PROOF INFINITY DOES NOT EXIST! Not even ONE type
in [Math]
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From: FredJeffries on 10 Jul 2010 19:22 On Jul 9, 10:17 am, Dan Christensen <Dan_Christen...(a)sympatico.ca> wrote: > I can't imagine that you would be able to do very much using > "finitist" methods. How do they handle such basic concepts as the > square root of 2? > Terence Tao in "A computational perspective on set theory" http://terrytao.wordpress.com/2010/03/19/a-computational-perspective-on-set-theory/ in which he explores the question "what is the finitary analogue of statements such as Cantors theorem or the Banach-Tarski paradox?" summarizes thus: <quote> The above discussion suggests that it is possible to retain much of the essential mathematical content of set theory without the need for explicitly dealing with large sets (such as uncountable sets), but there is a significant price to pay in doing so, namely that one has to deal with sets on a "virtual" or "incomplete" basis, rather than with the "completed infinities" that one is accustomed to in the standard modern framework of mathematics. Conceptually, this marks quite a different approach to mathematical objects, and assertions about such objects; such assertions are not simply true or false, but instead require a certain computational cost to be paid before their truth can be ascertained. This approach makes the mathematical reasoning process look rather strange compared to how it is usually presented, but I believe it is still a worthwhile exercise to try to translate mathematical arguments into this computational framework, as it illustrates how some parts of mathematics are in some sense "more infinitary" than others, in that they require a more infinite amount of computational power in order to model in this fashion. It also illustrates why we adopt the conveniences of infinite set theory in the first place; while it is technically possible to do mathematics without infinite sets, it can be significantly more tedious and painful to do so. </quote>
From: Vesa Monisto on 10 Jul 2010 20:01 "Marshall" <marshall.spight(a)gmail.com> wrote: > >> On Jul 10, 2:22 pm, Don Stockbauer <don.stockba...(a)gmail.com> wrote: >> You can save yourself a lot of bother if you just go with the >> cybernetic interpretation of infinity: >> >> http://pespmc1.vub.ac.be/INFINITY.html > > I felt myself getting stupider with each sentence of that > article that I read, so I stopped early. I expect a complete > recovery. "To find his stupidity is the first step to get rid of it". ;) The link Don gave is worth to read! Curt gave the idea of infinity in the form "10 goto 10". I gave the idea in Basic; here even in more conventional notation: Deduction called "mathematical induction": T_(n+1)_ = T_n_ + 1 Read: "Let the next term T_(n+1)_ be the earlier term T_n_ plus 1 in-finitely = continuously without Exit from the loop". In Basic: A = 0 Step: A = A+1 : Print A : Goto Step End Fibonacci sequence: T_(n+1)_ = T_n_ + T_n-1_ Read: "Let the next term T_(n+1)_ be the sum of two earlier Terms T_n_ and T_n-1_ in-finitely = continuously without Exit". In Basic: A = 0 B = 1 : Print B Fibo: C = A+B : Print C A = B B = C Goto Fibo End Homework: Write the same for fractals "zoomable ad infinitum". The idea: The programms are giving the stepping potential (= possibility) for to actualize the stepping process without Exit (= halting) from stepping. Even the programms are finite (Ending), the stepping process is infinite (never-ending), i.e., -> impossible. You can step endlessly, first forward, then shortening steps for to step at your place, and with negative steps even backwards ..... if you need a re-covery. -- Halting helps, too! V.M. (Just 'fun for the road'.)
From: Curt Welch on 10 Jul 2010 20:30 George Greene <greeneg(a)email.unc.edu> wrote: > On Jul 7, 2:36=A0pm, c...(a)kcwc.com (Curt Welch) wrote: > > You have to be careful in such debates to understand what "exists" > > means and how you are using it. > > You MEANT to say that ONE has to be careful. That's exactly right. I have a bad habit of writing "you" when I'm meaning "one". > *I* certainly do not have to be any more careful than usual. > I am already aware of the point you are trying to make here. > You are in no position to lecture me personally. Well, this is Usenet. We are both in a fine position to lecture anyone we damn well feel we want to here! As you have done to me now as well! > You have to be careful in these debates to understand what "you" > means, and how you are going to be perceived as using it. Yes, I do agree! -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/
From: Curt Welch on 10 Jul 2010 20:35 George Greene <greeneg(a)email.unc.edu> wrote: > On Jul 7, 2:31=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > c...(a)kcwc.com (Curt Welch) writes: > > > But math isn't about simple physical existence. =A0It's about words > > > and their definitions. =A0It's a study of formal langauge and what > > > can be _said_ using a formal language. > > > > No it's not. > > I wouldn't've been that simplistic about it, but I'll take your side > on this, > even though I usually defend that math is formal. NOTHING CAN be said > with a formal language. That's THE WHOLE POINT; it's FORMAL. > It doesn't MEAN ANYthing. It doesn't even NEED to mean anything. > To the extent that math DOES mean something, there is more going on > than "formal language". > ANYthing "can" be said using a formal language, once you admit the > possibility that formal languages > can say things. You can just stipulate (don't ask ME how -- the SAME > way the DICTIONARY does it; > THAT'S how) -- that this or that formal gibberish MEANS whatever. Well, that's a good point. The axioms of meaning on which the language is built has to come from the real world. Without it, the language can have no meaning at all. But even though they originate from reality, they are often twisted to a point that doesn't exist in reality, such as this point about absolute truth, instead of "nearly absolute" truth. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/
From: Curt Welch on 10 Jul 2010 21:27
"K_h" <KHolmes(a)SX729.com> wrote: > "Curt Welch" <curt(a)kcwc.com> wrote in message > news:20100708174523.499$Zc(a)newsreader.com... > > "K_h" <KHolmes(a)SX729.com> wrote: > >> "Curt Welch" <curt(a)kcwc.com> wrote in message > >> news:20100708093928.442$LY(a)newsreader.com... > >> > "|-|ercules" <radgray123(a)yahoo.com> wrote: > >> >> "Curt Welch" <curt(a)kcwc.com> wrote... > >> > > >> >> I realize the difficulty in confirming a rock exists. But all you > >> >> have to do is confirm *something* exists. Even if you're in error > >> >> the conclusion is still true. E x > >> >> > >> >> Herc > >> > > >> > That's an interesting point. I don't see any argument against the > >> > idea that something exists is an absolute truth. I think therefore > >> > I am. That might be the one and only absolute truth. > >> > >> Mathematical truth exists. To my recollection, I have never seen > >> anybody claim that 2x7=14 is false or fails to be true after somebody > >> dies. > > > > That's generally true. The question is not normally thought about. > > Humans tend to think and talk as if there are always humans around. > > > > But what happens if _everyone_ that understands the language died? > > What does it mean to suggest the "truth of the statement lives on" at > > that point? All that is left is ink marks in books at that point. > > Since when > > The truth embodied in the statement lives on. Of course, the symbols > 2x7=14 are conventions. Then if you believe that, tell me how it lives on without doing what you have done so far, which is simply by you saying, "because I said so". What exactly do you mean by "it lives on"? In what form does it exist which allows it to "live on"? Explain to me what you believe is "living on" here. The problem here is that it's very easy for us to think just like you seem to be thinking and talk as if "truth" can "live on". Especially a very fundamental truth of logic and language and space, such as what is represented in the language "2x7=14". Even if the Earth and the human race was exterminated, it's a "truth" that could be rediscovered by some other alien life form a million years from now. In that sense, we can say it's a truth that exists "out there" an not "in us". But what exactly is the truth that "lives on" here which we are talking about? Let me show another "truth". When I push the key that has a label of "e" on my keyboard, the symbol "e" shows up on my screen. This is a fundamental "truth" of the universe. The "truth" embodied in the language of this paragraph "lives on" even after I die, and even after all copies of this paragraph have been destroyed. But does it "live on" after the Earth and everything on it, and everything we have produced, and every creature that might have some memory of our language and our computers is gone from the universe? No, it doesn't. The truth expressed in the language would be gone from the universe at that point. So what physical thing are you making reference to when you say truth embodied in the statement 2x7=14 lives on? My point is that the physical universe is the only existence here. And if we are suggesting something actually exists, and "lives on" we _must_ be making reference to some physical attribute of the universe. So what physical attribute of the universe do you think is the truth embodied in that particular lagniappe statement? When we talk about ideas living on, we are making reference to some aspect of human brains (their behavior maybe), or maybe the language we write in books, but we are always making reference to one or more physical things in the end. Because of the abstract nature of mathematical language, it can be a bit hard to nail down what we are making reference to when we use the language. I'm open to arguments about what it might be that we are making reference to - it's valid to argue a few different stances there I suspect. But no matter what the position, it MUST be a reference to some physical aspect of the Universe, or we have to express some valid argument about why it doesn't have to be. I suggest it's a reference to a fundamental aspect of how brains (and machines like brains - such as our computers) work. Before we can have simple math, we need the ability to delineate objects or concepts. We need to know that the big rock, is _separate_ _from_ the small rock. This ability to slice up our sensory environment into "thinks" is so fundamental to how or brain works, that it becomes a fundamental nature of how we understand the universe. We see the universe not as some large continuum of energy, but instead, as a world full of separate "things". This ability to see A as different from B, is needed, before the axioms of mathematical language can be written. And it's an important part of the "fundamental truths" that lives on the language "2x7=14". But it's not a fundamental truth of the universe, as much as it's a fundamental truth of MACHINES that produce a sensory representation of the universe and divide that sensory representations into some internal representation of separate "objects". Once you have one of these machines, we get closer to the truth embodied in 2x7=14,. But without that machine, that "truth" really doesn't exist at all. If the universe goes back to being a large cloud of plasma, where does the truth in the math of 2x7=14 live on? > > But what do you think you are making reference to when you say it > > exists? Where exactly does it exist? What form does it exist in? How > > do truths (absolute or otherwise) even exist at all? > > This question doesn't need to be answered in order to know that these > truths always exist. How does anything exist? How does the galaxy > exist? People don't need to have those answers to know that they exist. That's just not true at all. What people in fact don't need, is to know the truth. They say things, simply because they have been conditioned to say them, whether they understand the meaning of what they so or not. They say "God exists" and they believe with all their heart that this is true, but yet, they don't have a clue what "God" actually is, or what "exists" actually means. They don't need to. The reasons they say it go far beyond "truth". It in fact has nothing to do with truth. And they don't for a second need to understand the reasons they say it. Likewise, you don't need to understand why you say "The truth embodied in the statement lives on". You also don't need to know what it means. All you need to know, is that it is the "right thing to say". That is, you have been conditioned to talk, and think that way, by the society you live in. But, as you might guess, I find it interesting to try and understand the real meaning of these sorts of things. I find it interesting to try and understand why we throw around a word like "exist" so haphazardly. It all grows from my interest in AI and building machines that act like humans - including talking in the fun and interesting way humans talk about these things (and everything else). Where does our view of reality come form that allows us to talk in these ways? Why do we so easily talk as if something is "living on" about these marks we scratch on paper when the "something" is not something obvious that lives on, like our sun lives on after we die? None of these questions and issues are the least bit important for doing mathematics. We can do math just fine by assuming the fundamental "truths" of math and logic always "lives on" without or without a universe full of mathematicians. But to fully understand the universe, and humans, so that we can do things like build AI machines, these little philosophical points become interesting (and somewhat important) debating points. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/ |