Re: Skolem's Paradox and why is math the way it is?
From: J.E. (troubled6man_at_yahoo.com)
Date: 10/08/04
- Next message: Jon Slaughter: "Re: Minimum modulus of a polynomial"
- Previous message: Jon Slaughter: "Re: Minimum modulus of a polynomial"
- In reply to: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
- Next in thread: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
- Reply: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
- Messages sorted by: [ date ] [ thread ]
Date: 7 Oct 2004 21:29:38 -0700
"Shmuel (Seymour J.) Metz" <spamtrap@library.lspace.org.invalid> wrote in message news:<41622952$29$fuzhry+tra$mr2ice@news.patriot.net>...
> In <39d6e584.0409291042.75243cea@posting.google.com>, on 09/29/2004
> at 11:42 AM, troubled6man@yahoo.com (J.E.) said:
>
> >I'm saying that it does land everywhere.
>
> OK, so you adhere to Many Worlds[1]. It's viable, but it also has
> issues.
No I don't adhere to Many Worlds, or to be more precise I've never
agreed with anyone I've met that claimed to agree with Many Worlds. I
have similar views in that I think the ONE wavefunction that exists in
configuration space has geographically different parts that correspond
to different versions of me that observe each of the results, but I
don't talk about many worlds or splitting or interfering or creating
of worlds, just smoot continuous wavefunction evolution like any
no-collapse quantum person would use. I follow Mermin more than
Everrett (if that's even how you spell his name).
> >the >wave-function splits
>
> Please describe the apparatus that measures the splitting.
No apparatus measures it perfectly, but any hamiltonian that causes
correlations between the wavefunction part corresponding to the
particle in a location on the cross section of the incoming bean and
and the wavefunction part corresponding to that cross section would be
adequate enough to see a coarse approximation to the pattern after
many trials. Any Bohmian model should describe this to you in
whatever language you like, since I'm unlike to happen to pick word
you already understand.
> >Thank you for telling me your opinion about the ZF axioms.
>
> Actually, I haven't. How do you know that I don't prefer GBN?
Well I've been asking why people like ZF axioms, and you haven't been
disparaging them in favor of GBN axioms. You could have any opinion
at all, I will only know the opinions you choose to share.
> >So it is interesting to you that there is no bijection from
> >the naturals to the power set of the naturals, right?
>
> Yes.
Can you describe how this is anything other than an ambiguous result
to have?
> >Are you concerned if that is bothersome to other people?
>
> No. Are you concerned that SR[2] is bothersome to other people?
I am concerned that SR bothers some people, but that is probably
because in my experience it comes from having bad teacher of SR, and I
disapprove of bad teaching. It is possible that I had bad teachers
for math. An over-reliance on the Moore method by my instructors
might be the culprit, but I don't like to place blame on my own
teachers, seems too easy and cheap. As a teacher I'll happily blame
myself for my students' misunderstandings, but as a study it seems
unjust to blame my teachers.
> >It is bothersome to the people who are not making further axioms to
> >add to ZF and instead are merely using ZF as is, because there
> >doesn't seem to be an uncountable number of things described by
> >ZF, given the faithful countable model by Skolem.
>
> What do you mean by "be"? Are you taking a Platonic position that
> every time you prove (Exists x)P(x), that there is a "real" object
> described by that x? Platonism is not a necessary part of Mathematics.
The ZF axioms fail to describe an uncountable number of things,
because they are incapable of proving that that many distinct things
exist.
> >I want to get rid of unobservables from math because I think it
> >would *help* to remove them from math first and *then* from physics.
>
> It would make more sense to first remove unobservables from Physics.
If I new how to do that first, I would. It seems hard when physics
uses math that already has elements that are independant of proof to
exist.
> >I was trying to say that the fact one can prove theorems with axioms
> >doesn't say anything about the axioms themselves.
>
> Why would you want to say anything about the axioms beyond stating the
> theorems that can be derived from them?
Incompleteness is a problem, because I want to use mathematics to
describe things, and the mathematical language of ZF is descriptively
incomplete (which means among other things (I'd bet), that I can
describe things that I cannot prove whether or not they exist).
> >I still don't understand these other criteria,
>
> That's the problem; Mathematics is an art form. One of the key terms
> is "elegant". The criteria for judging Mathematics are more intuitive
> to a musician or a sculptor than to a Physicist.
Descriptive completeness isn't beautiful or elegant to mathematicians?
> >I'm willing to consider that, but then I
> >want to know why you think the power set is interesting.
>
> Because it generalizes. Because it leads into other results. Because
> the proof is so simple.
It seems vague, like if someone claimed there was some deep symmetry
on the floor of a room that was mostly covered with a bland rug. If
the deep symmetries are forever unobservable, then what is the point.
In ZF, all the big parts of the big sets cannot be proven to exist.
So why consider them? Why not stick to definite results about things
we have observed and statistical claims about the things we haven't?
> >I understand talking about the collection of all subsets that exist
> >in a model of a particular set,
>
> But you seem to be equating the model with the theory that it is a
> model of. They are not the same.
A theory can only describe things that exist in all it's models,
right? So if consistency is independant of the theory, then it can't
be described in the theory. If truth is independent of the theory,
then it can't be described in the theory.
> >but to talk about there being more subsets when you
> >haven't yet introduced any further axioms is weird.
>
> What further axioms do you need? ZF already includes the requisite
> axioms. ZF is not a constructive theory; proving (Exists x)P(x) does
> not mean that we have show a construction for such an x, but only that
> the statement is a theorem of the system.
I don't know what your definition of construct is, I'm talking about
sets that you can prove exist from the axioms. The class of such
subsets of the naturals is clearly not different in kind than the
class of naturals themselves. The fact that no bijection exists from
the two classes, is an inadequacy of the thoery to describe it's
provability, it says NOTHING about the alleged larger class of subsets
that COULD have been proved with MORE axioms.
> >But from physics we want a "finished" mathematical theory
>
> That's impossible, if you want it to be consistent. It will always be
> incomplete.
Then a theory that is clear about the incompleteness being in the
parts we don't use should be good enough then. ZF doesn't seem set up
to make it clear where descriptive completeness starts and stops,
that's a failing in my opinion.
> >it's cumbersome to add another mathematical
> >axiom and then go do all our physics again,
>
> It's also unnecessary. ZF is perfectly adequate for the tasks to which
> physicists have applied it to date.
That statement is not a theorem of ZF. How can we tell that the
incomplete parts of the theory of ZF don't interest with the
physically interesting parts of our physical models based on ZF.
> >I wanted to know why uncountability is desirable,
>
> LIttle things like the MVT.
A single point is less than a drop in the bucket. Besides doesn't the
countable model have a mean value for every function that can be
proven to exist? So why do we need more points if their only purpose
is to be the mean value for functions that we can't prove exist
anyway?
> >What is your operational definition? Show me a power set in the lab
> >and say what is in it.
>
> The lab is a notebook. What is in it are marks on a piece of paper,
> forming inferences from the axioms in accordance with the rules of
> inference.
Then surely you can see how the class of sets you prove theorems about
is not different in kind than the class of natural numbers. There are
not MORE sets you prove theorems about than numbers you can write
down.
> >The problem is that WHICH elements are in a
> >power set depend on OTHER axioms than ZF
>
> No. You can only talk about a set being an element of another set in
> the context of a specific theory.
In that case there, the existance of some sets, that if they existed
would be elements of the power set depends on the OTHER axioms.
> >and you haven't CHOSEN those other axioms YET,
>
> If I chose other axioms then it wouldn't be ZF any more. If you find
> it more convenient to work with, e.g., inaccessible cardinals, that
> doesn't change the theorems of ZF.
I don't need inaccessible cardinals, there are missing sets in the
power set of the naturals. Why isn't this a problem to anyone else?
I don't publish proofs of this because it seems obvious and trivial.
It can't possibly be that no one else knows this except me, I find
that extremely hard to believe.
> >so you just insist that there ARE more subsets than can be proven
> >to exist
>
> No. I note that it is a theorem in ZF that they exist; I don't draw
> any metaphysical conclusions from that. If you want a constructive
> theory, they're down the hall ---->
Show me this theorem. I have only seen theorem about lack of
bijections not about existances of many sets. The two concepts are
different, as I HOPE the skolem paradox's "resolution" has already
made clear to you.
> >since IF another axiom was made to count the previously existing ones
>
> Are you talking about inaccessible cardinals?
No.
> >You toss around the word all in a very UNoperational way in ZF.
>
> Not at all; you simply don't understand what the relevant operations
> are. They're rules of inference.
There aren't proofs that all things that COULD consistently be added
to the theory ARE already in the theory. And if I mention that you
sometimes hide behind the tree that you can't add axioms, and
othertime behind the incompleteness tree. You can toss the word all,
but you haven't proven all, and I can describe things that should be
sets that you can't prove are.
> >The THEORY is about things you can measure
>
> That hasn't been true for a century. The theories used in Physics
> involve gauge fields that are quite removed from things that we can
> measure.
I'm sorry that we will have to disagree on this (I would have
preferred for you to understand my arguement and either adopt it
yourself or convince me of a better one, but you don't seem to either
not want to or not be able to understand it). The theory is about the
lab results and the calculation results, how an individual chooses to
solve it is not part of the theory. I could take some numerical
algorithms and translate these gauge theories into a program that
quizes the lab reporters about the experiment in terms that are
entirely observable and that gives a two printouts, one of observable
results and second rules to compare the experimentally observable
results to the first printout. I could them printout the source code
of the program and call THAT my theory. We don't do that all the time
in practise because mathematics allows us to exploit symmetries that
make the computations much more efficient than the brute force machine
I'd write. But there is nothing inherent about unobservable parts
about a theory for it to work. Nothing at all.
> >If I don't PUT IN absolute phases,
>
> What do you mean by that? Theories involving local gauge invariance
> require absolute phases, but the dynamics are invariant under gauge
> transformations.
Those are DESCRIPTIONS of the theory. There is more than one way to
describe a thoery. See the helpful computer program mentioned above
as an alternative.
> >so approximate solutions are fine if they are accurate enough.
>
> Then why do you care if the axiom systems aren't accurate in some
> vague metaphysical sense if the results are accurate enough?
Each logical problem from math is serious because I'd have to trace it
through every simplication I used to see if the problem infected my
results.
> >For instance, I know I guy in England who is spending his career
> >looking for closed timelike curves that are solutions to Einstein's
> >field equations, and he can't seen to prove that they either exist
> >or can't exist.
>
> Huh? Wasn't that solved a long time ago? Or do you mean that he is
> looking for solutions subject to a constraint that he is interested
> in?
I already said "Closed timelike curves," do you consider that to be
"merely" a constraint that he is interested in, or should I tell him
that it's already been done? I'm sure he'd eventually admit that he
wanted to be told, because if it's true then he'll find out eventually
and it's better sooner than later.
> >It would be nice to have a decidable model
>
> Kurt Gödel demolished that hope a long time ago. And you're confusing
> model with theory. Models aren't decidable or undecidable; only
> theories are.
How about a decriptively complete model?
> >If a small enough model now would avoid things like that being ambiguis,
>
> A small *theory* would leave more statements undecidable than a larger
> theory.
ZF seems to have it's undecidable statements buried in infinite sets,
so it seems like a theory with smaller sets could remove the
undecidable sets. Just say "no" more often instead of reaching for as
many yeses as possible that catches many maybes in the net.
> >but some results are independant of teh ZF axioms,
>
> What are you trying to say? The results are inferences from the
> axioms; they can't be independent of them. The best that you can say
> is that certain results exist in multiple theories. The results that
> appear to be relevant to Physics exist in both ZF and GBN.
There are things that should be sets, but that you can't prove if they
are sets in ZF. ZF is descriptively incomplete. Therefore the
existance of these sets is independant of the axioms of ZF.
> >I don't want an infinite regress based on a mathemtaically undecidable
> >proposition. It may be that no matter which axioms I choose that will
> >happen anyway, but I can't know unless I try the axioms out.
>
> You know that the theory will be incomplete. You can't tell whether
> that incompleteness is relevant to a physical theory until you have
> the physical theory. And if it is relevant, the solution is to add
> axioms, not to take them away.
Axioms haven't been well ordered. Adding an axiom that makes the
system inconsistent doesn't seem very helpful, and what if an existing
axiom is inconsistent with the axiom we really should add, how are we
supposed to know in advance. Isn't the prudent thing to do, to start
out with as few as possible so that the new ones have the freedom to
be what they need to be? If not, what's wrong with what I said?
> >You describe your inial conditions as a wavefunction, and your
> >observations as sections of a wavefunction.
>
> What do you mean by "sections of a wavefunction"? And is isn't a wave
> function rather unphysical?
Observations are local, the wavefunction is not. Clearly only a
portion is applicable to the verifyable parts of any particular
experiment. The other parts can be filled in with anything that makes
the computation easier and can be thrown away when you are done. So
basically you only describe the parts of the wavefunction that affect
your experiment and only those parts that affect the experiments
affect the portions of the calculations you care about. I don't
consider the wavefunction to be unphysical, but I consider anything
else to be, luck for me there IS nothign else in my models.
> >Huh? You do your calculations, no one walks in and forces you to
> >pretend things commute when they don't.
>
> The fact that they don't commute has some observable properties. In
> fact, some physicists start with the commutation relations.
Some physicists might start by counting their toes and drinking
coffee, it doesn't mean that affects me. What matters is the initial
conditions and the potentials and the evolution equations. Nothing
else matters, full stop.
> >I'm not an idiot,
>
> Perhaps, but when you write things like "Who cares about
> commutations?" and "You need an infinite space because you have an
> infinite number of potential particles" then it's not obvious,
> especially when you contradict yourself.
You ask such details questions sometimes that I forget you may not
understand how multiparticle quantum mechanics works. If I have an
upperbound on energies, and the size granality of my equipment such,
then I can get by with a finite number of equivalence classes of
states that will look and act the same for the experiment that is
complete in the sense that everything I could (and hence do) observe
is modelled. The simplest mathematical models that the members of
these equivalence classes live in that is probably familiar to you
have uncountable cardinality and infinite dimension. But that's more
about how I talk to YOU, than about how *I* do physics. You keep
assuming that I do my physics badly just because I don't do it like
you. That's more than annoying. I'm trying to get your perspective
and I treat you with respect in that I assume that you have good
reasons for having the opinions you do and for doing the things you
do. You seem to assume that I'm dishonest just because you aren't
familiar with my work.
> >What experiment are you performing,
>
> I'm measuring the position and momentum of an electron.
I have trouble believing you. What equipment are you using?. What
results are you capable of measuring and distguishing, and what theory
are you using to interpret your results? Everyone I meet does
position measurements, there's this constraint called locality and it
applies to all the experiments I can do with my funding. Maybe you
have better funding than me. And to be honest if your funding is THAT
much better, then maybe I shouldn't make models for you, because
that's probably out of my league.
> >what is the tolerances and the precisions.
>
> Arbitrary. Except that I can't, BECAUSE OF THOSE COMMUTATION
> RELATIONS.
I have trouble believing you about arbitrary precision. What
equipment are you using to get these results?. What theory are you
using to interpret your results? Everyone I meet does finite
precision measurements, there's this constraint called finite
accessible energy and it applies to all the experiments I can do with
my funding. Maybe you have better funding than me. And to be honest
if your funding is THAT much better, then maybe I shouldn't make
models for you, because that's probably out of my league.
> >Why are you computing something like that?
>
> Maybe I want to know the physical limits on what I can measure. There
> are observable consequences.
There are observable consequences to how much funding you have? Well,
I agree with you, that usually the biggest case made for asking for
more funding. But what does this have to do with science
specifically.
> >The results you need/want are the shape (location) of the wavefunction.
>
> How do I measure the wave funtion?
You subject your finger to a hamiltonian from your brain that
correlates the brain state with the position of your finger such that
the position of you finger is correlated with the parts of the
experimental apparratus that you decide to observe. This causes a
correlation between marks on your lab book and the wavefunction of the
experimental apparratus. If you repeat the process many times with
similar wavefunction but move you pen to a different page of the lab
book each time, then after a while the correlations between pages on
the lab books will approach the autocorrelation of the wavefunction of
the apparatus. The whole process assumes that you have a reliable
process to produce things to observe and to make things to observe
them.
> >I'm REALLY upset with people pretending that there is a complete
> >basis all the time.
>
> It may be a crooked game, but it's the only game in town. QFT is a
> mess, but we don't yet know how to clean it up, and meanwhile we get
> some very accurate predictions.
If it's gonna be approximate anyway, why not find a better basis then?
> >If we only wanted to calculate the ones we make in
> >the lab, then it's OK, but then we don't need an infinite model,
> >like you claim.
>
> You need an infinite state space even for a single particle system.
> What do you think the wave functions are? There is no finite basis for
> them.
For bounded energy bound states there is.
> >Especially because you don't like models,
>
> There you go again. If I tell you that I don't want raspberries in my
> chili, that doesn't mean that I dislike raspberries. It just means
> that they don't belong in chili, possibly because it's a waste of
> perfectly fine raspberries. What I don't like is confusing models with
> things that aren't models.
Models tell you things about your axioms. In this case they pointed
out the fact that lack of bijections and size are not intrinsically
related.
> >I understand that isomorphism, you are the one claiming we have to
> >use ONE particular model with an infinite algebra even though it
> >makes distinctions between events and states that we can't
> >distinguish between.
>
> No, I am the one that realizes that we've known for decades that the
> commutation relations can not be satisfied with a finite dimensional
> representation. If you're not aware of it, it's not my job to remedy
> the lack. That has nothing to do with the fact that we use
> approximations in our calculations.
But these lack of satisifactions only occur "under the rug" where we
can't observe them anyway with our finite precision. As long as the
basis is big enough for your current experiment, then it's good
enough. And if you need a bigger one for another one, that's OK too.
The fact that we use approximations in our calculations forgives us
from using an incomplete basis. If two states are differnt in your
BIG model, but can't be experimentally distinguished in a particular
experiment, then you can put them in an equivalence class. For
instance if there is much lasers being superimposed, but all you are
measuring is the color you observe with your eye, then instead of an
basis as large as the number of lasers you had, you only need a basis
of three elements.
- Next message: Jon Slaughter: "Re: Minimum modulus of a polynomial"
- Previous message: Jon Slaughter: "Re: Minimum modulus of a polynomial"
- In reply to: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
- Next in thread: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
- Reply: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
