1/ Words such as "absurd" and "existance" or "doesn't work" do not have much place in my own understanding of mathematics.
Absurdity of a construct is the fact that it leads to a contradiction. The actual word absurdity has been used by many mathematicians who wanted to describe a certain construct as being "wrong". Furthur more, I make no claims that my use of natural language will properly communicate my thoughts.
2/ The proof you gave, at best, proves that the 'x' solution of the above is not a natural number. The exact same reasoning you gave proves that there is no solution to x+1 = 0. Do you deny the "existence" (whatever meaning you give to that) of the number -1?
The proof of such extensions is fairly immediate, as was the one I gave. I mean, the proofs are so trivial in the first place, that the inductive proof that I gave can be easily seen as working for negative numbers. I am not going to be pull a Bertrand Russell and write a 10 page proof of formalisms for something so immediate. I made the point that the existance of such a number x such that x*0=1 is absurd - absurdity meaning that it is contradictory. Note that no real mathematician would ever knowingly claim that an absurd construct is math. The generalization of my proof to fractional numbers, negative numbers... they are trivial and nearly immediate.
3/ You use a recursive proof which you claim as being invalid in some other posts. This is internal inconsistency. While this has no mathematical value whatsoever, this puts in my "brain" a serious doubt regarding the rest of your "reasoning."
I assume you are referring to my explanation of the existance of the natural numbers. I blame this on the failure of natural language. I was attempting to describing something which is a priori (having always existed), as long as I have existed. This a priori intuition (of time) is concrete, precise, undeniable, etc... it needs no justification, because no matter what name I wish to give it, no matter how hard I deny it, it always exists. Basing mathematics on something this understandable, something this exact and precise, is what gives intuitionism certainty of truth. In no sense, with regards to itself, can such a system be wrong.
Now, if you are referring to my lame attempt at an example of a formal system which turns out to be paradoxical... well, it was just supposed to be a simple example. I guess it didn't get across the point. If not, go read Godel's incompleteness papers.
As a last comment, the use you make of "absurd" or "beauty" makes *you* behave as a religious zealot who owns The Only Real Trush of The Saint Intuitionism. And, frankly, I don't see why the fact that Cantor was Jewish has anything to do with the quality of his maths. Oh, or maybe it is because 42 is the answer?
For me to be a religious zealot, I would have to have claimed the basis of mathematics as that which is faith based. I did not. "Absurdity" is a commonly used comcept for contradictory constructions. My use of the word "beauty" with regards to mathematics is purely in the sense that something which is part of my Self, something which is precise, noncontradictory, certain, decidable, etc... I see these things as beautiful. If your idea of beauty is different, well then, sorry. I will claim that many mathematicians associate parsimony with beauty. Now, I would regard things which are wrong and uncertain as lacking the beauty of that which is correct and certain.
Remember, religion involves faith. I am encouraging the basis of mathematics on that which doesn't require faith. Its the faith based foundations that have been causing the problems.
Applied mathematics, which you describe is a useful thing. It helps us do stuff in order to obtain a predicted outcome. However, applied mathematics is a faith based occupation as I have already described (mainly in previous posts). Anything based on faith cannot have certainty and garrenteed freedom from contradiction, but it can have faith in such properties. Without a developed "way of knowing" which is certain as is mathematical Intuitionism, you will be lost in a sea of uncertainty. However, if your mind is like mine (in the basic sense), which I have faith in such a thing being true, then you will always have your innate intuition of time. You were born with it and you will die with it. Turn to it if you desire to construct that which is certain. Think of it as your anchor for your boat, which you are sailing through the sea of uncertainty. (use of metaphor, even if bad use, doesn't imply religious zealotry, so don't even start)
The claims of Cantor's religious tendencies influencing his "theories" are well documented. For example this book is a good starter for you.
I realize that according to the majority of religions in the world, I am sinful by requiring the use of as much non-faith based reasoning as possible. Christianity for instance preaches such stubborness as a sin.
...oh, and there has to be some way around the marking of such posts as the one I am replying to, as "Offtopic". The post is ontopic to the post that it is replying to, therefore it is not offtopic for its context.
All discussions twist and turn. Its unnatural to limit that.
x * 0 = 1 is absurd. The existance of such a number x is absurd, for such a number x should be greater than 0, and 1 doesn't work and neither does n+1 for arbitrary natural number n. Yeah, I agree, it took my brain about 1/2 second to realize that.
Historically, much of the mathematics have been built by defining something as the result of an equation with no result. For instance, x*x=-1 yields the complex numbers, x*x=2 yields the non-rational numbers, etc.
Ok, and this historical practice somehow justifies something as being mathematical. Throughout time, many people have done questionable things, and we should not use historical use as sole justification. I mean, reality used to consist of ether, according to historical account, and any number system based on absurd numbers is absurd itself.
It is not necessary to do something an infinite number of times; this is the diffrence between mathematics and accounting. Mathematics, as a science, is about proving things.
We are talking about completely different things. Mathematics is anything but a science, for being a science would make it uncertain and prone to numerous foundational problems such as: contradiction, incompleteness, undecidability, etc... I also do not believe that mathematics should be based on religious or faith based foundations, which science relies on.
The set of all dead presidents surely doesn't exist as a mathematical concept, and neither does the set of "all" natural numbers. I can't create the entire set, so how could the concept exist in a mathematical sense? It can exist through phantasy and faith, which is erroneously implied by many a mathematician to mathematics. This is what I refer to as the pollution of mathematics.
Leibniz was another religious mathematician. He invented binary numbers in the western tradition, in fact, he invented then for religious reasons. Your computer operates based upon his principals of binary addition and subtraction. His being religious did not negate the truth of his mathematics; why should Cantor be any diffrent.
It doesn't negate nor does it verify. However, religion and faith have no business in mathematics. Its a dangerous mix. It sends many mathematicians astray into uncertainty.
Intuitionism has shown that you can have freedom from paradox, contradictions, undecidability and incompleteness in mathematics. You are right though... formalist foundations for mathematics are broken or limited in that they are incomplete.
Formalist mathematics relies on the belief that the symbols written on a piece of paper and the finite set of inference rules and axioms can be used to derive truth by applying such rules. This belief cannot be be justified without an appeal to faith or metaphysics (read Arend Heyting's "Introduction to Intuitionism" book for a full explanation).
Infinity might be needed, as a concept, same as many concepts like God, but you can no more claim the existance of God than you can claim the existance of infinity - without the use of faith.
Now, as far as us imagining infinity, you are correct that the way that we imagine infinity is similar to how we imagine tooth fairies. Tooth fairies are mental constructions, not basic core, inherently a piori concepts. Tooth fairies are in a sense, phantasy. Now, the natural numbers 1,2,3,4,5,... (use a different name, symbols, its the same concept) are a different kind of mental construction than infinities and tooth fairies. The natural numbers can directly be built from our intuition of time. We all have an inherent a priori base construct to our thinking, which we cannot escape from: the present is different and comes after the past. This Present soon becomes part of the past, as the next present is experienced (in the most direct form of experience). This a priori understanding is called the natural numbers by some people. Now, other people believe the natural numbers to be something akin to a form of Plato's ideal realm, but that is religious, not mathematical.
I realize that I am just a geek with a poor ability to communicate what I mean, but I hope you see how our mind/self/conciousness/brain/whatever-you-call-it has a direct and a priori understanding of the natural numbers. We are only taught names for them, among other formalized rules for the natural numbers. When we are young, before we "can count", we still have an a priori understanding of the natural numbers... we just don't know how to communicate that understanding. Its kind of like going to another country, where they speak a completely different language than us... we could say "one", "two", "three", etc... when asked to count, but we will not be communicating our understanding well enough to let the witnesses understand.
True mathematics, the mathematics which is certain, complete, decidable, and free from contradction is the mathematics which is based on this a priori understanding (Brouwer uses the word intuition) of time - our natural numbers.
Note that by understanding of time, I am not refering to anything about physics or relativity or anything outside of the self. I simply mean how we are fundamentally aware (in pure thought), of thought that comes now and thought that came before. Its because of this squential intuition, that we precieve the world outside of us, as also flowing through time. Now, some interesting philosophical discussion can proceed on time not actually existing outside of us - in reality as part of, say, physics. Einstein's theories make good arguements for such... but I don't wanna go there and I will just admit that with regards to the outside world/physical reality... I don't know.
Finally, my point is that while a large part of popular mathematics can be created from our intuition of natural numbers, a similar concrete, undeniable intuition does not exist for the creation of infinity.
"Infinity", while defined better than, say, a "tooth fairy", is still more phantasy than mathematics.
I erge you guys to check out the original works of L.E.J. Brouwer for more insight on these topics. You won't regret it. Oh, and be careful of the many different interpretations of "intuition". Some uses mean "ways of knowing which are completely fuzzy or ambiguous". I use it, as Brouwer did, to refer to an undeniable (you can't deny it to yourself) a priori knowledge possessed by our minds, which to yourself individually, is so immediate that it doesn't need explanation or justification. It is absolute!
"Now if we build mathematical models of reality (eg, Quantum Chromodynamics, Superstring Theory, General Relativity), then there can be "religious" or "faith based" aspects, namely that you have faith that your model conforms to reality. Yet we have no assurances that "reality" can be mapped into a formal system. "
On that above quote of yours, I couldn't agree more! Its sad though, that others cannot see the truth. Whoever it was that said that "science is the religion of the day" (I think that it was someone in this thread), I am in great debt towards. I am going to use that quote, a plenty.
Infinity should also not be considered a mathematical construct, as it cannot be based on anything besides faith. Sure, infinity is needed, just as is love, god, etc... but these things are not mathematical.
You are the one that doesn't understand what intuitionism is. I truely hope that your major isn't mathematics, because otherwise, your schools have failed you!
First of all, Brouwer's Intuitionism and Intuitionistic Logic are two different, but related things. You seem to not be aware of the difference. Most likely because you haven't read Brouwer's original papers or lecture notes. Here goes a little history for you, since whereever you got your education from neglected to make you read the works of those "better minds" that you speak of.
During the end of the 19th century, Cantor polluted popular mathematics with his Jewish tendencies and beliefs about infinity. Another popular mathematician of Cantor's time objected to the transfinite crap of Cantor. This man's name was Leopold Kronecker. Of course, those times consisted of people mainly like you, so they laughed Kronecker out of public influence.
A little while later, Bertrand Russell and others started discovering paradoxes in popular mathematics. This was a problem, because a little digging showed that mathematics was without a sound foundation, which we could use to fend of fear of paradox and other errors of a mathematical system. Anyway, Russell preached formal logic as a foundation of mathematics and Hilbert preached a more general formalist foundation of mathematics.
Around the same time, a mathematician named L.E.J. Brouwer objected to both foundations proposed by Russell and Hilbert, pointing out that those foundations required metaphysical claims, faith, uncertainty, etc... The foundations also had demonstratable inconsistancies, which Brouwer showed. (go read his original works or translations if you don't believe me). Brouwer wasn't correct because I believe what he said. No, it just plain makes sense to anyone with enough independent thought to check out/interlibrary loan originals. Brouwer went on to denounce logic and all other formal mathematical systems, claiming that logic "is an interesting but irrelevant and sterile exercise." Yeah, 3am, really sounds like Intuitionism is a variant of logic. Whatever led you to believe so is misleading and incorrect. However, back to your history lesson...
Ok, so Hilbert and Russell's foundations are shown to be flawed, mainly by: Church, Turing, Godel. Don't believe me? Great. I don't want you to, I can only show you the door. You must walk through it. How? By reading the originals!!! Most popular books/professors on Intuitionism, Godel's Incompleteness theorems, Church and Turing's undecidability proofs - misrepresent what the original "better minds" have shown us Why would they misrepresent the truth?...simply because they do not understand and are confused, just like you.
Time passes and history shows that Brouwer's Intuitionism works and is not flawed, but it has one drawback, best described by Hermann Weyl:
"Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes."
What Weyl describes is the frustration that arises with Intuitionism, because it does not allow for formal synbol manipulations in proofs, as does popular mathematics. Classical and intuitionistic logics use symbols with axioms and inference rules, in order to formally derive proofs. Brouwer's Intuitionism does NOT claim that such things are mathematics, so their use is limited to documentation and communication, not derivation.
Now, the intuitionistic logic that you speak of, is not Brouwer's Intuitionism, and it was never directly approved of by Brouwer (though he was tolerant of it and grew more appreciative of it as he aged). However, it was his student, Arend Heyting, who formalized a logic which is not contradictory to Intuitionism. Gentzen, another famous logician, also contributed to the intuitionistic logic that you refer to, but I will leave the details of your history education up to you. Libraries will help you for free. You just have to ask.
Now, just to give you another modern example pointing out your ignorance, think about this quote (in relation to your comments) about intuitionism, from another "better mind than yours":
"...intuitionism is first and foremost a philosophy of mathematics..."From Brouwer to Hilbert: The Debate on the Foundations of mathematics in the 1920s by Paolo MancosuThe specific quote is from Walter van Stigt, and appears on the begining of page 4.
Now, 3am, don't you feel like the ignorant one? Don't you feel like your education has mislead you? If not, then read the original works of the "better mind than yours", and you will feel like your education has mislead you. Hopefully you are not so blind as to pull yourself from your ignorance.
Finally, note that my use of the word infinity is as used by Cantor, which is the common interpretation of the word. Boundless and no upper bound are different from infinity (and its not just me making this up, the "better minds" seem to agree too). Infinity implies a definite concrete existance of something which is "infinite" (yeah, a recursive definition, but I can't think of a better way to describe such nonsense without recursion). "Without bound" and "boundless" refer to something that can be repeated or continued as long as desired, but the use doesn't necessarily mean that the repetition goes on "forever". It literally means, as extreme as you want, but still finite. A variable without bound could never be "infinite", but it could be as big as you want to make it. So, as you should see, infinite is something which exists (supposedly), while something without bound is something which can be chosen as artbitrarily large/small/etc as you want. Subtle but important difference to mathematicians.
You argue for Cantor, yet against his infinity.
You argue for Hilbert, with disregard to the failure of his foundations.
You argue against philosophy, but you cannot escape it, no matter how hard you try - if you wish to have proper foundations for your mathematics.
You believe that my description of Intuitionism implies or leads to the destruction of mathematics. Yet all I describe is Brouwer's Intuitionism.
You claim that Hilbert was an intuitionist, when in fact, he was a formalist.
You argue for Brouwer and Hilbert, but Hilbert was the one to claim the loudest, that Brouwer was destroying mathematics, and Hilbert acted as such through arguement and through coercion (he kicked Brouwer out of a famous mathematics committee).
Tell me, who is confused? It appears that you have not even read the original papers and works of Hilbert, Brouwer, Heyting, Kleene, etc...
Do so, and you might save yourself from ignorance.
Godel proved that general mathematics cannot be verified to be free from paradox, inconsistancies, etc...
Bertrand Russell, among many other mathematicians have discovered the existance of paradoxes and inconsistancies in various mathematical systems. Note, as Godel has proved, even if we don't currently know of the existance of any of these paradoxes, we cannot be sure that they do not exist. Huge parts of mathematics could easily turn out to be bunk, simply because, overall, there are paradoxes or inconsistancies in the mathematical system. I realize that you have faith that such inconsistancies don't exist, but mathematics isn't a religion.
In addition, Church and Turing showed how certain parts of formalist "mathematics" are undecidable.
Now, you can continue to disagree with me, but history has shown that the formalist system which you describe mathematics to be, is nothing more than a faith based phantasy.
"Have you ever had a dream, 3am, that you were so sure was real"
"Some can never be free from the Matrix."
My point is I am not telling you how to truely understand infinity, because that is impossible. Infinity cannot exist in a mathematics which is not based on faith.
I do not care when people use infinity, but I do care when they claim that it absolutely exists, in a mathematical sense. Just because Plato and Cator and other injected religion into mathematics, doesn't mean that we have to use such broken uncertain foundations.
Mathematics is the only thing which is not based on faith and is therefore an achor in a see of uncertainty.
Now, as I eat my breakfast, I do so, mainly with faith alone. Its a damn good thing that I don't try to completely mathematically understand "eating breakfast", because such a thing is impossible... I would die from lack of nutrition.
Don't be so naive. If the foundations for mathematics are not chosen correctly, then you do not end up with the require properties of math: certainty, free from paradox, completeness, and decidable.
See, you build your foundations on mist, air, nothing... for what is an axiom? What is an inference rule? Do you have the required attributes of mathematics? You use these things as your foundations, but there use is not math (certainly free from paradox and completely decidable).
Here is a story for you, about an educated man that existed a long time ago, when civilzations were just begining to form. He knew how to read and write, among many other things that the masses didn't know. People of the small village that he went to starting trusting him, because his teachings, his wisdom worked as promised.
Then, shortly after the educated man's arrival, the previous village leader took a stand out in the center of the village, fearing that he was losing control of the people - this educated man was becoming more popular. A croud formed around the original leader, who has the educated man at his side. The original leader spoke out to the croud, "If this man is so great, then let him write a snake in the dirt on the ground." The educated man then proceeded to spell out the word "snake" in the written language (cuniform or an ancient arabic language). The original village leader then laughed, as right next to the educated man's "snake" writing, the village leader drew a long squiggly line - a picture of a snake. He then said, "...and who succeeded in writing a snake in the dirt?" The croud then turned on the educated man, being illiterate, they couldn't grasp the concept of a written language based on a small alphabet and grammar. The couldn't see through their ignorance. They could no longer see the value of the educated man's teachings.
Ok, now let me disprove your supposed foundations with a reduction to absurdity.
System Gamma
AXIOM OMEGA: A is true
AXIOM ALPHA: if A then B is true
AXIOM BETA: if B then C is true
AXIOM DELTA: if C then A is false
So a proof in this system could go:
OMEGA and ALPHA imply B,
which implies C,
which implies not A which contradicts OMEGA
What did I do, 3am? I used your arguement against you. I used your misunderstanding of Godel's incompleteness proofs against you. Godel proved that a formalist method, as you describe, is incomplete, and therefore the consistancy of a formal system based on axioms, cannot always be verified to be free from such contradictions. Sure, in my case, you can see that a contradiction arises from the axioms, but in general mathematical systems, a proof of consistancy is impossible! Your system could be as valuable as pure garbage, pure nonsense.
So, join the croud of the illiterate. Denounce those who bring you the truth, simply because you are too ignorant to see the error of your ways.
One more thing, just to help out a fellow slashdotter (or maybe I am just trying to brain wash you into agreeing with me). Since you seem to be a man more convinced by arguement of science, as opposed to an appeal to philosophy, check out this book: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being
The book is written by two cognitive scientists who use cognitive science to explain the origination of mathematics, as a phenomenon of our squash... *erghh I mean* brain.
Now, because of my philosophical standings (uncertainty of science), I cannot accept such arguement as a foundation for mathematics (which is supposed to be certain)... but I do enjoy the book and think that it is inspiring because it feeds our human need of scientific evidence, and I am sure that the arguement of science rings louder for you than it does me.
Supposedly the authors have insued allot of anger because of their counter-intellectual stance on biological origins of mathematics.
Actually, by definition of the philosphy of mathematical intuitionism, if you disagree and claim Platonic ideals like Cantor's infinities, you obviously don't understand. Now, if you define mathematics as a faithed based way of knowing, then we are arguing about two different things.
Happy with Cantor's infinities? Then you are stuck with a mathematics that is uncertain, inexact, full of paradoxes, contradictions, incompleteness, and undecidables. All because it is based on faith: "that which is hoped for".
With a mathematics so ambiguous, you can not be sure that you truely know what you believe you know. Your sciences, your theories, your infinities... They turn to mist as you realize that such a baseless mathematics is nothing more than confusion, when put to use as that which is obsolute/concrete.
I still recommend the "Brouwer to Hilbert" book, even if you disagree, so that you can see for yourself, the importance of at least one aspect of your understanding, not being based on faith. The book is a weekend's reading, and you can only benefit from such a solid understanding of the foundations of mathematics. Think of it as an anchor in the sea of uncertainty.
Dunno bout you, but I want to know.
I just wish that schools would cut the crap and start teaching a more open-minded mathematics program. We need to stop teaching our kids that platonic mathematics is absolutely the truth, because history has shown that it is not necessarily so with created paradoxes, undecidables, and incompleteness.
If you don't limit the length of the programs, then you end up with programs of unbounded size, not infinite programs. Infinity is a phantasy, not a mathematical construct.
Yeah, the basic Palm Pilot features are all I need too. The REX6000 supposedly has them, and since the only things that I want to change about my Palm Pilot are: its size, weight, durability, and battery life... the REX6000 looks like a tool that I would like to own. The 6 month battery life of the REX6000 is too good to be true.
Now, they just need to make a REX6000 which is shock proof and water proof (sometimes I think best in the shower).
Finally, someone who realizes that I am not trolling. Sheesh! However, there is a small misconception about my points about mathematics (try to checkout the stuff that I have been linking too, especially the book "From Brouwer to Hilbert")... mathematics is the one thing that doesn't have to be taken on faith. It is the only thing that you can know directly and exactly, because it is constructively built from ideas which are based on our a priori intuition of the continuum (our intuition of time). So, my point is that mathematics is absolutely true and real, but only to the mathematician. Math doesn't exist outside of us. Without people, math does not exist. Mathematics are concrete ideas, which are exact, free from paradox, etc... I know they exist in my mind because I directly experience them, but the claim that they exist outside of the mind, independent of it... that is a religious claim. Because of this, it can be seen that mathematics is only used to understand our world - this is what science does, its the use of our concrete ideas to describe the world outside of us, but the world does not necessarily obey mathematical rules or laws.
Finally, if you cannot directly create a concept, constructively, from the a priori intuition, then the concept is not mathematical... its fantasy or confusion or something else. Mathematics is this way because otherwise, it would be full of paradoxes, contradictions, uncertainty, etc...
Notice how the popular understanding of and use of and foundations of mathematics are full of paradoxes, contradictions, uncertainty, etc...? Its because the popular foundations for mathematics are faith based.
I disagree. I recently read through Milner's new book on the pi-calculus, which is very readable. You don't need to know anything about the CSP or CCS. The pi-calculi and cousins are far more generalized in the aspect of concurrency, than CSP or CCS. I suggest Milner's new book. Its short and to the point. From that book, you can branch out into reading all of the papers listed at the Mobile Calculi link I gave in the above comment.
I am placing my bets on the Ambient Calculi and family, as the canonical mobile/concurrent calculi. Need some proof? Check out Girard's recent Linear Logic and Ludics, and be reminded of the lambda-calculus/Intuitionistic Logic isomorphism. Its happening all over again, except this time, we are dealing with completely concurrent computation, as opposed to completely sequential.
Do you see a pattern?
Yeah, I disagree with the status quo, the popular misconceptions of mathematics, and I am a troll, with posts that should be modded down? Have you ever considered that not only that you could be wrong, but an entire institution that you had faith in was also wrong? If your mathematics is so pure, so great, then why does it contain paradoxes, contradictions, undecidability and incompleteness? These things make your mathematics anything but exact, sure, true, or anything that it is supposed to be. It becomes nothing more than a religion.
Oh, and don't forget to mark one of the greatest founding fathers of modern computing as a troll too, because Kleene was also backed intuitionist ideas.
Some of you guys are so blind in your ignorance that you destroy good things - all the while, believing that you are fighting for the just.
I hate ignorance. Listen, I am not trolling, please trust me.
The concept used in "The Matrix" the movie is not new or original, but it makes a good point. Its based around solipsism - the realization that you cannot know anything outside of your mind. With a little contemplation, you can see that you accept the theories of science on faith alone. Physics is just man's use of his mental concepts (mathematical concepts) to describe the world. If infinity is not mathematical, then the claim that its use in physics justifies it is naive at best.
Stop. Think. How do you know? Ask yourself that. Ask yourself, "What is the foundation of mathematics?" Many men have, and could not answer. Some chose faith, while others choose to understand, to know.
If you are confused, then let me help you, since your schools could not: Constructive Mathematics Intuitionism From Brouwer to Hilbert : The Debate on the Foundations of Mathematics in the 1920s
Trust me, you are confused. Many people are. They have accepted mathematics on faith alone, but you do not have to. The way to understand is easy! I highly recommend checking out/buying the book From Brouwer to Hilbert to truely understand what Intuitionism is. The book consists of a series of papers which carried on a drawn out arguement about the foundations of mathematics. These weren't joe blows arguing. These are arguements of legendary mathematicians, so there is some serious insight in their work that I cannot give you. Finally, the papers in that book are strung together through use narrative of the editors. Its a great buy and you will not regret it.
Now, when you are done checking out all of those links and the book, if you really want to know more, then check out "Brouwer's Collected Works". Its a two volume set. The books are big and out of print, so ask your library. Its worth it.
Amen. Wish I had mod points. Someone want to chip in? These guys are a bit naive in their attempts. Attempting to solve the ills of language based programming by inventing a language based programming language. It is so hypocritical its funny.
An algorithm is not necessarily a language dependent concept. In fact, mathematical intuitionists would claim that algorithms exist independently of language, as do all mathematical concepts.
Slashdot Mirror
The proof of such extensions is fairly immediate, as was the one I gave. I mean, the proofs are so trivial in the first place, that the inductive proof that I gave can be easily seen as working for negative numbers. I am not going to be pull a Bertrand Russell and write a 10 page proof of formalisms for something so immediate. I made the point that the existance of such a number x such that x*0=1 is absurd - absurdity meaning that it is contradictory. Note that no real mathematician would ever knowingly claim that an absurd construct is math. The generalization of my proof to fractional numbers, negative numbers... they are trivial and nearly immediate.
I assume you are referring to my explanation of the existance of the natural numbers. I blame this on the failure of natural language. I was attempting to describing something which is a priori (having always existed), as long as I have existed. This a priori intuition (of time) is concrete, precise, undeniable, etc... it needs no justification, because no matter what name I wish to give it, no matter how hard I deny it, it always exists. Basing mathematics on something this understandable, something this exact and precise, is what gives intuitionism certainty of truth. In no sense, with regards to itself, can such a system be wrong.
Now, if you are referring to my lame attempt at an example of a formal system which turns out to be paradoxical... well, it was just supposed to be a simple example. I guess it didn't get across the point. If not, go read Godel's incompleteness papers.
For me to be a religious zealot, I would have to have claimed the basis of mathematics as that which is faith based. I did not. "Absurdity" is a commonly used comcept for contradictory constructions. My use of the word "beauty" with regards to mathematics is purely in the sense that something which is part of my Self, something which is precise, noncontradictory, certain, decidable, etc... I see these things as beautiful. If your idea of beauty is different, well then, sorry. I will claim that many mathematicians associate parsimony with beauty. Now, I would regard things which are wrong and uncertain as lacking the beauty of that which is correct and certain.
Remember, religion involves faith. I am encouraging the basis of mathematics on that which doesn't require faith. Its the faith based foundations that have been causing the problems.
Applied mathematics, which you describe is a useful thing. It helps us do stuff in order to obtain a predicted outcome. However, applied mathematics is a faith based occupation as I have already described (mainly in previous posts). Anything based on faith cannot have certainty and garrenteed freedom from contradiction, but it can have faith in such properties. Without a developed "way of knowing" which is certain as is mathematical Intuitionism, you will be lost in a sea of uncertainty. However, if your mind is like mine (in the basic sense), which I have faith in such a thing being true, then you will always have your innate intuition of time. You were born with it and you will die with it. Turn to it if you desire to construct that which is certain. Think of it as your anchor for your boat, which you are sailing through the sea of uncertainty. (use of metaphor, even if bad use, doesn't imply religious zealotry, so don't even start)
The claims of Cantor's religious tendencies influencing his "theories" are well documented. For example this book is a good starter for you.
I realize that according to the majority of religions in the world, I am sinful by requiring the use of as much non-faith based reasoning as possible. Christianity for instance preaches such stubborness as a sin.
...oh, and there has to be some way around the marking of such posts as the one I am replying to, as "Offtopic". The post is ontopic to the post that it is replying to, therefore it is not offtopic for its context.
All discussions twist and turn. Its unnatural to limit that.
Emphasis should be on the word "Crash".
Ok, and this historical practice somehow justifies something as being mathematical. Throughout time, many people have done questionable things, and we should not use historical use as sole justification. I mean, reality used to consist of ether, according to historical account, and any number system based on absurd numbers is absurd itself.
The set of all dead presidents surely doesn't exist as a mathematical concept, and neither does the set of "all" natural numbers. I can't create the entire set, so how could the concept exist in a mathematical sense? It can exist through phantasy and faith, which is erroneously implied by many a mathematician to mathematics. This is what I refer to as the pollution of mathematics. It doesn't negate nor does it verify. However, religion and faith have no business in mathematics. Its a dangerous mix. It sends many mathematicians astray into uncertainty.
Formalist mathematics relies on the belief that the symbols written on a piece of paper and the finite set of inference rules and axioms can be used to derive truth by applying such rules. This belief cannot be be justified without an appeal to faith or metaphysics (read Arend Heyting's "Introduction to Intuitionism" book for a full explanation).
Infinity might be needed, as a concept, same as many concepts like God, but you can no more claim the existance of God than you can claim the existance of infinity - without the use of faith.
Now, as far as us imagining infinity, you are correct that the way that we imagine infinity is similar to how we imagine tooth fairies. Tooth fairies are mental constructions, not basic core, inherently a piori concepts. Tooth fairies are in a sense, phantasy. Now, the natural numbers 1,2,3,4,5,... (use a different name, symbols, its the same concept) are a different kind of mental construction than infinities and tooth fairies. The natural numbers can directly be built from our intuition of time. We all have an inherent a priori base construct to our thinking, which we cannot escape from: the present is different and comes after the past. This Present soon becomes part of the past, as the next present is experienced (in the most direct form of experience). This a priori understanding is called the natural numbers by some people. Now, other people believe the natural numbers to be something akin to a form of Plato's ideal realm, but that is religious, not mathematical.
I realize that I am just a geek with a poor ability to communicate what I mean, but I hope you see how our mind/self/conciousness/brain/whatever-you-call-it has a direct and a priori understanding of the natural numbers. We are only taught names for them, among other formalized rules for the natural numbers. When we are young, before we "can count", we still have an a priori understanding of the natural numbers... we just don't know how to communicate that understanding. Its kind of like going to another country, where they speak a completely different language than us... we could say "one", "two", "three", etc... when asked to count, but we will not be communicating our understanding well enough to let the witnesses understand.
True mathematics, the mathematics which is certain, complete, decidable, and free from contradction is the mathematics which is based on this a priori understanding (Brouwer uses the word intuition) of time - our natural numbers.
Note that by understanding of time, I am not refering to anything about physics or relativity or anything outside of the self. I simply mean how we are fundamentally aware (in pure thought), of thought that comes now and thought that came before. Its because of this squential intuition, that we precieve the world outside of us, as also flowing through time. Now, some interesting philosophical discussion can proceed on time not actually existing outside of us - in reality as part of, say, physics. Einstein's theories make good arguements for such... but I don't wanna go there and I will just admit that with regards to the outside world/physical reality... I don't know.
Finally, my point is that while a large part of popular mathematics can be created from our intuition of natural numbers, a similar concrete, undeniable intuition does not exist for the creation of infinity.
"Infinity", while defined better than, say, a "tooth fairy", is still more phantasy than mathematics.
I erge you guys to check out the original works of L.E.J. Brouwer for more insight on these topics. You won't regret it. Oh, and be careful of the many different interpretations of "intuition". Some uses mean "ways of knowing which are completely fuzzy or ambiguous". I use it, as Brouwer did, to refer to an undeniable (you can't deny it to yourself) a priori knowledge possessed by our minds, which to yourself individually, is so immediate that it doesn't need explanation or justification. It is absolute!
On that above quote of yours, I couldn't agree more! Its sad though, that others cannot see the truth. Whoever it was that said that "science is the religion of the day" (I think that it was someone in this thread), I am in great debt towards. I am going to use that quote, a plenty.
Infinity should also not be considered a mathematical construct, as it cannot be based on anything besides faith. Sure, infinity is needed, just as is love, god, etc... but these things are not mathematical.
First of all, Brouwer's Intuitionism and Intuitionistic Logic are two different, but related things. You seem to not be aware of the difference. Most likely because you haven't read Brouwer's original papers or lecture notes. Here goes a little history for you, since whereever you got your education from neglected to make you read the works of those "better minds" that you speak of.
During the end of the 19th century, Cantor polluted popular mathematics with his Jewish tendencies and beliefs about infinity. Another popular mathematician of Cantor's time objected to the transfinite crap of Cantor. This man's name was Leopold Kronecker. Of course, those times consisted of people mainly like you, so they laughed Kronecker out of public influence.
A little while later, Bertrand Russell and others started discovering paradoxes in popular mathematics. This was a problem, because a little digging showed that mathematics was without a sound foundation, which we could use to fend of fear of paradox and other errors of a mathematical system. Anyway, Russell preached formal logic as a foundation of mathematics and Hilbert preached a more general formalist foundation of mathematics.
Around the same time, a mathematician named L.E.J. Brouwer objected to both foundations proposed by Russell and Hilbert, pointing out that those foundations required metaphysical claims, faith, uncertainty, etc... The foundations also had demonstratable inconsistancies, which Brouwer showed. (go read his original works or translations if you don't believe me). Brouwer wasn't correct because I believe what he said. No, it just plain makes sense to anyone with enough independent thought to check out/interlibrary loan originals.
Brouwer went on to denounce logic and all other formal mathematical systems, claiming that logic "is an interesting but irrelevant and sterile exercise." Yeah, 3am, really sounds like Intuitionism is a variant of logic. Whatever led you to believe so is misleading and incorrect. However, back to your history lesson...
Ok, so Hilbert and Russell's foundations are shown to be flawed, mainly by: Church, Turing, Godel. Don't believe me? Great. I don't want you to, I can only show you the door. You must walk through it. How? By reading the originals!!! Most popular books/professors on Intuitionism, Godel's Incompleteness theorems, Church and Turing's undecidability proofs - misrepresent what the original "better minds" have shown us Why would they misrepresent the truth?
Time passes and history shows that Brouwer's Intuitionism works and is not flawed, but it has one drawback, best described by Hermann Weyl:
What Weyl describes is the frustration that arises with Intuitionism, because it does not allow for formal synbol manipulations in proofs, as does popular mathematics. Classical and intuitionistic logics use symbols with axioms and inference rules, in order to formally derive proofs. Brouwer's Intuitionism does NOT claim that such things are mathematics, so their use is limited to documentation and communication, not derivation.
Now, the intuitionistic logic that you speak of, is not Brouwer's Intuitionism, and it was never directly approved of by Brouwer (though he was tolerant of it and grew more appreciative of it as he aged). However, it was his student, Arend Heyting, who formalized a logic which is not contradictory to Intuitionism. Gentzen, another famous logician, also contributed to the intuitionistic logic that you refer to, but I will leave the details of your history education up to you. Libraries will help you for free. You just have to ask.
Now, just to give you another modern example pointing out your ignorance, think about this quote (in relation to your comments) about intuitionism, from another "better mind than yours":
Now, 3am, don't you feel like the ignorant one? Don't you feel like your education has mislead you? If not, then read the original works of the "better mind than yours", and you will feel like your education has mislead you. Hopefully you are not so blind as to pull yourself from your ignorance.
Finally, note that my use of the word infinity is as used by Cantor, which is the common interpretation of the word. Boundless and no upper bound are different from infinity (and its not just me making this up, the "better minds" seem to agree too). Infinity implies a definite concrete existance of something which is "infinite" (yeah, a recursive definition, but I can't think of a better way to describe such nonsense without recursion). "Without bound" and "boundless" refer to something that can be repeated or continued as long as desired, but the use doesn't necessarily mean that the repetition goes on "forever". It literally means, as extreme as you want, but still finite. A variable without bound could never be "infinite", but it could be as big as you want to make it. So, as you should see, infinite is something which exists (supposedly), while something without bound is something which can be chosen as artbitrarily large/small/etc as you want. Subtle but important difference to mathematicians.
Tell me, who is confused? It appears that you have not even read the original papers and works of Hilbert, Brouwer, Heyting, Kleene, etc...
Do so, and you might save yourself from ignorance.
Godel proved that general mathematics cannot be verified to be free from paradox, inconsistancies, etc...
Bertrand Russell, among many other mathematicians have discovered the existance of paradoxes and inconsistancies in various mathematical systems. Note, as Godel has proved, even if we don't currently know of the existance of any of these paradoxes, we cannot be sure that they do not exist. Huge parts of mathematics could easily turn out to be bunk, simply because, overall, there are paradoxes or inconsistancies in the mathematical system. I realize that you have faith that such inconsistancies don't exist, but mathematics isn't a religion.
In addition, Church and Turing showed how certain parts of formalist "mathematics" are undecidable.
Now, you can continue to disagree with me, but history has shown that the formalist system which you describe mathematics to be, is nothing more than a faith based phantasy.
"Have you ever had a dream, 3am, that you were so sure was real"
"Some can never be free from the Matrix."
My point is I am not telling you how to truely understand infinity, because that is impossible. Infinity cannot exist in a mathematics which is not based on faith.
I do not care when people use infinity, but I do care when they claim that it absolutely exists, in a mathematical sense. Just because Plato and Cator and other injected religion into mathematics, doesn't mean that we have to use such broken uncertain foundations.
Mathematics is the only thing which is not based on faith and is therefore an achor in a see of uncertainty.
Now, as I eat my breakfast, I do so, mainly with faith alone. Its a damn good thing that I don't try to completely mathematically understand "eating breakfast", because such a thing is impossible... I would die from lack of nutrition.
Don't be so naive. If the foundations for mathematics are not chosen correctly, then you do not end up with the require properties of math: certainty, free from paradox, completeness, and decidable.
See, you build your foundations on mist, air, nothing... for what is an axiom? What is an inference rule? Do you have the required attributes of mathematics? You use these things as your foundations, but there use is not math (certainly free from paradox and completely decidable).
Here is a story for you, about an educated man that existed a long time ago, when civilzations were just begining to form. He knew how to read and write, among many other things that the masses didn't know. People of the small village that he went to starting trusting him, because his teachings, his wisdom worked as promised.
Then, shortly after the educated man's arrival, the previous village leader took a stand out in the center of the village, fearing that he was losing control of the people - this educated man was becoming more popular. A croud formed around the original leader, who has the educated man at his side. The original leader spoke out to the croud, "If this man is so great, then let him write a snake in the dirt on the ground." The educated man then proceeded to spell out the word "snake" in the written language (cuniform or an ancient arabic language). The original village leader then laughed, as right next to the educated man's "snake" writing, the village leader drew a long squiggly line - a picture of a snake. He then said, "...and who succeeded in writing a snake in the dirt?" The croud then turned on the educated man, being illiterate, they couldn't grasp the concept of a written language based on a small alphabet and grammar. The couldn't see through their ignorance. They could no longer see the value of the educated man's teachings.
Ok, now let me disprove your supposed foundations with a reduction to absurdity.
System Gamma AXIOM OMEGA: A is true AXIOM ALPHA: if A then B is true AXIOM BETA: if B then C is true AXIOM DELTA: if C then A is false
So a proof in this system could go:
OMEGA and ALPHA imply B, which implies C, which implies not A which contradicts OMEGA
What did I do, 3am? I used your arguement against you. I used your misunderstanding of Godel's incompleteness proofs against you. Godel proved that a formalist method, as you describe, is incomplete, and therefore the consistancy of a formal system based on axioms, cannot always be verified to be free from such contradictions. Sure, in my case, you can see that a contradiction arises from the axioms, but in general mathematical systems, a proof of consistancy is impossible! Your system could be as valuable as pure garbage, pure nonsense.
So, join the croud of the illiterate. Denounce those who bring you the truth, simply because you are too ignorant to see the error of your ways.
One more thing, just to help out a fellow slashdotter (or maybe I am just trying to brain wash you into agreeing with me). Since you seem to be a man more convinced by arguement of science, as opposed to an appeal to philosophy, check out this book: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being
The book is written by two cognitive scientists who use cognitive science to explain the origination of mathematics, as a phenomenon of our squash... *erghh I mean* brain.
Now, because of my philosophical standings (uncertainty of science), I cannot accept such arguement as a foundation for mathematics (which is supposed to be certain)... but I do enjoy the book and think that it is inspiring because it feeds our human need of scientific evidence, and I am sure that the arguement of science rings louder for you than it does me.
Supposedly the authors have insued allot of anger because of their counter-intellectual stance on biological origins of mathematics.
Actually, by definition of the philosphy of mathematical intuitionism, if you disagree and claim Platonic ideals like Cantor's infinities, you obviously don't understand. Now, if you define mathematics as a faithed based way of knowing, then we are arguing about two different things.
Happy with Cantor's infinities? Then you are stuck with a mathematics that is uncertain, inexact, full of paradoxes, contradictions, incompleteness, and undecidables. All because it is based on faith: "that which is hoped for".
With a mathematics so ambiguous, you can not be sure that you truely know what you believe you know. Your sciences, your theories, your infinities... They turn to mist as you realize that such a baseless mathematics is nothing more than confusion, when put to use as that which is obsolute/concrete.
I still recommend the "Brouwer to Hilbert" book, even if you disagree, so that you can see for yourself, the importance of at least one aspect of your understanding, not being based on faith. The book is a weekend's reading, and you can only benefit from such a solid understanding of the foundations of mathematics. Think of it as an anchor in the sea of uncertainty.
Dunno bout you, but I want to know.
I just wish that schools would cut the crap and start teaching a more open-minded mathematics program. We need to stop teaching our kids that platonic mathematics is absolutely the truth, because history has shown that it is not necessarily so with created paradoxes, undecidables, and incompleteness.
If you don't limit the length of the programs, then you end up with programs of unbounded size, not infinite programs. Infinity is a phantasy, not a mathematical construct.
"Do you think that your Wu Tang sword can defeat me?"
...
*shreek* En guard! I will let you try my Wu Tang style!
Yeah, the basic Palm Pilot features are all I need too. The REX6000 supposedly has them, and since the only things that I want to change about my Palm Pilot are: its size, weight, durability, and battery life... the REX6000 looks like a tool that I would like to own. The 6 month battery life of the REX6000 is too good to be true.
Now, they just need to make a REX6000 which is shock proof and water proof (sometimes I think best in the shower).
Finally, someone who realizes that I am not trolling. Sheesh! However, there is a small misconception about my points about mathematics (try to checkout the stuff that I have been linking too, especially the book "From Brouwer to Hilbert")... mathematics is the one thing that doesn't have to be taken on faith. It is the only thing that you can know directly and exactly, because it is constructively built from ideas which are based on our a priori intuition of the continuum (our intuition of time). So, my point is that mathematics is absolutely true and real, but only to the mathematician. Math doesn't exist outside of us. Without people, math does not exist. Mathematics are concrete ideas, which are exact, free from paradox, etc... I know they exist in my mind because I directly experience them, but the claim that they exist outside of the mind, independent of it... that is a religious claim. Because of this, it can be seen that mathematics is only used to understand our world - this is what science does, its the use of our concrete ideas to describe the world outside of us, but the world does not necessarily obey mathematical rules or laws.
Finally, if you cannot directly create a concept, constructively, from the a priori intuition, then the concept is not mathematical... its fantasy or confusion or something else. Mathematics is this way because otherwise, it would be full of paradoxes, contradictions, uncertainty, etc...
Notice how the popular understanding of and use of and foundations of mathematics are full of paradoxes, contradictions, uncertainty, etc...? Its because the popular foundations for mathematics are faith based.
I disagree. I recently read through Milner's new book on the pi-calculus, which is very readable. You don't need to know anything about the CSP or CCS. The pi-calculi and cousins are far more generalized in the aspect of concurrency, than CSP or CCS. I suggest Milner's new book. Its short and to the point. From that book, you can branch out into reading all of the papers listed at the Mobile Calculi link I gave in the above comment.
I am placing my bets on the Ambient Calculi and family, as the canonical mobile/concurrent calculi. Need some proof? Check out Girard's recent Linear Logic and Ludics, and be reminded of the lambda-calculus/Intuitionistic Logic isomorphism. Its happening all over again, except this time, we are dealing with completely concurrent computation, as opposed to completely sequential.
Do you see a pattern?
Yeah, I disagree with the status quo, the popular misconceptions of mathematics, and I am a troll, with posts that should be modded down? Have you ever considered that not only that you could be wrong, but an entire institution that you had faith in was also wrong? If your mathematics is so pure, so great, then why does it contain paradoxes, contradictions, undecidability and incompleteness? These things make your mathematics anything but exact, sure, true, or anything that it is supposed to be. It becomes nothing more than a religion.
After you mod my posts down, don't forget to mod down the entries of the Encyclopædia Britannica on Intuitionism.
Oh, and don't forget to mark one of the greatest founding fathers of modern computing as a troll too, because Kleene was also backed intuitionist ideas.
Some of you guys are so blind in your ignorance that you destroy good things - all the while, believing that you are fighting for the just.
I hate ignorance. Listen, I am not trolling, please trust me.
The concept used in "The Matrix" the movie is not new or original, but it makes a good point. Its based around solipsism - the realization that you cannot know anything outside of your mind. With a little contemplation, you can see that you accept the theories of science on faith alone. Physics is just man's use of his mental concepts (mathematical concepts) to describe the world. If infinity is not mathematical, then the claim that its use in physics justifies it is naive at best.
Stop. Think. How do you know? Ask yourself that. Ask yourself, "What is the foundation of mathematics?" Many men have, and could not answer. Some chose faith, while others choose to understand, to know.
If you are confused, then let me help you, since your schools could not:
Constructive Mathematics
Intuitionism
From Brouwer to Hilbert : The Debate on the Foundations of Mathematics in the 1920s
Trust me, you are confused. Many people are. They have accepted mathematics on faith alone, but you do not have to. The way to understand is easy! I highly recommend checking out/buying the book From Brouwer to Hilbert to truely understand what Intuitionism is. The book consists of a series of papers which carried on a drawn out arguement about the foundations of mathematics. These weren't joe blows arguing. These are arguements of legendary mathematicians, so there is some serious insight in their work that I cannot give you. Finally, the papers in that book are strung together through use narrative of the editors. Its a great buy and you will not regret it.
Now, when you are done checking out all of those links and the book, if you really want to know more, then check out "Brouwer's Collected Works". Its a two volume set. The books are big and out of print, so ask your library. Its worth it.
For those who were lied to in school: Foundations of Mathematics? (preview is still screwy)
For those who were lied to in school: Foundations of Mathematics?
Amen. Wish I had mod points. Someone want to chip in? These guys are a bit naive in their attempts. Attempting to solve the ills of language based programming by inventing a language based programming language. It is so hypocritical its funny.
Ahhh!!! Good recommendation, but the more recent theory can be found here (CSP is "so last week"): Calculi for Mobile Processes
An algorithm is not necessarily a language dependent concept. In fact, mathematical intuitionists would claim that algorithms exist independently of language, as do all mathematical concepts.