A central problem in the interpretation of non-relativistic quantum mechanics is to relate the conceptual structure of the theory to the classical idea of the state of a physical system. This paper approaches the problem by presenting an analysis of the notion of an elementary physical proposition. The notion is shown to be realized in standard formulations of the theory and to illuminate the significance of proofs of the impossibility of hidden variable extensions. In the interpretation of quantum mechanics that (...) emerges from this analysis, the philosophically distinctive features of the theory derive from the fact that it seeks to represent a reality of which complete knowledge is essentially unattainable. (shrink)

Solving numeric, logic and language puzzles and paradoxes is common within a wide community of high school and university students, fact witnessed by the increasing number of books published by mathematicians such as Martin Gardner, Douglas Hofstadter [in one of the best popular science books on paradoxes ], inspired by Gödel’s incompleteness theorems), Patrick Hughes and George Brecht and Raymond M. Smullyan, inter alia. Books by Smullyan are, however, much more involved, since they introduce learning trajectories and strategies across several (...) subjects of mathematical logic, as difficult as combinatorial logic, computability theory, and proof theory. These books provide solutions to their suggested exercises. Both statements and their solutions are written in the natural language, introducing some informal algorithms. As an exercise in Mathematics we wonder if an easy proof system could be devised to solve the amusing equations proposed by Smullyan in his books. Moreover, university students of logic could well train themselves in constructing deductive systems to solve puzzles instead of a non-uniform treatment one by one. In this paper, addressing students, we introduce one such formal systems, a tableaux approach able to provide the solutions to the puzzles involving either propositional logic, first order logic, or aspect logic. Let the reader amuse herself or himself! (shrink)

A non-monotonic logic, the Logic of Plausible Reasoning (LPR), capable of coping with the demands of what we call complex reasoning, is introduced. It is argued that creative complex reasoning is the way of reasoning required in many instances of scientific thought, professional practice and common life decision taking. For managing the simultaneous consideration of multiple scenarios inherent in these activities, two new modalities, weak and strong plausibility, are introduced as part of the Logic of Plausible Deduction (LPD), a deductive (...) logic specially designed to serve as the monotonic support for LPR. Axiomatics and semantics for LPD, together with a completeness proof, are provided. Once LPD has been given, LPR may be defined via a concept of extension over LPD. Although the construction of LPR extensions is first presented in standard style, for the sake of comparison with existing non-monotonic formalisms, alternative more elegant and intuitive ways for constructing non-monotonic LPR extensions are also given and proofs of their equivalence are presented. (shrink)

In the the passage just quoted from theDialogues concerning Natural Religion, David Hume developed a thought-experiment that contravened his better-known views about chance expressed in hisTreatise and firstEnquiry.For among other consequences of the eternal-recurrence hypothesis Philo proposes in this passage, it may turn out that what the vulgar call cause is nothing but a secret and concealed chance.

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a (...) metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables. (shrink)

The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel's theorems without getting mired in syntactic or (...) computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions, on the provability predicate in a formal system which are jointly sufficient to yield the Gödel's second incompleteness theorem for it. A key role in Löb's analysis is played by what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere's [7] category-theoretic account of incompleteness phenomena. (shrink)

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view of a (...) non-standard development. MSC: 03E30, 03E35. (shrink)

Comparative constructions form two classes, those that permit direct comparisons (comparisons of measurements as in Seymour is taller than he is wide) and those that only allow indirect comparisons (comparisons of relative positions on separate scales as in Esme is more beautiful than Einstein is intelligent). In contrast with other semantic theories, this paper proposes that the interpretation of the comparative morpheme remains the same whether it appears in sentences that compare individuals directly or indirectly. To develop a unified account, (...) I suggest that all comparisons (whether in terms of height, intelligence or beauty) involve a scale of universal degrees that are isomorphic to the rational (fractional) numbers between 0 and 1. Crucial to a unified treatment, the connection between the individuals being compared and universal degrees involves two steps. First individuals are mapped to a value on a primary scale that ranks individuals with respect to the gradable property (whether it be height, beauty or intelligence). Second, the value on the primary scale is mapped to a universal degree that encodes the value’s relative position on the primary scale. Direct comparison results if measurements such as seven feet participate in the primary scale (as in Seven feet is tall). Otherwise the result is an indirect comparison. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice ( $${\mathfrak{I}}$$ -lattice), which can be modeled by an algebraic structure built in quasi-set theory $${\mathfrak{Q}}$$. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that ‘naturally’ arises is non distributive.

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

The book provides a historical and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas as well as points out various applications of the semantic theory of truth to philosophical (...) problems. (shrink)

The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has a (...) nonstandard character. It will be shown that equality can be characterised in a way which satisfies Došen’s criteria of logicality. In the case of Hacking’s approach the fully satisfying result can be obtained only for languages with a nonempty, finite set of predicate constants other than equality. Otherwise, cut elimination theorem fails to hold. (shrink)

A grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed. A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame (...) is not in the form of $\langle W, R, R'\rangle$ , or more generally, in the form of $\langle W, R, R', R'',...\rangle$ , which consists of a non-empty set with several relations defined on it. Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger (see [4] and [3]). On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames. However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment V M and some relevance truth-value assignments V R . V M (V R ) defined in this paper is not the same as the assignment contained in a modal (relevance) model. Second, in a fibred model each relevance world is associated (or fibred) with a modal model and each modal world with a relevance model. To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality □, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic. (shrink)

In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as `at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) (...) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames. (shrink)

G. Priest's anti-consistency argument (Priest 1979, 1984, 1987) and J. R. Lucas's anti-mechanist argument (Lucas 1961, 1968, 1970, 1984) both appeal to Gödel incompleteness. By way of refuting them, this paper defends the thesis of quartet compatibility, viz., that the logic of the mind can simultaneously be Gödel incomplete, consistent, mechanical, and recursion complete (capable of all means of recursion). A representational approach is pursued, which owes its origin to works by, among others, J. Myhill (1964), P. Benacerraf (1967), J. (...) Webb (1980, 1983) and M. Arbib (1987). It is shown that the fallacy shared by the two arguments under discussion lies in misidentifying two systems, the one for which the Gödel sentence is constructable and to be proved, and the other in which the Gödel sentence in question is indeed provable. It follows that the logic of the mind can surpass its own Gödelian limitation not by being inconsistent or non-mechanistic, but by being capable of representing stronger systems in itself; and so can a proper machine. The concepts of representational provability, representational maximality, formal system capacity, etc., are discussed. (shrink)

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

In set theory without the Axiom of Choice (), we investigate the problem of the placement of Łoś's Theorem () in the hierarchy of weak choice principles, and answer several open questions from the book Consequences of the Axiom of Choice by Howard and Rubin, as well as an open question by Brunner. We prove a number of results summarised in § 3.

In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection (...) of set-theoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper. (shrink)

We present an axiomatisation for the first-order temporal logic with connectives Until and Since over the class of all linear flows of time. Completeness of the axiom system is proved.We also add a few axioms to find a sound and complete axiomatisation for the first order temporal logic of Until and Since over rational numbers time.

The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.

This article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy (...) of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions. (shrink)

In this paper the structure of pretemporal algebra is used to study some aspects of temporal algebras and modal algebras. The paper contains a computational approach to finite temporal algebras and a general splitting theorem for finite temporal algebras.

The logic TK was introduced as a propositional logic extending the classical propositional calculus with a new unary operator which interprets some conceptions of Tarski’s consequence operator. TK-algebras were introduced as models to TK . Thus, by using algebraic tools, the adequacy (soundness and completeness) of TK relatively to the TK-algebras was proved. This work presents a neighbourhood semantics for TK , which turns out to be deductively equivalent to the non-normal modal logic EMT4 . DOI:10.5007/1808-1711.2011v15n2p287.

Proofs of Gödel's First Incompleteness Theorem are often accompanied by claims such as that the gödel sentence constructed in the course of the proof says of itself that it is unprovable and that it is true. The validity of such claims depends closely on how the sentence is constructed. Only by tightly constraining the means of construction can one obtain gödel sentences of which it is correct, without further ado, to say that they say of themselves that they are unprovable (...) and that they are true; otherwise a false theory can yield false gödel sentences. (shrink)

This article begins by outlining some of the history—beginning with brief remarks of Quine's—of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. (...) The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it. (shrink)

I show that the central notion of Constructive Empiricism, empirical adequacy, can be expressed syntactically and specifically in the Received View of the logical empiricists. The formalization shows that the Received View is superior to Constructive Empiricism in the treatment of theories involving constants or functions from observable to unobservable objects. It also suggests a formalization of ‘full empirical informativeness’ in Constructive Empiricism.

ABSTRACT Reiter's default logic is one of the most prominent and well-studied approaches to nonmonotonic reasoning. Its evolution has resulted in diverse variants enjoying many interesting properties. This process however seems to be diverging because it has led to default logics that are difficult to compare due to different formal characterizations—sometimes even dealing with different objects of discourse. This problem is addressed in this paper in two ways. One the one hand, we elaborate on the relationships between different types of (...) default logics and discuss their differences wrt some basic properties. In particular, we show how two recently proposed variants, namely rational and CA-default logic, are related to each other and existing default logics. As an interesting byproduct, we obtain a cumulative counterpart to Lukasiewicz' variant of default logic. The resulting system is insofar surprising since it necessitates to refine the canonical way of assuring cumulativity in default logics. On the other hand, we pursue an integrating approach. We give a translation embedding rational and CA-default logic into a recently developed framework for default logics. This allows us not only to classify these variants, but it moreover provides us with semantical underpinnings that have been lacking so far. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

According to the revision theory of truth, the binary sequences generated by the paradoxical sentences in revision sequence are always unstable. In this paper, we work backwards, trying to reconstruct the paradoxical sentences from some of their binary sequences. We give a general procedure of constructing paradoxes with specific binary sequences through some typical examples. Particularly, we construct what Herzberger called “unstable statements with unpredictably complicated variations in truth value.” Besides, we also construct those paradoxes with infinitely many finite primary (...) periods but without any infinite primary period, those with an infinite critical point but without any finite primary period, and so on. This is the first formal appearance of these paradoxes. Our construction demonstrates that the binary sequences generated by a paradoxical sentence are something like genes from which we can even rebuild the original sentence itself. (shrink)

This paper examines the question of the extensional correctness of Tarskian definitions of logical truth and logical consequence. I identify a few different informal properties which are necessary for a sentence to be an informal logical truth and look at whether they are necessary properties of Tarskian logical truths. I examine arguments by John Etchemendy and Vann McGee to the effect that some of those properties are not necessary properties of some Tarskian logical truths, and find them unconvincing. I stress (...) the point that since the hypothesis that Tarski's definitions are extensionally correct is deeply entrenched, the burden of proof is still on the shoulders of Tarski's critics, who have not lifted the burden. (shrink)

The Research Assessment Exercise was introduced in 1986 by Thatcher, and was continued by Blair. So it has now been running for 21 years. During this time, the rules governing the RAE have changed considerably, and the interval between successive RAEs has also varied. These changes are not of great importance as far as the argument of this paper is concerned. We will concentrate on the main features of the RAE which can be summarised as follows.

The semantic, or model-theoretic, approach to theories has recently come under criticism on two fronts: (i) it is claimed that it cannot account for the wide diversity of models employed in scientific practice—a claim which has led some to propose a “deflationary” account of models; (ii) it is further contended that the sense of “model” used by the approach differs from that given in model theory. Our aim in the present work is to articulate a possible response to these claims, (...) drawing on recent developments within the semantic approach itself. Thus, the first is answered by utilizing the notion of a “partial structure”, first introduced in this context by da Costa and French in 1990. The second claim is undermined by consideration of van Fraassen's understanding of “model” which corresponds well with that evinced by modem mathematicians. This latter discussion, in particular, has an impact on the continuing debate regarding the relative merits of the semantic and syntactic views and the developments presented here can be taken to provide further support to the former. (shrink)

Tarski apresentou sua definição de operador de consequência com a intenção de expor as concepções fundamentais da consequência lógica. Um espaço de Tarski é um par ordenado determinado por um conjunto não vazio e um operador de consequência sobre este conjunto. Esta estrutura matemática caracteriza um espaço quase topológico. Este artigo mostra uma visão algébrica dos espaços de Tarski e introduz uma lógica proposicional modal que interpreta o seu operador modal nos conjuntos fechados de algum espaço de Tarski. DOI:10.5007/1808-1711.2010v14n1p47.

Mathematicians and physical scientists depend heavily on the formal symbolism of mathematics in order to express and develop their theories. For this and other reasons the last hundred years has seen a growing interest in the nature of formal language and the way it expresses meaning; particularly the objective, shared aspect of meaning as opposed to subjective, personal aspects. This dichotomy suggests the question: do the objective philosophical theories of meaning offer concepts which can be applied in psychological theories of (...) meaning? In recent years cognitive scientists such as Chomsky [1980], Fodor [1981] and MacNamara [1982] have used philosophical approaches to the meaning of formal language expressions as the basis for their psychological theories. Following this lead it seems appropriate to review some of the main treatments of meaning with a view to their transferability. (shrink)

It is well known that Tarski proved a result which can be stated roughly as: no sufficiently rich, consistent, classical language can contain its own truth definition. Tarski's way around this problem is to deal with two languages at a time, an object language for which we are defining truth and a metalanguage in which the definition occurs. An obvious question then is: under what conditions can we construct a definition of truth for a given object language. Tarski claims that (...) it is necessary and sufficient that the metalanguage be "essentially richer". Our contention, put bluntly, is that this claim deserves more scrutiny from philosophers than it usually gets and in fact is false unless "essentially richer" means nothing else than "sufficient to contain a truth definition for the object language.". (shrink)

There are several open problems in the study of the calculi which result from adding either of Hilbert's ϵ- or τ-operators to the first order intuitionistic predicate calculus. This paper provides answers to several of them. In particular, the first complete and sound semantics for these calculi are presented, in both a “quasi-extensional” version which uses choice functions in a straightforward way to interpret the ϵ- or τ-terms, and in a form which does not require extensionality assumptions. Unlike the classical (...) case, the addition of either operator to intuitionistic logic is non-conservative. Several interesting consequences of the addition of each operator are proved. Finally, the independence of several other schemes in either calculus are also proved, making use of the semantics supplied earlier in the paper. (shrink)

This thesis considers the following problem: What methods should the epistemology of science use to gain insight into the structure and behaviour of scientific knowledge and method in actual scientific practice? After arguing that the elucidation of epistemological and methodological phenomena in science requires a method that is rooted in formal methods, I consider two alternative methods for epistemology of science. One approach is the classical approaches of the syntactic and semantic views of theories. I show that typical approaches of (...) this sort are inadequate and inaccurate in their representation of scientific knowledge by showing how they fail to account for and misrepresent important epistemological structure and behaviour in science. The other method for epistemology of science I consider is modeled on the methods used to construct valid models of natural phenomena in applied mathematics. This new epistemological method is itself a modeling method that is developed through the detailed consideration of two main examples of theory application in science: double pendulum systems and the modeling of near-Earth objects to compute probability of future Earth impact. I show that not only does this new method accurately represent actual methods used to apply theories in applied mathematics, it also reveals interesting structural and behavioural patterns in the application process and gives insight into what underlies the stability of methods of application. I therefore conclude that for epistemology of science to develop fully as a scientific discipline it must use methods from applied mathematics, not only methods from pure mathematics and metamathematics as traditional formal epistemology of science has done. (shrink)