Math "Disproofs"
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13-09-2012, 05:18 PM (This post was last modified: 14-09-2012 01:16 PM by Chas.)
RE: Math "Disproofs"
(12-09-2012 10:28 PM)Starcrash Wrote:  
(11-09-2012 07:25 AM)Chas Wrote:  I suggest you study up on infinite series and induction to aid your perception of countable infinity.

Also, you are still missing how a reductio ad absurdum proof works.

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.

Let me put this in a logical proof. Please, point out where it is invalid or unsound.

Premise 1: Cantor's diagonal argument creates a number, N.
Premise 2: If a number is not found in the set of all real numbers, then it is not a real number.
Premise 3: N is not found in the set of all real numbers.
Conclusion: Therefore, N is not a real number.

Good luck with that.


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 one-to-one 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.

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14-09-2012, 06:53 PM (This post was last modified: 14-09-2012 10:52 PM by GirlyMan.)
RE: Math "Disproofs"
(13-09-2012 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.

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15-09-2012, 03:14 PM
RE: Math "Disproofs"
(13-09-2012 05:18 PM)Chas Wrote:  
(12-09-2012 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.

Let me put this in a logical proof. Please, point out where it is invalid or unsound.

Premise 1: Cantor's diagonal argument creates a number, N.
Premise 2: If a number is not found in the set of all real numbers, then it is not a real number.
Premise 3: N is not found in the set of all real numbers.
Conclusion: Therefore, N is not a real number.

Good luck with that.


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 one-to-one 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.

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.

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15-09-2012, 03:20 PM
RE: Math "Disproofs"
(14-09-2012 06:53 PM)GirlyMan Wrote:  
(13-09-2012 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.

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 re-iterate 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 0-9, 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.

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15-09-2012, 03:27 PM
RE: Math "Disproofs"
(15-09-2012 03:14 PM)Starcrash Wrote:  
(13-09-2012 05:18 PM)Chas Wrote:  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 one-to-one 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.

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.

And I am going to conclude that you do not understand reductio ad absurdum proof.

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15-09-2012, 03:29 PM
RE: Math "Disproofs"
(15-09-2012 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.

(15-09-2012 03:27 PM)Chas Wrote:  
(15-09-2012 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.

And I am going to conclude that you do not understand reductio ad absurdum proof.

I think that's a sound and valid conclusion.

As it was in the beginning is now and ever shall be, world without end. Amen.
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Your shadow at morning striding behind you
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15-09-2012, 03:33 PM
RE: Math "Disproofs"
(15-09-2012 03:20 PM)Starcrash Wrote:  
(14-09-2012 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.

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 re-iterate 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 0-9, 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.

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.

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15-09-2012, 03:54 PM (This post was last modified: 15-09-2012 03:58 PM by GirlyMan.)
RE: Math "Disproofs"
(15-09-2012 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?

As it was in the beginning is now and ever shall be, world without end. Amen.
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Or your shadow at evening rising to meet you;
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15-09-2012, 04:12 PM
RE: Math "Disproofs"
(15-09-2012 03:54 PM)GirlyMan Wrote:  
(15-09-2012 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?

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.

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15-09-2012, 05:03 PM
RE: Math "Disproofs"
(15-09-2012 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.

As it was in the beginning is now and ever shall be, world without end. Amen.
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Your shadow at morning striding behind you
Or your shadow at evening rising to meet you;
I will show you fear in a handful of dust.
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