the system. In the unlikely case that ZF+AoC is not enough to prove Riemanns Hypothesis, mathematicians will step outside that formal system. In fact, if this is the case, we should get some more set theorists real soon.
I personally like the Turing proof of Godels theorem. The set of all theorems is recursively enumerable, but the set of all true statements about the natural numbers is not. The latter is a reduction from the complement of the halting problem, using computation histories of turing machines. It's really cool, the details can be found in Kozen's "Automata and Computability" which by the way is an excellent textbook on introductury computability theory.
Although I know that Platonism (art for art's sake) is regarded as naive, I suspect that a lot of pure mathematicians are Platonists at heart. I don't know if it is such a bad thing.
Most mathematical advances at the theoretical level (starting at the concept of zero and then the negative integers) have been considered "way too abstract" at some time or another. Time and time again, the rest of science catches up and finds use for these things.
Just to continue the timeline, consider complex numbers (electrical engineering) and group theory (chemistry). Even things that were considered abstract nonsense even a few years ago are now finding application (for example 2-category theory in physics). So I'm more of the opinion that we should let mathematicians do whatever they find interesting, it's worked up to now and I think that it will continue to do so.
As for the AoC, I think that most mathematicians don't consider it right or wrong, natural or unnatural. As Hilbert famously once said, "It's not mathematics, it's theology". But they are happy to use it if it will help them prove theorems in their mathematical world which consists of real things like the natural or the complex numbers. To most mathematicians, questions like the validitity of the continuum hypothesis are simply not interesting. I'm a computer scientist, but I do have some knowledge of maths departments. How many set theorists do you know of? I have never met one.
The Riemann Hypothesis (RH) is either true or false. If it is false if and only if there is a counterexample. It is the mathematician's job to show if it true or false.
RH is not on the same plane as the Axiom of Choice, which is independent of the standard (ZF) axioms of set theory. This means that there are set theories (mathematical objects) with the Axiom of Choice true and others with it false. However, there is only one set of complex numbers (or the natural numbers).
The complex numbers or the natural numbers are mathematical objects. Godel's (second?) incompleteness theorem says only that we will never be able to come up with finitely many (actually, recursively enumerable) axioms that will have the natural numbers or the complex numbers as their only model. This is a failure of first-order logic, not of complex number theory or number theory.
>Theorys are rock solid
Spelling is shaky.
Doesn't anyone remember the article by Jon Katz in which he was emailing a ``geek'' just outside of Kabul who was downloading and playing movies on a C64?
I have absolutely no idea of what are you talking about.. :)
Still, I'm worried that when it comes to science people have a generally closed minded approach about "accepted" theories (see most of the posts above). After all, all the really large breakthroughs in science came from people who disagreed in what was "accepted" knowledge at the time. This can only be bad for science...
This is not as much a problem in Maths of CS as a proof is either correct or it isn't; there isn't much ground to discussion.
I'm still not convinced that evolution answers the question of where we come from and I am of course really curious. I just hope that there are some biologists who share that curiosity.
Come on, this is bullshit. You argument has degenerated into using another definition of the work "evolving". Surely you're not saying that the kernel hackers are applying random changes! The changes are (usually) well thought out and they try to fix a problem, surely not a random mutation...
I agree with everything you say, except you don't refute my main point. You agree with me that the linux kernel thing wouldn't work. So where exactly do we come from then?
Humans have developed intelligence because it gives a better chance of survival. Why is it then so inconceivable that a full linux kernel wouldn't magically appear after a few million generations of semi random code-tweaking?
Why? Because such a system gets incredibly complicated and small mutations just don't cut it. Evolution (this way defined) simply doesn't scale.
I am not a biologist. I am a computer scientist. However, I can explain to a 10 year old why P=NP would mean the death of RSA.
I would like to have a laymans explanation of how complex systems can be brought about by small mutations? Especially since importing the idea into something a
a coder would understand simply would not work. Maybe my idea is too simple, if so why don't we get a biologist to tell us how macro evolution works, get a Beowulf cluster and make some cool complex software by starting with a "Hello World" program and introducing small random changes?