CHAPTER THREE: VARIATIONS ON THE NON-STATEMENT VIEW OF SCIENCE
In Chapter 1, the Introduction, I have discussed the differences between the statement and the non-statement approaches. The advocates of the statement approach depict scientific theories as axiomatised deductively closed sets of sentences within some appropriate syntactic system, and discuss the « empirical interpretations » of these theories in terms of some set of correspondence rules or bridge principles. The defenders of the non-statement approach, on the other hand, do not view the formal formulation of scientific theories in some appropriate language as the most useful characterisation of « theories•. Rather, they depict these « theories » in terms of sets of mathematical structures that are the models of the theory in question. They do not, however, escape the usual realist questions in this way, the problem is simply pondered in a different context with the help of a set of more workable – in the sense of adaptability to different situations – tools.
In what follows I shall discuss a few of the main non-statement programmes offered by current philosophy of science. I focus on the non-statement approach to science since its defenders’acknowledgement of the role of mathematical models in science’sprocesses is of paramount importance to my model-theoretic view of science. Although I have the notion of a scientific theory as a deductively closed set of sentences in common with the defenders of the statement approach to science, the rest of their (absolutist) views do not feature in my approach at all, and therefore I shall not discuss their approach in any more detail.
Patrick Suppes’sset-theoretic approach to science
Suppes offers one of the first viable alternatives to the « received (statement) view »of scientific theories and so brings about a radical turn in philosophy of science. Other than the structuralists who stress the use of formal semantics and meta science to appeal to the structural aspects of theories, Suppes finds the axioms of set-theory sufficient, and claims mathematics, rather than meta-mathematics to be the language of science.64
Since to define a class of set-theoretic structures of which a theory is true (relative to these structures) it is irrelevant to know really what the theory is about, he seems to be not overly interested in the problem of identifying or limiting the intended applications of scientific theories in the classic « statement » sense. I try to emphasise as often as possible that there are at least two kinds of truth relation to be examined when asking questions about the « truth of a theory ». As I have pointed out in Chapter 2, the first kind of truth relation that theories typically enter into is purely formal in the sense of Tarskian satisfiability, and here, obviously, it is the case that the content of theories is not necessarily a deciding factor. The second kind of truth relation (between models of theories and empirical models depicting the behaviour of phenomena in real systems) is also formal in this sense, but also far more complicated, given that these relations are the relations determining empirical adequacy and all that it implies. And, obviously, in the second case, that « about which » theories are, becomes extremely relevant.
According to Suppes (1967: 57) the problem with the statement approach’sco-ordinating correspondence rules is that they do not in the sense of modern logic offer an adequate semantics for the axiomatic calculus of the theory. Wojcicki (in Humphreys, 1994: 127) explains that » … if for the logical positivists the right way to define an empirical theory Twas to define a set of axioms from which all the other sentences valid in T are logically derivable, Suppes suggests that to define T is to define a set-theoretical predicate that denotes all the set-theoretical structures [semantical models) of which T is true in the Tarski sense ». Suppes does not so much emphasise the non-statement approach versus the statement approach though. He rather stresses the advantages of analysing empirical theories within a set-theoretical framework rather than a meta-mathematical one65 , and (Suppes, 1954: 244) writes:
… Why axiomatise?, I may briefly say that axiomatisation is one constructive way of obtaining the sort of intellectual clarity and precision for which philosophers are always striving with respect to the foundations of the various sciences. Unfortunately a good many philosophers seem to labour under the misimpression that to axiomatise a scientific discipline … one needs to formulate the discipline in some well-defined artificial language …. this kind of linguistic viewpoint is, in my opinion, seriously in error, and the predominance of this attitude has perhaps been one of the major reasons for the lack of substantial positive results in the philosophy of science …. Luckily we can pursue a programme of axiomatisation without constructing any formal languages. The viewpoint I am advocating is that the basic methods appropriate for axiomatic studies in the empirical sciences are not metamathematical (and thus syntactical and semantical) but set-theoretical. To axiomatise the theory of a particular branch of empirical science in the sense I am advocating is to give a definition of a set-theoretical notion, such as that of a system of classical particle mechanics (see McKinsey, Sugar, and Suppes (1953)), or that of a system of rigid body mechanics (see Adams (1959)), or that of a system of Mendelian genetics (see Rubin (1954)).
The class of structures (systems) under consideration is thus described by giving one « generic » structure, with parameters, which can be specified to deliver all the systems in the class. Suppes (in Morgenbesser, 1967: 60) acknowledges this when he points out that one of the simplest ways in which to provide an extrinsic characterisation of a theory is to define the intended class of models of the theory; and then asking if the theory can be axiomatised, merely comes down to asking if a set of axioms can be stated such that the models of these axioms are precisely the models in the defined class. He (in Morgenbesser, 1967: 61, 62) remarks however that » … the problem of intrinsic axiomatisation of a scientific theory is more complicated and considerably more subtle … . Fortunately, it is precisely by explicit consideration of the class of models of the theory that the problem can be put into proper perspective and formulated in a fashion that makes possible consideration of its exact solution. »
And, in model-theoretic terms, even more positively, such consideration of the class of models of a given theory shows the continuous character of science (see Chapter 2). The underdetermination of theories by models and their underdetermination by data are more problematic from a non-statement point of view perhaps, precisely because from such a point of view the issue of underdetermination seems untouchable (and so insoluble), given the non-statement aversion to theories as linguistic entities, so that the notion of a theory as some kind of overarching organising notion does not really exist for defenders of this view. In a model-theoretic approach underdetermination is more « natural » and even somehow forms part of scientific progress.00 I shall elaborate on the meaning of underdetermination in model-theoretic realist terms in Chapter 5.
Suppes addresses the philosophically problematic relations between empirical systems and theories (i.e. my « second set » of interpretational relations) in terms of a hierarchy of models that focuses on the complex nature of the experimental process.67 He (Suppes, 1954:243) already points out very early on in his work that progress in foundational studies of philosophy of science requires distinction between theory and experiment, since the reconstruction of the experimental stage of science is rather more problematic in comparison to the theoretical stage which may be axiomatised « quite easily »with the help of set-theoretic predicates. He (Suppes, 1954:246) wants to provide philosophy of science with » … a kind of algebra of experimentally realisable operations and relations » and emphasises that discussion of the empirical interpretations of the primitive notions for certain defined notions of some empirical theory imply interpretations of quantitative notions, which necessitates some systematic theory of measurement. 68 He is not interested in the classic notion of absolute objective truth, nor is he interested in the kind of framework offered by the instrumentalists, rather he wants to speak about truth in terms of modern statistical decision theory.
Thus, one of the most important issues in Suppes’s philosophy of science is the emphasis he puts on the « experimental stage » of science.69 Empirical interpretations of the primitive notions for certain defined notions of some empirical theory are interpretations of quant~ative notions, which necessitates some systematic theory of measurement, as already mentioned .
. .. the point of a theory of measurement is to lay bare the structure of a collection of empirical relations which may be used to measure the characteristics of empirical phenomena corresponding to the concept. Why a collection of relations? From an abstract standpoint a set of empirical data consists of a collection of relations between specified objects. For example, data on the relative weights of a set of physical objects are easily represented by an ordering relation on the set; additional data, and a fortiori an additional relation, are needed to yield a satisfactory quantitative measurement of the masses of objects » (Scott & Suppes, 1958: 113).
Thus, as far as the co-ordinating principles or bridge principles of the statement approach are concerned, Suppes stresses (in Morgenbesser, 1967:62) that the practice of testing scientific theories is a much more complicated issue than is implied by the usual comment about these issues.70 I agree with this, but I do not see the philosophical need for turning almost exclusively to the statistical methodology to examine these relations that Suppes (in Morgenbesser, 1967, and also Suppes, 1969, Suppes, 1989, and Suppes, 1993) insists on.71 I think that for the purposes of philosophy of science, it is sufficient – and a more philosophically challenging prospect, I might add – to look to the various model-theoretic relations involved, and to be able to point out all of (or as many as possible of) the factors involved in these connections. 72
Suppes and Dana Scott in their artide Foundational aspects of theories of measurement (1958) ground the foundational analysis of measurement in general model theory. Suppes (in Morgenbesser, 1967:58) points out that the essential characteristic of a theory of measurement is that it can study (in a precise way) the transformation or development of « qualitative observations » into the « quantitative assertions » characteristic of the more theoretical stages of the scientific process. He approaches this problem in terms of representation theorems, mainly because he views the models of the theory and the models of the data (see below) to be of different logical types: « Given an axiomatised theory of measurement of some empirical quantity such as mass, distance, or force, the mathematical task is to prove a representation theorem for models of the theory which establishes, roughly speaking, that any empirical model is isomorphic to some numerical model of the theory. The existence of this isomorphism between models justifies the application of numbers to things .
… What we can do is to show that the structure of a set of phenomena under certain empirical operations is the same as the structure of some set of numbers under arithmetical operations and relations » (Suppes in Morgenbesser, 1967:58). 73 Although I would read « conceptual model » for his « empirical model » and « empirical model » for his « numerical model », this is essentially my view of the « verification » of the models of scientific theories too. In my approach it is however not necessary to use a separate language – from the one talking about the content of a theory’sconceptual models – to talk about the empirical models of theories – although of course it can be done, and then Suppes’suse of representation theorems will become applicable too. 74
Suppes (1954: 245) sets out the various stages of formulating a set-theoretic predicate for (or axiomatising) a particular branch of empirical science as follows:
- In the beginning some kind of statement of what other theories are assumed (e.g. in axiomatising rigid body mechanics, one would assume the standard branches of mathematics and particle mechanics) is needed
- Then the « primitive » notions of the theory are listed, and their set-theoretic nature (in particle mechanics, notions like « set of particles », the « interval of elapsed time », the « position function »,the « mass function », and so on) is indicated.
- The set-theoretic definition can then be completed by listing the axioms which have to be satisfied, because one will then be able to examine the deductive consequences of the definition. Obviously one of the main tasks here is to rationally reconstruct within set theory the standard theorems of the branch being studied. One will also then be able to ask some of the questions of modem mathematics that have obvious implications for the structure of empirical theories, such as questions concerning the formulation of representation theorems75 which may be linked for instance to studies directed towards the problem of reduction between theories
Then finally, one will be in a position to give an empirical interpretation of the axiomatised theory, which will have to take the complexity of the entire experimental enterprise into account.
Chapter One: Introduction
1.1 The statement and non-statement accounts of science
1.2 The interpretation and use of the notion of « model » in philosophy of science
Chapter Two: A model-theoretic account of science
2.2 Terminological note
2.3 The formulation of scientific theories
2.4 The interpretation of scientific theories
2.5 The process of science
Chapter Three: Variations of the non-statement view of science
3.2 Patrick Suppes’s set-theoretical approach to science
3.3 The structuralist programme
3.4 The semantic approaches of Beth, Van Fraassen, and Suppe
Chapter Four: Nancy Cartwright and the lying laws of physics
4.2 Phenomenological and fundamental laws
4.3 The role of models in science and Cartwright’s « simulacrum » account of science
4.4 The process of science revisited
4.5 The « abstract » and the « concrete »
4.6 Nature’s capacities causally explained
4.7 The « abstract » and the « concrete » revisited
Chapter Five: A model-theoretic realism
5.2 Reality and science
5.3 A modified image of science
5.4 The empirical interpretations of scientific theories
5.5 The succession of theories, verisimilitude, underdetermination, and other intertheoretic issues
Chapter Six: Conclusion
6.1 The meaning of a model-theoretic realism for philosophy of science
6.2 Historical note
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