A more general treatment of the philosophy of physics and the existence of universes
aa r X i v : . [ phy s i c s . h i s t - ph ] J un A more general treatment of the philosophy of physics andthe existence of universes
Jonathan M. M. Hall ∗ University of Adelaide, Adelaide, South Australia 5005, Australia
Abstract
Natural philosophy necessarily combines the process of scientific observation withan abstract (and usually symbolic) framework, which provides a logical structure to thedevelopment of a scientific theory. The metaphysical underpinning of science includesstatements about the process of science itself, and the nature of both the philosophicaland material objects involved in a scientific investigation. By developing a formalism foran abstract mathematical description of inherently non-mathematical, physical objects,an attempt is made to clarify the mechanisms and implications of the philosophical toolof
Ansatz . Outcomes of the analysis include a possible explanation for the philosoph-ical issue of the ‘unreasonable effectiveness’ of mathematics as raised by Wigner, andan investigation into formal definitions of the terms: principles, evidence, existence anduniverses that are consistent with the conventions used in physics. It is found that theformalism places restrictions on the mathematical properties of objects that represent thetools and terms mentioned above. This allows one to make testable predictions regardingphysics itself (where the nature of the tools of investigation is now entirely abstract) justas scientific theories make predictions about the universe at hand. That is, the mathemati-cal structure of objects defined within the new formalism has philosophical consequences(via logical arguments) that lead to profound insights into the nature of the universe, whichmay serve to guide the course of future investigations in science and philosophy, and pre-cipitate inspiring new avenues of research. ∗ [email protected] ontents I -extantness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Introduction
The study of physics requires both scientific observation and philosophy. The tenants of sci-ence and its axioms of operation are not themselves scientific statements, but philosophicalstatements. The profound philosophical insight precipitating the birth of physics was thatscientific observations and philosophical constructs, such as logic and reasoning, could bemarried together in a way that allowed one to make predictions of observations (in science)based on theorems and proofs (in philosophy). This natural philosophy requires a philosophi-cal ‘leap’, in which one makes an assumption or guess about what abstract framework appliesmost correctly. Such a leap, called
Ansatz , is usually arrived at through inspiration and an inte-grated usage of faculties of the mind, rather than a programmatic application of certain axioms.Nevertheless, a programmatic approach allows enumeration of the details of a mathematicalsystem. It seems prudent to apply a programmatic approach to the notion of Ansatz itself andto clarify its process metaphysically, in order to gain a deeper understanding of how it is usedin practice in science; but first of all, let us begin with the inspiration.
In this work, a programme is laid out for addressing the philosophical mechanism of Ansatz. Inphysics, a scientific prediction is made firstly by arriving at a principle, usually at least partlymathematical in nature. The mathematical formulation is then guessed to hold in particularphysical situations. The key philosophical process involved is exactly this ‘projecting’ or‘matching’ of the self-contained mathematical formulation with the underlying principles ofthe universe. No proof is deemed possible outside the mathematical framework, for proof,as an abstract entity, is an inherent feature of a mathematical (and philosophical) viewpoint.Indeed, it is difficult to imagine what tools a proof-like verification in a non-mathematicalcontext may use or require.It may be that the current lack of clarity in the philosophical mechanism involved in apply-ing mathematical principles to the universe has implications for further research in physics. Forexample, in fine-tuning problems of the Standard Model of particle interactions (such as for themass of the Higgs boson
1, 2 and the magnitude of the cosmological constant ) it has been spec-2lated that the existence of multiple universes may alleviate the mystery surrounding them, in that a mechanism for obtaining the seemingly finely-tuned value of the quantity would nolonger be required- it simply arises statistically. However, if such universes are causally dis-connected, e.g. in disjoint ‘bubbles’ in Linde’s chaotic inflation framework,
9, 10 there is a greatchallenge in even demonstrating such universes’ existence, and therefore draws into questionthe rather elaborate programme of postulating them. Setting aside for the time being the useof approaches that constitute novel applications of known theories, such as the exploitation ofquantum entanglement to obtain information about the existence of other universes, a moreabstract and philosophical approach is postulated in this paper. Outside our universe, one is at a loss to intuit exactly which physical principles continue tohold. For example, could one assume a Minkowski geometry, and a causality akin to our cur-rent understanding, to hold for other universes and the ‘spaces between’, if indeed the universesare connected by some sort of spacetime? Indeed, such questions are perhaps too speculativeto lead to any real progress; however, if one takes the view of Mathematical Realism, whichoften underpins the practice of physics, as argued in Section 3, and the tool of Ansatz, onecan at least identify mathematical principles as principles that should hold in any physicalsituation- our universe, or any other. This viewpoint is more closely reminiscent of Level IVin Tegmark’s taxonomy of universes. One may imagine that mathematical theorems andlogical reasoning hold in all situations, and that all ‘universes’ (a term in need of a carefuldefinition to match closely with the sense it is meant in the practice of physics) are subject tomathematical inquiry. In that case, mathematics (and indeed, our own reasoning) may act as a‘telescope to beyond the universe’ in exactly the situation where all other senses and tools aredrawn into question.To achieve the goal of examining the process of the Ansatz- of matching a mathematicalidea to a non-mathematical entity (or phenomenon), one needs to be able to define a non-mathematical object abstractly, or mathematically. Of course, such an entity that can be writtendown and manipulated is indeed not ‘non-mathematical’. This is so in the same way that, indaily speech, an object can be referred to only by making an abstraction (c.f. ‘this object’,‘what is meant by this object’ ‘what is meant by the phrase ‘what is meant by this object’3, etc.). This nesting feature is no real stumbling block, as one can simply identify it as anattribute of a particular class of abstractions- those representing non-mathematical objects.Thus a rudimentary but accurate formulation of non-mathematical objects in a mathematicalway will form the skeleton outline for a new and fairly general formalism.After developing a mathematics of non-mathematical objects, one could then apply it toa simple test case. Using the formalism, one could derive a process by which an object isconnected or related somehow to its description, using only the theorems and properties knownto hold in the new framework. The formalism could then be applied to the search for otheruniverses, and the development of a procedure to identify properties of such universes. Indoing so, one could make a real discovery so long as the phenomenological properties are notintroduced ‘by hand’. This follows the ethos of physics, whereby an inspired principle (orprinciples) is followed, sometimes superficially remote from a phenomenon being studied, butwhich has profound implications not always perceived contemporaneously (and not introducedartificially), which ultimately guide the course of an inquiry or experiment.There is an additional motivation behind this programme beyond addressing the mech-anism of the Ansatz, which is to attempt to clarify philosophically Wigner’s ‘unreasonableeffectiveness’ of mathematics itself. It is the hope of this paper to identify this kind of ‘ef-fectiveness’ as a kind of fine-tuning problem, i.e. that it is simply a feature that naturally arisesfrom the structure of the new formalism. In the special situation where one uses mathematical constructs exclusively, the type of evi-dence required for a new discovery would also need to be mathematical in nature, and testingthat it satisfies the necessary requirements to count as evidence in the usual scientific sensecould be achieved by using mathematical tools within the new framework. To explain how thismight be done, consider that evidence is usually taken to mean an observation (or collection ofobservations) about the universe that supports the implications of a mathematical formulationprescribed by a particular theory. Therefore, it is necessary to have a strict separation betweenobjects that are considered ‘real/existing in the universe’, and those that are true mathematicalstatements that may be applied or projected (correctly or otherwise) onto the universe.Note that, for evidence in the usual sense, any observations experienced by the scientist4re indeed abstractions also. For example, in examining an object, photons reflected from itssurface can interact in the eye to produce a signal in the brain, and the interpretation of such asignal is an object of an entirely different nature to that of the actual photons themselves- muchobservational data is, in fact, discarded, and most crucially, the observation is then fitted intoan abstract framework constructed in the mind. In a very proper sense, the more abstract is themore tangible to experience, and the more material is the more alien to experience. Therefore,it seems reasonable to suggest that a definition of evidence in familiar scientific settings isalready equivalent to evidence in an entirely mathematical framework; in fact, the distinctionbetween the two is purely convention.
In the ‘world of ideals’ (as developed from the notion of Plato’s ‘universals’, rather thanBerkeley’s Idealism ) there are certain abstract objects (‘labels’ or ‘pointers’), which refer tomaterial objects. The Ansatz arises by guessing and then assuming a particular connectionbetween those pointers and other abstract objects. This allows material objects to be entirelyobjective, (to avoid solipsism), but also entirely subservient in some sense, to abstract ob-jects. An observer can only indirectly interact via interpretation. Thus, the abstract affects theabstract, and abstract the physical.One might argue, contrarily, that entities existing in the abstract mind are altered via naturalor material means. Certain mental states are invoked upon interpretation of the empirical,regardless of the existence of patterns, which are abstractions (and that this would be trueeven if the universe were ‘unreasonable’- not in general amenable to rational inquiry). It isthe point of view expounded in this treatise, however, that it is not true that material objectscan interact so directly with other material objects, but only indirectly, since the interactionsthemselves would otherwise need to be materials. Yet an interaction is necessarily an abstractlink (i.e. it has to follow some pattern, rule or law) even when at rest. That is, the notion of‘interaction’ is necessarily abstract. It is the feature of positing only indirect relation amongphysical objects that takes the form of a view opposite to that of epiphenomenalism.
16, 17
The5ifference in viewpoint may appear to hinge on the semantics of the terms ‘interaction’ and‘abstract’, but the goal is simply to characterise the oft-proved successful methods of physics as already applied in practice, through a particular choice in philosophy. Our goal is, by simplyidentifying (and thus labelling) the salient features of the philosophy, an investigation into themore general (and philosophical) aspects of the practice of physics can be conducted. On theother hand, the discussion of the soundness of such a philosophy on other grounds constitutesa tangent topic, and the presentation of a complete enumeration of various emendations orcontrary viewpoints on the choice of philosophy will be left for further investigation.
The evolution of metaphysics from rudimentary Cartesian Dualism to that proposed byBohm demonstrates the usefulness of a mathematical viewpoint in clarifying an enrichingphilosophical ideas as they pertain to physics, and the universe as a whole. Abstract relation-ships are centralised, and underlying principles of matter, rather than a catalogue of the im-mediate properties, are interpreted to have the greater influence in accessing the fundamentalnature of the physical world. The shift in perspective is that the simple and elegant descrip-tions of a physical system are those which incorporate its seemingly disparate features into anintegrated whole. One then goes on to postulate a relationship between the physical object inquestion and the machinery of its abstract description.Similar philosophical views have found success in the field of neuroscience, such as thework of Dam´asio in characterising consciousness.
20, 21
Instead of treating the mind and body asseparate entities, one postulates an integrated system, whereby mechanisms in the body, suchas internal and external stimuli, result in neurological expressions such as emotion. Emotionand reason are thus brought together on equal footing, since both actions are the result of com-paring and evaluating a variety of stimuli, including other emotions, to arrive at a response.That is, from a modern perspective, just as relationships between physical objects are funda-mental in characterising the intangible properties of their whole, it is the abstract relationshipsbetween faculties in the body and brain, such as interactions and stimuli, that characteriseconsciousness.The identification of phenomena with abstract descriptions, such as behaviour and interac-tions, was formalised in Putnam and Fodor’s Functionalism.
22, 23
Functionalism provides the6bility to consider indirect or ‘second order’ explanations for the nature of objects. UnlikePhysicalism, which identifies the nature of objects with the instances themselves that occur inthe real-world, Functionalism entails the generalisation of the objects in terms of their func-tional behaviour. These more general classes of object are identified by the features that allrelevant instances of the object have in common, and so the nature of the object becomes moreubiquitous, even if more abstract. This may be nothing more than a semantic shift, in caseswhere one is at liberty to allow the definitions of certain abstractions more scope as needed(such as pain or consciousness), so that they more closely match their use in daily humanendeavours. Further abstractions, such as ‘causes’, an integral part of many areas of science,follow naturally.Taken on its own, Functionalism represents a deprecated metaphysic, insufficient for acomplete account of internal states of a physical system, and is therefore commonly em-ployed simultaneously with another metaphysic (such as Physicalism). By not providing fora ‘real’ or material existence independent of an object’s functional behaviour, a bare Func-tional philosophy is not wholly suitable for describing the process of physics, which involvesidentifying material objects whilst projected upon them an Ansatz from some (abstract) theory.As an example of the shift in perspective that Functionalism brings, consider the followingscenario of an abstract entity based on real-world observations, such as an emotion/state ofaffairs, etc., whose cause is in want of identifying. Let this object be denoted a ‘feature’.Instead of the cause simply being identified specific phenomenon, or a mechanism based onthe real world, the cause is characterised by an abstract object, G , which represents a collectionof pointers to the relevant parts of the mechanism. If we posit, for the moment, an abstractinterpreting function, i : R → A , relating the real-world, R , to the world of ideals, A , andsimilarly, a pointer function, r : A → R , one can establish a relationship between the causeand the feature. For a mechanism, M , a set of features, F , and an element g ∈ G , define M = r ( g ) , which resides in r ( G ) , and then i ( r ( g )) = i ( M ) = F . In this (fairly loose)symbolic description, the features and the cause are related in the statement i ( r ( G )) = F . The‘reverse-epiphenomenal’ philosophy, akin to Interactionism, is to realise that the relation itselfbetween the two entities is an abstract one, whose attributes it will benefit us to characterise.7 Formalism
In this section, a new formalism is outlined in order to capture the essential features of thephilosophical problem at hand. A set of very general abstraction operators are defined, suchthat they may act upon each other in composition. By introducing another special kind ofoperation, the projection P , general objects may be constructed such that they, at the outset,obey the basic principles expected to hold for objects and attributes used in a recognisablecontext, such as in language. To avoid semantic trouble, when one is free to assume or assigna property in a given context, the choice made is that which is most closely aligned with‘what is commonly understood’ by a term. Note that other definitions are (unless logicallynon-viable) completely acceptable also- it is simply a choice of convenience to try to alignthe concepts chosen to be investigated with those of a language (such as a spoken or writtenlanguage). In fact, it is judicious to do so, given that any philosophical problems one may wishto address are usually cast in such a language. Though Cantor’s Theorem prohibits a consistent scheme classifying the space of all suchabstract entities, (as echoed by Schmidhuber ), the abstractions considered here are limitedto a set, W , of ‘world objects’, representing a set of a specific type of object with certain(very general) properties. Very little mandatory structure for the objects, w , inhabiting W isassumed, and they may be represented by a set, a group or other more specific mathematicalobjects. Thus, one may define W in a consistent fashion using an appropriate axiomization,such as that of ZFC, so long as none of the properties of the formalism is contravened.In Section 4.2.3, the properties of the real-world objects w are clarified. Some basic rules ofcomposition are assumed, but the spaces mapped-into in doing so are simply definitions ratherthan theorems; the tone of this work is not to impose any more specific details on the frame-work than is required in order to fulfill the aforementioned goals, namely, the construction ofa mathematical-like theory in order to address the mechanism of Ans¨atze. Other mathematicalformulations for obtaining general information about a system, such as Deutsch’s ConstructorTheory , take a similar approach in determining suitable definitions for objects required forcertain tasks in an inquiry. 8 .1.1 The labelling principle Firstly, the projection operator, P , obeys what shall be known as the labelling principle : P ◦ P = P . (1)A direct consequence of the labelling principle is that P has no inverse, P − . Proof:
Assume P − exists. Then: P ◦ P − = P6 = 1 . Therefore, P − DNE.where is the identity operator. An equivalent argument follows for an operator P − actingon the left of P .The projection operator may be applied to a world object w , and the resultant form, P ( w ) ,constitutes a new object, inhabiting a different space from that of w . Firstly, the consequencesof the lack of inverse of P directly affect the projected space P W , which will be interpretedphilosophically in the next section. Suffice to say, the judicious design of P W lends itself toa particular view of abstractions, whereby very little information can be gained from an objectin the real world directly , as expostulated regarding the definition of evidence, in Section 2.2. The notion of ‘abstraction’ is codified by postulating a certain operator A , which may acton objects residing in a space W , much like the projection operator. It will have, however,different properties to those of the projection operator. Using the abstraction operator, oneis able to go ‘up a level’, ( A ◦ A ( w ) = A ( w ) ), and establish new features of the object w .The sequential application of the abstraction operator creates a chain, in a reminiscent fashionto that of (co-)homologies; however, the properties of the abstraction operators are moregeneral. 9ne may define the abstraction classes, Ω i , as ∈ Ω , (2) A ∈ Ω , (3) A ◦ A ∈ Ω , (4)... (5)For the moment, the properties of the classes are no more extensive than, say, a collection ofelements (the operators). The range of A , namely Ω ≡ {A i } , is a class of any type of A .(What is meant by the set symbols { } will be discussed in Section 4.2.2.) It follows that, for w ∈ W , Ω ( w ) = w ∈ W .The sequential actions of the projection and the abstraction operators do not cancel eachother, and it can be shown that A ◦ P ( w ) = w : Theorem 1.
A ◦ P ( w ) = w .Proof: Assume
A ◦ P ( w ) = w . Then: A ◦ P ∈ Ω ⇒ A ◦ P = 1 ⇒ A = P − , DNE . Thus,
A ◦ P ( w ) ∈ Ω ( P ( w )) , for some w ∈ Ω ( w ) .Consider the complex of maps: Ω → Ω → Ω · · ·↓ց χ ↓ ↓P Ω P Ω P Ω · · · It is possible to design a function, χ ≡ P ◦A , which exists, and will be utilised in Section 4.2.3.However, it is important to note that our constraints on P do not allow the construction of afunction ϕ : P Ω → P Ω , or any other mapping between projections of abstraction classes.Mathematically, ϕ would have the form P ◦ A ◦ P − , which does not exist; but philosophically,10t is supposed of P ( w ) that it encode the behaviour of the actual world object meant by w . Oneinterprets the ‘non-mathematical’ object, P ( w ) ∈ P W , knowing that the fact it is necessarilyan abstraction is already encoded in the behaviour of P by construction. Note that the failureto construct a function ϕ as a composite of abstraction operators and their inverses does notmean that such a mapping does not exist. However, the principal motivation for supposing thenon-existence of such a function is to encode the features one expects in an abstract modellingof non-abstract objects.This view of the general structure of abstraction is an opposite view to the metaphysicof epiphenomenalism,
16, 17 in that, colloquially speaking, changes to ‘real-world’ objects canonly occur via some abstract state, and it does not make sense to set up a relationship betweennon-mathematical entities and insist that such a relationship must be non-abstract.Different instances of w cannot be combined in general, but their abstractions can be com-pared by composition. The objects A ( w ) and A ( w ) can also be defined to be comparable,via use of the commutators , in Section 4.1.3.In considering the properties of Ω ( W ) , one finds that Ω (Ω ( W )) = W (6) ⇒ Ω ◦ Ω = Ω . (7)Generalising to higher abstraction classes, we find the level addition property : Ω i ◦ Ω j = Ω i + j . (8)The non-uniqueness of A means that many abstract objects can describe an element of W . Ingeneral, A i ◦ A j ( w ) = A j ◦ A i ( w ) , so Ω ( A i ( w )) = A i (Ω ( w )) , though both Ω ( A i ( w )) and A i (Ω ( w )) are in Ω ( w ) . The set Ω includes the identity operator , but also containselements constructed from abstractions and other inverses, e.g. A − L,i ◦ A j ( w ) , to be discussedin Section 4.1.4. 11 .1.3 Commutators Define the commutator as an operator that takes the elements of the i th order abstraction space,acting on a world object w , to the same abstraction space acting on another world object w , Φ iW =Ω ( W ) : Ω i ( w ) → Ω i ( w ) . The subscripts on the commutator symbol indicate the spaceinhabited by the objects whose abstractions are to be commuted, and the labels of the discardedand added objects, respectively. The superscript denotes the order of abstraction (plus one) atwhich the commutation takes place. As a simple example, Φ W, , A ( w ) = A ( w ) . In general,let Φ W,i,j A ( w i ) = A ( w j ) , (9) Φ ( w ) ,i,j A k ◦ A i ( w ) = A k ◦ A j ( w ) , (10)and Φ b +1Ω b ( w ) ,i,j A ◦ · · · ◦ A bi ◦ · · · ◦ A | {z } a ( w ) = A ◦ · · · ◦ A bj ◦ · · · ◦ A ( w ) . (11)In order to construct the new object from the old object, one must successively apply ‘inverse’operations of the relevant abstractions to the left of the old object (as discussed in the nextsection), and rebuild the new object by re-applying the abstractions. This is not possible ingeneral, where objects may include operators that have no inverse, such as the projectionoperator. Define the left inverse A − L ◦ A = 1 , (12)or more generally, A − L ◦ A ( w ) = w . A right inverse is not assumed to exist in general, whichwill be important in establish certain kinds of properties in Section 4.3.As a generalization, one can define a chain of negatively indexed abstraction classes Ω −| i | .The level addition property can accommodate this scenario. The elements of Ω are populatedby objects of the form A − L,i ◦ A j , or A i ◦ A − L,j ◦ A k ◦ A − L,n , etc. That is, successive abstractionsand inverses in any combination such that the resulting abstraction space is order zero. Thisincludes the identity operator. 12y using the left inverse, it can be shown that the following theorem holds (which comple-ments Theorem 1), as a consequence of the choice of the philosophical properties of P : Theorem 2.
P ◦ A ( w ) = w .Proof: Assume Ω − = Ω − , and P = A − L . Then: P ◦ P ( w ) = A − L ◦ A − L ( w )= P ( w ) ( Eq. (1) )= A − L ( w ) ⇒ A − L ◦ A − L = A − L ⇒⇐ As a corollary, it also follows that
P ◦ A ( w ) = A ◦ P ( w ) . (13)In summary, this non-commutativity property of the abstraction operators in Eq. 13 isan important consequence of the reverse-epiphenomenal philosophical motivation behind thelabelling principle, and it will be the starting point for the construction of the generalisedobjects in Section 4.2.3. In order for a successful description of the relationships among different objects in general, adefinition of the mapping between objects of the form AP ( w ) and AP ( w ) is sought.Up until now, maps of the following types have been considered: • w → A ( w ) , where the notion of the map itself is an abstraction of A of the form: A map ◦ A ∈ Ω ( w ) ; • A ◦ A ( w ) → w , where the map is now of the form A map ◦ A ◦ A ∈ Ω ( w ) , and • A ( w ) → w , identical to the first case, except that the direction is reversed. By convention,let the definition of map be chosen such that the direction of mapping is not important, butsimply the relationship between the two objects. Therefore, the map is always taken such thatit exists in a space Ω i ≥ , that is, we define it as the modulus of the map. Note that each of thesemaps is constructed as a composite of other abstractions, and shall be denoted literal maps .13t cannot, in general, be attested that there is no such mapping between some w and w in W . However, one is free to define to exist a non-composite type of map, denoted auxiliarymaps , which relate objects of the form AP ( w ) and AP ( w ) , or APA ( w ) and APA ( w ) ,etc. The map itself exists in the space Ω ( w ) , that is, the application of what is meant by themap does not change the order of the objects to which it is applied.Using the concept of auxiliary maps, a generalized version of the commutator, denoted ˆ Φ ,may be defined in a way not possible for the na¨ıve construction of the commutator ˆ Φ ( w ) ,i,j A PA i ( w ) = A PA j ( w ) . (14)The auxiliary commutator will be important for the neat formulation of the general conditionfor a specific kind of existence, in Sec 4.3. One may define the total set of an object w as S ( w ) = ∞ [ i =0 Ω i ( w ) , (15) S ( P ( w )) = ∞ [ i =0 Ω i ◦ P ( w ) . (16)The S symbol denotes the range of the multiple objects, indexed by an integer i , j , etc. overwhich the set should be specified. Here, one postulates a certain supposition of physics , that P S ( w ) spans at least W . Supposition 1. P is surjective, i.e. ∀ w i ∈ W, ∃ σ i ∈ S ( w i ) such that P ( σ i ) = w i . This condition represents the ‘working ethos’ of the practice of physics. It is expected thatthere is some abstract description, however elaborate or verbose, to describe every real-worldobject.
In this section, the tools introduced in the preceding section will be used to define more generalrelationships between objects. In addition, a generalised object notation will be defined, andthe nature of the real-world objects w will be clarified.14 .2.1 Ans¨atze An Ansatz is formed by adding a structure, or additional layer of abstraction, and imposingit on what one considers ‘the real world’. Cast in the new framework, this is simply thesuccessive application of an abstraction and a projection operator upon some object Z i ( w ) = P ◦ A i ( w ) . (17)As a consequence of the labelling principle in Eq. (1), the Ansatz of an object, w , and whatis meant by the real-world object corresponding to w , cannot be related directly, recalling that ϕ : P ◦ A ( w ) → P ( w ) does not exist. This simply means that there is no way of generatingthe label of an object directly from the object itself; it is a free choice. The Ansatz is akin tothe statement: ‘let this new label (with possible additional information) be linked to the realobject’. The notion of Ansatz, particularly the special examples considered in Sec. 4.3, will beuseful in understanding the formal structure of existence as it pertains to real-world objects. A collection or set of objects, { w i } (indexed by integer i ), in the formalism, is simply treatedas an abstraction, A set , used in conjunction with a commutator: { w , . . . , w N } = N [ j =1 A set ◦ Φ W,i,j w i . (18)By further imposing that there should be a relationship (other than the collection itself) amongthe objects w i , the addition information is simply added by another abstraction, say, A ′ , andwhat is meant be this particular relationship is simply: r ( w i ) = P A ′ ◦ N [ j =1 A set ◦ Φ W,i,j w i . (19)A relationship in general, r = A ′ ◦ Φ A ∈ Ω , does not have to specify that there be aparticular relationship among objects w i . 15 xample: In identifying a ‘type’ of object, such as all objects that satisfy a particularfunction or requisite, one means something slightly more abstract than a particular instanceof an object itself. In order to produce a notion similar to the examples: ‘all chairs’ or ‘allelectrons’, one must construct a relationship among a set of w i ’s, each of which is a set of, say, n observations: w A , w B , . . . ∈ W ′ ⊂ W . Let w A = N A [ j =1 A set ◦ Φ W,i,j w ( A ) i , (20) w B = N B [ j =1 A set ◦ Φ W,i,j w ( B ) i , etc. (21) w ( A ) i , w ( B ) i , . . . ∈ W. (22)Then, { w A , w B , . . . } = n [ j ′ =1 N j ′ [ j =1 A set ◦ Φ W ′ ,i ′ ,j ′ A set ◦ Φ W,i,j w ( i ′ ) i . (23)The relationship itself that constitutes the ‘type’ thus takes the form: r type = A ′ ◦ n [ j ′ =1 N j ′ [ j =1 A set ◦ Φ W ′ ,i ′ ,j ′ A set ◦ Φ W,i,j w ( i ′ ) i ∈ Ω ( W ) (= Ω ( W ′ )) , (24)for some A ′ , and W ′ = Ω ( W ) † . This formula represents the notion of ‘types’ of object in afairly general fashion, in order to resemble as closely as possible the way in which objects aretypically characterised and subsequently handled in the frameworks of language and thought. Up until now, discussion of the nature of the real-world objects { w i } ∈ W has been avoided.However, in order to incorporate them in the most general way into the framework of theabstraction algebra, one may posit that the real-world objects are simply a chain of successiveprojection or abstraction operators. In general, one can construct ‘sandwiches’, such as: A ◦ · · · ◦ A i ◦ P ◦ A ◦ · · · . (25) † The form, W ′ = Ω ( W ) , makes sense, in that the notion of ‘being a subset’ is a single-level abstraction,residing in Ω . P (anything) ∈ Ω P (anything). Due to the corollary in Eq. (13), the projection operatorscannot be ‘swapped’ with any of the abstraction operators, and so the structure of the object isnontrivial. Let c denote a generalised object, living in the space: c ∈ Ω i P Ω i P · · · P Ω i n ( W ) ≡ C i i ...i n W . (26)The space W here could stand for any other general space constructed in this manner, notnecessarily the space inhabited by c itself; thus Eq. (26) is not recursive as it may initiallyappear. Because the internal structure, so-to-speak, of c contains a collection of a possiblemany abstractions, it may be expressed in terms of type. Here are two examples:Let c (1) = P{P ◦ A ( w ) , . . . , P ◦ A n ( w ) } ∈ Ω P Ω P Ω ( w ) = C w (27) = P n [ j =1 A set ◦ Φ ,i,j PA i ( w ) . (28)Or c (2) = P{P ( w ) , . . . , P ( w n ) } ∈ Ω P Ω P Ω ( W ) = C W (29) = P n [ j =1 A set ◦ Φ W,i,j P ( w i ) , (30)where, in the first case, Ω ( w ) ∈ Ω ( W ) .Consider the behaviour of an Ansatz Z = P ◦A Z acting on a generalised object c ∈ C i ...i n W : Z ◦ c = P ◦ A Z ◦ A ◦ · · · ◦ A i ◦ P ◦ · · · . (31) Z maps c into a space C i +1 ...i n W . If we define rank( c ) = n , then rank( Z ◦ c ) = n + 1 . Note thatthe rank of Z ∈ C c can also be read off easily: rank( Z ) = 2 . Objects of the form of Z are theprincipal rank Ans¨atze. Note that other rank objects besides Z exits, such as objects of theform P ◦ A ◦ A .A more general description of Ans¨atze also exist, analogously to the generalised objects.By constructing an object of the form: χ = A ◦ · · · ◦ A | {z } j ◦ P ◦ A ◦ · · · ◦ A | {z } j ◦P ◦ · · · , that is,for an object residing in a space C j j ...j m c , the composition χ ◦ c lies in C j ... ( j m + i ) ...i n W , whichis of rank n + m − . 17y convention, an Ansatz must contain a projection operator. Therefore, there is no rank Ansatz, and we arrive at our general definition of Ansatz:Any object acting on c , with a rank > , is an Ansatz. (32)In addition, there are no objects with rank ≤ . Proof:
Let ξ exist such that rank( ξ ) ≤ , and c ∈ C i ...i n W . Then:rank( ξ ◦ c ) = rank( ξ )+rank( c ) − < rank( c ) = n ⇒ ξ ◦ c ∈ C i ′ ...i ′ n − W ⇒ ξ is of the form X ◦ k [ j =1 P − ◦ i j [ i =1 A − i,L , where rank( X ) ≤ k ⇒ ξ DNE, for any X .For example, in the case k = 1 , X is a rank Ansatz, and ξ = X ◦ P − ◦ S i i =1 A − i,L , suchthat ξ ◦ c ∈ C i ...i n W ∼ = C i ′ ...i ′ n − W . The rank of ξ ◦ c is n − , and the rank of ξ is therefore .Because of the usage of the operation P − , such an object is inadmissible.We would like to use the Ans¨atze to investigate the properties of generalised objects. How-ever, there are a variety of properties in particular, discussion of which shall occupy the nextsection. The notion of ‘existence’ is a key example that urgently requires clarification, andit will be found that such a property (and those similar to it), when treated as an abstraction,must have additional constraints. I -extantness Firstly, one must make a careful distinction between what is meant by ‘existence’ in the senseof mathematical objects, and in the sense of the ‘real world’. In the former case, one mayassume that an object exists if it can be defined in a logically consistent manner. In the lattercase, it is a nontrivial property of an object, which must be investigated on a case-by-case basis,and the alternative word ‘extantness’ will be used in order to avoid confusion. The goal of theformalism is to relate the two terms- that an object’s extantness can be tested by appealing tothe existence (in the mathematical sense) of some construction.18e begin by assuming that extantness is an inferred property of an object, and thus addedby an Ansatz. Define its abstraction, A E , such that an object c = P ◦ A E ( w ) is extant ifsuch a construction exists; i.e. c is extant if it can be written in this form (for any w ). For c = P ◦ A E ◦ A ◦ · · · ∈ C i ...i n W ; the operator A E must occur in the left-most position of allthe abstractions in c . Clearly then, it must not necessarily be the case that P ◦ A E ( w ) exists, ifthis abstraction is to be equivalent to how extantness (or existence in the conventional sense)is understood. Example:
Consider the object
P ◦ A E (1) , where is the abstraction identity. A E (1) = A E is the extantness itself, and P ◦ A E is ‘what is meant’ by extantness, which is itself extant- itis the trivial extant object.This leads us to the first property of A E , that its right inverse, A − E,R , does not exist, asanticipated in Section 4.1.4.
Proof:
Consider c = P ◦ A ◦ · · · ( w ) such that it is not extant. Assume A − E,R exists also. Then: c = P ◦ A E ◦ A − E,R ◦ A ◦ · · · ( w )= P ◦ A E ( w ′ ) , where w ′ ≡ A − E,R ( w ) ⇒ c is extant. ⇒⇐ It is not necessary at this stage to suppose that the left inverse of A E does not exist either;however, if that were the case, then A E would share a property with P , in lacking an inverse.The two are unlike, however, in that A E ◦ A E = A E . Since A E lives in a restricted class ˜Ω ⊂ Ω , indicating the additional constraint of lacking an inverse, then the level additionproperty of Eq. (8) means A E ◦ A E ∈ ˜Ω ˜Ω . A further consequence of the non-existenceof A − E,L is that the statement A E ( a ) = A E ( b ) does not mean that a = b . One may interpretthis as the fact that two abstractions may simply be labels for the same extant object. Note thatthe definition of the literal commutator requires the existence of an inverse of each abstractionoperator that occurs in sequence to the left of the object being commuted, but that is not thecase for auxiliary commutators.Recalling the supposition of physics, that P S ( w ) spans at least W , a further clarificationmay now be added: 19 upposition 2. All extants have Ans¨atze, but not all elements of P ( W ) or P S ( w ) are extants. From the point of view of Mathematical Realism, one would argue that projected quanti-ties,
P ◦ · · · , are those which are ‘real’ (and not dependent on their extantness), since such adefinition of ‘real’ would then encompass a larger variety of objects, regardless of their par-ticular realisation in our universe. Such a semantic choice for the word ‘real’ seems to alignbest with the philosophy of Mathematical Realism. Nevertheless, it is still important to have amechanism in the formalism to determine the extantness of an object.Although extantness has been singled out as a key property, a similar argument may bemade for the truth of a statement, whose abstraction can be denoted as A T . Like extantness,the object P ◦ A T ( w ) may not exist for every w , and the trivially true object is P ◦ A T (1) . Letus label all properties of this sort, ‘ I -extantness’, since their enumeration in terms of commonwords is not of interest here. For any I -extant abstraction A I , we call ˜ C i ...i n W the restrictedclass of generalised objects, c I .A formula is now derived, which is able to distinguish between objects that are I -extant andthose that are not, by virtue of their mathematical existence. Consider the case that P ◦ A I ( w ) exists, but P ◦A I ( w ) does not. w must contain an addition property, A I ′ , that is not present in w . Unlike A I , it is not required that A I ′ occur in a particular spot in the list of abstractions thatcomprise w . Nor is there a restriction in the construction of an inverse, which would prevent acommutator notation being employed. Let w be represented by a collection of objects definedby: w = {A I ′ ◦ A ◦ · · · , A ◦ A I ′ ◦ · · · , etc. } . That is, w takes the form of a set of generalisedobjects, c , but for the replacement of an operator, A , with A I ′ . It is important to note thatthe A I ′ that distinguishes w from a non-extant object, such as w , is particular to w . Foran object c to be extant, it would have to include an abstraction A cI ′ , specific to c ; otherwise,any object related to c in any way would also be extant, which doesn’t reflect the behaviourexpected of extant objects in the universe.In commutator notation, one would need to write out a geometric composition of the form w = A set i − [ m =0 Φ i − m +1Ω i − m ( w ) ,m +1 ,I ′ H ◦ P ◦ i − [ m ′ =0 Φ i − m ′ +1Ω i − m ′ ( w ) ,m ′ +1 ,I ′ H ◦ · · · , (33) = A set ◦ H ◦ n [ p =2 P i p − [ m =0 Φ i p − m +1Ω ip − m ( w ) ,m +1 ,I ′ H p , (34)20or c = H ◦ P ◦ H ◦ · · · . A more elegant formula may be defined simply in terms of c itself, without the need of introducing new symbols, H , . . . , H n . One can achieve this usingauxiliary commutators w = n ˆ Φ ( P nj =1 i j )+1 C i ...inW , , I ′ c, ˆ Φ P nj =1 i j C i − ...inW , , I ′ c, . . . , ˆ Φ ( P n − j =1 i j )+1 C ...inW , i , I ′ c, . . . , ˆ Φ ( P n − j =1 i j )+1 C ...inW , i , I ′ c, . . . o (35) = i − [ m =0 A set ◦ ˆ Φ ( P nj =1 i j )+1 − m C i − m...inW , m +1 , I ′ c ∪ i − [ m ′ =0 A set ◦ ˆ Φ ( P n − j =1 i j )+1 − m ′ C i − m ′ ...inW , m ′ +1 , I ′ c ∪ · · · (36) = n [ p =1 i p − [ m =0 A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 − m C i ...ip − m...inW , m +1 , I ′ c. (37)It follows then, that a generalised object that is I -extant takes the form c I = P ◦ A I ◦ n [ p =1 i p − [ m =0 A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 − m C i ...ip − m...inW , m +1 , I ′ c, (38)where c I is of the form P ◦ A I ( w I ) . This is a powerful formula, as it represents the conditionfor I -extantness for a generalised object, c . Note that it would be just as correct to define c I as an element of a set characterised by the righthand side (i.e. using ‘ ∈ ’ instead of ‘ = ’), butbecause the notion of a ‘set’, A set , is simply an element of Ω , it can be incorporated into thegeneral form of C i ...i n W .One might wonder how to relate the properties of a proof (i.e. verifying the truth of a state-ment) with the existence of an abstraction, A T ′ . In a example, consider the object representingthe existence of truth, w T . The validity of the ‘excluded middle’ in this situation means thatthe proof is very simple: Proof: w T ⇒ w T ¬ w T ⇒ ( ¬¬ w T ) = w T . Since w T is the statement of truth itself, i.e. w T = P ◦A T , the inconsistency of P ◦A T ( w T ) means the inconsistency of P ◦ A T . Such a statement is not true by construction. One can nowidentify the abstraction, A w T T ′ as being A T itself. Thus, this exercise demonstrates that the21roof of a statement has consequences for the abstract form of the statement, allowing oneto identify more specific properties. Note that this does not, at this stage, provide extra proofmethods, since there is no procedure, as yet, for acquiring knowledge of the form of an object’srelevant A T ′ in advance. The content of the proof must rely on standard means. In the derivation of the general condition for an object c to be I -extant (Eq. (38)), one arrivesat a set of elements. In this notation, the set is not intended to specify all the possibilities thateach abstraction operator, A , can take. Rather, the set can be thought of as being ‘the set ofalterations from a general c ’ that encompass the required condition.If one seeks an absolute measure of the ‘size’ of the object, in terms of the overall pos-sibilities, one may define a type of cardinality, | c | , in terms of the total possible number ofabstractions. Recalling Cantor’s Theorem, there is no consistent description of such a uni-versal class. However, since the formalism accommodates the imposition of restrictions onthe kind of objects that can be represented, let the number of possibilities for A be assumedconsistently definable, and denote as L . L need not be finite, nor even countable, however, itcan be used to obtain formulae for the cardinality of an object.Define the number of abstractions, A , in c ∈ C i ...i n W as ¯ n ≡ P nj =1 i j . Thus one finds that | c | = L ¯ n , and (39) | c I | = ¯ n L ¯ n − . (40)The latter formula is simply a consequence of there being ¯ n possibilities for restricting oneabstraction operator to be A I ′ . If one enforces N restrictions on the set of A ’s, then it followsthat | c N | = (cid:18)
12 (¯ n − N ) + 1 (cid:19) L ¯ n − N . (41)This formula will become relevant in the next section.22 Unreasonable effectiveness
The goal is to use the general framework, described in Section 4, is to encapsulate the essenceof describing phenomena using a theory, in the sense used in physics. Thus, the issue ofWigner’s ‘unreasonable effectiveness’ of mathematics to describe the universe may be ad-dressed by transporting the problem to a metaphysical context. There, the tools from philoso-phy, such as logic and proof theory, can be directed at the question that involves not so muchthe behaviour of the universe, as the behaviour of descriptions of the universe (i.e. physicsitself). It is important to be able to transport certain features of physics into a context wherean analysis may take place, and such a context is, by definition, metaphysics.The notion of ‘effectiveness’ is that, given a consistent set of phenomena, v i ∈ V , One canextend V to include more phenomena such that (general) Ans¨atze able to explain the phenom-ena satisfactorily can still be found. In this general context, what is meant by an ‘explanation’will be taken to be a relationship among the phenomena, v i , in the form of abstractions. Theessence of the mystery of the effectiveness of mathematics is not whether one can always‘draw a box’ around an arbitrary collection of objects, or that laws and principles (of any kind)are obeyed, but the identification of particular principle(s) such that phenomena v , v , . . . areconsequences of them; and that via the principles, the whole of V may be obtained, indicatinga more full explanation of the phenomena. That is, the phenomena are extant because of thetruth of the underlying principles, rather than being identified ‘by hand’ (which would hold nopredictive power in the scientific sense). Note that the set V may, in fact, only include a subsetof the possible phenomena to discover in the universe, and so would represent a subset of theset, W , as discussed in Section 4.Let v , v . . . have descriptions A v , A v , . . . ∈ V ⊂ W , which are extant. Let there besome principle, (or even collection of principles with complicated inter-dependencies), de-scribed by the general object, c princ , such that each element of V may be enumerated. It is ourgoal to investigate under what conditions the following statement holds: c princ is true ⇒ v , v , . . . are extant , (42)i.e. h c princ = P ◦ A T ( w princ ) i ⇒ h A v i = P ◦ A E ( A y i ) ∈ V i . (43)If there is a principle that implies such a statement, we wish to identify it, and investigatewhether or not it is true. 23he circumstances of the truth of Eq. (43) depends on how w princ is related to the phe-nomena, v i . w princ itself represents principle(s) whose truth is not added by hand in Ansatzform. This does not mean that it is not true, since the form of w princ is as yet unspecified. Themost general way of relating w princ and all v i ’s is to use the method of substituting abstractionsinto the formula for a generalised object, such as that used to derive the general condition of I -extantness in Eq. (38). In the same way that the set of all possible locations of A I ′ in c was considered, here, all possible combinations of locations of abstractions describing v i in c must be considered, such that each A v i occurs at least once. This formula can be developedinductively.Consider only two phenomena, v and v , with corresponding abstract descriptions definedas A v and A v . For a generalised object, c ∈ C i ...i n W , one finds w ( N =2)princ = n [ p =1 i p − [ m ′ =0 m ′ = m i p − [ m =0 A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 − m ′ C i ...ip − m ′ ...inW ,m ′ +1 ,v ˆ Φ ( P n − p +1 j =1 i j )+1 − m C i ...ip − m...inW ,m +1 ,v c. (44)In the case of N phenomena, it is assumed that N ≤ i p : the number of abstractions avail-able in the general formula for c may be made arbitrary large to accommodate the number ofphenomena. One may make use of the following formula i p − [ m ( N − m ( N − = any other m ’s · · · i p − [ m (2)=0 m (2) = m (1) i p − [ m (1) =0 = N − [ k =0 [ m ( k,ip ) ∈ [0 ,i p − \ S kµ =0 { m ( µ ) } . (45)Here, [0 , i p − is the closed interval from to i p − in the set of integers, and for brevity, wedefine { m (0) } as the empty set: ∅ . The most general form of w princ may now be written as w Gprinc = n [ p =1 N − [ k =0 [ m ∈ [0 ,i p − \ S kµ =0 { m ( µ ) } A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 − m C i ...ip − m...inW ,m +1 ,v k +1 c. (46)In order for c princ to be true, w princ must contain information about the objects y i , such that A v i = P ◦ A E ( A y i ) . Therefore, we seek only those elements of Eq. (46) such that the phe-nomena v i take this form. This is a more restrictive set, as each abstraction of y i must beapplied to the right of A E , which must be applied to the right of P . There are only n − such24ccurrences of P in c , so in making this restriction, we are free to choose • N ≤ n − , (47) • N ≤ i p . (48)The form of the more restricted version of w princ is thus w Rprinc = N − [ k =0 [ p ∈ [2 ,n ] \ S kπ =0 { p ( π ) } A set ◦ ˆ Φ ( P n − p +2 j =1 i j )+1 C i ...ip...inW , ,E ˆ Φ P n − p +2 j =1 i j C i ...ip − ...inW , ,y k +1 c, (49)where { p (0) } = ∅ .If c princ can be constructed consistently, i.e. if it exists, then the form of w princ must berestricted to include an abstraction, A w princ T ′ , that ensures the existence of c princ . This uses thesame argument as in deriving Eq. (38), with c T = P ◦ A T ( w T ) , and involves the union ofEq. (49) with the object w T . Thus, the condition under which Eq. (43) is true can now bewritten. Theorem 3.
The condition under which the principles of a theory describe certain phenomenatakes the form w princ ⊆ w R ,T ′ princ = N − [ k =0 [ p ∈ [2 ,n ] \ S kπ =0 { p ( π ) } A set ◦ ˆ Φ ( P n − p +2 j =1 i j )+1 C i ...ip...inW , ,E ˆ Φ P n − p +2 j =1 i j C i ...ip − ...inW , ,y k +1 c ∪ w T . (50) Proof:
The statement of the theorem, that ‘ w princ ⊆ w R ,T ′ princ constitutes the condition for whichEq. (43) is true’, is only fulfilled if the general form of w princ (in Eq. (46)) includes a descriptionof extant phenomena explicitly, which takes the form shown in Eq. (49). That is, one mustshow that w R ,T ′ princ ⊆ w G ,T ′ princ . This entails that the elements of w R ,T ′ princ and w G ,T ′ princ are of the sameform, differing only by use of a restriction. Therefore, in this case, the abstraction A w princ T ′ issufficient to ensure the truth of the elements in both sets. Note that the inclusion of A w princ T ′ takesthe same form for both w R ,T ′ princ and w G ,T ′ princ . Therefore, it is sufficient to show that w Rprinc ⊆ w Gprinc .25xpress Eq. (49) in terms of abstractions, A v i , recalling that P ◦ A E ( A y i ) = P ◦ P ◦ A E ( A y i ) = P ◦ A v i : w Rprinc = N − [ k =0 [ p ∈ [2 ,n ] \ S kπ =0 { p ( π ) } A set ◦ ˆ Φ ( P n − p +2 j =1 i j )+1 C i ...ip...inW , ,v k +1 c. (51)Choosing the value m = 0 in w Gprinc yields w G ,T ′ princ ,m =0 = n [ p =1 N − [ k =0 A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 C i ...ip...inW , ,v k +1 c. (52)The only difference between Eqs. (51) and (52) is the choice of values of the iterator p . Toobtain w Rprinc ⊆ w Gprinc , it is sufficient to show that [2 , n ] \ k [ π =0 { p ( π ) } ⊆ [2 , n ] ∀ k ∈ [0 , N − . (53)Recalling the restrictions of Eqs. (47) and (48), take N ≪ n . Now, S kπ =1 { p ( π ) } is a finite setof integers that is a subset of [2 , n ] k [ π =1 { p ( π ) } ⊆ [2 , n ] , where { p (0) } = ∅ . ∴ k [ π =0 { p ( π ) } ⊆ [2 , n ] ⇒ w R ,T ′ princ ⊆ w G ,T ′ princ . Note that the fact that w R ,T ′ princ is a more restrictive set than w G ,T ′ princ does not mean that it is‘smaller’ in the sense of cardinality. Assuming a sufficiently large n value to accommodate all N + 1 restrictions, one finds that | w G ,T ′ princ | = (cid:18)
12 (¯ n − ( N + 1) ) + 1 (cid:19) L ¯ n − ( N +1) = | w R ,T ′ princ | (54) ⇒ w R ,T ′ princ ∼ = w G ,T ′ princ . (55)26he above observation provides a possible explanation for the appearance of the ‘unreason-able effectiveness’ of mathematics. The set of relationships among extant phenomena, w R ,T ′ princ ,is not smaller, in any strict sense, than the general set of relationships among abstractions, w G .T ′ princ . The countability of sets of phenomena filtering into a more restrictive and still count-able form, w R ,T ′ princ , combined with the formalism for describing non-mathematical objects in amathematical way, constitutes the metaphysical explanation for the ‘unreasonable effective-ness’ of mathematics. In other words, there is no unreasonableness at all, but it is simply amathematical consequence of the countability of phenomena, and the abstract description ofobjects that are not innately abstract.This demonstrates the power of metaphysical tools, in the form of principles and proofs,to address key philosophical issues in physics. That is, the process employed here was notphysics itself, but philosophical argumentation applied to the abstractions of objects used inthe practice of physics. The definition of evidence relies on the connection between a set of phenomena (called evi-dence), and the principles of a theory that the evidence supports. It is assumed here that thesense in which the phenomena support or demonstrate an abstraction, such as the theory, is thesame sense in which a theory can be said to entail the extantness of the phenomena. The sym-metry between the two arguments has not been proved, however, since it relies on the precisedetails of often-imprecisely defined linguistic devices.The description of evidence, using the formalism of Section 4, takes a similar form tothat of the description of Ansatz for phenomena in Eq. (43), except that the direction of thecorrespondence is reversed A v , A v , . . . are extant ⇒ c princ is true , (56)i.e. h A v i = P ◦ A E ( A y i ) ∈ V i ⇒ h c princ = P ◦ A T ( w princ ) i . (57)27he lefthand side of Eq. (57) restricts the form of the object, w princ (representing the setof principles) through w princ ⊆ w Rprinc . For the form of w princ to entail the righthand side ofEq. (57), it must also include the abstraction, A T ′ . Thus, the condition under which Eq. (57)is true, where phenomena constitute evidence for a set of principles, is h w princ ⊆ w R ,T ′ princ i (= c cond ) . (58)This is the same condition obtained for the examination of principles entailing extant phenom-ena, in Eq. (43).There is a duality between the two scenarios, which can be expressed in the followingmanner. If the condition of Eqs. (50) and (58) is true, then h c princ = P ◦ A T ( w princ ) i ⇔ h A v i = P ◦ A E ( A y i ) i . (59)That is, relationship between principles and evidence is symmetrical in a sense. The sort ofphenomena entailed by a theory is of exactly the same nature as the sort of phenomena thatconstitutes evidence for such a theory. This leads one to postulate an object, c D = P ◦A T ( w D ) ,which represents this duality, and one may identify a Duality Theorem , which takes the form w D = nh P ◦ A T ( w cond ) i ⇒ h(cid:16) c princ = P ◦ A T ( w princ ) (cid:17) ⇔ (cid:16) A v i = P ◦ A E ( A y i ) (cid:17)io . (60)The theorem is a consequence of the fact that the application of the restrictions acting upon w princ commute with each other in the formalism.Note that, in attempting to clarify a term ill-defined in colloquial usage, we have arrived atquite a strict definition of evidence: if w princ is to constitute a set of principles describing theelements, v i , it must at least take the form of a description based explicitly on all v i elements.Any part of w princ that does not lie in w R ,T ′ princ is not relevant for consideration as being supportedby the evidence. 28 .2 The relating theorem and the fundamental object Since the I -extantness of some c I has been related to the mathematical existence of an object A I ( w ) , a primary question to investigate would be the I -extantness of the statement of this re-lation itself. The statement of ‘the tying-in of the mathematical and non-mathematical objects’has certain properties that should deem the investigation of its own I -extantness a nontrivialexercise.Denote the above statement, which is an Ansatz, as Z I ( c I ) , and let c I be I -extant. That is,for c I ≡ P ◦ A I ( w ) , A I ( w ) exists; and let Z I ( c I ) ≡ P ◦ A Z ◦ c I , for some A Z . Recall thatthe assumed existence (in the mathematical sense) of the statement, Z I ( c I ) , does not triviallyentail I -extantness, under Supposition 2. To show that Z I is I -extant, it is required that it canbe put in the form Z I ( c I ) = P ◦ A I ( w Z ) , (61)which implies that P ◦ A Z ◦ c I = P ◦ A I ( w Z ) . (62)We would like to attempt to understand under what conditions this holds.Consider the scenario in which the I -extant form of Z I does not exist. In this case, it is notpossible to say that Z I is not I -extant, since the statement relating existence and I -extantnesshas not been proved, and no information about I -extantness can be gained using this method.If, however, the I -extant form of Z I does exist, then it is indeed certain that Z I is I -extant. Inother words, there is a logical subtlety that entails an ‘asymmetry’: the demonstration of the ex-istence of an object is enough to prove it, but the equivalent demonstration of its non-existenceis not enough to disprove it, since the relied-upon postulate would then be undermined. There-fore, in this particular situation, unless further a logical restriction is found to be necessary toadd in later versions of the formalism, it is sufficient to show that the I -extant can exist, for Z I to be I -extant. This is not true in general, due to Supposition 2. Theorem 4. Z I ( c I ) is I -extant. The above theorem, denoted the relating theorem , may be verified in proving Eq. (62). Itis enough to show that A Z ◦ c I = A I ◦ w Z for any c I , where there exists an A Z such that A I obeys the property: A − I,R
DNE, which is, in our general framework, the only distinguishing29eature of A I at this point. The demonstration is as follows: Proof:
Let Z I exist, such that Z I = A Z ◦ P ◦ A I ◦ n [ p =1 i p − [ m =0 A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 − mC i ...ip − m...inW , m +1 , I ′ c = A I ◦ w Z , for any w Z . Due to the labelling principle, this can only be true if w Z ≡ c I and A I ≡ A Z ;that is, the abstraction of c I (above) is an I -abstraction: A Z ∈ ˜Ω ( W ) . This is a valid choice,since the existence of A − Z,R was not assumed.Therefore, the form of Z I is now known: Z I ( c I ) = P ◦ A I ( c I ) (63) = PA I ◦ PA I ( w ) . (64)In words, what has been discovered is that the Ansatz of I -extantness is equivalent to theAnsatz in the statement ‘the I -extantness of c I is related to existence’. That is, the operationassociated with, say, ‘it is I -extant’ ( P ◦ A I ( w ) ), when applied twice, forms the statement‘its I -extantness is related to existence’ ( PA I ◦ PA I ( w ) ); and it is the same operation . Thisneedn’t be the case in general, and so it is a nontrivial result that Z I = P ◦ A I . (65)Note that Eq. (65), in this case, is not a definition, but a theorem , to be known as the corre-spondence corollary to the relating theorem.In a sense, Z I is the fundamental I -extant object, in that it is the most obvious starting pointfor the analysis of the existence of I -extant objects in general. It also constitutes the first ex-ample of an object demonstrated to exist in a universe (though, a clarification of distinguishingdifferent universes is still required, and investigated in Section 6.3).Recall, in construction of ‘types’ in Eq. (24), that familiar notions such as ‘chair’, or othersuch objects, are brought into a recognisable shape using this formula. Though the types may30ot appear more recognisable at face value, the properties of such a construction align moreclosely with what is meant phenomenologically but such objects. In a similar fashion, the typeof Z I can be established, to created a more full, complete, or ‘dressed’ version of the object. Z I is an example of a c I . Since projected objects cannot be related directly, a type will beconstructed from instances of A I (of which A E is one), and the dressed fundamental objectwill be a projection of the type. Using the same argument used in deriving Eq. (24), the set of n observed instances of A I takes the following form (acting on some set W ′ ) n [ j ′ =1 A set ◦ ˆ Φ ( W ′ ) ,i ′ ,j ′ A i ′ I ( W ′ ) . (66)If each observed instance may be identified as the set of a certain N characteristics residing in W , then our intermediate set W ′ can be dropped, and we find A I ◦ N i ′ [ j =1 A set ◦ ˆ Φ W,i,j w ( i ′ ) i ∈ A i ′ I (Ω ( W )) . (67)In this case, an instance of A i ′ I contains more information than just a set of characteristics, sinceit is also known that it is I -extant. Therefore, it contains an additional abstraction operator.What is meant by the fundamental type therefore takes the form R Z ≡ P r Z = P A ′ ◦ n [ j ′ =1 N j ′ [ j =1 A set ◦ ˆ Φ W,i ′ ,j ′ A I ◦ A set ◦ ˆ Φ W,i,j w ( i ′ ) i ∈ P Ω ◦ ˜Ω ◦ Ω ( W ) , (68)where A ′ is the abstraction of relationship. Note that R Z is, in general, an element of P Ω ( W ) . In this section, we address the issue of classifying universes by their properties in a generalfashion. An attempt can then be made to identify features that distinguish universes from oneanother, and thus clarify the definition of our own universe in a way that is convenient in thepractice of physics.Suppose the definition of a universe, U , to be the ‘maximal’ list of objects that have thesame character, that is, obeying the same list of basic properties. The list need not necessarily31e finite, as each collection of properties could, in principle, represent a collection of infinitelymany objects themselves. In the language of the formalism developed so far, a formula maybe constructed from a generalised object c by ensuring that each element of U is related to thecontent of the underlying principles, w princ . This formula will be analogous to Eq. (46). Supposition 3. ∃ U , such that a list of underlying principles may be configured to be enu-merable as a countable set, w princ = u ∪ · · · ∪ u N , for N elements. (It is not required that u ∪ · · · ∪ u N ∈ U ). A universe based on these principles is the object represented by thelargest possible set of the form: U = P ◦ n [ p =1 N − [ k =0 [ m ∈ [0 ,i p − \ S kµ =0 { m ( µ ) } A set ◦ ˆ Φ ( P n − p +1 j =1 i j )+1 − m C i ...ip − m...inW ,m +1 ,u k +1 c, (69) with respect to a generalised object, c . Note that extantness is a universal property, in that it can be defined in the formalismregardless of the universe in which it is extant. It may, but it is not required that it constituteone of the N underlying principles of a universe.The distinction between different universes is largely convention, based on the most con-venient definition in practising physics. One such convenience is the ability to arrive at a consistent definition of the universe. This is not the case for a na¨ıvely defined universe re-quired to contain all possible objects, due to Cantor’s Theorem. In order to establish twouniverses as distinct, the following convention is adopted:
Supposition 4.
Consider two consistently definable universes U and U ′ . If the universe definedas the union, X ≡ U ∪ U ′ is inconsistent, then the universes U and U ′ are distinct. Introducing a square-bracket notation, where U [ . . . ] indicates that the underlying principlesto be used in defining U are listed in [ . . . ] , one may write U = U [ w princ ] and U ′ = U [ w ′ princ ] .Define the following lists of principles to be consistent: w princ = u ∪ · · · u N − , (70) w ′ princ = u N ∪ u ′ ∪ · · · u ′ N ′ , (71)32ut suppose the inclusion of both principles u N − and u N to lead to inconsistency. It thenfollows that X ≡ U [ u ∪ · · · u N − ] ∪ U ′ [ u N ∪ u ′ ∪ · · · u ′ N ′ ] is inconsistent. (72) Example:
The inconsistency of a set of principles can emerge in the combination of negationand recursion, as clearly demonstrated by G¨odel and Tarski. If u N − were to express anegation, such as ‘only contains elements that don’t contain themselves’, and u N were toenforce a recursion, such as ‘contains all elements’, then Russell’s paradox would result. The final denouement is to demonstrate that the fundamental type constitutes evidence for auniverse distinct from our own. Consider N abstract objects, A v k , each of which representsan element of the fundamental type in Eq. (68). They may be expressed analogously to theoperator A i ′ I defined in Eq. (67), for N sub-characteristics: A v k ≡ P ◦ A I ◦ N [ j =1 A set ◦ ˆ Φ W,i,j w ( k ) i . (73)Though the sub-characteristics themselves are not vital in this investigation, one can simplysee that the objects are I -extant (by construction), by expressing them in the form: A v k = P ◦ A I ( A y k ) ∈ V ′ , (74)where V ′ denotes a set that contains at least all N elements, A v k . By determining the under-lying principles describing V ′ , one may denote its maximal set as U ′ . Since V ′ is an abstractobject existing as a subset of the objects that comprise our formalism, F , it follows that U ′ isthe set of all abstracts, and cannot be consistently defined. That is, by taking the maximalset of objects obeying this restriction, one arrives at a set containing itself, and all possibleabstract objects, which is Cantor’s universal set. This does not mean, however, that V ′ itself isinconsistent; since V ′ ⊂ F ⊂ U ′ , we are free to assume that V ′ is constructed in such a wayas to render it consistent, just as was assumed for F .33ow consider our universe, U , which takes the form of Eq. (69), with the restriction thatone of its underlying principles must be that all elements are extant. That is, let U only containelements of the form w = P ◦ A E ( c ) ∈ U . U may be still be consistently defined. Since U ′ isinconsistent, the definition of a composite universe X ≡ U ∪ U ′ is also inconsistent. Therefore,by the convention established in Supposition 4, U and U ′ are distinct . It also follows that V ′ ⊂ U ′ is also distinct from U .The pertinent underlying principle for V ′ is simply that it be consistent; or more specifi-cally, that an extant description of it be consistent: w ′ princ = Y ′ , such that V ′ = P ◦ A E ( Y ′ ) , (75)for some consistently definable Y ′ . It is reasonable to expect that the elements that comprise V ′ constitute evidence for the extantness of V ′ ∈ U ′ . This can be checked by determining thatthe condition for evidence is satisfied: w ′ princ ⊆ w R ,T ′ princ [ v k ] . (76)This condition, however, is not satisfied in general, due to the fact that Y ′ must contain an ab-straction, A E ′ , which is not present in the general formulation of w R ,T ′ princ . This makes sense thatthe truth of a statement does not necessarily entail an extantness. In the specific case above,though, we have considered the fundamental type of the general I -extantness, encompassingall abstractions of a specific form, as described in Section 4.3. In this special case, the ex-tantness required by both sides of Eq. (76) is the same, and we are required to demonstratethat w ′ princ = Y ′ ⊆ w R ,I ′ princ [ v k ] , (77)for V ′ = P ◦ A I ( Y ′ ) . (78) Proof: Y ′ is a set containing N + 1 abstractions, A v k , with A N +1 ≡ A I ′ : Y ′ = {A v , . . . , A v N , A I ′ } = N [ k =0 A set ◦ ˆ Φ V ′ ,i,k A v i . (79)34he righthand side of Eq. (77) takes the form: w R ,I ′ princ = N − [ k =0 [ p ∈ [2 ,n ] \ S kπ =0 { p ( π ) } A set ◦ ˆ Φ ( P n − p +2 j =1 i j )+1 C i ...ip...inW , ,v k +1 c ∪ w I . (80)The left term can be pared down by choosing n = 2 , i = 0 and i = 1 , and the right term, w I ,by choosing n = 1 , i = 1 : ⇒ (cid:16) N [ k =1 A set ◦ ˆ Φ P Ω ( W ) , ,v k +1 c (cid:17) ∪ (cid:16) A set ◦ ˆ Φ ( W ) ,m +1 ,I ′ c (cid:17) = {PA v ( w ) , . . . , PA v N ( w ) } ∪ {A I ′ ( w ) } = N [ k =0 A set ◦ ˆ Φ V ′ ,i,k A v i , for A N +1 ≡ A I ′ . The final line follows from the labelling principle,
P ◦ P = P , and the fact that each A v k is in extant form, P ◦ A I ( A y k ) . This manuscript has attempted to address key issues in physics from the point of view of phi-losophy. By adopting a metaphysical framework closely aligned with that used in the practiceof physics, the philosophical tool of Ansatz was examined, and the process of its use was clar-ified. In this context, a mathematical formalism for describing intrinsically non-mathematicalobjects was expounded. In examining the consistency of such a framework, a careful dis-tinction between existence (in the mathematical sense) and ‘extantness’ (in the sense of phe-nomena existing in the universe) was made. Using the formalism, a general condition forextantness was derived in terms of a generalised object, which incorporates the salient fea-tures of abstraction and projection to the non-mathematical world in a way easily manipulated.In principle, the formalism may make verifiable predictions, since properties (and the conse-quences of combinations of properties in the form of theorems) can be arranged to make strictstatements about the behaviour or nature of a system or other general objects.35s an example, a possible explanation of Wigner’s ‘unreasonable effectiveness’ of math-ematics was derived. This demonstrates the ability of a metaphysical framework to addressimportant mysteries inaccessible from within the physics itself being described.Lastly, an attempt was made to classify other universes in a general fashion, and to clarifythe characteristics and role of evidence for theories that provides at least a partial descriptionof a universe. The connection between phenomena that constitute evidence and the theoryitself was established in a Duality Theorem. Instead of focusing on attempting an ad-hocidentification of extra-universal phenomena from experiment, the formalism was used to derive basic properties of objects that do not align with our universe. As a first example towardsuch a goal, a fundamental object was identified, which satisfies the necessary properties ofevidence, and whose extantness does not coincide with our universe. This paves the way forfuture investigations into more precise details of the properties (perhaps initially bizarre andunexpected) that objects may possess outside our universe.
References John F. Donoghue, Eugene Golowich, and Barry R. Holstein.
Dynamics of the StandardModel (Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology) .Cambridge University Press, New Ed Edition, 1992, 1996. Stephen P. Martin. A Supersymmetry primer. 1997. Nima Arkani-Hamed, Savas Dimopoulos, and G.R. Dvali. The Hierarchy problem and newdimensions at a millimeter.
Phys.Lett. , B429:263–272, 1998. John A. Wheeler.
Foundational problems in the special sciences . D. Reidel PublishingCompany, Dordrecht, Holland, 1977. J¨urgen Schmidhuber. A computer scientist’s view of life, the universe, and everything. 1999. Richard J. Szabo.
An Introduction to String Theory and D-Brane Dynamics . Imperial Col-lege Press, 2004. Maurizio Gasperini.
Elements of string cosmology . Cambridge University Press, 2007.36
Brian Greene.
The Hidden Reality . Alfred A. Knopf, a division of Random House, Inc.,New York, 2011. Andrei D. Linde. Chaotic Inflation.
Phys.Lett. , B129:177–181, 1983. Andrei D. Linde. Eternal Chaotic Inflation.
Mod.Phys.Lett. , A1:81, 1986. Frank J. Tipler. Nonlocality as Evidence for a Multiverse Cosmology.
Mod.Phys.Lett. ,A27:1250019, 2012. Max Tegmark. Parallel universes. 2003. Eugene P. Wigner. The unreasonable effectiveness of mathematics in the natural sciences.
Communications on Pure and Applied Mathematics , 13:1–14, 1960. Plato.
The Republic . circa 380 BC. George Berkeley.
A Treatise Concerning the Principles of Human Knowledge . AaronRhames, for Jeremy Pepyat, Bookseller in Skinner-Row, 1710. Frank C. Jackson. Epiphenomenal qualia.
The Philosophical Quarterly , 32:127–136, 1982. Thomas H. Huxley. On the hypothesis that animals are automata, and its history.
TheFortnightly Review , 16:555–180, 1874. Ren´e Descartes. Meditations on first philosophy, in the philosophical writings of ren´edescartes.
Cambridge University Press , 2:1–62, 1984 (1641). David J. Bohm.
Wholeness and the Implicate Order . Routledge, Great Britain, 1980. Ant´onio R. Dam´asio.
Looking for Spinoza: Joy, Sorrow and the Feeling Brain . Orlando,Fla.: Harcourt, 2003. Ant´onio R. Dam´asio.
Self Comes to Mind: Constructing the Conscious Brain . PantheonBooks, 2010. Hilary W. Putnam.
Psychological Predicates . University of Pittsburgh Press, 1967. Ned Block.
What is functionalism?, in Readings in Philosophy of Psychology, 2 vols. , vol-ume 1. Harvard University Press, 1983. 37 John R. Searle. Minds, brains and programs.
Behavioural and Brain Sciences , 3:417–424,1980. Georg F. L. P. Cantor. ¨Uber eine elementare Frage der Mannigfaltigkeitslehre . GesammelteAbhandlungen mathematischen und philosophischen Inhalts, 1891. Ernst F. F. Zermelo. Untersuchungen ¨uber die Grundlagen der Mengenlehre I.
Mathematis-che Annalen , 65:261–281, 1908. David E. Deutsch. Constructor theory. 2012. Raoul Bott and Loring W. Tu.
Differential Forms in Algebraic Topology . Springer-VerlagNew York Inc., 1982. (W. D. Ross trans.) Aristotle. The Works of Aristotle: Metaphysics 4.4 . Oxford UniversityPress, 1953. Kurt F. G¨odel. ¨Uber formal unentscheidbare s¨atze der principia mathematica und verwandtersysteme, i.
Monatshefte f¨ur Mathematik und Physik , 38:173–98, 1931. Alfred Tarski. Der Wahrheitsbegriff in den formalisierten Sprachen.
Studia Philosophica ,1:261–405, 1936. Bertrand A. W. Russell.