John responds to me:
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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.
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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.
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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.
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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.
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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.
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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."
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Rrhain
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