vanesch said:
Well, these "laws of nature" could simply be a catalog of arbitrary 3D slices, no ? What is the "law of nature" describing the 2D slices of a ballpoint ?
The laws of nature are found in the correlations between the slices.
If the slices are indeed arbitrary as you suggest, with no correlation between them, then this suggests absence of law.
In the ballpoint case, the "law of nature" of the ballpoint tells you how one slice changes to another slice as you move along the length of the ballpoint. If the ballpoint slices are arbitrarily random then this would imply no law (or randomness), but if there is correlation between the slices (as indeed there is in the case of a ballpoint) then the law of nature of the ballpoint tells you how these slices correlate with each other.
vanesch said:
I know what you mean: you mean that there are some differential equations that relate these 3D slices in a very simple way, "locally" and "in a reductionist way".
No, I mean the slices are correlated with each other, they are not simply arbitrary.
vanesch said:
But that's a VERY PECULIAR KIND of determinism !
What I have described is exactly deterministic. If the slices were arbitrary then one could claim this would lead an observer in this world to conclude that the world is indeterministic (the slices are not crrelated, there is no law), but if the slices are correlated then the determinsitic laws which decribe the correlations are the laws of nature.
vanesch said:
This is a very peculiar "symmetry" rule that this 4-D shape has to satisfy. Because there could be a very complicated "law of nature" which somehow reads:
"look up the current 3D slice in the Big Catalog", turn the catalog 1 page further, and this is the prediction of the next 3D slice.
Yep, this would work. This would be a (very large, infinite?) look-up table of the slices.
vanesch said:
This is a "law of nature" with the right "big catalog", that always works.
It is impossible for us to know it, of course, but there could be such a "catalog" and that would then be the "law of nature", which deterministically joins each 3D slice to the next.
In a determinsitic world we observe the laws of nature to be apparently fixed with time (the law of gravity for example seems to be invariant). This means that we do not need to have a big catalog which contains a unique description of every slice - instead we can reduce the whole problem down to having just one page (description of one slice) plus a one-time description of the laws of nature which allows us (in principle) to calculate all of the other slices.
vanesch said:
In the case of our ballpoint, the "big catalog" is simply its technical drawing in 3D, sliced up in 2D slices. That technical drawing can be as simple, or as complicated, as the designer of the ballpoint decided.
Yes, again this is the look-up table version. But if the slices are correlated we can actually reduce the information, so that rather than having an infinite number of cross-sectional drawings, we have instead a small number of key cross-sections plus a series of equations (the "laws of the ballpoint") which tell us how the other (undrawn) cross sections are correlated with the drawn ones. In the ballpoint case, we may only be able to do this by postulating that the laws suddenly "change" are some points (ie there are discontinuities and the laws of the ballpoint are not in fact fixed) - but nevertheless we can still describe the ballpoint using a limited series of cross-sections plus a limited number of laws, rather than an infinite number of slices.
vanesch said:
So given this pathetic example of "law of nature" which determines the next 3D slice from the former, you see that ANY shape in 4D is possible.
Any shape in 4D is possible in principle. But in a deterministic world with fixed laws, then specifying just one slice allows us to construct the entire world.
vanesch said:
There is of course only ONE such shape, THE shape of the deterministic universe, but a priori, there's no reason why there should be SIMPLE laws of nature linking the different 3D slices, no ?
There is a reason, I believe - and that reason (IMHO) is that the emergence of life would have been extremely unlikely in a world with variable laws (ie if the laws of nature were random or not fixed) - in fact we DO see that the laws of nature appear to be fixed - hence we may live in a deterministic universe.
Note : even if the laws are not fixed, if they vary in a deterministic way then this variation becomes just another "law", and determinism is still possible.
MF
