The Persistence of Memory: Can We Reconstruct the Past? The Role of Time in Emergent Systems

John E. Gray

The problem of time (in its many guises), as the sage has said, will always be with us. In its many disguises, it effects discussions in all the hard sciences, biology to some extent, as well as psychology and to some extent philosophy as well. While discussions of emergence have primarily been devoted to discussions of the phenomena itself [Holland], does it exist and how do we characterize it, the issue of time has been ignored in most discussion of the phenomena. Note the definition we have taken from Dyson does not mention time at all. We propose a remedy of this by taking an approach that, while analytically based in terms of the origin of the phenomena discussed, remains to a large extent philosophical in nature.

What motivates this investigation of the problem of time in its relationship to the study of emergent systems is the problem of inference of properties that we term emergent. While a system may appear to be emergent, how do we distinguish between its natural evolution relationship between inverse problems is that causes an effect. It is possible to construct a computer model that demonstrates emergence, but this is only one aspect of the triad that is science these days. One should also be able to construct theoretical models (equations) that have provable properties (if it cannot be stated mathematically at some point, it isn't science but rather postage stamp collecting). Finally, a connection must be made to the physical world in terms of a system that actually exists in the external world that must demonstrate the property of emergence. It is this third aspect that we are concerned with. How does one demonstrate that a real system is emergent, rather than naturally evolving?

In order to answer this question, we must be able to take current data that we have about a complex system and infer its past. This leads us into considering two fairly technical area of physical mathematics, namely the 'time evolution' of partial differential equations and an area of scattering theory known as inverse problems [Newton, Colton,]. Given either a differential or a difference equation and boundary conditions, one would either like to infer what the data should be at some later time (the evolution problem which is the direct problem of emergence) or invert a given set of data and infer the 'forces' which created it which is termed the inverse problem of emergence. The direct problem has received a great deal of attention, while the inverse problem has not. Fortunately, there is some guidance to be gained from scattering theory.

In any scientific endeavor that has phenomenological aspects to it, it is not sufficient to consider the direct problem (e.g. the data is required to construct some of the unknown primitives of the theory which is an inverse problem.) Additionally, the data are not direct functions on the space that the differential equation is defined upon; so there can be no direct relationship between the two. There is a delicate problem of relating the primary experimental primitives of the experiment to the primitives of the theory. When one is trying to solve the inverse problem when one has not completely characterized the direct problem, this becomes a particularly acute problem both in the technical sense as well as conceptually. The direct problem of emergence has a difficult problem to deal with when considering data, the data problem for the inverse problem of emergence is even worse. The inverse problem has the five major components which are relevant to our studies [Newton].

1. Uniqueness: 'The first question is whether the map from to is one to one, or if perhaps several points in have the same image. If is not one to one then the inverse problem is ill-posed and our first task is to enlarge the number of quantities to be included as data until it becomes one to one, if that is possible.'

2. Reconstruction: 'Here we are given a set of data that is known to be the image of some member of , and the problem is to find a construction procedure to recover, assuming that the uniqueness problem has been solved.'

3. Construction: 'Suppose we are given data that are not known to be the image under of some member of. Then the first problem is to decide whether underlying forces exist, i. e., whether the data are admissible. We are thus led to the question of characterizing the admissible class. Naturally, we mean by this a characterization other than that is the image of under . One wants to be able to recognize in some way if given data are admissible. In addition to the characterization, of course, as in the reconstruction we require uniqueness and a construction procedure, it is necessary to make sure that the constructed functions when used as forces are actually the pre-images of the given data under.'

4. Stability: 'From a practical point of view it is of great importance to know if the inversion is stable with respect to small perturbations, i. e., if in some useful topology -1 is continuos. Since real data always have errors, an unstable procedure is of little use. Part of the stability problem is connected to the characterization problem. If the boundary of is hard to recognize most small perturbations of all admissible data may make them inadmissible. This problem has received little attention.'

5. Consistency: When the characterization problem is beset by an over abundance of data, it may over-determine the forces, they are redundant. In this case, a subset of the data is sufficient so that the problem can be simplified. This is the consistency problem.

All of these are technical concerns are to be dealt with when using data to solve the inverse problem of emergence. Even for theoreticians who are not concerned with the relationship of data to theory have their own version of the problem to deal with. In differential equation theory this is called the well-posed problem which is essentially a study of the properties of differential equations that evolve their initial conditions into the future in a manner that maintains uniqueness of the solution and continuity. This desired property is not present in many systems that evolve forward in time, so techniques have been devolved that help minimize the problems so that some of the desired characteristics are maintained [Payne]. The problem of uniqueness becomes more acquit moving backwards in time as one tries to determine the conditions that gave rise the present. If one only has data from the present, the past is in some since inaccessible to us, because we lack sufficient information to guarantee uniqueness of its reconstruction. Additionally, there may be a problem with stability, that is common to all emergent systems. The incompleteness of the problem of accessing the past that is inherent in emergent systems, leads one to consider what is the underlying ?cause? of the acausal problem of emergence.

At the heart of the problem is the nature of time and its relationship to what we have traditionally meant by emergent phenomena. Since the past is not necessarily discernible from the present, where are we left in our understanding? One is left to reconsider the role of time and its relationship to inversion problems. This lead to the discovery of some paradoxes about the nature of tense, which is at the heart of the problem. Discussions of tense and it relationship to our understanding of time were brought forth by McTaggart. He propertied to prove that time was unreal.

McTaggart begins his argument [Oaklander] for the unreality of time by arguing that there are two ways we can naturally talk about time. Either we take time to be moving or passing from the far future to the near future, from the near future to the present, and finally from the present to the past. In this case the delineation of events from past to present to future are termed to be located in an A-series. The other possibility is to order events as being earlier than, later than, or simultaneous with other events. Furthermore, these events are unchanging and fixed like geometry. This ordering of events does not change, once an order has been established, it remains well ordered. This ordering of events is referred to as a B-series. Since the order of events cannot change, B-series cannot come into and out of existence. Thus, if a B-series is to exhibit the characteristics of a time series, it must be associated with an A-series.

The argument about the unreality of time is directed at the properties of an A-series. He argues as follows [Oaklander]:

1. If the application of a concept of reality implies a contradiction, then that concept cannot be true of reality.

2. Time involves (stands or falls with) the A series and temporal becoming; that is if the A-series involves a contradiction, then time involves a contradiction.

3. The application of the A-series and temporal becoming to reality involves a contradiction.

4. Therefore, neither the A-series nor time can be true of reality, thus time is unreal.

The crucial point is (3). McTaggart argues (wrong tense?) that events pass through time; from the future to present to the past, thus every event must be past, present, and future. These are incompatible properties, an event if present is not past or future, an event that is past cannot be present and future, likewise if it is future, it cannot be past or present. Any sense of motion entails a contradiction, every event is and is not both past, present, and future---time is unreal.

When we contemplate these type of paradoxes associated with the tense of time, we are caught in the same web when we talk of emergence, for any sense of system demonstrating the phenomena of emergence is caught in the spider web that is tense. Therefore we are lead to the conclusion that emergence cannot truly exist because its existence would involve a paradox that is identical to the paradox associated with tense. Thus, we can only assume a position similar to Heidegger's Dasein [Heidegger]in which being and time become manifest. Emergence becomes a property of the universe, not to be explained but rather to be observed in the same way that we observe the effects of Darwinian evolution.

Once postulated, the principle of evolution is accepted as a meta-scientific principle that functions as the ultimate cause of all data. Much like Einstein's reply to a young lady as to what he would do if the expedition to confirm the bending of light predicted of general relativity, in fact, failed to confirm it: namely, blame the data. When there is incomplete data instead of resorting to a Jaynes maximum entropy [Jaynes] approach of minimal explanation that fits the facts, one is forced to resort to the retort that 'then I feel sorry for the dear Lord, for the theory is correct'. Embedding a principle of evolution in physics is now somewhat fashionable among some cosmologists and relativists. Based on the work associated with Gell-Mann and Hartle [Hartle] who have tried to establish a version of quantum mechanics that applies to the whole universe, the notion of quantum evolution and cosmology is being discussed [Smolin in Brockman, Hartle, Keifer]. One is then lead to the question of how did complexity come into the universe assuming this principle of cosmological evolution. To answer this question, we must be able to reconstruct the initial conditions of the universe and propagate them forward in time to reconstruct the present. It has be assumed by the cosmologists that this is possible, but we question the implicit assumption buried in this belief. Namely it assumes the solution to an inverse problem that we have already discussed. We are thus lead to the question: Given the present, can we reconstruct the past? We hope to provide a partial answer to this question, its relation to the inverse problem of emergence, and the tie between emergence and the tense-less theory of time.

Acknowledgments: The author would like to thank Andy Vogt, Frank Reifler, and Don Hilliard for many conversations about the nature of the universe, physics, and the subject of time. The discussions were, of course, timeless.

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