Philosophical Reflections in Physics and Math

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Philosophical Reflections in Physics and Math

The Concept of A Field

Up until the time that Michael Faraday introduced his concept of the field into nineteenth century thinking, physicists believed the most fundamental description of physical/material phenomena was a function of the manner in which discrete substances or pieces of matter were arranged. However, Faraday argued that the most fundamental description of the events of physics should be rooted in continuous, rather than discrete, processes.

H. C. Oersted, a Danish physicist, had made an interesting discovery in 1820. He found that the moving charges of an electrical current were capable of deflecting the needle of a compass which had been placed in a position perpendicular to the direction of motion of the moving electric charge.

This finding was noteworthy for two reasons. (1) It suggested there was a connection of some sort between electrical and magnetic phenomena. (2) Unlike the cases of gravitational and electrostatic forces, in which forces were transmitted between interacting objects along lines that linked the centers of these objects, moving electrical charges generated forces which were perpendicular to the usual direction of the transmission of forces.

Faraday believed Oersted's observations meant electricity and magnetism were different manifestations of one and the same force. The illusion of the existence of separate forces was more an artifact of the experimental situation in which relative motion was used to induce the underlying, single force to manifest itself in primarily an electrical or magnetic mode of expression.

This linking of electricity with magnetism was the staging area from which Faraday launched his revolutionary concept of the field. He jettisoned the traditional idea of discrete bodies acting on one another in terms of the Newtonian notion of 'action-at-a-distance'.

Faraday replaced that idea with his formulation of a potential field of force. In other words, he believed objects were linked by means of a field of force that continuously manifested itself in the space which permeated and surrounded the objects being linked by the field.

Later on in his career, Faraday proposed that the idea of a potential field of force be extended to cover the manifestation of all forms of physical force, not just those of electricity and magnetism. By suggesting such an extension or generalization of the field concept, Faraday became the first physicist to advocate using a unified field theory approach to account for all physical or material phenomena.

Continuity: an Integral aspect of the field concept

When Faraday used the term 'continuous field of force', he had something particular in mind. He believed a sphere of influence surrounded every charge.

The properties of this sphere of influence were a function of the character of the charge which generated it. However, irrespective of the particular properties of the sphere of influence that were generated by a given charge, all such spheres of influence manifested themselves in a continuous fashion.

Suppose one were to use a test charge to engage the sphere of influence at some point 'p'. According to Faraday, one should be able to anticipate that the properties of a given sphere of influence have the potential to affect the test charge in a determinate way at the point of engagement.

When considered as a whole, the sphere of influence of a given electrical charge will give expression to a field whose strength of intensity of electrical charge will vary from point to point in that field in a way that reflects the character of the electrical charge which generates the field in question. Thus, if one were to consider some other point, 'x', at some distance, 'd' from the point, 'p', at which one initially had engaged that field by means of a test charge, then, according to Faraday, one would find that the sphere of influence of the field generated by the electrical charge would affect the test probe with a strength of electrical field intensity which was characteristic of the field at that point of engagement. In fact, such fields are said to be continuous because one should be able to select any point in the interval 'd' between 'p' and 'x', or between any other points that might be selected, and determine the strength of electrical intensity with which the sphere of influence of an electrical charge's field will affect a test probe that is introduced at such intermediate points.

The idea of a continuous field requires that there can be no point within the sphere of influence of a given electrical charge which does not have the potential to affect, with some manifestation of strength of electrical intensity, a test probe which engages the field at that point. In short, the potential capability of a field to exert a force of variable strength of electrical intensity at each and every point of the field renders the field continuous.

One of the up-shots of the foregoing position is as follows. The idea of atomism is rejected since such an idea necessarily carries with it a discrete perspective in which the phenomena of the physical universe are expressions of interacting particles that are distinct and separate from one another in certain ways.

Instead, the atomistic properties which various phenomena seem to possess are only apparent and are not real. Underlying these discrete-appearing surface features is a smooth or continuous distribution of field variables manifesting themselves in ways which are sometimes intense and concentrated or localized.

At other times, these field variables are dispersed and not localized. The combination of these concentrated and dispersed manifestations of a continuously varying set of field variables gives rise to the illusion there are discrete events.

Thus, from Faraday's perspective, there are no fundamental entities such as elementary particles or atoms. Everything is an expression of a single unified field which manifests itself on a continuous basis by means of transitions in the way various field variables are given expression through the field. These field variables are not individual, distinct, discrete features. They are, in a sense, abstractions or samples which have drawn from one of the smooth distributions of values which characterize a given field’s manner of manifesting itself.

Although all of the experimental evidence available to physicists in the 1800s supported Faraday's idea of a field, Faraday's position was not unassailable. For example, on some exceedingly small level of scale, there could be one, or more, points which fall within the sphere of influence of an electrical charge and, yet, do not manifest a strength of electrical intensity capable of affecting a test probe inserted at that point.

In this case, the variable distribution of the strength of electrical intensity that characterizes the field at such points would fall off to zero. As a result, the field would be manifesting discontinuous properties. However, the level of sophistication of experimental methodology may not be able to detect the presence of such points of discontinuity and, consequently, would produce experimental results that indicated the field in question was continuous.

One could approach the test charge issue from a perspective that is somewhat similar to Weierstrass' epsilon/delta format. In other words, the neighborhood of these points can be explored on varying levels of scale.

Within the limits of one's instrumentality and methodology one could challenge the assumption of continuity in such neighborhoods as much as one likes. The idea of continuity stands as long as one can meet any test challenges which are made in a neighborhood whose outer boundaries are marked by the two points, 'p' and 'x', and which fall within the parameters of the sphere of influence of an electrical charge.

Another possibility is to get entirely away from approaches requiring one to construe continuity in terms of a series of inexhaustible points that occupy the space within a certain set of parameters. For example, continuity might be construed as an expression of the integrity of the phase relationships (For now, one might characterize phase relationships as expressions of the way different aspects of ontology interact with one another while in certain states, conditions, and cycles of manifestation. These states, conditions, and cycles constitute the phases of an object or process during particular modes of being that give expression to various dimensions of possibility inherent in an object’s or process’ manner of being) which are preserved among the neighborhoods that constitute the 'point-structures' of a field latticework being probed by a test charge or force of some sort.

From the perspective of the foregoing position, a field is not infinite. It is finite.

What makes it continuous is the network of phase relationships which link one neighborhood with another, or which link the different, internal aspects of a neighborhood with one another. As long as there is some minimal set of phase relationships that permit a latticework, or a given neighborhood, to manifest one or more of the ratios of constraints and degrees of freedom which are encompassed by the spectrum of ratios that constitute the structural character of the latticework, or neighborhood, then continuity has been maintained.

Given the foregoing, if one found 'holes' (that is, non-active areas which were not manifesting field properties) in the vicinity of a neighborhood, or somewhere in a latticework, these 'holes' would not necessarily represent disruptions in the continuous character of the neighborhood or latticework. For example, conceivably, the character of a field could involve a complex structure such that the field is defined as being wherever it manifests itself.

If one finds a 'hole', in the foregoing sense, one has merely located one of the parameters or boundary markers of the field. The more holes of this sort there are, the more complex the boundary structure of the field becomes. As such, the field becomes a topological object comparable to a very complex torus.

Thus, a field manifests itself continuously, but not necessarily in the sense that every point of a given space is under the sphere of influence of that field. The field is continuous because one, or more, of the ratios of constraint and degrees of freedom which characterize that field is (are) being manifested at any given instance of time.

Continuity is a function of how a certain latticework of order manifests itself and preserves itself across time. This does not necessarily require the latticework to be able to express itself at any given point of space. Moreover, if a given field is capable of withstanding any sort of epsilon/delta-like challenge which might be thrown at it, this is a special case that does not violate the more fundamental property of continuity as characterized in terms of order, as opposed to being characterized in terms of space.

A field may have a dialectical relationship with the dimension of space through which it is manifested, but the field is not reducible to space. Other dimensions must interact with space to generate a field, and when the field is generated, it need not occupy all of space to be continuous. The field is continuous by virtue of the set of phase relationships to which the latticework that constitutes the field gives expression.

Entropy as a ratio of constraints to degrees of freedom

If one characterizes entropy in terms of the ratio of constraints to degrees of freedom in a given context, then, one can speak of the entropy spectrum for a structure. Such a spectrum constitutes the envelope of ratio values which are possible for that structure under a variety of circumstances- whether induced or spontaneously manifested.

In general terms, if there is a change in the ratio of constraints to degrees of freedom for a given structure, then, there has been a change in the entropy character of that structure. Or, said slightly differently, another aspect of the structure's entropy spectrum has been manifested.

If the nature of the ratio change is to shift the manifestation of the structure's entropy spectrum in the direction of more constraints, relative to degrees of freedom, such a change is said to constitute an increase in the entropy of the structure. This is the case since, relative to the entropy state prior to the change in question, the structure is less able to give expression to its degrees of freedom.

Neither an increase in entropy, nor a decrease in entropy, affects the orderedness of the structure or system undergoing a transition in the way the entropy spectrum is being manifested. Orderedness is a reflection of the fact there is some kind of ratio of constraints to degrees of freedom being given expression though a set of phase relationships that are bound together to form a particular point, neighborhood, or latticework.

Very rarely, if ever (at least in the created realm), would one find cases of pure constraint, without degrees of freedom (e.g., even at, or near, Absolute Zero, there are a variety of strange phenomena which have been observed to occur and, therefore, this state does not constitute a realm of pure constraint as once was thought), or pure degrees of freedom without constraint. Usually, constraints and degrees of freedom pair off to form a source of tension of a dialectical nature.

Therefore, as far as the issue of orderedness is concerned, what the character of the associated ratio is doesn't make any difference. As long as a ratio is present, then, the degree of order doesn't fluctuate even if the character of that ratio does change.

This is in direct contrast to the way modern thermodynamics and information theory tie orderedness to the idea of entropy. On the other hand, the present position is resonant with certain aspects of Sheldrake's views on these issues which were outlined in the essay on morphogenetic fields and the hypothesis of causative formation which can be found elsewhere in this folder.

Approached from the foregoing perspective, the idea of a smooth distribution can be construed in terms of an envelope of values or a set of parameters. This set describes how the entropy spectrum manifests itself through an overlapping sequence of transitions in the ratio of constraints and degrees of freedom governing the dialectic of two or more points, neighborhoods, or latticeworks of coherent phase relationships.

Any individual expression of a given ratio of constraints to degrees of freedom is, in point of fact, a phase state. Consequently, the envelope of values which gives expression to the set of ratios that make up the entropy spectrum governing the dialectical interaction between two or more neighborhoods, etc., constitutes the bundle of phase relationships which mark the different facets of the way the neighborhoods, etc., are, or can be, linked with one another.

If there is a disruption in the phase relationships connecting different neighborhoods, etc., such that the ratio of constraints to degrees of freedom which gives expression to this connectivity drops to zero, then, there is no longer any connection between the neighborhoods. Continuity has been broken.

When the ratio is zero, this means, effectively, none of the neighborhoods, etc., which previously had been linked are capable of constraining one another. Furthermore, they are not capable of entering into dialectical engagement with one another in accordance with some range of degrees of freedom.

In short, the phase relationships that had connected the neighborhoods and which had been given expression in the form of shifting ratios of constraints and degrees of freedom of dialectical interaction, no longer exist. The minimal condition for continuity - namely, that the neighborhoods in question be linked through some on-going manifestation of an entropy spectrum, is no longer capable of being satisfied.

In the context of hermeneutics, one of the ways shifts in the character of the entropy spectrum manifest themselves is the manner in which such shifts affect the activity of the hermeneutical operator ( which, briefly speaking, can be characterized as a set of operations consisting of: (1) identifying reference, (2) reflexive awareness, (3) characterization, (4) the interrogative imperative, (5) inferential mapping, and (6) congruence functions) engage the various horizonal considerations and work toward a hermeneutical orientation. Any increase or decrease in the constraints that are placed on the activity of the hermeneutical operator which is not congruent with the structural character of that aspect of ontology or experience to which identifying reference is being made is, generally speaking, construed as an increase in the entropy of the hermeneutical system. Similarly, any increase or decrease in the degrees of freedom which occur with respect to the activity of the hermeneutical operator and which are incongruent with the structural character of the aspect of ontology or experience to which identifying reference is being made is, generally speaking, to be construed as an increase in the entropy of the hermeneutical system.

Thus, increases in entropy are a function of what brings distortion, deviation or error into the activity of the hermeneutical activity. Consequently, the hermeneutical analog for high entropy concerns those instances of hermeneutical operator activity in which there is either: (a) an insufficient number of constraints or degrees of freedom of the right character available, or (b) there is an excess of constraints and/or degrees of freedom of the wrong character. The 'rightness' and 'wrongness' of character alluded to in the foregoing depends on whether or not a given instance of hermeneutical operator activity is capable of being used in a constructive, positive, heuristically valuable fashion so that progress toward establishing full analogical congruence can be achieved (i.e., one approaches the truth of something as a limit).

There appear to be a variety of inferences one might make with respect to educational issues on the basis of the foregoing discussion linking entropy and the activity of the hermeneutical operator. Perhaps, the most fundamental of the points which might be made in this regard, however, concerns the following consideration. The educational process should provide the individual with a means of learning how to go about constructing, generating, acquiring and/or searching for an entropy ratio which will maximize the heuristic value of one's dialectical engagement of experience and/or reality.

Lines of force

For Faraday, a line of force is inherently characterized by two polarized ends of opposite charge. That is to say, he believed one could not have either kind of polarity in isolation. From the perspective of this essay, a line of force could be characterized as a manifestation of the phase relationship which arises between polar ends.

Faraday demonstrated that if one had a wire of a certain conductivity, then, the total current which could be induced in that wire was entirely dependent on the number of lines of force which were cut. This was the basic rule of electromagnetic induction.

Faraday did not commit himself to any particular view as to the specific identity of a line of force. They could be lines of vibration, or they could be lines of ether flow, or they could be lines of action at a distance. However, Faraday did feel that whatever their actual character may be, they stood for the physical means by which, or through which, force was transmitted in nature.

Unfortunately, Faraday was never able to demonstrate that lines of force had a physical reality of their own. Nevertheless, he did believe his notion of lines of force had a heuristic value. More specifically, the idea of ‘lines of force’ helped lend the sort of concreteness and form to a theory which would help one to develop ways of testing the theory, making deductions with respect to the theory, and so on.

For Faraday, the fundamental - indeed, the only - physical substance responsible for natural phenomena is force. Thus, a field is, from Faraday's perspective, an expression of the presence of force. Another way of stating the same thing is to say that forces generate fields which can be described in terms of lines of force.

According to Faraday, force acts exclusively on the contiguous points of force with which it comes into contact as it manifests itself. This concept of force was quite different from the kind of force which was operative in Newton's theories.

For Newton, force was something which any given particle exerted on other particles that were at a distance from the particle exerting the force. As such, forces did not manifest themselves through a field - whether it be a field of force or some other kind of field. Newtonian forces manifested themselves directly on the other body in an instantaneous fashion.

Within the context of Newtonian theory, one might speak of measuring field intensities in terms of force per unit of mass. Moreover, in such a context, one might speak of the force exerted on a body which encounters a region of given field intensity as being equal to the product of the mass of that body and the level of field intensity which it encounters. However, the role played by the use of field terminology in Newton's theories amounted to little more than a mathematical means for arriving at an answer in relation to questions concerning the amount of force which was being exerted on a particular body in a given set of circumstances.

One of the major stumbling blocks one encounters in attempting to construct a testable theory built around Faraday's concept of a field, as expressed in terms of lines of force, is to establish the laws which govern the way forces interact. Without such laws, one is in no position to attempt to account for how material phenomena can be construed as expressions of an underlying set of interacting forces.

Faraday had proposed only one such law: the conservation of force. Essentially, this law indicated there is no change in the total amount of force in the universe. However, this law was not clearly stated.

In other words, Faraday never clearly spelled out how the field of force giving expression to one body exerts its effect on the field of force which gives expression to some other body being acted upon. Consequently, one does not know how the changes in force occurring in one region are to be compensated for by changes in force in contiguous areas, and such knowledge is essential for any proposed law of the conservation of force.

Whatever specific content one decides to introduce into the laws which govern the interaction of forces, they must retain three features inherent in Faraday's notion of force if such laws are to be consistent with what Faraday may have had in mind. To begin with, forces have a definite location. Secondly, Faraday's idea of force has an element of directionality to it. Finally, the rate at which a force is transmitted through a field is not only finite, but this rate also is dependent on the character of the contiguous forces that it encounters during the process of being transmitted through the field with which the force is associated.

The idea of lines of force may fit in nicely with the idea of a latticework as an expression of the set of constraints and degrees of freedom giving expression to the dynamics of phase relationships that arise as a result of the dialectic of dimensions which have been set in motion by the underlying order-field. On the other hand, in the context of phenomenology and hermeneutics, these lines of force are not linear in character, nor are they limited to three or four dimensions.

These 'lines' are, instead, complex, multi-dimensional manifolds which are shaped by a variety of chaotic attractors. Nonetheless, like their counterparts in the physical world, hermeneutical and phenomenological lines of force have a capacity to affect, alter, shape, orient, transform or operate on 'objects' which come within, and are receptive to, the sphere of influence of the field that makes possible lines of force of such structural character.

For example, one can speak of the lines of force which are established phenomenologically and hermeneutically between focus and horizon. These components can be considered as two polarized ends of opposite 'charge'.

Obviously, however, the nature of a hermeneutical or phenomenological charge is something quite different than an electrical charge. However, the important consideration here is the aspect of polarity that exists between focus and horizon since there is a dynamic tension between them which generates phase relationships or lines of force.

The structural character of any given pair of focal/horizonal phase relationships will be a function of the intensity, orientation, and so on, of the 'charge' character that arises whenever one brings a focus and a horizon of such determinate nature into dialectical engagement of one another. Moreover, as is the case with electrical charge, one cannot treat either focus or horizon in isolation from one another. Where one is, one also will find the other.

One measures the continuous mapping of the lines of force between oppositely charged poles in an electrical field by inserting a test probe into the field, thereby deriving an indication of the electrical potential which has been created at the point of insertion. One also can sample something of the flavor or character of the continuous mapping of the lines of force which have been generated between a given focus and horizon by inserting into the hermeneutical or phenomenological field a test probe. This probe is rooted in one, or the other, of the poles, thereby, permitting one to derive an indication of the hermeneutical or phenomenological potential which has been created at the point of insertion.

In the context of hermeneutics and phenomenology, the character of the test probe will come in the form of questions, emotionally charged issues, beliefs, values, ideas and so on. Anything which is capable of eliciting, evoking or inducing various kinds of phenomenological response is capable of serving as a hermeneutical probe.

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