Theory to Data and Phenomena
Given the complexity of scientific method as a phenomenon, methods that are capable of generating insight into the feasibility of scientific knowledge must be capable of elucidating complex phenomena. This requires the screening out of details that are not relevant to a given epistemological task. The way that this is handled in applied mathematics is to use systematic means of abstraction from the full detail of the phenomenon being examined. Batterman(2002b) andWilson(2006) have shown how
mathematical modeling techniques used in applied mathematics accomplish this kind of abstraction to gain insight into the dominant behaviour of a phenomenon. This abstraction process,inter alia, serves to make information about a phenomenon more feasibly accessible. The results of this study show that the processes of data handling and numerical computation play a formally identical role in theory application. The argument for this proceeds by developing a technique of epistemological abstraction from the details of the case examples, the use of which reveals interesting structural patterns throughout the application process.
The epistemological abstraction method just described is modeled on the mathe- matical model reduction methods used in applied mathematics. A characteristic fea- ture of these reduction methods is that they begin with a more complex description and then generate a simpler description that still contains the information required to describe the dominant behaviour of the phenomenon. In a similar way, I treat the detailed consideration of the case examples as the complex description that requires simplification. I then use a systematic means of abstracting out detail from the ex- amples in a way that preserves dominant features of the inferential process involved in the application process. This systematic abstraction method is based on a model of theory application that I develop through the study in part II.
In chapter 2, I argue that an adequate formal method for studying feasible theory application requires the representation of both the syntax and the semantics of a theory or model together with how they arecorrelated; in particular how theycovary
in the sense that syntactic modification of a model implies certain semantic modifi- cations and vice versa. The ability to represent syntactic and semantic covariation allows a faithful representation of the different methods of variation or transformation of models used in the process of feasibly applying a theory, including an account of the error such methods introduce. In chapter 3, I present the basic structure of a model of theory application that is capable of representing application processes in terms of the syntax and covarying semantics of a theory, and in such a way that can model the effects of the kinds of error that are introduced in the process of application. The details of this model are extraordinarily complex, which requires the development of a way of presenting the model that brings out the epistemologically most significant, or dominant, behaviour. For this purpose I develop a conceptual language that can,
in a qualified sense, be regarded as generalizing concepts from mathematical logic by introducing imprecision and descriptions that are local to a given context. I present the details of this conceptual language as they are needed to clarify structural and behavioural features that are found in different parts of the application process.
Chapter 2has a dual purpose for the study as a whole. In part, it provides clear examples of logical approaches to scientific epistemology, facilitating demonstration of the limitations of logical approaches by showing those features of the case examples that are omitted or misrepresented. But it also functions to set up the epistemo- logical modeling approach I develop in this study, since there are respects in which the methods I use evolve out of the logical approaches. In particular, this helps to motivate and specify the details of the model of theory application in terms of co- varying syntax and semantics in mathematical modeling. A major difference in my approach, however, is that it is designed to result in a way of clarifying the structure and behaviour of the application process by using an abstraction process to eliminate the details of the full model of applying a theory that are irrelevant to the dominant inferential processes involved in application.
The consideration of the case examples is spread throughout the three chapters in part II. In chapter 3, I consider in detail the process of constructing a mathematical model in the cases of the double pendulum and in near-Earth object (NEO) modeling. I show that this process relies heavily both on abstraction and approximation, and explain the ways in which logical approaches fail to accurately or adequately represent this structure. I also consider the main kinds of error that are introduced in the model construction process and I show how my epistemological model clarifies the ways that error affects the structure of a model.
In chapter 4, I consider in detail the process of handling data in astronomy, par- ticularly in the case of solving the orbit determination problem for NEO tracking. I show that this process relies heavily on a complex variety of mathematical models in a way that is not adequately accounted for in typical presentations of the theory- world relation in terms of a “hierarchy of models” connecting theory with data or phenomena. In particular, I show that, at least in the case of astronomy, the pro- cess of using data in scientific inference is connected to the structure of the models used in data handling, a structure which is not hierarchical. I also consider what
is involved in ensuring that inference based on data is reliable and I show how my epistemological model accounts for this reliability in terms of the structural stability of transformations between mathematical models.
In chapter 5, I consider in detail the process of computing numerical solutions to mathematical models. I show that this process is not a “merely pragmatic” concern since it is required for us to actually know the predictions of a theory in many cases and I show how my epistemological model, unlike the usual logical approaches, is able to faithfully represent methods used in numerical computing. I consider the main kinds of error introduced in numerical computing and how they affect the structure of a mathematical model. And I also consider what is involved in ensuring that inference based on numerical computations is reliable and show how my epistemological model accounts for this reliability in terms of a kind of structural stability that is identical in essence to the structural stability of inference using data.
Finally, in chapter6, I summarize the results of the study and show how my episte- mological model represents the entire application process. The model depicts theory application in terms of recursive methods of inference that, after a small number of steps, terminate in either accessible data or accessible solutions to a model. The steps in inferences of this kind, whether used in modeling, data handling and computing, all share a common structure of translations between modeling frameworks that allow certain information to be more easily accessed. The recursive inference process ter- minates when the information sought is feasibly accessed in some, possibly encoded, form. My epistemological model also provides a way of clarifying what is involved in the stability of inference under the various kinds of error introduced in the process and can be a useful tool in the analysis of inferential stability.
Therefore, I conclude that the logical approaches are ill-suited to the investiga- tion of feasible scientific inference and method. A suitable approach must be able to faithfully represent actual scientific methods in detail and provide a means of sim- plifying this description in order to clarify epistemologically important structure. I therefore conclude that for an epistemological theory to adequately account for actual scientific inference it must be modeled on the methods of applied mathematics and not pure mathematics. Given the insight into large-scale patterns in feasible scientific inference seen to come out of the epistemological tools that I develop through this
study, I conclude that these methods are effective for feasible epistemology, at least in scientific contexts that use methods similar to those used in the case examples. Given that accurate description of feasible scientific methods requires logical concepts that generalize those of mathematical logic, I conclude that the picture of the theory-world relation in terms of a derivation from theory provides a badly distorted and false rep- resentation of how we actually gain knowledge from scientific theories, obscuring the actual extent and limitations of our knowledge.
Chapter 2
Applying Theories in an Ideal World:
Classical Accounts of Epistemological
Access to the World
A central aim of this study is to determine the characteristics that an epistemological method must have to be able to gain insight into the complexity of method in real sci- ence. This relates to the syntactic and semantic views of theories in two main ways. The first relation concerns my claim that that there are numerous ways in which typical syntactic and semantic approaches to theories and their relation to data or phenomena are either inadequate or inaccurate in terms of how they represent how theories are applied to gain actual knowledge in real science. To argue effectively for this claim requires clearly demonstrating those features of the syntactic and se- mantic approaches that result in inadequate or inaccurate representations of scientific method. This, in turn, requires a clear specification of how syntactic and semantic approaches represent theories and their application.
The second relation of this central aim to the syntactic and semantic approaches concerns the approach to modeling the epistemology of feasible scientific inference that I develop through the study of the two central case examples in part II. This epistemological modeling approach that I develop can be seen, in certain respects, to evolve out of syntactic and semantic approaches to theories and their application. Showing how certain advantageous features of the syntactic and semantic approaches are preserved in a modeling approach suited to feasible scientific inference will provide partial motivation for the epistemological modeling method.
develop fully as a scientific discipline it must adapt methods from applied mathematics to the purposes of investigating feasible scientific inference, rather than relying solely on methods from pure mathematics and metamathematics. One of the ways that I argue for this thesis is by showing how an epistemological modeling method that adapts methods from applied mathematics, developed throughout part II, provides insight into feasible scientific inference that the syntactic and semantic approaches typically used in epistemology of science do not provide and are not naturally capable of providing. Indeed, this will be seen to provide an argument that in order to address these limitations, syntactic and semantic approaches must adopt an approach that is, in certain fundamental respects, essentially the same as the basic modeling framework I develop. This line of argument also relies on having a clear exposition of certain important features of the syntactic and semantic approaches to the theory-world relation.
It is with these two central aims in mind, then, that we consider the syntactic and semantic approaches in this chapter. We will begin with the consideration of the “re- ceived view” of the logical empiricists. I will argue that certain basic features of this approach to reconstructing scientific theories are incompatible with the aim of investi- gating feasible scientific inference. Since this view is often used as the representative of syntactic approaches and it has not been amended since the 1960s, I also show how a more up-to-date syntactic approach is capable of usefully modeling important structural and behavioural features of feasible scientific inference. This is followed by a consideration of the semantic approaches to theories developed by Suppes and van Fraassen. Although these approaches are seen to fare far better in their ability to accurately represent scientific practice than the received view, they nevertheless also have basic features that make them incompatible with investigating feasibility in science. In fact, I argue that one of these is a feature typically regarded as an ad- vantage of the semantic approach, viz., that it characterizes a theory independently of a particular linguistic formulation. I argue that the actual linguistic formulation of a scientific theory must be represented for a logical reconstruction to explain the reliability of the methods actually used in the practice of applied mathematics to gain knowledge of aspects of the behaviour phenomena. A general reason for this, which we will discuss, is that because many mathematical techniques used in modeling are
applied as rules of symbolic manipulation on particular symbolic constructions, the explanation of the reliability of these techniques requires appeal to 1) those particular symbolic constructions, 2) their mathematical meaning and 3) how the prior two as- pects are related. This is a problem for the syntactic and semantic views because the explanation is provided neither in purely semantic terms nor in terms of a canonical linguistic reconstruction.1 The chapter concludes with a brief consideration of the
form a basic formal framework must have to adequately represent and account for feasible scientific inference.
2.0.1
Abstraction and Formal Reconstructions of Science
The formalizations of scientific theories that the so-called received view provided are considered as rational reconstructions of theories, putting them on a canonical rational foundation with the intention of screening out uncertain or metaphysical notions or concepts. In this way, these formalizations can be seen as models of scientific theories in essentially the same sense used in applied mathematics, viz., rational reconstructions can be construed as abstract representations of scientific theories that can be studied independently in order to gain insight into their structure and content.
Suppe (1974) discusses this explicitly:
The Received View begins by specifying a canonical formulation for theories in terms of an axiomatic calculus and rules of correspondence. This canonical formulation is claimed to stand in the following relation to a scientific theory: Any given scientific theory could be reformulated in this canonical manner, and such a canonical formulation would capture and preserve the conceptual and structural content of the theory in such a manner as to reveal most clearly and
illuminatingly the conceptual or structural nature of that theory (op. cit., 60).
This is a strong claim concerning what sorts of structure the received view is able to represent faithfully. Not only that, it implies that one canonical axiomatic presenta- tion of a theory will best elucidate the conceptual nature of a theory and its relation
1Another reason for this that we will not discuss is that different languages have different com-
putational complexity from the point of view of making certain kinds of inference about a given domain, which is very important for feasibility considerations. As a simple example, consider arith- metic calculations performed using positional notation, as we do, versus using Roman numerals. The former drastically reduces the computational complexity of the former, which had significant historical consequences.
to the world. There are general issues with this claim, in that given that very few theories have been presented in this way and, as a result, the state of the evidence for such a generalin principle claim is quite weak. Our concern here, however, is not so much whether the received view, or any other approach to the formal reconstruction of theories, can accomplish such a task in principle, but rather what such approaches can accomplish in terms of practical investigation of science.
Although reconstructions using mathematical logic do not typically make such strong claims about their epistemological capabilities, they nevertheless claim to be elucidating epistemologically significant features of real theories and their relation to the world. If, then, we are to regard a successful formal reconstruction of a theory and its relation to the world as an abstraction from the details of actual scientific theories and their relation to the world that preserves and elucidates the essential content, then (at least) two things must be the case:
(A) The formulation of a real scientific theory (and its relation to the world) in a reconstruction must faithfully abstract and elucidate the essential structural and conceptual content from a fully detailed description of the theory (and its relation to the world); and
(B) The structural and conceptual content eliminated in the abstraction must be irrelevant or insignificant in relation to the nature of a theory (and its relation to the world).
In order to definitively answer the question of whether any reconstruction process meets these conditions, it is necessary to be able to assess the structural relationship between the reconstructions and fully detailed descriptions of real theories and their relation to the world. Without a characterization of the fully detailed description of a theory, these conditions are impossible to assess rigorously; but we may still assess an approach in relation to these conditions.
Since logical reconstructions are typically formulated a priori, in one manner or another, and not through a process of investigation that abstracts content from fully detailed descriptions, a given approach makes a decision a priori about what concep- tual and structural content is and is not epistemologically relevant. Nevertheless, for a given approach we can assess condition (A) by determining whether the approach
faithfully captures epistemologically important content and we can assess condition (B) in terms of what content the approach evidently determines is not epistemologi- cally relevant.
I will argue over the next two sections that neither typical syntactic approaches nor typical semantic approaches meet these two conditions adequately. What these logical approaches do well is provide a means of studying general conceptual or struc- tural properties of theories, particularly where particular real scientific theories can be studied axiomatically. What I will argue that these approaches do not do ad- equately, however, is elucidate the relationship between scientific theories and the phenomena they are used to describe or represent. A major reason for this is that logical approaches to the epistemology of science typically assume that there is some clear, more or less direct, relationship between theoretical models and phenomena or the world and, consequently, that the details of the methods actually used to gain knowledge about phenomena using scientific theories have little or no relevance to the