In light of the observations and research design presented in the previous reports, the current technical report is focused on the relationship between the quality and specificity of the content of the gambling sites and the site’s SEO and marketing policy. This relationship is dependent upon the category of the gambling site and the difference in content quality, and the degree to which the mathematical dimension of gambling is reflected in this content is explained by this dependence.

The current technical report of the research project investigating how the mathematical dimension of gambling is reflected in the communication and texts associated with the gambling industry raises the problem of the adequacy of sampling and proposes a new approach in this respect. The qualitative analysis of the reviewed websites is extended to a deeper analysis of language and also to the organization and structure of websites’ content. Although not stated as a goal of the initial project, the research (...) will also try to identify relationships between the specificity of content relative to the mathematical dimension of gambling, and the management and marketing of the website. An overview of the results of the statistical analysis of the current sample is also presented. (shrink)

The current technical report presents the partial results of the quantitative analysis of the research project, after the review of 247 gambling websites. It is focused on and discusses the usage of the math terms specific to gambling in the reviewed sample. In particular, the fifth technical report discusses the usage of math terms associated with the game of slots, as found in the reviewed sample.

The present article seeks to provide an overview of the general characteristics of the cultural and scientific climate in the Duchy of Urbino. Three of the Duchy’s milieus seem to have been particularly important for scholars who were engaged in the study of mathematics: the so-called “School of Urbino”, the environment of the court, and the world of the technicians and engineers. While the Urbino School has already been the object of previous studies, the other two milieus and their (...) effect on the mathematicians’ work have been rather neglected and are, consequently, addressed here. The paper’s final section presents some documents that attest the importance of Renaissance scholars’ interaction with these environments to their actual scientific activity, taking the case of Guidobaldo dal Monte as an example. (shrink)

The current study evaluates qualitatively how the mathematical dimension of gambling is reflected in the content of gambling websites. A number of gambling websites have been reviewed for their content in that respect. A statistical analysis recorded the presence of the mathematical dimension of gambling and its forms in the content of the participating websites, and a qualitative research study analyzed and assessed the quality of the content with respect to that dimension. The technical reports associated with this study (...) describe the methods used in the processes of review and content analysis, details of the processes, and the specific challenges and difficulties that they posed. (shrink)

Employing Searle’s views, I begin by arguing that students of Mathematics behave similarly to machines that manage symbols using a set of rules. I then consider two types of Mathematics, which I call Cognitive Mathematics and Technical Mathematics respectively. The former type relates to concepts and meanings, logic and sense, whilst the latter relates to algorithms, heuristics, rules and application of various techniques. I claim that an upgrade in the school teaching of Cognitive Mathematics (...) is necessary. The aim is to change the current mentality of the stakeholders so as to compensate for the undue value presently attached to Technical Mathematics, due to advances in technology and its applications, and thus render the two sides of Mathematics equal. Furthermore, I suggest a reorganization/systematization of School Mathematics into a cognitive network to facilitate students’ understanding of the subject. The final goal is the transition from mechanical execution of rules to better understanding and in-depth knowledge of Mathematics. (shrink)

This second technical report shows some partial results for the variables of the proposed statistical analysis and a discussion about some changes in sampling. In what concerns the qualitative analysis of content, the report presents the general predominant tendencies that get contoured with the first two samples.

Over the years, Skolem’s Paradox has generated a fairly steady stream of philosophical discussion; nonetheless, the overwhelming consensus among philosophers and logicians is that the paradox doesn’t constitute a mathematical problem (i.e., it doesn’t constitute a real contradiction). Further, there’s general agreement as to why the paradox doesn’t constitute a mathematical problem. By looking at the way firstorder structures interpret quantifiers—and, in particular, by looking at how this interpretation changes as we move from structure to structure—we can give a technically (...) adequate “solution” to Skolem’s Paradox. So, whatever the philosophical upshot of Skolem’s Paradox may be, the mathematical side of Skolem’s Paradox seems to be relatively straightforward. (shrink)

This is a technical supplement to "Abstraction and Grounding", forthcoming in /Philosophy and Public Affairs/.

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. (...) Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured (...) previous discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

The onset of the electronics-based information revolution will augur changes in the sociological perceptions of 'suitable careers' for women. This phenomenon is particularly evident in rural Appalachia. A planned, systematic delivery system was designed, developed, and implemented by Southwest Virginia Community College to introduce women to the challenges and possibilities of technical careers. This was accomplished through a gradualized phase-in to Technological Literacy, followed by in-depth involvement, culminating in an industrial internship experience. A special curriculum was designed to ease (...) the transition of the women, most of whom were homemakers, into the fast-paced mainstream of the academic environment. Duration of the introductory segment was one academic quarter. All project participants were clustered in a peer group, led by the project director, a trained counselor. In addition to a central course on Career Directions for Women, the program design emphasized Industrial Dynamics, the variety of uses possible with "user-friendly" microcomputers, Computer-Assisted Drafting, Electronics Instrumentation, Mathematics and Communications. Field experiences and visits to area industries were integrated into the program plan, as were visits from women who had achieved success in technical careers. Particular efforts were made to arrange for visits by the limited number of women alumni from the college's technical programs. Students were then mainstreamed into a regular program in Electronics or Computer Aided Drafting in the intermediate segment. Students are given a closer look at the industrial sector in the final segment. An examination of the social structure in Appalachian Virginia is presented, together with an albeit tentative prediction of the implications for female wage earners in this highly traditional region, so dependent heretofore on coal mining. Finally, the concept of whether this effort will lead to evolution, rather than revolution, is discussed. While covering a somewhat homogeneous and unique population cluster, the action blueprint outlined can be applied to other population groups in the U.S. and abroad. (shrink)

In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of argumentative products, but little attention has been paid to the creative process of arguing. Mathematics can be used as a clear example to illustrate some significant theoretical differences between mathematical practice and the products of it, to differentiate the distinct components of the arguments, and to emphasize the need to address the different types of argumentative discourse and argumentative situation in the practice. (...) I consider some issues of recent papers associated with mathematical argumentation in an attempt to contribute to the discussion about the role of arguing in mathematical practice and in the evaluation of the products of this practice. I apply this discussion to learning environments to defend the thesis that argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity. (shrink)

Technical chronology establishes the structure of calendars and the dates of events; it is, as it were, the foundation of history, particularly ancient history. The chronologer must know enough philology to interpret texts and enough astronomy to compute the dates of celestial phenomena, above all eclipses, which alone provide absolute dates. Joseph Scaliger, so we are told, was the first to master and apply this range of technical skills: Of the mathematical principles on which the calculation of periods (...) rests, the philologians understood nothing. The astronomers, on their side, had not yet undertaken to apply their data to the records of ancient times. Scaliger was the first of the philologians who made use of the improved astronomy of the sixteenth century to get a scientific basis for historical chronology. So Mark Pattison. This verdict can be challenged on a number of grounds. The one relevant at present is simple: Scaliger himself claimed far less. He certainly said that technical chronology had been untouched in modern times — not an entirely fair judgement — but in antiquity it had been practised in exactly the manner he considered proper, or so he believed. In particular he singles out Censorinus, whose De die natali drew extensively on Varro's lost Antiquitates rerum humanarum, books 14–19, for information on chronology. Students of Varro have long appreciated the importance of Censorinus. His dry and compact treatise offers Varronian views on etymology, the human life-span, and the course of history itself, all couched in language so jejune as to suggest that he added little or nothing to what he read. (shrink)

Technical chronology establishes the structure of calendars and the dates of events; it is, as it were, the foundation of history, particularly ancient history. The chronologer must know enough philology to interpret texts and enough astronomy to compute the dates of celestial phenomena, above all eclipses, which alone provide absolute dates. Joseph Scaliger, so we are told, was the first to master and apply this range of technical skills: Of the mathematical principles on which the calculation of periods (...) rests, the philologians understood nothing. The astronomers, on their side, had not yet undertaken to apply their data to the records of ancient times. Scaliger was the first of the philologians who made use of the improved astronomy of the sixteenth century to get a scientific basis for historical chronology. So Mark Pattison. This verdict can be challenged on a number of grounds. The one relevant at present is simple: Scaliger himself claimed far less. He certainly said that technical chronology had been untouched in modern times — not an entirely fair judgement — but in antiquity it had been practised in exactly the manner he considered proper, or so he believed. In particular he singles out Censorinus, whose De die natali drew extensively on Varro's lost Antiquitates rerum humanarum, books 14–19, for information on chronology. Students of Varro have long appreciated the importance of Censorinus. His dry and compact treatise offers Varronian views on etymology, the human life-span, and the course of history itself, all couched in language so jejune as to suggest that he added little or nothing to what he read. (shrink)

This fundamental and straightforward text addresses a weakness observed among present-day students, namely a lack of familiarity with formal proof. Beginning with the idea of mathematical proof and the need for it, associated technical and logical skills are developed with care and then brought to bear on the core material of analysis in such a lucid presentation that the development reads naturally and in a straightforward progression. Retaining the core text, the second edition has additional worked examples which users (...) have indicated a need for, in addition to more emphasis on how analysis can be used to tell the accuracy of the approximations to the quantities of interest which arise in analytical limits. (shrink)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, (...) and there are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

Ekphrasis is familiar as a rhetorical tool for inducing enargeia, the vivid sense that a reader or listener is actually in the presence of the objects described. This book focuses on the ekphrastic techniques used in ancient Greek and Roman literature to describe technological artifacts. Since the literary discourse on technology extended beyond technical texts, this book explores 'technical ekphrasis' in a wide range of genres, including history, poetry, and philosophy as well as mechanical, scientific, and mathematical works. (...) Technical authors like Philo of Byzantium, Vitruvius, Hero of Alexandria, and Claudius Ptolemy are put into dialogue with close contemporaries in other genres, like Diodorus Siculus, Cicero, Ovid, and Aelius Theon. The treatment of 'technical ekphrasis' here covers the techniques of description, the interaction of verbal and visual elements, the role of instructions, and the balance between describing the artifact's material qualities and the other bodies of knowledge it evokes. (shrink)

Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary (...) philosophy, and mathematically trained philosophers continue to contribute to the literature in logic. For instance, modal logics were first investigated by philosophers and now have important applications in computer science and mathematical linguistics. (shrink)

Introduces the technical tools and concepts employed in advanced work in philosophy. Beginning with the fundamentals of set theory, the author examines relations, functions and the theory of arithmetic before using these tools to clarify the metatheory of the predicate calculus.

A Non-Technical Introduction to Ethnomethodological Investigations of the Foundations of Mathematics through the Use of a Theorem of Euclidean Geometry* I ...

Mathematics is a highly specialised arena of human endeavour, one in which complex notations are invented and are subjected to complex and involved manipulations in the course of everyday work. What part do these writing practices play in mathematical communication, and how can we understand their use in the mathematical world in relation to theories of communication and cognition? To answer this, I examine in detail an excerpt from a research meeting in which communicative board-writing practices can be observed, (...) and attempt to explain the observed exchanges with reference to relevance theory (a cognitivist theory of pragmatics), in combination with the situated cognition paradigm. Successful communication in the excerpt appears well described by the notion of metacognitive acquaintance, as described in Sperber & Wilson (2015), especially when a situated view of cognition is adopted. Though this example is taken from a relatively informal face-to-face communicative situation, characteristics of such situations extend even into formal written mathematics, since even the technical terminology developed in the course of the meeting includes details that may provide readers with a kind of access to the mathematicians’ cognitive processes. (shrink)

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors (...) and many others. The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

Taken at face value, a programming language is defined by a formal grammar. But, clearly, there is more to it. By themselves, the naked strings of the language do not determine when a program is correct relative to some specification. For this, the constructs of the language must be given some semantic content. Moreover, to be employed to generate physical computations, a programming language must have a physical implementation. How are we to conceptualize this complex package? Ontologically, what kind of (...) thing is it? In this paper, we shall argue that an appropriate conceptualization is furnished by the notion of a technical artifact. (shrink)

„Zeit“ ist seit einigen Jahren ein intensiv debattiertes Thema in der Geschichtswissenschaft. Auch in der Technikgeschichte finden zunehmend Überlegungen dazu statt. In den historischen Forschungen zu Infrastrukturen spielt der Aspekt allerdings noch eine geringe Rolle. In diesem Aufsatz möchte ich die jüngsten Ansätze aufgreifen und das Verhältnis netzgebundener Infrastrukturen zur Zeit als ein doppelseitiges Produktionsverhältnis darstellen: In Infrastrukturen lagern sich unterschiedliche Epochen mit ihren zeitlichen Kontexten als Zeitschichten ab. Dies schlägt sich nicht nur in technischen Komponenten unterschiedlichen Alters nieder, sondern (...) auch in Organisationsstrukturen oder Handlungsroutinen. Zugleich produzieren Infrastrukturen aber auch Zeit, indem sie Art und Geschwindigkeit beeinflussen, mit der Stoffe, Personen, Informationen oder Güter durch sie zirkulieren. In diesem Aufsatz schlage ich die Nutzung von folgenden Metaphern für die Analyse dieser Zusammenhänge vor: „Zeitschicht“ und „Palimpsest“ für die in den Systemen materialisierten Folgen ihrer Geschichtlichkeit, „Einschreibung“ und „Überschreibung“ für die zugrundeliegenden Prozesse. Die Produktion von Zeitlichkeit durch die so zusammengesetzten Infrastrukturen soll in Form von „Rhythmen“ beschrieben werden, welche die Zirkulation strukturieren. In der Summe ergibt sich ein komplexes Bild: Die unterschiedlichen Zeitschichten einer Infrastruktur tragen zur Ausbildung von Rhythmen bei. (shrink)

It is proposed that the physical structure of an observer in quantum mechanics is constituted by a pattern of elementary localized switching events. A key preliminary step in giving mathematical expression to this proposal is the introduction of an equivalence relation on sequences of spacetime sets which relates a sequence to any other sequence to which it can be deformed without change of causal arrangement. This allows an individual observer to be associated with a finite structure. The identification of suitable (...) switching events in the human brain is discussed. A definition is given for the sets of sequences of quantum states which such an observer could occupy. Finally, by providing an a priori probability for such sets, the definitions are incorporated into a complete mathematical framework for a many-worlds interpretation. At a less ambitious level, the paper can be read as an exploration of some of the technical and conceptual difficulties involved in constructing such a framework. (shrink)

This book is intended to fill a gap between popular 'wonders of mathematics' books and the technical writings of the philosophers of mathematics.

This essay discusses ways in which contemporary mathematical logic may be reconciled with John Dewey’s logical theory. Standard formal techniques drawn from dynamic modal logic, situation theory, generative grammar, generalized quantifier theory, category theory, lambda calculi, game theoretic semantics, network exchange theory, etc., are accommodated within a framework consistent with Dewey’s Logic: The Theory of Inquiry (1938). This essay outlines some basic features of Dewey’s logical theory, working in a top-down fashion through various technical notions pertaining to existential and (...) ideational aspects of inquiry, from contextual notions of situations and ideologies, to primitive operational and empirical notions of modes of action and particular qualities. The aim here is to display the scope of Dewey’s logical theory and thus establish a framework relative to which details may be explored without losing sight of their larger significance. (shrink)

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without (...) appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories. (shrink)

In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse (...) Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ₁. (shrink)

The generative and governing "idea" of radical modernity is spawned by the technique of mathematical construction deployed and interpreted by the major early-modern thinkers and their legatees. ;Chapter I is a survey of this legacy as it appears in Vico, Kant, Fichte, Marx and Nietzsche and in the post-Nietzschean inheritance of contemporary philosophy, hyperbolic in the case of Derrida et al., elliptical, in the case of Carnap and Goodman. ;In Chapter II I try to show how the pre-modern mathematical tradition, (...) represented by Euclid, aimed at keeping the enticements of technical facility in check by means of didactic phronesis and how the post-Kantian interpretation of "existence" in Euclid as constructibility betrays his usage and self-understanding. I suggest that his focus in the postulates and elsewhere is on the undistorted iterability of graphic evocations of the items already intelligible thanks to the definitions or to the pre-understanding shared by the teacher and student. ;In Chapter III, devoted to Descartes the principal claims of modern constructivism are brought to sight. After examining Descartes' fabulous autobiography and its emphasis on self-origination, I turn to the style, contents and under-pinnings of the Geometry in an effort to extract from that text what he once referred to as "the metaphysics of geometry." The latter yields the conditions of successful problem-solving, i.e., dimensional homogeneity and kinematic continuity. These conditions, in turn, find their justification in Descartes' theses in the Rules concerning order, measure and the uniformity of "mental" activity. In the final section I apply the lessons learned from the Geometry and the Rules to one critical issue in the later Meditations, the transition from essence to existence. Descartes' "solution" generates a sequence of perplexities with Hobbes, Leibniz, Kant and other radical moderns continue to wrestle. (shrink)

The "mathematization" of knowledge is a historically inevitable process governed by two circumstances. In the first place there is the need for the further extension of knowledge in all areas of human activity, whether it be the study of natural phenomena or the theory of taking decisions in the economic or social sphere. Marx pointed out long ago that a science reaches its highest levels only when it succeeds in making use of mathematics. The second circumstance rendering the process (...) of deepening and broadening the mathematization of knowledge not only necessary but possible lies in the constant development both of mathematics itself and of the technical means broadening the sphere of its application. (shrink)

To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On (...) the philosophical side, it is argued that any mentalist-based radical constructivism suffers from a kind of neo-Kantian apriorism. It would be at best a lucky accident if objective spacetime structure mirrored mentalist mathematics. the latter would seem implicitly committed to a Leibnizian relationist view of spacetime, but is it doubtful if implementation of such a view would overcome the objection. As a result, an anti-realist view of physics seems forced on the radical constructivist. (shrink)

This book provides a systematic exposition of mathematical economics, presenting and surveying existing theories and showing ways in which they can be extended. One of its strongest features is that it emphasises the unifying structure of economic theory in such a way as to provide the reader with the technical tools and methodological approaches necessary for undertaking original research. The author offers explanations and discussion at an accessible and intuitive level providing illustrative examples. He begins the work at an (...) elementary level and progessively takes the reader to the frontier of current research. This second edition brings the reader fully up to date with recent research in the field. (shrink)

The IOth International Congress of Logic, Methodology and Philosophy of Science, which took place in Florence in August 1995, offered a vivid and comprehensive picture of the present state of research in all directions of Logic and Philosophy of Science. The final program counted 51 invited lectures and around 700 contributed papers, distributed in 15 sections. Following the tradition of previous LMPS-meetings, some authors, whose papers aroused particular interest, were invited to submit their works for publication in a collection of (...) selected contributed papers. Due to the large number of interesting contributions, it was decided to split the collection into two distinct volumes: one covering the areas of Logic, Foundations of Mathematics and Computer Science, the other focusing on the general Philosophy of Science and the Foundations of Physics. As a leading choice criterion for the present volume, we tried to combine papers containing relevant technical results in pure and applied logic with papers devoted to conceptual analyses, deeply rooted in advanced present-day research. After all, we believe this is part of the genuine spirit underlying the whole enterprise of LMPS studies. (shrink)

From classical times to the present, stories have been the constant companions of mathematical studies. Though seemingly simple in structure, these tales have both defined and expressed the nature of the mathematics, its relation to the world, and the roles of its practitioners. As popular tales, mathematical stories are shaped by the mores of their time and place, while at the same time they inform abstract and highly technical mathematical practices. Poised between the popular world of storytelling and (...) the rarefied air of advanced mathematics, these stories are a key to relating mathematical practices to their cultural and historical setting. The three essays in this section explore different ways in which stories can be used in historicizing mathematical practices. Focused on the century between 1730 and 1830, they combine to tell the story of Enlightenment mathematics and its transformation in the nineteenth century. (shrink)

Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his (...) own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development. (shrink)

We count apples and divide a cake so that each guest gets an equal piece; we weigh galaxies and use Hilbert spaces to make amazingly accurate predictions about spectral lines. It would seem that we have no difficulty in applying mathematics to the world; yet the role of mathematics in its various applications is surprisingly elusive. Eugene Wigner has gone so far as to say that “the enormous usefulness of mathematics in the natural sciences is something bordering (...) on the mysterious and that there is no rational explanation for it” (1960, p. 223). The issue is not much discussed under the heading “applied mathematics,” yet it is pivotal to several philosophical debates. In recent years three rather general questions have been central: (1) Just how does mathematics “hook onto” the world? This is the main concern of a rather technical branch of philosophy of science known as measurement theory (see measurement). (2) Are some of the objects referred to in various theories merely mathematical objects, or do they have some other status? This problem often comes up in the philosophy of the special sciences. For example, do space‐time and the quantum state exist in their own right, separate from their mathematical representations; or are they nothing but mathematical entities? (3) Is mathematics essential for science? Following Hartry Field's work, this has become a focal point in the debate between realists and nominalists in the philosophy of mathematics. (shrink)

In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim (...) which the argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments, uncovering flaws in each one. In the second half of the paper, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND Argument, contrary to the claims of its supporters. (shrink)

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)