marc.geddes.domain.name.hidden wrote:
>
>
> 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.
How can Everett's "every possibility is realized" be logically compatible with Bohm's "there's only one, deterministic outcome", we just don't know which one" and Griffith's "it's a probabilistic theory so some things happen and some don't". I can hardly imagine less compatible interpretations of the same mathematics. I could add Cramer's transactional interpretation and Feynmann's zig-zag in time interpretation. Are all those maps or territories?
>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.
Of course every physicist from Newton to Einstein would have said there are generally empirically established features of mechanics like locality, determinism, independence of momentum and position variables, and flow in phase space that must remain the same.
Brent Meeker
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Received on Wed May 09 2007 - 01:57:18 PDT