Tegmark's Multiverse, classification of sciences, maths
(Adapted from "What of these other fours?"). Recentest significant change: October 29, 2013.

There appears to be some structural correlation between my tetrastic classification of the fields of research and Max Tegmark’s theory of a four-level multiverse in which every possibility is actualized (“everything exists”) and in which mathematical existence is real existence. I’m not saying that I think that Tegmark’s four-level multiverse picture is true (or that any multiverse picture is true). Tegmark claims that it is at least testable. (I am not a physicist and feel unprepared to evaluate his claims of testability.) What I’m saying is that there is a correlation between my tetrastic classification of the fields of research and aspects of the structure of physics-relevant ideas built by Tegmark as the structure of his claimed four-level multiverse. As for the reality of Tegmark’s four-level multiverse, your guess is as good as and maybe better than mine.

Tegmark’s Multiverse	My classification of research areas into families	Particle possibilities (feasibles/optima), probabilities, detection, & experimentation as correlating to Tegmark levels (I haven't seen this correlation, or whatever it is, posed by anybody else, but it seems implicit.)
Level IV: includes other mathematical structures, different fundamental physical equations.	1. Pure mathematics (analytic equations, extremization, topology, graph theory, integration, measure, enumeration, differentiation, calculation, limits, order, etc.).	(1st) IV. Feasibles & optima Variation across mathematical structures. The particle's (wave packet's) evolution involves contributions from classically absurd (but quantum-feasible) potential trajectories and counterfactual circumstances, which mostly cancel each other out. Do they correspond, as "would-have-beens," to potential variations of a Level-I experimental setup? Yet how meaningful would that correspondence be if they involve variations in shortest distances, spacetime metrics, and mathematical structure generally, well beyond variations that we could actually carry out in our Level I experimental setup? This raises a question of mathematical equivalences between seemingly dissimilar scenarios.
Level III: includes alternate-outcome worlds (quantum branching); same features as Level II. Works only if quantum evolution is unitary.	2. Applied yet mathematically deep mathematics (deductive maths of: optimization, probability, information, and logic).	(2nd) III. Probabilities Variation across quantum branches, corresponding to aprioristic distribution of probabilities for what the experimental particle will do at a given point in its evolution.
Level II: includes other post-inflation bubbles, same fundamental equations of physics, but possibly different particles, constants, and dimensionality.	3. Abstract yet positive-phenomenally deep sciences/studies (inverse optimization, statistical theory, descriptive and ampliatively-inductive areas of information theory, and philosophy).	(3rd) II. Information Variation along one quantum branch (repeated experiment with same setup).
Level I: includes regions beyond our cosmic horizon, “universes” or Hubble volumes in a single given inflationary bubble.	4. Concrete empirical sciences/studies (physical, material, biological, and behavioral/social/human).	(4th) I. Logic Variation of experimental setup, an actual history (establishment of a hypothesis, theory, etc.)

Tegmark correlates Level IV with mathematics; he takes mathematics as being the world from a “bird’s eye view”. He correlates Level I with the world at the level of our “frog’s eye view” (though it’s much too huge for us to observe most of it), the world of concrete empirical physics and chemistry as we know them. Ergo what about Levels III and II? If Tegmark’s picture were to prove true, then it would be exceedingly strange if there were a one-to-one correlation between research families and only two of four multiverse levels. Let me put it informally and as a question: we’re talking about the grand system of everything, right?

In other words, one would expect that the “city of research,” in its evolved broad layout, would naturally come consistently, if it came at all, to resemble the “sky” of constellated multiverse structures “above” it. I mean that a resemblance that goes half-way and then simply quits seems rather unsatisfying.

Another question is, of course, whether our civilization's “city of research” has evolved sufficiently for a systematic resemblance between it and multiverse structures to emerge. Whatever the case may be in that regard, I think I do see a correlation between the multiverse structures and the layout, as I see it, of research fields.

However, in the correlation, fields such as deductive logic, which Tegmark associates with Level IV, are associated instead with Level III. Deductive logic is about the structures of alternatives among predicates or propositions which, according to the quantum Many Worlds view, are all actualized thanks to quantum branching into alternatives. Deductive logic is one of a family of fields, including also the deductive mathematics of optimization, probability, and information, studying such alternatives. They are considered mathematically deep, yet are not usually called “pure” mathematics, but “applied.” (One is stuck with their distinction’s being made with the terms “pure” and “applied”; one can see how it came about, but it’s neither the most illuminating way nor even true in every relevant sense. And as Dieudonné points out in his mathematics article in the Encyclopedia Britannica Fifteenth Edition, the rubric “applied” jumbles deep and trivial areas of math together. “Pure” does not.)

Now, Tegmark follows tradition in regarding formal deductive logics as the most basic area in maths. I discuss issues of this kind at greater length in my post “Logical quantities, categories of research, and categories”. To summarize here, such deductive logics are about proof, and to put them as most basic within mathematics is to order the maths in the order of knowledge and of how we know things. Yet tradition also puts physics as more basic then chemistry, biology, etc., yet that is not in the order of knowledge but in the order of being. Tradition, on these points, is inconsistent, and the neat inter-family alignment of members of the research families tends to bear this out (see table below). If Tegmark on some level liked an element of research-classificational traditionalism as “leavening” his cosmological radicalism, I’d say he should have been even more radical instead. (Skip tables ►)

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Simple & conjoined logical quantities for terms (subject, predicate, etc.)

	Universal:	Special:
General:	1. Universal general.	3. Non-universal general.
(Multi-)singular (monadic, polyadic, etc.):	2. Universe; 'grand' polyad, gamut, etc.; total population & its parameters.	4. (Multi-)singular (monadic, polyadic, etc.) in a larger world, i.e., special (multi-)singular.

Tetrastic 4x4 classification of the sciences & mathematics.

CLASSES (columns)          BANDS OF AFFINITY (rows)		Universal generals.	Universes, gamuts, total populations   & their parameters.	Non-universal generals (non-singular specials).	Singulars (monadic, polyadic, etc.) in a larger world.
		Pure maths (at least some are classified below):	Applied yet mathematically often-deep, mathemtically duductive theories:	General ('domain-independent') studies of positive phenomena:	'Special' sciences:
Universal generals.	Rules, constraints, equational balances:	Extremization, equations, graph theory, many-to-many relations.	Mathematics of optimization.	Inverse optimization (descriptive & inductive phases) & its mathematical formalisms.	Sciences of motion, forces.
Universes, gamuts, total popu­lations & their para­meters.	Elements, compo­sitions, full comple­ments:	Integration, measure, enumeration, one-to-many relations.	Mathematics of probability.	Statistics.	Sciences of matter.
Non- universal generals (non- singular specials).	Differenti­ations, speciali­zations, kinds, special parts:	Differentiation of functons, calculation (algebra as theory of calculation), many-to-one relations.	Mathematics of information (historical overlap into abstract albegra).	Info/ communication theory (descriptive, inductive) & its mathematical formalisms.	Sciences of life.
Singulars (monadic, polyadic, etc.) in a larger world.	Singular- izations, unique relations, orderings, hierarchies:	Limits, ordering, inference conditions, one-to-one relations.	Mathematics of logic.	Philosophy.	Sciences / studies of mind, intelligence, intelligent life.

To put logic first among maths is an inclination of many people, usually anti-Platonistic, who regard the existence of mathematical objects as a fiction, at best a convenient fiction - for them, there is no order of being, but only order of knowledge, in mathematics. That's not a constraint which Tegmark needs to heed in his theory that mathematical existence is real existence.

What about the Comtean idea that the field which supplies basic principles to another field is the prior field? That's an ordering according to being, not according to learning, since the principles in question, involving laws, facts, entities, etc., are explanatory, descriptive, predictive, or verificative as contents of inferences. Deductive logic supplies principles or methods for inferring things and therefore for showing or accounting for or learning and knowing about things - not principles that, as inference factors, show or account for things in other areas of mathematics, areas and principles which are not about inference or its factors. Consider the concrete parallel: a study of actual methods of inference to concrete fact and even to laws covering concrete particulars would be neither physics nor prior to physics, but instead a social study, including sociology about science, also with some overlap into psychology, and would involve applications of philosophy and logic.

Now, two families of mathematics are regarded as deep, and one of them as pure and deep, and the other as applied yet (mathematically) deep. Pure mathematics includes such areas as simultaneous equations, topology, matrices, extremization, graph theory, integration, measure, enumeration, differentiation, calculation (algebra), groups, limits, and kinds of ordering e.g. well ordering. Conclusions drawn in these fields tend to be “reversibly” a.k.a. “equivalentially” deductive (in mathematical induction, the minimal case and the heredity, conjoined, are equivalent to the conclusion) and structures of equivalences are rife throughout pure mathematics. Applied yet mathematically deep mathematics consists of deductive mathematical theories of optimization, probability, information, and logic; conclusions in these fields tend to be non-reversibly deductive (though to the extent that deductive mathematical theory of information has “re-invented” group theory, it has developed pure-mathematical interests and, presumably, has drawn many conclusions through mathematical induction.) All of these applied yet deep mathematics are about structures of alternatives. They are about the structures of those alternatives which all are actualized across Tegmark’s Level III, the Many Worlds of quantum physics, and they deduce from totalities to parts.

What about Level II? Now, Level III and Level II are each other’s “inverses,” Level III actualizing alternate outcomes across quantum branchings, and Level II actualizing alternate outcomes in various times and places along a single branch, so that the two levels come out the same in their features. Likewise is there a family of abstract yet positive-phenomenally deep areas of research, such as statistical theory, areas each of which deals with the inverse problem of a correlated area of applied yet deep mathematics, and each of which deals in a general way with gathering data from various actual places and times and drawing ampliatively-inductive conclusions from parts, samples, etc., to totalities. These areas pertain to phenomena in general rather than to any special class, any single sample of the concrete real (and thus are all cenoscopic in the Peircean sense). They include inverse optimization problems (which comprise a young research field though some of the problems are old), statistical theory, descriptive and ampliatively inductive areas of information theory, and the descriptive and ampliatively inductive study of logic and intelligent processes — I mean philosophy, not AI or computer research. This family of research seem to stand to Level II as the deductive maths of optimization, probability, information, and logic stand to Level III.

Finally of course, correlated to Level I, there are the concrete empirical or “special” sciences — physical, chemical, biological, behaviorial/social/human, which tend to conclude in surmises, as cogent as they can make them.

I had kind of hoped to discuss some of this with the folks at the "everything" mailing list, but the arguments there tend to revolve around computationalism (most of the active participants are genial, e.g., Bruno Marchal, and they're all intelligent). Also, they don't think that much can be said about Tegmark’s Level IV. I suspect that this is because they haven’t yet been able to incorporate extremal principles into their work as they would like, but I don't think that I convinced them that there's any particular reason to think that there's a connection.

Note: How to say “everything exists.” In standard first-order logic, the phrase “everything exists” would be taken to trivially mean “that, that is, is,” or the like. Is there a way to say it in Tegmark’s sense in first-order logic at all? Is it an idea that can be logically expressed at that basic level? What would it mean if it can’t? Well, there does appear to be a way to say it in a specially restricted kind of first-order logic, by use of a special kind of quantificational functor. As for whether this leads to a coherent logical idea in less restricted logic, you be the judge. The result is, at least, a kind of statement which seems to lead to an area of logical issues raised by Tegmark’s picture, in any case, with regard to saying that every “potential” particular definite individual is actualized somewhere and somewhen, or, on the other hand, that the world altogether lacks some particular definite individual(s). The objectual version of the formalism sharpens the problem by allowing the individual(s) in question to be unspecified and even unspecifiable.
Now, in defining the existential particular quantification, one may start with a finite universe of objects named by constants “ a ” through “ h ”, and say “There is a such that...a...or there is b such that...b... ... ...or there is h such that...h....” and agree to write this as “ ∃ x ...x....” Then one drops the substitutionalist requirement that x ranges over only named objects a, b, c, etc. Then the variable x is no longer substitutional but instead is objectual. To get to our new special functor will be a matter of replacing the repeated “or” with a repeated “and”.
Let’s define a functor “ Æ ” such that “ Æ x ...x....” is equivalent to “There is a such that...a...and there is b such that...b... ... ...and there is h such that...h....” In effect one is saying that every name names something. Now, what happens when the substitutionalist requirement is dropped? In considering just what it is that x now ranges over, and whether the objectual statement “ Æ x ” is contingently or formally true or contingently or formally false or formally or contingently undecidable or (despite its fraternal-twin relationship with the existential particular) just plain ill-defined, one is led to consider some of the logical problems which arise in any case in entertaining the general idea that “everything exists.” In other words, we seem to arrive at some of the right problematics.
(Note: “ Æ x ” should NOT be called the “existential universal” which would instead be properly applied to whatever is equivalent to the conjunction or predicative combination of the existential particular and the hypothetical universal, where you say, e.g., “there’s some food that’s good, and any food is good” or “there’s some food that’s good such that any food is good” or “there’s food and any food is good” or “ ∃ x ( F x ) & ∀ x ( F x → G x ) ” or “ ∃ x ∀ y ( [ F x ] & [ F y → G y ] ) , ” etc. I suppose that “ Æ x ” could be called the “omniexistential.” - posted by The Tetrast @ 2:26 PM [top of post] [top of page] Copyright © BenjaminðA.ðUde11.