Observers, observations and referencing in physics theories
OObservers and observations in physics theories
Uri Ben-Ya’acov
School of Engineering, Kinneret Academic College on the Sea of Galilee,D.N. Emek Ha’Yarden 15132, IsraelE-mail: [email protected]
Abstract.
The rˆole of observers and observations in physics theories is considered in the lightof G¨odel’s incompleteness theorem. Incompleteness arises in G¨odel’s theorem withself-referential propositions, when the system asks to define itself in its own terms.Self-referencing occurs in physics whenever the observer is recognized as being part ofthe observed system, so the acts of observation are also observables and should becomepart of the phenomena considered by the theory. This is emphasized by the fact thatin many instances the same physical phenomenon may be viewed in more than oneway, hence its interpretation depends on the mode of observation.Observations of observations imply a potentially endless hierarchy of levels ofobservations. Each higher level suggests a larger overview with more profound insightinto the universe, empirically implying essentially new discoveries and realizations.New discoveries imply new first principles in the foundation of the theory, so thetheory remains open and cannot be complete.
Keywords : observers and observations; self-referencing; self-negation; G¨odel’sincompleteness theorem; logical paradoxes; theory of everything; participating universe “A human is the universe’s way of knowing the universe”Anonymous
Any scientist is, first of all, a human being, acting as a scientist as part of the humanmission.
1. Introduction
G¨odel’s incompleteness theorem [1, 2, 3, 4] implies that any formal structure, based ona finite number of first principles and inference rules, which is broad enough, cannot beat the same time both consistent and complete – there can always be propositions, wellformulated within the theory, that are undecidable . G¨odel’s theorem concerns arithmeticsand logic, but since physical theories use mathematics and are organized as formalstructures then naturally comes the question: Does G¨odel’s theorem apply to physics? a r X i v : . [ phy s i c s . h i s t - ph ] J u l bservers and observations in physics theories e.g. geometry[4], and that this is the type of mathematics that physics uses. Therefore we should notexpect that G¨odel’s theorem applies to physics [7, 8, 9].However, close inspection of G¨odel’s theorem reveals that the core principle itrelies upon is self-reference [10]: If self-reference is possible then there can always bewell formulated propositions that are undecidable – cannot be proven or refuted. Theseundecidable propositions are characterized by self-negation, appearing when the systemasks to define itself negatively in its own terms (in other words – refute itself), leadingto logical conflicts and paradoxes in a manner similar to the liar paradox or Russel’sparadox [11]. Not all self-referencing is paradoxical, and not all negation is paradoxical,but self-referencing allows paradoxical self-negation.The present article proposes that self-reference is the decisive component of G¨odel’stheorem where physics is concerned. When it exists or when it is possible it indicatesincompleteness. Focusing on referencing and self-referencing liberates us from themathematical argument – it is not important any more what kinds of mathematicsare used by physics.Self-reference is found in physics when we realize that observers and observationsare participating in physical phenomena. In classical physics, till the end of 19th century,the common view was that the human-observer-scientist is a separated, not involved,by-standing witness to all universal phenomena. But 20th century physics made usrealize that in many instances the observer is capable of influencing the outcome ofexperiments. Therefore, the observer should be recognized as a full participant, anintegral part of the observed system, with the acts of observation being also observablesthat should become part of the phenomena considered by the theory. Observation is bservers and observations in physics theories .. we are not angels, who view the universe from the outside. Instead, we andour models are both part of the universe we are describing. Thus a physicaltheory is self referencing, like in G¨odels theorem. One might therefore expectit to be either inconsistent or incomplete. The theories we have so far are bothinconsistent and incomplete. [12]The purpose of the present work is to elaborate on these ideas. The applicabilityof G¨odel’s incompleteness theorem to physics was initially discussed along these linesin [13]. In the present article we further the study of the rˆole of referencing and self-referencing in physics.The questions that the article follows are “How to describe referencing in physics?Does it lead to incompleteness?”We start by discussing referencing and self-referencing, first in principle, followingG¨odel’s theorem, then in physics and the theories formed from our observation of theuniverse.It is argued that in physics self-reference is like self-testimony, with observersobserving their own act of observation. Moreover – in many instances the samephysical phenomenon may be viewed in more than one way, so that the mode of theobservation determines its consequences. Self-reference and referencing in observationslead to identifying levels of observation, between observer and observations. Each levelof observation requires a higher one for the former to be observed, thus creating a(potentially infinite) hierarchy of levels of observation.Observations are followed by interpretations, therefore each higher level suggestsa more profound insight which empirically implies an essentially new discovery, andtogether a potentially infinite hierarchy of levels of interpretation. New discoveriesimply new first principles in the foundation of the theory, so the theory remains openand cannot be complete.
2. Observations, consistency and completeness in physical theories
Let X be the space, realm, of all physical observables – all the physical phenomena in thewhole universe, from sub-atomic particles via rocks and oceans to galaxies and clustersof galaxies, not only the objects but including all the processes and interactions, fromatomic and molecular configurations via rains and hurricanes to dynamics of galaxies.The result of an observation depends upon the point of view ( e.g. , a reference frame)from which we choose to perform the observation, and in many instances also uponthe way we choose to interpret it. Let a denote a typical point of view from which weobserve such phenomena, and let A be the sum-total of all the possible points of view, bservers and observations in physics theories A = { a } . An observation is an act from some point of view a ∈ A to some physicalphenomenon x ∈ X , which we denote in the following a (cid:38) x .According to the accepted scientific paradigm, the highlight of scientific research isthe ability to summarize the results of the study in a finite set of insights, as simple aspossible, that will explain familiar phenomena and predict related phenomena that havenot yet been observed. A physical theory is therefore a statement about the totality ofour observations of the physical world, { a (cid:38) x } .The expectation is that it is possible to identify in this totality of observationscommon fundamental principles that can be grasped by human cognition. Thesefundamental principles stand as axioms at the basis of scientific theory, and fromthem are deduced properties, statements and conclusions corresponding to the objectof the research. The combination of logical inferences with the results of observationsmakes it possible to examine the correctness of the basic principles that have beenidentified as the foundation of the theory.Two essential characteristics are expected from any physical theory – consistencyand completeness, both logically and physically: • Logical consistency – the theory does not produce conflicting predictions. • Physical consistency the theory does not produce predictions that contradictphysical observations. • Logical completeness all the predictions of the theory are uniquely concludable. • Physical completeness given initial data, the future can be predicted with anydesired accuracy.The last feature is the requirement of determinism.G¨odel’s incompleteness theorem casts doubt on the possibility of the existencetogether of these characteristics for sufficiently broad physical theories.
3. Referencing and self-referencing
The crux of G¨odel’s proof is the ability to formulate, in arithmetic terms, a formula G that says “ G cannot be proven”. Such a definition may look very odd, due to itscircularity, but G¨odel managed to cast it in precise arithmetic context, so it can get anumerical value. The sentence “ G cannot be proven” is about arithmetics (since G isan arithmetic formula) so G is also a meta-arithmetic formula, and since it refers backto G it is self-referential. The self-negation in G is reminiscent of the self-negation inlogical paradoxes such as the liar paradox or Russel’s paradox, and similarly G is anundecidable arithmetic formula, demonstrating that arithmetics is incomplete.Self-negation is possible when referencing and self-referencing are possible. Let ususe an analogy to illustrate and clarify referencing vs. self-referencing . Imagine a groupof children playing in the yard. Then an adult calls them from a balcony, which is somemeters above the yard. The height difference puts the balcony position at superiority bservers and observations in physics theories reference . G¨odel’s theorem deals with the meta-arithmetic statement in arithmeticterms, which is like the balcony being in the yard’s level, so that both children and adultmay now refer to each other on the same footing. This is self-reference .Referencing and self-referencing may be formally represented as follows. Theanalogy above suggests introducing the concept of reference levels , with the relativestatus of referrer level and referent level . Let a denote the referent and b denote thereferrer, with a unidirectional reference relation b (cid:38) a between them (we use the samenotation as for observations above because, as is argued in the following, an observationis an act of reference). If the two levels – referrer level and referent level – are distinct,than the referrer level may be regarded as higher or superior (‘balcony’) to the referentlevel (‘yard’). This is just referencing. But if referrer and referent levels are notdistinguished then referencing is possible in both directions, b (cid:38) a and a (cid:38) b , andthat is self-referencing.For instance, a may be some statement S while b is a statement about S , asin G¨odel’s theorem. Or, examples corresponding to some logical paradoxes, e.g. , “ b declares attribute a ” (liar paradox) or “ a is/isn’t a member of set b ” (Russel’s paradox).Also “ b shaves/doesn’t shave a ” (as in the barber paradox, see Appendix in [13]).Self-referencing may also be demonstrated graphically, as in works of the Dutch artistEscher † , in particular “Drawing hands” (Figure 1) with “ b draws figure a ”, manifestingparadoxical self-referencing. Figure 1. “Drawing Hands”M.C. Escher (1948)
It is important to note that self-reference is a necessary condition for logicalparadoxes, but not sufficient. The children and the adult being in the same (yard)level doesn’t necessarily mean that there’s a conflict. But if the adult calls the childrento end their playing and come home for dinner then a conflict is likely to occur. Similarly,consider Russel’s paradox. Along the same lines we may create a set N which is “theset of all sets that are members of themselves”. There is no paradox here. But if sucha construction is allowed, then negation may also be introduced and Russel’s paradoxensues. † bservers and observations in physics theories { p } , and thedoublets:(i) p ≡ ‘ q is true’ q ≡ ‘ p is true’(ii) p ≡ ‘ q is false’ q ≡ ‘ p is false’(iii) p ≡ ‘ q is true’ q ≡ ‘ p is false’The building blocks (‘ p is true’, e tc.) are perfect logical statements in all three cases,and each combination is self-referential. The first two are tautologies, even though (ii)contains negation, while (iii) is paradoxical negation.It follows, therefore, that when we scan the whole spectrum of possibilities allowedby self-referencing, self-negation may be found there as part of the possibilities – ifaffirmation is possible, so is negation.
4. Participating observers and self-referencing
The principle of relativity manifests the fact that measurements are observer-dependent.In quantum experiments, the way the experiment has been set up and the chosen modeof observation may determine the outcome of the experiment – whether the observedobject is detected as a wave or a particle, or which path it follows in traveling from onepoint to another, etc. . The mere act of observation may therefore influence the universeat the most fundamental levels.Hence the view, that a quantum measurement creates an actuality out of a merepotentiality. Then it is not only the value of the measurement that is observer-dependent, but the very nature of the empirical end-result. Then the observer isnot a by-stander, uninvolved, separated witness of physical phenomena but an activeparticipant involved in the occurrence of physical experience. The mode of observation– arbitrarily chosen by the observer – determines, even in small, the way the universeevolves.When the act of observation, being influential in physical phenomena, becomes anintegral part of the phenomenon, then with it, necessarily, also the human observers,which become participating observers. When the observer, the observation and thesubject all are part of the physical phenomena, this is a manifestation of self-reference(Figure 2).In the following sections we explore referencing observations, first in principle andthen in current physics.
5. Observers, observations and referencing
In arithmetics, G¨odel’s theorem demonstrates that the possibility of self-referencing –arithmetic statements that are also meta-arithmetic – allows paradoxical self-negation bservers and observations in physics theories Figure 2.
Self-reference in physics: the observer, the observation and the subject allare part of the physical phenomena which in turn implies undecidability, and therefore incompleteness. In physics, so far, wedon’t have such physical-metaphysical statements or formulae. However, we observe theuniverse, then construct theories of physics from the interpretation of our observations.These observations are like the referencing in G¨odel’s theorem.To consider the observation processes, let us start, as an indicative example, withthe drawing known as “Rubin’s vase” (Figure 3). The first thing to notice and emphasizeis the fact that the same drawing may be interpreted in more than one way, and themode of interpretation depends entirely on our choice: If we focus our mind on thewhite area we see a cup; but if we focus our mind on the black area we see two faces.We can even switch our attention from the black to the white area, and back, thusskipping between cup and faces. Here the drawing is the subject, or referent, and thestate of mind which focuses on either the white or black areas is the referrer level. Butthen, there is a higher state of mind that monitors our observation of the drawing andappreciates both options. This higher state of mind is a referrer level for the state ofmind which sees either the cup or the faces, which in this relation becomes the referent.
Figure 3.
Rubin’s vase: Cup or faces?
Similarly, consider Escher’s “Drawing hands” (Figure 1). In first-level observationswe see (focus on) either one hand or the other as the drawing hand but not both together bservers and observations in physics theories x ∈ X be a physical object or phenomenon, the subject of observation (more precisely,the image of the subject of observation in the observer’s mind). Let a ∈ A be thepoint of view of the observer while observing x , with a (cid:38) x denoting the act ofobservation. A = { a } is then the totality of points of view for direct observation ofphysical phenomena.With the observer being also part of the universe, the act of observation becomes anobservable phenomenon. Observers (necessarily human) have the ability to observe theirown process of observation. Then the observer can develop another, higher or superior ,point of view, b , from which he observes a (cid:38) x . So now we also have b (cid:38) ( a (cid:38) x ), with B = { b (cid:38) ( a (cid:38) x ) | a ∈ A} the totality of observations of A -level observations . Level A is referrer level for X , but referent level for B .Hence, for Escher’s “Drawing hands” (Figure 1), while A -level viewing sees each ofthe hands either as drawing or as being drawn, B -level viewing sees both hands together,drawing and being drawn: Let a be viewing the right hand drawing the left hand and a be viewing the left hand drawing the right hand . Then, with x being the drawingitself, the B -level viewing is b (cid:38) { a (cid:38) x, a (cid:38) x } .Similarly, for “Rubin’s vase” (Figure 3), with a being focusing at the white area(cup) and a being focusing at the black area (faces) , and x being the drawing itself, A -level viewings { a (cid:38) x, a (cid:38) x } are seeing either the cup or the faces, while B -levelviewing b (cid:38) { a (cid:38) x, a (cid:38) x } is seeing both at the same time (monitoring the possibleswitching of our focusing between the two).Another example is the well-known story of the elephant in the village of blindpeople: The elephant itself is the subject x . It is the referent for the blind persons,each touching a different member of the elephant’s body, being at different positions inlevel A which for them is the referrer level. But the wise man, who understands thatthey have touched different members of the same elephant, he is in level B which is thereferrer level relative to the villagers.There may be more than just two reference levels in observation: From any levelof observation it is possible to observe only lower-order observations, not its ownobservations. Hence, using short-hand notation A (cid:38) x for “observing a phenomenon x from some point of view in A ”, etc. , then from B it is possible to observe A (cid:38) x , i.e. , B (cid:38) ( A (cid:38) x ), but B (cid:38) ( A (cid:38) x ) itself cannot be observed from B . Still, theobserver may develop a higher-order level of view C from which it is possible to observe B (cid:38) ( A (cid:38) x ); namely, C (cid:38) ( B (cid:38) ( A (cid:38) x )) is possible, etc. . bservers and observations in physics theories
6. Referencing observations in physics
An observation is our interpretation of some input into our mind and consciousness ‡ .No matter which devices and how many are used for measurement and registration, atthe end it is human interpretation. Science is the end result that we make of theseinterpretations.“Rubin’s vase” (Figure 3) is a simple graphical illustration: The drawing itself isjust raw data – a set of pixels. When these pixels enter our mind, their interpretationdepends on what we choose to focus upon – the black or the white area.Classical physics corresponds to the base level of observation – direct observationsof physical phenomena, corresponding to level A in the notation above. A characteristicof the classical level is localizability (local reality) – it can only accept that any objectcan be at any moment in only one place, or that a physical system can be in onlyone, definite, state, defined by the detecting devices. A -level observations A (cid:38) x areconfined to detecting either particle-like properties or wave-like properties, becausethese are detectable observables. This corresponds to viewing each of the hands inFigure 1 either as drawing or as being drawn, and to viewing either the cup or thefaces in Figure 3. Classical physics theories thus refer to A -level observations only, {A (cid:38) x } , uniquely determined by the system or method of detection.Quantum mechanics manifests non-localizability. The classically educated mind,demanding localizability and confined to A -level observations only, cannot acceptnon-localizability. But many quantum delayed-choice experiments indicate, quiteconvincingly, that a photon is wave-particle – an entity that combines both propertiesbut is neither this nor that exclusively – and it is only the nature of the detection devicethat exposes either the wave or particle aspect [14].Non-localizability necessarily introduces higher levels of observation: A higher-than- A level is needed to be able to appreciate that a physical entity can besimultaneously in two or several places or in more than one state (in the classicalsense); e.g. , a photon being wave-particle. This is very much like seeing both handssimultaneously drawing and being drawn in Figure 1, or observing that we can seeboth cup and faces in Figure 3, as from level B above. While the subjects of A -levelobservations ( A (cid:38) x ) are directly detectable physical phenomena, the subjects of B -level observations are the A (cid:38) x observations themselves. The understanding that wecan arbitrarily choose between detecting the photon as a particle and detecting as awave is a B -level appreciation. Therefore, the base A (cid:38) x observations become alsoobservables – they become participating observations – and higher, B -level observations B (cid:38) ( A (cid:38) x ) are necessarily involved.Moreover – recent delayed-choice experiments reviewed in [14] indicate that even thedistinction between ‘entanglement’ and ‘separability’ depends on the detecting device ‡ Mind and consciousness are meant here, in a broad sense, as the domain where mental processes takeplace and we interpret our experiences, whether internal or external, and find meaning and significancefor them. bservers and observations in physics theories B -level observations, therefore has to be a C -level observation, namely C (cid:38) ( B (cid:38) ( A (cid:38) x )), following the notation of the previous section.Classical physics realizes only A -level observations, of physically detectablephenomena. New empirical (quantum mechanical) evidence, unexplainable by classicalphysics, led to realization of B -level observations. While these B -level observations –observations of observations, of the kind B (cid:38) ( A (cid:38) x ) – are recognized empirically,they are not accounted for, so far, theoretically. Physics theories don’t know yet howto include observers and observations and how to refer to participating observations A (cid:38) x as part of the subject matter.The higher B -level observations {B (cid:38) ( A (cid:38) x ) } call for insights and understand-ings above and beyond the lower A -level observations {A (cid:38) x } , and these have to bepart of a new, futuristic theory. To successfully account for quantum phenomena thetheory must refer explicitly to (at least) two-stage processes of observation, i.e. , tothe totality of {B (cid:38) ( A (cid:38) x ) }§ (this also explains why local hidden variables meth-ods couldn’t succeed to explain quantum mechanics – such methods are classical-like, A -level methods).Similarly, new empirical evidence regarding B -level observations, such as the non-essentiality of the distinction between ‘entanglement’ and ‘separability’, call for newinsights and understandings above and beyond the B -level understandings, which arethen contained in C -level understandings.Another feature of classical physics is time-symmetry. Classical physics, with A -level observations, cannot account for the arrow of time, neither in thermodynamics norin electrodynamics or cosmology. The observation of time-asymmetry is therefore atleast B -level – the arrow of time is certainly observed, but cannot be accounted for bycurrent physics theories.
7. Hierarchies of observation levels and physical incompleteness
The preceding sections explain the levels of observation and how they relate to oneanother. Recalling that observation is a unidirectional act, from observer to observed, wemay distinguish between the referrer (observer) and referent (observed) levels, puttingthe former at superiority relative to the latter. Hence, in observing the universewe construct hierarchies of observation levels (points or levels of view). Since froman observation level it is possible to observe lower-order observation levels, these arereference levels.As discussed above, we currently experience in physics only very few levels ofobservation – A -level, B -level, and possibly start of C -level observations. But whena new phenomenon is observed that the current theory cannot explain, we seek for new § See also Zwick [17] for a two-stage proposition for quantum measurement. He ignores, though, therˆole of the observer in the process. bservers and observations in physics theories etc. ) occur.Self-reference is therefore an indication for squeezing separate levels into one.Paradoxes are indications of misconceptions, and self-reference paradoxes appear whenlevels that should be separated are instead squeezed (in our mind) into just one commonlevel.
8. Concluding remarks
For millennia humans wish to unlock and decipher the secrets of the universe, includingalso our human part in it. If the universe is the realm where everything happens, thenwe, humans, are part of the big happening. Not only with our physical body, but withthe whole of us, and that includes our mind and consciousness as well.This is important because our observations and their interpretations, from which bservers and observations in physics theories from and within our consciousness. Therefore, questionsregarding the universe are asked from within it, and any theory that aspires toencompass the whole universe, with all the phenomena in Nature, is the universe, viaour consciousness, inquiring itself.The idea that the observers are active participants in the evolution of the universeis certainly not new, and was proposed and acknowledged by many. J.A. Wheeler,for instance, a prominent proponent of this idea, suggested a series of delayed-choiceexperiments (as mentioned above) to corroborate it [14, 15, 16].It is important to emphasize, though, that we don’t propose here that the universeevolves only when humans observe and inquire in and upon it. On the contrary –the universe evolved for a very-very long time without the presence of humans. Onlyafter the earth reached a sufficiently evolved state it could sustain homo sapiens whichstarted inquiring. We cannot also exclude the possibility of existence of other intelligentbeings somewhere else in the universe, also inquiring it and participating in its evolution.With the big-bang a huge amount of power was unleashed, which has been since thenthe moving force for the evolution of the universe. We, humans, are expressions of thismoving power, and inquiring the universe is part of our human mission. Homo sapiens– the inquiring human – allows the universe to be aware of itself.While G¨odel’s incompleteness theorem deals with arithmetics and logic, it servedhere as a guide for possible implications to physics and the natural sciences. The primefeature of the theorem is that self-referencing allows paradoxical self-negation, whichimplies incompleteness.In wishing to encompass the entire universe by the natural sciences, we haveto realize that observers and observations should be included as observables, part ofthe subject-matter of the theory, thereby introducing self-referencing into the theory.The self-referencing may be circumvented by recognizing the hierarchy of levels ofobservations, as discussed in previous sections, but nevertheless incompleteness ensuesdue to the endless hierarchy; and in physics incompleteness implies non-determinism –there are processes whose outcome cannot be predicted by the theory, at least not fully.The persuasion in the existence of a ‘theory of everything’, with a finite set offundamental principles, has been for many years part of the central paradigm of science,and physics in particular. The expectation for a finite TOE rests on the ancient beliefthat the universe is fixed and eternal. But modern physics demonstrates that someempirical results are observer-dependent, so that our inquiries and observations affectthe evolution of the universe. Therefore the universe does not follow a fixed plan, butconstantly evolves openly, with always the possibility of appearance of essentially newphenomena, unexplainable by former theories, with new discoveries that offer new first bservers and observations in physics theories
Acknowledgements.
I wish to thank Noson Yanofsky for fruitful communications. [1] G¨odel K 1931 ¨Uber formal unentscheidbare S¨atze der Principia Mathematica und verwandterSysteme I
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Pontifical Academy ofSciences
Acta 18
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