Re: On Tegmark: Different most-suitable forms of mathematical logic, information, probability, uncertainty across Level III?

From: Benjamin Udell <budell.domain.name.hidden>
Date: Wed, 23 Apr 2003 18:34:59 -0400

So far I've received two off-list responses regarding my having attributed to Tegmark the idea that Level IV represents variation in mathematical laws (an editing error on my part, actually.) One disagreed with the idea of variation of mathematical laws, the other supported the idea. My response to the latter, in which I incorporated my response to the former, was: :

> 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]

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.
Received on Wed Apr 23 2003 - 18:35:22 PDT