Math "Disproofs"



13092012, 05:18 PM
(This post was last modified: 14092012 01:16 PM by Chas.)




RE: Math "Disproofs"
(12092012 10:28 PM)Starcrash Wrote:(11092012 07:25 AM)Chas Wrote: I suggest you study up on infinite series and induction to aid your perception of countable infinity. It's not condescension. You are arguing against a premise, an assumption, and the method of proof. You seem to agree with the assumption, but try to construct your own mapping. It doesn't need constructing  it's a premise. It is assumed as truethat we have a onetoone mapping of the natural numbers into the real numbers between 0 and 1. The creation of the 'diagonal number' is valid for each step of its construction, and true due to induction. It is not in the mapping by the nature of the mapping. We assumed that every real was there and mapped to a natural number. The created number is not there, therefore the mapping does not exist. The proof is valid. If you don't accept it, then I think you should create a different proof based on your methodology, which I said elsewhere is very clever. Edit: Your conclusion "Therefore, N is not a real number" is incorrect. The conclusion is that the real number N is not in the mapping, therefore the mapping is incorrect. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

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14092012, 06:53 PM
(This post was last modified: 14092012 10:52 PM by GirlyMan.)




RE: Math "Disproofs"
(13092012 05:18 PM)Chas Wrote: You are arguing against a premise, an assumption, and the method of proof. Think I've pointed this out before but I'll repeat it just for the hell of it. There are only 2 reasonable ways to attack Cantor's argument. One is to reject the premise of infinity altogether and the second is to deny the validity of proof by contradiction. Might just be my little pea brain, but I see no other lines of attack. And I don't see either of those as being vulnerable. There is only one really serious philosophical question, and that is suicide. Camus 

15092012, 03:14 PM




RE: Math "Disproofs"
(13092012 05:18 PM)Chas Wrote:(12092012 10:28 PM)Starcrash Wrote: Can we stop with the condescension? Stick to the arguments at hand. Do you really want to keep getting told "you don't get what I'm saying"? We keep talking past each other  you insist on the argument being "reductio ad absurdum", as if that means that it can break the laws of math and/or logic. It doesn't. I'm going to guess that you don't understand logical deduction. If the premises are true, and the structure is valid (it's a syllogism  if A, then B. A, therefore B  and thus has been proven a valid structure uncountable times), then the conclusion has to follow from the premises. If you were going to argue that my logic was wrong, the conclusion is the one place that isn't reasonable to do such a thing. My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her nonstick pan. 

15092012, 03:20 PM




RE: Math "Disproofs"
(14092012 06:53 PM)GirlyMan Wrote:(13092012 05:18 PM)Chas Wrote: You are arguing against a premise, an assumption, and the method of proof. These are just assertions without evidence. Just because you can't think of other ways doesn't mean they don't exist. I've "reasonably" attacked Cantor's argument in many ways, but there is obviously no arguing with the dogmatic belief in Cantor's argument here. The method I've used here (since I can't get anyone to agree that Cantor's argument doesn't actually produce a number), is prove through logic that even if it could produce a number, that number would not be a real number by definition  thus, it couldn't be added to the set of real numbers in order to increase the size of that set. Let me reiterate it again, for the nth time. When we say that the set of real numbers contains "all" real numbers, what we mean is that it contains "every possible" real number. All real numbers are made up of the digits 09, spread out in a single horizontal line from the right of a decimal point to many finite and infinite lengths. The set of all real numbers contains every possible iteration of those digits in every possible configuration, right through infinity. Thus, if Cantor's argument could produce a number (and I still believe that it can't), then that number is on the list (because it would be among "every possible iteration of those digits") or it is not a real number. Either possibly precludes it from adding to the size of the set of all real numbers. My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her nonstick pan. 

15092012, 03:27 PM




RE: Math "Disproofs"
(15092012 03:14 PM)Starcrash Wrote:(13092012 05:18 PM)Chas Wrote: It's not condescension. You are arguing against a premise, an assumption, and the method of proof. And I am going to conclude that you do not understand reductio ad absurdum proof. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

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15092012, 03:29 PM




RE: Math "Disproofs"
(15092012 03:20 PM)Starcrash Wrote: I've "reasonably" attacked Cantor's argument in many ways, ... No StarCrash, you haven't. Your line of attack is quixotic, not reasonable. I don't think I can help you to understand what part of proof by contradiction you don't get because that form of reasoning is so clear and obvious to me. I do appreciate how constructivists can reject proof by contradiction, but that is not what you are arguing. There are only 2 reasonable ways to attack the argument, all others are irrational and illogical. (15092012 03:27 PM)Chas Wrote:(15092012 03:14 PM)Starcrash Wrote: I'm going to guess that you don't understand logical deduction. If the premises are true, and the structure is valid (it's a syllogism  if A, then B. A, therefore B  and thus has been proven a valid structure uncountable times), then the conclusion has to follow from the premises. If you were going to argue that my logic was wrong, the conclusion is the one place that isn't reasonable to do such a thing. I think that's a sound and valid conclusion. There is only one really serious philosophical question, and that is suicide. Camus 

15092012, 03:33 PM




RE: Math "Disproofs"
(15092012 03:20 PM)Starcrash Wrote:(14092012 06:53 PM)GirlyMan Wrote: Think I've pointed this out before but I'll repeat it just for the hell of it. There are only 2 reasonable ways to attack Cantor's argument. One is to reject the premise of infinity altogether and the second is to deny the validity of proof by contradiction. Might just be my little pea brain, but I see no other lines of attack. And I don't see either of those as being vulnerable. You have not proved that your construction of the mapping of natural numbers to real numbers is complete, that it is a bijection. To assert that a constructed number (the diagonal) is in the mapping, you must first prove that you have a complete mapping. Cantor's proof has no such requirement because the mapping is assumed, and then a contradiction is derived, which denies the assumption. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

15092012, 03:54 PM
(This post was last modified: 15092012 03:58 PM by GirlyMan.)




RE: Math "Disproofs"
(15092012 03:33 PM)Chas Wrote: Cantor's proof has no such requirement because the mapping is assumed, and then a contradiction is derived, which denies the assumption. We can argue about whether or not that formally falsifies the assumption, but that's all I can see to argue about. And I don't see any particularly strong arguments that it doesn't. But this whole business with StarCrash and Cantor brings up another question, TeacherChas. I'm frustrated that he doesn't seem to able to grok proof by contradiction. As a mentor myself (fucking work requires it of me, I don't do it willingly 'cause I'm basically an asshole not a teacher), how do I get someone to grok something which is clear and obvious to me that they just can't seem to be able to grasp? Do I fall back to more elementary examples? What do you do? There is only one really serious philosophical question, and that is suicide. Camus 

15092012, 04:12 PM




RE: Math "Disproofs"
(15092012 03:54 PM)GirlyMan Wrote:(15092012 03:33 PM)Chas Wrote: Cantor's proof has no such requirement because the mapping is assumed, and then a contradiction is derived, which denies the assumption. In my experiences as a tutor, I would keep trying different approaches, different wording, different analogies, different metaphors, and so on. I usually found an approach  but then I could see the student's eyes and body language for feedback. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

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15092012, 05:03 PM




RE: Math "Disproofs"
(15092012 04:12 PM)Chas Wrote: In my experiences as a tutor, I would keep trying different approaches, different wording, different analogies, different metaphors, and so on. I usually found an approach  but then I could see the student's eyes and body language for feedback. Thanks, I think the analogies and metaphors are key. There is only one really serious philosophical question, and that is suicide. Camus 



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