posted
This is a good example of what to look for axioms. An axiom is something that can’t be proved, and is thus, assumed. For instance, a commonly used axiom in high school geometry is from Euclid, that for a given line, and a point not on that line, there is only one line that goes through the point and does not intersect the other line. In other words, there’s only one line parallel to another that goes through a given point. You can not prove this, it must be assumed.
So, some words on axioms, the best axioms are orthogonal axioms, ones that don’t reference themselves or other axioms. You example doesn’t have this, the two axioms (below is false, above is true) make assumptions about each other, and this leads both to paradoxes (as these do) and to unnecessary axioms, overlap in assumptions. That’s not necessarily a bad thing, but it means excess data.
Axioms are a way of creating the world we perceive, when we take the first example, Euclid’s (5th, I believe) axiom we’ll find a perfectly flat space, and one that represents a common view of the universe. This is one version of reality. But if we assume differently, that there are an infinite number of lines that could cross that point and be parallel to the other line we’ll find ourselves in a different reality, with a whole new set of rules and theorems built on this new axiom.
Finally, when you get a paradox, what you’re getting is proof that the world you base axioms created can not exist. This is actually very helpful when trying to prove something, in math for instance. It called “proof by contradiction”, the way it works is that you take whatever it is you’re trying to prove, and then start off with the assumption that it’s not true. Then you work out a series of logical steps that end up contradicting each other (for instance, you may show that 3*7 = 0 ), and once you’ve done this you’ve proved that the world in which the statement you’re trying to prove is false can not exist, and there for what you’re trying to prove must be true.
So we see from the logical extension of your axioms that the reality that they create can not exist, and thus at least one of them must be false.
quote: For instance, a commonly used axiom in high school geometry is from Euclid, that for a given line, and a point not on that line, there is only one line that goes through the point and does not intersect the other line.
They only use that for 2D space. There are many non-parallel, non-instersecting lines in 3D space.
Posts: 16551 | Registered: Feb 2003
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posted
It's only true in flat 2D space. If I rembember right, in hyperbolic 2D space, there's more than one parallel line through a given point, and in spherical 2D space there are no parallel lines through a given point.
posted
OK, this is true, but I didn't really want to give an "n-dimensional" axiom.
Just for you MPH: For any non-abstract space [this is so we don't have disscuss function spaces, I'm not sure there is a function space corrallary of Euclid's 5th axiom, and I'm still just guessing the 5] of dimensions n, there exists only one object of dimensions n-1 that contains a specified point, but does not interesect with another object of dimension n-1, which does not cocntain that point.
Of course our current understanding of the universe shows this axiom to be false, dues to space's curvature.
posted
Right Dag, our space is non-Euclidian, but the interesting thing is that within a Euclidian world there is no contradiction. So a Euclidian space is perfectly logically valid, it just doesn't happen to be ours.
posted
So you are saying that in 3D space, (assuming space is flat), that if you have a plane and a point not on that plane, there is only one plane that goes through the point and does not intersect the original plane?
Yeah, I figured there would be corrolary axioms for spaces of different dimensions.
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posted
Has it been conclusively discovered which way space curves in our universe? Do you know how they did it?
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posted
Can you logically prove anything back to "reality"? Or does all logic eventually rest on sets of assumptions or interpretations we've made based on our observations of reality?
Posts: 5957 | Registered: Oct 2001
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posted
You can't logically prove an axiom, that's what makes it an axiom. You can show that a series of axioms are contradictory and don't describe a viable universe, but I know that's not what you're asking. So if you want to find out about reality what you have to do is try your best to discover if the axioms you currently have are contradictory (you can't prove that they aren't remember, but you can at least try out a few different ideas in them to test). If they aren't (you think), then you can do some tests to see if they match the way the world works. This wont be proof either, but it can be dis-proof. Assume that all grass is green, seeing one million blades of green grass and nothing else would make it a likely axiom, but not prove it. Seeing a single blad of grass that's not green would disprove it.
quote:So you are saying that in 3D space, (assuming space is flat), that if you have a plane and a point not on that plane, there is only one plane that goes through the point and does not intersect the original plane?
Yeah, I figured there would be corrolary axioms for spaces of different dimensions.
Yep.
quote:Has it been conclusively discovered which way space curves in our universe? Do you know how they did it?
Not the fogiest, sorry.
quote:Yep, that's right. Now try to imagine that in 4-D space.
*migraine*
It's just easier to talk about 'n' dimensions and not think about it. At some point though you have to discuss function space, just try not to vomit.
posted
I believe by measuring the angles of the corners of very large triangles - let me see if I can find a link.
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posted
Well we can't prove it (see above) but so far all experiments match with that idea (that it's curved). This has to do with the speed of light being constant, and it can be shown that if we assume the speed of light to be a constant, space and time must then be non-constant, or be able to curve and change. So far it appears that the speed of light is constant, and so it appears that space is curved.
Ohh right, we can prove it's not flat, but we can't prove what it actually is. You can always prove something is wrong (assuming it is of course ) but can't ever prove a scientific theory, because you can't ever prove the axioms it is based on.
posted
I recently read about a reaction Samuel Johnson had to Berkeley. I thought it was pertinent to the discussion:
"After we came out of the church, we stood talking for some time together of Bishop Berkeley's ingenious sophistry to prove the nonexistence of matter, and that every thing in the universe is merely ideal. I observed, that though we are satisfied his doctrine is not true, it is impossible to refute it. I never shall forget the alacrity with which Johnson answered, striking his foot with mighty force against a large stone, till he rebounded from it -- 'I refute it thus.'" Boswell: LifePosts: 5957 | Registered: Oct 2001
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posted
You guys have heard the story about the explorer and the "This statement is a lie" paradox, right?
The explorer is captured and told he will be executed in the morning. If the next statement he makes is a lie, he will be strangled. If the next statement he makes is the truth, he will be thrown off the mountain.
He says, "I will be strangled."
What most people don't know is what happened to him.
posted
I have a t-shirt that says on the front: "The sentence on the back of this shirt is false." and the back says: "The sentence on the front of this shirt is true."
posted
Hobbes, as I recall, a study on cosmic background radiation angular variations has suggested that spatial geometry is flat, not curved. This points to a universe which has a critical energy density, meaning that even including estimates on dark matter, there needs to be about 70% of the energy density of the universe in vacuum energy.
Or, I may be cobbling unrelated things together, I don't know.
Posts: 1769 | Registered: Feb 2000
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posted
You have to distinguish between local and global curvature of space-time. Locally, space-time is curved due to mass and energy. Globally, I think you're correct, space-time seems to be flat or near-flat, though that view has changed before and might well change again.
edit: yes, I think that's right, the prevailing view is that we are at or near critical mass-energy density in the universe, meaning there is just enough mass-energy to slow down expansion after the Big Bang forever (also meaning space-time is globally flat). If there were any more the universe would contract after a finite period, the Big Crunch.]
posted
According to the latest observations, the expansion of the Universe is actually accelerating. I don't quite know what sort of geometry this corresponds to. On the other hand, listening to cosmologists is a good way to go wrong. Ten years later they always reverse their predictions.
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quote:So we see from the logical extension of your axioms that the reality that they create can not exist, and thus at least one of them must be false.
If at least one of them is false then one of the premises is violated, and thus one of them cannot be false.
Also, who said anything about creating alternate realities? We just want to know which of the statements, existing in our own reality, are true.
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quote: According to the latest observations, the expansion of the Universe is actually accelerating. I don't quite know what sort of geometry this corresponds to.
Actually, that doesn't correspond to any geometry. They think there is some other reason for that accceleration.
<--- doesn't understand the other reasons for that acceleration.
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posted
I don't believe the universe expands and contracts in many big bangs... There would need to be a large force to pull the whole universe as far spread as it is to do it again. I don't think there could be a force that great in the center of the universe to do that.
Posts: 1831 | Registered: Jan 2003
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posted
I just love that idea, though. The universe expanding and contracting, back and forth, like a cosmic 3 dimensional yo-yo. Wouldn't the cumulative gravitational pull of all the rest of the stars in the glaxy that are on the other side of the universe do it? Is the universe revolving? Can an angel escape the gravitational pull of a black hole? If so, how many of them could dance on the head of a pin?
I love Hatrack. So many smart people.
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quote: Wouldn't the cumulative gravitational pull of all the rest of the stars in the glaxy that are on the other side of the universe do it? Is the universe revolving?
It depends on how much mass there is in the universe. Not enough, and the universe expands forever. Just the right amount (coincidentally, the same amount required to make 3D space flat), and it stops expanding, but doesn't contract. Any more mass, and it will contract again, for the big crunch.
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posted
It's accelerating. That's what they know now. And everything is expanding at the same rate. Space itself gets bigger. Sort of analogous to how polka dots on the surface of a balloon being blown up all stretch apart at the same rate. (Thinking of the surface distance between them.)
That's a 2 dimensional analog to galaxies in 3 dimensional space.
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posted
If the rate of expansion is the same, how is there an acceleration? Constant velocity = no acceleration.
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There is really only one answer to this that takes in all it's angles and meets all its logical complexities. That answer is, quite simply, to give you a melvin (whether or not it's atomic is really a matter of personal preference).
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posted
The rate of expansion is increasing over time. Over the age of the universe it's sped up appreciably.
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posted
I think the prevaling view is it accelerated rapidly, then decelerated up until recently, and now a force that is most likely represented by the vacuum density is causing a secondary acceleration, but one that levels off, and will not lead to a big rip (runaway rapid expansion) or a big crunch.
M Theory contends that this vacuum density is determined by the topology of the Calabi-Yau manifold, a 6 dimentional space curled up into plank length sizes. The exact topology of this space, and the branes and flux lines inhabitied by it would determine the exact vacuum density. There is a fascinating article about it in this month's Scientific American.
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posted
You're talking about inflation, yes. They think there was an inflationary period in the early early universe. I just meant since then. Posts: 6246 | Registered: Aug 2004
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posted
Oooh, I love self referential sentences, paradoxical and non-paradoxical!
This sentence contradicts itself -- or rather -- well, no, it actually doesn't!
It goes without saying that
This sentence will end before you can say "Jack Rob
This sentence contains exactly threee erors.
This sentence contains only one nonstandard English flutzpah.
This is to be or actually not two sentences to be, that is the question, combined.
This sentence every third, but it still comprehensible.
The whole point of this sentence is to make clear what the whole point of this sentence is.
Let us make a new convention: that anything contained in triple quotes -- for example '''No, I have decided to change my mind; when the triple quotes close, just skip directly to the period and ignore everything up to it''' -- is not even to be read (much less paid attention to or obeyed).
Edit to say that I got all these from Douglas Hofstadter who collects them.
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