On May 9, 3:22 pm, Brent Meeker <meeke....domain.name.hidden> wrote:
> marc.ged....domain.name.hidden wrote:
>
> > Infinite sets and infinitesimals are a lot more than 'mathematical
> > conveniences'. There are precise logical theories for these things
> > (As I mentioned before - Cantor worked out the theory of infinite
> > sets, Robinson/Conway worked out the theory of infinitesimals).
>
> Being logically consistent and/or precise doesn't imply existence. There are consistent theories in which there is a cardinality between the integers and the reals. But there are also consistent theories which deny such a cardinality. Does that mean some set exists with that cardinality or not?
It depends on which theory has the most explanatory power when it
comes the concepts in need of explanations. Which explanation best
simplifies or *integrates* our knowledge? That is the criteria one
should use. It's true that consistency/precise alone doesn't imply
existence, but they are factors that one can take into account.
Empirical measurement isn't the defining factor either. Esoteric math
concepts by definition are far removed from direct empirical data but
progress is still made.
>
> > It's true that infinite sets are not used in comptuer science (which
> > is all about discrete/finite math) but beware of making assumptions
> > about reality purely on the basis of what can be measured ;)
>
> If it can't be measured even indirectly, like an infinitesimal, then whether it is kept in your model of reality is mostly a matter of convenience.
See above. Math seems to be a branch of knowledge which falsifies
this view. Mathematicians don't regard differences between theories
as 'mere convenience', even though the theories under definition may
be far removed from direct empirical observations. The view you've
just stated seems to be a logical positivist/intrumentalist view of
science. Again, I don't think the criteria for existence is not
whether something can be measured (which by the way would itself
always involve some theoretical judgement calls- since I think you
yourself said that all observations are theory laden). The criteria
for existence should be absed on the explanatory power of the concept
and the extent to which it *integrates* knowledge.
> >> But QM assumes a fixed background spacetime, which is inconsistent
> >> with general relativity - so one of them (or more likely both) are
> >> wrong.
>
> >> Brent Meeker
>
> > There are *degrees* of rightness/wrongness. Later successful
> > theories of reality will still have to have some of the same features
> > of the earlier theories in areas where the earlier theories were
> > empirically proven.
>
> But the earlier theories were NOT empirically proven - they were found to hold over the observable domain. Later they were disproven in a wider domain and replaced by another theory, e.g. thermodynamics was replaced by statistical mechanics. Where they overlapped they agreed on the observations, what could be measured, but they didn't agree on the ontology. So it is the facts, the observations, that are the aspects of reality that are preserved as theories change. Not the mathematics and not the ontology.
Not sure it's as bad as you are making out. The mathematics of the
earlier successful theory can still be shown to be a special case of
(or apporximations to) the later theory. Further, it's not clear that
all the ontology from the earlier theories is being thrown away.
>
> >For instance it's been proven from the EPR
> > experiments that any theory that replaces current QM still has to
> > have some of the same general features such as a 'wave of
> > possibilities/sum over histories', non-locality or uncertainties and
> > so on.
>
> There are already at least three formulations of QM, Griffiths', Bohm's, and Everett's that are logically incompatible at the level of fundamental ontology - and yet they are described by exactly the same mathematics.
>
> Brent Meeker
It's not so clear that the different interpretations really are
logically incompatible. Further, all of them actually use some of the
same concepts they just ascribe different ontological status to them.
On the wiki:
http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics
Of course it wouldn't be surprising if QM were modified in the
future. But all interpretations make use of something like a 'wave of
possibilities', by the Alain Aspect experiements, it's known that
future theories would have to have either non-locality or
indeterminism. And there are other general empirically established
features of QM that would have to remain the same.
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Received on Wed May 09 2007 - 00:16:32 PDT