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Because you don't understand the rules of multiplication.
A positive times a positive is a positive. A negative times a negative is a negative.
i requires that a negative times a negative is a negative or that a positive times a positive is a negative.
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What makes you think mathematical operators in the Complex number system are going to behave the same way they do in the Reals?
I didn't say that mathematical operators within the complex number system would be the same as within the real number system. What I said is that [/i]i[/i] itself is a problematic number, as exlained above.
You've got one system with a contradictory number, or two systems incompatible with one another. The 'bridge' between the two is a number that can't exist in one system. Either way, the point is the same, there are holes in mathematics. It isn't complete. It isn't absolute.
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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.
Yeah, I know that. And, as you say...
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...the Incompleteness Theorems cannot be applied to things other than axiomatic number theories sophisticiated enough to model arithmetic.
Fine. Why are you avoiding the question?
When you refer to geometry and mathematics and logic, are you not talking about systems sophisticated enough that the incompleteness theorems apply?
In other words, the 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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Neither do I...you cut out so much context that I'd have to trace the thread back by hand to find out.
More whining... Sorry. I assumed you knew what you were talking about.
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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.
The argument itself is true, else it would be a silly argument. You are talking about the right side of the if/then statement-- the conclusion. I am talking about the truth value of the whole statement, or the whole argument.
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In reference to something else entirely. Why are you equivocating?
Why are you changing the rules to suit yourself?
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Do I really need to go through the proof of the non-existence of a largest prime?
You need to dig yourself out of math and into the real world.
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And that doesn't happen in mathematics because of what, precisely?
Because we made up the rules. The universe is not bound by our rules.
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You mean it is impossible to deduce anything?
Yes, in an absolute sense. You can always assume a starting point and go from there, which is exactly what we do. There is no other choice. But it is dishonest to make absolute statements when there are assumptions at the foundation.
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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.
You just don't listen do you? I give you definition I used and you still don't get it? This is getting to be ridiculous. Half your argument has come to special pleading for your preferred definition.
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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?
Jesus Christ!!! I rephrased the damn question for you!!!
So you agree that if there is a largest prime then our system of mathematics is contradictory? That was the question.
How could you miss that?
BTW, there is a little button right at the botom of the message that will take you back to all the context you could want.
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If there is a largest prime, our system of mathematics would be contradictory, yes.
Do you have proof of such?
Do I have proof that this statement is true? Yes, and you know what that proof would be, or you wouldn't have agreed that the statement is true.
Do I have proof that there is a largest prime and thus mathematics is contradictory? Do I have proof that any number exists? Nope. Kind-of a silly question really. We made them up to record quantities. They are mnemonics. But what does it matter? The point is that your 'absolutes' are conditional upon our understanding of mathematics.
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No, the incompleteness theorems again. Statements that can be decided will not be contradictory.
And? That still leaves the parts that cannot be decided. And you've already equated 'incomplete' and 'inconsistent.' Looks like you've cut your own throat.
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No, axiomatic number systems complex to model arithmetic will be either incomplete or inconsistent.
Can we stop playing games here? When I say 'math' do you think I mean some minimal system like Presburger arithmatic which only models addition? Or do you think I mean the mathematics kids are taught in grade school, high school and college? Why are you trying to wiggle around this by continually referencings some interesting but virtually non-functional mathmatical system?
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The moment you leave the abstract, you have only proven the specific.
Until you leave the abstract you haven't really proven anything. Why do you think physicists look for observational evidence to support what has been proven in theory?
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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.
Read you statement carefully and think 'generalization from the specific to the universal.' I could't have said it more plainly. Thanks.
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For myself and the vast majority of mathematicians, the objects of mathematics are real.
I am aware that many mathematicians share your belief. For the vast majority of people on Earth, God-- of some variety-- is real. Is that adequate reason for the belief?
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You mean the number of fingers on your hands changes if you're not paying attention?
Don't know. If I have no sensory input of, or from, my fingers, I can't really say what they are doing or how many they are. I assume they stay the same, but I know that is assumption.
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If you were to think really, really hard, you might actually have six on one and four on the other?
Precisely where did I say I believe that by thinking really really hard I can change the number of fingers on my hand? This is pitiful, Rhhain.
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Incorrect.
But since you cut out so much of the context, I can't really say more than that.
This little mantra is becoming very tiresome. You seem to pull it out when you don't have an response. Sad...
Now....
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You: Just because observation is consistent with prediction doesn't mean the theory is "true."
Me: Nor does it mean contrary or variant theories are false, as you appear to be arguing.
( Copy your part ) Just because observation is consistent with prediction doesn't mean ( add my part ) contrary or variant theories are false.
Why is that so hard to put together? And why is it so hard to answer?
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That's because, once again, you are responding to what you wish I had said and not what I actually did.
BS. You are back-pedaling.
Look at the context. My post # 53:
[qs]Same with circles, same with triangles. And we end up with something we call geometry. It is the generalization from specific cases to the universal and as such will always be questionable.[/b][/quote]
To which you respond in post # 64:
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.
You respond to my analysis of the creation of geometry by saying that Euclid made up definitions. What does this tell me? Hint: You consider Euclid's work to be analogous with but contrary to my analysis. You consider Euclid's systematizing geometry to somehow negate the source of those axioms-- the experience of countless forgotten engineers stretching deep into the past.
Despite the rant, you don't actually touch the issue. The point has been quoted above, but perhaps I should restate? The axioms did not come from nowhere. Euclid, and whomever he drew from, simplified known geometric figures into axioms-- into abstractions. This is why he constructed Euclidian geometry, instead of some other geometry. And this is why no one questioned these axioms for centuries. Plane geometry is blatantly obvious to someone living on a 'flat' surface, building square houses, etc.
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No, the postulates are separate from the definitions. Haven't you read the books?
Don't be a child.
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It's impossible to have only four points. If you have more than one, you have infinitely many.
Mathematically, but we were talking about a singularity in the real world. The universe appears to be jump-y. This is the essense of quantum theory. The world doesn't follow this mathematical law. Care to try again?
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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.
Now you just being stubborn. Did you miss the part where I explained that...
[qs]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.[/b][/quote]
Grow up.
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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.
Equivocation. This is absurd. Two grids of squares with numbers in them. Compare them. If the numbers match up, the two squares are the same. Don't pretend this proves the principle of identity. It is an application of that principle.
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If so, then those Incompleteness Theorems apply.
If not, well, then they don't.
So much for a straight answer.
Lets see if we can't construct what you won't state plainly.
"The various geometries taken individually are internally consistent."
Fine. Taken individually, they are also inadequate. Each is very limited in scope, applying as each does to a particular type of surface. They are also mutually incompatible and there is no bridge between them, which there must be if we are to get a complete description of the real world, or even a complete ( colloquial ) abstract geometry for that matter.
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Because there is no other way for them to behave.
How do you know that everything must behave logically? This is nothing but 'I can't imagine it to be different.' Incredulity at its finest.
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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.
Certainly. And it is horribly ill-named. I found the following definition:
In other words, chaos theory deals with systems so complex they appear random. It is the search for pattern in apparent chaos. It doesn't follow that what appears chaotic actually has a pattern. We won't know that until a pattern is found.
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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.
In the dream world of mathematics where you get to make up all the rules, you might be able to do this. The real world doesn't have to cooperate. Why can you not see this?
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But 1 + 1 = 2 isn't an axiom. It's a conclusion.
I didn't say it was an axiom. Pay attention. It is a conclusion-- more of an observation really-- based on experience. Axioms are/were generalized from this experience. Think science. You make a bunch of observations and generalize to a rule or a mechanism-- an 'axiom.' The same process underlies the origin of mathematics.
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And how can you claim you don't perceive five if you count five?
Because I don't see 'five.' I see five fingers. I see four books. I don't see 'four.'
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"Yes, I am sensing red, but I'm not seeing red."
Your color analogy is ridiculous. 'Red' can be analyzed in several different ways. It has measurable properties-- like wavelength, energy, etc. 'Five' has none of those measurable properties. Build a pure 'raw number' detector then get back to me. Pass 'five' through a detector and give me a mass, frequency, particle size-- something.
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No webpage found at provided URL: www.hells-handmaiden.com
. Perhaps perceptions of people can only exist in each other's minds in that format. But can you find any way to demonstrate which, if either, of these 2 views is correct -
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