I should also point out that in addition to your claim of independence of Church and Turing's works and that Turing wasn't one of Church's doctoral students... your claim that Turing didn't show the equivalence of computability (using his machines) and lambda-definability is also wrong. Just read the original papers.
In my opinion, McCarthy just reinvented the lambda-calculus, or more accurately popularized a formal language to software engineers, that is roughly the same as the lambda-calculus.
Why is Alonzo Church so lacking in popularity and Turing and McCarthy both so popular? Oh well, the actual works of Turing and McCarthy are definitive evidence of the contribution of Church... even if people today turn a blind eye towards this evidence.
I guess this is thing happens all the time: Skolem and the incompleteness of formal mathematics, the Chinese and the printing press, Hitler and the extermination of semites. Godel, Gutenburg, and Bush are given credit for these things... but someone else did it long before them.
You are the one that is wrong. Church was the teacher of many of the first great computer scientists. Including Turing and Kleene. Your Wiki is wrong. Read Kleene's "Introduction to Metamathematics" or "Mathematical Logic" textbooks. I assure you that Kleene, one of Church's students and a founding father of computer science knows better than whoever threw together those Wiki pages.
Turing's work was somewhat independent from Church's, but Church had his paper published first, and this prevented Turing from getting his work published... as someone beat him to the punch. Turing later revised his paper, including references to Church's paper, after communication with Church, and then got it published. Turing then became one of Church's doctoral students. Church's work was, at the time, more mathematical, while at the time Turing's was more humanistic. Church approached computability in terms of how we formally describe calculated functions, while Turing dscribed computability in terms of an ideal mathematician - something similar to Brouwer's earlier work on Mathematical Intuitionism. Read Turing's original paper! His machine is model after a human. Read Brouwer's account of constructive mathematics, where he defines mathematics as that which can be computed by the ideal mathematician.
I really find it interesting that people time and again disassociate Turing's work with his teacher's. People confuse "Church's Thesis" with the "Church-Turing Thesis". It is either ignorance or people's inherent tendency to fail to see that the accomplishes of great people are due to a continuum of great people and great ideas. Where did Church get his motivation? David Hilbert and L.E.J Brouwer. Hilbert got his motivation from Brouwer, etc...
In a recent paper Alonzo Church[2] has introduced an idea of "effective calculability", which is equivalent to my "computability", but is very differently defined. Church also reaches similar conclusions about the Entscheidungsproblem.[3] The proof of equivalence between "computability" and "effective calculability" is outlined in an appendix to the present paper.
Is version 4 of ReiserFS done yet and in the new kernel? I am interested in seeing benchmarks of that puppy. Reiser seems to be interested in revolutionary improvements for the filesystem.
The significance of Turing's "machines" is not about inventing software. It was about re-affirming his PhD Supervisor's thesis of effectively computable. I don't expect non-CS people to understand this, but roughly 100 years ago in the field of metamathematics, there was a question as to the definition of computability. It had to be precise so that you could formally prove things to be computable or uncomputable.
Originally there were attempts such as primitive recursive functions, but they were shown to not encompass all that is computable because they wouldn't allow you to "program" certain fuctions, e.g. Ackermann's function.
Then came partial recursive functions and the lambda-calculus. Church formulated his thesis: if something is computable it could be "programmed" in either the lambda-calculus programming language or the partial recursive functions programming language. (note I am trying to dumb this down with modern computer terms) Both computational models or programming languages were equivalent, as an interpreter could be written for one, in the other.
Church's student, Turing, wanted to come up with a more humanistic model of effectively computable. He decided to abstractly and formally define a human mathematician calculating something. Humans do this by using some amount of paper, a writing/erasing tool for creating/destroying symbols on the paper, and a finite set of rules for carrying out the calculation. Thus you have a Turing machine.
Turing went on to show that you could write an interpreter/compiler for the lambda-calculus that ran on a Turing machine. This showed that the Turing machine was just as computationally powerful as the lambda-calculus and therefore general recursive functions. The other direction was also shown to hold: Turing machines could be emulated in the lambda-calculus.
Therefore, Turing re-affirmed in a more humanistic terms that his teacher, Church was right: something is computable if and only if it can be programmed in the lambda-calculus.
These kinds of questions: "what is computable"? Are extremely important in computer science and mathematics. You have to know what you are working with, which involves knowing when it starts and where it ends.
Making computers and software is also an important thing, but Turing et al are famous for the metamathematics questions that they answered... not for computer or software engineering.
That movie should be required viewing for Freshmen CS and SE students. It is a classic, and anyone that has worked during the dot-com in an IT companies will watch the movie and say "thats so true", "I knew a guy like that", etc...
I have used Wine on my desktop for about a year now, and I have noticed that while some applications work flawlessly with Wine, many other applications require copying DLLs from a Windows install - which dramatically increases compatibility, but is legally ambiguous. Is a 100% OSS implementation an absolute goal for the Wine developers, or is their a tendency to stop short of implementing full compatibility because the user can just "borrow" DLLs to cover the compatibility gap?
A great free NES emulator has been available for the GBA for years now. Back then Nintendo would even claim that emulation was illegal... now they are cashing in? I still like the idea of a NES-themed GBA. I will buy that and a nice flash cart to use with the PocketNES emulator.
There is a difference between actual infinity and potential infinity. The distinction has been brought up time and again in the study of mathematical foundations. Constructivists, for example, deny actual infinity, and only allow for things as large as unbounded in size. Classicists do allow actual infinities, but these tend to lead to paradoxes. For example, the set of all functions on naturals can be proven to be both uncountably infinite and countably infinite.
So in short, infinite tape implies that at any given moment you have an infinite amount of tape. While unbounded tape implies that at any given time you have a _finite_ amount of tape... but at any given time you can increase the amount of tape by a finite increment.
Now, whether or not the universe is finite in storage capacity is a question that I don't think we can answer.
Set theory and formal logic are branches of mathematics. You say math isn't important, but then name two fields of mathematics that are important. Which one is it? Important or not important?
A Turing machine doesn't have an infinite tape. It has a tape of unbounded size, and at any given time, it has only used a finite amount of tape. Hence, a computer could be considered a Turing machine as long as more memory can be added to the computer as needed. This could be accomplished via a connection to an ever expanding internet.
Maybe there is no dark matter. Science only describe predicted observations. Reality doesn't necessarily obey the laws of science. Belief in such is similar to belief in a deity. Maybe the universe is governed by the laws of science, but then again, maybe it is governed by such-n-such a deity.
So if a theory isn't cutting it, then create a new model of whatever observation that you are trying to describe. It seems silly to try to fit nature to the theory, and not the theory to nature.
Yeah, like I saw this guy, and like he was playing this really old game called "chess". What a total loser, like, everyone knows that it is only cool to play Playstation 2 games. Like really, don't ya know? Old games aren't any fun.
Yeah, really insightful post. I just installed a beta of the upcoming Redhat Fedora distro, and all I did was click "next" in the installer. Everything worked and I was surfing the web immediately after install.
I guess some people will never be happy with Linux until it becomes Windows, OSX, and includes everyting plus a kitchen sink.
However, even after Linux reaches that level... people will still claim that it is hard to install and use.
WTF?
So you have a story of when you tried to install Linux on unsupported hardware, and stuff didn't work right? Hell, I can try to install Windows on a Sparcstation, and it won't even boot the installer, let alone run the OS.
Does that mean that Windows is hard to install? Does that mean that Windows is hard to use?
About a year ago, I did the switch. Cold turkey, never used Windows since that day, never looked back since the switch. All of my desktops and servers run Redhat/Fedora. In fact, right now I have a box with Redhat 9, a laptop with Fedora Core 1, and the computer I am typing this comment from is a Fedora Core 2 test 3 install... just finished the install today, btw. Each install is a mostly default workstation install.
With each release, there have been obvious dramatic improvements, from more useful features to performance improvements to bug fixes. Just to give an example of the improvements, I have recently been toying with Debian Sarge Beta 3... I was getting sick of Gnome 2.4, the slowness and buginess of Nautilus, etc... I also didn't like the small Fedora apt repositories.
I was planning on switching to Debian and KDE....
Today I downloaded and installed Fedora Core 2 test 3, just to give Redhat one last chance. Wow! Nautilus is really frickin fast! In fact, the entire desktop is extremely fast! The Evolution email client opens instantly, Nautilus windows open instantly, its very impressive.
Is it the new 2.6.x kernel included in Fedora Core 2? Is it the new Gnome 2.6 desktop? I don't care what it is, the fact is that I have a very coherent "desktop experience" with this latest Fedora Core 2 release candidate from install to posting on Slashdot:) The fact that I have been accustomed to the Redhat Bluecurve Gnome desktop and the fact that such huge improvements have been made have convinced me to stick with Redhat......well... as with everything in the OSS world, I will stick with it as long as there isn't a better free alternative. Hence the beauty of OSS. It is good to be critical of the distros, and it is healthy to consider alternatives. Try not to be biased, and use the distro that works for you.
If you need rock hard stability, go with Debian stable. If you want a coherent desktop experience, then one good option is Redhat's Fedora. Yes there are others, but at least from my experiences... Fedora is a damn good choice!
FLAC is great! I rip all of my store bought CDs to FLAC. Sure the file sizes are larger than lossy formats, but if you are going to pay for the music, shouldn't it be stored in a high quality format?
Penrose probably doesn't even understand what Godel's Incompleteness Theorem means. Hell, the fact that most people credit Godel with proving the incompleteness of formal axiomatic mathematics is proof that most people don't know math. Skolem's paradox can be seen as the first incompleteness proof and it predates Godel's proof.
Skolem's paradox basically boils down to the fact that within axiomatic set theory you can prove the existence of uncountable sets, but if a model exists for the theory then it is countable. Hence according to the axiomatic system, there exist uncountably many sets, but the system can really only manifest countably many sets. This is similar to how Godel showed that formal arithmetic cannot manifest proofs for every proposition the formal axiomatic system claims exist.
While Skolem's paradox is held amongst by many mathematicians as the first incompleteness proof, in my opinion, this proves something much more significant... it proves that axiomatic set theory is semantically inconsistent.
Set theory has been problematic from the start, and even after axiomatizing it in order to avoid simple inconsistency... set theory is still plagued with inconsistency of a different form: the formalisms don't hold any meaning - they are semantically inconsistent. What the formalisms say contradicts what the formalisms mean.
The problem is that lots of people don't understand math, and even many of those that do understand it, love it so much that they are unwilling to give up the flawed parts.
For me, mathemathics is constructive recursive mathematics:) No inconsistency here. No incompleteness here. All math is computable!
So does this imply that all physical computers are finite state machines? Even when connected to the internet, their total number of computational states are finite, though extremely large, and therefore Universal Turing Machines are only a mathematical construct.
Actually, scientific evidence shows that the New Testament didn't exist until around 300c.e. Furthermore, the "bibles" found during these times contained books no longer included in the bibles of today. Furthermore, the Gospels are different from today's. In fact, the Gospels changed drastically from 50c.e, until the Gospel of John was written in roughly 100c.e. This is a good source for a scientific account of the Gospels.
Should I even get into the fact that many of the writings of Paul outright contradict the Gospels?
The PAL slowness is actually quite common amongst old PAL console games, and all of the other "implementation mistakes" for SMK are harmless. The being able to move after losing is actually quite charming as it lets the loser vent some frustration of losing by getting a few red shells and shooting their opponent, who is invunerable.
There is one interesting bug that I see happen on a relatively regular basis, maybe once every 20 battle mode matches... but I can't seem to reproduce on demand. This bug involves the fact that one player can sometimes not be hit by the other player's kart or items.
The most obvious case of this bug was when I hit my opponent with a red shell, then I immediately picked up another red shell and shot it at my opponent who still had one balloon left.
This red shell didn't hit them, but instead started to vibrate inside of their player's sprite... for a couple seconds.
The problem is that I picked up a star quickly after shooting the second red, and I activated it and was driving straight at my opponent. I figured if the red wouldn't work, then I would finish the job myself. I drove my kart straight into my opponent's, but instead of hitting him, I passed straight through him and the red vibrating inside of him was destroyed by my star.
Note that this all took place within 3-5 seconds of time. So it all happened really fast (yes I am a combo fiend), but there were 4 people watching the game and they all agree on what happened. My opponent's avatar wasn't solid after being hit by the first red shell.
This bug doesn't last very long though. An avatar is only not solid for a few seconds, after which they can be hit. Sometimes reds are shot at such players and they vibrate inside them for a few seconds, and if the player doesn't somehow shake the red in that time, it hits them.
When I say that the red shell vibrates inside the avatar, it is just like when a red is fired at a player that has already won the game.
OO.o's main weakness is documents with lots of mathematical symbols. The fonts get all messed up when printed or converted between Microsoft formats and OO.o's formats. Of course, LyX is a better option for such documents, as it is based on LateX... which was designed with mathematical and scientific documents in mind.
Gnome's file manager, which I have been using for a year now, as I have been using Gnome on Linux as my only desktop for a year... is complete crap. It is slow, crashes often, and often fails to render the contents of a directory.
Gnome cannot browse SSH shares, while KDE can work with any filesystem, both local and remote, as if the filesystem was local. For example, I can open up a text file on an FTP server in a KDE text editor, modify it, and save it just as I would if the file existed in my local home directory! In fact, this generality of file management exists in all KDE apps.
Gnome on the other hand. Well I would have to use an FTP application such as gFTP to download the text file to my local filesystem. Then I would have to use Gedit to open the copy residing on my local filesystem. After modifying the file, I would save it back to the local filesystem, and then I would have to use gFTP again to upload the modified copy up to the FTP server.
With KDE, this kind of thing can be done with any file type, any KDE app, and any remote filesystem! A desktop should help a user manage their files. KDE does a better job of this than Gnome.
I know this because I use Gnome on a daily basis. I know this because I am experimenting with KDE on another computer. Yes I am days away from switching to KDE 3.2, but I am waiting for it to make its way into Debian testing. Right now it is only in Debian unstable.
I have been using Gnome on Linux as my only desktop for a year now. Gnome has many problems, and I have been experimenting with KDE on spare computer. One very simple example of Gnome's problems is that files on NFS, SMB, FTP, etc... are functionally different from files on the local disk. So if I want to play an MP3 on that resides on my file server, I have to copy it to my desktop, under Gnome, and then play it... while with KDE I can simple browse to the file and play it.
And even for browsing the local filesystem, Nautilus is a usability nightmare! It is slow, often crashes, and many times fails to render the contents of a directory.
While the SMK tracks exist in some ported form in Super Circuit, the physical gameplay is different. Battle mode is different. The way the cars handle, the shells handle, etc... they are all different. Of course, I have been playing this game for over 10 years, so I wouldn't doubt that there are things that stick out like a sore thumb to me, which other gamers can't see.
In short, no other Mario Kart game has come close to SMK's battle mode. I could go into detail as to why, but I will leave that for a later time.
No, my argument is founded in logic. I am aware of my Self. It needs no more proof to me. As far as I know, this is the definition of being self-aware. I can prove it to myself, just as I can prove that 1=1. There is no need for passion to prove such a thing to myself. However, to prove that other people have souls, or that god made my soul, etc... those things are based on passion.
Almost by definition, science cannot measure the Self. It can only measure outward side-effects that we associate with the presence of our own Self. However, there is a chance that in the future we could somehow confirm the existence of "Selfs" other than our own.
The problem is that proof is a subjective thing. Proof of the existence of my Self is immediate, axiomatic - the first undeniable law... however to other people it is not immediate. As far as readers of my post know, I am just another person claiming that they have a soul... we could all be lying, we could all just be chemical machines that spout such verbal claims, even though no such thing exists. Hell, I could be a chat-bot written in C++.
So while I cannot prove the existence of my Self to you, it is proven to me. I don't know why it is axiomatic, I don't believe it, but I know it is true... again, just as I know that 1=1. Could you prove that 1=1?
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I should also point out that in addition to your claim of independence of Church and Turing's works and that Turing wasn't one of Church's doctoral students... your claim that Turing didn't show the equivalence of computability (using his machines) and lambda-definability is also wrong. Just read the original papers.
In my opinion, McCarthy just reinvented the lambda-calculus, or more accurately popularized a formal language to software engineers, that is roughly the same as the lambda-calculus.
Why is Alonzo Church so lacking in popularity and Turing and McCarthy both so popular? Oh well, the actual works of Turing and McCarthy are definitive evidence of the contribution of Church... even if people today turn a blind eye towards this evidence.
I guess this is thing happens all the time: Skolem and the incompleteness of formal mathematics, the Chinese and the printing press, Hitler and the extermination of semites. Godel, Gutenburg, and Bush are given credit for these things... but someone else did it long before them.
Turing's work was somewhat independent from Church's, but Church had his paper published first, and this prevented Turing from getting his work published... as someone beat him to the punch. Turing later revised his paper, including references to Church's paper, after communication with Church, and then got it published. Turing then became one of Church's doctoral students. Church's work was, at the time, more mathematical, while at the time Turing's was more humanistic. Church approached computability in terms of how we formally describe calculated functions, while Turing dscribed computability in terms of an ideal mathematician - something similar to Brouwer's earlier work on Mathematical Intuitionism. Read Turing's original paper! His machine is model after a human. Read Brouwer's account of constructive mathematics, where he defines mathematics as that which can be computed by the ideal mathematician.
I really find it interesting that people time and again disassociate Turing's work with his teacher's. People confuse "Church's Thesis" with the "Church-Turing Thesis". It is either ignorance or people's inherent tendency to fail to see that the accomplishes of great people are due to a continuum of great people and great ideas. Where did Church get his motivation? David Hilbert and L.E.J Brouwer. Hilbert got his motivation from Brouwer, etc...
I mean, did you even read the online version of Turing's paper that is linked from your Wiki page? I quote Alan Turing:
Is version 4 of ReiserFS done yet and in the new kernel? I am interested in seeing benchmarks of that puppy. Reiser seems to be interested in revolutionary improvements for the filesystem.
The significance of Turing's "machines" is not about inventing software. It was about re-affirming his PhD Supervisor's thesis of effectively computable. I don't expect non-CS people to understand this, but roughly 100 years ago in the field of metamathematics, there was a question as to the definition of computability. It had to be precise so that you could formally prove things to be computable or uncomputable.
Originally there were attempts such as primitive recursive functions, but they were shown to not encompass all that is computable because they wouldn't allow you to "program" certain fuctions, e.g. Ackermann's function.
Then came partial recursive functions and the lambda-calculus. Church formulated his thesis: if something is computable it could be "programmed" in either the lambda-calculus programming language or the partial recursive functions programming language. (note I am trying to dumb this down with modern computer terms) Both computational models or programming languages were equivalent, as an interpreter could be written for one, in the other.
Church's student, Turing, wanted to come up with a more humanistic model of effectively computable. He decided to abstractly and formally define a human mathematician calculating something. Humans do this by using some amount of paper, a writing/erasing tool for creating/destroying symbols on the paper, and a finite set of rules for carrying out the calculation. Thus you have a Turing machine.
Turing went on to show that you could write an interpreter/compiler for the lambda-calculus that ran on a Turing machine. This showed that the Turing machine was just as computationally powerful as the lambda-calculus and therefore general recursive functions. The other direction was also shown to hold: Turing machines could be emulated in the lambda-calculus.
Therefore, Turing re-affirmed in a more humanistic terms that his teacher, Church was right: something is computable if and only if it can be programmed in the lambda-calculus.
These kinds of questions: "what is computable"? Are extremely important in computer science and mathematics. You have to know what you are working with, which involves knowing when it starts and where it ends.
Making computers and software is also an important thing, but Turing et al are famous for the metamathematics questions that they answered... not for computer or software engineering.
That movie should be required viewing for Freshmen CS and SE students. It is a classic, and anyone that has worked during the dot-com in an IT companies will watch the movie and say "thats so true", "I knew a guy like that", etc...
I have used Wine on my desktop for about a year now, and I have noticed that while some applications work flawlessly with Wine, many other applications require copying DLLs from a Windows install - which dramatically increases compatibility, but is legally ambiguous. Is a 100% OSS implementation an absolute goal for the Wine developers, or is their a tendency to stop short of implementing full compatibility because the user can just "borrow" DLLs to cover the compatibility gap?
A great free NES emulator has been available for the GBA for years now. Back then Nintendo would even claim that emulation was illegal... now they are cashing in? I still like the idea of a NES-themed GBA. I will buy that and a nice flash cart to use with the PocketNES emulator.
There is a difference between actual infinity and potential infinity. The distinction has been brought up time and again in the study of mathematical foundations. Constructivists, for example, deny actual infinity, and only allow for things as large as unbounded in size. Classicists do allow actual infinities, but these tend to lead to paradoxes. For example, the set of all functions on naturals can be proven to be both uncountably infinite and countably infinite.
So in short, infinite tape implies that at any given moment you have an infinite amount of tape. While unbounded tape implies that at any given time you have a _finite_ amount of tape... but at any given time you can increase the amount of tape by a finite increment.
Now, whether or not the universe is finite in storage capacity is a question that I don't think we can answer.
Set theory and formal logic are branches of mathematics. You say math isn't important, but then name two fields of mathematics that are important. Which one is it? Important or not important?
A Turing machine doesn't have an infinite tape. It has a tape of unbounded size, and at any given time, it has only used a finite amount of tape. Hence, a computer could be considered a Turing machine as long as more memory can be added to the computer as needed. This could be accomplished via a connection to an ever expanding internet.
Maybe there is no dark matter. Science only describe predicted observations. Reality doesn't necessarily obey the laws of science. Belief in such is similar to belief in a deity. Maybe the universe is governed by the laws of science, but then again, maybe it is governed by such-n-such a deity.
So if a theory isn't cutting it, then create a new model of whatever observation that you are trying to describe. It seems silly to try to fit nature to the theory, and not the theory to nature.
Yeah, like I saw this guy, and like he was playing this really old game called "chess". What a total loser, like, everyone knows that it is only cool to play Playstation 2 games. Like really, don't ya know? Old games aren't any fun.
Yeah, really insightful post. I just installed a beta of the upcoming Redhat Fedora distro, and all I did was click "next" in the installer. Everything worked and I was surfing the web immediately after install.
I guess some people will never be happy with Linux until it becomes Windows, OSX, and includes everyting plus a kitchen sink.
However, even after Linux reaches that level... people will still claim that it is hard to install and use.
WTF?
So you have a story of when you tried to install Linux on unsupported hardware, and stuff didn't work right? Hell, I can try to install Windows on a Sparcstation, and it won't even boot the installer, let alone run the OS.
Does that mean that Windows is hard to install? Does that mean that Windows is hard to use?
About a year ago, I did the switch. Cold turkey, never used Windows since that day, never looked back since the switch. All of my desktops and servers run Redhat/Fedora. In fact, right now I have a box with Redhat 9, a laptop with Fedora Core 1, and the computer I am typing this comment from is a Fedora Core 2 test 3 install... just finished the install today, btw. Each install is a mostly default workstation install.
...
:) The fact that I have been accustomed to the Redhat Bluecurve Gnome desktop and the fact that such huge improvements have been made have convinced me to stick with Redhat... ...well... as with everything in the OSS world, I will stick with it as long as there isn't a better free alternative. Hence the beauty of OSS. It is good to be critical of the distros, and it is healthy to consider alternatives. Try not to be biased, and use the distro that works for you.
With each release, there have been obvious dramatic improvements, from more useful features to performance improvements to bug fixes. Just to give an example of the improvements, I have recently been toying with Debian Sarge Beta 3... I was getting sick of Gnome 2.4, the slowness and buginess of Nautilus, etc... I also didn't like the small Fedora apt repositories.
I was planning on switching to Debian and KDE.
Today I downloaded and installed Fedora Core 2 test 3, just to give Redhat one last chance. Wow! Nautilus is really frickin fast! In fact, the entire desktop is extremely fast! The Evolution email client opens instantly, Nautilus windows open instantly, its very impressive.
Is it the new 2.6.x kernel included in Fedora Core 2? Is it the new Gnome 2.6 desktop? I don't care what it is, the fact is that I have a very coherent "desktop experience" with this latest Fedora Core 2 release candidate from install to posting on Slashdot
If you need rock hard stability, go with Debian stable. If you want a coherent desktop experience, then one good option is Redhat's Fedora. Yes there are others, but at least from my experiences... Fedora is a damn good choice!
FLAC is great! I rip all of my store bought CDs to FLAC. Sure the file sizes are larger than lossy formats, but if you are going to pay for the music, shouldn't it be stored in a high quality format?
Penrose probably doesn't even understand what Godel's Incompleteness Theorem means. Hell, the fact that most people credit Godel with proving the incompleteness of formal axiomatic mathematics is proof that most people don't know math. Skolem's paradox can be seen as the first incompleteness proof and it predates Godel's proof.
:) No inconsistency here. No incompleteness here. All math is computable!
Skolem's paradox basically boils down to the fact that within axiomatic set theory you can prove the existence of uncountable sets, but if a model exists for the theory then it is countable. Hence according to the axiomatic system, there exist uncountably many sets, but the system can really only manifest countably many sets. This is similar to how Godel showed that formal arithmetic cannot manifest proofs for every proposition the formal axiomatic system claims exist.
While Skolem's paradox is held amongst by many mathematicians as the first incompleteness proof, in my opinion, this proves something much more significant... it proves that axiomatic set theory is semantically inconsistent.
Set theory has been problematic from the start, and even after axiomatizing it in order to avoid simple inconsistency... set theory is still plagued with inconsistency of a different form: the formalisms don't hold any meaning - they are semantically inconsistent. What the formalisms say contradicts what the formalisms mean.
The problem is that lots of people don't understand math, and even many of those that do understand it, love it so much that they are unwilling to give up the flawed parts.
For me, mathemathics is constructive recursive mathematics
So does this imply that all physical computers are finite state machines? Even when connected to the internet, their total number of computational states are finite, though extremely large, and therefore Universal Turing Machines are only a mathematical construct.
Actually, scientific evidence shows that the New Testament didn't exist until around 300c.e. Furthermore, the "bibles" found during these times contained books no longer included in the bibles of today. Furthermore, the Gospels are different from today's. In fact, the Gospels changed drastically from 50c.e, until the Gospel of John was written in roughly 100c.e. This is a good source for a scientific account of the Gospels.
Should I even get into the fact that many of the writings of Paul outright contradict the Gospels?
The PAL slowness is actually quite common amongst old PAL console games, and all of the other "implementation mistakes" for SMK are harmless. The being able to move after losing is actually quite charming as it lets the loser vent some frustration of losing by getting a few red shells and shooting their opponent, who is invunerable.
There is one interesting bug that I see happen on a relatively regular basis, maybe once every 20 battle mode matches... but I can't seem to reproduce on demand. This bug involves the fact that one player can sometimes not be hit by the other player's kart or items.
The most obvious case of this bug was when I hit my opponent with a red shell, then I immediately picked up another red shell and shot it at my opponent who still had one balloon left.
This red shell didn't hit them, but instead started to vibrate inside of their player's sprite... for a couple seconds.
The problem is that I picked up a star quickly after shooting the second red, and I activated it and was driving straight at my opponent. I figured if the red wouldn't work, then I would finish the job myself. I drove my kart straight into my opponent's, but instead of hitting him, I passed straight through him and the red vibrating inside of him was destroyed by my star.
Note that this all took place within 3-5 seconds of time. So it all happened really fast (yes I am a combo fiend), but there were 4 people watching the game and they all agree on what happened. My opponent's avatar wasn't solid after being hit by the first red shell.
This bug doesn't last very long though. An avatar is only not solid for a few seconds, after which they can be hit. Sometimes reds are shot at such players and they vibrate inside them for a few seconds, and if the player doesn't somehow shake the red in that time, it hits them.
When I say that the red shell vibrates inside the avatar, it is just like when a red is fired at a player that has already won the game.
What about donations? Heck, the BitTorrent creator made some good money on donations alone. Why wouldn't the same thing work with Gentoo?
OO.o's main weakness is documents with lots of mathematical symbols. The fonts get all messed up when printed or converted between Microsoft formats and OO.o's formats. Of course, LyX is a better option for such documents, as it is based on LateX... which was designed with mathematical and scientific documents in mind.
Gnome's file manager, which I have been using for a year now, as I have been using Gnome on Linux as my only desktop for a year... is complete crap. It is slow, crashes often, and often fails to render the contents of a directory.
Gnome cannot browse SSH shares, while KDE can work with any filesystem, both local and remote, as if the filesystem was local. For example, I can open up a text file on an FTP server in a KDE text editor, modify it, and save it just as I would if the file existed in my local home directory! In fact, this generality of file management exists in all KDE apps.
Gnome on the other hand. Well I would have to use an FTP application such as gFTP to download the text file to my local filesystem. Then I would have to use Gedit to open the copy residing on my local filesystem. After modifying the file, I would save it back to the local filesystem, and then I would have to use gFTP again to upload the modified copy up to the FTP server.
With KDE, this kind of thing can be done with any file type, any KDE app, and any remote filesystem! A desktop should help a user manage their files. KDE does a better job of this than Gnome.
I know this because I use Gnome on a daily basis. I know this because I am experimenting with KDE on another computer.
Yes I am days away from switching to KDE 3.2, but I am waiting for it to make its way into Debian testing. Right now it is only in Debian unstable.
I have been using Gnome on Linux as my only desktop for a year now. Gnome has many problems, and I have been experimenting with KDE on spare computer. One very simple example of Gnome's problems is that files on NFS, SMB, FTP, etc... are functionally different from files on the local disk. So if I want to play an MP3 on that resides on my file server, I have to copy it to my desktop, under Gnome, and then play it... while with KDE I can simple browse to the file and play it.
And even for browsing the local filesystem, Nautilus is a usability nightmare! It is slow, often crashes, and many times fails to render the contents of a directory.
While the SMK tracks exist in some ported form in Super Circuit, the physical gameplay is different. Battle mode is different. The way the cars handle, the shells handle, etc... they are all different. Of course, I have been playing this game for over 10 years, so I wouldn't doubt that there are things that stick out like a sore thumb to me, which other gamers can't see.
In short, no other Mario Kart game has come close to SMK's battle mode. I could go into detail as to why, but I will leave that for a later time.
No, my argument is founded in logic. I am aware of my Self. It needs no more proof to me. As far as I know, this is the definition of being self-aware. I can prove it to myself, just as I can prove that 1=1. There is no need for passion to prove such a thing to myself. However, to prove that other people have souls, or that god made my soul, etc... those things are based on passion.
Almost by definition, science cannot measure the Self. It can only measure outward side-effects that we associate with the presence of our own Self. However, there is a chance that in the future we could somehow confirm the existence of "Selfs" other than our own.
The problem is that proof is a subjective thing. Proof of the existence of my Self is immediate, axiomatic - the first undeniable law... however to other people it is not immediate. As far as readers of my post know, I am just another person claiming that they have a soul... we could all be lying, we could all just be chemical machines that spout such verbal claims, even though no such thing exists. Hell, I could be a chat-bot written in C++.
So while I cannot prove the existence of my Self to you, it is proven to me. I don't know why it is axiomatic, I don't believe it, but I know it is true... again, just as I know that 1=1. Could you prove that 1=1?