Figure 66: Definitions of logical, mathematical and metamathematical models

 

content

Encyclopaedia Britannica. 1964

Paul Edwards (Ed.): The Encyclopedia of Philosophy. 1967

Encyclopaedia Britannica. 1973

Edward Craig (Ed.): Routledge Encyclopedia of Philosophy. 1998

 

 

 

Vol. 15, Chap.: Mathematics, Foundations of, 83

 

Further Perspectives

 

... Using the classical law of the excluded middle, Goedel in 1930 proved a completeness theorem for the first-order functional calculus F. In one form, this says that if a formal system obtained by adjoining mathematical axioms to F is consistent, it has a model, i. e., there is an interpretation which makes the axioms true. Indeed, the model can be constructed using the natural numbers as the objects. In another form, the theorem says that in such a system every formula is provable which is true under all the interpretations of the undefined terms which make the axioms true.

 

In view of this, when S is based an first-order functional calculus, the unprovability in S of ~A_q implies that, while ~A_q is true under the intended interpretation of the undefined terms, there is some other interpretation which makes the axioms true (a "non-standard model") under which ~A_q is false. This illustrates the theorem of Skolem (1933) that no list of axioms in the symbolism of the first-order functional calculus can characterize the natural numbers categorically.

If higher functional calculus is used, the deductive apparatus will be incomplete. It thus appears that the logistic method is inadequate to characterize the natural numbers categorically. Peano's axioms characterize them categorically, but only through an interpretation which cannot be rendered fully through the deductive possibilities.

Also we see the Goedel formal undecidability of A_q as a phenomenon of the same kind as the undecidability of Euclid's parallel Postulate from the other postulates of Euclidean geometry.

 

Stephen Cole Kleene

 

 

New York, London: Macmillan, vol. 5

Article „Models and Analogy in Science“, 354-359

 

 

Logical models.

Formal logic is concerned with sets of axioms and their deductive consequences and also with the interpretations of these axioms and theorems in "models" - that is, sets of entities that satisfy the axioms. These relationships are most easily exemplified in terms of elementary geometry.

Suppose a formalized geometry contains as an axiom the sentence "Any two points lie an one and only one straight line." In a fully formalized system there will be no definition of the terms "point" and "straight line" apart from this axiom and others in which these terms appear. As far as the formal system is concerned, the use of these terms is wholly defined by their relationships as given in the axioms and their deductive consequences. If we ask what the axioms are about, the only answer that can be given is that they are about just those sets of entities that satisfy the axioms.

 

One such set of entities clearly consists of the points and straight lines drawn in geometrical diagrams, or, rather, idealizations of these, which accurately reproduce the relationships specified in the axioms.

However, it does not follow that this obvious interpretation of the axioms is the only possible one. In the geometric example the axiom may otherwise be interpreted in terms of certain sets and their members, so that the axiom would read "Any two individuals are comembers of one and only one set." Similarly, a formalized Boolean algebra can be interpreted as a calculus of classes, as a calculus of propositions, or in terms of spatial areas as in the Venn diagrams. Any set of entities that constitutes an interpretation of all the axioms and theorems of a system and in which those axioms and theorems hold true is called a model (in the logician's sense) of that system.

 

Such an informal characterization of this sense of "model" is, of course, entirely inadequate for the logician's purposes. But it is sufficient to indicate, first, how the term "model" has become attached to certain semiformal and nonformal systems in the empirical sciences and, second, what crucial differentes exist between these logical models and those that are of interest in science. Some uses of "model" in science, such as nineteenth-century mechanical "models" of the ether, antedate the logician's use; others are consequences of it. It can be said as a preliminary that most uses of "model" in science do carry over from logic the idea of interpretation of a deductive system. Most writers on models in the sciences agree, however, that there is little else in common between the scientist's and the logician's use of the term, either in the nature of the entities referred to or in the purposes for which they are used.

 

... Semiformal or mathematical models.

The logical sense of „model“ has led to widespread use of the word in connection with a variety of mathematical theories developed in the sciences. No element of a replica is involved in these theories, and their interpretation is in terms of mathematical concepts such as probability or the elements of a geometry.

It has become common to speak of "probabilistic models" of, for example, psychological learning theory or population dynamics. In this context "model" refers to a mathematical theory containing the axioms of probability together with an interpretation of all or some of the nonlogical constants and variables of the theory into empirical observables. In these cases use of the word "model" seems to borrow most of its appropriateness from the logical sense, and the analogy involved is almost wholly fojmal. Insofar as it is merely formal, some writers, such as Max Black, have denied that it has any causal or explanatory forte, since the theories involved are no more than convenient mathematical expressions of the.empirical data.

 

In some cases, however, these theories do seem to have some element of material as well as formal analogy. For example, the "probability" that is a limiting-frequency interpretation of the axioms of probability exhibits some material analogy with the logical or range model of probability, for the similarity or correlation of the two notions in, for instance, games of chance exists not only by virtue of formal analogy with the same axiom system but also might well be apprehended independently of knowledge of that system. Where such material analogy does exist in connection with theoretical models, we may well say that a mathematical model does have causal, predictive, and explanatory force as an interpretation of a formal system.

 

Bibliography

Black, M., Models and Metaphors. Ithaca, N. Y., 1962. See Chs. 3 and 13 for an important analysis of linguistic metaphor and application to theoretical models.

 

Mary Hesse

 

 

Vol. 14, Chap.: Mathematics, Foundations of, 1103

Identical in Encyclopaedia Britannica, Macropaedia, vol. 11, 1976, 639

 

Current Directions.

 

From around 1960, work an the foundations of mathematics, or mathematical logic, has been often described as falling mainly in four areas, called recursion theory, proof theory, model theory and set theory.

Recursion theory is another name for the theory of computability and decidability. This name came into use from treatments based an the general recursive functions.

Proof theory is another name for metamathematics. One of the newer developments in proof theory is the singling out of portions of analysis formalized so as to use only properties of sets up to a given level in the recursion-theoretic hierarchy.

Model theory deals with the interpretations, or models, which satisfy the axioms of a given formal system. For example, by Goedel's completeness theorem (1930), if a formal system based an the first-order functional calculus F is consistent, there is a model in which the objects are the natural numbers. But the axioms might be those of an axiomatic set theory, which purport to describe an uncountably infinite totality! The model is then a "nonstandard model" of set theory (first observed to exist by Skolem in 1922-23). The Opposition of Goedel's completeness and incompleteness theorems, as remarked above, reveals that there is a nonstandard model of arithmetic.

Set theory, as a branch of foundational investigations, received great impetus from the construction in 1963 by Cohen of models which falsify the continuum hypothesis of Cantor. Cohen's and related methods are being used to show that various conjectures in set theory, analysis and topology cannot be settled an the basis of the usual axioms of Zermelo and Fraenkel, sometimes augmented by new axioms.

 

Stephen Cole Kleene

 

 

London, New York: Routledge, vol. 6

Article „Model“, 443-447.

 

Of the many kinds of things that serve as 'models', all function fundamentally as representations of what we wish to understand or to be or to do.

Model aeroplanes and other scale models share selected structural properties with their originals, while differing in other properties, such as construction materials and size.

Analogue models, which resemble their originals in some aspect of structure or internal relations, are important in the sciences, because they can facilitate inferences about complicated or obscure natura! Systems. A collection of billiard balls in random motion is an analogue model of an ideal gas; the interactions and motions of the billiard balls are taken to represent - to be analogous to - the interactions and motions of molecules in the gas.

 

In mathematical logic, a model is a structure - an arrangement of objects - which represents a theory expressed as a set of sentences. The various terms of the sentences of the theory are mapped onto objects and their relations in the structure; a model is a structure that makes all of the sentences in the theory true.

This specialized notion of model has been adopted by philosophers of science; an a 'structuralist' or `semantic' conception, scientific theories are understood as structures which are used to represent real systems in nature. Philosophical debates have arisen regarding the precise extent of the resemblances between scientific models and the natural systems they represent.

 

1 Types of model

 

... Mathematical models are even more abstract than analogue models, in that the objects and their interrelations in the model are all mathematical entities. In the sciences, mathematical models are used to represent a wide variety of real objects and situations.

Finally, there are metamathematical models, which are technical entities used to understand formal systems, such as mathematical systems themselves.

 

2 Models in metamathematics

 

In metamathematics (or `metalogic', the study of the features of formal systems), a 'model' is a structure that makes all of the sentences in a theory true, where a 'theory' is a set of sentences in a language, the various terms of which are mapped onto objects in the system and their relations.

Even in metamathematics, though, ‚model' is used in several ways: most generally, a model is a special kind of interpretation, where an interpretation of a theory consists in both the assignment or mapping of terms in the theory to objects, and to a structure, which consists of objects and their relations to one another. More specifically, an interpretation of a language specifies:

(1) a domain (universe of discourse), for example, the range of any variables that occur in any sentence in the language;

(2) a designation (denotation, bearer, reference) for each name in the language;

(3) a function f, which assigns a value in the domain for any sequence of arguments in the domain, for each function symbol in the language;

(4) a truth-value for each sentence letter in the language; and

(5) a characteristic function for each predicate letter (Boolos and Jeffrey [1974] 1980). In these cases, models are defined as those interpretations under which all the sentences in the theory are true.

 

Alternatively, models are simply the structures themselves under a specified mapping assignment (where the mapping assignment is taken to be external to the model; Robinson 1965; Tarski 1941). Models can be any kind of structure; in metalogic, models are usually expressed in terms of set theory, but technically any group of objects and their relations could serve as the model for a theory, provided that it displayed the right structure.

For example, take as a theory the sentences: 'object A is touching object B'; 'object C is touching object B'; and 'object C is not touching object A’ (where A, B and C are terms in the theory, and 'is touching' is a relation between two or more of the terms). We can easily construct or imagine a structure which `satisfies' or makes true all of these sentences in the little theory: it could consist of three objects in a row, 1, 2, 3, each one touching only the next. Notice that they could be any kind of object, including cats, jars of jam, and so on, or some mixture of these. The mapping assignment for the theory might map A onto 1, B onto 2 and C onto 3. (Equally, it could map A onto 3, B onto 2, and C onto 1.) On the usual logical definition, then, the objects 1, 2, 3 constitute one model of the theory, because a 'model' is a structure which can be interpreted so as to make all the sentences in a specific theory true.

 

3 Models in science

 

... One link between the metamathematical and scientific uses of models lies in the notion of interpretation. In the sciences, models are sometimes used as tools for making sense of theories which are not otherwise immediately comprehensible; a model can thus help the community of scientists articulate and pursue scientific theories. The wave and particle interpretations of light and of quantum mechanics are well-known examples; both the wave and particle models offer ways to understand the theoretical (and well-confirmed) equations, but they present different and incompatible features.

 

4 Models in philosophy of science

 

While discussion of the above uses of models in the sciences occupied philosophers of science for many decades, one of their basic assumptions was that models are essentially different from scientific theories. This distinction has come under scrutiny since the 1950s, and much work in recent philosophy of science going under the narre of 'structuralist' or 'semantic' approaches has centred an analysing scientific theories in terms of metamathematical models.

 

For the middle third of the twentieth century, the reigning logical positivist approaches to science understood scientific theories as 'sets of deductively connected sentences in a formal language' combined with rules for interpreting some of the terms of that language. In contrast to this linguistic view of theories, some philosophers advocated viewing theories as structures, 'which are propounded as standing in some representational relationship to actual and physically possible phenomena' (Suppe 1979: 320). More specifically, scientific theories are understood as presenting models in the metamathematical sense, that is, arrangements of objects and their relations. These structures function as iconic models for the scientists using them: they characterize, in idealized circumstances, the systems that they represent.

 

Take evolutionary population genetics as an example. Population geneticists tend to present their theories in the form of mathematical models. This means that, given the mathematical models, it is possible to examine the structures that instantiate that theory. Under this 'semantic approach' to theories, the focus is an the structures themselves, rather than an an attempt to reconstruct, in some theoretical language, the sentences of the theory, as demanded by the axiomatic and positivist approaches to theory structure.

(It remains a matter of debate whether there is an important difference between viewing a theory as a set of sentences and viewing it as a set of models. Some philosophers have suggested that the difference is merely pragmatic or heuristic: scientists sometimes find it more natural to regard a theory one way rather than another, but the two views are in principle intertranslatable. See Schaffner 1993: 99-125.)

 

This use of metamathematical models is an extension from their use in interpreting formal systems. When models serve as icons, some aspects of the model's entities and their interrelations are taken to be similar to the phenomena in the natural world that are being investigated.

In metamathematics, the more precise notion of isomorphism is central to evaluating the similarity of models. Two models are isomorphic if it is possible to make a one-to-one mapping from each element in one model to an element in the other model, and to make a one-to-one mapping from each relation among elements in one model to a relation among corresponding elements in the other model. While the notion of similarity between model and nature may be intuitive, precision demands that the natural system be represented, abstracted or measured in some fashion, if evaluations of isomorphism between the theoretical model and nature itself are to be attempted. Typically, measurements or data from the natural System are arranged into a 'data model', which provides the basic entities and relations to which the theoretical models are compared (Suppes 1962).

 

References and further reading

 

Boolos, G., and Jeffrey, R. C. (1972) Computability and Logic, New York: Cambridge University Press; 2nd edn 1980.

Robinson, A (1965) Introduction to Model Theory and to the Metamathematics of Algebra, Amsterdam: North Holland.

Schaffner, K. F. (1993) Discovery and Explanation in Biology and Medicine, Chicago, IL: University of Chicago Press.

Suppe, F. (1979) ‚Theory Structure’,  Current Research in Philosophy of Science, East Lansing, MI: Philosophy of Science Association.

Suppes, P. (1962) ‚Models of Data’, in E. Nagel, P. Suppes and A. Tarski (eds) Logic, Methodology, and Philosophy of Science: Proceedings of the 1960 International Congress, Stanford, CA: Stanford University Press.

Tarski, A. (1941) Introduction to Logic and to the Methodology of the Deductive Sciences, New York: Oxford University Press; 2nd end, 1946.

 

ELISABETH A. LLOYD