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Post by speakertoanimals on Jan 18, 2011 21:40:43 GMT 1
Wrong. Utterly and totally wrong. Typically you totally confuse maths and physics.
SO, the axioms of euclidean geometry are as good as they ever were.
The axioms of euclidean geometry are as good as they ever were.
Euclidean geometry is only out-dated in terms of applying the maths of euclidean geometry to the real world (physics), NOT in terms of the 'logical structure of reality'.
Axioms aren't established, because they AREN'T propositions of physics that need top be proved, they are axioms (go look up what that means!).
And more of the same utter nonsense!
You have very, very obviously failed to understand the basics of maths, and this is particularly relevant in the case of godel, since the whole point and method of proof was reducing maths to a sequence of statements, expressed in terms of abstract symbols (and in fact reduced to NUMBERS), that had NO MEANING beyond their own formal system. It was maths and the generation of true statements that could be done totally mechanistically, no need for human intuition, you just applied the laws of the system.
That is the great thing about Godel -- it shows that discovering maths can't be reduced to something machine-like, but not for any of the reasons you espouse.
Nope, you really have totally failed to understand what an axiom is, tyo distinguish between the axioms, and the rules used to generate true statements from axioms. In a word, you haven't even started the Introduction to a summary for non-specialists of Godels theorem, let alone started on Godels theorem itself.
WHY don't you accept you need to LEARN something about maths in order to have a sensible discussion about maths? you seem to think ANYTHING and EVERYTHING can usefully be discussed from a position of almost total ignorance of anything relevant................
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Post by abacus9900 on Jan 23, 2011 16:24:59 GMT 1
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Post by carnyx on Jan 23, 2011 17:35:26 GMT 1
Abacus,
That is fairly conclusive, that;
Or even, and to the chagrin of Dawkins et al, ... that the concept of God is actually a necessity.
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Post by abacus9900 on Jan 23, 2011 17:53:05 GMT 1
Abacus, That is fairly conclusive, that; Or even, and to the chagrin of Dawkins et al, ... that the concept of God is actually a necessity. Doesn't this really show that people who like to think of themselves as impartial 'scientists' are in reality distorting things in favour of their personal view of things? Like the guy in the video said, we have to make assumptions about stuff we call 'true' but that cannot be proven and this is a problem no matter how much knowledge we think we have. STA does not seem to be able to grasp this at all.
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Post by eamonnshute on Jan 23, 2011 18:14:04 GMT 1
The video is a load of nonsense - confusing Physics with Maths, which Speaker warned about. What does it mean to say that Euclid's postulates are true? The fifth postulate can be replaced to give non-Euclidean geometries, which are just as consistent as Euclidean Geometry.
It is like saying that the laws of Chess are "true", when they are merely arbitrary starting points, which can be changed as much as you wish to give other variants of the game. All that matters are that your chosen rules are compatible. And the rules of Chess tell us nothing about the physical world!
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Post by abacus9900 on Jan 23, 2011 18:37:49 GMT 1
The video is a load of nonsense - confusing Physics with Maths, which Speaker warned about. What does it mean to say that Euclid's postulates are true? The fifth postulate can be replaced to give non-Euclidean geometries, which are just as consistent as Euclidean Geometry. It is like saying that the laws of Chess are "true", when they are merely arbitrary starting points, which can be changed as much as you wish to give other variants of the game. All that matters are that your chosen rules are compatible. And the rules of Chess tell us nothing about the physical world! You have not understood a thing, just a blind loyalty to whatever Speaker says. As the guy in the video said, everything in what we call the 'universe' (including maths) cannot be proved, even though we regard it as 'true' for all practical purposes. Non-Euclidean geometry does not alter this because that too cannot be proved.
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Post by Progenitor A on Jan 23, 2011 18:49:28 GMT 1
Well a sort of Christian refraction of God -(oh el!)
Thanks for the links Abacus
A polemic, an argument for God based on Godel's theorem!
Fascinating!
This is how science/maths should be!
Knowledgeable people constructing without dogma, without hubris, logical steps leading to a conclusion that demands questioning, demands reasoned opposition
Loved it
Wish I knew the proof of Godel's theorem that all postulates have an assumption that lays outside the postulate, and that if we attack the 'outside' bit then that assumption will have an assumption that lays outside the assumption that we are attacking
An infinite regression
God does not exist!
Outside that axiom insights an assumption that someone else 'did-it'
It was a quantum fluctuation what done it!
With that hypothesis Godel demands that something else must insight outside
And so it does
Who created the Quantum fluctuation?
Every regression leads us away from science into the realms of metaphysics
In the end only 'metaphysics' is true
But what is truth?
The more we know,the more we realise we do not know
That is the truth
That Godel deserves a kick in the Gonads!
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Post by carnyx on Jan 23, 2011 19:05:55 GMT 1
Eammon,
Try this one; in any system of thought, there will aways be a situation that cannot be proved within the given rules .. and to resolve it requires reference to something outside the system to arbitrate .... i.e to make up an additional rule.
And as you see in the case where Euclid's 5th postulate is discovered not to work when the surface is curved.
So, we have to introduce a new definition to limit Euclid's geometry ... that the surface on which Euclid's geometry is done, must be a flat plane. This in turn gives rise to a proliferation of different kinds of geometry for various types of curvature , each of which will contain their own hidden 'undecidability' .. and when discovered will require new 'rules' ... and so on. So, we started with one Geometry, which was supposed to be a mathematical system that encompassed all shapes, but we ended up with hundreds of them! So in essence, Goedel showed that no system can be complete ... and there will always be 'undecidable propositions' or snags lurking there ..
And, IF this applies to mathematics, then it must also apply to any other kind of systems of thought, including physics.
I think you would agree that nothing of man can ever be perfect, and Goedel shows that this is true even in theory. So there never can be a 'watertight theory of everything' unless you include the concept of God.
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Post by eamonnshute on Jan 23, 2011 19:25:05 GMT 1
As the guy in the video said, everything in what we call the 'universe' (including maths) cannot be proved, even though we regard it as 'true' for all practical purposes. Non-Euclidean geometry does not alter this because that too cannot be proved. What do you mean " 'true' for all practical purposes"? Is Euclid's 5th postulate "true for all practical purposes", or is it untrue, as in non-Euclidean geometries? You can't have it both ways! The fact is that it is a given, an arbitrary starting point. It can be true, it can be false, whatever you choose, so talking about proving it one way or the other is nonsense.
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Post by eamonnshute on Jan 23, 2011 19:35:22 GMT 1
And, IF this applies to mathematics, then it must also apply to any other kind of systems of thought, including physics. That does not follow, it is merely an unjustified leap of faith on your part.. Godel's theorem does not apply to Physics or the physical world, only to certain mathematical systems, regardless of whether or not they apply to the real world.
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Post by Progenitor A on Jan 23, 2011 19:46:57 GMT 1
The video is a load of nonsense - confusing Physics with Maths, which Speaker warned about. But, Eamon, modern physics is almost entirely based upon mathematics therefore the'confusion' is part and parcel of modern physics? It is like saying that the laws of Chess are "true", when they are merely arbitrary starting points, which can be changed as much as you wish to give other variants of the game. Yes indeed but other games outside the rules of chess are evidently not chess, therfore the rules of chess are 'true' to chess. .. the rules of Chess tell us nothing about the physical world! But the rules of chess are of the physical world and therefore must tell us something of the physical world
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Post by abacus9900 on Jan 24, 2011 10:09:24 GMT 1
The video is a load of nonsense - confusing Physics with Maths, which Speaker warned about. This is the point naymissus. We model physics through mathematics, therefore, our representation of reality is only as good as the validity of mathematical axioms which Godel showed to be unprovable, though true in practice. We always have to make assumptions about mathematical postulates that we can never prove so that we can never be sure our models will always work in all circumstances.
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Post by speakertoanimals on Jan 24, 2011 13:50:01 GMT 1
Wrong. The DEFINITION of a flat space is that the geometry is euclidean. Euclidean itself is defined perfectly well by the axioms underlying EUclidean geometry.
Utterly wrong. Godels theorems apply to not a system of thought, but a purely mechanistic system, that from axioms applies rules as to how symbols can be manipulated, in order to derive true statements from those axioms. It's not THOUGHT at all, which is part of the whole point of the effort -- showing that mathematicians who think are reequired, we can't just construct a machine, turn the handle, and generate all true statements that way. Maths is richer than that.
It most specifically DOES NOT apply to any system other than the ones defined in the proof.
Utter, UTTER bollocks! shows that you havent' grapsed even the basics of maths. The point being, Godel didn't show a damn thing about axioms being unprovable, the point being axioms are not defined as being provable or unprovable.
So when it comes to geometry, if we have the axiom that no parallel lines meet, then we have Eucliden geometry. If we drop this axiom, we get non-Euclidean geometry.
Hence the statement is neither true nor false, but instead a CHOICE. If we choose it to be true, we get one type of geometry. If we chose it to be false, we get another.
Which geometry the universe uses is another matter, it's not a matter for mathematical proof or dis-proof, but experimental verification. A different matter.
TO put it another way, we could wriote down rules of chess where knights move a usual, or we could write down rules of chess where knights moved slightly differently. BOTH would be valid rules for a game, although one set might be more interesting than another. But there is no proof of the way a knight MUST move.
So, to decide which game we are in, knight moves normally or not, we just have to watch a game, and see which way knights move. Same with the universe, we watch stuff and see how it moves, to try and work out what rules the game of the universe is using.
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Post by Progenitor A on Jan 24, 2011 15:18:44 GMT 1
This is the point naymissus. We model physics through mathematics, therefore, our representation of reality is only as good as the validity of mathematical axioms which Godel showed to be unprovable, though true in practice. We always have to make assumptions about mathematical postulates that we can never prove so that we can never be sure our models will always work in all circumstances. Yup that about sums it up for me. If physics is based upon maths (and it sure is) then physical hypotheses are unprovable, quite nicely illustrated by QM 'explanations' (constructed from mathematical models) that are self evidently unprovable , such as, for example, the hypothesis that a photon (e.g) is in more than one place at once. To me the delight of that unprovable hypothesis is that those who 'believe' it (rather than those who just take it as unprovable working hypothesis), tell us, with straight face, that when we look to see it in two places a once it 'collapses' into just one place, rather like imaginary fairies at the bottom of the garden that disappear when you look at them Another prime example (cited in your referenced videos) is the claim that the universe was created from a 'quantum fluctuation. Now this hypothesis comes about from cosmologists following through mathematical constructions. Godel tells us that the hypothesis, because it is mathematically based, is unprovable, something we all know very well without the assistance of Godel, but it is nice to know that our intuition is in agreement with the widely accepted Godel theories Incidentally, as Godels theories mathematical theories then they are unprovable! Isn't that just lovely? And did anyone notice from the videos that one explanation used the ISO OSI 7-layer Communication Protocol? Astonishing that the model my type uses every day at work was used to show there is probably a God! ;D
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Post by carnyx on Jan 24, 2011 15:26:19 GMT 1
STA,
YOu have given us a classic example of circular reasoning at it's best!
And what do we do with the discoveries of these natural rules?
Well, bless me if we don't create model 'games' in our heads, ... bound by rules ( aka laws .. aka axioms)....and call them theories.
And we give these 'games' names ... such as 'physics', or 'chemistry', or even 'astrology.'
But, in our games, or models or simulacrums of reality, we find undecidable situations .... and we then know that all our 'games' ... and hence our knowledge of nature, is incomplete.
And .. can never be complete.
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