Handling Complex Structure

In the vast majority of cases where one needs to model a physical system using mathematics the structure of the system as it would be rendered within our most advanced or fundamental theories is vastly more complicated than we can feasibly handle. Any physical system with macroscopically many particles, for example, falls in this class of cases. For this reason alone we require techniques that allow us to simplify the model description to a such a degree that a mathematical analysis becomes feasible. Even if we could handle the full detail feasibly, however, we would still require methods to reduce complexity so that we could focus in on the dominant behaviour pertaining to the question or problem at hand, and to gain insight into why this behaviour is dominant. In this subsection we will consider ways in which the methods of abstraction from the complexity of a phenomenon are epistemologically significant. This will provide evidence that typical logical accounts of the theory- world relation miss epistemologically important structure from the point of view of how we actually relate theories to phenomena and in terms of the epistemological

significance of the methods used.

There are two main types of procedures that are essential for abstraction from complex structure:

1. The selection of modeling assumptions motivated by a consideration of the phenomenon that avoid the need to deal with the complex descriptions of our most advanced or fundamental theories; and

2. The application of analytic model reduction techniques in order to simplify the form of the under- or partially constrained equations of a theory or to reduce the model descriptions of a well-constrained system of equations.15

Both kinds of abstraction are epistemologically significant, as we will see, and reveal the manner in which the picture of theory application as a “derivation from theory” is woefully inadequate in terms of giving insight into the epistemology of theory application.

Consider how a model of a physical double pendulum is constructed. Given the simplicity and high symmetry of a double pendulum system (see figure 3.3) there is quite a bit that can be done using modeling assumptions in order to generate a simple yet epistemically useful model. There are a number of macroscopically stable assumptions that can be made that can effect a rather miraculous reduction in structure over how we would be compelled to describe the system in terms of its microscopic constitution. As a first approximation it is valid to treat the pivot point as rigid, the two joints as frictionless, the rods as rigid, the rods as one-dimensional, the rods as massless, the rods of equal length `, and the weights as sizeless. How good these are as approximations depends on the details of the constitution of the system (respectively, how stable the pivot is, how small the joint-friction is, how rigid the rods actually are, how thin/homogeneous the rods are, how large the ratio between the weights and the rods is, and how small the weights are). So, depending on these details, one or more of these assumptions may be better than just a first approximation. From a complexity reduction point of view the use of symmetry assumptions in this way is extremely powerful. Indeed, over an atomic-constitutive model, this method reduces the degrees of freedom from over 1025 _{to 4!}

15_{There is a third possibility that reduces to the latter sort, viz., the reduction of the under-}

constrained equations of on theory to underconstrained equations of another, which is discussed in

α

β

Figure 3.3: An ideal four-dimensional double pendulum system.

An interesting epistemological feature of this approach is that it effects a quite extraordinary reduction in the structure (analytically and geometrically) of the model

before one has decided what theories to use in the modeling. The decision to treat the pendulum in terms of rigid bodies, frictionless joints,etc., can be made independently of any particular system of mechanics used to model the pendulum. Thus, we may think of this complexity reduction procedure as the translation of phenomenological constraints into constraints within some theoretical framework. The result, then, is that using macroscopic constraints imported from experience it is possible to obtain a useful model without any explicit consideration of a classical base model. Quite far from being a “derivation from theory”, then, the model is essentially formulated by direct abstraction from experience into a given theoretical framework. Further- more, the role of experience in model construction seen here is not accounted for in typical logical approaches, including Suppes’s “hierarchy of models” picture of the theory-world relation. Thus, even at the earliest phase of model construction there is already epistemologically significant structure that logical approaches miss, viz., pre-theoretical constraints.

The construction of a model of the orbit of a NEO also uses symmetry and rigidity assumptions. The major bodies in the solar system are very close to being spherically symmetric and their rotation is very close to that of a rigid body. They are so close that the deviations are comparable in size with only the second order of relativistic corrections to the newtonian equations of motion (Kopeikin et al., 2011, §6.3). Ac- cordingly, in most solar system modeling, the bodies are assumed to be spherically symmetric and rigid bodies. A significant feature of this is that rather than having

to deal with the complexity of treating the bodies in terms of the consitutent parti- cles, or with the complexity of partial differential equations (PDE) to treat non-rigid motion, we can use hamiltonian mechanics or post-newtonian vector mechanics, or some combination of the two. This shows that the abstraction of constraints from the phenomenon can allow the use of a theoretical framework that yields simpler model descriptions than a theoretical framework capable of a more detailed and ac- curate treatment of the behaviour. Provided that the constraints are effectively true of the phenomenon, where by ‘effectively true’ I mean that they provide an accurate description of the behaviour of the phenomenon within a reasonable error tolerance given the scale of the modeling, the use of the simpler theoretical framework can still result in a valid model.

Unlike the double pendulum case, taking the simplifications that follow from sym- metry and rigidity assumptions into the context of the constraints of an appropriate theory is not straightforward. Our most advanced theory for the treatment of gravi- tational phenomena is general relativity. The equations of general relativity, however, are very complex and the mathematics is difficult to handle directly. This, together with the fact that newtonian models are much easier to handle both conceptually16

and computationally,17 _{means that it is more effective to use apost-newtonian model,}

i.e., a newtonian model with relativistic corrections, when relativistic effects need to be taken into account. This is not a straightforward process, because there are a number of different post-newtonian equations of motion that one could use depend- ing on what details need to be taken into account in the modeling. In any case, analytic simplification techniques are required in order to find the post-newtonian corrections (first-order relativistic corrections) to the newtonian equations of mo- tion. Consequently, the symmetry and rigidity assumptions made will determine which post-newtonian limiting reduction of general relativity one will need to con-

16_{Incidentally, an important part of what I mean by conceptual accessibility in the context of}

mathematics is available reasoning based on the formalism, on the mathematical interpretation, and the ability to move back and forth between the two. If one can only think in one of those two main modes, then one will be limited in the level of understanding one can obtain. The more opaque the mathematics, therefore, the less conceptual accessibility. So, there is some aspect of this that is relative to an individual’s capability, but there is also an “objective” side of it that is relative to human capacities for reasoning in general.

17_{There is also the fact that there is considerable technological and human resource investment in}

sider. Generally one wishes to make the strongest modeling assumptions possible in the context, because this simplifies the post-newtonian corrections that need to be included in the model. Furthermore, feasibility considerations also require that one use the simplest set of equations possible, both for greatest insight and for fastest computations.

We see here that constraints of feasibility drive the need to work with the strongest simplifying assumptions possible while still allowing for an accurate model. The reasons for this are somewhat subtle, since this not only achieves a combination of ease of analytic manipulation and computation, but it also promotes insight by eliminating all information from the model that does not contribute significantly to the behaviour of interest. The basic idea here being that if it is possible to get an accurate description using a highly abstract model, then it shows that only those features preserved in the abstraction are responsible for the dominant behaviour. By the same token, if very strong simplifying assumptions yield a model that nearly captures the behaviour, this can point to a small effect that needs to be taken into account. For example, one could see that the computed orbit of an object has a small systematic error that is known to not be due to perturbing bodies. This could indicate a significant Yarkovsky effect, viz., where anisotropic thermal emission of a rotating object produces a non-negligible force, meaning that rotational motion must be taken into account.

Let us now turn our attention to the analytic part of the model construction process. In the case of modeling the double pendulum system using hamiltonian mechanics, the symmetry and rigidity assumptions allow a quite simple hamiltonian to be written down for the system, for which solutions may be sought in order to study the behaviour of the system under different conditions. The interest in the double pendulum system comes largely from its providing a very simple example of a chaotic system. As a result, hamiltonian mechanics is a useful theory to use for the purposes of studying the chaotic behaviour of the pendulum. As we will see later in the chapter, modeling chaotic behaviour is particularly problematic for the logical derivation view of the theory-world relation.

The ability to feasibly construct mathematical models in the context of a physical theory is obviously quite crucial for our ability to use theories to gain insight into

the nature of phenomena. And the need to be able to construct models efficiently is reflected in many ways in the syntactic tools that physical theories provide. Indeed, most physical theories, and hamiltonian mechanics is no exception, provide us with analgorithm for the construction of a model on the basis of given modeling assump- tions. As we discussed in the first chapter, the application algorithm of hamiltonian mechanics involves the following steps:

1. Specify a set of generalized coordinates q needed to track the motion of the system;

2. Calculate the kineticT(q,q˙)energy and potentialV(q,q˙)energy of the system in terms of the generalized velocities ˙q, the time derivatives of the generalized coordinates;

3. Calculate the lagrangian L(q,q˙), the difference between the kinetic and poten- tial energy;

4. Calculate the generalized momenta p by taking partial derivatives of the la- grangian;

5. Calculate the hamiltonian H(q,p) from the lagrangian L(q,p) expressed in terms of the generalized coordinates and momenta;

6. Calculate the equation of motion by substituting the hamiltonian in the hamil- ton equations: ˙ q= ∂H ∂p ≡f(q,p) p˙ = − ∂H ∂q ≡g(q,p).

In terms of the specification of a model from the general theory, the most signifi- cant steps are the first, second, and sixth. The first step of the algorithm determines the dimension of the configuration space, which thus determines the dimension of the phase space. Thus, even though the precise dynamics has not been specified, the Euler-Lagrange equations or Hamilton’s equations equations become partially specified. Constructing a model from this then requires specifying the form of the dy- namics. The specification of the dynamics occurs as a result of specifying the kinetic and potential energy in terms of a finite number of parameters, though the exact

form of the dynamics on phase space is not known until the last step of the algo- rithm. The result of the process is a hamiltonian model. This shows, clearly, then, that the constraints input to the mathematical framework from the phenomenological considerations, e.g., symmetry and rigidity constraints, come in the first two steps of the algorithm: specifying the generalized coordinates, and specifying the kinetic and potential energy of the system. The rest is an algorithmic process internal to the theory, resulting in the equations of motion.

The model that results from this process is the feasible base model referred to in the introduction to the section. Given our interest in the studying of the chaotic behaviour of double pendulum systems in general, rather than in the study of a par- ticular system, the actual value of the length ` of the rods is of little consequence. As a result, it is natural to alsonondimensionalize the hamiltonian in order to study the scale invariant behaviour of the system. This reduces the number of data-type parameters in the system by one, since ` is removed from the hamiltonian. This is not the most powerful application of nondimensionalization. The real power of nondimensionalization is that it can reduce the number of solution-type quantities in a problem, leading to a genuine simplification of the problem. The only partic- ularly significant simplification it does offer in this case is that when studying the behaviour of solutions using numerical methods, there is no need to arbitrarily select a value of the length`. This shows why the nondimensionalization step is particularly appropriate for a computer study.

Although the specification of the number of degrees of freedom and the kinetic and potential energies both require input from the consideration of the phenomenon, the specification of the kinetic and potential energies also requires mathematical input from outside of the basic theoretical constraint imposed by Hamilton’s equations, since other equations from physics must be employed to compute these quantities. This requires an expansion of the system of equational constraints, introducing a number of other quantities, like g and ` in the case of the pendulum models. The specification of the kinetic and potential energies specify the dynamical behaviour of the system, and so are analogous to the specification of constitutive equations or equations of state in other areas of physics.

assumptions that are experimentally-motivated, i.e., models drawn from experimen- tal data, or theoretically-motivated modeling assumptions, i.e., models constructed on the basis of some more-or-less principled theoretical considerations of constitution or behavioural phenomenology. In either case, the result is a set of equations added to the constraint system. In the experimental case, behavioural phenomenology must be translated into equations interpretable in the theoretical framework. And in the theoretical case, constitutive equations must be chosen or constructed from the theo- retical consideration of special cases of modeling contexts. An example of this is the general treatment of small oscillations in analytical mechanics. In the present case, however, it is just basic kinematics that is used to compute the lagrangian. But the more significant point is that in more complex modeling situations the computation of the lagrangian or hamiltonian may require modeling considerations that go outside of the bounds of theory, which is the case when behavioural equations require actual measurements of behaviour for their specification. This shows yet another way in which a representation of model construction as logical deduction is inadequate.

There is yet another case where input is required from outside the framework of a given theory. This is when another theory entirely is required for an adequate treatment of the behaviour of the phenomenon. If we needed to treat electrodynamic behaviour in addition to mechanical behaviour, we would have to add equations from electrodynamics to the constraint system of whichever classical theory we are using. Where such combination is common new theories are created from the merging of the basic equations of different theories. An example of this in astrophysics is magnetohydrodynamics, which combines fluid mechanics and electrodynamics for the treatment of electrically charged fluids. There are also less sharp combinations of theories, such as climate models and semi-classical theories where part of the system is treated quantum mechanically and classical assumptions are used to treat the effective behaviour of some other part. We are not, however, considering such cases in the present study.

The reason that empirical equations specifying dynamical behaviour are often required in feasible modeling is that our knowledge of how to specify both consti- tutive behaviour (microscopic theory) and phenomenological behaviour (macroscopic theory) of real systems is quite limited. In the case of our knowledge of constitu-

tive behaviour, in so far as the constraints on behaviour at this level are complete, they are usually of limited use for systems that have large numbers of microcom- ponents. Particularly when the numbers of microcomponents reaches macroscopic levels, the models of the system specified constitutively become both analytically and computationally intractable. This often forces the use of infinite idealizations in order to leverage the descriptive power of constitutive theory. This is why consideration of intertheoretic relations in scientific explanation, like those studied by Batterman

(2002b), are feasibly necessary, whether or not they are necessary in principle, as Batterman argues. In the case of our knowledge of phenomenological behaviour, the constraints on behaviour that we can obtain usually only penetrate so far into the nature of the phenomenon. Even the Navier-Stokes equations of fluid dynamics are not complete and interesting generalizations have been developed and studied, such as the equations of Ladyzhenskaya (1967). Part of this is due to the fact that there is an underlying discrete phenomenon. But it is also the case that more fundamental phenomenological equations can be determined locally, and it is not clear that there is any unique phenomenological foundation that could be found, even in quite re-