Actually, I run a Windows98SE box, for a gaming machine. USB support under Windows95,98, and 98SE is a crappy hack at best. It takes a few patches before you can get most devices to work correctly. Note that the patches are hidden deep down in the rats nest of Microsoft.com, which makes for great fun when initially setting up a Windows95,98,98SE box. Can't speak for WindowsME, as I wouldn't touch such an OS... its not as good as Windows98SE for gaming (gotta have win3.1 and DOS support).
You can peer review anything, but it doesn't make it a science just because you peer review it. Peer review is not an intrinsic part of most mathematical systems.
Well, most mathematicians would like to have an absolute undeniably correct system of knowledge, and not lasting forever would be a problem. Intuitionism solves these problems, but causes problems of difficulty of complex high-level proof and it causes problems when communcating between two mathematicians. If you ever study Brouwer's Intuitionism, you will see why it works, and why it has the mentioned flaws.
Dude, every library has interlibrary loans. You wouldn't believe how cool libraries really are. Working the system is worth it when it comes to libraries. Many libraries take interlibrary loan requests online. So pop a couple requests off, you get an email a week later, and then you pick up the books from your local library. Dope stuff, eh?
His development of a set theory and a measure theory that do not use the principle of the excluded middle are interesting, but they are simply alternative theories, no more and no less valid than those in popular usage. Believing blindly that his theories are correct and everything else is crap is simply foolish.
Who said anything about blind belief? Heyting's introductory book on Intuitionism gives a nice arguement for why Platonic and formalist mathematical philosophies are flawed. Basically, religious belief is what destroys the Platonic idealism as a foundation for mathematics, and the formalist foundations have been proved to be flawed by Church, Turing, and Godel, to name a few.
I do not blindly believe in Brouwer's Intuitionism. I justify its correctness through direct mental construction.
Also, to say that the intuitionistic theories are nothing more than alternative theories is an exclamation of stubborn ignorance. Things must be proved in mathematics, and if the method of proof is flawed at its foundations, then the traditional theories to which Brouwer developed alternatives are in fact wrong and in need of replacement. So, many theories in popular usage are less valid than alternative intuitionistic theories.
Now moving onto a comment on the begining of your post. After Brouwer finished his graduate dissertation on the foundations of mathematics, he realized that in order to get people to listen to his radical ideas, he would have to establish himself. Therefore, for several years, he did feverish work in topology, and even made use of flawed principles. Of course, he was just prostituting himself out in order to gain recognition. Once he had that recognition, he returned to his interest in his intuitionistic foundations for mathematics, ready to complete his goal of reconstructing mathematics. This reminds me of the old saying, "Do as I say, not as I do."
*cough* case in point *cough*
Greed implies taking more than your share (see Marxism). Knowledge is created within a person's mind, and therefore labelling such activity as greed is an abuse of the word. If I create it within myself, then how could it be taking away from somebody else's share?
I also never implied that I wanted to stop people from living their lives, as long as them doing so does not harm others. However, if gaining monopolistic power over an industry makes a person happy, yet harms thousands of people... I say that is a "wrong" way to live. If running around killing people makes someone happy, well I am sorry, but that is a "wrong" way to live.
Most people call my philosophy "libertarianism". I claim that the greedy violate the libertarian law by taking away from others.
Actually, not all of constructive mathematics is as you say. You have described a small subset of constructive mathematics called "constructive recursive mathematics and Markov's School" (page 25 in Constructivism in Mathematics, An Introduction by Troelstra and van Dalen). Constructive recursive mathematics formulates everything as an algorithm. However, there are other constructive mathematical systems such as the most famous Brouwer's Intuitionism and lesser systems such as Finitism, Bishop's Constructive Mathematics, etc...
You do bring up a good point though, that there are alternative mathematical foundations which start from a different philosophy and therefore aren't plagued with such problems pointed out by this topic. The problem is that your average joe blow never questions what he is taught, and therefore, we end up with masses of sheep who whole-heartedly believe that Platonic mathematics is absolutely and undeniably correct. What is even better is when you try to point out people's dogmatic mathematical beliefs, they call you a troll and mod you down into nothingness. It is too overwhelming for some people to realize that the mathematics that they have been taught is horribly flawed, and that better foundations/philosophies have been around for quite some time.
... and what did he say the number was exactly or did he never give an exact number because he made reference to non-constructive methods which have been questioned by mathematicians for 100s of years (Kronecker and Brouwer for example).
The academic definition of science is: that which follows the philosophy of positivism and uses the scientific method. Mathematics does not fit this definition.
Now, if you use the "joe blow" defintion for many different words, you will end up with totally different things. Take the joe blow definition of "hacker" for exmaple. How about the joe blow definition of "operating system", or "computer", or the "internet"? Do you get my point? Pulling out the dictionary only verifies that joe blow uses the particular word in that way.
Furthermore, I figured that we are discussing an academic topic, and therefore, I assumed the academic definition of "science".
Who ever said that the good books were cheap. One of the reasons that the book is so expensive is because it uses a sew-threw-the-fold binding, which is the most high quality book binding and the most desired too because the book lays flat when opened to any page. It is also the strongest binding.
In addition, considering that there are only 1 or 2 different books available on Brouwer's Intuitionism, other economic factors come into play.
If you disagree with the price, then ask your library for a copy. If they don't have it, they can interlibrary loan it for free. Remember, knowledge doesn't necessarily come easily.
Finally, Intuitionism is not a hoax. Search the net for info on "Brouwer Intuitionism". It is a very well established mathematical philosophy/system, which has been around for almost 100 years.
Nice troll. While you may believe these things, and I am not saying they are not true (your beliefs)... I am saying that the Platonic Idealism is a belief system... a religion. Just as the existance of God cannot be proved, neither can an extrenal objective absolute reality of ideal forms.
I have been studying the foundations of mathematics as a special person interest of mine. I am a CS major, so we aren't forced to learn such interesting things, which is sad.
Well, if you really think that I am right, can you spare some mod points;-) These guys are putting me into a hole. This always happens when mathematical topics come up.
Actually, you couldn't be more wrong. If integer arithematic cannot be proven to be consistant (free from contradiction), then there is the possibility that you could wake up one day and have 2+2=5. I am not claiming that any of this will happen, because I have no mathematical proof, but the whole problem with a lack of consistancy proof is the problem you have mentioned... waking up only to realize that your math was nothing more than a "Matrix" (in the movie sense) so to speak.
Time and time again, the chosen philosophy of mathematics used for a foundation of mathematics as shown to cause huge differences in the actual mathematical system. Bertrand Russell's paradoxes (A set which contains all sets that do not contain themselves. This set both contains itself and doesn't contain itself.), Brouwer's Intuitionism (mathematics is complete, and undeniably consistant in with this philosophy), the Platonic Idealism (you have foundations that are of the quality of the foundations of Christianity), Formalism (Godel's Incompleteness Theorem says that we can never know if this system is flawed or not), etc...
Saying that philosophy does influence mathematics down to a consistancy level ignores hundreds of years of mathematical history! Philosophical foundations can lead to actual contradictions. This is why philosophy has an extremely important role in mathematics.
Yeah, but try convincing the majority of people in the USA that greed is a bad thing, and that understanding is a good thing.
Look at Universities for instance. Most people are in Univeristy so that they can make more money in the industry. Most of these people can't even see another good reason for being in a University, besides: "It will get me more pay."
Why oh why did you have to get me started on this topic. I better stop typing before I get modded down even more. (If you go against the norm, you risk getting modded down.)
Please tell me I didn't have a point! I can't believe that I got modded down for making a good point that there are many things that are not science.
The moderation is really bad on for this article. Earlier I saw something get a +3, which was complete and total garbage about some prophecy of this by a writer.
This could be used to argue against the principle of Occam's Razor (which says that the simplest theory that fits the facts of a problem is the one that should be selected), because science is based on the belief that mathematical concepts can be usefully projected onto the perception of nature. If it turns out that the mathematics used is horribly complex and disconnected, then Occam's Razor could cause a scientist to turn from the truth more often than he/she is turned towards the truth by the principle.
Note that I use the term "truth" with regards to scientific "truth", realizing that science can never in fact portray any absolute truth, as is the normal definition of truth (i.e. undeniable truth). This is why science has evolutionary mechanisms built in like peer review and disproving old theories.
Of course it has helped throw more doubt on the current popular foundations of mathematics, just as Godel, Church, and Turing threw doubt on the quality of our popular mathematical foundations.THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES!
What you have just touched on is a philosophical issue. You for some reason believe in the Platonic idealism, that mathematical concepts exist independently of the mind. Without the existance of humans, you believe that mathematics still exists.
However, this belief has never been proven. It is nothing more than a belief, just like many people agree that their exists a God. Therefore, just as the belief in the existance of a God turns something into a religion; the belief in the Platonic idealism turns mathematics into a religion - rife with all of the problems associated with religions!
Great mathematicians such as L.E.J. Brouwer argued that such dogmatic beliefs should not be used within mathematics, because it causes horrible foundational problems of paradox, undecidability, and incompleteness. Brouwer went on to establish the mathematical philosophy of intuitionism, and then built an entire mathematical system ontop of that. In effect, he created mathematical intuitionism, just as each mathematician creates (or recreates depending on how you look at it) mathematical concepts in their mind.
The Platonic idealism has been a cancer on the foundation of mathematics for thousands of years. Please, stop and realize that the Platonic idealism is nothing more than a belief system, and witness how it has partially destroyed mathematics.
THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES!
If you liked Tentacle, take a look at this...
on
Leisure Suit Unix
·
· Score: 1
... Sam and Max, by LucasArts. Such a funny game, that you will wet-your-pants-laugh.
You know how some geeks try to be more elite by using old school hardware? I can just picture it now, someone trying to port Linux to a vaccuum tube computer, and having trouble writing drivers for their oscilloscope commandline display.
Anyway, vaccuum tools do look cool.
Re:Feynman Lectures on Computation
on
Quantum Computers
·
· Score: 1
Sorry, bad grammer usage. I didn't mean to say that the book was used by Feynman in his lectures. I meant to say that the lecture notes used to write the book, are the same notes that Feynman used during his famous 1980 lectures. Dangling modifiers always get me.
Slashdot Mirror
Actually, I run a Windows98SE box, for a gaming machine. USB support under Windows95,98, and 98SE is a crappy hack at best. It takes a few patches before you can get most devices to work correctly. Note that the patches are hidden deep down in the rats nest of Microsoft.com, which makes for great fun when initially setting up a Windows95,98,98SE box. Can't speak for WindowsME, as I wouldn't touch such an OS... its not as good as Windows98SE for gaming (gotta have win3.1 and DOS support).
You can peer review anything, but it doesn't make it a science just because you peer review it. Peer review is not an intrinsic part of most mathematical systems.
Well, most mathematicians would like to have an absolute undeniably correct system of knowledge, and not lasting forever would be a problem. Intuitionism solves these problems, but causes problems of difficulty of complex high-level proof and it causes problems when communcating between two mathematicians. If you ever study Brouwer's Intuitionism, you will see why it works, and why it has the mentioned flaws.
In the voice of Homer Simpson, "I'm just a man"!
Dude, every library has interlibrary loans. You wouldn't believe how cool libraries really are. Working the system is worth it when it comes to libraries. Many libraries take interlibrary loan requests online. So pop a couple requests off, you get an email a week later, and then you pick up the books from your local library. Dope stuff, eh?
Ever heard of a library? They let you use their books for free.
Sometimes the only way to get people to get someone's attention is to slap them around a little.
I do not blindly believe in Brouwer's Intuitionism. I justify its correctness through direct mental construction.
Also, to say that the intuitionistic theories are nothing more than alternative theories is an exclamation of stubborn ignorance. Things must be proved in mathematics, and if the method of proof is flawed at its foundations, then the traditional theories to which Brouwer developed alternatives are in fact wrong and in need of replacement. So, many theories in popular usage are less valid than alternative intuitionistic theories.
Now moving onto a comment on the begining of your post. After Brouwer finished his graduate dissertation on the foundations of mathematics, he realized that in order to get people to listen to his radical ideas, he would have to establish himself. Therefore, for several years, he did feverish work in topology, and even made use of flawed principles. Of course, he was just prostituting himself out in order to gain recognition. Once he had that recognition, he returned to his interest in his intuitionistic foundations for mathematics, ready to complete his goal of reconstructing mathematics. This reminds me of the old saying, "Do as I say, not as I do."
*cough* case in point *cough*
Greed implies taking more than your share (see Marxism). Knowledge is created within a person's mind, and therefore labelling such activity as greed is an abuse of the word. If I create it within myself, then how could it be taking away from somebody else's share?
I also never implied that I wanted to stop people from living their lives, as long as them doing so does not harm others. However, if gaining monopolistic power over an industry makes a person happy, yet harms thousands of people... I say that is a "wrong" way to live. If running around killing people makes someone happy, well I am sorry, but that is a "wrong" way to live.
Most people call my philosophy "libertarianism". I claim that the greedy violate the libertarian law by taking away from others.
Actually, not all of constructive mathematics is as you say. You have described a small subset of constructive mathematics called "constructive recursive mathematics and Markov's School" (page 25 in Constructivism in Mathematics, An Introduction by Troelstra and van Dalen). Constructive recursive mathematics formulates everything as an algorithm. However, there are other constructive mathematical systems such as the most famous Brouwer's Intuitionism and lesser systems such as Finitism, Bishop's Constructive Mathematics, etc...
You do bring up a good point though, that there are alternative mathematical foundations which start from a different philosophy and therefore aren't plagued with such problems pointed out by this topic. The problem is that your average joe blow never questions what he is taught, and therefore, we end up with masses of sheep who whole-heartedly believe that Platonic mathematics is absolutely and undeniably correct. What is even better is when you try to point out people's dogmatic mathematical beliefs, they call you a troll and mod you down into nothingness. It is too overwhelming for some people to realize that the mathematics that they have been taught is horribly flawed, and that better foundations/philosophies have been around for quite some time.
... and what did he say the number was exactly or did he never give an exact number because he made reference to non-constructive methods which have been questioned by mathematicians for 100s of years (Kronecker and Brouwer for example).
The academic definition of science is: that which follows the philosophy of positivism and uses the scientific method. Mathematics does not fit this definition.
Now, if you use the "joe blow" defintion for many different words, you will end up with totally different things. Take the joe blow definition of "hacker" for exmaple. How about the joe blow definition of "operating system", or "computer", or the "internet"? Do you get my point? Pulling out the dictionary only verifies that joe blow uses the particular word in that way.
Furthermore, I figured that we are discussing an academic topic, and therefore, I assumed the academic definition of "science".
Who ever said that the good books were cheap. One of the reasons that the book is so expensive is because it uses a sew-threw-the-fold binding, which is the most high quality book binding and the most desired too because the book lays flat when opened to any page. It is also the strongest binding.
In addition, considering that there are only 1 or 2 different books available on Brouwer's Intuitionism, other economic factors come into play.
If you disagree with the price, then ask your library for a copy. If they don't have it, they can interlibrary loan it for free. Remember, knowledge doesn't necessarily come easily.
Finally, Intuitionism is not a hoax. Search the net for info on "Brouwer Intuitionism". It is a very well established mathematical philosophy/system, which has been around for almost 100 years.
*cough* *cough* *choke*
:-) it doesn't mean that they don't know their math.
Ok, while readers of the book may be crappy programmers
Nice troll. While you may believe these things, and I am not saying they are not true (your beliefs)... I am saying that the Platonic Idealism is a belief system... a religion. Just as the existance of God cannot be proved, neither can an extrenal objective absolute reality of ideal forms.
I have been studying the foundations of mathematics as a special person interest of mine. I am a CS major, so we aren't forced to learn such interesting things, which is sad.
;-) These guys are putting me into a hole. This always happens when mathematical topics come up.
Well, if you really think that I am right, can you spare some mod points
Actually, you couldn't be more wrong. If integer arithematic cannot be proven to be consistant (free from contradiction), then there is the possibility that you could wake up one day and have 2+2=5. I am not claiming that any of this will happen, because I have no mathematical proof, but the whole problem with a lack of consistancy proof is the problem you have mentioned... waking up only to realize that your math was nothing more than a "Matrix" (in the movie sense) so to speak.
Time and time again, the chosen philosophy of mathematics used for a foundation of mathematics as shown to cause huge differences in the actual mathematical system. Bertrand Russell's paradoxes (A set which contains all sets that do not contain themselves. This set both contains itself and doesn't contain itself.), Brouwer's Intuitionism (mathematics is complete, and undeniably consistant in with this philosophy), the Platonic Idealism (you have foundations that are of the quality of the foundations of Christianity), Formalism (Godel's Incompleteness Theorem says that we can never know if this system is flawed or not), etc...
Saying that philosophy does influence mathematics down to a consistancy level ignores hundreds of years of mathematical history! Philosophical foundations can lead to actual contradictions. This is why philosophy has an extremely important role in mathematics.
Yeah, but try convincing the majority of people in the USA that greed is a bad thing, and that understanding is a good thing.
Look at Universities for instance. Most people are in Univeristy so that they can make more money in the industry. Most of these people can't even see another good reason for being in a University, besides: "It will get me more pay."
Why oh why did you have to get me started on this topic. I better stop typing before I get modded down even more. (If you go against the norm, you risk getting modded down.)
Please tell me I didn't have a point! I can't believe that I got modded down for making a good point that there are many things that are not science.
The moderation is really bad on for this article. Earlier I saw something get a +3, which was complete and total garbage about some prophecy of this by a writer.
Don't mod posts unless you understand the topics!
This could be used to argue against the principle of Occam's Razor (which says that the simplest theory that fits the facts of a problem is the one that should be selected), because science is based on the belief that mathematical concepts can be usefully projected onto the perception of nature. If it turns out that the mathematics used is horribly complex and disconnected, then Occam's Razor could cause a scientist to turn from the truth more often than he/she is turned towards the truth by the principle.
Note that I use the term "truth" with regards to scientific "truth", realizing that science can never in fact portray any absolute truth, as is the normal definition of truth (i.e. undeniable truth). This is why science has evolutionary mechanisms built in like peer review and disproving old theories.
Of course it has helped throw more doubt on the current popular foundations of mathematics, just as Godel, Church, and Turing threw doubt on the quality of our popular mathematical foundations.THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES!
What you have just touched on is a philosophical issue. You for some reason believe in the Platonic idealism, that mathematical concepts exist independently of the mind. Without the existance of humans, you believe that mathematics still exists.
However, this belief has never been proven. It is nothing more than a belief, just like many people agree that their exists a God. Therefore, just as the belief in the existance of a God turns something into a religion; the belief in the Platonic idealism turns mathematics into a religion - rife with all of the problems associated with religions!
Great mathematicians such as L.E.J. Brouwer argued that such dogmatic beliefs should not be used within mathematics, because it causes horrible foundational problems of paradox, undecidability, and incompleteness. Brouwer went on to establish the mathematical philosophy of intuitionism, and then built an entire mathematical system ontop of that. In effect, he created mathematical intuitionism, just as each mathematician creates (or recreates depending on how you look at it) mathematical concepts in their mind.
The Platonic idealism has been a cancer on the foundation of mathematics for thousands of years. Please, stop and realize that the Platonic idealism is nothing more than a belief system, and witness how it has partially destroyed mathematics.
THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES!
... Sam and Max, by LucasArts. Such a funny game, that you will wet-your-pants-laugh.
You know how some geeks try to be more elite by using old school hardware? I can just picture it now, someone trying to port Linux to a vaccuum tube computer, and having trouble writing drivers for their oscilloscope commandline display. Anyway, vaccuum tools do look cool.
Sorry, bad grammer usage. I didn't mean to say that the book was used by Feynman in his lectures. I meant to say that the lecture notes used to write the book, are the same notes that Feynman used during his famous 1980 lectures. Dangling modifiers always get me.