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How applied mathematics became pure
 Review of Symbolic Logic
"... Abstract. This paper traces the evolution of thinking on how mathematics relates to the world— from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics ..."
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Abstract. This paper traces the evolution of thinking on how mathematics relates to the world— from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics in contemporary science. My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one
Similarity in Programs
"... Abstract. An overview of the concept of program similarity is presented. It divides similarity into two types—syntactic and semantic— and provides a review of eight categories of methods that may be used to measure program similarity. A summary of some applications of these methods is included. The ..."
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Abstract. An overview of the concept of program similarity is presented. It divides similarity into two types—syntactic and semantic— and provides a review of eight categories of methods that may be used to measure program similarity. A summary of some applications of these methods is included. The paper is intended to be a starting point for a more comprehensive analysis of the subject of similarity in programs, which is critical to understand if progress is to be made in fields such as clone detection.
Of numbers and electrons
 In Proceedings of the Aristotelian Society
, 2010
"... The sciences are full of theories which, in the course of making detailed claims about the physical world, say things which entail that there are mathematical entities like numbers and sets. According to an influential tradition stemming from Quine (1948) and Putnam (1972), good scientific reasoning ..."
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The sciences are full of theories which, in the course of making detailed claims about the physical world, say things which entail that there are mathematical entities like numbers and sets. According to an influential tradition stemming from Quine (1948) and Putnam (1972), good scientific reasoning—induction, broadly construed—requires us to believe some such theory, or some disjunction of such theories. And it is because of this that we ought to believe that there are mathematical entities. The belief that there are numbers is, according to this tradition, on a similar epistemological footing to the belief that there are electrons, viruses, quasars, etc. 1 Some will regard this analogy as unhelpful because they think that we can know that there are mathematical entities in the same way—whatever it is—that we know that all dogs are dogs, or that all bachelors are unmarried. 2 Others may regard this analogy as unhelpful because they think that we can directly perceive that there are mathematical entities—such as sets of
Once Upon a Spacetime
, 2005
"... This dissertation concerns the nature of spacetime. It is divided into two parts. The first part, which comprises chapters 1, 2, and 3, addresses ontological questions: does spacetime exist? And if so, are there any other spatiotemporal things? In chapter 1 I argue that spacetime does exist, and i ..."
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This dissertation concerns the nature of spacetime. It is divided into two parts. The first part, which comprises chapters 1, 2, and 3, addresses ontological questions: does spacetime exist? And if so, are there any other spatiotemporal things? In chapter 1 I argue that spacetime does exist, and in chapter 2 I respond to modal arguments against this view. In chapter 3 I examine and defend supersubstantivalism— the claim that all concrete physical objects (tables, chairs, electrons and quarks) are regions of spacetime. Fourdimensional spacetime, we are often told, ‘unifies ’ space and time; if we believe in spacetime, then we do not believe that space and time are separately existing things. But that does not mean that there is no distinction between space
Modest Evolutionary Naturalism
"... I begin by arguing that a consistent general naturalism must be understood in terms of methodological maxims rather than metaphysical doctrines. Some specific maxims are proposed. I then defend a generalized naturalism from the common objection that it is incapable of accounting for the normative as ..."
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I begin by arguing that a consistent general naturalism must be understood in terms of methodological maxims rather than metaphysical doctrines. Some specific maxims are proposed. I then defend a generalized naturalism from the common objection that it is incapable of accounting for the normative aspects of human life, including those of scientific practice itself. Evolutionary naturalism, however, is criticized as being incapable of providing a sufficient explanation of categorical moral norms. Turning to the epistemological norms of science itself, particularly those governing the empirical testing of specific models, I argue that these should be regarded as conditional rather than categorical and that, as such, can be given a naturalistic justification. The justification, however, is more cognitive than evolutionary. The historical development of science is found to be a better place for applying evolutionary ideas. After briefly considering the possibility of a naturalistic understanding of mathematics and logic, I turn to the problem of reconciling scientific realism with an evolutionary picture of scientific development. The solution, I suggest, is to understand scientific knowledge as being “perspectival ” rather than absolutely objective. I first argue that scientific observation, whether by humans or instruments, is perspectival. This argument is extended to scientific theorizing which is regarded not as the formulation of universal laws of nature but as the construction of principles to be used in the construction of models to be applied to specific natural systems. The application of models, however, is argued to be not merely opportunistic but constrained by the methodological presumption that we live in a world with a definite causal structure even though we can understand it only from various perspectives.
VISUALIZATION OF ORDINALS
, 2007
"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. ..."
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We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics.
Evolution without Naturalism
 STUDIES IN PHILOSOPHY OF RELIGION
"... Does evolutionary theory have implications about the existence of supernatural entities? This question concerns the logical relationships that hold between the theory of evolution and different bits of metaphysics. There is a distinct question that I also want to address; it is epistemological in ch ..."
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Does evolutionary theory have implications about the existence of supernatural entities? This question concerns the logical relationships that hold between the theory of evolution and different bits of metaphysics. There is a distinct question that I also want to address; it is epistemological in character. Does the evidence we have for evolutionary theory also provide evidence concerning the existence of supernatural entities? An affirmative answer to the logical question would entail an affirmative answer to the epistemological question if the principle in confirmation theory that Hempel (1965, p. 31) called the special consequence condition were true: The special consequence condition: If an observation report confirms a hypothesis H, then it also confirms every consequence of H. According to this principle, if evolutionary theory has metaphysical implications, then whatever confirms evolutionary theory also must confirm those metaphysical implications. But the special consequence is false. Here‟s a simple example that illustrates why. You are playing poker and would dearly like to know whether the card you are about to be dealt will be the Jack of Hearts. The dealer is a bit careless and so you catch a glimpse of the card on top of the deck before it is dealt to you. You see that it is red. The fact that it is red confirms the hypothesis that the card is the Jack of Hearts, and the hypothesis that it is the Jack of Hearts entails that the card will be a Jack. However, the fact that the card is red does not confirm the hypothesis that the card will be a Jack. 2 Bayesians gloss these facts by understanding confirmation in terms of probability raising: The Bayesian theory of confirmation: O confirms H if and only if Pr(H│O)> Pr(H). The general reason why Bayesianism is incompatible with the special consequence
The cognitive basis of arithmetic
"... Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities ..."
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Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities
The Mathematician as a Formalist
 in Truth in Mathematics (H.G. Dales and
, 1998
"... Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next mil ..."
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Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)