Question.... (Processes of Logic)

Well I am obviously out of my field I was writing with vague memory of a book by D. Hofstadter godel escher bach sure you no the one . I went and got it off the shelf to see where I screwed up. Hofstadter writes his interpretation all consistent axiomatic formulations of number theory include undecidable propositions is this a good representation of the theory? In any case I see the problem with what I wrote.Plus, the two concepts of incompleteness and inconsistency are the same thing...the difference is just in how you write things down.As I read it incompleteness in theory ie. a incomplete set of truths, is different to a set which has inconsistency’s ie untruths and truths?Any thoughts?

Autocatalysis responds to me:

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Well I am obviously out of my field I was writing with vague memory of a book by D. Hofstadter godel escher bach sure you no the one .

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Yep...it's one of those books that a lot of people admire but few have actually read all the way through, every word, beginning to end. I've tried and it's tedious (of course, the fact that I don't own a copy and thus have to read it in snatches from other people's copies doesn't help.)

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Hofstadter writes his interpretation all consistent axiomatic formulations of number theory include undecidable propositions is this a good representation of the theory?

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Yes, because it deals with both aspects. The first is that we're dealing with systems that are somewhat sophisticated (number theory) and the second is that he establishes that if you insist upon consistency, then you end up with undecidable. If you don't insist upon consistency, then you can end up with every statement being decidable.

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Plus, the two concepts of incompleteness and inconsistency are the same thing...the difference is just in how you write things down.

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As I read it incompleteness in theory ie. a incomplete set of truths, is different to a set which has inconsistency’s ie untruths and truths?

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Any thoughts?

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An inconsistent set can be turned into an incomplete set simply by changing the way you process the work.That is, you can write things down such that you can prove both A and ~A, or you can write things down such that neither A nor ~A are provable.For example, take the Continuum Hypothesis. Godel proved that if you assume it to be true (c = aleph-one), you don't get a contradiction. Cohen then came along and showed that if you assume it to be false (c != aleph-one), you don't get a contradiction, either.So is this incompleteness or inconsistency? Depends on how you decide to look at it. Since most mathematicians would much rather have an incomplete system rather than an inconsistent system, we tend to look at things as incomplete and thus they're called "incompleteness" theorems rather than "inconsistency" theorems.------------------
Rrhain
WWJD? JWRTFM!

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Right, but if you assume that the map is the same as the physical reality, then you confuse the model with the reality. You reify, in other words.

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Hmm, thats not quite what I meant. We have no choice but to treat the map as if it were reality, because the map is our method for comprehending reality. It is not a case of reifying the map, becuase we have no input but the map. You only see a spot about half a centimeter across at any given moment; your perception of your environment is wholly illusory, a synthetic "image" knitted from spot observations, and yet pretty accurate. If you can see a cliff in front of you, you stop moving forward.

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If this wasn't true, how could maps be wrong? We know that some maps are wrong, however, suggesting that maps are simply another kind of mental model. Based on reality, sure, but no more real than any other mental models.

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It is indeed possible to fool the eye and seeing is not believing. However, that does not imply that our perception of reality is wholly false; it is merely qualified. But given the Uncertainty principle, huge amounts of entirely non-illusory physical processes are also known to be understandable only in qualified ways.I can accept that our representation of mathematics is conctructed. But I do not accept that the physical processes we describe are therefore also constructs. Whether you call X times objects on a table X or Y has no impact, no relevance, to the objects.

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I've never heard an argument for the ontological existence of mathematics that couldn't be applied to Monopoly, as well.

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Yes that was rather the point. The only serious position from which this claim can be advanced is the solipsistic fallacy that denies the existence of a material environment.

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I can accept that our representation of mathematics is conctructed. But I do not accept that the physical processes we describe are therefore also constructs.

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The trick is that we understand the physical processes largely through our constructs. I think this is what Kant glimpsed when he divided the world into noumena and phenomena. Anyway... not argument-- just wanted to point out that most of us here seem to be defending that first sentence, while Rhhain does the reverse.------------------
No webpage found at provided URL: www.hells-handmaiden.com

No, John. You've misrepresented my position.------------------
Rrhain
WWJD? JWRTFM!

You have repeatedly claimed that mathematical entities are real things and that mathematics were discovered, not invented. Invented == constructed. Now you are going to say it ain't so?------------------
No webpage found at provided URL: www.hells-handmaiden.com

No, I'm not going to deny that, John.However, you weren't talking about mathematics. You were talking about our representation of mathematics and that, indeed, is invented.------------------
Rrhain
WWJD? JWRTFM!

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You were talking about our representation of mathematics and that, indeed, is invented.

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What of mathematics do we have that IS NOT our representation of it?------------------
No webpage found at provided URL: www.hells-handmaiden.com

John responds to me:

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What of mathematics do we have that IS NOT our representation of it?

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The mathematics that exists in reality.Surely you're not saying that an actual apple is dependent upon us calling it an "apple," using those specific letters, pronounced in that specific way, are you? An apple exists regardless of the representation we use to describe it.You confuse the map for the terrain.------------------
Rrhain
WWJD? JWRTFM!

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The mathematics that exists in reality.

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The question, Rhhain, is which part is the REAL part?You cannot answer the question, "Which is the real part?" with "The part that is real." Come on. You can do better than that.

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An apple exists regardless of the representation we use to describe it.

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Yes, AN apple does exist regardless of what we choose to call it. But Apple with a capital 'A' is another story altogether. There is no such animal, no matter what we choose to call it. It is an abstraction. Math is in the later category, not the former.

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You confuse the map for the terrain.

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If this is the case I am sure you can point to a mathematical entity-- a five, for example. Not five things, which is an abstraction. Not a character or a word which are mere representations of an idea, but to the real thing-- Five in its unadulterated glory, in the same sense as you can point to an apple.------------------
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John responds to me:

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The mathematics that exists in reality.

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The question, Rhhain, is which part is the REAL part?

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Oh, you're not about to equivocate on the world "real" are you?

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You cannot answer the question, "Which is the real part?" with "The part that is real." Come on. You can do better than that.

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But what else is there? I've given you plenty of examples. For example, the five fingers on your hand. Or are you saying that if you think about it really hard, you can change it to six?

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An apple exists regardless of the representation we use to describe it.

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Yes, AN apple does exist regardless of what we choose to call it.

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Precisely. AN apple.As in ONE apple.Or are you saying that if you think really, really hard, you can turn AN apple into SOME apples?

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But Apple with a capital 'A' is another story altogether.

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Huh? Didn't we just agree that it didn't matter what you called it? An apple is an apple?

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There is no such animal, no matter what we choose to call it. It is an abstraction.

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Um...last time I checked, my friends all used Apples. They don't like Windows.

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Math is in the later category, not the former.

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You mean you don't have five fingers on your hand?

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You confuse the map for the terrain.

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If this is the case I am sure you can point to a mathematical entity-- a five, for example. Not five things, which is an abstraction.

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No, five things is a concrete example of five.Let me see if I can understand your logic. You want me to show you a real version of an "apple," but you will not accept a piece of fruit because that's an "abstraction"?Are you saying you don't have five fingers?

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Not a character or a word which are mere representations of an idea, but to the real thing--

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I have never been referring to the symbology. That's crash's problem. He seems to think that if we call it "six," it really is six.Are you saying you don't have five fingers on your hand? That if you think really, really hard, you can actually make another finger appear?

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Five in its unadulterated glory, in the same sense as you can point to an apple.

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I've been pointing to your hand for months and the fingers that are on it.Or do you have some other number of fingers on your hand? Can you make the number change just by thinking about it?------------------
Rrhain
WWJD? JWRTFM!

You've asked this about a hundred times, and it's been a straw man every time - changing five to six is not a necessary result of math being simply another kind of language or conceptual framework instead of a really real thing. It's as dumb as saying you can change who's in the Oval Office now by voting really hard. It's meaningless. I can't change the meaning of words just by thinking hard, even though language is just a mental concept. Why should numbers change under the same circumstances? It's only "really" six (in my defense) because "six" isn't real. A fruitful (pardon the pun) question might be - would numbers exist if there was nothing to count? If so, where would they be?

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Oh, you're not about to equivocate on the world "real" are you?

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No. I am not nearly as equivocative as yourself. Nor do I avoid the question by suggesting, without cause, that someone is about to make a logical fallacy.

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But what else is there?

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Lol..."What observable events are God's doing?" "Why, the part that God did.""Which elf shot Santa?" "Why, the elf that shot Santa, of course.""Which number is the largest of them all?" "Why, the largest one, obviously."Can't you see how stupid that answer is?

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I've given you plenty of examples. For example, the five fingers on your hand.

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So, you've proven that I can group objects. Where is the proof that number exists outside the brain that is doing the grouping?

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Or are you saying that if you think about it really hard, you can change it to six?

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This isn't mathematics, Rh. It is physics. We know that one cannot make an extra finger grow by thinking really hard, but this has nothing to do with math. It is a red herring.

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As in ONE apple.

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Congratulations! You've labeled an object! It isn't possible, in English, to refer to an object without refering to a number-- one or many. Let's not trip over language, eh?

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Or are you saying that if you think really, really hard, you can turn AN apple into SOME apples?

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Where did you get this? No one has made any such claim.

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Huh? Didn't we just agree that it didn't matter what you called it? An apple is an apple?

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You seem to have no idea of the meaning of Platonist, which is what you have labeled yourself. Strange...apple == an object on the kitchen table
Apple == the essence of apple, its perfect formSee, Platonism works like this. Everything we observe, every object is a reflection of its perfect Form. ( Capitalizing the Form's name has been traditional in Philosophy for quite some time. Can't imagine how a Platonist like yourself could have missed that. ) The Forms are real, the objects we see are merely reflections. Pointing to the 'reflections' DOES NOT prove the existence of the Forms, but only the existence of those reflections. The problem Platonism has is that there is no logical way to get from the 'reflections' to the existence of the Forms. You just have to assume the whole weltanschauung. That leaves us with just the reflections, and reflections without Forms makes no sense. Thus we ditch the Forms and accept what we can actually observe.

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Um...last time I checked, my friends all used Apples. They don't like Windows.

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Cute. And OS X does kick much butt. However, you used to joke to avoid the issue. You are arguing for the existence of abstractions. And are defending the claim by pointing to physical opjects. It doesn't work.

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No, five things is a concrete example of five.

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Nope. Five things is a concrete example of five things. The 'five' is a label, an abstraction-- so is 'one', 'red', 'hard', etc. though they can all be related to something physical. With no one to do the labelling the things would remain-- we assume -- but the label would not, any more than the label 'apple' would exist if there were no English speakers. The things are real, math is a way to group them-- like kinship is a way to group people. It might be based on certain physical relationships but the kinship system is nonetheless an abstraction and does not exist except in the minds of those who know about it. Even such concepts as 'mother' and 'father' are fluid across cultures ( but becoming less so as all the nifty cultures are getting drowned out by the industrial age ).

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Let me see if I can understand your logic. You want me to show you a real version of an "apple," but you will not accept a piece of fruit because that's an "abstraction"?

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But this isn't what is being asked of you. I'll accept an apple as an apple. I won't accept an apple as ONE, which is what you want to do with it. Nor would I accept an apple as Red, Sweet, or as proof of any other abstraction.

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I have never been referring to the symbology.

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Sure you have been. Your arguments are nothing but references to the symbology.The funny thing is that even if we assume the truth of Platonism, your arguments through most of this thread don't work. The only mathematics we can use are our representations of mathematics and this is different from mathematics. See, your post # 156. Unless our representation of math is identical to math then we cannot have the certainty about it that you have insisted we have.------------------
No webpage found at provided URL: www.hells-handmaiden.com

John responds to me:

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Or are you saying that if you think about it really hard, you can change it to six?

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This isn't mathematics, Rh.

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Yes, it is. You are saying that mathematics is something that doesn't exist outside of a person's thoughts.If that were the case, then you could think really hard and change the number of something since number is a mathematical property.But if you can't change the number of something simply by thinking about it, then number must necessarily be something that exists outside of a person's mind.

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It is physics.

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Thus showing that number is a property that has physical existence. If number, which is a mathematical property, can only be changed by physical processes, then number must physically exist.And thus, mathematics physically exists.Thank you for finally coming around.------------------
Rrhain
WWJD? JWRTFM!

crashfrog responds to me:

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Or are you saying that if you think about it really hard, you can change it to six?

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You've asked this about a hundred times

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And it's been avoided every single time.I know you don't like it. I know you don't think it's important. But it is, however, the entire point: If you can't change it simply by thinking about it, then it isn't a mental construct but a physical one.

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I can't change the meaning of words just by thinking hard

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Sure you can. That's exactly how meanings of words change: People think they mean something different. That's how new words come into existence: People think them into being. "Spam" didn't exist as a word until the Hormel company came along and then a bunch of computer geeks started using it to mean not a registered trademark referring to canned pork and pork shoulder but to junk email.It all happened with a thought.

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Why should numbers change under the same circumstances?

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Um, you've got it backwards. You can't change number simply by thinking about it. You can change language simply by thinking about it.

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I've been pointing to your hand for months and the fingers that are on it.

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A fruitful (pardon the pun) question might be - would numbers exist if there was nothing to count?

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Of course.The empty set is still a set, you'll recall.

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If so, where would they be?

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In the nothingness.Now you can understand why it took so long for many cultures to come to terms with zero.And you're learning why many people who don't spend a lot of time in cosmological origins have a hard time conceptualizing the "nothingness" out of which the Big Bang happened...to the point that there was no time, no space, no matter, no energy, etc. Absolutely nothing.------------------
Rrhain
WWJD? JWRTFM!