A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in (...) discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time. (shrink)

Wilson [Dialectica 63:525–554, 2009], Moore [Int Stud Philos Sci 26:359–380, 2012], and Massin [Br J Philos Sci 68:805–846, 2017] identify an overdetermination problem arising from the principle of composition in Newtonian physics. I argue that the principle of composition is a red herring: what’s really at issue are contrasting metaphysical views about how to interpret the science. One of these views—that real forces are to be tied to physical interactions like pushes and pulls—is a superior guide to real forces than (...) the alternative, which demands that real forces are tied to “realized” accelerations. Not only is the former view employed in the actual construction of Newtonian models, the latter is both unmotivated and inconsistent with the foundations and testing of the science. (shrink)

This book develops new forms of logic: Operator Logic, Probabilistic Operator Logic and Quantum Operator Logic. It then proceeds to create a new view of metaphysics, Relativistic Quantum Metaphysics, for physical Reality. It then derives the form of The Standard Model of Elementary Particles. In particular it derives the origin of parity violation, the origin of the Strong interactions, and the origin of its peculiar symmetry. Also developed are new formalisms for Logic that are of interest in themselves. (...) While mathematics is essential in the latter stages of the book it is presented with sufficient text discussion to make what it is doing understandable to the non-mathematical reader. Generally the jargon of Philosophy, Logic and Physics is avoided as much as possible. But the second part of this book is very mathematical of necessity. In it the form of The Standard Model is derived in detail from a fundamental set of principles. (shrink)

The metaphysics of early embryos is a hotly debated topic in contemporary bioethics and metaphysics. Many contemporary Aristotelians believe that a human being is present from the moment of conception. At the same time, certain findings in modern embryology about the formation of identical twins challenge this belief. It becomes much harder when these theories are taken into account to understand the continued identity over time of the embryo(s) given the twinning process. In this article, I will consider (...) the philosophical implications of two models of monozygotic twinning within an Aristotelian metaphysical schema one of which is the standard, or traditional, model. The other of which is a new model recently put forward by Herranz. For the sake of completeness, I will also consider the philosophical implications of chimeras for the Aristotelian position. I will explain how Aristotelians can understand the process of twinning while holding on to their belief that a human being is present from the moment of conception. Summary: I will argue that a human being is present from the moment of conception. I will argue for this on Aristotelian grounds, and I will then defend this claim from criticisms based on a number of findings in modern embryology. (shrink)

Since the time of Aristotle, metaphysics has been an ill-defined term. This paper defines it as a logically idempotent metalinguistic identity of reality which couples the two initial ingredients of awareness: perceptual reality (the basis of physics), and cognitive-perceptual syntax, a formalization of mind. The explanation has been reduced to a few very simple, clearly explained mathematical ingredients. This paper contains no assumptions or arguable assertions, and is therefore presented as an advanced formulation of logic which has been (...) updated for meaningful reference to the structure of reality at large. This structure, called the Cognitive-Theoretic Model of the Universe or CTMU, resolves the problems attending Cartesian dualism by replacing dualism with the mathematical property of self-duality, meaning (for reality-theoretic purposes) the quantum-level invariance of identity under permutation of objective and spatiotemporal data types. The CTMU takes the form of a global coupling or superposition of mind and physical reality in a self-dual metaphysical identity M:L<-->U, which can be intrinsically developed into a logico-geometrically self-dual, ontologically self- contained language incorporating its own medium of existence and comprising its own model therein. (shrink)

The book examines issues related to the way modeling and simulation enable us to reconstruct aspects of the world we are investigating. It also investigates the processes by which we extract concrete knowledge from those reconstructions and how that knowledge is legitimated.

The essence of number was regarded by the ancient Greeks as the root cause of the existence of the universe, but it was only towards the end of the 19th century that mathematicians initiated an in-depth study of the nature of numbers. The resulting unavoidable actuality of infinities in the number system led mathematicians to rigorously investigate the foundations of mathematics. The formalist approach to establish mathematical proof was found to be inconclusive: Gödel showed that there existed true propositions (...) that could not be proved to be true within the natural number universe. This result weighed heavily on proposals in the mid-20th century for digital models of the universe, inspired by the emergence of the programmable digital computer, giving rise to the branch of philosophy recognised as digital philosophy. In this article, the models of the universe presented by physicists, mathematicians and theoretical computer scientists are reviewed and their relation to the natural numbers is investigated. A quantum theory view that at the deepest level time and space may be discrete suggests a profound relation between natural numbers and reality of the cosmos. The conclusion is that our perception of reality may ultimately be traced to the ontology and epistemology of the natural numbers. (shrink)

Materialism, epiphenomenalism, dualism, idealism, and dual-aspect theories may all be represented by an appealing abstract mathematical device called a commutative diagram. Properties of the components of such diagrams characterize and, to some extent, even parameterize these systems and attendant metaphysical concepts (such as causal closure and supervenience) in a unified framework; process thought is of particular interest in this connection. In many cases we can even exemplify the theories typified by these diagrams in explicit graphical models. All of this (...) tends to clarify the relationships among key philosophical positions and to sharpen our sense of the effective domain and principal limitations of each. Systematic variation of these abstract diagrams may even suggest cogent metaphysical systems yet to be examined. (shrink)

We have two main aims: to construct mathematical models for analysing determinism, causality and supervenience; and then to use these to demonstrate the possibility of constructing an ontic construal of the operation of free will - one requiring both the presentation of genuine alternatives to an agent and their selecting between them in a manner that permits the attribution of responsibility. Determinism is modelled using trans-temporal ontic links between discrete juxtaposed universe states and shown to be distinct from predictability. (...) Causality is defined on a temporal sequence of δ-algebras and quantified using a measure. The measure leads to definitions of causal overdetermination and epiphenomena. Proofs are constructed to demonstrate deterministic universes must carry their properties essentially but not necessarily locally. We argue determinism and causality are separate doctrines. These models and results are marshalled to put the case that a counterfactual construal of ontic choice cannot work. In response we propose ‘immanence’ - a modified form of indeterminism whereby a universe can present choices to its denizens. We prove that beings subsumed within a universe cannot pilot their own actions. We then argue these beings can exercise free will only when selecting between choices inhering within immanent relata. A being is responsible for its selections if and only if it is constituted of a temporally evolving deterministic substructure. Our proposal is novel: it avoids injecting indeterminism into the decision process. Topological models for property supervenience are developed and used to reconstruct standard definitions from the literature. These are then used to demonstrate considerations of supervenience do not affect our arguments. We have demonstrated that a model of the exercise of free will involving both genuine choices and responsibility is possible but can only operate within a non-deterministic universe possessing specific traits. (shrink)

Abstract.At the heart of the most radical proposals in Stuart Kauffman's Investigations is his attempt to show that we find in evolutionary biology some configuration spaces—the sets of possible developments for any given system—that (unlike those in traditional physics of Newtonian, relativistic, and quantum stripes) cannot be completely described in advance. We bring Charles Peirce's work on the philosophy of continuity to bear on the problem and discover, first, that Kauffman's arguments do not succeed; second, that Peirce's metaphysics provide (...) new and sounder arguments for the same propositions; third, that Peirce's rigorous but nonstandard treatment of mathematical continuity shows great promise for modeling the unpredictability and growth we find in evolutionary biology; fourth, that it also strengthens a development only hinted at by biologists thus far—the inevitable involvement of the observer's mind in constituting the objects of science. We close with a logical argument for the surprising relevance of metaphysical hypotheses in the natural sciences and with suggestions for future work that will connect these questions to what Kauffman terms the “narrative stance” in biology. (shrink)

In this paper, I distinguish scientific models in three kinds on the basis of their ontological status—material models, mathematical models and fictional models, and develop and defend an account of fictional models as fictional objects—i.e. abstract objects that stand for possible concrete objects.

The Pythagorean idea that numbers are the key to understanding reality inspired philosophers in late Antiquity (4th and 5th centuries A.D.) to develop theories in physics and metaphysics based on mathematical models. This book draws on some newly discovered evidence, including fragments of Iamblichus's On Pythagoreanism, to examine these early theories and trace their influence on later Neoplatonists (particularly Proclus and Syrianus) and on medieval and early modern philosophy.

The notion of truth is a central subject both in Philosophy and Mathematical Logic. The logical approach on the one side and the philosophical one on the other, however, mostly deal with problems which, apparently, require different tools to be tackled. In this paper I argue that such a separation can and should be overcome, and, in order to build a bridge, I focus on the philosophical issue of the insubstantiality of truth, which is a crucial topic to distinguish (...) inflationist from deflationist proposals. Elaborating on the interpretation of insubstantiality in terms of the sparse/abundant classification of properties, I put forward a refined version in which certain flaws afflicting other formulations are solved. Then, I show how, using this improved variant, the philosophical notion of abundance can be fruitfully related to the formal notion of expandability of models, if a logical framework is adopted. Among other virtues, the obtained link can shed new light on the debate of deflationism and conservativity. (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended (...) that this endeavor can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

role in scientific theories, and their relation to time. ;Chapter 1, "Why Apply Mathematics?" argues that scientific theories are not about the mathematics that is applied in them, and defends this thesis against the Quine-Putnam Indispensability Argument. ;Chapter 2, "Scientific Ontology," is a critical study of W. V. Quine's claim that metaphysics and mathematics are epistemologically on a par with natural science. It is argued that Quine's view relies on a unacceptable account of empirical confirmation. ;Chapter 3, "Prior and (...) the Platonist," demonstrates the incompatibility of two popular views about time: the "Platonist" thesis that some objects exist "outside" time, and A. N. Prior's proposal for treating tense on the model of modality. ;Chapter 4, "What has Eternity Ever Done for You?" argues time), and presents a defense of the rival view that they exist sempiternally. (shrink)

The form in which the metaphysical quality appears most definitely is in the character of matter that cannot be experienced itself and yet is regarded without question by the scientist as there. There is an illustration of a possible metaphysical inquiry, in a study of the matter and form of electrons. We can set up models of them but we know that no model we set up can have the character that the electrons themselves have; yet these electrons themselves are (...) the conditions for the experience that we do have, and they can be presented to us in terms of that experience of which they are the condition. Such a field of investigation, as to their reality, their nature, would be a metaphysical field. It is a field of reality that cannot be brought inside of experience itself, because it is a condition of that experience, and yet it is a field which the experience itself implies. That would be, I presume, that which would appear in our own thought as the most evident example of a metaphysical discipline. Of course if it is approached from metaphysics as distinct from the physical standpoint, we are not distinguishing what the scientist is particularly interested in. He postulates something that is there, and is able to determine relations of uniformity, logical patterns, as appearing in that field, and he is liable to set up apparatus of the experiential type to occupy the field and to serve as relata. But he is not himself particularly interested in distinguishing what the matter of those relata is in itself. He is interested in relations, in laws of matter, in uniformities, in the mathematical functions that can be made use of for expressing those laws. These logical patterns are what he is seeking and he constructs the models which will enable him to grasp them. The metaphysical investigation would ask what the natures of these forms are that constitute the relata between which these relations obtain. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and (...) that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely (...) at mathematicians; they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

Husserl’s philosophy of mathematics, his metatheory, and his transcendental phenomenology have a sophisticated and systematic interrelation that remains relevant for questions of ontology today. It is well established that Husserl anticipated many aspects of model theory. I focus on this aspect of Husserl’s philosophy in order to argue that Thomasson’s recent pleonastic reconstruction of Husserl’s approach to essences is incompatible with Husserl’s philosophy as a whole. According to the pleonastic approach, Husserl can appeal to essences in the absence of a (...) positive metaphysical account of their nature. I show, using central results from recent model theory, that the pleonastic approach undermines Husserl’s approach to formalization and categoricity, an effect that will ripple out from Husserl’s philosophy of mathematics into Husserl’s metatheory and transcendental phenomenology. The result is that Husserl cannot appeal to formal essences without metaphysical commitments. However, the very observations Thomasson makes about the nature of eidetic intuition in Husserl lead to a general strategy for responding to the problem. The article thus illustrates that the pleonastic and the model-theoretic routes for making Husserl relevant to present-day ontology are competing approaches, but I conclude that Husserl scholars seeking to set Husserl’s present-day relevance into sharp relief could do worse than to emphasize the model-theoretic nature of Husserl’s enterprise. (shrink)

Theory and Reality is about the connection between true theories and the world. A mathematical framefork for such connections is given, and it is shown how that framework can be used to infer facts about the structure of reality from facts about the structure of true theories, The book starts with an overview of various approaches to metaphysics. Beginning with Quine's programmatic "On what there is", the first chapter then discusses the perils involved in going from language to (...) metaphysics. It criticises contemporary intuition-driven metaphysics, comments on naturalistic approaches, and then presents the main proposition put forward in the thesis: we should base metaphysics on model theory. In chapters 2 to 5, mathematical treatments are given of concepts that we need: theories, metaphysics, necessitation and semantics. These are used in chapters 6 and 7 to prove that, seen from a certain informative view point, any true theory will give rise to an isomorphism between that theory and the world. This conclusion is similar to Wittgenstein's in the Tractatus, but differs in that it places the structural relationship on the level of whole theories, rather than single propositions. (shrink)

Model transfer refers to the observation that particular model structures are used across multiple distinct scientific domains. This paper puts forward an account to explain the inter-domain transfer of model structures. Central in the account is the role of validation criteria in determining whether a model is considered to be useful by practitioners. Validation criteria are points of reference to which model correctness for a particular purpose is assessed. I argue that validation criteria can be categorized as being mathematical, (...) theoretical or phenomenological in nature. Model transfer is explained by overlap in validation criteria between scientific domains. Particular emphasis is placed on overlap between phenomenological criteria. Overlap in phenomenological criteria can be explained through the notion of universal patterns. Universal patterns are abstract structures that can be made to refer to multiple distinct phenomena when coupled with phenomena-specific empirical content. I present the case study of the Yule Process, in which universal patterns play a crucial role in explaining model transfer. This paper provides an account of model transfer that stays close to modelling practice and expands existing accounts by introducing the notion of universal patterns. (shrink)

Trust and monitoring are traditionally antithetical concepts. Describing trust as a property of a relationship of reliance, we introduce a theory of trust and monitoring, which uses mathematical models based on two classes of functions, including _q_-exponentials, and relates the levels of trust to the costs of monitoring. As opposed to several accounts of trust that attempt to identify the special ingredient of reliance and trust relationships, our theory characterizes trust as a quantitative property of certain relations of reliance (...) that can be quantified and expressed as a scalar quantity. Our theory is applicable to both human–human and human–artificial agent interactions, as it is agnostic with respect to the concrete realization of trustworthiness properties, and is compatible with many views differing on which properties contribute to trust and trustworthiness. Finally, as our mathematical models make the quantitative features of trust measurable, they provide empirical studies on trust with a rigorous methodology for its measurement. (shrink)

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general (...) adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful. (shrink)

The philosophy of mathematics has long been concerned with deter- mining the means that are appropriate for justifying claims of mathemat- ical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary math- ematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The as- sociated values are often loosely classied as (...) aspects of \mathematical understanding." Meanwhile, in a branch of computer science known as \formal ver- ication," the practice of interactive theorem proving has given rise to software tools and systems designed to support the development of com- plex formal axiomatic proofs. Such eorts require one to develop models of mathematical language and inference that are more robust than the the simple foundational models of the last century. This essay explores some of the insights that emerge from this work, and some of the ways that these insights can inform, and be informed by, philosophical theories of mathematical understanding. (shrink)

What is the proper metaphysics of quantum mechanics? In this dissertation, I approach the question from three different but related angles. First, I suggest that the quantum state can be understood intrinsically as relations holding among regions in ordinary space-time, from which we can recover the wave function uniquely up to an equivalence class (by representation and uniqueness theorems). The intrinsic account eliminates certain conventional elements (e.g. overall phase) in the representation of the quantum state. It also dispenses with (...) first-order quantification over mathematical objects, which goes some way towards making the quantum world safe for a nominalistic metaphysics suggested in Field (1980, 2016). Second, I argue that the fundamental space of the quantum world is the low-dimensional physical space and not the high-dimensional space isomorphic to the ``configuration space.'' My arguments are based on considerations about dynamics, empirical adequacy, and symmetries of the quantum mechanics. Third, I show that, when we consider quantum mechanics in a time-asymmetric universe (with a large entropy gradient), we obtain new theoretical and conceptual possibilities. In such a model, we can use the low-entropy boundary condition known as the Past Hypothesis (Albert, 2000) to pin down a natural initial quantum state of the universe. However, the universal quantum state is not a pure state but a mixed state, represented by a density matrix that is the normalized projection onto the Past Hypothesis subspace. This particular choice has interesting consequences for Humean supervenience, statistical mechanical probabilities, and theoretical unity. (shrink)

In this paper, the (possible) role of model theory forstructuralism and structuralist definitions of ``reduction'' arediscussed. Whereas it is somewhat undecisive with respect tothe first point – discussing some pro's and con's ofthe model theoretic approach when compared with a syntacticand a structuralist one – it emphasizes that severalstructuralist definitions of ``reducibility'' do not providegenerally acceptable explications of ``reducibility''. This claimrests on some mathematical results proved in this paper.

Exploration of a hypothetical model of the structure of the Emergent Event. -/- Key Words: Emergent Event, Foundational Mathematical Categories, Emergent Meta-system, Orthogonal Centering Dialectic, Hegel, Sartre, Badiou, Derrida, Deleuze, Philosophy of Science.

I argue that informational models of consciousness, including those proposed by the Integrated Information Theory, don’t presuppose or entail any particular view about the physical or metaphysical nature of consciousness. Such models only tell us how certain properties of consciousness can be mathematically described, thus providing a quantitative characterization of the phenomenon of consciousness that may contribute to the development of new methods of assessment and guide the explanatory project by supplying additional constraints on theoretical proposals. While informational models are (...) in principle compatible with multiple physical interpretations, a preliminary argument in favour of the continuity between informational approaches and the mechanistic project in neuroscience can be made based on historical precedents. (shrink)

A mathematical model in science can be formulated as a counterfactual conditional, with the model’s assumptions in the antecedent and its predictions in the consequent. Interestingly, some of these models appear to have assumptions that are metaphysically impossible. Consider models in ecology that use differential equations to track the dynamics of some population of organisms. For the math to work, the model must assume that population size is a continuous quantity, despite that many organisms are necessarily discrete. This means (...) our counterfactual representation of the model can have an impossible antecedent, giving us a counterpossible. Analogous counterpossibles arise in other sciences, as we’ll see. According to a prominent view in counterfactual semantics, the vacuity thesis, all counterpossibles are vacuously true, that is, true merely because their antecedents are necessarily false. But some counterpossible formulations of differential equation models in science are not all vacuously true—some are non-vacuously true, and some are false. I go on to show how an alternative semantics, one that employs impossible worlds, can deliver this judgment. (shrink)

At the end of the 19th century, Josiah Royce participated in what has come to be called the great debate (Royce, 1897; Armour, 2005).1 The great debate concerned issues in metaphysical theology, and, since metaphysics was primarily idealistic, it dealt considerably with the relations between the divine Self and lesser selves. After the great debate, Royce developed his idealism in his Gifford Lectures (1898-1900). These were published as The World and the Individual. At the end of the first volume, (...) Royce added a Supplementary Essay. The Essay, continuing themes from the great debate, developed an intriguing mathematical model of the relations between the divine Self and lesser selves. The second volume went on to.. (shrink)

SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. Dummett (...) draws the general conclusion that accounts of meaning must traffic only in decidable circumstances. I suggest that Dummett can be right about platonism but wrong in this general conclusion: the centrality of decidable circumstances in competent use of language is a special feature of mathematical language. This means that someone who recoils from the anti‐realism constituted by Dummett's generalized anti‐platonism, in the case of, say, statements about other minds, need not be recoiling into a close analogue of platonism, as Dummett suggests. We can reinstate the intuitive idea that platonism goes wrong by inappropriately modelling the epistemology and metaphysics of mathematics on the epistemology and metaphysics of the natural world. And we make room for the suggestion that anti‐realism makes a converse mistake; in this vein, I propose a picture of Dummettian anti‐realism as a novel expression of familiar and suspect epistemological and metaphysical thoughts. (shrink)

This chapter explains the relationship between metaphysics and science as expounded by Descartes. The topic includes the effect of concepts of God and soul into the discovery of the scientific method. Metaphysical certainty grounds the rule of evidence in God, who is the source of all truth, and then extends from mathematics to everything so demonstrated in physics, and the knowledge that material things exist. The chapter examines in particular the stages of science's subordination to God, and some of (...) its consequences: What are the meaning and limits of the “fable” of The World? What is the difference between hypotheses, which move toward a conjunction with the real, and the “comparisons” employed in the Essays, which are not based on “principles”? Is the modern notion of a model ruled out, and with what legitimacy? (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. (...) Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

Two of Hilary Putnam's model-theoretic arguments against metaphysical realism are examined in detail. One of them is developed as an extension of a model-theoretic argument against mathematical realism based on considerations concerning the so-called Skolem-Paradox in set theory. This argument against mathematical realism is also treated explicitly. The article concentrates on the fine structure of the arguments because most commentators have concentrated on the major premisses of Putnam's argument and especially on his treatment of metaphysical realism. It is (...) shown that the validity of Putnam's arguments is doubtful and that realists are by no means forced to accept the theses Putnam ascribes to them. It is concluded that Putnam fails to give convincing arguments for rejecting mathematical or metaphysical realism. Furthermore, Putnam's internal realism is discussed critically. (shrink)

“Modal logic was conceived in sin: the sin of confusing use and mention.” So quips Quine. The stigma stuck with modal logic for a while. But by the mid-sixties, a whole cluster of mathematically elegant interpretations of modal logic became available. All are natural extensions of the classical Tarskian semantics of predicate logic. By the mid-seventies, Quine’s criticisms seemed obsolete. Today, we teach the model theory of modal logic as a matter of course. Quine’s “interpretive problem” is just forgotten. The (...) purpose of my dissertation is to revamp Quine’s thesis: the very foundations of modal logic—simple propositional modal logic, even before quantifiers are added—are shaky. Quine holds that the natural interpretation of the operator “necessarily” is in terms of a full-blooded metaphysical notion of real necessity. Such a notion, Quine conjectures, is committed to the “metaphysical jungle of Aristotelian essentialism”. Hence, the intended interpretation is to be rejected. Thus Quine: so much the worse for real necessity. An alternative comes to mind: a meta-level, formal rather than real, semantic interpretation of the operator of necessity. In the late forties, Carnap developed this line of interpretation which culminates in the aforementioned model theory (“Kripke’s possible world semantics”) in the sixties. On this basis a formal interpretation is introduced: it grounds necessity in the notion of model theoretic validity. As pointed by friends of the formal interpretation (Marcus, Parsons and Kaplan), no invidious essentialist claim is verified. But have Quine’s concerns been answered? Yes and no. Technical problems regarding quantification across modal operators, substitutivity, matters of scope, and the basis of cross-world identification have indeed been solved. However, no real interpretation of “necessarily” has been provided. I argue that the seeds of such an interpretation are present in Kripke’s philosophical discussion of de re necessities in Naming and Necessity. It is not model theoretic, and it does not reduce necessity to some non-modal notion, just as Quine predicted. Even more in his vein, the real interpretation is (i) metaphysically, grounded indeed in essentialist theses, and (ii) epistemologically, forced to give up Kant’s ideal that all necessities are known a priori. (shrink)

In the ordinary way of representing relations, the order of the relata plays a structural role, but in the states themselves such an order often does not seem to be intrinsically present. An alternative way to represent relations makes use of positions for the arguments. This is no problem for the love relation, but for relations like the adjacency relation and cyclic relations, different assignments of objects to the positions can give exactly the same states. This is a puzzling situation. (...) The question is what is the internal structure of relations? Is the use of positions still justified, and if so, what is their ontological status? In this paper mathematical models for relations are developed that provide more insight into the structure of relations “out there” in the real world. (shrink)

Time is fascinating and important to everyone. This book does two things: it reviews a great range of images of time, that is, theories of time, from a diversity of perspectives: historical, religious, biological, mathematical, and scientific. In addition to a wide-ranging discussion of many aspects of time, the book shows how a spreadsheet can be used easily to explore various aspects of time such as time travel and light cones in relativity. But it goes further. It introduces the (...) concept of generalized proposition, which can be used by the reader to distinguish those images of time that are metaphysical (which means they cannot be scientifically validated) and those that could in principle be put to empirical test. This is of particular importance in this day and age, when we are flooded by a plethora of competing images of time. Many of these have no scientific basis, empirical support, or content but are presented as if they do. (shrink)

At the beginning of the 20th century, Gestalt psychologists put forward the concept of psychoneural isomorphism, which was meant to replace Fechner’s obscure notion of psychophysical parallelism and provide a heuristics that may facilitate the search for the neural correlates of the mind. However, the concept has generated much confusion in the debate, and today its role is still unclear. In this contribution, I will attempt a little conceptual spadework in clarifying the concept of psychoneural isomorphism, focusing exclusively on conscious (...) visual perceptual experience and its neural correlates. Firstly, I will outline the history of our concept, and its alleged metaphysical and epistemic roles. Then, I will clarify the nature of isomorphism and rule out its metaphysical role. Finally, I will review some epistemic roles of our concept, zooming in on the work of Jean Petitot, and suggest that it does not play a relevant heuristic role. I conclude suggesting that psychoneural isomorphism might be an indicator of robustness for certain mathematical descriptions of perceptual content. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can (...) be also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

The semantic view of theoriesis one according to which theoriesare construed as models of their linguisticformulations. The implications of thisview for scientific realism have been little discussed. Contraryto the suggestion of various champions of the semantic view,it is argued that this approach does not makesupport for a plausible scientific realism anyless problematic than it might otherwise be.Though a degree of independence of theory fromlanguage may ensure safety frompitfalls associated with logical empiricism, realism cannot be entertained unless models or (abstractedand/or idealized) (...) aspects thereof are spelled out in terms of linguistic formulations (such as mathematical equations),which can be interpreted in terms of correspondencewith the world. The putative advantage of thesemantic approach – its linguistic independence – isthus of no help to the realist. I consider recent treatmentsof the model-theoretic view (Suppe, Giere, Smith), and find that although some of these accounts harbour the promiseof realism, this promise is deceptive. (shrink)

There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider (...) two models for such orderings, (i) an implicational model and something I call (ii) a mosaic model. -/- Both models fail to do what such orderings should do. Moreover, in both cases, the trouble arises because the class of truths to be ordered is taken to be closed under various weakening implications. This is a result of their employing a global conception of logic. (shrink)

This book presents a detailed analysis of three ancient models of spatial magnitude, time, and local motion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in (...) its discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments. (shrink)

Mathematical models and their modeling frameworks developed to advance knowledge in one discipline are sometimes sourced to answer questions or solve problems in another discipline. Studying this aspect of cross-disciplinary transfer of knowledge objects, philosophers of science have weighed in on the question of whether knowledge about how a mathematical model is previously applied in one discipline is necessary for the success of reapplying said model in a different discipline. However, not much has been said about whether the (...) answer to that epistemological question applies to the reapplication of a modeling framework. More generally, regarding the nature of the production of knowledge in science, a metaphysical question remains to be explored whether historical contingencies associated with a mathematical construct have a genuine impact on the nature—as opposed to sociological practices or individual psychology—of advancing scientific knowledge with said construct. Focusing on this metaphysical question, this paper analyzes the use of mathematical logic in the development of the Chomsky hierarchy and subsequent reapplications of said hierarchy; with these examples, this paper develops the notion of “spillovers” as a way to detect cross-disciplinary justifications for better understanding the relations between reapplications of the same mathematical construct across disciplines. (shrink)

The omnipresence of the same basic equations, function forms, algorithms, and quantitative methods is one of the most spectacular characteristics of contemporary modeling practice. Recently, the emergence of the discussion of templates and template transfer has addressed this striking cross-disciplinary reach of certain mathematical forms and computational algorithms. In this paper, we develop a notion of a model template, consisting of its mathematical structure, ontology, prototypical properties and behaviors, focal conceptualizations, and the paradigmatic questions it addresses. We apply (...) this notion to three widely disseminated and powerful model templates: the Sherrington-Kirkpatrick model of spin glasses, scale-free networks, and the Kuramoto model of synchronization. We argue that what appears to be an interdisciplinary model transfer between different domains turns out, from a broader perspective, to be the application of transdisciplinary model templates across a multitude of domains. We also point out a further feature of template-based modeling that so far has not been discussed: template entanglement. Such entanglement enhances and makes manifest the conceptual side of model templates. (shrink)

What are scientific models? Philosophers of science have been trying to answer this question during the last three decades by putting forward a number of different proposals. Some say that models are best understood as abstract Platonic objects or fictional entities akin to Sherlock Holmes, while others focus on their mathematical nature and see them as set theoretical structures. Although each account has its own strengths in offering various insights on the nature of models, several objections have been raised (...) against these views which still remain unanswered, making the debate on the ontology of models seem unresolvable. The primary aim of this paper is to show that a large part of these difficulties stems from an inappropriate reading of the main question on the ontology of models as a purely metaphysical question. Building on Carnap, it is argued that the question of the ontology of scientific models is either an internal theoretical question within an already accepted linguistic framework or an external practical question regarding the choice of the most appropriate form of language in order to describe and explain the practice of scientific modelling. The main implication of this view is that the question of the ontology of models becomes a means of probing other related questions regarding the overall practice of scientific modelling, such as questions on the capacity of models to provide knowledge and the relation of models with background theories. (shrink)

This paper argues that a successful philosophical analysis of models and simulations must accommodate an account of mathematically rigorous results. Such rigorous results may be thought of as genuinely model-specific contributions, which can neither be deduced from fundamental theory nor inferred from empirical data. Rigorous results provide new indirect ways of assessing the success of models and simulations and are crucial to understanding the connections between different models. This is most obvious in cases where rigorous results map different models on (...) to one another. Not only does this put constraints on the extent to which performance in specific empirical contexts may be regarded as the main touchstone of success in scientific modelling, it also allows for the transfer of warrant across different models. Mathematically rigorous results can thus come to be seen as not only strengthening the cohesion between scientific strategies of modelling and simulation, but also as offering new ways of indirect confirmation. (shrink)

The apparent chasm between two camps in metaphysics, analytic metaphysics and scientific metaphysics, is well recognized. I argue that the relationship between them is not necessarily a rivalry; a division of labour that resembles the relationship between pure mathematics and science is possible. As a case study, I look into the metaphysical underdetermination argument for ontic structural realism, a well‐known position in scientific metaphysics, together with an argument for the position in analytic metaphysics known as (...) ontological nihilism. I argue that we can ascribe the same schema to both arguments, which indicates that analytic metaphysics can offer an abstract model that scientific metaphysics may find useful. (shrink)