Formal approaches to Metaphysics
Overview
I am at present thinking about physics from two distinct but connected points of view. I am first of all trying to understand the world, as a philosopher might, as it is described by modern physics. Secondly I am trying to understand how to formalise physics and also perhaps to make some progress towards achieving a formalisation. As befits a primarily philosophical enterprise, my conception of it is vague and fluid. This is my best attempt to present the enterprise as I think of it at present.
It may be argued that fundamental theories in physics go beyond the facts in various ways. We discuss here some ideas about how they do that and how one might attempt to separate out the hard content of a theory from these more difficult areas.
Preliminaries
Learning as Re-discovery

Because of the possibility of using geometric algebra for the formalisation of physics I have recently read some papers by David Hestenes, who has played a prominent role not only in developing geometric algebra for application in physics but also in arguing the case for changes to the way physics is taught to take advantage of the power and simplicity of geometric algebra.

The geometric algebra is not relevant to what I have to say on this page, but there are two other topics addressed by Hestenes which are. One is the idea that physics should be regarded as engaged in the construction of models, of which more later. The other is the idea of learning as re-discovery. Hestenes argues that when a student learns about science he must (or should?) be engaged in a creative act of rediscovery.

Now I have no idea how true this is in general, or to what extent this is a practical way of thinking of physics teaching, but it is very relevant to the way in which I approach learning, and I think if I say a few more words about this here, it might help explain the process of which this page is an evolving snapshot.

I am approaching the study of physics in two complementary ways, both more typical of philosophers than of physicists. This is typical of my approach to almost any subject matter, but I speak here specifically of physics. The two "particular" interests which I bring to bear, are first in the formalisation of physics, by which is meant its formalisation in a manner similar to that in which Russell and Whitehead formalised mathematics in Principia Mathematica, but benefiting from the subsequent development of information technology and a few decades of research into the use of computers to assist in producing formal proofs.

This perspective immediately transforms the learning of physics into exercise in reformulation or recasting which is itself a creative process, as is illustrated I think by the long period of time which David Hestenes and others have spent developing geometric algebra and geometric calculus for applications in physics. The formalisation can of course take advantage of this work, but even in these purely mathematical areas the process of formalisation invites further development, of which Rob Arthan's construction of GA(∞,∞) is an early example.

This process of formalisation is of independent interest, but not quite enough for me. The approach to physics via formalisation is undertaken for ulterior motives, of a philosophical character. It is in part an exploration of the limits of deductive methods, and in part an exploration of what there might be in this particular direction (if anything) which transcends those limits. The particular direction being that of physics, that of trying to understand the world from a physical perspective. Beyond physics here, hints at metaphysics.

This second motivation provides another provocation to reformulation, and in this case reformulations which may be more controversial than those which arise in formalisation. In physical theories, at least those which might be called fundamental, intertwine two distinct kinds of element which are not readily separable. They provide accurate models of the material world, which are couched in terms which go beyond what might be called the empirical content of the models, and hence which go beyond what can experimentally be verified. There seems to be in the practice of physics an acceptance that one may infer to the "best explanation" and that whatever gains general acceptance must then be a true picture of reality. Sure enough, some physicists, like Hestenes, do propound a modelling philosophy, but mostly physicists take literally what I am inclined to call the metaphysical content of their theories.

Beyond the Facts
It may be argued that fundamental theories in physics go beyond the facts in various ways. We discuss here some ideas about how they do that and how one might attempt to separate out the hard content of a theory from these more difficult areas.
Positive Science

I'd like to mention first, to put this discussion into philosophical context, the notion of "positive science" which originates with Comte and has variants in many kinds of positivism. In David Hume we see first the praise of science and the castigation of metaphysics. In Comte we see an awareness that not all that passes for science is good (some entire disciplines are regarded by him as worthless), and that even in domains which are productive, the manner of doing science and the conclusions which scientists draw may be inappropriate.

In positive science the scientist is engaged in undertaking experiments and in reporting the results of his experiments. His theories should not go beyond the experimental data, they should merely summarise it.

What I want to do here is keep this idea in mind, but moderate it somewhat so that we end up with some sense of how a theory of physics may go beyond its experimental mandate, discover some way to separate out the definite from the speculative, and then consider ways in which the issues which then remain uncertain can be progressed.

Moderating Positivism

I don't know exactly what Comte had in mind when he talked about theories not going beyond the experimental data on which they are based, but there is one very basic way in which we do normally expect them to do so. We expect a scientific theory to be a generalisation, of which the experimental observations provide particular instances. The merit of such a theory lies precisely in its going beyond the experimental observations to tell us what the results would be if many other similar experiments which fall within its scope.

A second way in which positivists seek to limit science is phenomenalistically. Thus, one approach to formalisation of physics attempted by Carnap and others in the most recent major phase of positivism was to do so using a language which spoke only of the phenomena or sense data. This is not what I have in mind here. We need to think and talk about the things which cause these phenomena.

The Logical Positivists had another approach to separating science from metaphysics. This was to be liberal about the language of science but insist that a theory be empirically "verifiable". They ultimately failed to define this notion in a satisfactory way, but even if they had succeeded it would not suffice for my present purposes. For this tests the theory as a whole and I am interested in taking a single theory, say for example General Relativity, and distinguishing physical content from verbal or metaphysical content.

Positivist Residue

What remains may not qualify as positivist, it is so only in the limited sense of recognising that there may be differences in the character of different aspects of a physical theory which allow further discussion of certain aspects of a physical theory which has solid observational support. The distinctions include something which might reasonably be described as the distinction between physics and metaphysics, though in this case the attitude towards the metaphysics is not so negative as is typical for positivists. I cannot offer any kind of definition of the distinctions which are at stake here, it is part of the enterprise to make the distinctions clearer, but I have no expectation that they will become sharp.

The distinctions of interest here are between physical and verbal content (in which no great weight should be attached to the word "verbal") and that between metaphysical and other content.

Historically it is natural to think of the structure of space and time as metaphysics. In the twentieth century they have come to be viewed as physics, because first special and then general relativity put forward theories which depended upon changes to our ideas about space and time, and turned out to be better models of physical reality. However, though the theories are better models than their predecessors, there may be alternatives which are equally good in accounting for the observational data but which do not involve changes in our conception of space and time. In the case of special relativity, there are equivalent theories which retain Newtonian space-time and are equally consistent with experimental data. I'm not intimate enough with the detail but lets suppose that Newtonian mechanics with absolute motion and a lorentz contraction on bodies according to their absolute velocities is one such theory and call it NFL. Not sure whether there is a stronger relationship between SR and NFL that their joint consistency with the empirical data. Suppose that they are equivalent in a strong way.


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