In this paper I revisit the plausibility of the all possible worlds hypothesis, and suggest an approach 1 using a Kantian starting point and a generalisation of the concept of all possible worlds that leads to an overcoming of the problem posed by Leslie. We obtain the bonus of a prediction that our world adheres to the simplest possible physical laws consistent with the presence of advanced, self-aware life.
The first section outlines the Kantian split and the version of the Principle of Sufficient Reason that points the way forward to the concept of all possible states, whose implications are explored in section two. Following this I use the idea of formal systems to illustrate how to apply a natural order to all states, which leads away from exploding cows and towards our lawful world. The final section clears up some loose ends and briefly touches on the problem of subjective experience.
Our starting point is effectively Kant's '... we can therefore have no knowledge of any object as thing in itself, but only in so far as it is ... an appearance' (Bxxvi). We can ascribe a certain kind of reality to subjective experience, in the sense it is all we can be immediately sure about, but we also suspect that it is only appearance - there must be a hidden, or underlying, reality responsible for it. We can reinforce this suspicion with the application of the Principle of Sufficient Reason, a variant of which will be important here: the Non-arbitrariness Principle (NAP) states that if it is logically possible for an entity to have existed in another state, or form, there must be a reason why that entity has its actual state, or form. We will henceforth use the term 'state' when referring to a distinguishable entity, with the proviso that 'copies' of states will also be mentioned later. We will also use the term 'u-reality' to refer to the underlying reality responsible for experiences, and how they are bound together. 2 (Later we will briefly consider and discard the possibility that for us the only appearances are elemental experiences.) Finally as part of our starting position, we note that we cannot yet say anything about the form or content of this hidden (underlying) reality. For Kant this is a consistent standpoint, for us the starting position only.
So, to account for the existence of our world, we have to explain how u-reality comes about, and also how it gives rise to our experience. We will concentrate on the first part, together with a derivation of how u-reality is involved in the particular forms of experience; this means that we will only briefly mention the remaining problem of how u-reality gives rise to the 'felt' nature of subjective experience. We should also note here a further restriction immediately imposed by NAP: u-reality cannot only give rise to our particular world of experiences, without good reason, and this is true even if we succeed in explaining how u-reality itself comes about.
We are given that something (experience, or whatever underlies it) exists. Ex nihilo creation tends to be controversial at best, but there is no particular reason to assume 'nothing' as a starting point anyway, and NAP can readily be generalised to demand such a reason. 3 Furthermore, NAP would require a reason for u-reality, or experience, to exist in any particular state. Intermediate explanations for u-reality or experience abound (a Supreme Creator, a set of physical laws, and so on), but we always fall short on a follow up reason to support the new entities invoked. In this paper we confront NAP directly: the only non-arbitrary starting point is not one particular state, or any particular combination, but rather all (logically) 4 possible states.
The first task must be to begin to examine what is meant by 'all possible states'. These states should surely include those describable by formal systems 5 ; other possibilities will be considered later. As discussed by Tegmark, formal systems can certainly describe a material universe, and one where life, and perhaps intelligent life, evolves. But we cannot yet say that their experiences would have a qualitative nature to them. Formal systems describing material universes can be relatively straightforward but laborious: most of the axioms could separately describe the nature of each elementary particle and their space-time co-ordinates; or they could be more compressed, where axioms refer to universal properties of elementary particles; or maximally compressed, where one would suspect that they would resemble the physicists' hypothetical 'Theory of Everything' (TOE), if such is attainable.
A second task that will be useful to undertake is to establish the number and relative frequency, if appropriate, of identical copies for each state. One version of each state intuitively seems closest to conforming to NAP, but other possibilities should be considered. Certainly a finite number of copies of each runs foul of NAP (why that particular number?), as does an infinite number of each, but at different relative frequencies (compare the ratio of all the odd numbers to all those divisible by four - why the particular ratio 2:1 for an analogous pair of states?). Fortunately we don't have to even consider the next case, where we have an infinite number of copies of each state but at the same relative frequency: in all later analyses it is effectively equivalent to the case of one version per state. The final possibility where we have an infinite number of occurences of each state, but at a completely indeterminate relative frequency, appears counter-intuitive for really existing states, and this feeling is reinforced once we reconsider the case of Leslie's exploding cows: to make any progress at all in attempting to refute the existence of all possible worlds, Leslie has to assume a determinate relative frequency of these worlds, if they were to exist, whether we are comparing ordered and disordered worlds, or copies thereof. And this approach appears inevitable for any all possible worlds/states hypothesis (except for certain exceptionally skeptical proposed solutions): if there is more than one possible future, our repeated samplings of that future give clear evidence that certain classes of future are more likely than others. (Lewis (p115-123) actually does attempt to use the indeterminate relative frequency of worlds to support his modal realism, citing lack of salient linear ordering to defend the indeterminacy, despite the paradox he accepts this will entail. We shall see that an adequate ordering is in fact possible for distinguishable states: by extension the nature and frequency of copies need not be so fundamentally different as to introduce indeterminacy. Lewis also thinks that the occurence of copies should not be a relevant factor (p119), and certainly one can quite reasonably claim that any complete indistinguishability renders meaningless their plurality.)
Having now established the initial plausibility of 'all possible states', and given the strong empirical support for at least one example of u-reality, it cannot be implausible to infer that one (at least) of the possible states is the example of the u-reality that is responsible for our experiences.
Even if this proposal is accepted, we still cannot impose the restriction that only our particular representation of u-reality exists, unless we can find some reason for this. Three further points: if there exist different examples of u-reality, there is nothing to say that some of them may not cause identical experiences (rough analogy: the supervenience of the mental on the physical); secondly, there is also nothing to say that there need be a close resemblance (say a one to one mapping) between elements of experience and hypothetical elements of u-reality; thirdly, just as one formal system can specify more than one material universe, it could well be the case that some examples of u-reality (that is, certain states) may give rise to more than one world of experiences.
If we permit ourselves room for some constructive speculation at this stage, we can, by way of parallel with the case of formal systems, consider the possibility that there might be one element (whatever this might be) of u-reality corresponding to the equivalent of one generalised electron, or else one for each electron that makes a contribution to our experiences as detailed by current physics. Note that we do not have to ontologically commit to electrons as components of a hypothetical physical world to do this - they are merely conceptual devices to enable us to begin to get a handle on how u-reality might be constituted. What we can say though, is that if this form of decomposition is valid, then the former example of u-reality is objectively more 'compressed' or 'descriptively shorter' than the one element per electron case.
The briefest of recaps at this point: we have the existence of all possible states, a substantial number if not all of which are modellable by formal systems, and some of which are examples of u-reality.
We need to find a way to impose some form of framework on this burgeoning anarchy of states and worlds. The method adopted here is to find a natural order for formal systems, and then use the same principles to establish a similar or identical one for examples of u-reality. It turns out that both orders only need be sufficiently well defined, objective and impartial to contrast with the intuitively constructed relative frequency of types of material universe that we tend to imagine from our own parochial, experiential standpoint.
We firstly observe that all formal systems are compressible into their constituent axioms and rules of inference. Any exceptions to the reducibility of rules of inference to modus ponens and substitutivity (see Tegmark) should not be relevant at our level of approximation, which leaves an ordering by number of axioms (in non-conjunctive form), or perhaps by the total number of symbols comprising the axioms in some standard minimal system. This provides a rough and ready ordering of all formal systems defining (inter alia) material universes. The appendix details how under such a scheme, and certain other similarly motivated schemes, we would expect more simply specified physical universes (that is, those requiring the least relevant axioms/units in their corresponding formal systems) to predominate.
Having speculated earlier on different types of elements for u-reality, we can now begin to sense how there might be 'compressed' examples of u-reality which also outnumber their less compressed counterparts. We note that in order to even consider the hypothesis of 'all possible states', we have to accept some uniquely distinguishing elemental factors for each state, excluding possible copies.
In order to give credance to the idea of differently compressed examples of u-reality, we can start by saying that if formal systems do uniquely represent all of this reality, then the compression concept is legitimate as part of an ordering strategy, and the analysis of the appendix is immediately applicable to u-reality. But we do not have to assume that each possible formal system corresponds to an example of u-reality; the following example illustrates why.
Consider the following three pairs in terms of the most direct possible correlation (a one
to one mapping of elements, if applicable) between the two sides separated by each 'AND',
where 'X' is either null if subjective experience is only part of a physical reality, or
else is the hypothetical extra ingredient needed to fully account for subjective experience.
(A) Some minimal formal specification of a physical world identical to the one that appears to us (perhaps a TOE or other minimal specification of physical laws and boundary conditions) AND a state that is an example of the u-reality behind the experience which appears to be of an identical physical world, but excluding X.
(B) A phase space (position and momentum) specification of the elementary physical particles comprising a world identical to the one that appears to us AND a state that is an example of the u-reality behind the experience which appears to be of an identical world, but excluding X.
(C) A formal specification of all the experiences of one or more communicable individuals AND a state that is an example of the u-reality for the identical experiences, which immediately appear as images and other sources of those experiences, but excluding X.
Note that the direct correlation between the two sides (even if it is only by number of elements, say) enables the right hand side of each pairing to be ordered if the left hand side is. We assume that the example of u-reality that is responsible for our experiences has some specification that most directly correlates with it. This is surely reasonable because it must hold in some form the information contained in our experience, so that our particular experience results. But we also have to cater for the strong likelihood, if all possible examples of u-reality exist, that different examples give rise to identical experiences (for example if different configurations or types of elements would create differences outside our experience), which must also be reflected in an identifying specification, so the right hand side of (A), and something at least crudely approximating to that of (B) and (C), and/or some other forms of non-minimal specification, should all exist: an accurate formal specification must uniquely define all distinguishing elements of a u-reality, not only those that only uniquely define any experience that might result. Where we have all possible such elemental combinations in examples of u-reality, as 'all possible states' implies, then the appendix indicates that the one directly correlated with the minimum formal specification (whatever that might be) will occur most frequently.
So we should inhabit a world most closely corresponding to a type (A) specification (leaving the 'X' problem aside). Note that (A) should be shorter than (C) (if it exists), if we bear in mind the complexity of thought, memory, perception, feelings and how they are bound together. If we try to deny a u-reality for the appearance of complex experiences and their bindings, by permitting only elemental experiences, then we are starting to deny what we are setting out to explain: the world as it appears to us. If we deny complex experiences, why not simple ones as well?
Once the idea of multiple examples of u-reality giving rise to separate but identical experiences is accepted, then it should be clear that there is no requirement for there to be identical numbers of indistinguishable experiences - it is the occurence of different examples of u-reality that determines the relative frequencies of experiences. The reasoning of the appendix shows not only that our experiences are most likely to be the product of an example of u-reality that is directly specified by something akin to a physicists' TOE, but also explains why alternative kinds of experiences like exploding cows do not occur for us: inductive success and Ockham's Razor for the physical realm become obvious consequences.
There remains the epistemological possibility that there could be states not accurately
describable by formal systems as defined here, including some not describable at all, or
else only by formal approaches so complex or alien that they are not subject to the
analysis of the appendix. I make three points in response.
(a) States must be distinguishable in some way to conform to the requirements of NAP (apart from the possible case of copies, which we covered in section 2). This implies some form of information based distinction, so, it would at least superficially appear, a fit to the logical unit analysis of the appendix via say, a binary digit unit.
(b) Only approximate fitting to the appendix analysis is needed for it to be applicable. Extraneous factors may cancel through, or be treated as 'extra' logical units (say extra axioms), so excluding these cases as being frequently represented among worlds.
(c) Even if there are states that do not conform to the analysis of the appendix in a way already described, it is still possible that they may fit to its 'simplest worlds are more numerous' conclusion for other reasons; or not swamp cases that do fit the analysis; or not be of the kind that are examples of u-reality for experiences of a self-aware nature.
The essential point of this article is that because u-reality cannot be said to directly resemble the content of our experience, there is no necessity for a 'one to one' type mapping between any hypothetical elements of both: elements of u-reality could be fewer or greater, and all of these possibilities are to be expected to occur in the 'all possible states' scenario demanded by NAP. From this, straightforward analysis using all possible combinations (a further NAP requirement) of the u-reality elements concludes that it is reasonable to suppose that we are most likely to be the outcome of an example of u-reality that is minimally representable: cows will not explode, pigs will not sprout wings. So the underlying reason for the success of induction can involve the existence of all possible worlds, but their relative frequency would relate directly to objectively accurate units of description of their underlying reality rather than to any kind of feature of experience.
There remains the question of how u-reality actually gives rise to the qualitative nature of experience. We can now say that there is good reason to suppose that some progress has been made in characterising hidden (underlying) reality, and that the essential 'qualia' problem is at least no worse than that confronted by any other approach that seriously tries to account for the regularities that we encounter in experience.
This appendix shows that where all states can be uniquely defined in terms of each logically possible combination of some form of elemental logical units (such as the axioms of a descriptive formal system), then out of all those states that include a physical universe, those collections of states directly specifiable by a minimum number of units relevant to that universe will be most frequently represented, compared to those containing the identical, non-minimally specified universes. It further shows that under the same conditions, identical copies of any given universe will occur more frequently than those of any other universe that is identical to the first save for an additional event that requires extra defining elemental logical units (such as an exploding cow).
Let the state that is our universe (or its closest material equivalent if we don't wish
to assume a physical universe) be minimally specified by n logical units.
Let a state that is a replica of our universe (or its material equivalent) be non-minimally specified by n+d logical units (d always positive).
Then, for all logically possible states specifiable by m units, where m is any number beyond n+d, it is evident that there will be far more possible combinations of units containing the particular minimal n units relevant to our universe (with the excess m-n units specifying combinations of entities and/or space-time regions elsewhere), than those containing the non-minimal n+d units. (Note that any statistical bias due to possible exclusions of particular combinations of units on logical inconsistency grounds will be insignificant: other axiom/unit combinations representing separate entities/space-times (or just rubbish) will always far outnumber these cases. Similar considerations apply to additional relevant universes within states.)
So, if we are in a physical universe, and states are built from all logically possible combinations of some form of elemental units, we will almost certainly be in a universe minimally specifiable in terms of these units. The same reasoning clearly applies to any physical universe.
The foregoing analysis can equivalently proceed where the second state referred to above is instead a replica of our universe except for the sudden explosion of a cow, with a more interesting conclusion:
Let the state that is our universe (or its closest material equivalent if we don't wish
to assume a physical universe) be minimally specified by n logical units.
Let a state that is a replica of our universe (or its material equivalent) except for an exploding cow event be minimally specified by n+e logical units (e always positive).
Then, for all logically possible states specifiable by m units, where m is any number beyond n+e, it is evident that there will be far more possible combinations of units containing the particular minimal n units relevant to our universe (with the excess m-n units specifying combinations of entities and/or space-time regions elsewhere), than those containing the particular minimal n+e units relevant to a universe that is identical to our own except for an additional exploding cow event. (The same comments apply to possible statistical bias as those given above).
So, with the same premises of occupancy of a physical universe and one state per unique combination of elemental logical units, we can say that for us paranormal events like exploding cows are not to be expected: we are most likely to be in a universe with minimal (but comprehensive) physical laws consistent with the existence of advanced self-aware life.
1. I would like to gratefully acknowledge the important part played by Russell Standish in overcoming a central difficulty to this approach, permitting the solution detailed in the appendix. His approach, which has close parallels with the one presented here, is identified in the references, and our correspondence is archived at http://www.escribe.com/science/theory July 1999.
2.The suggestion here of two distinct meanings to reality is deliberate, but will not be pursued because it is too great a digression, and not central to the main thrust of this paper. Crudely, subjective experience is real in the sense that it is indisputable evidence that something exists; u-reality is real in the sense that it is that something.
3.Our generalised NAP would be: if it is logically possible for an entity or the absence of an entity to have been in another state or form, there must be a reason why that entity has its actual state or form, or why there is no entity. It is the concept of logical possibility that is fundamental here - this is naturally generalisable to the absence of entities.
4.I would just like to clarify the use of the restriction '(logically) possible' here, by making three points. It cannot be denied that it is possible that there are states to which no accurate description can be applied, even in principle. This does not make the state illogical. If a description implies inconsistent statements (like 'round square'), then it cannot be a faithful description, so it is the description that is excluded as a possibility, not any state. Finally, 'all possible states' is not intended to include states that affect all other states - this would lead to logical impossibility at another level.
5.We will confine our definition of a formal system here to be: a consistent set of theorems that comprises axioms, and propositions derivable from them by inferential rules. They may well be non-finitary.
6.In each case of the right hand side of the three pairings, the ontological relevance of what u-reality gives rise to, if it is not yet subjective experience, can be a matter of taste. It need not have ontological significance, but if it is deemed to, then one may well wish to include these amongst our all possible states (that is, what would have been u-reality products but not yet, or ever, subjective experience). But if we do this, then because subjective experience arises from some of them, then functionally they are the same as u-reality itself - either the same analysis applies as for the right hand side of the three pairings, or they will all be deemed equivalent to (C), and similarly excluded as highly represented among worlds conducive to self-aware life. In a similar vein, if we wish all possible states to directly include examples of subjective experience itself (so attempting to remove any metaphysical split between subjective experience and u-reality as semantically different realities - a debatable point), then one can still substitute examples of subjective experience for the right hand side of pairing (C), and follow the logic to exclude this possibility as for the previous case equated with (C).
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6th July 2002