|
Post by speakertoanimals on Jan 25, 2011 2:13:40 GMT 1
You have random insults, I have properly referenced quotations from scientific papers, wins every time!
And don't it annoy the heck out of you!
As regards grassmann numbers:
But many people already know of an thing that multiplis but doesn't commute -- a matrix. So for matrices A + B = B + A, but in general AB is not the same as BA.
Anticommuting is a bit special, in that ab = -ba.
|
|
|
Post by abacus9900 on Jan 25, 2011 10:44:03 GMT 1
Physics is ahead of you there! Way ahead! First, these aren't LAWS they are axioms -- so, if we say, we have numbers, and addition and multiplication are commutative, you get one type of number system -- the real numbers. If you say you have numbers of some sort, but multiplication is no longer commutative, then you have grassmann numbers. Which are used in quantum theory for fermions. Which doesn't mean that you could have a universe where the real numbers weren't commutative - they are commutative by definition! If they don't commute, they aren't the real numbers! So, I can imagine (sort of) a universe that never used the real numbers. What numbers the universe uses is a question for physics, not maths! They exist, whether the universe uses them or not! I should add that what godel did was a theorem NOT a theory. So, Godels result is TRUE (and proven to be true), under conditions stated. Whether or not those conditions apply to anything in the physics of our universe is a different question, and even if they didn't, doesn't make godels theorem any less true! Don't confuse maths and physics! And don't confuse theories in physics, with theorems in maths. And the next person who says 'it's JUST a theory' gets sent to the naughty step with the creationists (who occupy it permanently, and are very boring companions!)................... This is simply more waffling, Speaker. Fact is, no matter what system of maths you are looking at it can never be proved in any absolute sense and has to be based on intuition. Why are you trying to obscure and complicate what has been a settled issue since 1931? Playing with words does not help at all, it just tries to make you look as though you know what you are talking about. I'm sorry, it doesn't. And how you can pretend maths and physics aren't intrinsically wrapped up with one another is simply staggering. Kindly tell me how we are able to examine physical reality without using maths. You attended Oxford? Really?
|
|
|
Post by speakertoanimals on Jan 25, 2011 14:54:29 GMT 1
More random bollocks!
I'll go off and prove pythagoras' theorem, which according to this poster doesn't actually exist...........................
We go from grabled Godel, to the usual nonsense put out by those who have no actual idea about maths or science, but like to think they do, and like to try and put both down. Odd then they have to use a mathematical theorem to try and demonstrate the supposed lack of proof in maths --- or could be their argument has as much consistency as a melted Mars bar...........................
|
|
|
Post by abacus9900 on Jan 25, 2011 19:44:44 GMT 1
Pythagoras' theorem may be true but that only applies to cases to which so far it satisfies. Nobody can possibly prove that it would always be true to all cases, hence the distinction between 'true' and 'proveable.' And it's no good pointing out that we can always produce new axioms and theorems because that is in exactly the same situation as the foregoing.
A simple example of this is the centre of a black hole. Inside the centre of a black hole conventional maths breaks down, so here we can see that maths and physics are really one and the same thing since we need to come up with a new set of theorems with which to predict the behaviour of the inside of a black hole.
|
|
|
Post by speakertoanimals on Jan 25, 2011 19:55:34 GMT 1
Utterly wrong. Pythagoras' theorem is TRUE, provably true, within the system of euclidean geometry.
Which makes no sense. What I think you've taken is that within other possible geometries (ie ones with different sets of axioms, non-euclidean geometry), pythagoras theorem is demonstrably NOT true.
this is NOT the distinction between true and provable that Godel meant!
And this is just gibberish!
Utter piffle! SOmeone who doesn't know the difference between maths and physics, and between theories (physic s), and theorems (maths).
As I keep saying, Pythagoras is true and provably true within the mathematical system of Euclidean geometry. And provably not true within non-euclidean geometry. Those are FACTS about maths.
Whether or not the universe uses Euclidean or non-euclidean geometry is not a matter of mathematical proof or dis-proof. It is something that CAN be disproved by demonstrating a PHYSICAL case where Pythagoras is untrue (i.e. drawing triangles), but it cannot be proved to be Euclidean (because we'd need to show that nowhere and nowhen in the universe was non-euclidean).
What we need to describe the interior of a black hole is not new theorems, or new axioms (probably), because we have plenty of maths that talks about geometry of various types, real manifolds, complex manifolds, discrete manifolds, non-commutative geometry etc etc. What we need (probably) is new physical theories, which MAY use some of the above.
really, how hard can this be, the distinction between maths and physics, between mathematical proof, and whether or not the universe uses that maths...................
I think you're so far from having ANY idea what ANY of this is about, that's there's no point going on. You can't handle the basic terms in maths or physics, so no wonder you've no idea at all what Godels theorem means for either.........................
|
|
|
Post by abacus9900 on Jan 25, 2011 20:10:38 GMT 1
Yes, I know they're true. They are not proveable. What do you mean by 'proveably true?' If you are using 'proveable' in the context used by Godel you are in error. What you probably mean is that they can be shown to be true within a limited set of cases, this is something you can't seem to grasp.
What Godel meant when he used the word 'prove' was that there is no system of axioms that can possibly be shown to be proveable because they inevitably have to be self-referential. Take the phrase 'I am a fibber.' Now, this statement is obviously contradictory since if I am a fibber then I must be telling the truth but if I'm telling the truth I can't be a fibber and just leads to circular reasoning getting nowhere. This is an example of a self-referential system and all reasonably complete sets of axioms are in the same category.
|
|
|
Post by speakertoanimals on Jan 25, 2011 20:25:47 GMT 1
Uh, yes they are! Godel theorem does NOT say that nothing is provable, just that in any sufficiently complex system, there are statements which are undecidable, doesn't mean that ALL statements are undecidable!
Anyway, Euclidean geometry ISN'T a sufficiently complex system for Godel to apply, since Euclidean geometry doesn't include 'enough' for Godel to apply! So you're scuppered twice!
WHat sort of statements are undecidable? Well, one example is:
which is talking about infinite sets, and is undecidable within the context of set theory (or rather the usual axiomisation of set theory).
Nope, this isn't what I meant, and it actually makes no sense!
|
|
|
Post by abacus9900 on Jan 25, 2011 20:46:31 GMT 1
You are interpreting Godel much too narrowly.
If you look at the Pythagoras example then it can be seen that it is only true for triangles existing in 4 dimensional spacetime, not for more complex triangles that involve more than 4 dimensions.
|
|
|
Post by speakertoanimals on Jan 25, 2011 21:28:38 GMT 1
Laugh? I nearly wet myself (again!).
Pythagoras applies to any triangle in a Euclidean space of dimension two or greater.
Says nothing about spacetime, it talks about space only.
The point being, if I draw two straight lines that meet at their ends in ANY dimensional Euclidean space greater than 2, then those two lines define a plane that contains those lines. Then we just have Pythagoras in the plane, for that plane. Works whether we have two dimensions ton start with, or six million.
You really don't have ANY idea do you?
ANy idiot can claim -- you are interpreting Godel too narrowly -- doesn't mean anything (as your statement doesn't godel doesn't apply to Euclidena geometry since it is not rich enough to contain arithmetic). godel specifies ANY system rich enough to contain arithmetic.
But the 'too many dimensions' random statement (and the addition of spacetime), just shows that you have no idea what euclidean geometry actually IS...................
|
|
|
Post by abacus9900 on Jan 25, 2011 21:31:19 GMT 1
Speaking of idiots, what makes you think the sides of a triangle are perfectly straight?
|
|
|
Post by speakertoanimals on Jan 25, 2011 21:46:28 GMT 1
As regards triangles in euclidean spaces of one dimension (ignoring the fact that triangles as plane figures don't exist strictly speaking in one dimension!), what does it reduce to?
Well, suppose I have line 1 goes from O to A, and line 2 goes from 0 to B.
'hypotenuse' goes from A to B. Let lengths be a and b. Then length of hypotenuse is either a+b (points in order A O B), or |a-b| for points in the order O A B or O B A.
So, analog of pythagoras becomes:
If (a+b), then (a+b)^2 which is always greater than a^2 + b^2
If |a-b|, then (|a-b|)^2 = a^2 + b^2 - 2ab
So, in 1D, the square of the 'hypotenuse' can be either greater or less than the sum of the squares on the other two sides, but the one thing it can't be (as long as all sides have non-zero length!), is EQUAL to the sum of the squares! which is kind of freaky weird, and perhaps a little stretching of Pythagoras..................
Which you can see from the plane, by squishing a triangle onto a line. To get the A O B case, the right angle that gives pythagoras has to be opened out, hence hypotenuse gets too long. To get O A B case, the angle has to be reduced, and the triangle folded down to give a zero angle where right angle was. Hence hypotenuse gets too short!
ONLY case where the 1d case 'works' is when the original planar triangle had one side zero anyway, so was already a line, in which case you have just discovered that (a+0)^2 = a^2.....................
|
|
|
Post by speakertoanimals on Jan 25, 2011 21:54:48 GMT 1
Speaking of idiots, what makes you think the sides of a triangle are perfectly straight? They are by definition in euclidean spaces, that is what 'straight' means in that context. In a non-euclidean space, they aren't 'straight' in the euclidean sense, but the shortest distance (geodesic). And surprise, surprise, Pythagoras doesn't hold. Really, stop making random daft statements -- the first few were slightly amusing, and you sometimes throw up weird conjunctions of ideas that way. But basing your whole posting strategy on it gets a little wearing............................
|
|
|
Post by carnyx on Jan 25, 2011 22:10:27 GMT 1
STA,
Have you done the history of geometry? Doesn't it show a proliferation of geometries? And these new geometries came into being as a result of undecidable propositions ( aka apparent paradoxes) in the latest, greatest, geometry?
Could you comment on what held up the building of Milan Cathedral for so long?
And could you let us know what the significance of the square root of 2 was, to the Pythagoreans?
|
|
|
Post by Progenitor A on Jan 25, 2011 22:33:22 GMT 1
You are interpreting Godel much too narrowly. If you look at the Pythagoras example then it can be seen that it is only true for triangles existing in 4 dimensional spacetime, not for more complex triangles that involve more than 4 dimensions. Inded Abacus the link that you provided showed that Euclid's fifth postulate was unproveable. As that postulate is part of the set that defines his geometry then it surely follows that if one postulate is unproveable then all postulates must fall in to that category? What I like about this is that Godels theorem must also be unproveable! (By unproveable we do mean, I think , that the mathematical axioms or postulates make assumptions without proof therefore the whole construct is unproved. That does not invalidate deductions made upon the basis of the assumption, but does question the truth of any such deductions. To prove the axiom, the assumption must be proved, but in mathematics there are always unproved assumptions - great 'what-ifs' made, therefore, although mathematics may produse useful results and give insights that otherwise might not be made, they are in the final analysis based on guesswork.) The important result, to me, is that mathematical models of the universe make very big assumptions and that mathematics is only guessing at the origin of the universe. Another very obvious expose of mathematics is infinity - that is based on assumptions that are impossible to verify. in the case of pythagerous if the fifth postulate cannot be proved, then what are straight line that stretch out in both directions? If it cannot be proved that they continue indefinitely then we are unsure about certain lengths of lines,and if we are unsure of the lengths of lines, how can Pythagerous be 'true' outside certain (quite small) limits of line length? For example does pythagerous apply to a line length of 10 67 metres? How can that be proved? Who knows? I think Godel adds some overdue humility to some aspects of science/mathematics - science relies almost totally on mathematics today and therefore canot escape Godels incompleteness. Seems intuitively correct to me
|
|
|
Post by speakertoanimals on Jan 26, 2011 2:38:04 GMT 1
No,what it shows is that the fifth postulate cannot be proven using the other 4 axioms. Which means you either add the fifth postulate as an axiom (and create Euclidean geometry), or add its converse (and create non-euclidean geometry).
No, because the whole point about axioms is that you DON'T prove them, you prove what follows on from those axioms, assuming the axioms to be true. Hence we have in euclidean geometry, Pythagoras, and Pythagoras not true in non-euclidean geometry.
Hence we have TWO possible classes of geometries, Euclidean or non-euclidean, both with various theorems which can be proved (such as Pythagoras or not). The job of PHYSICS is then to determine which geometry nature happens to be using (if either, it might be that space and time are discrete in which case BOTH continuum geometries are out of the window).
Wrong. When it was discovered that the fifth postulate could not be proved form the other 4 (NOT a paradox), then it was realised that we had a choice -- either add it as an axiom or not, and hence get two different types of geometry.
I think no one else on here has grasped yet what Godel actually means, what undecidable means -- no surprise given that people are still unable to get their heads rounds the basics, such as what an axiom is, what is provable and what is not, and what a paradox is...............
Easy -- they didn't like the fact that they could prove that root 2 was not a rational number (of the form p/q where p and q both integers). Hence rational numbers had to be extended to the reals, IF you wanted to include square roots in your number system.
you DON'T prove axioms, you numpty. That is the whole point of axioms! You can have DIFFERENT sets of axioms (like geometry with or without the fifth), and they are ALL valid geometries. What nature uses DOES NOT make other systems untrue, they are still equally valid, just that nature doesn't use them.
The same confusion over maths and physics..................
Wrong again! You DON'T prove axioms (else they wouldn't be axioms, they would be theorems that follow from axioms). And infinity just means you have to extend set theory, which was done by Cantor, to include various sizes of infinity. Rather than an expose, it instead shows the great power of maths, in that it can deal with infinite numbers as well as finite numbers. Hence mathematically, infinity is as good a number as any finite one.
Infinity in physical theories is a DIFFERENT matter (physics not maths, do keep up!), in that having physical quantities take infinite values is usually assumed to mean that the physics isn't complete. But mathematically, no problem, infinity is as good a number as 2, or pi, or zero.
|
|