### Models and Representation: Why Structures Are Not Enough

- MEASUREMENT IN PHYSICS AND ECONOMICS PROJECT DISCUSSION PAPER SERIES, DP MEAS 25/02, LONDON SCHOOL OF ECONOMICS
, 2002

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### Frege’s correlation

"... 1. Let me remind you of an old puzzle about applied arithmetic: How can knowing anything about numbers be relevant to one’s knowledge of the natural world? Why is it, for example, that knowing that the number of the apples is identical to the number of the children can be relevant to knowing that no ..."

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1. Let me remind you of an old puzzle about applied arithmetic: How can knowing anything about numbers be relevant to one’s knowledge of the natural world? Why is it, for example, that knowing that the number of the apples is identical to the number of the children can be relevant to knowing that no apples will be left if each of the children eats an apple? It is not unusual to think that the right answer to this puzzle is implicit in Frege’s Grundlagen (see, for instance, Steiner 1998: 21–22). The Fregean answer is supposed to be this: There is a correlation between arithmetical truths, on the one hand, and certain second-order truths, on the other. It is embodied in the following principle: Hume’s Principle The number of the Fs is identical to the number of the Gs if and only if the Fs are in one-one correspondence with the Gs. 1

### unknown title

, 2013

"... This paper is an effort to extract some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase-functions for the language of arithmetic, and suggest a way around the pr ..."

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This paper is an effort to extract some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase-functions for the language of arithmetic, and suggest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases. Although much of the material in this paper is drawn from the book—and from an earlier paper (Rayo 2008)—I hope the present discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. 1 Nominalism Mathematical Nominalism is the view that there are no mathematical objets. A standard problem for nominalists is that it is not obvious that they can explain what the point of a mathematical assertion would be. For it is natural to think that mathematical sentences like ‘the number of the dinosaurs is zero ’ or ‘1 + 1 = 2 ’ can only be true if mathematical objects exist. But if this is right, the nominalist is committed to the view that such sentences are untrue. And if the sentences are untrue, it not immediately obvious that they would be

### The Info-Computation Turn in Physics

"... Modern Computation and Information theories have had significant impact on science in the 20th century- in theory and application. This influence is tracked (through a generalized, Information-laden scientific Style of Reasoning denoting the Information-theoretical and Computational turn in science) ..."

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Modern Computation and Information theories have had significant impact on science in the 20th century- in theory and application. This influence is tracked (through a generalized, Information-laden scientific Style of Reasoning denoting the Information-theoretical and Computational turn in science), with a focus on the information processing and transfer metaphors and descriptive tools prevalent in current physics. Implementation of Information-Theoretical concepts produces such mathematical physical developments as Black-Hole Thermodynamics (BHTD) and the Black-Hole War. The treatment of physical systems as information processing systems drives such branches of physics as Quantum Information Theory (QIT). The common Informational basis of computation and communication brings about a foundational shift in scientific reasoning with deep – potentially problematic as well as intriguing – philosophical ramifications. Models of computation and of physics 1 Scientific Revolutions through a New Style of Reasoning Conceptions (Rolf Landauer, 1991) of physical reality as an Information 1-processing system permeate current science-- from rigorous statistical analysis (thermodynamics of black holes), through semantic aspirations to world-views such as Extreme Digital Ontology. Information processing ontologies are heralded as new Galilean world-systems of Info-Computationalism. 2 As pervasive (Gleick, 2011) and perhaps overused a concept as it is – Information is heralded by some as 'the new language of science ' (Baeyer, 2004). Even if accepted, this role requires a finer tuned description, offered by Hacking’s Styles of Reasoning. This brand of metaconcepts (Hacking, 1992, 2004) enables a description of a science imbued with information processing and transmission terminology. Hacking’s Styles (adopted from Crombie 3) come about through scientific revolutions, quieter than paradigm-shifts or the great scientific revolutions of the 20 th century, but with deep impact on science and society. 1 The term “Information ” will – unless explicitly defined otherwise – refer to the leanest technical definition in binary

### AN EPISTEMIC STRUCTURALIST ACCOUNT OF MATHEMATICAL KNOWLEDGE

, 2003

"... This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures ..."

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This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures or patterns, has become an increasingly prominent theory of mathematical ontology in the later decades of the twentieth century. The epistemically driven version of structuralism that is advocated in this thesis takes structures to be primarily physical, rather than Platonically abstract entities. A fundamental benefit of epistemic structuralism is that this account, unlike other accounts, can be integrated into a naturalistic epistemology, as well as being congruent with mathematical practice. In justifying mathematical knowledge, two levels of abstraction are introduced. Abstraction by simplification is how we extract mathematical structures from our experience of the physical world. Then, abstraction by extension, simplification or recombination are used to acquire concepts of derivative mathematical structures.

### Complex states of simple . . .

, 2008

"... A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including lig ..."

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A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities (”the phase problem”), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow. When

### Informal Prologue

, 2000

"... This paper is born out of many years of thinking about the issues relating science and theology. I started out as an undergraduate, pondering such questions as those regarding the nature of physical law and the puzzle of why mathematics works in describing nature. Over the years of reflection, I hav ..."

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This paper is born out of many years of thinking about the issues relating science and theology. I started out as an undergraduate, pondering such questions as those regarding the nature of physical law and the puzzle of why mathematics works in describing nature. Over the years of reflection, I have come to believe that if Christians are to take their faith seriously into the realm of science, then a thorough re-examining of the relation between theology and science is warranted. In particular, as we learn from the philosophy of science and from the Dutch Reformed tradition that we cannot avoid our presuppositions when theorizing about science, for Christians it becomes all the more obvious that whatever lies at the foundation of faith commitments for any scientist cannot be avoided. Thus we must ask the pro-active question: just how does our faith give a foundation for our own way of understanding science? This paper is an attempt to address that question from the perspective of the Reformed tradition. My task is of course a highly integrative effort, combining ideas from science with those from philosophy, theology and history. As a physicist without formal training in these other disciplines, I fully expect that my story is incomplete; I expect that there are important sources I have missed while writing this paper which would provide an even fuller picture. On the other hand, integration by its very nature should be viewed as a community effort, so I welcome comments and suggestions which might serve to add to the story and

### 1 Clifford Algebraic Computational Fluid Dynamics: A New Class of Experiments.

, 2010

"... Though some influentially critical objections have been raised during the ‘classical ’ pre-computational simulation philosophy of science (PCSPS) tradition, suggesting a more nuanced methodological category for experiments2, it safe to say such critical objections have greatly proliferated in philos ..."

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Though some influentially critical objections have been raised during the ‘classical ’ pre-computational simulation philosophy of science (PCSPS) tradition, suggesting a more nuanced methodological category for experiments2, it safe to say such critical objections have greatly proliferated in philosophical studies dedicated to the role played by computational simulations in science. For instance, Eric Winsberg (1999-2003) suggests that computer simulations are methodologically unique in the development of a theory’s models3 suggesting new epistemic notions of application. This is also echoed in Jeffrey Ramsey’s (1995) notions of “transformation reduction,”—i.e., a notion of reduction of a more highly constructive variety.4 Computer simulations create a broadly continuous arena spanned by normative and descriptive aspects of theory-articulation, as entailed by the notion of transformation reductions occupying a continuous region demarcated by Ernest Nagel’s (1974) logical-explanatory “domain-combining reduction ” on the one hand, and Thomas Nickels ’ (1973) heuristic “domain-preserving reduction, ” on the other. I extend Winsberg’s and Ramsey’s points here, by arguing that in the field of computational fluid dynamics (CFD) as well as in other branches of applied physics, the computer plays a constitutively experimental role—supplanting in many cases the more traditional experimental methods such as flow-visualization, etc. In this case, however CFD algorithms act as substitutes, not supplements (as the notions “simulation ” suggests) when it comes to experimental practices. I bring up the constructive example involving the Clifford-Algebraic algorithms for modeling singular phenomena (i.e., vortex

### Reflections on Mathematics

, 2007

"... Though the philosophy of mathematics encompasses many kinds of questions, my response to the five questions primarily focuses on the prospects of developing a unified approach to the metaphysical and epistemological issues concerning mathematics. My answers will be framed from within a single concep ..."

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Though the philosophy of mathematics encompasses many kinds of questions, my response to the five questions primarily focuses on the prospects of developing a unified approach to the metaphysical and epistemological issues concerning mathematics. My answers will be framed from within a single conceptual framework. By ‘conceptual framework’, I mean an explicit and formal listing of primitive notions and first principles, set within a well-understood background logic. In what follows, I shall assume the primitive notions and first principles of the (formalized and) axiomatized theory of abstract objects, which I shall sometimes refer to as ‘object theory’. 1 These notions and principles are mathematics-free, consisting only of metaphysical and logical primitives. The first principles assert the existence, and comprehend a domain, of abstract objects, and in this domain we can identify (either by definition or by other means) logical objects, natural mathematical objects, and theoretical mathematical objects. These formal principles and identifications will help us to articulate answers not only to the five questions explicitly before us, but also to some of the other fundamental questions in the philosophy of mathematics raised below.