Modern Physics, Newtonian Paradigm, and the notion of State

[Tim Gwinn <***>]

My thoughts on Modern Physics, Newtonian Paradigm, and the notion of State....

 

The Newtonian paradigm is a very fundamental and pervasive construct for science and physics in particular. In Life Itself (ch. 4), Rosen details the mechanistic nature of the Newtonian paradigm by using the basic deterministic form as it comes from Newton. However, the Newtonian paradigm is not at all limited to this particular version, and presuppositions in the Newtonian paradigm are found implicit in its structure, as well as in its explicit laws. The best description I've found that encapsulates this is in Rosen's chapter in Theoretical Biology and Complexity (1985):

    "In this section we shall briefly review the salient features of the class of mathematical or formal systems that are now accepted as models of material reality (what we have called "natural systems"). That is, these are the formal systems that can sit on the right-hand side of a commutative diagram such as Fig. 1. [the Modeling Relation - TG] As we shall see, our basic unchanged ideas on this subject go back, in one way or another, essentially unchanged, to the mechanics of Newton's Principia. Despite enormous technical variations in mathematical language (e.g., from classical to relativistic to quantum; from continuous to discrete time; from continuous state to discrete state; from deterministic to stochastic; from autonomous to forced; from finite-dimensional to infinite-dimensional; etc.), the basic epistemological presuppositions remain the same, untouched and all but unnoticed.

    This basic presupposition, as we shall see, is that systems have states and that upon these states some kind of dynamical laws, or equations of motion, are superimposed. The states represent in a sense what is intrinsic, while the dynamical laws reflect the nature of the impinging environment in acting on what is intrinsic. Thus the dichotomy between states and dynamical laws embodies a distinction between system and environment. Also, in a formal way, the dualism of states and dynamical laws exactly parallels the purely mathematical dichotomy between propositions and inferential laws, or production rules, which is nowadays considered as the anatomical foundation of any mathematical formalism whatsoever.

    But this basic presupposition, so familiar and axiomatic to us all, involves tacit hypotheses, not just one but several, about the natural world and its mathematical images. We will examine these in detail in the following sections. For the moment, simply review some of the salient formal and historical roots of what we shall call the Newtonian paradigm." [p. 181]

He continues later on the next page:

    "Newton recognized that the arbitrary specification of configuration at an instant placed no restriction on the velocities of the constituent particles; i.e., both configuration and the first temporal derivative of configuration could be chosen completely arbitrarily. One might think that the same would be true for second time derivatives of configuration (i.e., acceleration) and for all higher time derivatives. But here Newton interposed his deep insight. He said, in effect, that the rest of the world (i.e., the environment of our system of mass points) exerts forces on the particles. What these forces are, intrinsically, cannot be (and need not be) specified, but the effect of these forces is to determine the acceleration of the particles of our system. That is, insofar as the "force" experienced by a particle is determined by where it is and how fast it is going, the acceleration of the particle, and hence all higher temporal derivatives of configuration, are then completely determined by configuration, rate of change of configuration, and the "forces" then imposed by the rest of the world. Mathematically, this amounts to expressing acceleration recursively as a function of the lower temporal derivatives. This _expression_ is the dynamical law governing the system; by a mathematical process of integration we can convert this dynamical law into a relation giving configuration as a function of time."

 

The term 'Newtonian paradigm' then applies to any formal encoding framework in which systems are characterized by their "state" (spatial configuration and rate of change of configuration) and by a separate characterization of their trajectory (e.g., equation of motion) from that state to another one in temporal increments as specified above.

 

Note that there is no specification of the details of the spatial configuration, so that probabilistic (e.g., quantum) as well as deterministic (e.g., classical) configurations fall under this classification, as do deterministic or probabilistic (e.g., Feynman) trajectories. It also does not matter how many dimensions of space we are specifying, nor the exact size of the temporal increments. And so on. In short, any kind of an encoding framework which follows these general rules will fall under the Newtonian paradigm.

 

The effect of Newton's designating that some causes are intrinsic (i.e., from state) and some are extrinsic (i.e., from dynamical laws) also implies an absolute partitioning of causes in Aristotelian terms: the state information contains a material cause in its initial conditions, formal cause in its parameters (that mediate between system and environment), and efficient cause in its dynamical laws. [LI p. 102-103] The key thing is that this partitioning of causal categories, once specified for a given system, is then fixed. This encoding thereby stipulates a kind of implicit metaphysical principle. It also implies that all systems adhere to this fixed partitioning of causal categories. The class of systems which do adhere to such a fixed partitioning is the class of mechanisms. [TB&C p. 190]

 

All this is irrespective of the details of the variations of the encoding, whether they be variations in the specifics of 'state' and 'trajectory' as noted above or variations in the encoding specifics, such as Hamiltonians, Lagrangians, Feynman path integrals, Hilbert spaces, Riemannian manifolds, etc. Also, any kinds of extraneous principles (e.g., symmetry principles) which apply to encodings within a Newtonian paradigm do not thereby change the paradigm, but instead operate within it.

 

Practically speaking, this Newtonian paradigm pervades all of physics which rests on some kind of encoding of systems in a spacetime. From within this paradigm, it is possible to see symptoms of complexity, such as the inability to solve the N-body problem in closed form (i.e., as a model rather than as a simulation), the partial differential activation-inhibition networks of Rosen [EL ch.23], etc. The paradigm itself, however, does not allow us to go beyond these symptoms to more complete encodings of these complex systems (aside from approximations), because that would require an encoding paradigm which allows the intertwining of causal categories.

 

Even topological quantum field theory (TQFT), which is a theory which portrays certain invariants of a QFT, does not seem to take us out of the Newtonian paradigm. Because what it is that comprises the content of the invariants are collections of quantum states in Hilbert space. From Baez (http://math.ucr.edu/home/baez/symmetries.html):

"In any event, a quantum field theory that is invariant under all diffeomorphisms of spacetime is called by physicists a "topological quantum field theory," or TQFT. It has only been a few years since people have been seriously studying worked-out examples of TQFTs. The understandable examples so far have been in 2- and 3-dimensional spacetimes, not our own lovely 4-dimensional spacetime. But these examples are still amusing and perhaps enlightening. They also have a lot to do with KNOTS - but that's another story.

 

What's a topological quantum field theory, mathematically? It's a functor. Namely, it is a functor from the category Cob to the category Hilbert. The category Cob is the category whose objects are (n-1)-dimensional manifolds ("space") and whose morphisms are n-dimensional manifolds ("spacetime") having one (n-1)-dimensional manifold as "incoming" and another as "outgoing" boundary. We say that the n-manifold is a cobordism between the two (n-1)-manifolds.

 

...snip...

 

In any event, a TQFT is a functor from the cobordism category Cob (for some dimension n) to the category Hilb of Hilbert spaces (with not necessarily unitary operators as morphisms!). That is, a TQFT would assign to each (n-1)-dimensional manifold a Hilbert space of states representing the states the system can take on that manifold, which represents space at a given time. And given a cobordism between (n-1)-dimensional manifolds we obtain a linear "time evolution" operator between the corresponding Hilbert spaces. Since there is no such thing as waiting "a certain amount of time" in a general TQFT, this time evolution operator only depends on the topology of the cobordism between two manifolds.

 

...snip...

 

Note that a TQFT is a "representation of a category," that is, a functor from a category to a category of vector spaces (actually Hilbert spaces). The symmetries in topological quantum field theories generalize the symmetries of earlier theories, thus, since earlier theories only dealt with group representations, while TQFTs are category representations."

 

It does seem to me possible, though, that just as there are systems of dynamical equations such as the activation-inhibition networks of Rosen which can display the symptoms of complexity, that there might be TQFT representations which are complex (e.g., impredicative), and whose counterparts in Hilbert space might be symptomatic of complexity.

 

One way to step outside the Newtonian paradigm, then, is to step outside of a preoccupation with spatial configurations, and their temporal evolution, entirely.  The functional relational models of Rosen are of this nature. They are atemporal and they do not map into states. I also think it is a misnomer to call these relational models "synchronic", since that insinuates a temporal sense to them but they are atemporal. In TB&C, Rosen recounts his failed attempts to work toward a realization of an (M,R)-system by interpreting it as a sequential machine or finite automata with states. In light of these failures, Rosen remarks:

"In particular, the very fact that the same mathematical formalism (e.g. a network) could be interpreted in so many disparate physical ways ultimately led me to suspect that something crucial might be missing from the mathematics itself. In other words, I began to entertain the possibility that our conventional mathematical descriptions of physical reality, which have essentially gone unquestioned for three centuries, might themselves be fundamentally deficient, that it was this deficiency that was responsible for the problems posed by an attempt to realize physically an abstract functional organization." [TB&C p. 178]

These functional relational models are in an entirely different formal universe of discourse than a formal universe of discourse built around spatiotemporal relations. It is, metaphorically speaking, like a parallel formal universe - a rather alien one to the one in which we are used to doing physics in. In TQFT, the relations represented are still between spatiotemporal quantum states: the encoding has gone from the natural system to a spatiotemporal encoding and then to a topological one. In Rosen's relational models, the encoding goes directly from the natural system to the functional relational model. This allows for representations of organizational qualities that cannot be encoded into the spatiotemporal encodings of the Newtonian paradigm. (Of course, the opposite is also true: these relational models have abstracted away state information entirely - they have "thrown away the physics".)

 

I would imagine that if these two very different formal universes of discourse can both provide us with commuting models for various natural systems, then it is highly likely that we can create yet more formal universes of discourse (ones that provide model types that cannot be reduced to ones of the first two types of formal universes) that could even further expand on our model making abilities. Probably such formal universes would provide very strange kinds of models as compard with those we normally associate with physics.

 

Regards,

Tim