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Author Topic:   Question.... (Processes of Logic)
crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 76 of 210 (40964)
05-22-2003 2:52 AM
Reply to: Message 74 by Rrhain
05-21-2003 10:21 PM


Presburger arithmetic. It models addition, but not multiplication. Presburger showed that there is an algorithm that can decide of any given statement is true or not. Fischer and Rabin then showed that all algorithms that can decide such statements have a runtime of 22cn for some c and n being the length of the statement.
But if it can model arithmetic, isn't it sufficiently complex as to be incomplete? And how could you have addition without multiplication? And what would prevent the insertion of a self-contradicting statement into that algorithm?
Perhaps it's best to open a new thread.
It's the same question as asking what color they are. If they have color, they have number.
Why? To determine their color, all I have to do is look at them. As a non-color blind person their color is immediately apparent. But to know how many there are I have to count them. Are you really saying that color and number are this related? That's a bold statement.
Anyway, you said that I would see "five". Not five fingers, just five. Five in it's unitary nature apart from describing the set of my fingers.
All I see are fingers. Explain to me how I'm supposed to just see "five".

This message is a reply to:
 Message 74 by Rrhain, posted 05-21-2003 10:21 PM Rrhain has replied

Replies to this message:
 Message 77 by compmage, posted 05-22-2003 7:06 AM crashfrog has not replied
 Message 81 by Rrhain, posted 05-24-2003 1:54 AM crashfrog has replied

compmage
Member (Idle past 5153 days)
Posts: 601
From: South Africa
Joined: 08-04-2005


Message 77 of 210 (40985)
05-22-2003 7:06 AM
Reply to: Message 76 by crashfrog
05-22-2003 2:52 AM


crashfrog writes:
Why? To determine their color, all I have to do is look at them. As a non-color blind person their color is immediately apparent. But to know how many there are I have to count them.
Speaking out of turn and with incomplete knowledge I expect this is more a result of how your brain and eyes work. You automatically 'see' the colour (assuming you're not colour blind) while you have to be taught the concept of numbers.
However, I think what Rrhain is trying to get at (correct me if I am wrong) is that different frequencies of light (colour) exist regardless of weather we are aware of it or not, in the same way that you have 5 fingers on each hand (assuming you are normal in this regard) regardless of weather you are aware of number systems or not.
I tend to agree with him.
------------------
He hoped and prayed that there wasn't an afterlife. Then he realized there was a contradiction involved here and merely hoped that there wasn't an afterlife.
- Douglas Adams, The Hitch Hiker's Guide to the Galaxy

This message is a reply to:
 Message 76 by crashfrog, posted 05-22-2003 2:52 AM crashfrog has not replied

Chavalon
Inactive Member


Message 78 of 210 (41008)
05-22-2003 10:52 AM
Reply to: Message 74 by Rrhain
05-21-2003 10:21 PM


Hi Rrhain -
Yes, I do see 5 fingers when I look at my hand, but I don't accept this as evidence that 'fiveness' exists anywhere except in people's minds.
Evidence of colour exists even in a world of colour blind people - assuming they can make things like spectrometers. But how can evidence of 'pure fiveness' exist independent of our conception of it?

This message is a reply to:
 Message 74 by Rrhain, posted 05-21-2003 10:21 PM Rrhain has replied

Replies to this message:
 Message 79 by crashfrog, posted 05-22-2003 1:23 PM Chavalon has not replied
 Message 82 by Rrhain, posted 05-24-2003 1:57 AM Chavalon has replied

crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 79 of 210 (41025)
05-22-2003 1:23 PM
Reply to: Message 78 by Chavalon
05-22-2003 10:52 AM


And why is math "discovered", while baseball and Monopoly were "invented"? (To bring in Schraf's point again.) What's the difference? How do we tell?

This message is a reply to:
 Message 78 by Chavalon, posted 05-22-2003 10:52 AM Chavalon has not replied

Replies to this message:
 Message 83 by Rrhain, posted 05-24-2003 1:58 AM crashfrog has replied

John
Inactive Member


Message 80 of 210 (41071)
05-23-2003 2:17 AM
Reply to: Message 64 by Rrhain
05-21-2003 9:08 PM


quote:
You're not about to invoke Godel, are you?
Well, no, not really. Just explain how i does not violate the basic rules of multiplication.
quote:
Be careful, because the Incompleteness Theorems do not say that all axiomatics systems are necessarily incomplete or inconsistent.
Now, why should I be careful? Godel's proof may not apply to all systems but it damn sure does apply to those systems in question-- those very systems you rely upon. So why should I be careful?
quote:
Not all axiomatic systems are that sophisticated and, indeed, they are both complete and consistent.
Godel's proof works for systems that are almost trivially simple, but you know this. Why are you pretending that something significant can be constructed that does not violate that proof? Sounds to me like you are just being diversionary, and effectively just skipping around the issue-- which, if you've still missed it, is that logic and mathematics which you have made out to be absolute and infallible are the very things which get screwed by Godel's proof.
quote:
When did we agree on the axiomatic system?
What are you talking about? Seriously, I don't know where this came from.
quote:
But by being true, it necessarily results in certain other things being false.
Let me put it this way, when you attempt to prove a conclusion, or formulate an argument, do you attempt to create true argument, or a false one? If you attempt to create a true argument, you are by default proving a positive. Reductio ad absurdam works similarly. You have an argument you hope to be true. You reverse the conclusion, thereby creating a false argument. If indeed it does prove to be contradictory, you can infer that the original arguement was true. And you are in the same position-- that of proving a positive. But even when you have reversed the conclusion and found a contradiction, you still have a true argument-- false premises, false conclusion, true argument.
quote:
And that statement just might be "It is true that this is false."
Yes. Godel again. Thank you for pointing that out. But why? I am not claiming that math, or logic, is an unquestionable arbiter of truth. This doesn't hurt me.
quote:
We're not talking about my sister's hair. We're talking about mine.
Prove any statement, about your hair, that is not true-- the statement as a whole, not just 'the conclusion has a ~ in it."
quote:
But sometimes the only way to show something to be true is by showing something else to be false.
Why, if something is true, is the ONLY way to prove it true by proving something else false? Doesn't make sense. The only practical way, perhaps. It can certainly be easier, but it is all conditional upon whatever system you used to make the inference. There is no guarantee that that system is correct.
quote:
I know. But you're arguing a completely different point.
Different from your point, perhaps, but germaine to the issue that started this in the first place.
quote:
Who said anything about drawing anything?
ummmm.... I believe that I did. And, in fact, you brought up drawing with a compass and straight edge in an earlier post. It was just up in post #48.
quote:
You do understand the difference between necessary and sufficient, yes?
And do you understand what is sufficient support for a universal statement such as X can or cannot exist? You need have absolute knowledge of the universe. Or have a guaranteed system for working out answers, which would be the equivalent of absolute knowledge. You have neither, we have neither, but you still insist on making absolute claims about existence.
quote:
Excuse me? We're talking about existential mathematical statements and somehow the existential operator isn't relevant?
I never said we were using a existential mathematical operator. You assumed that and have been tripping over it ever since. I, as I have explained repeatedly, used another meaning of the term 'existential.' It is a meaning introduced to this discussion by crash or maybe Mr. P. and, I believe, first used in this way by a logician named Adler. It is comical that you can't grasp this.
quote:
Indeed.
So you agree that if there is a largest prime then our system of mathematics is contradictory? That was the question.
Now, how is it you are sure that our system of mathematics is NOT contradictory? Assumption? Ya just can't imagine it being otherwise?
You have a modus tollens. P = largest prime. M = math is contradictory.
So: if P, then M.
~M
therefore ~P.
The trick is getting that not-M. You have valiantly avoided the fact that it simply isn't true. Math is contradictory. Lots and lots of people have been struggling with the consequences for a nubmer of years.
quote:
We do not even attempt to "fill in the blank" with a specific number because the specific number is not necessary.
The proof depends upon the fact that you COULD do so. That is what variables are all about. No amount of semantic BS will change that. The first conclusion drawn in your proof is that there is a number larger than the assumed prime, whatever that assumed prime may be. That is an infinite series of positives, which you then translate into English as a negative.
quote:
There is me and the complement of me, "not-me." What are the elements of "not-me"? They are the infinite possible things that are not in the set of "me."
All that math has given you a false sense of precision. In the real world, you'll never pin down 'me' or 'not-me.' There are infinite elements to the definition. Of course, you will object that not all elements need to be included. For practical purposes, this is true, and you end up with grossly truncated definition. But for statements that are universal absolutes, it is not. Every time you claim something can or cannot be because math says so, you are making a statement about a universal absolute.
quote:
or do you claim that the objects of mathematics aren't real?
Geez!!!! I've said so more times than I remember. There is no reason to believe that math is anything more than a symbolic system we made up, and of course we made it match the world we observe but there is no guarantee that we got it right or that it applies to the whole universe.
quote:
Just because observation is consistent with prediction doesn't mean the theory is "true."
Nor does it mean contrary or variant theories are false, as you appear to be arguing.
quote:
So instead, science sets out to chip away the false things so that what is left behind is the most consistent with observation.
Science is the attempt to reduce the complex world to formulas simple enough that we can comprehend them. We adopt the simple an elegant as 'the truth' but the absurdly complex might actually be true.
quote:
You have hacked my post down to such small statements, including single words, eliminating all context, that I am having a hard time following you.
Admin(s), not long before you arrived here, requested we keep the size of messages down. Drive space costs money. Sorry. I know it can be hard to follow.
quote:
No. Do I need to crack open Elements and give you the definition?
Do you actually believe that Euclid invented geometry? Euclid died in 265 BC. The Egyptians had knowledge of geometry 2500 years earlier. So did the Babylonians, so did the Sumerians, so did Greeks long before Euclid. Do you actualy believe the Egyptians, who made damn near square bases for there pyramids, had no knowledge of geometry? Come on! This statement is staggering in its ignorance.
quote:
I hate to break it to you, but there was this guy named Euclid and he collected the works of other mathematicians, as well as coming up with some work on his own, and came up with an axiomatic system of geometry by which circles, triangles, and squares were not a "generalization from the specific to the universal" but rather were definitions.
Lol... Where do you think Euclid, and the mathemmaticians from whom he draws, first got these definitions? Pulled them out of thin air? Got drunk and made up stuff? Nope. The definitions were matched to what they observed around them. That is why euclidian geometry is that of a plane. The world Euclid could observe was planar. That is also why no one questioned Euclid until fairly recently. That is also why euclidian geometry ain't universally applicable the real world-- the real world, on a large scale, isn't what Euclid observed. He didn't have the technology to observe anything different, so he created an abstraction of his tabletop, essentially. And it is a generalization from the specific to the universal-- assuming, of course, that you consider it to have a relationship with the real world. Euclid drew from, say, 20 percent of the world's mathematical knowledge-- or from a small chunk of one planet-- and created a geometry that supposedly applies everywhere. How is this not generalization from the specific to the universal?
Now, you want to call Euclids postulates 'definitions.' How does this help? It only makes the whole structure one more step removed from reality. And you want to argue that one can determine truths in the real world via systems such as this? If geometry is only definition, why should anyone care? Definitions can be made to prove anything. There are no rules. Invoke observation and you are right back to over-generalization.
quote:
Of course not. That's why they're called assumptions. If they were derived from some place else, they'd be conclusions.
Bud, you are making my case. And you are contradicting yourself. You know there are unproven and/or unprovable assumptions and yet still insist that you can get ABSOLUTE answers about the real live observable universe out of axiomatic system.
quote:
Are you saying planes don't exist in the real world?
Perhaps not. Assuming Einstein's conception of gravity, there might not be any true plane anywhere. But, no, that isn't what I was getting at.
Geometry, any axiomatic system, is conditional upon its assumptions. If ( assumptions are true ), then ( system is true ). You know this, which is why it is so hard for me to understand why you feel that you can make absolute determinations based on an axiomatic system. No matter how well the system works, the fact remains that it rests on assumption. What you appear to be doing is arguing that since something is contradictory, or impossible, within an axiomatic system, it is absolutely impossible. This amounts to 'proving' one of your assumptions-- non-contradiction--, which is circular, which is invalid.
quote:
Nope. A circle is contained by a line. A line has more than one point. If space results in a single point, we have no circles, no squares, no lines. Simply a point.
hmmm... then back up from that singularity until you have four points. Is that a circle or a square?
quote:
Things that exist have definition, but not all things that have definition exist.
I think it is fair to say that you have frequently argued otherwise.
quote:
See, this is what I mean about you hacking my post to shreds, completely devoid of context.
Do you think you might have it in you to stop whining about things of which you are also guilty? Just a thought.
quote:
Ad hoc.
Do you actually know what this means? The way you use it makes me think that you don't. Ad hoc is a statement used to brush aside facts that refute an argument. Producing additional evidence or stating previously unstated evidence is not ad hoc. If it were, all of science would be ad hoc. Nor is attempting to explain a previous statement, ad hoc. Nor is introducing additional argument or producing a modified version of an argument, ad hoc. And pointing out the blatantly obvious fact that we do not know everything is damn sure not ad hoc.
Now, my statement was:
Could be. Of two things which appear to us to be the same, or make no difference, one or the other may well be true and the other false. Whether we are ever able to sort this out is another thing.
Do you actually deny that this is an accurate statement? Do you deny that appearances may deceive us? That our current understanding could be wrong? If you do deny these things, then... wow... what can I say? How do you wake someone from that dogmatic slumber? If you don't deny these things, it is absurd to insist on absolutes as you do.
quote:
I gave you examples. That you refused to read them is not my problem.
ummm... when I asked for examples did it not occur to you that I WAS ASKING FOR EXAMPLES? If you gave some and I missed them, sorry. I certainly didn't do it on purpose. More likely, you made some statement or statements which you feel qualify as the examples I am looking for, but which don't look that way to me to the point that I don't even know which ones it is to which you refer.
quote:
And why is it impossible to have absolute and infallible knowledge?
Is that not what definitions are for?

Are you serious? Definitions constitute infallible knowledge? A definition is a tag. It is shorthand. Definitions are names. I can pick any combination of letters and make up a definition. How is that even knowledge, much less infallible knowledge?
Need I direct you attention to where you claim that you do not confuse definitions with reality?
quote:
Then why do I have so many textbooks that ask you to prove that A = A? Surely they think there is a point to it.
I know that some textbooks have this proof in them. I don't know how many such textbooks there are, but I don't think it matters. And not having those books, I can't say why the question is asked. Am I sure, they think there is a point. But I don't think that matters either. Here is why.
A postulate is "A statement, also known as an axiom, which is taken to be true without proof." ( mathworld.wolfram.com ) Yes? Now, ever heard of the identity postulate? Also know as the reflexive postulate? Now put the two together. Identity is a fundamental assumption of mathematics and logic. Proving it isn't possible within the system. The argument would be circular.
quote:
By turning around, you risk running into a different result.
I realize that. Which is why I asked the question. This is the point.
quote:
Once again, you have hacked my post to shreds so small that all context has been lost.
Funny... I posted the full response you gave me. All you said was "How is that insufficient?" What kind of context do you want? Maybe the thread itself? Surely a smart guy like you can keep track? Personally, I think you are avoiding the fact that we invented math.
quote:
Because you apparently do not understand the Incompleteness Theorems.
Oh? Tell me. When you've been talking about geometry, is that a geometry to which Incompleteness Theorems apply? If so, it isn't me who is misunderstanding, it is you who are in denial.
quote:
Because there is no other way for them to behave. Are you saying they behave according to the rules of external logic? That we can think our way into forcing the behaviour of a black hole to conform to those thoughts?
How do you know they have any logic at all?
quote:
Therefore, there is an internal logic to that behaviour.
Doesn't follow. There may be no pattern at all, and hence no logic.
quote:
Even randomness and chaos behave in a logical manner.
??????? Then it isn't randomness and chaos. A truly random series has no pattern. Logic is all about pattern.
quote:
So you are saying that because I can't disprove an infinite number of things, I am incapable of disproving a finite number of things.
No. This isn't the argument. It isn't that you are incapable of disproving a finite number of things. It is that to make a statement that encompasses the whole universe you would have to disprove an infinite number of things. You can limit your set, but then you are not making a statement about the whole universe-- it is no longer a universal statement. This is what you did with the car key example. It is no longer a statement about the absolute existence, or non-existence, of the car keys. It is a conditional.
quote:
Who's jumping from the specific to the universal? I'm certainly not.
I'm afraid you are. Notice every example you give of counting. Two apple + 2 apples = four apples. hmmm... try again. Same result. And again. Again. Same result. Eventually this pattern was generalized into a law, a maxim, an axiom or whatever you want to call it. This is a jump from the specific-- lots of specifics actually-- to the universal. This was the creation of mathematics.
quote:
I'm not talking about induction, though. I'mm talking about deduction.
Deduction has the same failing as induction for the reasons given above. The axioms of the system had their origins in the observation of events. Those observations were generalized to universals to create the axiomatic system.
quote:
Assuming you have a typical hand, take a look at it.
You'll see five.

A joke based on the Latin word for finger, perhaps?
No. I don't see 'five.' I see fingers and I count five of them. I do not percieve any element of 'fiveness.'
quote:
In the sense of the incompleteness theorems, yes. You can choose which way you want to go. You can write it out such that you get A and ~A or you can write it out such that you cannot say either A or ~A. They're the same thing.
I get.
A · ~A
And.
~(A v ~A) which gives ~A · A
Ok. Got it.
------------------
No webpage found at provided URL: www.hells-handmaiden.com

This message is a reply to:
 Message 64 by Rrhain, posted 05-21-2003 9:08 PM Rrhain has replied

Replies to this message:
 Message 86 by Rrhain, posted 05-24-2003 3:04 AM John has replied

Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 81 of 210 (41186)
05-24-2003 1:54 AM
Reply to: Message 76 by crashfrog
05-22-2003 2:52 AM


crashfrog responds to me:
quote:
quote:
Presburger arithmetic. It models addition, but not multiplication. Presburger showed that there is an algorithm that can decide of any given statement is true or not. Fischer and Rabin then showed that all algorithms that can decide such statements have a runtime of 22cn for some c and n being the length of the statement.
But if it can model arithmetic,
Stop right there. It isn't modeling all of arithmetic...just addition.
quote:
isn't it sufficiently complex as to be incomplete?
No, because it isn't modeling all of arithmetic.
quote:
And how could you have addition without multiplication?
Look it up and find out. There is more to multiplication than "repeated addition."
quote:
And what would prevent the insertion of a self-contradicting statement into that algorithm?
Read the proof and find out.
quote:
quote:
It's the same question as asking what color they are. If they have color, they have number.
Why? To determine their color, all I have to do is look at them. As a non-color blind person their color is immediately apparent. But to know how many there are I have to count them.
How is "counting them" fundamentally different from "looking at them"?
quote:
Are you really saying that color and number are this related? That's a bold statement.
I'm saying they are fundamentally equivalent, yes. Just as an object has size, shape, texture, color, it also has number.
quote:
Anyway, you said that I would see "five". Not five fingers, just five. Five in it's unitary nature apart from describing the set of my fingers.
But just as many objects can be red, many sets can be five. Surely you wouldn't say that only a red apple can be red, would you? So why are you making a distinction that your fingers are somehow unique to being five?
quote:
All I see are fingers. Explain to me how I'm supposed to just see "five".
How many are there?
You want to see red? Look at this apple (though you may need special equipment to see it).
You want to feel smoothness? Touch this apple.
You want to see one? Count this apple.
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 76 by crashfrog, posted 05-22-2003 2:52 AM crashfrog has replied

Replies to this message:
 Message 84 by crashfrog, posted 05-24-2003 2:25 AM Rrhain has replied

Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 82 of 210 (41188)
05-24-2003 1:57 AM
Reply to: Message 78 by Chavalon
05-22-2003 10:52 AM


Chavalon responds to me:
quote:
Yes, I do see 5 fingers when I look at my hand, but I don't accept this as evidence that 'fiveness' exists anywhere except in people's minds.
Evidence of colour exists even in a world of colour blind people - assuming they can make things like spectrometers. But how can evidence of 'pure fiveness' exist independent of our conception of it?
Would the number of fingers on your hand change if nobody was around to count them?
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 78 by Chavalon, posted 05-22-2003 10:52 AM Chavalon has replied

Replies to this message:
 Message 92 by Chavalon, posted 05-24-2003 10:12 AM Rrhain has replied

Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 83 of 210 (41189)
05-24-2003 1:58 AM
Reply to: Message 79 by crashfrog
05-22-2003 1:23 PM


crashfrog responds to me:
quote:
And why is math "discovered", while baseball and Monopoly were "invented"? (To bring in Schraf's point again.) What's the difference? How do we tell?
Because if it's still there when there's nobody around, it's a discovery.
Math is still there, even when there's nobody around to think about it.
Or do the number of fingers you have on your hands change when you're not looking?
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 79 by crashfrog, posted 05-22-2003 1:23 PM crashfrog has replied

Replies to this message:
 Message 85 by crashfrog, posted 05-24-2003 2:53 AM Rrhain has replied

crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 84 of 210 (41190)
05-24-2003 2:25 AM
Reply to: Message 81 by Rrhain
05-24-2003 1:54 AM


Stop right there. It isn't modeling all of arithmetic...just addition.
Sorry, I'm not trying to argue these points. I really just don't understand. I was just curious.
How is "counting them" fundamentally different from "looking at them"?
To look, all I have to do is point my eyes.
To count, I have to go "One, two, three, four, five."
Seems fundamentally different to me.
I'm saying they are fundamentally equivalent, yes. Just as an object has size, shape, texture, color, it also has number.
But then:
But just as many objects can be red, many sets can be five.
So, it's sets that have number, not objects. That's what I've been arguing all along.
They why:
So why are you making a distinction that your fingers are somehow unique to being five?
No, like you said, it's not my individual fingers that have five-ness. It's the set of my fingers that has the quantity "five".
This distinction between objects and sets containing objects seems to be a tough one for you. Why is that? The teapot is not the water it contains.

This message is a reply to:
 Message 81 by Rrhain, posted 05-24-2003 1:54 AM Rrhain has replied

Replies to this message:
 Message 87 by Rrhain, posted 05-24-2003 3:13 AM crashfrog has replied

crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 85 of 210 (41192)
05-24-2003 2:53 AM
Reply to: Message 83 by Rrhain
05-24-2003 1:58 AM


Math is still there, even when there's nobody around to think about it.
So is Monopoly. Prove me wrong.
Or do the number of fingers you have on your hands change when you're not looking?
Does the mortgage value of Boardwalk change when we're not playing?
Just because the rules don't change each time you play the game doesn't mean the game itself wasn't made up. Constancy is no argument, in this case, for independant existence.

This message is a reply to:
 Message 83 by Rrhain, posted 05-24-2003 1:58 AM Rrhain has replied

Replies to this message:
 Message 88 by Rrhain, posted 05-24-2003 3:17 AM crashfrog has not replied

Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 86 of 210 (41193)
05-24-2003 3:04 AM
Reply to: Message 80 by John
05-23-2003 2:17 AM


John responds to me:
quote:
quote:
You're not about to invoke Godel, are you?
Well, no, not really. Just explain how i does not violate the basic rules of multiplication.
Because you don't understand the rules of multiplication. What makes you think mathematical operators in the Complex number system are going to behave the same way they do in the Reals? i is not a Real.
quote:
quote:
Be careful, because the Incompleteness Theorems do not say that all axiomatics systems are necessarily incomplete or inconsistent.
Now, why should I be careful?
Because false premises can lead to any conclusion you desire.
quote:
Godel's proof may not apply to all systems but it damn sure does apply to those systems in question-- those very systems you rely upon. So why should I be careful?
Because they don't apply to the system in question.
quote:
quote:
Not all axiomatic systems are that sophisticated and, indeed, they are both complete and consistent.
Godel's proof works for systems that are almost trivially simple,
Irrelevant. The question is not whether the system is "trivial. It is whether it can model arithmetic.
The axioms of Presburger arithmetic, for example, are not that sophisticated. You can get addition, but you can't get multiplication. And as Presburger, himself, showed, they are both complete and consistent.
quote:
Why are you pretending that something significant can be constructed that does not violate that proof?
I am not. I am accurately stating that the Incompleteness Theorems have very specific meanings and applications and attempts to use them outside of those specific areas is inappropriate. Just as those trying to use the Heisenberg Uncertainty Principle to explain human behaviour under observation, the Incompleteness Theorems cannot be applied to things other than axiomatic number theories sophisticiated enough to model arithmetic.
Sounds to me like you are just being diversionary, and effectively just skipping around the issue-- which, if you've still missed it, is that logic and mathematics which you have made out to be absolute and infallible are the very things which get screwed by Godel's proof.
quote:
quote:
When did we agree on the axiomatic system?
What are you talking about? Seriously, I don't know where this came from.
Neither do I...you cut out so much context that I'd have to trace the thread back by hand to find out.
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But by being true, it necessarily results in certain other things being false.
Let me put it this way, when you attempt to prove a conclusion, or formulate an argument, do you attempt to create true argument, or a false one?
Depends on what I'm trying to do.
Again, sometimes it's easier to draw the negative space than the positive space. The definition of an infinite set is that it is not finite.
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If you attempt to create a true argument, you are by default proving a positive. Reductio ad absurdam works similarly. You have an argument you hope to be true. You reverse the conclusion, thereby creating a false argument. If indeed it does prove to be contradictory, you can infer that the original arguement was true.
Bingo. You just proved a negative.
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And you are in the same position-- that of proving a positive.
By negating a falsity.
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And that statement just might be "It is true that this is false."
Yes. Godel again.
No, now you're getting into Russell. "This statement is false," and all that.
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Thank you for pointing that out. But why? I am not claiming that math, or logic, is an unquestionable arbiter of truth. This doesn't hurt me.
You're the one saying you can't prove a negative. But "This is false" is a negative and "It is true that this is false" is proof of that negative.
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But sometimes the only way to show something to be true is by showing something else to be false.
Why, if something is true, is the ONLY way to prove it true by proving something else false? Doesn't make sense.
That's because you're reading what you wish I would have said and not what I actually did. Try again.
But sometimes the only way to show something to be true is by showing something else to be false.
You see that word "sometimes"? What do you think it means?
Infinite sets are, by definition, not finite. In order to show a set to be infinite, you must show it not to be finite.
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Who said anything about drawing anything?
ummmm.... I believe that I did. And, in fact, you brought up drawing with a compass and straight edge in an earlier post. It was just up in post #48.
In reference to something else entirely. Why are you equivocating?
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You do understand the difference between necessary and sufficient, yes?
And do you understand what is sufficient support for a universal statement such as X can or cannot exist?
Yes.
Do I really need to go through the proof of the non-existence of a largest prime?
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You need have absolute knowledge of the universe.
And that doesn't happen in mathematics because of what, precisely? Are you saying there might be a largest prime?
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Or have a guaranteed system for working out answers, which would be the equivalent of absolute knowledge. You have neither, we have neither, but you still insist on making absolute claims about existence.
You mean it is impossible to deduce anything?
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Excuse me? We're talking about existential mathematical statements and somehow the existential operator isn't relevant?
I never said we were using a existential mathematical operator.
What is an existential mathematical statement without the existential operator? That's the defining characteristic of an existential statement: It uses the existential operator. This is in contrast to a universal statement that uses the universal operator. They are negations of each other. The negative of the universal is the existential and vice versa.
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Indeed.
So you agree that if there is a largest prime then our system of mathematics is contradictory? That was the question.
See...this is what I mean by your complete destruction of context. What was I saying "Indeed" to? Any idea? Why did you remove all the context? What on earth are you talking about?
If there is a largest prime, our system of mathematics would be contradictory, yes.
Do you have proof of such?
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Now, how is it you are sure that our system of mathematics is NOT contradictory? Assumption? Ya just can't imagine it being otherwise?
No, the incompleteness theorems again. Statements that can be decided will not be contradictory.
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Math is contradictory.
No, axiomatic number systems complex to model arithmetic will be either incomplete or inconsistent.
Not all of math is such a system.
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We do not even attempt to "fill in the blank" with a specific number because the specific number is not necessary.
The proof depends upon the fact that you COULD do so.
No, the proof depends on the fact that you don't. The moment you leave the abstract, you have only proven the specific. We need to get this out of the specific and into the general.
That's how inductive proofs work. Yes, you need to show a specific case, but the inductive step is that you generalize to all others.
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or do you claim that the objects of mathematics aren't real?
Geez!!!! I've said so more times than I remember.
Then there really is no need to continue. We have a fundamental difference. For myself and the vast majority of mathematicians, the objects of mathematics are real. I do not say that as if that is justification. I am simply pointing out that I am not an isolated instance.
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There is no reason to believe that math is anything more than a symbolic system we made up, and of course we made it match the world we observe but there is no guarantee that we got it right or that it applies to the whole universe.
You mean the number of fingers on your hands changes if you're not paying attention? If you were to think really, really hard, you might actually have six on one and four on the other?
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Just because observation is consistent with prediction doesn't mean the theory is "true."
Nor does it mean contrary or variant theories are false, as you appear to be arguing.
Incorrect.
But since you cut out so much of the context, I can't really say more than that.
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No. Do I need to crack open Elements and give you the definition?
Do you actually believe that Euclid invented geometry?
Of course not. He didn't even discover most of it. He was compiling the work of others.
But, since you cut out all the context, you're missing the point. It is that a circle is not as you think it within the realm of plane geometry.
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Come on! This statement is staggering in its ignorance.
That's because, once again, you are responding to what you wish I had said and not what I actually did.
Go back and read it again. Notice the part where I mention Euclid compiling the work of others. What do you think that might imply?
And considering that I am arguing from the position that the objects of mathematics are real, what do you think that means? Hint: Would it matter to me if Euclid existed at all?
You even responded to it. Why didn't you pay attention? "...there was this guy named Euclid and he collected the works of other mathematicians." Hmmm...does that statement indicate that I think Euclid invented geometry?
Think carefully, now. Answer based on what I actually said, not what you wish I would have said.
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Now, you want to call Euclids postulates 'definitions.'
No, the postulates are separate from the definitions. Haven't you read the books?
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Nope. A circle is contained by a line. A line has more than one point. If space results in a single point, we have no circles, no squares, no lines. Simply a point.
hmmm... then back up from that singularity until you have four points. Is that a circle or a square?
It's impossible to have only four points. If you have more than one, you have infinitely many.
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Things that exist have definition, but not all things that have definition exist.
I think it is fair to say that you have frequently argued otherwise.
And you'd be wrong.
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I gave you examples. That you refused to read them is not my problem.
ummm... when I asked for examples did it not occur to you that I WAS ASKING FOR EXAMPLES?
Non sequitur.
If you ask for examples and I give you examples, how is that not giving you the examples you asked for?
Look, if you don't want to read them, that is not my fault, but for you to claim that I didn't give you the examples you requested is disingenuous at best.
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Then why do I have so many textbooks that ask you to prove that A = A? Surely they think there is a point to it.
I know that some textbooks have this proof in them.
Then is it not possible that these people know something you don't? That there is a point to asking the question?
For example, one of the things in linear algebra is that a matrix is equivalent to a given matrix if each corresponding element is equivalent (Two m x n matrices A = aij and B = bij are equal if aij = bij for i = 1, 2, ..., m and j = 1, 2, ..., n.)
Thus, you prove that A = A by showing that each element within it is equivalent.
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By turning around, you risk running into a different result.
I realize that. Which is why I asked the question. This is the point.
But that doesn't mean the forward direction is invalid.
All squares are rectangles. Not all rectangles are squares.
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Once again, you have hacked my post to shreds so small that all context has been lost.
Funny... I posted the full response you gave me. All you said was "How is that insufficient?" What kind of context do you want? Maybe the thread itself?
Like maybe the comment you made which prompted my response. With no antecedent, it's hard to say what "that" is that I seem to think is insufficient.
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Personally, I think you are avoiding the fact that we invented math.
And I think you're avoiding the fact that we discovered it.
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Because you apparently do not understand the Incompleteness Theorems.
Oh? Tell me. When you've been talking about geometry, is that a geometry to which Incompleteness Theorems apply? If so, it isn't me who is misunderstanding, it is you who are in denial.
Question: Are the axioms of geometry that of a number system sufficient to model arithmetic?
Think carefully...consider the fact that not all numbers are constructible in geometry....
If so, then those Incompleteness Theorems apply.
If not, well, then they don't.
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Because there is no other way for them to behave. Are you saying they behave according to the rules of external logic? That we can think our way into forcing the behaviour of a black hole to conform to those thoughts?
How do you know they have any logic at all?
Because there is no other way for them to behave.
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Therefore, there is an internal logic to that behaviour.
Doesn't follow. There may be no pattern at all, and hence no logic.
But even randomness and chaos behave in a logical manner.
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Even randomness and chaos behave in a logical manner.
??????? Then it isn't randomness and chaos. A truly random series has no pattern. Logic is all about pattern.
Incorrect.
Or have you not heard of Chaos Theory? Gleick wrote a wonderful layman's book about the subject. Perhaps you've heard of it: Chaos. The chaotic behavior of the logistic map was what I wrote my Sophomore Thesis on.
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So you are saying that because I can't disprove an infinite number of things, I am incapable of disproving a finite number of things.
No. This isn't the argument. It isn't that you are incapable of disproving a finite number of things. It is that to make a statement that encompasses the whole universe you would have to disprove an infinite number of things.
Not if it can be shown that all of the things that are not the thing I'm looking for are of a piece and that piece cannot be true.
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Who's jumping from the specific to the universal? I'm certainly not.
I'm afraid you are. Notice every example you give of counting. Two apple + 2 apples = four apples. hmmm... try again. Same result. And again. Again. Same result. Eventually this pattern was generalized into a law, a maxim, an axiom or whatever you want to call it. This is a jump from the specific-- lots of specifics actually-- to the universal. This was the creation of mathematics.
But 1 + 1 = 2 isn't an axiom. It's a conclusion.
Russell worked very hard to show that.
And mathematics wasn't created...it was discovered.
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Assuming you have a typical hand, take a look at it.
You'll see five.
A joke based on the Latin word for finger, perhaps?
Nope...a simple request.
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No. I don't see 'five.' I see fingers and I count five of them. I do not percieve any element of 'fiveness.'
If you have five of them, how can you have some other number of them? Do they change number when you're not paying attention? If everybody were to die and thus have nobody around to count them, would their number change?
And how can you claim you don't perceive five if you count five?
"Yes, I am sensing red, but I'm not seeing red."
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 80 by John, posted 05-23-2003 2:17 AM John has replied

Replies to this message:
 Message 98 by John, posted 05-26-2003 12:06 PM Rrhain has replied

Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 87 of 210 (41194)
05-24-2003 3:13 AM
Reply to: Message 84 by crashfrog
05-24-2003 2:25 AM


crashfrog responds to me:
quote:
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Stop right there. It isn't modeling all of arithmetic...just addition.
Sorry, I'm not trying to argue these points. I really just don't understand. I was just curious.
S'aright. It's complicated. The axioms of Presburger arithmetic are simple, but they are based in symbolic logic and hard to transcribe.
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How is "counting them" fundamentally different from "looking at them"?
To look, all I have to do is point my eyes.
To count, I have to go "One, two, three, four, five."
That's only because you're faster at figuring out red than you are at figuring out five.
If I give you a color that's very close to what you would consider to be the "border" between red and orange, you'd have to spend a bit more time thinking about what color it is, wouldn't you? Does that hesitation mean that color is something that is sometimes real and sometimes manufactured?
Things exist only if we can so quickly process the information that we don't do it consciously?
What about the people who can determine number as quickly as you can determine color? They do exist.
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But just as many objects can be red, many sets can be five.
So, it's sets that have number, not objects. That's what I've been arguing all along.
A set is an object.
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So why are you making a distinction that your fingers are somehow unique to being five?
No, like you said, it's not my individual fingers that have five-ness. It's the set of my fingers that has the quantity "five".
Yes...and? Are you saying you don't have a set of five fingers on the end of your hand? That if you stop paying attention, you might return to a set of six?
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This distinction between objects and sets containing objects seems to be a tough one for you. Why is that? The teapot is not the water it contains.
I would say you have the opposite problem.
I am well aware of the difference between a set and its elements. Russell did quite a lot of work in that area and Russell's Paradox is specifically about that.
But you seem to think that a set is not an object.
If you stop paying attention to your hand, does the number of fingers on your hand change? If everyone were to die and thus there would be nobody around to count them, would the number of fingers on your hand change?
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 84 by crashfrog, posted 05-24-2003 2:25 AM crashfrog has replied

Replies to this message:
 Message 90 by crashfrog, posted 05-24-2003 3:40 AM Rrhain has replied

Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 88 of 210 (41195)
05-24-2003 3:17 AM
Reply to: Message 85 by crashfrog
05-24-2003 2:53 AM


crashfrog responds to me:
quote:
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Math is still there, even when there's nobody around to think about it.
So is Monopoly. Prove me wrong.
You can play Monopoly without players?
There's a good trick.
Oh, the Monopoly set is there, all the pieces, the money, the board, the dice, etc., but with nobody to play it, it's nothing more than a bunch of brightly colored pieces of plastic and paper.
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Or do the number of fingers you have on your hands change when you're not looking?
Does the mortgage value of Boardwalk change when we're not playing?
With nobody to play it, how is there a mortgage?
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Just because the rules don't change each time you play the game doesn't mean the game itself wasn't made up. Constancy is no argument, in this case, for independant existence.
I'm not saying that they are.
But if everybody were to die, you'd still have five fingers on your hand.
If everybody were to die, Monopoly would never get played.
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 85 by crashfrog, posted 05-24-2003 2:53 AM crashfrog has not replied

Replies to this message:
 Message 89 by NosyNed, posted 05-24-2003 3:23 AM Rrhain has not replied

NosyNed
Member
Posts: 8996
From: Canada
Joined: 04-04-2003


Message 89 of 210 (41196)
05-24-2003 3:23 AM
Reply to: Message 88 by Rrhain
05-24-2003 3:17 AM


Geez you guys!!
I think the point might be that math would probably be very much the same developed (discovered) by an alien civilization but monopoly would be very, very unlikly to be invented by them.

This message is a reply to:
 Message 88 by Rrhain, posted 05-24-2003 3:17 AM Rrhain has not replied

crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 90 of 210 (41198)
05-24-2003 3:40 AM
Reply to: Message 87 by Rrhain
05-24-2003 3:13 AM


A set is an object.
I don't think it is. It has no physical existence. It's simply the expression of a relationship between objects.
Are you saying you don't have a set of five fingers on the end of your hand? That if you stop paying attention, you might return to a set of six?
No, I'm saying there's nothing about any of my fingers that contains "five-ness". That's a property of the set of my fingers (on one hand) that isn't inherited by the objects in that set.
If everyone were to die and thus there would be nobody around to count them, would the number of fingers on your hand change?
If I'm not around - in which case I certainly wouldn't have fingers, nor be able to count them - how could I even asnwer that question? It's meaningless.
Sets aren't objects because they have no independant existence. If you take the objects out of a set the set no longer exists (or becomes the null set, which is no difference.)
If I have a relationship between two objects, but those two objects cease to exist, so does the relationship. it has no independant existence. It's just a concept we create in our heads, based on certain rules we all agree to.

This message is a reply to:
 Message 87 by Rrhain, posted 05-24-2003 3:13 AM Rrhain has replied

Replies to this message:
 Message 91 by Rrhain, posted 05-24-2003 5:56 AM crashfrog has replied

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