Stabbing Toward a Mathematical Formalization of Euryphysics

This post attempts to summarize many of the themes in the prior posts into something at least pointing in the direction of a coherent formal model of the Eurycosm, and tries to address the emergence of something like quantum theory from this Eurycosmic model.

Eurycosm as a Process/Program/Proposition Space

As  conceptual experiment, let us provisionally (and, to be sure, partially) model the Eurycosm – from the perspective of a particular observer – as a multiset of processes.   (A “multiset” means that some processes may occur with a greater count, or density, or weight, or whatever than others…)

Let us assume that these processes form a non-foundational set, i.e. each process takes some other process as input and gives some other process as output, and this network of process I/O can form a graph with cycles.

So far this is quite general could be considered a sort of modern mathematical formulation of Whiteheadian philosophy (Whitehead, 1929).   But it’s also interesting to get more specific…   

Let us focus for now on processes that can be formalized as a program in some language; and we may as well make this a typed language.

By the Curry-Howard correspondence, a program type is isomorphic to a proposition, and a program inhabiting a type is isomorphic to a proof of that proposition.  

So from this view, we can view the Eurycosm as a set of propositions and proofs.

(Whether this view captures the WHOLE Eurycosm is an issue certainly open to discussion.   But even if “most” of the Eurycosm is uncomputable in some sense, the portion of the Eurycosm that is modelable via computable processes may also be interesting.)

Next I propose that: The propositions comprising this model of a portion of Eurycosm may be wildly contradictory to each other.

The consequences of this contradictoriness are interesting and I will unravel some of them below.   But first let us introduct pattern into this picture.

Patterns and Associated Properties

Let us assume that the observer, with respect to which these processes are being identified, has a certain (implicit or explicit) simplicity measure associated with them, so that some of these proofs and propositions are more simple and some are more complex.   From this we can derive a notion of (observer-dependent) pattern.   To wit: Given a program P that produces output O from input I, we may say that (I,P) is a pattern in O if (I,P) is simpler than O.

Now let us assume that the Eurycosm has the property that: propositions with more pattern in them, have a higher weight.   (We can call this the PPP or Pattern-Proposition Property.)

It’s interesting to note that this yields a certain form of “morphic resonance” as a consequence, because of the following reasoning.   Suppose we have two systems S1 and S2, both evolving over time, and suppose that in S1 (as compared to S2) it tends to be a little more true that when subsystems share some common pattern, they come to share more common pattern afterwards.   Then under reasonable assumptions, we can show S1 is going to have more pattern to it altogether; thus S1 will be more likely according to the PPP.

One way this could occur might be: there are multiple agents recognizing patterns in the Eurycosm, and the patterns they recognize become part of the Eurycosm.  These agents may then be recognizing patterns in each others’ activity as well as in whatever ambient structure existed in the Eurycosm otherwise.  This sort of mutual pattern-recognition activity generally creates a lot of noise but also may create some structure via collaborative dynamics.   The complexity is increased if one assumes that these pattern-recognition agents are themselves (modelable as) programs existing in the Eurycosm.  

In this approach, the PPP can be derived as a consequence of the assumption that a fair percentage of the programs in the Eurycosm are pattern-recognizing agents.  That is one, must simply assume that the pattern-recognizing programs get fairly high weights, according to the weight distribution.

Consistent Universes within the Contradictory Multiverse/Eurycosm

Now let us return to this contradictoriness issue.   While the posited lack of logical consistency in the Eurycosm is important (for reasons I will suggest below, and others), the identification of coherent consistent subsets of propositions within the Eurycosm is also important.

Let us define a Universe as: a Subset of the propositions in the Eurycosm that

a)    comprises a (multi-)set of mutually consistent propositions
b)   constitutes a “cluster subgraph” of the process graph of the Eurycosm, in the sense of maximizing a measure of homogeneity (the weight of the patterns within the set) versus separation (the weight of the patterns spanning processes in the set and processes outside the set)

What is a cluster subgraph depends on the choice of a precise measure of cluster quality, so this is an additional observer-dependence (the observer must choose a measure of cluster quality, just as they must choose a simplicity measure).

Note that in this definition Universes need not be disjoint; they can overlap.

This is a “multiverse” theory in a stronger sense than current quantum theory; it views different universes as related but different models of portions of a larger eurycosm; and it allows for potential probabilistic dependencies between different universes in a sense that quantum theory does not.

Complex vs. Real Probabilities

The PPP implies a set of conditional probabilities between propositions.   Given probabilities for a set of propositions, probabilities for other propositions are implied.   However, the contradictory nature of the propositions comprising the Eurycosm means that these can’t generally be real-number probabilities; they have to be complex-number probabilities (as complex-number probabilities can accurately model the relationships between mutually contradictory statements such as “A = ‘B is not true’; B = ‘A is not true’ “ (and more complex webs of inter-contradictory statements)).

The path from contradiction to complex truth values is laid out in some detail in (Nguyen et al, 1998), along with other considerations related to fuzzy truth values.   It is shown there, via elementary considerations, that if we want to say that

A and not-A = True

while assuming

A or not-A = True

then the only solution obeying basic symmetry properties is to allow truth values to be complex numbers.   As one part of the argument, note that if we let

x * (1-x) = 1

then the only numerical solutions we find are where x are complex numbers with an imaginary component.

The complex-number probabilities get collapsed to real-number probabilities when the observer’s perspective transitions from the Eurycosm to a specific Universe within the Eurycosm.    The difference between the complex-number view and the real-number view, is the difference between observing using a potentially self-contradictory model, and observing using a guaranteed-consistent model.

In quantum theory it is sometimes said that complex probabilities or amplitudes must be applied to entities that a given observer cannot, in principle, observe.  This seems basically equivalent to saying that complex probabilities must be applied to entities that a given observer cannot consistently observe (where e.g. “consistently” may be interpreted as “consistently assuming the commonly held assumption of the 4D spacetime continuum in which we appear to carry out our everyday lives”).   I.e. real probabilities are applied within a Universe, and complex probabilities are applied to propositions that span Universes.

But modeling propositions with complex probability values is one route to deriving quantum mechanics (e.g. according to Saul Youssef’s “quantum mechanics with exotic probabilities”; Youssef, 2001).   So we conclude that quantum mechanics must be applied to describe the Eurycosm, except in cases where attention is restricted to a consistent set of observations.

Another way to phrase this is to adopt the Possibility Principle (PP) that: About choices whose resolution we cannot observe, we must assume all possible options exist side-by-side in the Eurycosm.   That is, we must assume

Observe( system S with possible states A and B)

does not necessarily collapse to

Observe(system with state A) or Observe(system with state B)

This raises the question of how we know the system S has possible states A and B.   This must be because of similar systems S1 with possible states A1 and B1, so that

Observe( system S1 with possible states A and B)

does, for S1, collapse to

Observe(system with state A) or Observe(system with state B)

-- for which there is a very powerful pattern spanning S and other similar systems S1.   This powerful pattern has a high weight by the PPP, meaning that the probability of

Observe( system S with possible states A and B)

is evaluated as very high.   On the other hand, there must be other systems A*1, B*1, etc. so that A is very similar to A*1 and B is very similar to B*1, and neither

Observe(system with state A*1) or Observe(system with state B*1)

ever holds.   It must be a very powerful pattern that (A,B) are similar to many such systems (A*1, B*1).    Putting these pieces together, we arrive at the desired conclusion.

But why might this be the case?   From one perspective, this has to do with the underlying logic of the propositions in the Eurycosm.   There must be other propositions the observer holds, that are consistent with “X or Y” but not with X or Y individually (where here X = system with state A, “X or Y” = system with possible states A or B, etc.).   This simply means that the relevant logic for the Eurycosm does not have the rule “O and (X or Y) à (O and X) or (O and Y)”, where O is the proposition “what the observer observes during a certain time interval”.  

But this option is left open by the PP, since the PP implies that if the outcome of (X or Y) can’t be observed consistently by the observer, then all possibilities regarding (X or Y) should be considered open, including inconsistent ones.

The basic subtlety here is not assuming logical consistency of that which is unobserved.

Logic and Evidence

The relationship between logical consistency and assumptions of “unquestionable reality” for certain percepts, is interesting to note.

Logical consistency can be understood to emerge from a process of evidence-counting.  Inconsistency amounts to taking X as evidence for not-X, and so forth – and it is clear that this will not happen if one begins in a grounded way and starts with certain specific observations and calculated truth values based on that.    To get consistency one assumes certain propositions as ground truth, and then calculates probabilities of other propositions based on that.   Without assuming certain propositions as ground truth, one is left with a web of mutually contradictory propositions with interdependent probabilities (which then come out as complex-valued).

But if the observer cannot observe the details of (X or Y), the observer cannot disambiguate between X and Y based on evidence, and cannot use the “grounding in assumed ground truth” approach.

A community of observers who assumes a common set of “ground truth” of observations and a common simplicity metric and clustering criterion, can then do science together effectively.

Subjectively Irrefutable Evidence

But what is this “irrefutable ground truth”?   From the point of view of a human-like mind, subjectively perceiving the world, this comes down to some basic relationships among sense-perceptions.

Content like “the ball is red” or “the quale I just observed manifests redness” seems uncompelling for the “ground truth” role, not because it’s necessarily unreal in all senses, but because it maps very poorly into the logical and propositional realm we are considering.

On the other hand, various comparative judgments seem fairly sound candidates to be “irrefutable observations.”  

To give the discussion some context, let us suppose we are discussing a mind (like yours or mine) that has some perceptions and memories in it.   It’s OK if there are sometimes difficulties distinguishing a perception from a memory.    Let us also assume that this mind, during each interval of time, has some entities (perceptions/memories) that are especially high-focus or intense within it (the “attentional focus” of the mind).   Setting aside association-laden verbiage, basically all we are assuming here is the existence of some set of entities, some notion of (not necessarily 1D) time; and a mapping that, for each connected subset of time, assigns some entities values in an ordered set (these are the “attention values”).

In general, what seem to be “irrefutable” observations in this sense are statements of the form:

·      In terms of Property P, experience A is greater than experience B
·      In terms of Property P, experience A is closer to experience B than to experience C
·      Property P and Property Q are different

Among the properties that can be involved in such statements (this is definitely a non-exhaustive list) are

·      temporal and spatial proximity
·      some sensory characters like loudness, brightness and color
·      some proprioceptive characters like force and speed
·      some emotional-sensory characters like pleasure and pain
·      “origin”, to be described below

To make this more concrete, I will next give some specific examples.  However, it’s important to interpret these examples appropriately.  In the following examples, terminology like “the little duck” and “throwing of the red ball” is used.   However, we are talking at a level where there is no independent meaning to these labels (because we’re talking about, among other things, how such labels come to being).   Where “the little duck” is referenced in the following, one should read this as “the only experience, in my focus of attention at present, that feels especially closely associated to the experience of the label ‘the little duck’.”   That is: the perceived similarity between experience X and the experience of the label ‘the little duck’ is much greater than the perceived similarity between any other currently high-focus experience and the experience of the label ‘the little duck’ “.   The reason using these labels seems OK in the examples below is that this kind of association between labels and experiences can, in itself, be formulated as an “irrefutable observation” of the sort to be described below.

So: given these caveats, some good candidates for irrefutable observations seem to be:

·      The little duck and the little goose appear to be closer to each other than the little duck and the scary bear
·      The movement I just made feels more forceful than the movement I made a minute ago
·      The movement I just made feels faster than the movement I made a minute ago
·      The pain I just experienced, felt more severe than the pain I felt a minute ago
·      The joy I just felt today, was closer to the joy I felt as a child, than to the joy I felt yesterday
·      The yell I heard 5 seconds ago, was apparently louder than the whisper I heard 3 seconds ago
·      The event of me throwing the red ball, and the event of me sitting in the blue house, appeared to be closer together in time than the event of me eating the yellow duck
·      Suppose you have observed three colorful entities at around the same time, and: Entity 1 looked red, Entity 2 looked red, Entity 3 looked blue.    The irrefutable content here is: Entity 1 looked more similar to Entity 2, than to Entity 3, in terms of color.
·      The difference in apparent loudness between the yell I am hearing now, and my memory of the yell I heard a minute ago, is less than the difference in apparent loudness between the yell I am hearing now and my memory of the whisper I heard 5 minutes ago
·      Loudness is different than brightness
·      Brightness is different than color

Another interesting property, complementing the ones considered in these examples, is “origin” (which we can think of as “selfness”) -- a kind of “inner location.”    It may be the case that:

The origin of experience A and the origin of experience B, are much closer to each other than to the origin of experience C

“Selves,” in their most primitive form, may be viewed as clusters formed based on the property of origin.

Building Refutable Conjectures from Irrefutable Evidence

These sorts of irrefutable observations are enough to build clusters, where a cluster of experiences may be defined (for instance) as a set S so that: members of S tend to be closer to each other in terms of various properties, as compared to how close they are to experiences not in S.   One can build clusters based on individual properties, sets of properties, or all available properties.

Once it has clusters, a mind can build patterns like

In terms of Property P, experiences that lie in {both cluster B and either cluster C or D}, tend to be greater than experiences in cluster A

A mind can also create concepts like

X = (B and (C or D))

And then to save memory, given our limited resources, a mind can forget whether a given memory was in C or D, but remember that it was in X.

Observations like

In terms of Property P, experiences that lie in X, tend to be greater than experiences in cluster A


In terms of Property P, experiences that lie in X, are greater than experiences in cluster A, with probability .8

are the refutable ones.   They are built up by induction, and there is no guarantee that new experiences falling into X are going to obey the same rules as the previous ones that fell into X.    Induction requires some intuitive or explicit criterion of simplicity, and the refutable observations a mind builds will depend on the simplicity assumptions wired into its organism.

Science, it seems, may be reasonably well modeled as a complex set of interlocking refutable observations, built up from a set of observations accepted by a community as collectively irrefutable, based on a sense for simplicity  commonly accepted by that same community (see The Hidden Pattern (Goertzel, 2006) for a discussion of philosophy of science along these lines).   “Common sense” may be reasonably well modeled in a similar way, though here there is less formality involved in the definition and maintenance of the set of irrefutable observations and the agreement on what are their consequences.  Either in the case of science or of common sense, though, the refutable observations involved can always be invalidated via presentation of new data, or by (in the case of common-sense, often very fuzzy and uncertain) logical reasoning.

Abstract Feynman Sums

While it may seem we are in a very abstract and subjectivist realm here, it’s interesting to note that the formalism of quantum mechanics can be fairly straightforwardly captured in this context.

In the Feynman path-summation approach, the probability of a transition from one state A to another state B, is calculated by adding up the probabilities of the different ways of deriving B from A (this is the Feynman path sum).   We can apply this to proofs via positing that: The probability of a proof for deriving B from A, is gotten by normalizing the weight of that proof.   We can then use these proof-probabilities to derive an overall probability for transition from A to B.  (This corresponds to an eccentric choice of measure for the Feynman path sum; here the PPP is guiding the measure, which can lead to morphic resonance type phenomena.)  If these are complex probabilities then constructive or destructive interference phenomena may occur.  

This application of Feynman summation to logic can be applied in a physics context, but also in many other domains.    In standard quantum-mechanics cases, e.g. A might be “particle p is at location x at time T”, and B might be “particle p is at location y at time T+t”, and the proof may be done using a certain set of other observations as assumed ground truth regarding the physical situation.


While “objective reality” and physics result from assuming some irrefutable realities and then counting evidence based on these, subjective consciousness results otherwise – from accepting the contradictory and non-foundational nature of the Eurycosm (as reflected in the contradictory and non-foundational nature of the proposition/process Eurycosm model proposed here).

In (Goertzel, 2011) it is argued that “reflective consciousness” is at base a proposition of the form “X is looking at X”, which is no problem to construct in a non-foundational process space.   However, to assign a probability to this one must use a (real or complex valued) infinite-order probability distribution.   

For reflective consciousness with an object, we are looking at constructs like

X = “X is looking at both X and A”

which are also perfectly valid mathematical objects in an anti-foundational set theory (for example).  

The nature of these infinite-order constructs is that once one assumes ground truth, one can not ever get to such an infinite-order construct via a finite number of inference steps from one’s ground truth (unless the ground truth one assumes, includes such infinite-order constructs).   This yields the general confusion about “whether consciousness exists” – some folks want to assume such infinite-order constructs as part of the ground truth, and others do not.

Constructs of the form “X is looking at (Y or Z)” would generally need to be modeled using complex infinite-order probabilities.   In this sense we can have “quantum consciousness.”   These quantum-ish conscious processes can sometimes emerge as subpatterns in sets of (mutually contradictory) non-strange-loopy propositions, in the same way that the formula for the infinite series of integers is a subpattern in any reasonably finite long series of integers.

The issue of “qualia” (basic units of conscious experience) can also be addressed in this framework, in a way that differentiates it from reflective consciousness.

Given an irrefutable observation of the form

·      In terms of Property P, experience A is greater than experience B

we can form: the set of all experiences A so that, in terms of property P, A is greater than or less than something.   We may then say: the quale of P is equivalent to the characteristic function of this set.  

This is a basic observational quale – not the only kind, but an interesting kind.   For instance if P is “redness”, this is being equated with “the set of experiences whose redness can be compared.”

(If we want to introduce a sort of “univalence axiom” here and say “equivalence equals equality” (Awodey et al, 2013), as in homotopy type theory, then we will conclude that the quale of P is in fact the same thing as the characteristic function of the set….   But this is a major philosophical step, and in a general eurycosm-theory perspective must be viewed as one interesting perspective rather than as a definitive proclamation.  In Peircean terms (Peirce, 1867), the “univalence axiom” here is trying to map a First into a Thirdness of First.)

I cannot know that what you and I call “red” is the same thing.   However, I may be able to observe that the experiences with origin in cluster “me” and labeled “red” and the experiences with origin in cluster “you” and labeled “red”, tended to be in many of the same clusters, and many of the same useful cluster-combinations.  

The non-foundational patterns associated with reflective consciousness may be treated similarly, e.g. if P = “P is looking at both P and A”, we may then look at the collection of experiences that are describable as P (which will vary according to parameter A).    The characteristic function of this collection of experiences is then equivalent to the quale of P.   This is not the same as a basic observational quale, because it has a sort of foundational self-reference built into it.   But the existence of many different species of qualia is hardly news.


Awodey, Steve, Álvaro PelayoMichael A. Warren (2013).   Voevodsky's Univalence Axiom in homotopy type theory.  Notices of the American Mathematical Society

Goertzel, Ben (2011).  Hyperset Models of Self, Will and Reflective Consciousness.  International Journal of Machine Consciousness vol 3.

Goertzel, Ben (2006).   The Hidden Pattern.  BrownWalker.

Nguyen, Hung, Vladik Kreinovich, and Valery Shekhter (1998). "On the   Possibility of Using Complex Values in Fuzzy Logic For   Representing Inconsistencies", International Journal of   Intelligent Systems, 1998, Vol. 13, No. 8, pp. 683-714.

Peirce, C.S. (1867), "On a New List of Categories", Proceedings of the American Academy of Arts and Sciences 7 (1868), 287–298.

 Whitehead, A.N. (1929). Process and Reality. An Essay in Cosmology. 

Youssef, Saul (2001).  Physics with exotic probability theory, hep-th/0110253,

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