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on Full theories of time, existence and qualia

So, suppose there is no fully-communicable "full" theory of time. Call a theory T₁ about something full if it accounts for everything I know about the something, but I *necessarily* cannot communicate the entire theory to you, and you have an analogous theory T₂.

The initial idea was that I can't write down a full theory of time and thereby communicate all of it to you.

Let T₁ be person 1's full theory involving time, and suppose it has atomic elements/operations {x, y}. Let T₂ be person 2's full theory involving time with atomic elements {x, z, w}, so they're not isomorphic. In this case we can utter truths (or think of them), but sometimes we literally won't know what the other is talking about because we don't have the same referents (y versus z and w). T₁ could be isomorphic to a person 3's theory T₃ which has atomic elements {x, v}, but this de-emphasizes that they are not ontologically equal to each other so I'll ignore it. But suppose T₁ has a well-defined sub-theory T₁', and T₂ has a well-defined sub-theory T₂' such that:

(1) T₁' is isomorphic to T₂'

This isomorphism is the part of our theories where we can agree, in some sense, about "objectively true" stuff. Namely, the isomorphism, in practice, warrants the ontological assumption that T₁', T₂', and the definians or objects they use/assert to exist, such as the number 5, are independent of who or what system or whether there is anyone/thing at all instantiating/conceiving of it. Which interpretation you use for the number 5 doesn't matter, you can just as well take the set {0, {0}} as given by ZFC.

What's going on seems to be for a full T₁ the disjoint elements that "exist", according to T₁, are necessarily the only perspectives from which T₁ can be defined. A difference with set theory is that in set theory it's irrelevant what the elements {a, b, c, ...} of a set s "really are", you just assume the existence of the power set P(s), or at least some constructive schema independent of the content of s. On the other hand, suppose you have a set theory S that's "full". Suppose that, according to S, there exist two urelements a and b. Then there doesn't exist the set {a, b}, but rather the two set theories

(2) S_a = {a, 'b'} and S_b = {b, 'a'}

'b' is in a's theory (i.e. a's theory of a and b), 'a' is something in b's theory (i.e. b's theory of a and b). S_a exists in the same way a does, and S_b exists in the same way b does.

Suppose we start with elements a, b, and c. a and b exist, but c is a purple unicorn, and doesn't exist. The possible set theories are S_a, given in terms of {a, 'b', 'c'}, and S_b given in terms of {b, 'a', 'c'}, and that's all. There is no fact of the matter in terms of {a, b} or {a, b, c}. If c did exist it would only add S_c to the mix given in terms of {c, 'a', 'b'}. An iteration gives the four set theories from the perspective of a, b, S_a, and S_b, since each of these exists. Another iteration gives

(3) S_a(a, 'b', 'c', 'S_a(a, 'b', 'c')', 'S_b(b, 'b', 'c')'), ... S_Sb(S_b(b, 'a', 'c'), 'a', 'b', 'c', 'S_a(a, 'b', 'c'))

Existence

Since a full theory seems to have this ontologically-self-referential behavior it's reasonable you could have a "full" theory of ontological existence, independently of any considerations about time.

Let F_a, F_b, F_c, ... be full theories of existence of a, b, c, ... Then F_a's assertion that b exists is true only if it's symmetric in some way with F_b, or in the same equivalence class, or something.

Qualia.

That you can't communicate them is paradigmatic of qualia. One could argue the reason for this is the necessarily-exesitentially-self-referntial behavior again. Consider three theories of qualia

(4) the set s = {this experience of green}

(5) this experience of green is neuronal processes X, or else is correlated to neuronal processes X

If these theories are going to be full (and refer to qualia), then the variable this experience of green doesn't range over my experience of green *and* your experience of green (i.e. each instantiation of green in the universe). There is a different variable in some sense for each person, and there may not be, in the ontology of the theory, a variable in existence that ranges over both our greens. (It doesn't really matter you define a "person" to be, it could be a person, an information-bearing agent, a Super Universal Turning Machine, a donkey, or, on some accounts, a rock. The points are the same.)

Let green₁ be green as perceived by person1 and green₂ be green as perceived by person2. The set {green₁, green₂} is ontologically inconsistent, if the set is supposed to exist in the same way that green₁ does to person1, and green₂ does to person2, i.e. fully.

Instead the total information of the situation is given by two weaker theories with (2nd-order?) interrelations:

(6) s_person1 = {green₁, 'green₂'}, s_person2 = {green₂, 'green₁'},

w/relations between 'green₁' and green₁, etc. In (6) 'green₂' is person 1's theory of person 2's green₂. (A "theory" could be a conceptual referent, an unspecified representation, the concurrent physical instantiation of my theorizing, the set of possible interactions between person1 and person2, etc. The idea is that ) Since iterating the process never yields a legitimate assertion that green₁ = green₂, the ontology is underdetermined, in the sense that for all we know my green could be your red (spectrum inversion!). This is consistent with the laws of physics from each person's perspective. Schematically, the idea is

(7) person1(green₁, 'green₂', physics) [physics] person2(green₂, 'green₁', physics)

The physics in this case are the isomorphic subtheories mentioned in (1). This is a very different kind of model than either (physics) or _x(green₁, green₂, physics), where x ranges over everything in the universe.

So, there is a calculus of qualia there. But, more-or-less, it's based on the single property of the ineffability of qualia. A more sophisticated treatment would account for other properties. The place to start is Lormand's list of 6 properties of qualia: Intrinsicness, Directness, Reliability, Unanalyzability, Ineffability, Privacy. Other properties should be considered, whether they turn out to be properties of qualia or not. For example, for quale q,

(8) q is a property of q

and

(9) q is epistemologically objective and ontologically subjective (and exists)

I'm told Searle also considers (9) (references welcome to all the ideas of these posts.)

The question is, which of these 8+ properties do qualia have, does time have, and does existence have? For example, it is perfectly reasonable to wonder, as in (8), if existence exists, whatever the eventual consensus turns out to be. Also, is (8) true of time? Temporal flow seems to be a property of time itself. I would also argue that being green is an ontological property of green itself... In fact it's plausible that time and existence are special cases of qualia (ones that are more public than green, but still basically have most of the properties of qualia).

Full Physics Theories

What must full theories of physics look like?

The basic idea is that the following is *not* a valid conclusion

(1) There exists a particle p1 whose motion in R³ is given by a time parameter t

(2) There exists a particle p2 whose motion in R³ is given by a time parameter t'

therefore

(3) There exists a system or set of particles {p1, p2}, whose state in (R³crosst)cross(R³crosst') is given by a time parameter t''.

This is much stronger than the assertion that p1 and p2 can't live in the "same" space. This is the assertion that there is no fact of the matter as to "the state" of p1 and p2. Instead, under reasonable assumptions, the ontologically consistent model is that p1 is a function of a timeline/parameter t_p1 and p2 is a function of a related but ontologically distinct timeline/parameter t_p2. t_p1 and t_p2 could be thought of as locally parallel in the geometry of p1 and p2's potential interactions, as specified from the timeline t_p1 on one hand and t_p2 on the other. But there is no objective evolution of "the state of both particles" (unless they form one ontological system, contra hypothesis), because that would require a third, "more objective" time parameter t_thirdSystem, contra to the hypothesis there are exactly two particles existing in this model.

Suppose there is a single system S evolving according to a time parameter t. Now suppose that, according to the ontology of the theory, S splits into two distinct (disjoint?) systems p1 and p2. Necessarily, the only possible "theories of physics" in this universe are the theories instantiated by p1 as it moves through t_p1 on hand and theories instantiated by p2 on the other hand. It is not possible for there to be a theory of physics that is not had by either p1 or else p2. It's not possible for either p1 or p2 to possess a theory of physics in both full variables t_p1 and t_p2 in this universe.

I'll consider two cases. The first is where there are just a set of relations between p1 and p2, and the second where p1 and p2 are evolving "in space".

Case 1

In the first case, p1 will evolve in t_p1. By assumption p2 is an ontologically distinct system. So all p1 can do is specify the possible future states of its interactions with p2, and it can do this using only information of t_p1 and 't_p2', where 't_p2' is p1's mathematical theory (or instantaneous theoretical projection) of t_p2 at time t_p1. The analogous things hold for p2. In this universe there is no such thing as the (full) state of both systems at a time t_p1 or a time t_p2 or a combined time (t_p1 and t_p2) or some other such.

Of course things aren’t completely random in this model: presumably if p1 and p2 have full theories T_p1 and T_p2 they may suppose (predict) the other particle is moving according to some subtheory T'_p1 and T'_p2 respectively, which makes T'_p1 and T'_p2 isomorphic as ontological objects and equal when regarded as mathematically formulated theories. p1 and p2 later recombine as judged (experimentally) by either p1 or p2 then form one system again and evolve according to one temporal parameter.

This is *not* 2-dimensional time. In that scenario (known to be wrong in models of physics, as I understand it) you could have one or more particles' movements parametrized by a communal temporal parameter space t₁crosst₂. That suggestion suffers from all the problems mentioned above, and has nothing to do with my topic.

What's interesting is that there is seemingly quantum behavior (ontologically) just from being consistent about a full theory of time.

Case 2.

Suppose p1 evolves according to t₁ in R³. What ontologically exists in this model is either 1. just the particle p1, or 2. the particle p1 and the points of space given by R³. But it won't matter for the following arguments.

Now suppose we add particle p2 to the scenario. The *incorrect* thing to do is just let p1 and p2 both be in R³, and evolve the system according to t₁ or at least some new temporal parameter t. If time full (for both p1 and p2) there is no objective fact of the matter about "the state of p1 and p2 in R³". Instead there is the theoretical situation as given by p1 as p1 moves along t_p1. The "theoretical situation" as given by p1 can be in terms of the variables t_p1 and 't_p2', but not t_p1 and t_p2. So the only possible theories of physics are p1's theory of the combined system, call it T_p1(t_p1, 't_p2') and p2's theory of the combined system T_p2(t_p2, 't_p1'). It would be ontologically inconsistent to say there is a physical system that exists as a point (value) of some third time parameter t_p1andp2. Therefore, a theory of physics that uses t_p1andp2 or even assumes it is equal to either t_p1 or else t_p2 cannot be fundamental.

What's interesting is if you consider p1's theory. All spatial interactions with p2 are specifiable by the 2-dimensional coordinate system on the sphere at distance 1 meter from p1 itself. p1 can specify the evolution of p2 in terms of t_p1 and 't_p2', but 't_p2' is really just an element of p1's theory about p2, so I'll assume it is part of the theory to begin with. So p1's theory evolves according to t_p1, i.e. T_p1 (t_p1) is p1's theory of p2's evolution on p1's co-evolving sphere. The radial distance to p2 can be given by another variable. And similarly for p2. So, we have the theories

1. T_p1(t_p1) gives the predicted locations of p2 in the variables (angle 1, angle 2, radial distance)

2. T_p2(t_p2) gives the predicted locations of p1 (on p2's sphere) in the variables (angle' 1, angle' 2, radial distance')

We can model this on the 16-dimensional manifold

(t_p1)cross(angle1 values, angle 2 values, radial values)

cross

(t_p2)cross(angle' 1 values, angle' 2 values, radial distance' values)

with lots of dimensional reductions, depending on the physical laws and geometry between t_p1 and t_p2 et. al.

Another interesting thing is that "a physical state" on M is non-local. If the points of R³ are taken to ontologically exist in the same way the particles do, then each one will have a separate (but normally parallel) temporal evolution, and the collection of all such theories is equivalent to a classical theory of the system {p1, p2}, probably.

An interesting case (but only one case) is where p1, p2, ... are Super Universal Turning Machines. If they start out with representations of their environments, including the initial states of the other machines, then they can evolve into a Nash equilibrium in which their states are constant or cyclic. This has the interpretation that none of the pi will change their mind about the universe from then on. 

Theories fully involving Time are Necessarily inequivalent

Let T be any well-defined theory that involves time. T could be a temporal logic, a theory of temporal moments, or a theory of things that evolve in time, etc. Say

(1) T is full iff it accounts for everything I know about time

The idea is a "full" theory that involves time in some way can account for the Moving Point argument, i.e. it can account in all respects for the "meta-time" behavior of a point moving along (in "real time", so to speak) of the theory's own model of time. You could interpret a set of axioms (relevant to time) {A_i} in a model M, both given by a Kripke structure K. But you don't have to, so I'll just call any theory involving time T. The point (more or less) is that one can always form the singleton set s = {T}. Having done this, the Moving Point argument applies again, and T necessarily doesn't account for all temporal behavior. Thus T is not "full" because it cannot account for everything we know about time. Here are two responses to this (not necessarily the only two):

(2) There is no full theory T. Boring.

(3) I am in possession of a full theory T_1, but I can't communicate all of it to you, and you are in possession of a full theory T₂, but can't communicate all of it to me.

If we demand of a theory that it *realistically* incorporates time (3) is the avenue to explore. The idea is there is a full theory T₁, and a full theory T₂, and they are necessarily ontologically distinct. They are necessarily distinct in the ontology of the theories themselves. So (3) is a kind of ontological consistency condition on any full theory involving time. (It doesn't matter if your universe has only physical instantiations of what people think are full theories, or if your universe includes the existence of mental objects that are full theories: any representation of the of theory in the universe (of the theory) will do).

Presumably, there are isomorphic subtheories of T₁ and T₂ that allow us to compare notes on what we know about time (though T₁ and T₂ themselves need not be isomorphic). But they are necessarily not equivalent. Therefore they do not have the properties we usually ascribe to numbers (or to the set {0, {0}} as given by ZFC). We assume when we teach someone 2 + 3 = 5, this fact is true independent of who considers it (or knows it), and (usually) it's independent of whether or not anyone is around to consider the fact. These are true regardless of the formal definition of these numbers and operations, and regardless of whether numbers are supposed to "exist" in some sense or not. That's fine. But if T is full the ontological situation is necessarily different (assuming (3)).

The idea that 2 + 3 = 5, however it's ontologically instantiated, is an equivalence class of it's instantiations (regarding it as a potentially full theory). Happily, can be sloppy about this equivalence class: we can think of it as having a representative element that's true, or of all of them being true, or we can assume it's a true assertion about the equivalence class independently of whether there is anything in the universe, etc. ("true" means all past current and future behaviors and objects that exist in any way and their relations are invariant with respect to any ontological instantiation of 2 + 3 = 5). The point is for *numbers* all of these are valid. But a (necessarily) full theory of time requires more careful ontological distinctions.

I'll argue this has critical consequences for any theory of physics that wants to be realistic with regard to time (and which have not been incorporated yet). And I'll argue this idea of a full theory of time extends to the notion of a full theory of qualia and to the notion of a full theory of existence.

the Moving Point argument

the Moving Point argument



Suppose the location x of a particle on the real line R is given as a function of a time variable t, so x = x(t). The particle is moving in time.
The position of the particle can be plotted on a space-time diagram and it'll make some curve. This curve is a subset of the 2-dimensional manifold coordinitized by R and (T = all t). As such, it is unchanging/timeless/static in the sense all mathematical objects are. Therefore one can imagine the particle moving along the curve. This situation can be modeled as a curve in the manifold (RcrossT)crossT. But then the particle can be imagined to move on this curve, and etc.
At no stage have we incorporated the full behavior of time (or things in time), since each stage doesn't take into account the behavior leading to the stage after it.

Characterizing the Explanations for Existence that Don't Work

What would an "explanation" for existence look like?

Take "explanation" to mean you start without assuming the existence of anything, and end by concluding something exists.

Here's an attempt at an such an explanation.

Okay, suppose Sally is a purple unicorn, and consider

(1) Sally is tall and shy

(2) Sally is tall

So (1) implies (2) but not the converse. But this relationship between the assertions (1) and (2) exists whether or not Sally exists. Therefore, something exists.

This doesn't assume the 1st-order properties of Sally are instantiated. The claim is there exists a relationship between (1) and (2) even though Sally is a purple unicorn. The property (1) that Sally would have if she existed is necessarily related to the property (2) that Sally would have if she existed. So the relation exists.

This attempt at an explanation for the existence of something doesn't work (I assume), no doubt for several reasons. But what's needed is a characterization of the class(es) of explanations that don't work.

Existence and Numbers as Sets of Properties

Existence and Numbers as Sets of Properties

In 1^st-order number theory one proves, for example, “5 exists”, expressed

(1) (there exists x)(x = 5).

But what happens if

(2) an existence statement about a natural number is a strictly 2^nd-order property

(3) a natural number is defined as a set of properties (that the number would have had, so to speak)

Both are inconsistent with (1), or at least arguably so, and both have a surprising amount of philosophical motivation. To get an idea of what (2) might mean, “5 exists” might be replaced with “the properties of 5 are instantiated” (though this is only one interpretation of (2)). (3) is known as a bundle-theory, as opposed to a substance-theory. In a bundle-theory the number 5 is really just a bundle of properties, and there is no other entity that 5 refers to.

The question is, is there anything to be gained by implementing (2) or/and (3) in a mathematical theory?

I don't have the sophistication to answer this question on the mathematical side, or the philosophical side, so I'll speculate.

Part 1 (1) versus (2)

Let N1 be 1^st-order number theory with parameters {for all, etc.} as in (Enderton).

I'll stay with the relatively simple intuition above: “the number 5 exists” is to be replaced with “the properties of the number 5 are instantiated”. How do you define the set of properties of the number 5?

The set of properties of 5 could be defined as the (“provable”?) collection of closed formulas in N1 that contain at least one instance of 5. Call this “characteristic” set C_N1(5). This set is probably too big, at least in the sense that N1 contains the 1^st-order existence assertions we're trying to avoid. So, informally, define

N2 is just like N1 except it doesn't contain existence-statements (or for-all statements)

N3 is just like N1 except it doesn't contain existence-statements, and it does not have strong negation, but it does have for-all statements (?)

Anyway, there are many theories smaller than N1 one could start with.

Consider N2's 2^nd-order language. Does it have the existence-quantifier? I don't know. Does one need a new “instantiation” quantifier I, such that if p₁ is a property of 5,

Ip₁ iff one would normally prove 5 exists

But I might not be necessary, because 2^nd-order instantiation looks (to me) like it behaves like 2^nd-order existence.

The point is this. If you'd normally say something exists, you can now say there is a (maximal?) collection of properties (its characteristic set) that must have certain conditions on it. To begin with all of the properties of something that we would normally say “exists” must be consistent with each other, e. g. if p and q are in a characteristic set, then the property p and q must be in it. Call this set of conditions CondEx. Now potential “things” can be recognized by this condition on the corresponding set.

(Decidability issues.)

Consider the collection of all sets of 1^st-order properties. Which sets have CondEx? Are these exactly the ones that correspond to what we'd normally call (extant) numbers? Do the non-standards play a new role?

What is the definition of CondEx? Certainly there are more than just consistency criteria that characterize existential things. The properties a unicorn would have, if the unicorn existed, are consistent with each other.

The next question is how can the conditions CondEx be relaxed (or strengthened)? Suppose there are other sets that meet a condition weaker than CondEx, do they exist, but to a lesser degree? (This from the philosophical literature.)

One issue is that if a number exists in the sense of (2) you'd expect distributivity to hold for all of its properties. But it does not always hold in a quantum context. In one variant of quantum logic, it may be true that an electron

“has a y-spin up and has an x-spin up or down”

but it is not then true that the electron

“has a y-spin up and an x-spin up, or has a y-spin up and an x-spin down”

yet the electron exists. (I'm not saying this is a paradox, just that it's interesting.)

In fact, suppose that, in some generic sense of ontology, “a exists” and “b exits”. It is a further assumption on the nature of ontological existence that there are “two things” in any sense, or that a and b can be compared in some way. Physicists currently consider disjoint universes that have different, though related, physical laws. What are the philosophically motivated alternative definitions to CondEx? i.e. other conditions on sets of properties of something that might reasonably exist.

Let C_N2(5) be the properties of 5 starting with N2, properties being provable sentences (and “provably” so, if necessary) containing at least one 5. One could argue that a more complete characterization of 5 can be given by it's 3rd-order properties, call it C(C_L2(5) ). Either the 2^nd-order language has the existential quantifier, or it doesn't, whatever. This process can be continued indefinitely. The question is, is there a point at which the collection of n^th-order (or higher) properties of 5 starting with N2 is isomorphic to the collection starting with N1?

If so, then 1^st-order existence in arithmetic is in some strong sense dispensable, and it is immune to many philosophical objections.

Voevodsky at the IAS conjectured (http://video.ias.edu/voevodsky-80th ) that 1^st-order arithmetic is inconsistent. He explained how and why this could happen. He also points out that it would nevertheless remain true that if you have 2 physical apples, and you have 3 more physical apples, you will then have 5 physical apples. So there is some “physically justified” proper subset of arithmetic that is “necessarily” consistent, even if arithmetic itself is inconsistent. The question is: what is the extent of this physically justified subset?

A physically justified subset will have objects and operations. (I tried getting around that, thinking objecthood isn't so straightforward in quantum field theories, but the guys at MathOverFlow pointed out the flux integrals around disjoint sets sum additively. So far as I can see, that shows the quantum case is like the classical case, but doesn't settle what the classical case is.) The objects are supposed to be the things that the theory asserts exist. Given the notions of existence above it might be possible to delineate the maximal physically justified subset of arithmetic.

Part 2 (1) versus (3)

According to (3), 5 is to be defined by its properties. These properties are initially sentences in which 5 occurs. But some of these will be like 2 + 3 = 5. To be consistent, one wants 2 and 3 to be defined by sets of properties too. But some of the properties of 2 and 3 will include 5. Thus 5 will really be a set of properties of properties of properties of... This is non-wellfounded behavior (Aczel, Barwise and Moss). So far as I can tell, most “circular” phenomena like this can be modeled by non-wellfounded sets, though there are some would-be definitions that are inconsistent.

For example consider

(4) 2 + 3 = 5

and suppose one of the properties of 3 is

(5) not(2 + 5 = 3)

In view of (2), (4) and (5), and depending on the iteration process, informally speaking, stuff like this will come up in the beginning:

(6) {1st-order properties of 2} + {not(2 + 5 = 3), other 1st-order properties of 3} = 5

It looks to me like the non-wellfoundedness, at least, is not a problem.

Part 3

What happens when (2) and (3) are combined? Is it necessary to combine them for philosophical reasons?

What happens when they are combined in other mathematical contexts?

Set Theory. Is the definition of CondEx the same in set theory as in arithmetic? Do the universes of L, ZFC, whatever, get bigger?

Some version of Comprehension might be recoverable. The standard problem is one can define a set R that both contains and doesn't contain itself. But if R doesn't exist in the first place, it can't cause any problems for the universe.

Finally, if questions like these can be given definite answers they would impact the philosophical discussion. 

proving there is no proof of existence

Is it possible to give a proof there is no "explanation" of existence?

That is, what is the widest class of "explanations for existence" that can be ruled out?

Here are some thoughts.

1. you can prove
(there exists x) (x = y)
in usual 1st-order logic. This asserts something exists. Of course this is not helpful, because the only reason we can prove it is that we assumed the existence of something in the model of the language in the first place.

Still, this is not exactly a proof there is no sort of explanation whatsoever.

For example, call the language in which we can prove (there exists x) (x = y) L1, and call its model M1. Now take both (some unspecified representation of) L1 together with M1 as a new model, call it M2. Construct the theory of M2, L2. Let M3 be M2 together with L2, etc... Each stage in the sequence is justified by the one after it. The problem is, what justifies the sequence as a whole? But this is an odd problem: it seems to assume the existence of a completed infinity *before* it allows the conclusion there is no point along this sequence at which the existence of something may be said to be justified. But why would we need the existence, in whatever sense, of a completed infinity, to prove there no no explanation for existence? That seems like a it would be an obscure reason for the phenomena.

2. Suppose, for the sake of argument, that the universe is mathematical structure "all the way down". In this scenario, the stuff that the physical universe is ultimately composed of is mathematical structure. Mathematical structure can effectively be modeled by the interrelations of sentences of a formal language. So suppose what the universe is, ultimately, is (the sentences of) some formal language T. Now, some formal languages can refer to themselves. Suppose T is one of these languages. Suppose also that T incorporates the necessity modal operator.

The point is this. If T is such that it proves the theorem
T implies necessarily-exists-T
then it would be a *physical* circumstance that the universe necessarily exists.

The problem is, this interesting property of T does not actually get us anywhere in justifying T's existence in the first place. C. pointed out to me there are many things that necessarily exist, if only they exist to begin with.

But again, the fact that this attempt at an explanation for existence fails, is not the same a proof there is no explanation for existence, for a sufficiently defined class of "explanations".