Doubting Didymus

You guessed.

Seriously, the whole point of the dark matter thing is that we are desperately trying to detect the stuff. 'Detection' in this case can be practically anything, including theoretical prediction fulfillment. (i.e. if dark matter exists, then the universe will expand in a certain way, we see that this is so, therefore dark matter exists).

Cosimo

What tickles me is how we condemn accountants for trying to fool the public with slippery maths, but we'll read with all seriousness the articles in Scientific American, The Economist, et al. in which the revered cosmologist solemnly passes off the Mother of All Plug Numbers.

Clutch

This is a strange notion of restriction -- that is, one that I have never encountered in any logic text. It sound more like undefinedness than unrestrictedness.

Whatever. The more important point is that if that's supposed to be the sense in which "you can't prove a negative", then the contrast between negatives and positives again evaporates. "An undetectable thing exists" is in the same evidential boat as "No undetectable thing exists".

Feather

I agree that definitions/axioms can (and in fact must) be used in any proof.

What I meant, though, was that your "spherical cubes" are just non-existent because they've been defined to be so.

That is, a cube, by definition, is not the locus of all points equidistant from a given point. So to say that a "spherical cube" does not exist is to say that a cube is not the locus of all points equidistant from a given point. I.e. it really doesn't "prove" anything, as it's already taken to be axiomatic.

That's different from proving, say, the spherical law of sines and cosines from definitions related to the sphere (for example).

sentinel00

I can prove lots of negatives: I am not a giraffe, I am not female, etc.

Infinity Lover

In a different thread of mine, Tronvillain confronted me with a hypothetical sphere you can't get out off, because there's no beyond it's boundary to go to. Note the word 'because' linking the two elements.

The inability to get beyond the boundary makes it impossible to determine for certain, that there's no outside. That makes the 'the lack of an outside'a possibility that could never become a certainty. So how can it be considered the cause of the boundary being impossible to cross?

Isn't a posibility (?), that could never become a certainty (!), an impossibility (0) by definition?

[ September 05, 2002: Message edited by: Infinity Lover ]</p>

Doubting Didymus

I was not trying to say that that is THE sense in which a negative is unrestricted, just an attempt at a completely unrestricted statement. I agree that the positive form suffers the same problem.

Usually, unrestricted negatives are unrestricted in only a practical sense. 'there are no hippogriffs anywhere in the universe' is theoretically restricted, in that we COULD look everywhere in the universe, but in practical terms it is unrestricted.

A better example of what I consider an unrestricted, unprovable negative would be my earlier formulation 'magic hippogriffs do not exist'. You cannot prove this, but you could DISprove it, by producing a magic hippogriff.

Clutch

Didymus, could you define a "magic hippogriff" for me? I'm trying to get a handle on your notion of unrestrictedness. It still looks equivalent to ill-formedness.

tronvillain

Infinity Lover:
No. After all, you contend that the universe is infinite, a possibility that could never become a certainty. That it may be impossible to completely confirm that something is the case does not mean it is impossible for it to actually be the case.

Infinity Lover

In the case of a boundary that's impossible to get beyond, you know for 100% certain that the possibility could never be upgraded to a conclusive.

A possibility, by definition, is a possible certainty. So if you know for sure it could never become a certainty, it's no longer a possibility (a possible certainty) you're dealing with.
There is a distinction there.

The lack of beyond the boundary being mentioned as the cause, for the imposibility to get beyond the boundary, only makes the whole scenario more logicly contradictory.

Saying there's always something, however, because there's never nothing, may be considered a naively oversimplified axiom. But all it ultimately states, is that it would be impossible to have any given point in 'space' or 'time' where there's a situation of 'nonexistence'.

Now what happens when you combine the logical contradiction of the one scenario, and combine it with the logical consistency of the other?

That's the funny thing whith imposibilities. Contrary to possibilities and certainties they don't exist.
Everything that exists is possible.
Everything that's impossible doesn't exist.

[ September 07, 2002: Message edited by: Infinity Lover ]</p>