On Tegmark: Different most-suitable forms of mathematical logic, information, probability, uncertainty across Level III?
Benjamin Udell
I notice that he does not mention universes wherein unconventional (for our everyday purposes) forms of logic, information, or probability would hold as most suitable. He does put predicate logic at the base of his mathematical structure. But would variations in suitable logical structure - different numbers of truth values, for instance - be most likely to be found only across Level IV?
More generally, _wouldn't variations in the suitable form of mathematical logic, information, & probability & uncertainty, more plausibly reflect something pertaining to quantum coherence, decoherence, etc., which are at the basis for the idea of Level III, than anything pertaining to the ideas for the other Levels?_ (Not that I know a great deal about quantum coherence!) But Level III seems like the place for such variations, if anywhere. No?
Tegmark does discuss how different lightcone structures (reflecting variation across Level II) would affect one's capacity for inference, but inferential processes in an intelligent system are a different thing than the subjects of mathematical theories of logic, information, probability, & uncertainty. I mean that differences of lightcone structure would affect intelligent (including inferential & other philosophically interesting) processes as they would affect cybernetic, stochastic, & sensitive dynamic processes. But these differences would not necessarily reflect different forms of *mathematical* logic, information, or probability or (more generally) uncertainty.
I don't know whether variation in "most-suitable" forms of math-logic (e.g., different numbers of truth values), math-information, math-probability across a multiverse would tend to lead to the "high-energy messes" that Tegmark mentions. I don't know whether they would lead to qualitatively more universes in Level III than in Level II. It seems to me like they might, but I'm no physicist! Maybe it would amount to a violation of ergodicity as Tegmark warns, but perhaps for some quantum-related reason these particular alternate kinds of universes would be counted differently by the theory.
As for information, probability, & uncertainty, I don't know enough about mathematical theories of them, to know what element in them one might consider as perhaps varying across a multiverse. Sorry, I wish I could do better. As regards three- or higher-valued logics, where these prevail as suitable, there perhaps they would start to impact the view of mathematics itself in strange ways, I don't even know that much. For all I know, if alternate forms of logic, etc., hold as most suitable in some places, maybe this impacts those places' view of the structure of the four-level multiverse itself. Maybe some places would regard there as being nine or 27 rather than four levels? Should predicate logic, if the root of such radical variations, therefore be put, as Tegmark already puts it, at the base of the mathematical structure? (On the other hand, do the four levels have to have a rigidly Comtean hierarchy? They might have some kind of 4-chotomical structure with a subtler kind o
f!
hierarchicality. After all, some consider mathematical logic an applied field while others regard it as Day One 8 a.m.)
What got me asking these questions was that Tegmark's four-level multiverse picture struck me as seeming to reflect, somewhat, a structure of the fields of research as I had pictured them.
If all possible universes exist, & if the idiosyncrasies of histories are reflected at Level I, & the far reaches of abstract mathematics at Level IV, one might well wonder whether there is a reflection, a similarity to be seen, between the set of the four Levels of multiverses & the outlines of the most general kinds of research -- science, mathematics, & intermediate fields. A similarity between multiverse "constellations" & the "city" of research. This would require at the least that researches collectively have developed to a point where their "natural" (unplanned, but persistent & making some kind of sense) outlines could be seen. And of course who knows what multiverse-theory will be like (if it persists at all) in even only a few years. So in a way this is a lark.
I wanted to pursue the analogy, because it seemed to be roughly working for Levels I, II, & IV. Tegmark's III & my "III", if they do roughly match up, are less than obvious about it to me.
Tegmark's multiverse levels:
I. Different initial conditions, different histories.
II. Different constants.
III. Different quantum branches.
IV. Different mathematical laws.
Arrangement of areas of research:
I.. Empirical sciences/studies (from human/social studies to physics).
II. Mechanical/systems/process theoretical research with generalized applicability (e.g., philosophy itself?, cybernetics, statistical mechanics, complex (conditions-sensitive) mechanics).
III. Applied yet mathematically deep fields (math. theories of logic, info., probability, & perhaps some other mathematical theories of uncertainty, like possibility theory).
IV. Pure mathematics.
Actually I tend to put these fields of research in exactly the reverse order, it's become a habit. My habits are a unusual kind of subject, marked as they are by the fact that, about them, I actually know something.
Thank you for your patience, anybody who has read all this, & any responses will be appreciated.
Benjamin Udell,
Lifelong layperson who likes 4-chotomies, reads some philosophy, & enjoys articles on cosmology for the educated public.
Benjamin Udell
Tegmark's multiverse levels:
I. Different initial conditions, different histories.
II. Different constants.
III. Different quantum branches.
IV. Different mathematical STRUCTURES.
Benjamin Udell
Benjamin Udell
> I'd have trouble conceiving of really "different" mathematical laws that would not merely (if with great toil) be integrated into mathematics as representing different assumptions or postulates -- parallel lines do/don't meet, etc.-- which would be talked about as representing different structures. One could call them different "laws" but we might want to reserve that way of speaking for mathematical structures that really can't be integrated into the rest of mathematics.
> Arguably a situation approximating that scenario already exists for strictly constructive mathematics to the extent that such mathematics ends up omitting & not integrating into itself a good deal of modern mathematics. (I've noticed that there are over 4,000 messages at this discussion site, which it would be hopeless for me to try to read, but I've noticed some post titles mentioning "oracles" so maybe there's something there that would be relevant here.)
However, I haven't received a response (on-list or off-list) regarding the main questions which I posed (corrected version of original post reproduced below). Maybe I haven't waited long enough, or maybe I didn't pose the questions well -- I know that I am not my own best editor -- or maybe nobody feels they have anything helpful to say about it. List members' interests will not always intersect of course, so maybe that's all it is. Still, I'd like to ask: Does anybody happen to recall whether among the 4000 or so posts archived here, whether variations in structures of math-logic, math-probability, etc., were discussed? And, does anybody know of an additional forum where I might post my questions with hope for a response?
Thank you for your attention.
Benjamin Udell
Original post with correction:
===========================
I've read Tegmark's Scientific American article, & the related article "Is the 'theory of everything' merely an ensemble theory" at his Website, & they're mind-benders all right.
I notice that he does not mention universes wherein unconventional (for our everyday purposes) forms of logic, information, or probability would hold as most suitable. He does put predicate logic at the base of his mathematical structure. But would variations in suitable logical structure - different numbers of truth values, for instance - be most likely to be found only across Level IV?
More generally, _wouldn't variations in the suitable form of mathematical logic, information, & probability & uncertainty, more plausibly reflect something pertaining to quantum coherence, decoherence, etc., which are at the basis for the idea of Level III, than anything pertaining to the ideas for the other Levels?_ (Not that I know a great deal about quantum coherence!) But Level III seems like the place for such variations, if anywhere. No?
Tegmark does discuss how different lightcone structures (reflecting variation across Level II) would affect one's capacity for inference, but inferential processes in an intelligent system are a different thing than the subjects of mathematical theories of logic, information, probability, & uncertainty. I mean that differences of lightcone structure would affect intelligent (including inferential & other philosophically interesting) processes as they would affect cybernetic, stochastic, & sensitive dynamic processes. But these differences would not necessarily reflect different forms of *mathematical* logic, information, or probability or (more generally) uncertainty.
I don't know whether variation in "most-suitable" forms of math-logic (e.g., different numbers of truth values), math-information, math-probability across a multiverse would tend to lead to the "high-energy messes" that Tegmark mentions. I don't know whether they would lead to qualitatively more universes in Level III than in Level II. It seems to me like they might, but I'm no physicist! Maybe it would amount to a violation of ergodicity as Tegmark warns, but perhaps for some quantum-related reason these particular alternate kinds of universes would be counted differently by the theory.
As for information, probability, & uncertainty, I don't know enough about mathematical theories of them, to know what element in them one might consider as perhaps varying across a multiverse. Sorry, I wish I could do better. As regards three- or higher-valued logics, where these prevail as suitable, there perhaps they would start to impact the view of mathematics itself in strange ways, I don't even know that much. For all I know, if alternate forms of logic, etc., hold as most suitable in some places, maybe this impacts those places' view of the structure of the four-level multiverse itself. Maybe some places would regard there as being nine or 27 rather than four levels? Should predicate logic, if the root of such radical variations, therefore be put, as Tegmark already puts it, at the base of the mathematical structure? (On the other hand, do the four levels have to have a rigidly Comtean hierarchy? They might have some kind of 4-chotomical structure with a subtler kind o
f!
hierarchicality. After all, some consider mathematical logic an applied field while others regard it as Day One 8 a.m.)
What got me asking these questions was that Tegmark's four-level multiverse picture struck me as seeming to reflect, somewhat, a structure of the fields of research as I had pictured them.
If all possible universes exist, & if the idiosyncrasies of histories are reflected at Level I, & the far reaches of abstract mathematics at Level IV, one might well wonder whether there is a reflection, a similarity to be seen, between the set of the four Levels of multiverses & the outlines of the most general kinds of research -- science, mathematics, & intermediate fields. A similarity between multiverse "constellations" & the "city" of research. This would require at the least that researches collectively have developed to a point where their "natural" (unplanned, but persistent & making some kind of sense) outlines could be seen. And of course who knows what multiverse-theory will be like (if it persists at all) in even only a few years. So in a way this is a lark.
I wanted to pursue the analogy, because it seemed to be roughly working for Levels I, II, & IV. Tegmark's III & my "III", if they do roughly match up, are less than obvious about it to me.
Tegmark's multiverse levels:
I. Different initial conditions, different histories.
II. Different constants
III. Different quantum branches
IV. Different mathematical [structures]
Hal Finney
> However, I haven't received a response (on-list or off-list) regarding
> the main questions which I posed (corrected version of original post
> reproduced below).
I will respond in part; however, your message is too long to be the
basis for a practical discussion. There are too many different
ideas where we would not agree on the assumptions. I will just focus
on the first part.
> I notice that he does not mention universes wherein unconventional
> (for our everyday purposes) forms of logic, information, or probability
> would hold as most suitable. He does put predicate logic at the base of
> his mathematical structure. But would variations in suitable logical
> structure - different numbers of truth values, for instance - be most
> likely to be found only across Level IV?
Yes, since level 4 includes all possible mathematical structures, that
would include ones with non-standard logic.
> More generally, _wouldn't variations in the suitable form of mathematical
> logic, information, & probability & uncertainty, more plausibly reflect
> something pertaining to quantum coherence, decoherence, etc., which are
> at the basis for the idea of Level III, than anything pertaining to the
> ideas for the other Levels?_ (Not that I know a great deal about quantum
> coherence!) But Level III seems like the place for such variations,
> if anywhere. No?
Level 3 refers to our own universe's quantum physics as expressed in the
many-worlds interpretation. All of the level 3 worlds should share the
same concepts of logic, information and probability. Now, it's possible
that other level 4 worlds, which might have a different form of logic
or probability, might themselves have level 3 parallelism within them.
> Tegmark does discuss how different lightcone structures (reflecting
> variation across Level II) would affect one's capacity for inference, but
> inferential processes in an intelligent system are a different thing than
> the subjects of mathematical theories of logic, information, probability,
> & uncertainty. I mean that differences of lightcone structure would affect
> intelligent (including inferential & other philosophically interesting)
> processes as they would affect cybernetic, stochastic, & sensitive
> dynamic processes. But these differences would not necessarily reflect
> different forms of *mathematical* logic, information, or probability or
> (more generally) uncertainty.
I'm not sure what you're getting at here, but let me make one point.
It's not clear that you can say that our universe is based on a particular
form of mathematical logic. Does our universe in any sense rely on or
embody the principle that we can't have P and not-P? I don't see it.
Our universe is based on atoms and quarks and fields. When we say
that P and not-P can't both be true, the logic is in our heads, not in
the universe.
The sense in which our universe is a mathematical structure is something
like the following. Imagine that we develop a complete theory of
physics; we know all of the particles, all of the forces, all of the
laws. Everything fits together, there are no exceptions or unknowns.
And further let us suppose that we discover the exact initial conditions
of the universe (Tegmark hypothesizes that the universe is initially
in a state of perfect simplicity). Then these laws of physics define
a mathematical structure which is essentially identical (isomorphic)
to our physical universe.
I don't see that these laws would be particularly likely to embody a rule
that says p and ~p is impossible. Newton's laws don't have any such rule.
They just say Force = Mass times Acceleration. Einstein's equations
don't have such a rule; they say that particles follow geodesic paths
and that the shape of space is determined by the stress energy tensor.
Quantum mechanics doesn't have such a rule; it says that the time
evolution of the system is based on the exponential function applied to i
times the Hamiltonian. The point is, these are the kinds of mathematical
statements which have been the basis for physical laws in the past, and it
may be that in its final form, physics will be expressed in similar ways.
The equations won't necessarily build on mathematical logic at all.
For another view of the Level 4 multiverse, I recommend Juergen
Schmidhuber's description of it as the output of a super-computer that
runs all possible computer programs. He has a new page up at
http://www.idsia.ch/~juergen/computeruniverse.html, with the introductory
paper at http://www.idsia.ch/~juergen/everything/html.html. This is
compatible with Tegmark's view if we assume that to every mathematical
structure there corresponds a computer program, and vice versa.
Sorry I wasn't able to respond more substantively to the later portions
of your message. You might want to re-write some parts and describe
your ideas in new messages.
Hal Finney
Bruno Marchal
>I notice that he does not mention universes wherein unconventional (for
>our everyday purposes) forms of logic, information, or probability would
>hold as most suitable. He does put predicate logic at the base of his
>mathematical structure. But would variations in suitable logical structure
>- different numbers of truth values, for instance - be most likely to be
>found only across Level IV?
Tegmark is just very naive about what mathematical
reality can be. He has a 19th century philosophy of mathematics.
At least Tegmark, unlike Schmidhuber, is aware that we must
take into account some difference between point of views, but
he does not take those differences into account enough, and sweeps
(like so much "scientist") the mind-body problem under the rug.
>More generally, _wouldn't variations in the suitable form of mathematical
>logic, information, & probability & uncertainty, more plausibly reflect
>something pertaining to quantum coherence, decoherence, etc., which are at
>the basis for the idea of Level III, than anything pertaining to the ideas
>for the other Levels?_ (Not that I know a great deal about quantum
>coherence!) But Level III seems like the place for such variations, if
>anywhere. No?
Only if you presuppose physicalism (which, BTW, has been
shown incompatible with the computationalist hypothesis
in the cognitive science, Marchal 88, Maudlin 89, see the ref
in my thesis downloadable from my URL, below).
>Tegmark does discuss how different lightcone structures (reflecting
>variation across Level II) would affect one's capacity for inference, but
>inferential processes in an intelligent system are a different thing than
>the subjects of mathematical theories of logic, information, probability,
>& uncertainty. I mean that differences of lightcone structure would affect
>intelligent (including inferential & other philosophically interesting)
>processes as they would affect cybernetic, stochastic, & sensitive
>dynamic processes. But these differences would not necessarily reflect
>different forms of *mathematical* logic, information, or probability or
>(more generally) uncertainty.
This depends on many possible philosophical assumptions.
Mathematical logic is just a branch of math, like topology or
geometry, etc. The idea that logic should be used as foundations
is known to be silly since the work of Dedekind, Godel, etc.
To study "variant of logics" two related roads can be used: either
by a weakening of classical logic (you can get
intuitionistic logic, quantum logic, intuitionistic quantum
logic, etc.), or by extending (classical) logic with modal connectives.
Some weak logics can be mirrored by modal logics and can
give epistemic or pragmatic interpretations to those weak logic.
Modal logic is a powerful tool to study invariant discourses
with respect to "observer's possible trips". The main
problem is how to choose a modal logic. In my work I have
shown how extract the unique possible logic for the physical
propositions from the only possible psychological propositions
you get from Church thesis and some amount of arithmetical
realism, once you postulate the computationnalist hypothesis
in the cognitive science. In french: with the comp hypothesis
the aristotelian notion of substance evaporates and physical
realities emerge from the provable coherence of machine dreams,
where a machine dream is 2^aleph_0 computations seen
from inside, and where "seen from inside" is defined through the
complete (at some level) and sound godelian logic of
self-reference.
>What got me asking these questions was that Tegmark's four-level
>multiverse picture struck me as seeming to reflect, somewhat, a structure
>of the fields of research as I had pictured them.
>
>If all possible universes exist, & if the idiosyncrasies of histories are
>reflected at Level I, & the far reaches of abstract mathematics at Level
>IV, one might well wonder whether there is a reflection, a similarity to
>be seen, between the set of the four Levels of multiverses & the outlines
>of the most general kinds of research -- science, mathematics, &
>intermediate fields. A similarity between multiverse "constellations" &
>the "city" of research. This would require at the least that researches
>collectively have developed to a point where their "natural" (unplanned,
>but persistent & making some kind of sense) outlines could be seen. And of
>course who knows what multiverse-theory will be like (if it persists at
>all) in even only a few years. So in a way this is a lark.
I doubt it. Not only there are too many evidences for multiverse
theories, but there are no evidence for 1 universe theory,
and strictly speaking there is only 0 universes-theory or
many-universes theory in the literature. Of course a lot of
people believes that classical physics, for example, is a
1-universe theory. But this is logically untenable, and is
just a form of wishful thinking in naive cosmology.
Reality is like the Chinese TAO. Give it a name and then
it multiplies.
>I wanted to pursue the analogy, because it seemed to be roughly
>working for Levels I, II, & IV. Tegmark's III & my "III", if they do
>roughly match up, are less than obvious about it to me.
>
>Tegmark's multiverse levels:
>I. Different initial conditions, different histories.
>II. Different constants.
>III. Different quantum branches.
>IV. Different mathematical laws. STRUCTURES
>
>Arrangement of areas of research:
>I.. Empirical sciences/studies (from human/social studies to physics).
>II. Mechanical/systems/process theoretical research with generalized
>applicability (e.g., philosophy itself?, cybernetics, statistical
>mechanics, complex (conditions-sensitive) mechanics).
>III. Applied yet mathematically deep fields (math. theories of logic,
>info., probability, & perhaps some other mathematical theories of
>uncertainty, like possibility theory).
>IV. Pure mathematics.
>
>Actually I tend to put these fields of research in exactly the reverse
>order, it's become a habit. My habits are a unusual kind of subject,
>marked as they are by the fact that, about them, I actually know something.
>
>Thank you for your patience, anybody who has read all this, & any
>responses will be appreciated.
>
>Benjamin Udell,
>Lifelong layperson who likes 4-chotomies, reads some philosophy, & enjoys
>articles on cosmology for the educated public.
You are welcome, You will find links to more of
what I have said in my URL below, including some
contributions to this list.
Bruno