Oh my god, you actually like Windows Media Player? A much better media player for Windows, that can play every media format (windows, quicktime, real, divx, xvid, etc) is Media Player Classic. Also, Media Player Classic is both free and opensource. It also is also faster than Windows Media Player. In fact, there is really no reason to not use Media Player Classic.
I have been using apt and Redhat/Fedora as my exclusive desktop for a year now. There are some notable differences from apt and Debian:
1. Debian's apt repositories contain far more software than Fedora repositories. For example, Debian testing has roughly 14000 packages, while all of the Fedora repositories pulled together have 4000 packages.
2. Debian's repositories have more strict, stable, and reliable standards. So installing software from them is less likely to mess up your system. With Fedora, you usually have to combine repositories because one repository alone lacks needed packages. This often causes conflicts between repositories.
3. Debian repositories are not just standardized, but they are mirrored across around the world. Each Fedora repository (FreshRPMs for example), is only available from a few mirrors.
So while apt itself might work the same on both distros, the repositories are what makes the difference. Debian's repositories offer better quality and quantity of packages in an organized in heavily mirrored fasion.
I should be honest. I have been toying with Debian Sarge Beta 3 on a few computers, over the past few weeks. Once Sarge goes stable, I am going to migrate my desktops from Fedora Core 1 to Sarge. It really is all about the repositories for me. The distro is the package repository.
If you are using a standard Debian repository, and there is a dependency that cannot be resolved, then you should report that bug to the Debian project. Somehow I get the impression that you were using something like a harddrive install of Knoppix with all of Knoppix custom non-standard repositories, and NOT a standard Debian repository.
Since when does Joe Sixpack install new hard disk drives, swap IDE cables, and manipulate BIOS settings to detect the new hard disk? If they can handle that, I am sure they can handle fstab.
You do not know what you are talking about. On Redhat you have apt and yum, for managing, installing, updating, and removing software. For Debian you have apt and comprehensive standardized repositories. Manrake, Gentoo, and Suse also have similar standardized software package management utilities and repositories.
What crack are you smoking? Debian's apt repositories are standardized: stable, testing, and unstable. For almost every piece of Linux software, that is roughly 15000 software packages, are available through the unstable repository within days of the software's release.
You are obviously talking out of your ass. Please stop commenting on something you have obviously never used.
How do we prove that the system is simply consistent relative to those axioms? We would have to use a larger system of proof than the one we are proving simply consistent. So how do we know the system of proof used to prove our system consistent is consistent? We use an even larger system of proof to show that our meta system is simply consistent?
So proving simple consistency is impossible because it requires the assumption that the even larger system used to prove consistency is itself consistent. But what about intuitive-semantic consistency? How do you suggest we prove that our system of symbol shuffling actually generates proofs of meaningful concepts? Trying to do this will result in an even bigger failure than trying to prove simple consistency.
I guess schools don't teach meta-mathematics anymore?
Mathematics is the study of pure ideas. When a mathematician creates a proof, they make sure, hopefully, that it represents something meaningful... even if they use symbols and paper to help them along, they create the ideas that thes symbols represent in their head.
A computer, as far as we can tell, is not Self aware, and does not actually think. It just shuffles symbols around. Do to meta-mathematical critique, we know that formal axiomatic systems cannot be proven consistent, let alone meaningful. So in a sense, it is a mistake to trust such an important human endevor to symbol shuffling. This critique was made not of artificial computers, but of human computors. That is, people doing math by shuffling symbols around according to a set of axioms and inference rules. So if a human computer is the wrong way to go about doing math, then an electronic computer is just as bad.
Now, I am all for using computers to aid mathematicians, just as maths have used pencil and paper for hundreds of years. The important thing is that computers cannot replace humans with regards to math. Meta-mathematics has proven this: undecidability, incompleteness, inconsistency, uncountability, etc...
Creative thought is required to do math. Symbol shuffling is only a tool used to help our weak human memory.
I disagree. For example, various constructive schools of math are founded on the concept that something is only true when it can be proven. So A=A is true in the sense that A exists. If A exists as a finite concrete concept, then it is obviously itself. Therefore the existence of A is proof that A=A. Denying that is similar to denying one's own existence.
For example, realizing that 1 is 1 requires only that 1 exists. For infinite objects, yes, faith must come into play as infinite conceptual objects never exist. They can never be created in their entirety.
In fact, various philosophical foundations of math such as: finitism, constructive recursive mathematics do not allow for infinite mathematical objects, i.e. it ain't a mathematical concept unless it is finite. This might seem like a strong restriction, but then again, this is math, we can't take things based on faith... that is religion.
Note also under these philosophies, various unwanted phenomenon do not occur, such as the need for an "Axiom of Extension" taken on faith, as you have stated. In fact, these schools or programmes of mathematics were founded in order to deal with the various mathematical and philosophical problems of classical, formal, axiomatic mathematics.
Actually, Godel's theorem says, and I can confirm this as I have Kleene's Introduction to Meta-mathematics right beside me. Here is an accurate paraphrasing:
Assuming that the aforementioned formalized axiomatic system of number theory is simply-consistent and omega-consistent, then the system is incomplete.
He later goes on to prove that it is impossible to prove that the system is simply-consistent or omega-consistent, without using the aforementioned theory and additional theory. In other words, consistency proofs are not possible without using an even larger system of math, but such a larger system of math can only be proven to be simply and omega consistent with an even larger system of math and so on.
The problem is that we have no way of knowing that the formalisms represent meaningful ideas, which is the whole point of mathematics to begin with. So do doubt we can feed a list for symbols to a computer, the computer can verify that they are constructed using certain rules and restrictions, and it can spit out more formalisms.
But how do we know that these formalisms have any meaning? How do we know that the formal proof expounds meaningful ideas?
Mathematics is surely not a game of symbol shuffling.
What about co-inductively proved? You do know that mathematical induction is a form of deductive proof? Also, Godel's Incompleteness Theorem only applies to formal axiomatic mathematics. There are other philosophies of math that are not plagued by such problems.
Anyway, what you describe is the mutation of math into just another science. The problem is that such a thing is NOT math, by very definition. Math doesn't sort-kinda prove that something might be true. If math ever starts to use proofs of that kind, then we are all doomed, as mankind will have lost mathematics. Then again, it might not be a bad thing as most people understand very little about mathematics to begin with. Scientists obviously don't.
How do we know that the formal axiomatic system used to construct the proof is simply consistent? Omega consistent? Intuitive-semantic consistent? Mathematics is dead, if it is reduced to a game of symbol manipulation. Mathematics is about the study of pure ideas, not a game of shuffling symbols.
It is impossible to prove that the formal axiomatic system, let alone the automated theorem prover, is simply, omega, or intuitive-semantic consistent. So the computer could just spit out formalisms which can be grinded out using the formal mathematical system... yet the symbols might not represent anything meaningful.
there are also several levels of depth for proofs, ranging from "i've found a counter example so i can write the whole thing off as garbage" to "i have exhaustively and rigorously proved this starting with the basic axioms of number theory and worked my way on up"
Counter-examples are rigorous proofs. If I want to show "it is absurd that for all x P(X)", then a counter-example is extremely rigorous as it is a constructive proof!
Also, if your proof uses other lemmas and theorems... all in intuitionistic logic, then proof normalization can be used to get a direct proof. I think your concept of mathematical rigor is incorrect. Formal constructive proofs are the most rigorous type of proofs, yet your explanation claims otherwise.
So you are saying that the foundation of math should rest of faith? Sounds like a religion to me... not math. Anyway, A=A does not need to be assumed. It is self apparent. No axiom is needed. In fact, the formal axiomatic approach to mathematics is just silly.
I think that it should also be pointed out that the formal axiomatic approach to mathematics is plagued with imperfections such as lack of proofs of simple, omega, and intuitive-semantic consistency. In the event of simple and omega consistency, such approach to mathematics is incomplete. So for all we know, such an approach to math could be incomplete and intuitive-semantically inconsistent.
Hence it would only be an incomplete game of symbols, that lets us "prove" meaningless things. But history shows that people are simple creatures that don't like to question the way popular mathematics is practiced or taught.
Click around on koreanair.com, especially "Sky Pass" and "Flight Schedule Display". After clicking a link, let the target page load and wait for at least 10 seconds. The browser should crash. I reported this bug a while ago, and it crashes Mozilla on Windows and Linux.
From the email updates from the bug report, it looks like a collection of people have recreated the bug, created a patch, tested the patch, etc...
So I guess it will be fixed by 1.7. They take crash bugs VERY seriously.
Slashdot Mirror
Oh my god, you actually like Windows Media Player? A much better media player for Windows, that can play every media format (windows, quicktime, real, divx, xvid, etc) is Media Player Classic. Also, Media Player Classic is both free and opensource. It also is also faster than Windows Media Player. In fact, there is really no reason to not use Media Player Classic.
Puzzle Pirates is another great game that runs on Linux.
I have been using apt and Redhat/Fedora as my exclusive desktop for a year now. There are some notable differences from apt and Debian:
1. Debian's apt repositories contain far more software than Fedora repositories. For example, Debian testing has roughly 14000 packages, while all of the Fedora repositories pulled together have 4000 packages.
2. Debian's repositories have more strict, stable, and reliable standards. So installing software from them is less likely to mess up your system. With Fedora, you usually have to combine repositories because one repository alone lacks needed packages. This often causes conflicts between repositories.
3. Debian repositories are not just standardized, but they are mirrored across around the world. Each Fedora repository (FreshRPMs for example), is only available from a few mirrors.
So while apt itself might work the same on both distros, the repositories are what makes the difference. Debian's repositories offer better quality and quantity of packages in an organized in heavily mirrored fasion.
I should be honest. I have been toying with Debian Sarge Beta 3 on a few computers, over the past few weeks. Once Sarge goes stable, I am going to migrate my desktops from Fedora Core 1 to Sarge. It really is all about the repositories for me. The distro is the package repository.
If you are using a standard Debian repository, and there is a dependency that cannot be resolved, then you should report that bug to the Debian project. Somehow I get the impression that you were using something like a harddrive install of Knoppix with all of Knoppix custom non-standard repositories, and NOT a standard Debian repository.
Since when does Joe Sixpack install new hard disk drives, swap IDE cables, and manipulate BIOS settings to detect the new hard disk? If they can handle that, I am sure they can handle fstab.
You do not know what you are talking about. On Redhat you have apt and yum, for managing, installing, updating, and removing software. For Debian you have apt and comprehensive standardized repositories. Manrake, Gentoo, and Suse also have similar standardized software package management utilities and repositories.
What crack are you smoking? Debian's apt repositories are standardized: stable, testing, and unstable. For almost every piece of Linux software, that is roughly 15000 software packages, are available through the unstable repository within days of the software's release.
You are obviously talking out of your ass. Please stop commenting on something you have obviously never used.
I am not sure when Debian got the 2.6 kernel, but its been installed on my Debian computer for a while now. It works great!
Too bad Nautilus still sucks. Why can't Nautilus manage my files that are on sftp or ssh shares?
How do we prove that the system is simply consistent relative to those axioms? We would have to use a larger system of proof than the one we are proving simply consistent. So how do we know the system of proof used to prove our system consistent is consistent? We use an even larger system of proof to show that our meta system is simply consistent?
So proving simple consistency is impossible because it requires the assumption that the even larger system used to prove consistency is itself consistent. But what about intuitive-semantic consistency? How do you suggest we prove that our system of symbol shuffling actually generates proofs of meaningful concepts? Trying to do this will result in an even bigger failure than trying to prove simple consistency.
I guess schools don't teach meta-mathematics anymore?
Mathematics is the study of pure ideas. When a mathematician creates a proof, they make sure, hopefully, that it represents something meaningful... even if they use symbols and paper to help them along, they create the ideas that thes symbols represent in their head.
A computer, as far as we can tell, is not Self aware, and does not actually think. It just shuffles symbols around. Do to meta-mathematical critique, we know that formal axiomatic systems cannot be proven consistent, let alone meaningful. So in a sense, it is a mistake to trust such an important human endevor to symbol shuffling. This critique was made not of artificial computers, but of human computors. That is, people doing math by shuffling symbols around according to a set of axioms and inference rules. So if a human computer is the wrong way to go about doing math, then an electronic computer is just as bad.
Now, I am all for using computers to aid mathematicians, just as maths have used pencil and paper for hundreds of years. The important thing is that computers cannot replace humans with regards to math. Meta-mathematics has proven this: undecidability, incompleteness, inconsistency, uncountability, etc...
Creative thought is required to do math. Symbol shuffling is only a tool used to help our weak human memory.
I disagree. For example, various constructive schools of math are founded on the concept that something is only true when it can be proven. So A=A is true in the sense that A exists. If A exists as a finite concrete concept, then it is obviously itself. Therefore the existence of A is proof that A=A. Denying that is similar to denying one's own existence.
For example, realizing that 1 is 1 requires only that 1 exists. For infinite objects, yes, faith must come into play as infinite conceptual objects never exist. They can never be created in their entirety.
In fact, various philosophical foundations of math such as: finitism, constructive recursive mathematics do not allow for infinite mathematical objects, i.e. it ain't a mathematical concept unless it is finite. This might seem like a strong restriction, but then again, this is math, we can't take things based on faith... that is religion.
Note also under these philosophies, various unwanted phenomenon do not occur, such as the need for an "Axiom of Extension" taken on faith, as you have stated. In fact, these schools or programmes of mathematics were founded in order to deal with the various mathematical and philosophical problems of classical, formal, axiomatic mathematics.
Actually, Godel's theorem says, and I can confirm this as I have Kleene's Introduction to Meta-mathematics right beside me. Here is an accurate paraphrasing:
Assuming that the aforementioned formalized axiomatic system of number theory is simply-consistent and omega-consistent, then the system is incomplete.
He later goes on to prove that it is impossible to prove that the system is simply-consistent or omega-consistent, without using the aforementioned theory and additional theory. In other words, consistency proofs are not possible without using an even larger system of math, but such a larger system of math can only be proven to be simply and omega consistent with an even larger system of math and so on.
So formal axiomatic math is problematic.
The problem is that we have no way of knowing that the formalisms represent meaningful ideas, which is the whole point of mathematics to begin with. So do doubt we can feed a list for symbols to a computer, the computer can verify that they are constructed using certain rules and restrictions, and it can spit out more formalisms.
But how do we know that these formalisms have any meaning? How do we know that the formal proof expounds meaningful ideas?
Mathematics is surely not a game of symbol shuffling.
What about co-inductively proved? You do know that mathematical induction is a form of deductive proof? Also, Godel's Incompleteness Theorem only applies to formal axiomatic mathematics. There are other philosophies of math that are not plagued by such problems.
Anyway, what you describe is the mutation of math into just another science. The problem is that such a thing is NOT math, by very definition. Math doesn't sort-kinda prove that something might be true. If math ever starts to use proofs of that kind, then we are all doomed, as mankind will have lost mathematics. Then again, it might not be a bad thing as most people understand very little about mathematics to begin with. Scientists obviously don't.
This has to be a troll.
How do we know that the formal axiomatic system used to construct the proof is simply consistent? Omega consistent? Intuitive-semantic consistent? Mathematics is dead, if it is reduced to a game of symbol manipulation. Mathematics is about the study of pure ideas, not a game of shuffling symbols.
It is impossible to prove that the formal axiomatic system, let alone the automated theorem prover, is simply, omega, or intuitive-semantic consistent. So the computer could just spit out formalisms which can be grinded out using the formal mathematical system... yet the symbols might not represent anything meaningful.
Counter-examples are rigorous proofs. If I want to show "it is absurd that for all x P(X)", then a counter-example is extremely rigorous as it is a constructive proof!
Also, if your proof uses other lemmas and theorems... all in intuitionistic logic, then proof normalization can be used to get a direct proof. I think your concept of mathematical rigor is incorrect. Formal constructive proofs are the most rigorous type of proofs, yet your explanation claims otherwise.
So you are saying that the foundation of math should rest of faith? Sounds like a religion to me... not math. Anyway, A=A does not need to be assumed. It is self apparent. No axiom is needed. In fact, the formal axiomatic approach to mathematics is just silly.
If your language lacks referential transparency, you can have A=A to be false.
I think that it should also be pointed out that the formal axiomatic approach to mathematics is plagued with imperfections such as lack of proofs of simple, omega, and intuitive-semantic consistency. In the event of simple and omega consistency, such approach to mathematics is incomplete. So for all we know, such an approach to math could be incomplete and intuitive-semantically inconsistent.
Hence it would only be an incomplete game of symbols, that lets us "prove" meaningless things. But history shows that people are simple creatures that don't like to question the way popular mathematics is practiced or taught.
Tell that to constructivists, to whom there is no truth, only proof.
Click around on koreanair.com, especially "Sky Pass" and "Flight Schedule Display". After clicking a link, let the target page load and wait for at least 10 seconds. The browser should crash. I reported this bug a while ago, and it crashes Mozilla on Windows and Linux.
From the email updates from the bug report, it looks like a collection of people have recreated the bug, created a patch, tested the patch, etc...
So I guess it will be fixed by 1.7. They take crash bugs VERY seriously.
The ability to sell closed source software created using opensource software has nothing to do with being free. It has more to do with leeching.