Math "Disproofs"
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16-09-2012, 09:18 PM (This post was last modified: 16-09-2012 09:22 PM by Starcrash.)
RE: Math "Disproofs"
Girlyman, reasonable doesn't mean what you think it means. It means being open to reason, accepting arguments and possibly allowing them to change your mind.

Have you once doubted your position? Or do you read my posts thinking "let's see where he's wrong so I can attack it"?

I thought the Cantor argument was bullshit when I first saw it, but the argument about .999... = 1 convinced me for quite a while. How did I change my mind? Because the math didn't work. I actually prefer the idea that .999... = 1, because it means I could keep on surprising people with another counter-intuitive but clever thought. I prefer the truth always, though, so I've abandoned that line of thinking for my current one. Are we all agreed that the truth is our goal, or is this about "winning"?

I've been reasonable for a long time now. When I'm wrong, I admit it. I'd rather be as close to right as I can be now rather than to be found out wrong -- that's not embarrassing for me, because I can admit to be human and fallible. I'd be embarrassed if I was presented with a strong argument that I couldn't rationally refute and I stuck to my old beliefs.

Logic is the key to reason. The way you use the terms "valid" and "sound" (incorrectly) makes me doubt that you understand logic. You probably have some knowledge about what it is, and you could probably pick it out of a police lineup, but you don't know how it operates. And that's a problem... a logical argument won't work unless the person (or people) I'm arguing with accepts the rules of logic and abide by them.

My answers tend to be quite long, because I wish to explain my thinking as thoroughly as I can, and it's frustrating when I get back a short reply -- not just because of its length, but because it doesn't address my whole argument, and often doesn't even address any part of my argument.

Chas, you keep bringing up the phrase "reductio ad absurdum" over and over again (while simultaneously claiming that your arguments are made up of "different approaches, different wording, different analogies, different metaphors"... that actually sounds more like my approach, not yours) and claiming that I don't understand it. Here's a bit from my college textbook on logic about this subject.

Quote:You may have noticed something in particular about this argument. Looking at the consequent
of the conditional, “N → (P & ~ P)” is “P & ~ P”, you can see from the fourth row of the
truth table, “P & ~ P” is false on all possible truth-values. A sentence that is false under all
possible truth-values is called a contradiction; it is a sentence that cannot possibly turn out to
be true. This is why NewBrandToys is in trouble, of course: if they want to stay in business,
at least according to this argument, they have to do something that cannot be done (raise
prices and not raise prices); they have to make a contradiction be true, which it never is. In
fact this argument form is very old and very well known—old enough to be referred to by
the Latin term for “reduce to absurdity,” reductio ad absurdum. In this case, it is absurd to
raise and not raise prices; showing that it must do so to stay in business is to show that the
hope of staying in business under such conditions makes no sense, or is absurd.

Obviously I know what it is. Not only could I have looked it up in my textbook, but I could've just gone to Wikipedia. It's not a form of logical reasoning, or at least not a valid one. As the paragraph above makes clear, reductio ad absurdum is an attempt to make a contradiction true. I understand what it is. Do you? You never made an attempt to explain it to me, despite your belief that I didn't understand it. Perhaps you didn't because you didn't know what it was.

Now I know that you believe this contradiction is true, despite it being a contradiction. Did you ever consider that it might not be valid, and just a contradiction? The odds were in favor of that. Most contradictions turn out to be logically untenable.

But it feels like I'm just banging my head against a wall. I would really love to hear a good argument. You're not the only one I've debated this with, and I've had fruitful discussions on this topic. This isn't one of them. I strongly feel that there is literally nothing that I could present that would change your mind.

Now, you may think "that's not fair, because there's nothing that would change your mind!" Not only is that untrue, but I'll be happy to demonstrate things that would: Give me an example of a number created by Cantor's diagonal that is a real number but not found among the set of all real numbers. I don't just want an example of "how it works", because I understand the method. I want to see it work in practice, not just in theory.

Of course, you can't type out an infinitely long number (and any length of number will still easily be refuted), so why don't you actually address my argument? Tell me how a number that is not among the set of all real numbers could still hold the characteristics of a real number. Or explain why Cantor's diagonal can't be used in a mirror method with integers. You did claim it was a clever idea, but you never said why it wouldn't work. If it does, then even if Cantor's argument is true, it still doesn't create unmapped numbers because you can make a mirror mapping with the same method.

My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her non-stick pan.
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17-09-2012, 05:24 AM (This post was last modified: 17-09-2012 07:17 PM by Chas.)
RE: Math "Disproofs"
(16-09-2012 09:18 PM)Starcrash Wrote:  Chas, you keep bringing up the phrase "reductio ad absurdum" over and over again (while simultaneously claiming that your arguments are made up of "different approaches, different wording, different analogies, different metaphors"... that actually sounds more like my approach, not yours) and claiming that I don't understand it. Here's a bit from my college textbook on logic about this subject.

Quote:You may have noticed something in particular about this argument. Looking at the consequent
of the conditional, “N → (P & ~ P)” is “P & ~ P”, you can see from the fourth row of the
truth table, “P & ~ P” is false on all possible truth-values. A sentence that is false under all
possible truth-values is called a contradiction; it is a sentence that cannot possibly turn out to
be true. This is why NewBrandToys is in trouble, of course: if they want to stay in business,
at least according to this argument, they have to do something that cannot be done (raise
prices and not raise prices); they have to make a contradiction be true, which it never is. In
fact this argument form is very old and very well known—old enough to be referred to by
the Latin term for “reduce to absurdity,” reductio ad absurdum. In this case, it is absurd to
raise and not raise prices; showing that it must do so to stay in business is to show that the
hope of staying in business under such conditions makes no sense, or is absurd.

Obviously I know what it is. Not only could I have looked it up in my textbook, but I could've just gone to Wikipedia. It's not a form of logical reasoning, or at least not a valid one. As the paragraph above makes clear, reductio ad absurdum is an attempt to make a contradiction true. I understand what it is. Do you? You never made an attempt to explain it to me, despite your belief that I didn't understand it. Perhaps you didn't because you didn't know what it was.

A reductio ad absurdum proof is not an attempt to make a contradiction true.
It is a proof by contradiction. That is, we make an assumption P, then by logical deduction we arrive at ~P, or to Q & ~Q. This shows that the assumption P is not true.

Quote:Now I know that you believe this contradiction is true, despite it being a contradiction. Did you ever consider that it might not be valid, and just a contradiction? The odds were in favor of that. Most contradictions turn out to be logically untenable.

No, I don't believe the contradiction is true, I believe the contradiction makes the proof true.

Quote:But it feels like I'm just banging my head against a wall. I would really love to hear a good argument. You're not the only one I've debated this with, and I've had fruitful discussions on this topic. This isn't one of them. I strongly feel that there is literally nothing that I could present that would change your mind.

There is nothing I can imagine that will change my mind on this proof.

Quote:Now, you may think "that's not fair, because there's nothing that would change your mind!" Not only is that untrue, but I'll be happy to demonstrate things that would: Give me an example of a number created by Cantor's diagonal that is a real number but not found among the set of all real numbers. I don't just want an example of "how it works", because I understand the method. I want to see it work in practice, not just in theory.

You demonstrate over and over that you don't understand the method of proof. Your description of RAA, including the quote, show a complete misapprehension of it.

Quote:Of course, you can't type out an infinitely long number (and any length of number will still easily be refuted), so why don't you actually address my argument? Tell me how a number that is not among the set of all real numbers could still hold the characteristics of a real number. Or explain why Cantor's diagonal can't be used in a mirror method with integers. You did claim it was a clever idea, but you never said why it wouldn't work. If it does, then even if Cantor's argument is true, it still doesn't create unmapped numbers because you can make a mirror mapping with the same method.

The bolded sentence in the above again show a misunderstanding of the method of proof. The point is that the constructed number is a real number and therefore must be in the set of reals, but is not. That is the contradiction, that is what shows the premise (that we have mapped all the real numbers x, 0 < x < 1) to be false.

I have been thinking about how to use your mirror method to construct the equivalent proof. I haven't come up with it yet, but I'm no Cantor.Consider


But we need to address the other issue, as well. That 1 = 0.9999...

Do you accept that 1/9 = 0.1111... ? That is, the decimal expansion of 1/9 is zero point one repeating?

If so, then you also accept that 2/9 = 0.2222... and 3/9 = 0.3333... and so on.

Skepticism is not a position; it is an approach to claims.
Science is not a subject, but a method.
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17-09-2012, 06:08 PM
RE: Math "Disproofs"
(16-09-2012 09:18 PM)Starcrash Wrote:  Girlyman, reasonable doesn't mean what you think it means. It means being open to reason, accepting arguments and possibly allowing them to change your mind.

Fair enough, Starcrash, I've always considered reasonable and rational as synonymous but I can admit a looser definition. I will rephrase my position.

"There are only 2 rational ways to attack the argument, all others are unreasonable."

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