### The Use of Symbols in Mathematics and Logic

"... Abstract. It is commonly believed that the use of arbitrary symbols and the process of symbolisation have made possible the discourse of modern mathematics as well as modern, symbolic logic. This paper discusses the role of symbols in logic and mathematics, and in particular analyses whether symbols ..."

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Abstract. It is commonly believed that the use of arbitrary symbols and the process of symbolisation have made possible the discourse of modern mathematics as well as modern, symbolic logic. This paper discusses the role of symbols in logic and mathematics, and in particular analyses whether symbols remain arbitrary in the process of symbolisation. It begins with a brief summary of the relation between sign and logic as exemplified in Indian logic in order to illustrate a logical system where the notion of ‘natural ’ sign-signified relation is privileged. Mathematics uses symbols in creative ways. Two such methods, one dealing with the process of ‘alphabetisation ’ and the other based on the notion of ‘formal similarity’, are described. Through these processes, originally meaningless symbols get embodied and coded with meaning through mathematical writing and praxis. It is also argued that mathematics and logic differ in the way they use symbols. As a consequence, logicism becomes untenable even at the discursive level, in the ways in which symbols are created, used and gather meaning. The role of symbols in the formation of the disciplines of logic (particularly modern and symbolic logic) and mathematics is often acknowledged to be of fundamental importance. However, symbols have become so essential that their function in these disciplines is rarely queried. In the epoch of any discipline it is always worthwhile to periodically reconsider the foundational elements. It is in this spirit that I approach the reconsideration of the role of symbols in logic and mathematics. Signs, in the most fundamental sense of the word, can refer to anything which stands for something else (the signified). Thus, a word is a sign; for example, the word ‘cow ’ stands for the object cow. There are many ways by which a sign can come to stand for something else. There could be a natural relation which immediately suggests the relation between a sign and the signified. Or, the

### Conceptions of the Continuum

"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."

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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical

### 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

### 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

### 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

### 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

### 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

### Measurement in Physics and Economics Project Discussion Paper Series, DP MEAS 25/02, London School of Economics 2002. Models and Representation: Why Structures Are Not Enough.

"... Models occupy a central role in the scientific endeavour. Among the many purposes they serve, representation is of great importance. Many models are representations of something else; they stand for, depict, or imitate a selected part of the external world (often referred to as target system, parent ..."

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Models occupy a central role in the scientific endeavour. Among the many purposes they serve, representation is of great importance. Many models are representations of something else; they stand for, depict, or imitate a selected part of the external world (often referred to as target system, parent system, original, or prototype). Well-known examples include the model of the

### Philosophy of Mathematics: Making a Fresh Start

"... ABSTRACT: The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment ..."

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ABSTRACT: The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the effectiveness of mathematics in natural science.