This Week's Finds in Mathematical Physics (Week 146)
David Manheim
>But what if Wheeler was wrong? What if there is *not* a uniquely
>compelling principle or set of equations that governs our universe? For
>example, what if *all* equations govern universes? In other words, what
>if all mathematical structures have just as much "physical existence"
>(whatever that means!) as those describing our universe do? Many of
>them will not contain patterns we could call awareness or intelligence,
>but some will, and these would be "seen from within" as "universes" by
>their inhabitants. In this scenario, there's nothing special about
>*our* universe except that we happen to be in this one.
or what if no finite set of equations can fully describe the universe?
Ralph E. Frost
> On March 11, 2000, ba...@math.ucr.edu (John Baez) wrote:
>
> >But what if Wheeler was wrong? What if there is *not* a uniquely
> >compelling principle or set of equations that governs our universe? For
> >example, what if *all* equations govern universes? In other words, what
> >if all mathematical structures have just as much "physical existence"
Not a completely alien or new idea in math history...
"To Laplace mathematics were the accident and natural phenomena the
substance --- a point of view exactly opposite to that of Lagrange. To
Laplace mathematics were tools, and they were handled with extraordinary
skill, but any makeshift of a proof would do, provided that the problem
was solved."
[Turnbull, Herbert Westren, _The Great Mathematicians_, Barnes & Noble
Books, New York 1993 p.121]
> > [...] as those describing our universe do? Many of
> >them will not contain patterns we could call awareness or intelligence,
> >but some will, and these would be "seen from within" as "universes" by
> >their inhabitants. In this scenario, there's nothing special about
> >*our* universe except that we happen to be in this one.
That's the trick, isn't it? In all the many different descriptions, we
end up in them. How that turns out to be the general case is actually
not nothing special, but quite special.
> or what if no finite set of equations can fully describe the universe?
CMIIAW, but so far, in science, none do.
Also, if you find Godel's proposal meaningful -- that far less complex
functions are unfigurable -- doesn't it seems reasonable to elevate
one's sights on groking the universal math equation and shoot to come up
with helpful approximations?
--
Best regards,
Ralph E. Frost
Frost Low Energy Physics
http://www.dcwi.com/~refrost/index.htm ..Feeling is believing..
Aron Wall
material. This is because it addresses one of those questions in the
region where mathematics, physics, philosophy and theology all meet.
I read Max Tegmark's paper on mathematical and physical reality with
great interest (see Week 146, currently the most recent one). The paper
makes a lot of very interesting points and is a fun read. But it seems
to me that the basic idea behind his paper is very flawed.
Tegmark divides the possible theories about the relationship between
mathematical and physical reality as follows (more or less):
------------------------------------------
1. The physical universe is fully described by mathematics:
a. All mathematical structures have physical reality.
b. Some mathematical structures have physical reality.
c. No mathematical structures have physical reality.
2. The physical universe is not fully described by mathematics.
------------------------------------------
I hope the validity of 1c is not a live issue for most of us. Tegmark
dismisses 2 on the grounds that it basically admits defeat for the
physicist (I do not consider this quite fair. As a theist, I do not
think I could say that God is fully describable mathematically, but I
would still feel free to have the material universe fit possibility 1).
1b is probably the most traditional scientific approach. Tagmark's
thesis is 1a, that all mathematical structures are physically real.
An obvious counterclaim to this hypothesis is that it yields no testable
predictions and is therefore unscientific. Tegmark disagrees, and
provides a schema for falsifiable predictions. According to Tegmark, we
ought to use Bayesian analysis, starting with all mathematical
structures allowed. Then we reduce this probability distribution with
the observation that there is at least one Self Aware Substructure (SAS)
is the universe (that's me). Then we examine the resulting probability
distribution, to see whether we live in a "typical" SAS containing
universe. If we don't (i.e. fine-tuning unrelated to the existence of
SAS's), the theory is falsified.
But the above analysis has a big hole. We are using Bayesian analysis
here. So what is the prior that we use to determine the initial
probability distribution for each of the mathematical structures? There
seems to be no obvious canonical choice. One might very well try to
squirm out of this by saying that one doesn't need a prior for all
universes, just the ones with SAS's. But starting from the
SAS distribution is equivalent to giving up all hope of predicting this
probability distribution. And this means there are several choices for
this probability distribution.
The whole notion of using probability to determine which of a number of
physical structures *you* are strikes me as absurd. Probability theory
seems to me to be all about having a number of different mathematical
structures and not knowing which one is physically real. For example,
suppose I flip a coin. I can say that it will come up heads with 50%
probability and tails with 50% probability. But if I believed that
*both* happened in two separate universes, I could no longer say this
using the normal meaning of probability, which is that we tend to
observe high probability events more than low probability events, since
both heads and tails would be observed. (Do not confuse this with an
affirmation of the frequentist point of view. Probability is still a
reflection of certainty about which universe is "real", rather than a
measure of frequency.) Even if the coin were weighted do that heads
only had a 1% chance of occurring, it would still be as "observed" just
as tails is. This criticism applies to all forms of Many-Worlds
theories, by the way.
But supposing I am wrong about all this. There is still an additional
problem with Tegmark's hypothesis. Any theory of physics can be
modified so as to be unobservably different. You can add a particularly
wimpy WIMP, you could miraculously change the laws of physics in a few
arbitrarily determined small regions, you could... The sky's the limit!
And no reason to keep the theory covariant, pretty, or anything else,
either.
Speaking of ugliness, another consequence of the 1a theory is that one
has to discard Occam's razor, since even an intolerably complicated
universe must exist. And such a theory could be constructed so as to be
closely approximating, but observably different from, a universe with an
SAS. Science progresses based on the assumption that the universe is
fundamentally simple and understandable. But if all mathematical
universes have physical reality, that means the ugly ones also have
physical reality. And there are as many ugly possibilities as beautiful
ones.
For these reasons I am convinced that only some mathematical structures
have physical reality. Tegmark apparently thinks that this philosophy
is rather arbitrary; why should this one particular structure exist
physically while so many of the other ones don't exist? I do not know
how most physicists resolve this question, but I would say that this
particular universe derives its existence from God. This is why our
universe is one of the beautiful ones, because the universe which we
could not endure to live in is the universe that He could not endure to
create.
Despite my criticism of its ideas, I highly recommend this paper as
reading material. If it is wrong, it is at least *interestingly*
wrong. Plus it contains an nice discussion of how physics would be
different if its basic parameters were altered, which is interesting no
matter what you think of the question of physical reality.
Comments are welcome, although if you want to comment on this in a way
not very related to physics it might be best to email me privately.
Aron Wall
Chris Hillman
> Tegmark divides the possible theories about the relationship between
> mathematical and physical reality as follows (more or less):
>
> ------------------------------------------
> 1. The physical universe is fully described by mathematics:
> a. All mathematical structures have physical reality.
> b. Some mathematical structures have physical reality.
> c. No mathematical structures have physical reality.
>
> 2. The physical universe is not fully described by mathematics.
> ------------------------------------------
I have no comment on the "theological" implications (if any), but I also
felt that Tegmark's dichotomy was dubious. Indeed, I'm rather puzzled why
anyone would believe that -either- 1 or 2 are plausible.
My own feeling is that the behavior of the natural world can be modeled,
imperfectly, by mathematical models, each having some domain of
(approximate) validity. I've never understood why anyone might expect "a
final theory" of -anything-; it seems to me that history suggests rather
that science yields an overlapping mosaic of theories or models which
rarely fit together perfectly, or even agree perfectly on "overlaps in
their domains of (approximate) validity".
Indeed, I think the two dimensional metaphor of a mosaic is greatly
oversimplified: a better pictorial image for how "known mathematical
theories" relate to one another might be a gigantic cell complex with
fabulously complicated topology. Indeed, one should imagine this cell
complex to "flex" and "crease" rather like a bellows as one performs
"quotients" by "identifying concepts" or "liftings" by "distinguishing
previously conflated intuitive notions".
The additional demand made upon mathematical models in science, that the
model approximate certain phenomena in the natural world, further
increases the complexity of this pictorial image.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/personal.html
John Baez
>Warning: the following post contains a certain quantity of theological
>material.
... not suitable for children under 18 or easily excitable philosophers!
>I read Max Tegmark's paper on mathematical and physical reality with
>great interest (see Week 146, currently the most recent one). The paper
>makes a lot of very interesting points and is a fun read. But it seems
>to me that the basic idea behind his paper is very flawed.
Indeed, it falls into a class of ideas, like the anthropic principle,
that are a bit too grandiose and a bit too short on testable predictions
to become solid physics any time soon (if at all). I mentioned it just
because such things are fun to think about. They're no substitute for
honest toil. Tegmark himself spends most of his time doing ordinary
astrophysics and refers to this paper as "my nutty theory of everything" -
a sign that he's keeping things in perspective.
>Tegmark's thesis is 1a, that all mathematical structures are physically
>real.
>
>An obvious counterclaim to this hypothesis is that it yields no testable
>predictions and is therefore unscientific.
Right. To get any predictions out of it, we'd need to make some
supplementary hypotheses that will probably be almost as controversial
as his main thesis. And even if we do, it will be tough to get
actual "predictions". For the most part, as with the anthropic
principle, we'll get "postdictions" - explanations of stuff we already
know. Postdictions are never quite as satisfying as predictions.
>Tegmark disagrees, and
>provides a schema for falsifiable predictions. According to Tegmark, we
>ought to use Bayesian analysis, starting with all mathematical
>structures allowed. Then we reduce this probability distribution with
>the observation that there is at least one Self Aware Substructure (SAS)
>is the universe (that's me). Then we examine the resulting probability
>distribution, to see whether we live in a "typical" SAS containing
>universe. If we don't (i.e. fine-tuning unrelated to the existence of
>SAS's), the theory is falsified.
>
>But the above analysis has a big hole. We are using Bayesian analysis
>here. So what is the prior that we use to determine the initial
>probability distribution for each of the mathematical structures?
Excellent point! Picking a prior amounts to making up some supplementary
hypotheses.
>There seems to be no obvious canonical choice.
Probably not. I sure don't know any canonical probability measure
on the set of mathematical structures! It seems unlikely to exist,
at first glance, but who knows? - people probably haven't thought
about this issue very much. There could be whole disciplines
lurking out there in unexplored regions of thoughtspace, like
"stochastic axiomatic set theory", which would have something to
say about this. But this is getting pretty speculative.....
>The whole notion of using probability to determine which of a number of
>physical structures *you* are strikes me as absurd.
It doesn't strike me as inherently absurd; I can imagine using
probability theory to reason about any situation in which I have
insufficient information to deterministically crank out the answers.
But to reason probabilistically, we need a well-defined space of
alternatives and a probability measure on that space. *That* is
what we are missing here!
>But supposing I am wrong about all this. There is still an additional
>problem with Tegmark's hypothesis. Any theory of physics can be
>modified so as to be unobservably different. You can add a particularly
>wimpy WIMP, you could miraculously change the laws of physics in a few
>arbitrarily determined small regions, you could... The sky's the limit!
>And no reason to keep the theory covariant, pretty, or anything else,
>either.
Right. But....
>Speaking of ugliness, another consequence of the 1a theory is that one
>has to discard Occam's razor, since even an intolerably complicated
>universe must exist. And such a theory could be constructed so as to be
>closely approximating, but observably different from, a universe with an
>SAS. Science progresses based on the assumption that the universe is
>fundamentally simple and understandable. But if all mathematical
>universes have physical reality, that means the ugly ones also have
>physical reality. And there are as many ugly possibilities as beautiful
>ones.
This is a good point. Here's a nutty proposal to try to save the
day. Perhaps our probability measure on mathematical structures could
be designed to weight "simpler" or "more elegant" mathematical structures
more heavily. I.e., we could incorporate Occam's razor into our prior!
On the one hand, this is not quite as crazy as it sounds, because in
practice we all do this when dreaming up theories - as you say, science
progresses based on this assumption. But on the other hand, there have been
endless arguments about how you meausre the "simplicity" or "elegance" of
a theory, so this supplementary hypothesis opens yet another can of worms.
And if proposing that all mathematical structures physically exist is
metaphysical, unfalsifiable, overly Platonic and all those other bad
things, just WAIT and see all the flak you'll get when you add a footnote
saying that beautiful mathematical structures exist "more". The empiricists
will come down on you like a ton of bricks.
So I probably won't even post this article... in most universes, nobody
will ever see it.
Walter Kunhardt
On Thu, 16 Mar 2000, Aron Wall wrote:
> ...
>
> Tegmark divides the possible theories about the relationship between
> mathematical and physical reality as follows (more or less):
>
> ------------------------------------------
> 1. The physical universe is fully described by mathematics:
> a. All mathematical structures have physical reality.
> b. Some mathematical structures have physical reality.
> c. No mathematical structures have physical reality.
>
> 2. The physical universe is not fully described by mathematics.
> ------------------------------------------
>
> I hope the validity of 1c is not a live issue for most of us.
Well , if there were a vote, I'd vote for 1c.
Let me try to sketch my personal poit of view regarding these issues.
Honestly, I feel it very hard to understand the phrase "X has physical
reality", X being some "mathematical structure".
e.g. X = the natural number 3. Does it "have physical reality"?
The number 3 is the common characteristic feature of all collections
of three (different) things. In my opinion, the latter may have
"objective physical reality" (at least, if they are macroscopic
objects, such as three apples, ...). But the number 3 is an
abstraction of the human mind, and as such, I wouldn't say that is
has physical reality on its own.
or X = the exponential function, a mathematical structure which is
rather simple (at least in contrast to things such as weak
omega-categories... :-) ). It doesn't have physical reality, does
it?
or X = the euclidean plane. Does it exist somewhere in a museum?
or X = a free point particle in Minkowski spacetime
(Choose your favourite mathematical formulation of this concept.)
These and almost all examples I can think of lead me to think that no
mathematical structure has physical reality. Moreover the examples
tend to indicate that a mathematical structure has "more" physical reality
as it comes "closer" to the Theory of Everything (in the literal sense
of the word, not the sort of watered-down meaning of TOE in particle
physics!). But such a thing is not conceivable, in my opinion.
So, what is now the relation between mathematical structures and physical
reality?
In my opinion, physics is about describing the laws of nature and about
making predictions of what will happen in certain situations/experiments.
(*) To this end, one invents models which have a certain range of
validity. A model always involves some idealisation; in this process it is
necessary to make the distinction between the phenomena under study and
aspects which are to be neglected. Such a model, finally, is described by
the appropriate mathematical structures; this often goes along with the
invention of new mathematical concepts.
(QED is an example where the physical theory seems to be well-understood.
We all know the amazing accuracy of some of its predictions. But a
mathematically consistent formulation is still lacking!!)
To conclude, I'd put it like this:
Some mathematical structures turn out to be useful to the formulation of
physical models, and some physical models, in turn, yield (within their
range of validity) an accurate description of physical phenomena.
Regards,
Walter Kunhardt.
(*) In particular, I think that a question like what an electron
_really_is_ --- Is it a particle? Is it a wave? --- is beyond the scope of
physics. The relevant point is how it behaves.
Ralph E. Frost
>Aron Wall <ar...@wall.org> wrote:
>Postdictions are never quite as satisfying as predictions.
Never? Reinterpretation and necessary back-tracking come to mind.
> >Science progresses based on the assumption that the universe is
> >fundamentally simple and understandable. But if all mathematical
> >universes have physical reality, that means the ugly ones also have
> >physical reality. And there are as many ugly possibilities as beautiful
> >ones.
Drawing on imagery of overlapping mosaics given in one of Chris
Hillman's recent posts here, (or quasi-crystal topology-like things
fitting together), and the notion in the prior post (snipped) about
various mathematical approaches, what is called "beauty" within one
set of axioms most likely is meaningless complexity and unnecessary
abstract compensation when viewed from a more fundamental set -- the
next closest, more general mosaic -- a sub-quasi-crystal.
As a crude example, consider that the xyz axes of cubic relationship
were rationalized first and then several decades later, t, was added.
The alternative for "Descartes" (or whoever it was, if not him) would
have been to begin with a form having four axes and only use three to
start with, keeping one in reserve in case, you know, a need arose.
Initial conditions set the path and boundaries.
So universe is still "fundamentally simple and understandable" and
exceedingly rich in its great simplicity.
Amw
> My own feeling is that the behavior of the natural world can be modeled,
> imperfectly, by mathematical models, each having some domain of
> (approximate) validity. I've never understood why anyone might expect "a
> final theory" of -anything-; it seems to me that history suggests rather
> that science yields an overlapping mosaic of theories or models which
> rarely fit together perfectly, or even agree perfectly on "overlaps in
> their domains of (approximate) validity".
The reason why would one would expect such a TOE is that so
far, science has been pretty successful in explaining things. There is
at present no reason to assume that the search for the big TOE is
not a path we should tread.
Either there is one or there is not. A simple logical tautology. If there
is none, then it raises some nasty question such as, why even look?
Or more importantly, why should I fund your search for the big TOE?
Are these questions that you want raised? What would be the logical
consequences of such a position?
And also, what is your probability of finding such a theory if you
think/feel/believe it does not exist? How does your attitude toward
such an objective affect your success?
> The additional demand made upon mathematical models in science, that the
> model approximate certain phenomena in the natural world, further
> increases the complexity of this pictorial image.
I do not see how this is possible. By restricting mathematical models to
only those that correspond to reality, that would -reduce-, not increase
the complexity of such a map.
Ben
Amw
Walter Kunhardt wrote:
> On Thu, 16 Mar 2000, Aron Wall wrote:
> > Tegmark divides the possible theories about the relationship between
> > mathematical and physical reality as follows (more or less):
> >
> > ------------------------------------------
> > 1. The physical universe is fully described by mathematics:
> > a. All mathematical structures have physical reality.
> > b. Some mathematical structures have physical reality.
> > c. No mathematical structures have physical reality.
> >
> > 2. The physical universe is not fully described by mathematics.
> > ------------------------------------------
> >
> > I hope the validity of 1c is not a live issue for most of us.
> Well, if there were a vote, I'd vote for 1c.
I understood this slightly differently than you did, possibly
falling into the trap of confusing the map with the terrain. I
understood the question to be that, for 1c, No mathematical
structures -correspond- to physical reality, or at lest some
part of it. Interpreted the way you did, I would not argue.
<snipped for brevity>
> These and almost all examples I can think of lead me to think that no
> mathematical structure has physical reality. Moreover the examples
> tend to indicate that a mathematical structure has "more" physical reality
> as it comes "closer" to the Theory of Everything (in the literal sense
> of the word, not the sort of watered-down meaning of TOE in particle
> physics!). But such a thing is not conceivable, in my opinion.
This I do disagree with. A mathematical structure is just an idea. As
such it can be quite useful, in ways that physically real things cannot
be. A mental model of a rock can be tosses an infinite number of times
without having to be fetched after each toss. And thereby work out
the ballistics of the rock prior to physically tossing it. But the two
categories of real things should not be confused with each other.
There is a marked difference between what is physically real and
mentally real. (Or alternately, objectively real and subjectively real)
In short, one has to be very careful to avoid confusing the map with
the territory, the model from the thing being modeled.
> (*) In particular, I think that a question like what an electron
> _really_is_ --- Is it a particle? Is it a wave? --- is beyond the scope of
> physics. The relevant point is how it behaves.
I am not sure I agree here. It behaves the way it does because it
is what it is. If it were different, it would behave differently. I think
that the scientific epistemology is quite adequate to determining
just what an electron is, even if it is not completed yet.
Ben
Amw
John Baez wrote:
> In article <8aultv$s81$1...@pravda.ucr.edu>, Aron Wall <ar...@wall.org> wrote:
> >Warning: the following post contains a certain quantity of theological
> >material.
> ... not suitable for children under 18 or easily excitable philosophers!
You rang? :)
While I am well over 18, I should admit that I consider myself more
of a "natural philosopher" than a physicist. The math is easier in
philosophy. Maybe someday I will achieve the necessary mathematical
'maturity' to do some real physics, assuming I have enough time, but
until then....
I have not read the paper in question, just Wall's summary. However
I think there are a few points that should be addressed.
First off, what exactly is mathematics? It is a system of formalized
symbolic logic. When Tegmark says "1a.; All mathematical structures
have physical reality.", in effect he is saying that the universe
is logical -despite- the postulates one accepts. That is a very key
point from a philosophical perspective. To rule out one mathematical
model in favor or another, assuming both structures are completely
internally consistent, logically, requires an examination of the
original assumption, postulates and axioms used by that particular
mathematical structure. One has to examine very carefully what
those postulates are, where they come from, and any inversions
of those axioms that may or may not be physically real.
We observe the universe to be a logical place, even in those areas that
we find the logic rather confusing. That logic matches or corresponds
to reality as observed. But all logical structures are built on those
initial assumptions, assumptions that we cannot prove logically, but
must find another source for validation.
A really good example of what I am trying to say is the difference
between Euclidean and Riemannian geometry. Invert one postulate
and you have a completely different mathematical structure. You
can apply one to the question of space time, and come up with
GR. You can attempt to apply the other to space-time, and come
up with LET and its descendants. But if you compare the two,
mathematically they end up the same, are constrained to be
mathematically similar due to observation of the space-time
around us. LET is forced to use Lorentzian invariance, because
that is what is observed, despite the a priori assumption that
space-time is "really" Euclidean.
An examination shows that the Euclidean structure as proposed
by LET is not only unobservable, but unnecessary as well. While
one can use either theory to model the universe, it is effectively
saying that space-time is both static and Euclidean as well as
dynamic and Lorentzian. Mutually exclusive postulates, a logical
contradiction. (Note that static is a necessary property of
a Euclidean manifold, otherwise any deviation of that manifold
renders it non-Euclidean.)
Central to that debate, is the, again, a priori assumption that
the space-time and aether of LET are two separate and separable
entities. Yet again, this is not based on observation. It has no
basis whatsoever. (One writer says that he accepts the Euclidean
space-time idea because it "is simpler")
What is boils down to is this. If 1a is true, then that is effectively
saying that any logical structure corresponds to reality despite the
underlying axioms of that logical structure. Despite the fact that two
opposing axioms may be mutually exclusive and hence logically
inconsistent. Axioms have to come from observation, as that is
ultimately our only source of information and knowledge about
the outside world, the external world. If Occam's razor is true
then 1a is excluded.
That physicists in general do not study much in the way of philosophy
I think is a bit tragic. That most philosophers are confusing windbags,
which makes the physicist's position on the subject highly understandable,
I can hardly argue without presenting evidence for the opposition.
> >The whole notion of using probability to determine which of a number of
> >physical structures *you* are strikes me as absurd.
> It doesn't strike me as inherently absurd; I can imagine using
> probability theory to reason about any situation in which I have
> insufficient information to deterministically crank out the answers.
> But to reason probabilistically, we need a well-defined space of
> alternatives and a probability measure on that space. *That* is
> what we are missing here!
I am somewhat curious as to how chaotic "unpredictable determinism"
would fit into this scheme.
> >But supposing I am wrong about all this. There is still an additional
> >problem with Tegmark's hypothesis. Any theory of physics can be
> >modified so as to be unobservably different. You can add a particularly
> >wimpy WIMP, you could miraculously change the laws of physics in a few
> >arbitrarily determined small regions, you could... The sky's the limit!
> >And no reason to keep the theory covariant, pretty, or anything else,
> >either.
> Right. But....
This addresses the problem of unobservables. If an entity is unobservable,
in principle, what are you saying? You are saying that it does not affect
your senses, nor any other entity in its environment. If it affects another
entity, then you could observe that second entity and thereby deduce
the original. You can stick all kinds of unobservables into a theory.
However if they are unobservable, if they have no effect on their
environment, what do you mean when you say they are "real"?
The little I have seen in my life is that the positing of an unobservable
in effect does nothing but sweep a problem under the rug. Prime example
is Everett.
> >Speaking of ugliness, another consequence of the 1a theory is that one
> >has to discard Occam's razor, since even an intolerably complicated
> >universe must exist.
> This is a good point. Here's a nutty proposal to try to save the
> day. Perhaps our probability measure on mathematical structures could
> be designed to weight "simpler" or "more elegant" mathematical structures
> more heavily. I.e., we could incorporate Occam's razor into our prior!
>
> On the one hand, this is not quite as crazy as it sounds, because in
> practice we all do this when dreaming up theories - as you say, science
> progresses based on this assumption. But on the other hand, there have been
> endless arguments about how you measure the "simplicity" or "elegance" of
> a theory, so this supplementary hypothesis opens yet another can of worms.
Necessity seems essential to any measure of "simplicity". Unobservables are
definitely part of that equation. The more you have, the 'uglier' it is.
> And if proposing that all mathematical structures physically exist is
> metaphysical, unfalsifiable, overly Platonic and all those other bad
> things, just WAIT and see all the flak you'll get when you add a footnote
> saying that beautiful mathematical structures exist "more". The empiricists
> will come down on you like a ton of bricks.
I think that 1a is just that, metaphysically unfalsifiable, due to the whole
problem with first principles in the first place. Since axioms are based on
observations, and observations are done by 'local' entities, you have a focus
or selection problem in your data set. It can (and will) be argued that
your data set of all observations is simply too small to recognize all that
is out there.
As to whether it is overly Platonic, then the question arises "are ideas real?"
And whether you say yes or no, you will be forced to explain what 'real'
means and how you know what is and is not real. Define those terms first
and you may have a way out. However I don't see it as saving 1a.
> So I probably won't even post this article... in most universes, nobody
> will ever see it.
Well I have been working on trans-dimensional communications, which
is why I was able to read and respond. :)
Ben
Chris Hillman
> Drawing on imagery of overlapping mosaics given in one of Chris
> Hillman's recent posts here, (or quasi-crystal topology-like things
> fitting together),
No, no! I said a -cell complex-! -Quasicrystals- are something
-completely- different, and the visual image I suggested is one which is
-completely- different from a Penrose tiling (say).
John Baez
Charles Francis <cha...@clef.demon.co.uk> wrote:
>>[Moderator's note: Kunhart is correct. Nobody knows whether ordinary
>>QED is consistent. This is a famous unsolved problem. - jb]
>Thanks for clearing that up. I get a little fed up with being told that
>my work is not interesting because there was no problem to solve in the
>first place.
Some physicists shrug off the issue of whether quantum field
theories like QED are mathematically consistent, but this may
just be their way of avoiding a somewhat embarrassing issue.
It seems to me that the more intelligent ones admit there is a
problem, but believe that new physics at short distance scales -
e.g. a new concept of spacetime - will solve this problem. This
is certainly the belief of most people working on quantum gravity,
including both string theorists and folks working on loop quantum
gravity (which posits a kind of quantum discreteness of spacetime
geometry at short distance scales).
The advantage of the Wilsonian attitude to renormalization is that
it lets you reap the advantages of quantum field theory while
admitting the possibility that strange new things will happen
at short distance scales. But in fact, this attitude goes back
(in a less precise form!) long before Wilson's work on the
renormalization group. In fact, already in 1949 Feynman wrote:
"The philosophy behind these ideas [renormalization] might be
something like this: A future electrodynamics might show that
at very high our theory is wrong. In fact we might expect it
to be wrong because undoubtedly hih energy gamma rays may be
able to produce mesons in pairs, eg., phenomena with which we
do not deal in the current formualation of the electron-positron
electrodynamics. If the electrodynamics is altered at very short
distances then the problem is how accurately can we compute
things at relatively long distances. The result would seem to
be this: the only thing which might depend sensitively on the
modification at short distances is the mass and charge, but that
all observable processes will be relatively insensitive and we
are now in a position to be able to compute these real processes
fairly accurately without worrying about the modifications at
high frequencies."
From this pragmatic point of view, it scarcely matters whether
QED is consistent, because it turns into a consistent theory as
soon as we impose ultraviolet and infrared cutoffs.
Still, it's a fascinating question whether QED remains consistent
as we remove the cutoffs and suitably renormalize the electron
mass and charge. This is the question Kunhart had mentioned in
his post, which I agree is unsolved. This question has exercised
many smart mathematical physicists for many decades. It is not
quite as precisely formulated as, say, Fermat's Last Theorem,
but one could phrase it fairly sharply like this: "Is there a
theory whose n-point Green's functions satisfy the Garding-Wightman
axioms and which are close to those predicted by the first few
terms of the perturbative expansion for QED?" Whatever slack these
is in this formulation is compensated by the fact that *no*
interacting quantum field theories have been proven to satisfy
the Garding-Wightman axioms in 4d spacetime - so constructing
anything *close* to perturbative QED would already be a triumph.
Now, I suspect that you are not actually interested in this
famous question. Everything you write suggests that, on the
contrary, you are willing to modify the rules of the game quite
a bit:
>Now can we ask whether discrete qed amounts to a mathematically
>consistent formulation of ordinary qed? Personally I think it does. At
>any rate I see the question as a purely semantic one, so that a negative
>answer is likely to be the consequence of some arbitrary mathematical
>criterion, not a valid argument in physics. For the purpose, I would say
>that the criterion for it to be qed is that we have an interacting Dirac
>equation, and the predictive formulae of qed, not that we necessarily
>have, for example, axioms governing a continuum structure, such as are
>included in Von Neumann's axioms for quantum mechanics.
This is fine for physics, and I agree with you that doing new real
physics is more important than determining whether an axiomatized
version of a famous old theory of physics is consistent or not.
However, it would be better for your cause if you didn't claim to
have proved that ordinary QED is consistent, because nothing you
say suggests that you have done this. On the contrary:
>I prefer the theory to be discrete, but if you do not like discreteness,
>you can always eliminate it by taking limits. Then of course you recover
>ordinary qed after renormalisation. And in the process solve your famous
>unsolved problem, showing the model to be consistent, provided that
>limits are taken correctly.
When you write this, you make it sounds as if "taking the limits
correctly" is a fairly trivial matter. But in fact, everybody knows
that QED with cutoffs is consistent. Taking the limit as the cutoffs
go to infinity is precisely the hard part! It is very, very hard to
understand how this theory behaves as the cutoffs are removed to
the extent of being able to prove that the limiting theory has well-
defined n-point functions satisfying the Garding-Wightman axioms.
This is the famous unsolved problem.
It would be better to advertise your theories as the correct theory
of physics, than as the solution to this famous problem.
jackm...@yahoo.com
Antonio Ramirez <an...@turtle.mit.edu> wrote:
> ba...@galaxy.ucr.edu (John Baez) writes:
> > This is a good point. Here's a nutty proposal to try to save the
> > day. Perhaps our probability measure on mathematical structures
> > could be designed to weight "simpler" or "more elegant"
> > mathematical structures more heavily. I.e., we could incorporate
> > Occam's razor into our prior!
>
> It talks (IIRC) about a way to define a distribution on strings on
> symbols based on their Kolmogorov complexity, which is something like
> an inherent "prior" complexity of the string even without needing an
> interpretation for it.
The Kolmogorov complexity of a string is the length of the shortest
Turing machine (TM) program that outputs that string. There are some
issues regarding which TM to use but also some proposed solutions.
On the everything-list mailing list, which is for discussion of the
all-universes hypothesis (AUH) (which some of us conceived of
independently of Max Tegmark) (archive at
http://www.escribe.com/science/theory/ )
an often discussed model for the set of all possible physical laws is
the set of all Turing machine programs.
The best thing is that with this model it can be seen that you get
back Occam's razor for free. That's because short programs are more
likely than long programs to be the initial segment of a randomly
chosen string. The program ends with a loop back instruction, so for
example a program that takes n+m bits would be 2^m times less likely
than one that only needs n bits.
See http://parallel.hpc.unsw.edu.au/rks/docs/occam/ for the take of
another guy on the list.
Presumably this model should be generalized, but the proper way to
list all possible physical laws should also have this feature which
automatically gives high probability to simple laws of physics. Since
no one knows how to generalize it, some people think reality does just
consist of all TM programs running.
One final comment: This is real physics, with real mathematical
problems to tangle with and in principle the possibility to make some
interesting postdictions. If it works out, and I don't know if it will
or if it will be proven wrong, it would certainly be a hell of a lot
more interesting to know than all the previous physics knowledge
gathered to date.
If you want to know how seriously Tegmark takes it, I suggest you
ask him privately.
--------------
Jacques Mallah (jackm...@yahoo.com)
Physicist / Many Worlder / Devil's Advocate
"I know what no one else knows" - 'Runaway Train', Soul Asylum
My URL: http://hammer.prohosting.com/~mathmind/
Sent via Deja.com http://www.deja.com/
Before you buy.
Alastair Malcolm
news:8aultv$s81$1...@pravda.ucr.edu...
> Speaking of ugliness, another consequence of the 1a theory is that one
> has to discard Occam's razor, since even an intolerably complicated
> universe must exist. And such a theory could be constructed so as to be
> closely approximating, but observably different from, a universe with an
> SAS. Science progresses based on the assumption that the universe is
> fundamentally simple and understandable. But if all mathematical
> universes have physical reality, that means the ugly ones also have
> physical reality. And there are as many ugly possibilities as beautiful
> ones.
There are plausible explanations for why 'ugly' universes may not
predominate over 'neat' ones (see references below), without having to bring
in Occam's Razor or 'weightings' as a premise. Briefly, they boil down to
choosing measures for all possible theories which are as anthropically
unbiassed as possible (say by using axiom counts, or minimal information
content) from which one can simply establish that (SAS-including) 'ugly'
theories tend to be outnumbered by the 'neat' theory duplicates which are
appended by different irrelevant axioms, or bits, etc. (Hence we are far
more likely to be in such functionally 'neat' universes.)
Alastair Malcolm
References:
(1) 'Why Occam's Razor'. R. Standish.
http://parallel.hpc.unsw.edu.au/rks/docs/occam.
(2) In 'Fundamental Questions' starting at
http://www.physica.freeserve.co.uk/p105.htm
(Aimed at a non-scientific audience, but with a variety of potential
solutions to the 'ugly universes' problem.)
Charles Francis
<ba...@galaxy.ucr.edu>
>In article <eKz+0MAH...@clef.demon.co.uk>,
>Charles Francis <cha...@clef.demon.co.uk> wrote:
>
>>>[Moderator's note: Kunhart is correct. Nobody knows whether ordinary
>>>QED is consistent. This is a famous unsolved problem. - jb]
>
>Now, I suspect that you are not actually interested in this
>famous question. Everything you write suggests that, on the
>contrary, you are willing to modify the rules of the game quite
>a bit:
On the contrary I would say that most things I write on qed are
concerned with this question, in one way or other. But I do not
necessarily agree that the question is correctly stated by you or that
the Garding-Wightman axioms are even correct. Yes, I will modify the
rules if I can see that the rules are not the rules of nature, and I am
not interested in a mathematical expression which is not according to
the rules of nature, except in so far as it helps research in finding a
better expression of those rules.
>
>>Now can we ask whether discrete qed amounts to a mathematically
>>consistent formulation of ordinary qed? Personally I think it does. At
>>any rate I see the question as a purely semantic one, so that a negative
>>answer is likely to be the consequence of some arbitrary mathematical
>>criterion, not a valid argument in physics. For the purpose, I would say
>>that the criterion for it to be qed is that we have an interacting Dirac
>>equation, and the predictive formulae of qed, not that we necessarily
>>have, for example, axioms governing a continuum structure, such as are
>>included in Von Neumann's axioms for quantum mechanics.
>
>This is fine for physics, and I agree with you that doing new real
>physics is more important than determining whether an axiomatized
>version of a famous old theory of physics is consistent or not.
>However, it would be better for your cause if you didn't claim to
>have proved that ordinary QED is consistent, because nothing you
>say suggests that you have done this.
I did try to make it clear that I was only claiming to have resolved the
physical issues. That is why I gave criteria for what I meant by qed,
and the criteria I gave are not necessarily the ones you impose. You say
you think I am making a false claim, whereas I think you are putting
words in my mouth, and then falsifying something I have tried to make
clear that I am not saying.
> On the contrary:
>
>>I prefer the theory to be discrete, but if you do not like discreteness,
>>you can always eliminate it by taking limits. Then of course you recover
>>ordinary qed after renormalisation. And in the process solve your famous
>>unsolved problem, showing the model to be consistent, provided that
>>limits are taken correctly.
>
>When you write this, you make it sounds as if "taking the limits
>correctly" is a fairly trivial matter.
I don't think I can call discrete qed trivial, even in some show of
humility. It took me a very long time to work it out. In setting the
theory up there are numerous traps and much room for false expression.
The slightest inaccuracy seems to cause the model to break down, long
before you look at the perturbation expansion. But once the model is
developed that far, taking limits is easy.
> But in fact, everybody knows
>that QED with cutoffs is consistent.
I'm glad we agree on that much. It is often possible for me to sense
that you do not agree with me, or perhaps just do not believe me, and
yet I am unable to get a handle on why.
>Taking the limit as the cutoffs
>go to infinity is precisely the hard part! It is very, very hard to
>understand how this theory behaves as the cutoffs are removed to
>the extent of being able to prove that the limiting theory has well-
>defined n-point functions satisfying the Garding-Wightman axioms.
>This is the famous unsolved problem.
>
Certainly if you simply introduce cut offs into the formulae, it is not
easy to see what the correct limiting procedure is to remove them. But
if the theory is actually set up in a discrete mathematical structure,
and carefully developed stage by stage, then the manner in which limits
are taken is clear.
>It would be better to advertise your theories as the correct theory
>of physics, than as the solution to this famous problem.
>
I cannot see the difference. The problem, in so far as it has a right to
be famous, is the physical problem. The Garding-Wightman axioms are only
interesting in so far as they correctly describe the physical problem,
or form part of the solution to it. My interest in that is to give
Garding & Wightman credit if they are right and ignore them if they are
wrong. If you will tell me what their axioms are, I will tell you if
discrete qed satisfies them (I am interested enough to want to know if
it does). But I will continue to maintain that the solution of the
problem is the solution of the physical problem, and if the Garding-
Wightman axioms are not satisfied, then it is because they are not
correctly part of the problem at all.
Now if I may rephrase your expression of the problem without refering to
them,
"Is there a well-defined theory whose n-point Green's functions are
close to those predicted by the first few terms of the perturbative
expansion for QED?"
then I do claim to have solved the problem, and the answer is yes.
John Baez
Charles Francis <cha...@clef.demon.co.uk> wrote:
>[...] I do not
>necessarily agree that the question is correctly stated by you or that
>the Garding-Wightman axioms are even correct. Yes, I will modify the
>rules if I can see that the rules are not the rules of nature, and I am
>not interested in a mathematical expression which is not according to
>the rules of nature, except in so far as it helps research in finding a
>better expression of those rules.
That's fine. This is probably the best attitude to take when
doing physics! All I was saying is that the problem of "whether
one can rigorously construct QED" is a famous and fairly well-defined
mathematical problem, a bit like Fermat's Last Theorem or Goldbach's
Conjecture. Lots of people have worked on it and anyone who solves
it will become famous. This does not mean the problem is important;
indeed, in a certain sense it's a completely useless and silly problem,
just like Fermat's Last Theorem and Goldbach's Conjecture. Nothing
suggests you have solved this problem or that you even really care what
it is, so I was just advising you to avoid giving the impression that
you're claiming to have solved it.
>I did try to make it clear that I was only claiming to have resolved the
>physical issues.
Okay, good - it wasn't clear at first. Sometimes you say "QED" when
you mean "discrete QED", and then I get the impression you're making
false claims about a theory of physics I sort of understand, rather
than true claims about your own theory (which I do not understand).
>> But in fact, everybody knows
>>that QED with cutoffs is consistent.
>I'm glad we agree on that much. It is often possible for me to sense
>that you do not agree with me, or perhaps just do not believe me, and
>yet I am unable to get a handle on why.
Basically, the problem is that I rarely understand what you're saying.
Some of this turns on different ways of using words.
>>It would be better to advertise your theories as the correct theory
>>of physics, than as the solution to this famous problem.
>I cannot see the difference.
All the more reason for taking my advice. :-)
If you want to know more about exactly what problem I'm talking
about, try Glimm and Jaffe's "Quantum Physics: A Functional Integral
Point of View". These guys are the real experts on constructive
quantum field theory. This book has the Garding-Wightman axioms
in it and also the other major axiomatizations of quantum field theory.
>The problem, in so far as it has a right to be famous, is the physical
>problem.
Let us leave "rights" out of this. Problems are not famous because
they have a "right" to be! In my opinion Fermat's Last Theorem has
no right to be famous - I couldn't really give a damn about whether
there is any nontrivial integer solution to x^n + y^n = z^n for n > 2.
Nonetheless it is famous, so I would not say I'd solved this problem
when I had solved some other, modified version of the problem - even
if my modified version was more useful and interesting!
>Now if I may rephrase your expression of the problem without refering to
>them,
>
>"Is there a well-defined theory whose n-point Green's functions are
>close to those predicted by the first few terms of the perturbative
>expansion for QED?"
>
>then I do claim to have solved the problem, and the answer is yes.
But I already knew the answer to *this* problem is yes. You see, the
term "a well-defined theory" is sufficiently broad that I can just take
the first few terms of the perturbation expansion for QED and use
*that* as my well-defined theory.
Charles Francis
<ba...@galaxy.ucr.edu>
>In article <wiZjVfAJ...@clef.demon.co.uk>,
>Charles Francis <cha...@clef.demon.co.uk> wrote:
>
>>[...] I do not
>>necessarily agree that the question is correctly stated by you or that
>>the Garding-Wightman axioms are even correct. Yes, I will modify the
>>rules if I can see that the rules are not the rules of nature, and I am
>>not interested in a mathematical expression which is not according to
>>the rules of nature, except in so far as it helps research in finding a
>>better expression of those rules.
>
>doing physics! All I was saying is that the problem of "whether
>one can rigorously construct QED" is a famous and fairly well-defined
>mathematical problem, a bit like Fermat's Last Theorem or Goldbach's
>Conjecture. Lots of people have worked on it and anyone who solves
>it will become famous. This does not mean the problem is important;
>indeed, in a certain sense it's a completely useless and silly problem,
>just like Fermat's Last Theorem and Goldbach's Conjecture.
Actually it is a vital problem to anyone who requires realism in
physical theory. It is just that a given set of axioms may not properly
define the problem. Actually the chances are they don't. I am sure that
anyone who does serious research discovers that it is generally harder
to define a problem than it is to solve it.
[snip]
> It is often possible for me to sense
>>that you do not agree with me, or perhaps just do not believe me, and
>>yet I am unable to get a handle on why.
>
>Basically, the problem is that I rarely understand what you're saying.
>Some of this turns on different ways of using words.
This is a problem (a problem for me, anyway; you may not give a stuff).
But if you do not understand me, who will? You are expert in the right
areas, and have a greater flexibility of thought than most. When I first
made headway with the idea that quantum mechanics could be understood by
removing ontological space-time it seemed to me that those with no
physical training would find it easier to see what I am on about than
the experts, and by and large I have found this to be the case. There is
nothing particularly sophisticated or difficult about what I say, and
even the mathematical structure of discrete qed is fairly
straightforward, resting entirely on things one learns as an
undergraduate. But the concepts clash seriously with things to which you
have become accustomed. As I do not know exactly what these are (and
they are different for different physicists) it is difficult for me to
explain how to resolve these clashes, except if you will explain where
you have trouble with what I say.
Actually it puzzles me that you understand so little of what I say. You
are doing much the same thing, ditching the space-time continuum in your
spin foam stuff. But your spin foams are conceptually far more difficult
than the simple particle interactions of discrete qed. Now I admit I
haven't made a huge attempt to understand them, but that is because I
think I already know the answer to the question they are trying to
solve. On the other hand discrete qed is already fairly developed, and
has answers for questions you want to solve. I would have hoped you
would want to understand at least enough to be able to say where it is
wrong. Anyway I shall probably make a pain of myself on the NG until you
do.
>
>>>It would be better to advertise your theories as the correct theory
>>>of physics, than as the solution to this famous problem.
>
>>I cannot see the difference.
>
>All the more reason for taking my advice. :-)
You generally give good advice. But in cases like this, I obviously
cannot follow the advice until you have given it, and you have no reason
to give the advice until I have failed to follow it...
>
>If you want to know more about exactly what problem I'm talking
>about, try Glimm and Jaffe's "Quantum Physics: A Functional Integral
>Point of View". These guys are the real experts on constructive
>quantum field theory. This book has the Garding-Wightman axioms
>in it and also the other major axiomatizations of quantum field theory.
>
Oh no, yet another text book that I know of but haven't looked at. How
often do I wish I had easy access to a library. If you had given me that
advice a week ago I would have borrowed it from the UL at Cambridge, but
it's a five hour drive, so I guess I'll have to buy it. Amazon quote 4-6
weeks, so it'll be a while before I can comment on the Garding-Wightman
axioms.
>>The problem, in so far as it has a right to be famous, is the physical
>>problem.
>
>Let us leave "rights" out of this. Problems are not famous because
>they have a "right" to be! In my opinion Fermat's Last Theorem has
>no right to be famous - I couldn't really give a damn about whether
>there is any nontrivial integer solution to x^n + y^n = z^n for n > 2.
Bad example, unless I'm mistaken. Isn't it of vital significance in code
breaking, and worth billions to espionage organisations? But I agree
with your point. I haven't been interested in Fermat's last theorem
since I was fifteen either - my mother bought me the recent book, and I
am ashamed to say I haven't even opened it.
>Nonetheless it is famous, so I would not say I'd solved this problem
>when I had solved some other, modified version of the problem - even
>if my modified version was more useful and interesting!
>
But the mathematical problem you refer to is just an attempt at working
towards the solution of a much deeper and far more important physical
and philosophical problem. As such I feel justified in saying that it is
you who refer to a modified version of the problem.
>>Now if I may rephrase your expression of the problem without refering to
>>them,
>>
>>"Is there a well-defined theory whose n-point Green's functions are
>>close to those predicted by the first few terms of the perturbative
>>expansion for QED?"
>>
>>then I do claim to have solved the problem, and the answer is yes.
>
>But I already knew the answer to *this* problem is yes. You see, the
>term "a well-defined theory" is sufficiently broad that I can just take
>the first few terms of the perturbation expansion for QED and use
>*that* as my well-defined theory.
>
That would give you recipes for physical predictions, and I think more
or less summarises the status of this area of modern theoretical
physics. Starting with the "quantisation of matter fields" is no better
as metaphysics than simply doing as you suggest with the perturbation
expansion. But I would dispute that it actually constitutes a theory.
Nevertheless there is a case for finding a better expression of the
problem, and I can't think of one just now, except to say that to have a
theory of physics, we must abstract mathematical properties from
physical entities, and derive our predictions from them, just as we do
in classical physics. Moreover the physical entities must be "beables"
in the sense of Bell. It is not enough to establish mathematical
consistency, but we must also have a conceptual coherence which is less
easy to define.
--
Regards
Charles Francis
cha...@clef.demon.co.uk