A Finite Memory Argument for an Axiomatic Conception of Scientific Theories

Transcription

1 A Finite Memory Argument for an Axiomatic Conception of Scientific Theories Holger Andreas (Penultimate draft of a paper published in International Studies in the Philosophy of Science) Abstract This article concerns the split between syntactic and semantic approaches to scientific theories. It aims at showing that an axiomatic representation of a scientific theory T is a precondition of comprehending T if the models of T contain infinite entities. This result is established on the basis of the proposition that the human mind which is finitely bounded for all we know is not capable of directly grasping infinite entities. In view of this cognitive limitation, an indirect and finite representation of possibly infinite components of the models of a scientific theory proves to be indispensable. Sets of axioms and sets of axiom schemes provide such a representation. These considerations will be cast into an argument for an axiomatic conception of scientific theories. The article concludes with a case study of the ideal gas model. 1. Introduction As is well known, there are two major formal approaches to scientific theories: syntactic and semantic approaches. The syntactic approach was driven by the idea of a formal and universal calculus representing deductive reasoning in science, where the focus was on reasoning with well-established and axiomatizable theories. In the semantic approach, on the other hand, scientific reasoning with axioms dropped out of consideration. The focus moved 1

2 to reasoning with iconic, or analogical, models. In addition, proponents of the semantic approach believed they had a firmer grip on the identity of scientific theories, as well as relations of equivalence and reduction between such theories. The present article aims at showing that an axiomatic representation of a scientific theory T is a precondition of comprehending T if the models of T contain infinite entities. This result is established on the basis of the proposition that the human mind which is finitely bounded for all we know is not capable of directly grasping infinite entities. In view of this cognitive limitation, an indirect and finite representation of possibly infinite components of the models of a scientific theory is indispensable. Sets of axioms and sets of axiom schemes provide such a representation. This result refutes on certain further assumptions that are endorsed by the semanticist the view that the representation of a scientific theory in terms of models is more appropriate than an axiomatic representation. By means of these considerations, this article aims to establish a cognitive justification for an axiomatic conception of scientific theories, by which I mean any conception that considers axioms to be essential to a scientific theory. The axiomatization of the respective theory may be informal, semiformal, or be given in a fully formalized language. Logical theories of axiomatic systems may be viewed as theoretical accounts of certain properties of axiomatic systems being given in semiformal or informal language. And the analysis of scientific reasoning is one of the primary objectives of such logical theories. Furthermore, the notion of an axiom is understood as encompassing also axiom schemes. Most notably, axioms are composed of syntactic entities, and express universal statements. I do not propose, however, to identify a scientific theory with a particular axiomatization. The notion of an equivalence class of such axiomatizations seems to be a more promising candidate for a fully developed notion of a scientific theory. The core claim to be vindicated here is that some particular axiomatization is needed to identify and to comprehend a scientific theory. 1 The plan of the investigation is as follows. Section 2 explains, in some detail, the split into syntactic and semantic approaches in philosophy of science. Some elementary conceptual clarifications about human cognition, memory, and scientific reasoning will be made in section 3. Section 4 devel- 2

3 ops an argument for a finite representation of scientific theories. The case study of section 5 deals with the famous ideal gas model, and shows that our conception of this model is essentially axiomatic. 2. Semantic versus Syntactic Approaches to Science There are several ways of drawing the divide between semantic and syntactic approaches in philosophy of science: (1) The view that a scientific theory is a class of models versus the view that such a theory is a set of axioms. (2) The view that representation of scientific theories in terms of models is more appropriate than an axiomatic representation versus the opposite view. (3) Axiomatization by set-theoretic predicates versus axiomatizations without such predicates. The identification of a scientific theory with a class of models is the most explicit and straightforward way of expounding the semantic conception of scientific theories. Suppe (2000) has defended such an understanding of the semantic conception. Other accounts of the semantic conception carefully avoid a reification of scientific theories, and thus circumvent claims of identity between a class of models and a scientific theory. These accounts, however, maintain some prioritization of semantic entities over syntactic ones for the analysis of scientific theories. Da Costa and French (2003, ch. 2), for example, think that a representation of a scientific theory T in terms of models is more appropriate than an axiomatic representation. This gives us the second of the above ways of understanding the divide into semantic and syntactic approaches to scientific theories. The claim that representing a scientific theory T in terms of models is more appropriate than an axiomatic representation is made by Da Costa and French without further qualifications as to which contexts and aspects of science this superiority holds. Hence, we must understand it as applying to all contexts, or at least to those which are central concerns in our philosophical accounts of scientific theories. Undoubtedly, scientific reasoning is such a 3

4 concern. Thereupon proponents of syntactic and semantic approaches agree. The present article s argument aims at showing that the semantic conception leads us to the conclusion that the human mind is incapable of scientific reasoning with scientific theories, a conclusion that is plainly unacceptable. By contrast, a corresponding problem does not arise for the axiomatic conception. Hence, the semantic conception fails to be more appropriate for the analysis of scientific reasoning. This refutes the claim that representing scientific theories in terms of models is more appropriate tout court. Distinction (3) between set-theoretic and non-set-theoretic axiomatizations should be understood as a distinction within the syntactic approach. This is because set-theoretic predicates and their definitions are syntactic entities, as has been acknowledged by Da Costa and French (2003, 25). Notably, a clear prioritization of semantic entities over syntactic entities is difficult to attribute to Suppes, who has established the method of settheoretic predicates as a genuine formal approach to scientific theories. The axiomatic method of representing a scientific theory is rather at the core of his Representation and Invariance of Scientific Structures (Suppes 2002, 30). When explaining the merits of using set-theoretic predicates in comparison to other methods, the target of his criticism is not the syntactic approach in general but what he calls the standard formalization of a scientific theory, by which he means a formalization using first-order logic without set theory. The understanding of the syntactic approach that underlies the present investigation is not confined to first-order axiomatizations. Such a limitation would not be appropriate, either from a systematic or from a historical perspective. From a systematic point of view, the semantic approach is most generally characterized as setting forth some prioritization of semantic entities over syntactic entities in regard to the notion of a scientific theory. From a historical point of view, it is likewise inappropriate to attribute a confinement to first-order languages to proponents of syntactic approaches. Carnap s Aufbau (Carnap 1928), the most systematic account of logical empiricism, works in a type theoretic system. So do his later accounts of the dual level-level conception of scientific theories with its bipartition into an observational and a theoretical vocabulary (Carnap 1956, 1958). 2 The finite memory argument of the present article is directed against the semantic approach understood in the sense of (1) and (2). As explained, this argument aims at a cognitive justification of an axiomatic conception of scientific theories. It defends the syntactic approach insofar as axioms and 4

5 axiom schemes of whatever fashion (informal, semiformal, or formal) are syntactic entities or composed of such entities. Unless otherwise stated, the notion of a model is understood here in the model-theoretic sense, which concurs with the understanding of this notion in the most prominent variants of the semantic approach (Suppes 1960; van Fraassen 1980; French and Ladyman 1999; Suppe 2000; Da Costa and French 2003). 3 Recall that models in the model-theoretic sense are commonly conceived as sequences of sets of the following type: D 1,..., D k, R 1,..., R n where D 1,..., D k (k 1) are domains of interpretations and R 1,..., R n are relations. Model-theoretic models may furthermore contain functions as components, but these admit of an equivalent representation by relations. The semantic-syntactic split in philosophy of science has some bearing on our understanding of soundness and completeness theorems for first-order logic. As is well known, strong soundness and strong completeness mean that the set of semantic consequences of a set P of formulas is equivalent to the set of formulas being formally derivable from P. Despite this equivalence, it is an interesting question whether the semantic or the syntactic perspective on inference best captures certain patterns and methods of drawing inferences. There is quite a bit of literature in cognitive science on this question (see, e.g., Johnson-Laird 1983). In computer science, we can observe a competition between theorem proving and constraint processing, exploiting the syntactic and the semantic perspective respectively. 3. Cognition, Memory, and Scientific Reasoning Human cognition essentially consists in processing information. This dictum has served and continues to serve as the guiding principle of cognitive science (Bermúdez 2014). Notably, it is not bound to what is called the Turing account of cognition, according to which cognitive processes can be understood along the lines of manipulations of physical symbols. For neural networks are conceived to process information inasmuch as physical symbolic systems do. In cognitive science, both symbolic systems and neural networks are driven by the computational paradigm of human cognition. 5

6 We can thus distinguish between two components of cognition: first, the information, and, second, operations upon pieces of information, making up the information processing. The faculty of storing information is usually called memory. And it seems uncontroversial that only information that is stored, in some way or other, can actually be processed. We may understand the memory of the human mind in a very wide and liberal sense, which is not limited to items of information that we have memorized in the narrow psychological sense of the term. Taking inspiration from the extended mind thesis (Clark 2008), all the information contained in books, scientific journals, and electronic libraries may count as belonging to the extended memory of a single human mind. To put it more carefully, external information, such as library resources, count as belonging to the memory of a single human mind to the extent in which this mind is able to access these resources of information. This understanding of the memory of a human mind appears most natural on the computational paradigm of human cognition. How can we relate these very basic notions of cognition to our capacity of scientific reasoning with scientific theories? A scientific theory may simply be considered as a complex item of information. And steps of reasoning using a scientific theory certainly count as instances of information processing, where the information being processed consists, in part, of a scientific theory or some components of such a theory. This being said, we can recognize a finite bound inherent in human reasoning and cognition: the memory of the human mind is finitely bounded. To be more precise, the memory of the human mind which is used for storing scientific theories is finitely bounded. The finite character of the internal memory of the human mind comprising just information that is stored by the brain itself follows from the two following propositions, both of which seem to be fairly well established: (1) The number of states of a biological neuronal system with a finite number of neurones is finite. (2) The human brain contains only a finite number of neurones. The finite character of our external memory follows easily from the possibility and current practice of digitalizing books, journals, and further library 6

7 resources. A pdf copy of a book is certainly a finite entity in terms of the amount of information it contains. 4. A Finite Memory Argument for an Axiomatic Conception of Scientific Theories Having thus clarified the conceptual ground of the present investigation, we develop an argument that exploits the finiteness of the human mind for our understanding of scientific theories. This argument has the form of a reductio of the semanticist s claim that a non-axiomatic representation of scientific theories is appropriate tout court. The argument will immediately reveal that a similar contradiction cannot be derived if axioms are admitted as an appropriate means of representing a theory. Here it is: (1) A non-axiomatic representation of a scientific theory in terms of its models D 1,..., D k, R 1,..., R n is appropriate for the philosophical analysis of such a theory. (Premise) (2) The memory of the human mind is finitely bounded. (Premise) (3) If R is an appropriate representation of a scientific theory, R is cognitively accessible to human minds. (Premise) (4) Any finite representation of an infinite set S is axiomatic. (Premise) (5) If an infinite set S is not specified by some finite representation, S is not cognitively accessible to human minds. By (2) (6) Suppose, some theory T is represented by a class C of models D 1,..., D k, R 1,..., R n in a non-axiomatic way. (Premise) (7) Suppose, furthermore, that among the relations R 1,..., R n of the models of T there is at least one infinite set. (Premise) (8) C is an appropriate representation of T. By (1) and (6) (9) C is not cognitively accessible to human minds. By (4), (5), (6), (7) (10) C is not an appropriate representation of T. By (3) and (9) (11) Contradiction. From (8) and (10) 7

8 In sum, if some member of the class C of models (which is assumed to represent some theory T ) contains an infinite set without an axiomatic characterization, the proposition that C is an appropriate representation of T leads to a contradiction, in the context of rather uncontroversial premises. A similar contradiction cannot be derived if T is represented in an axiomatic way. This result refutes the semantic conception understood as claiming that the representation of a scientific theory in terms of models is more appropriate than an axiomatic representation. Premise (1) is implied by a weak form of the semanticist conception, as explained in section 1. Da Costa and French (2003) are committed to this premise for the following reasons. Recall from section 2. that the semanticist sets forth some prioritization of the models over the axioms of a scientific theory. In the case of Da Costa and French, this prioritization concerns the propriety of the style of representing a scientific theory: The question we have to answer, as philosophers of science, is what is the most appropriate representation of theories? In answering this question, we must acknowledge that a theory can be represented from various perspectives: A Suppes predicate, understood as a linguistic notion, determines a family of structures, which are nonlinguistic and (normally) conversely. Representationally, then, theories present us with two faces: the syntactical and the semantical. (Da Costa and French 2003, 25) Da Costa and French, eventually, make a decision in favour of the semantic face, i.e., they present an account of theories in terms of models (see, e.g., Da Costa and French 2003, 33). Hence, the semanticist position is captured in the above argument by proposition (1), according to which a non-axiomatic representation of a scientific theory in terms of models is appropriate. If the semanticist were to concede the indispensability of an axiomatic representation of the models, it would be extremely difficult, if not impossible, to say what the decision in favour of the semantic face of a theory consists in. Notably, Da Costa and French admit that language is needed to communicate the models in terms of which, in their eyes, a theory is to be represented: it is impossible to entirely cut models free of language, although, of course, here we are not talking about English or Portuguese 8

9 but the language of set theory (Da Costa and French 2003, 34). However, this statement on the role of language for the purpose of communicating models concerns only the language of set theory, but not the language of the specific axioms of the respective theory. It implies, for example, that set-theoretic language is needed to represent the models of classical mechanics, but not necessarily Newton s equations being understood as syntactic entities or equivalence classes of such entities. Finally, the seminal Reinflating the Semantic Approach by French and Ladyman (1999) highlights the relevance of human cognition in the semantic approach, and consequently maintains that models are accessible independently of any axiomatization: Leaving this issue [i.e., the issue of antirealism about mathematics, H. A.] aside, if models either involve representations of familiar objects or mathematical structures then we have argued that we do have access to them independently of any axiomatization, and that, therefore, the claims of the semantic approach to independence of linguistic formulation are to some degree justified. (French and Ladyman 1999, 118; emphasis added) Again, it is evident that models are conceived to be independent of an axiomatic characterization by the semanticist. Moreover, the explicit commitment to a cognitive dimension of the semantic approach justifies the ascription of premise (3) in a straightforward manner. Drawing on the conceptual clarifications in section 3., we can furthermore establish a more general and straightforward justification of premise (3). By way of very elementary reflections upon human cognition and scientific reasoning, we have concluded that scientific reasoning with a scientific theory requires cognitive access to this theory. At the very least, we need to have cognitive access to those parts of the theory which are relevant for the respective operations of reasoning. Joined with premise (1), this result implies that we do have cognitive access to scientific theories. The semanticist may object that partial access to a scientific theory suffices for being capable of scientific reasoning. Hence, we never need complete access to a scientific theory. Note, however, that a finite part of an infinite whole forms only an infinitely small part of this whole. It is difficult to understand how such an infinitely small part of a scientific theory may 9

10 in fact serve the purpose of scientific reasoning. The semanticist has not provided even a sketch how scientific reasoning is supposed to work with infinitely small fractions of a scientific theory being conceived as a set of models. In addition to these more technical considerations, we can point out in support of premise (3) that it is hardly doubted that scientists understand and comprehend the theories they are working with. And comprehension seems to imply cognitive access in the more technical sense of computational approaches to human cognition. Denying that scientists are capable of comprehending at least some substantial parts of the theories in their field would be a major departure from the common notion of a scientific theory, and so requires a justification that the semanticist has not given so far. Premise (2) has been established in section 3. Premise (4) follows from our account of sets in set theory. There, we have exactly two ways of specifying a set. First, by an open formula that designates a unary property. This kind of specification is expressed by the axiom scheme of unrestricted comprehension in naive set theory and some axiom scheme of specification also described as restricted comprehension in axiomatic systems of set theory, such as Zermelo-Fraenkel and von Neumann-Bernays-Gödel set theory. The second way of specifying a set consists in individually naming each member of this set. This second method can be explained in terms of the first method, but in informal expositions of set theory the two methods are often explained independently. We say that the set x is a component of the set y iff x is a member of y or a member of a member of y, etc. Let us call a set x infinite iff it has an infinite component set. When specifying the models of an axiomatic theory, we use some axiom of restricted or unrestricted comprehension in an iterated way. For we say that x is a model of T iff x is a sequence of sets the components of which satisfy such and such conditions. The axioms of T are used to specify the latter conditions. If we were to specify the models without reference to any axioms, we would need to enumerate all members of all sets being components of the respective model. Hence, the non-axiomatic specification of an infinite set is itself infinite. From this it follows that any finite representation, or specification, of an infinite set is axiomatic. Why is our set-theoretic understanding of sets decisive for the present investigation? This is because set theory serves as foundation of model theory, and we are 10

11 concerned with model-theoretic models in the semantic approach. The semanticist might doubt premise (4), but this forces her to explain what kind of finite and non-axiomatic specification of possibly infinite sets she has in mind. No such explanation has been given in the literature on the semantic approach. The discussion of the ideal gas model in the next section will reveal that we conceive its model-theoretic models in an axiomatic fashion. The claim of premise (6) can be made for any scientific theory according to the semantic conception of scientific theories. Premise (7) is satisfied for the majority of quantitative theories in modern science. This premise is particularly easy to establish for theories that involve differentiable functions since both the range and the domain of such functions must contain an interval of the real numbers. This point will also be exemplified with the case study of the next section. Note, finally, that we seem to frequently encounter the case where a theory has infinitely many models. This type of infinity is different from the infinity of a component set of a model but poses a similar problem for the semanticist position. One might argue that the state space of a physical system can be rendered finite by a complete quantization of all quantities. Even in this case, we are confronted with the problem of combinatorial explosion of the cardinality of the state space, i.e., the problem that this space grows exponentially with the number of particles in the physical system. Such an exponential growth is considered to be beyond the cognitive grasp of human minds and computers in the theory of computational complexity (see, e.g., Arora and Barak 2009). 5. Model-based versus Axiomatic Reasoning in Science: A Case Study The undeniable existence of iconic, or analogical, models in science has received much attention in the semantic approach. Suppes (1960) has argued that models in the model-theoretic sense also capture iconic models. Da Costa and French (2003) have suggested that the relation between an iconic model and the world may be captured along the lines of partial structures and partial truth. In the syntactic approach, by contrast, iconic models 11

12 were largely ignored. Carnap (1939, 67-69) went so far as to claim, with reference to quantum mechanics, that the request for an intuitive understanding of scientific concepts, of which an iconic interpretation is an instance, inhibits scientific progress. Iconic models are certainly important in science and play a crucial role in scientific reasoning. Undoubtedly, they deserve philosophical analysis. Are iconic models powerful enough to replace axiomatic reasoning? The general view in the semantic approach seems to be that models, whether iconic or not, are the primary vehicles of scientific reasoning. This is indicated, for example, by the subtitle of Da Costa s and French s Science and Partial Truth (2003), according to which this book aims at a unitary approach to models and scientific reasoning. That scientific reasoning is model-based in a non-axiomatic fashion is also implied by the semanticist s non-axiomatic conception of models. With a brief case study of the famous ideal gas model, I will show that model-based reasoning, important as it is, cannot replace axiomatic reasoning. Although we can recognize instances of model-based reasoning in the theory of ideal gases, these instances do not conform to the semanticist account of models in science. For the (model-theoretic) models of the ideal gas are clearly specified in an axiomatic fashion. The ideal gas model, therefore, provides us with a counter-instance to the semanticist prioritization of semantic entities over syntactic ones. The crucial result that could be established by means of the ideal gas model is the kinetic theory of heat, according to which the temperature of a real gas is a linear function of and proportional to the mean kinetic energy of the gas s molecules. For monatomic gases, this theory takes the form of the following equation: T = 2 3k E kin (1) where T designates the temperature function and E kin the function of mean kinetic energy. k is the Boltzmann constant. For this equation to obtain by means of an abductive inference, the following deductive inference is needed: 4 12

13 x is an ideal monatomic gas x satisfies equation (1) of the kinetic theory of heat Definition of pressure x satisfies the ideal gas law (2) To recall the ideal gas law: p V = N k T (3) where p designates pressure, V volume, and N the number of molecules of the gas. If joined with a hypothesis concerning the relationship between ideal gases and real gases, the ideal gas law allows us to deductively infer gas laws for real gases that are empirically well confirmed: Ideal gases satisfy the ideal gas law Real gases approximately behave like ideal gases with respect to their mechanical and thermodynamical quantities (4) Real gases satisfy Boyle s law, Charle s law and Gay-Lussac s law These two deductive inferences can be merged into a single deductive inference on the basis of which one can infer by means of an abductive inference - (i) that real gases approximately behave like ideal gases with respect to their mechanical and thermodynamical quantities, and (ii) that monatomic ideal gases satisfy equation (1). From these two propositions, in turn, it is deductively inferable that equation (1) approximately holds for monatomic real gases. This result establishes the kinetic theory of heat for real gases. The corresponding results for diatomic as well as further gases are easy to obtain (see Flach 2000 for a concise logical account of abduction). Do these inferences qualify as axiomatic or model-based? If such a distinction turns out inapplicable, can we separate model-based from axiomatic parts of reasoning? If such a separation is feasible, how much model-based and how much axiomatic reasoning is involved? To answer these questions, let us first describe the ideal gas model in an informal way: (1) Molecules are conceived as unextended particles that undergo collisions with the walls of a vessel. (2) There are no collisions other than those between a molecule and a wall of the vessel. 13

14 (3) All collisions are elastic. (4) Newton s equations hold for the trajectories of the molecules and the collisions. (5) The velocity distribution of the particles is isotropic. We can readily understand the concept of an ideal gas as a set-theoretic concept in defining the corresponding set-theoretic predicate: Definition. Ideal gas x is an ideal gas (x M(IG)) if and only if there are sets P, W, C, T, s, m, f, and sequences x and x of sets such that (1) x = P, W, C, T, R, s, m, f ; (2) P and W are finite non-empty sets; P : particles; W : walls of the vessel (3) T R and there are a, b s.t. T = [a, b]; T : temporal interval (4) C (P W ) (P W ) T ; C(x, y, z): x collides with y at time z (5) s : P T R 3 ; s: space function (6) d2 s exists at all u (a, b); dt2 (7) m : P W R; m: mass function (8) f : P T R 3 ; f: force function (9) y z t(c(y, z, t) y P z W y W z P ); all collisions are such that a particle collides with a wall (10) p p (p P p P m(p) = m(p )); (11) w(w W m(w) = ); all walls have infinite mass (12) ds x dt = ds y dt = ds z dt ; (13) x = P W, T, s, m, f ; velocity distribution is isotropic (14) x M(CPM); Newton s equations hold 14

15 (15) x = P W, T, s, m ; (16) x M(CCM); conservation of momentum holds (17) x M(ECCM). 5 all collisions are elastic Two observations concerning the models of an ideal gas can be made in a straightforward manner. First, most importantly, the models of an ideal gas are characterized in an axiomatic way. This is particularly obvious for the explicitly set-theoretic conception of an ideal gas. But even in the informal characterization of an ideal gas we can recognize general propositions whose logical form is axiomatic in the sense of being universal statements. A closer look at the textbook derivations by means of which it is established that ideal gases satisfy the ideal gas law reveals that this axiomatic characterization is indispensable for the use of the ideal gas model in physics. If the collisions between particles and walls were not elastic, for example, the conclusion of inference (2) could not be established. Satisfaction of Newton s equations is likewise essential to this inference. In brief, the use of the ideal gas model in physics is, to a large extent, of the axiomatic type. Second, the models of the ideal gas have components that are infinite sets. This is because the space function assigns a position to any particle for a non-infinitesimal interval of real numbers, which represents a temporal interval. Such intervals of real numbers are uncountable and, hence, infinite. (Real numbered intervals of time points, as well as real numbered vectors of space points, are needed for the space function in order to apply real analysis, which in turn is needed for the formulation of Newton s equations.) The extension of the space function in the models of an ideal gas is therefore an infinite set. Hence, the finite memory argument of the present article applies to the (model-theoretic) models of the ideal gas model in a straightforward way. Is there any place left for genuinely model-based reasoning? Yes, there is. The proposition that real gases behave like ideal gases with respect to their mechanical and thermodynamical quantities (which serves as a premise in inference (4)) is obtained using genuinely model-based reasoning. More precisely, this proposition is established by means of presumed similarities between point particles of ideal gases and molecules of real gases. Another instance of model-based reasoning is recognizable in the justification of the propositions (i) that collisions of the molecules with the 15

16 walls are elastic, and (ii) that such collisions do not diminish the value of the velocity of the molecules. These propositions are sometimes justified with reference to collisions of billiard balls with the borders of a billiard ball table. Note, however, that both the ideal gas model and the billiard ball model are specified in an axiomatic fashion. This has just been shown for the ideal gas model. For the billiard ball model, a similar demonstration could be given: the trajectories of billiard balls contain infinite sets and are therefore cognitively accessible only by way of some axiomatic characterization. The mere image of colliding billiard balls may well complement the axiomatic description of elastic collisions but is insufficient to replace the latter. To gain more clarity about the present instances of model-based reasoning, it seems advisable to distinguish between (i) iconic models, (ii) idealized models, and (iii) model-theoretic models. As indicated above, some semanticists thought to capture iconic and idealized models in terms of model-theoretic models. No doubt, one and the same model can be iconic, idealized, and of the model-theoretic type. In the present case we come to the following classification. The billiard ball model clearly is an iconic model in the sense of providing an analogy between the behaviour of billiard balls and that of molecules. The billiard ball model is not so much an idealization since real billiard balls are subject to friction and, hence, do only approximately satisfy the law of elastic collisions. The ideal gas model, by contrast, is an idealized model of real gases. It is not an iconic model in the sense of providing an analogy between two different domains. On the grounds just explained, neither the billiard ball nor the ideal gas model come in the form of naked model-theoretical models bare of an axiomatic characterization. Hence, the ideal gas model and the billiard ball model qualify as models only in the sense of being idealized and iconic, respectively. In sum, the use of the ideal gas model in physics displays axiomatic as well as model-based instances of reasoning. Model-based instances connect propositions about ideal gases with propositions about real gases. They hypothetically establish that certain assumptions concerning the submicroscopic particles of ideal gases are approximately true of real gases so that ideal gases largely behave like real gases with respect to their macroscopic properties. These instances of model-based reasoning qualify as model-based because they are concerned with an idealized model. However, they are not concerned with non-axiomatic model-theoretic models. For the inferences 16

17 that establish that ideal gases satisfy the ideal gas law are based on an essentially axiomatic characterization of ideal gases. The ideal gas model, therefore, does not conform to the semanticist account of models in science, which prioritizes semantic entities over syntactic ones. It is therefore less surprising that no properly semanticist demonstration has been given of the claim that ideal gases satisfy the ideal gas law. 6. Conclusion We commonly understand the notion of a scientific theory in such a manner that scientists are well capable of comprehending the theories in their fields. It seems fairly absurd to say that Niels Bohr, for example, developed a theory of the hydrogen atom without being able to comprehend it. Not accidentally, the Greek origin of our modern notion of a theory is an epistemic notion in the sense of being very closely tied to human cognition. θɛωρɛιν means to consider, to view, and to observe; the relevant translation of θɛωρια is consideration or view. How is a human mind, which is finitely bounded for all we know, capable of comprehending the models of a scientific theory if these models contain infinite components? An axiomatic conception of scientific theories can answer this question by pointing out that axioms and axiom schemes are the primary vehicles of human reasoning with scientific theories as well as the means of comprehending such theories and their models. The semanticist, by contrast, is committed to the view that we do have cognitive access to the models of a scientific theory independently of any axiomatization, a commitment that is explicitly endorsed by French and Ladyman (1999). Any direct access to infinite sets, however, is ruled out for human minds by the finite bounds of their cognitive capacities. The question thus arises whether there is a way of comprehending infinite sets in an indirect and yet non-axiomatic manner. Proponents of the semantic conception have failed to prove that this question can be answered in the affirmative. At the present stage of philosophical research, we must answer it in the negative. Therefrom we can infer that human minds are bound to use some axiomatization of a scientific theory, in order to comprehend such a theory and its models, if these models contain infinite sets. This result is inconsistent with the prioritization of models over axioms inherent in the semantic approach. 17

18 Acknowledgements This work was supported by the Alexander von Humboldt Foundation. I am also grateful to three unnamed referees of this journal for very helpful comments on earlier draft of this article. Notes 1 For a related, yet different line of criticism of the semantic view, see Halvorson (2012, 2013) and Lutz (2014). They attack the semanticist s claim that the representation of a scientific theory by model-theoretic structures can be understood independently of a vocabulary interpreted by such structures. 2 See also Lutz (2012) for further evidence that the received view in philosophy of science was not confined to first-order theories. 3 The presence of the framework of model theory is most obvious in the case of Suppes (1960), French and Ladyman (1999), Suppe (2000), and Da Costa and French (2003). The notion of embedding being crucial to van Frassen s semantic approach implies a commitment to the model-theoretic framework. Semanticists sometimes prefer to speak of configurated statespaces rather than model-theoretic models (Suppe 2000). However, such state-spaces contain infinite entities to the same extent as model-theoretic models of a scientific theory do. Hence, the central argument of the present article applies to state-spaces inasmuch as it does to model-theoretic models. 4 I omit the details of this deduction, because, first, they are not relevant to the finite memory argument of the present article, and, second, they can be found in standard undergraduate textbooks of physics (see, e.g., Feynman, Leighton, and Sands 2011, ch. 39). The ideal gas model continues to be used in statistical physics (see, e.g., Reif 2008, ch. 5). For the purpose of the present article it suffices to deal with the non-statistical use of this model. Note that the axioms of classical mechanics are not explicitly stated as a premise in the derivation of the ideal gas law, because satisfaction of these axioms by an ideal gas is, in the present account, part of the meaning of such a gas. This will become more obvious soon. 18

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