This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as (...) the explication of the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

The various papers and short "discussions" contained in this latest addition to the "Studies in Logic" series were presented at the 1965 International Colloquium in the Philosophy of Science, in London. Of the nine "problems" considered in this symposium, seven have directly to do with philosophy, one is an historical study of the origins of Euclid's axiomatics, and the last is an interesting—if one-sided—discussion of the "new math" controversy in the pre-college curriculum. Happily, this book demonstrates that the (...) important issues in the philosophy of mathematics somewhat outrun the usual ism-mongering debate between representatives of various schools. Unhappily, it also demonstrates that a good deal of undeserving material seems to find its way between the covers of scholarly books. After reading certain of the articles in this volume one is left with the feeling that the authors have not come to grips with the issues which they raise, that a lot of effort has been expended on peripheral matters, and that, in some instances, a sledge hammer has been brought down upon an egg. Abraham Robinson's article, "The Metaphysics of the Calculus," deals with a series of important linguistic and ontological problems concerning the foundations of analysis. Moreover, these are problems which can no longer be ignored, especially in view of the evolution of non-standard analysis. But, unfortunately, the interested reader would do far better to turn to Robinson's recent book on the subject, and pay special attention to the second, third, and last chapters therein. "On a Fregean Dogma" by Fred Sommers gives a rather curious non-quantificational algorithm for the monadic fragment of syllogistic. This Sommers does in the interest of "a conservative effort to restore categorical statements to predicative status." Despite the heat and the occasional interesting remarks which the discussion generated, one is inclined to ask, "So what?" The sequence of three papers by Mostowski, Bernays, and Körner is best considered as a group, since each of these is concerned with the philosophical significance of recent results in set theory and logic. Students of the philosophy of mathematics would do well to read this part of the book very carefully, although one wishes that the authors would elaborate a bit on their specific philosophical views. Finally, there is Georg Kreisel's difficult and amazing paper, "Informal Rigor and Completeness Proofs," in which is presented a mathematical treatment of the problem of selecting the primitive rules and concepts of mathematical systems. Kreisel's ideas are refreshing, unorthodox, dubious, and a genuine contribution to the philosophy of mathematics. In sum, this book is a mixed bag, uneven in the quality of its contents, but worth reading.—H. P. K. (shrink)

The predisposition of the Indispensability Argument to objections, rephrasing and versions associated with the various views in philosophy of mathematics grants it a special status of a “blueprint” type rather than a debatable theme in the philosophy of science. From this point of view, it follows that the Argument has more an epistemic character than ontological.

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important (...) part of mathematics, namely mathematics as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly (...) intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They (...) investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (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)

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of (...) mathematics are discussed more briefly in section 6. (shrink)

The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat (...) such objects in our discourse as though they are independent of time and space; finding a place for such objects in a broader framework of thought is a central task of ontology, or metaphysics. The rigor and precision of mathematical language depends on the fact that it is based on a limited vocabulary and very structured grammar, and semantic accounts of mathematical discourse often serve as a starting point for the philosophy of language. Although mathematical thought has exhibited a strong degree of stability through history, the practice has also evolved over time, and some developments have evoked controversy and debate; clarifying the basic goals of the practice and the methods that are appropriate to it is therefore an important foundational and methodological task, locating the philosophy of mathematics within the broader philosophy of science. (shrink)

The guiding idea behind formalism is that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess. This idea has some intuitive plausibility: consider the tyro toiling at multiplication tables or the student using a standard algorithm for differentiating or integrating a function. It also corresponds to some aspects of the practice (...) of advanced mathematicians in some periods—for example, the treatment of imaginary numbers for some time after Bombelli's introduction of them, and perhaps the attitude of some contemporary mathematicians towards the higher flights of set theory. Finally, it is often the position to which philosophically naïve respondents will gesture towards, when pestered by questions as to the nature of mathematics. (shrink)

Deductivism says that a mathematical sentence s should be understood as expressing the claim that s deductively follows from appropriate axioms. For instance, deductivists might construe “2+2=4” as “the sentence ‘2+2=4’ deductively follows from the axioms of arithmetic”. Deductivism promises a number of benefits. It captures the fairly common idea that mathematics is about “what can be deduced from the axioms”; it avoids an ontology of abstract mathematical objects; and it maintains that our access to mathematical truths requires nothing (...) beyond our ability to make logical deductions. Sections 1 and 2 define and motivate deductivism in more detail. Section 3 covers four authors (Russell, Hilbert, Pasch, Curry) who have endorsed deductivism at some point. Section 4 aims to clarify what semantic claim deductivists make. Sections 5–9 review objections to deductivism and possible replies. (shrink)

Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application constraint (...) within an abstractionist setting. (shrink)

Doctoral programs at U.S. universities play a critical role in the development of human resources both in the United States and abroad. This volume reports the results of an extensive study of U.S. research-doctorate programs in five broad fields: physical sciences and mathematics, engineering, social and behavioral sciences, biological sciences, and the humanities. Research-Doctorate Programs in the United States documents changes that have taken place in the size, structure, and quality of doctoral education since the widely used 1982 editions. (...) This update provides selected information on nearly 4,000 doctoral programs in 41 subdisciplines at 274 doctorate-granting institutions. This volume also reports the results of the National Survey of Graduate Faculty, which polled a sample of faculty for their views on the scholarly quality of program faculty and the effectiveness of doctoral programs in preparing research scholars/scientists. This much-anticipated update of such an essential reference will be useful to education administrators, university faculty, and students seeking authoritative information on doctoral programs. (shrink)

Since the turn of the century, however, a double upheaval has occurred, the formulation of the quantum theory and the theory of relativity, providing the ground for the development of modern physics. These theories issued from the problems of light that, in their strict forms, could not be assimilated by Newtonian physics. Before the turn of the century the wave theory had been victorious over the emission theory, and an hypothetical ether was assumed which was intended ultimately to represent absolute (...) space and to play a role similar to that of Anaximander's apeiron or Aristotle's prima materia. This last role seemed required since many philosophers of nature believed that they should see in matter and its elementary components the singularities, the exceptional peculiarities of the ether. However, since the ether held itself strictly incognito, Albert Einstein renounced it and declared that all systems in uniform, rectilinear motion are equivalent. The first portentous ontological consequence of this renunciation was that statements concerning spatial and temporal separation become relative. True relativity means, however, that objectivity and subjectivity enter equally into all statements, for to everything relative, as Max Planck once said, an absolute belongs. In this case the absolute was the combination of space and time into a four-dimensional spatio-temporal unity, which observers in different systems split into different space and time determinations. Our statements concerning space and time are thus neither merely subjective nor merely objective; rather a subject-object relation enters into them and makes them all relative. There is, to be sure, within the four-dimensional spatio-temporal unity a region of a spatial type within which the space-axes of the different observers lie, and the same is true for the temporal region. But even if there is no one universal time or simultaneity, temporality nevertheless plays its own role in contradistinction to spatiality, though both are expressible mathematically. We can say in the sense of critical realism that our spatial and temporal measurements are commensurate with the ordering relations of things and events. (shrink)

Suppose we are asked to draw up a list of things we take to exist. Certain items seem unproblematic choices, while others (such as God) are likely to spark controversy. The book sets the grand theological theme aside and asks a less dramatic question: should mathematical objects (numbers, sets, functions, etc.) be on this list? In philosophical jargon this is the ‘ontological’ question for mathematics; it asks whether we ought to include mathematicalia in our ontology. The goal of (...) this work is to answer this question in the affirmative, by drawing on considerations on the the applicability of mathematics to natural science. (shrink)

Julian Cole argues that mathematical domains are the products of social construction. This view has an initial appeal in that it seems to salvage much that is good about traditional platonistic realism without taking on the ontological baggage. However, it also has problems. After a brief sketch of social constructivist theories and Cole’s philosophy of mathematics, I evaluate the arguments in favor of social constructivism. I also discuss two substantial problems with the theory. I argue that unless (...) and until social constructivists can address the two concerns, we have reason to be skeptical about social constructivism in the philosophy of mathematics. (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 Brouwer, Weyl, Bernays and Hilbert, and 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)

The author presents a version of the ontological scheme of Newton’s mechanistic worldview based on both the study of previous versions of its understanding and the text of the “Mathematical Principles of Natural Philosophy”. Newton developed a model of new universality or a homogeneous and isotropic world in which uniform laws operate. This model is based on several ontological postulates Newton introduced, which can be isolated from several provisions of his classic work. The new mechanistic worldview is (...) based on the imputation of the world of universal simplicity. The quantitative “unit” of a simple, homogeneous, physical-geometric universe is an ambivalent corpuscle-point. The main constants of the “mechanistic universe” are the diversity of the numbers of masses, motions, and forces connected by clear reciprocal relationships. Newton also introduced theoretical space and time as a privileged, absolute reference system. Finally, in the Newtonian version of the mechanistic worldview, there are compelled metaphysical ingredients or inexplicable and transcendental qualities. They are gravity, ether, and God. Thus, the ontological scheme of Newton’s mechanistic worldview is a construction based on the sequential mental experiment of presenting the universe exclusively from the side of its “objectivity” and “sensory certification”. (shrink)

Mathematical explanation is a topic of great contemporary interest in the philosophy of mathematics. The question of whether mathematics can play an explanatory role in empirical science is thought by many to be the key to making progress on the realism versus anti-realism debate in the philosophy of mathematics. Questions about explanation within mathematics are also interesting and are important for the development of a general account of explanation. In a series of groundbreaking (...) papers from 1978 to 1983, Mark Steiner set the agenda for much of the modern debate about mathematical explanation. In the present paper we look at Steiner’s role in advancing mathematical explanation as a central topic in the philosophy of mathematics and his legacy for the modern debates about mathematical explanation. (shrink)

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended (...) further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics. (shrink)

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)

This text is organized around the distinction between finite and infinite. It includes a brief overview of what different philosophers have said about infinity, and looks at some of the arguments to the effect that one should adopt a pro-infinity attitude. Other chapters contain an exposition of the ontological schools; interactions among these schools and various theories of truth; the relationship between mathematics and values; a history of mathematics; an analysis of mathematical knowledge; the role of (...) class='Hi'>mathematics in eduction; the implications of religion for the philosophy of mathematics; and referenc eto mathematical objects. (shrink)

This book contains more than 15 essays that explore issues in truth, existence, and explanation. It features cutting-edge research in the philosophy of mathematics and logic. Renowned philosophers, mathematicians, and younger scholars provide an insightful contribution to the lively debate in this interdisciplinary field of inquiry. The essays look at realism vs. anti-realism as well as inflationary vs. deflationary theories of truth. The contributors also consider mathematical fictionalism, structuralism, the nature and role of axioms, constructive existence, and (...) generality. In addition, coverage also looks at the explanatory role of mathematics and the philosophical relevance of mathematical explanation. The book will appeal to a broad mathematical and philosophical audience. It contains work from FilMat, the Italian Network for the Philosophy of Mathematics. These papers collected here were also presented at their second international conference, held at the University of Chieti-Pescara, May 2016. (shrink)

There are two main camps in 20th-century philosophy of time: A-theorists who believe in the dynamic model of reality, and B-theorists who maintain a static model of reality. After the publication of Putnam’s influential article, “time and physical geometry”, the implications of the Special Theory of Relativity became serious in metaphysical discussions about temporal reality. Some philosophers argued that this theory contradicts the dynamic model and implies the ontology of the static model, namely, the objective reality of the present, (...) past and future events. In response, some advocates of the dynamic model argued that there can be dynamic models in relativistic setting, though they are different, radically, from old theories of dynamic reality. In this paper, first, I shall discuss how special relativity contradicts the traditional dynamic model and, then, how new dynamic models imply a relative concept of reality, in contrast to its old absolute concept and, so, why these strategies are very unintuitive. Finally, I shall argue that this radical conceptual shift about reality is, not only unintuitive, but unwarranted and, in some sense, inconsistent as well. (shrink)

Many long-standing problems pertaining to contemporary philosophy of mathematics can be traced back to different approaches in determining the nature of mathematical entities which have been dominated by the debate between realists and nominalists. Through this discussion conceptualism is represented as a middle solution. However, it seems that until the 20th century there was no third position that would not necessitate any reliance on one of the two points of view. Fictionalism, on the other hand, observes mathematical (...) entities in a radically different way. This is reflected in the claim that the concepts being used in mathematics are nothing but a product of human fiction. This paper discusses the relationship between fictionalism and two traditional viewpoints within the discussion which attempts to successfully determine the ontological status of universals. One of the main points, demonstrated with concrete examples, is that fictionalism cannot be classified as a nominalist position. Since fictionalism is observed as an independent viewpoint, it is necessary to examine its range as well as the sustainability of the implications of opinions stated by their advocates. (shrink)

The paper outlines a project in the philosophy of mathematics based on a proposed view of the nature of mathematical reasoning. It also contains a brief evaluative overview of the discipline and some historical observations; here it points out and illustrates the division between the philosophical dimension, where questions of realism and the status of mathematics are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to do with ways of organizing the mathematical (...) material. The paper’s concern is with the first. The grand tradition in the philosophy of mathematics goes back to the foundational debates at the end of the 19th and the first decades of the 20th century. Logicism went together with a realistic view of actual infinities; rejection of, or skepticism about actual infinities derived from conceptions that were Kantian in spirit. Yet questions about the nature of mathematical reasoning should be distinguished from questions about realism (the extent of objective knowledge– independent mathematical truth). Logicism is now dead. Recent attempts to revive it are based on a redefinition of “logic”, which exploits the flexibility of the concept; they yield no interesting insight into the nature of mathematics. A conception of mathematical reasoning, broadly speaking along Kantian lines, need not imply anti–realism and can be pursued and investigated, leaving questions of realism open. Using some concrete examples of non–formal mathematical proofs, the paper proposes that mathematics is the study of forms of organization—-a concept that should be taken as primitive, rather than interpreted in terms of set–theoretic structures. For set theory itself is a study of a particular form of organization, albeit one that provides a modeling for the other known mathematical systems. In a nutshell: “We come to know mathematical truths through becoming aware of the properties of some of the organizational forms that underlie our world. This is possible, due to a capacity we have: to reflect on some of our own practices and the ways of organizing our world, and to realize what they imply.. (shrink)

In the philosophy of mathematics, as in its a meta-domain, we find that the words as: consequentialism, implicativity, operationalism, creativism, fertility, … grasp at most of mathematical essence and that the questions of truthfulness, of common sense, or of possible models for (otherwise abstract) mathematical creations,i.e. of ontological status of mathematical entities etc. - of second order. Truthfulness of (necessary) succession of consequences from causes in the science of nature is violated yet with Hume, so that some (...) traditional footings of logico-mathematical conclusions should equally be falled under suspicion in the last century. We have in mind, say, strict-material implication which led the emergence of relevance logics, or the law of excluded middle that denied intuitionists i.e. paraconsistent logical systems where the contradiction is allowed, as well as the quantum logic which doesn't know, say, the definition of implication etc. Kant's beliefs miscarried hereafter that number (arithmetic) and form (geometry) would bring a (finite) truth on space and time, when they revealed relative and curvated, just as it is contradictory essentially understanding of basic phenomena in the nature: of light as an unity of wave – particle, or that both "exist" and "don't exist" numbers as powers of sets between 0א and c (the independence of continuum hypothesis) etc. Mathematical truths are ''truths of possible worlds'', in which we have only to believe that they will meet once recognizable models in reality. At last, we argue in favour of thesis that a possible representing "in relief" of mathematical entities and relations in the "noetic matter" (Aristotle) would be of a striking heuristic character for this science. (shrink)

The dissertation studies the mathematical strength of strict constructivism, a finitistic fragment of Bishop's constructivism, and explores its implications in the philosophy of mathematics. ;It consists of two chapters and four appendixes. Chapter 1 presents strict constructivism, shows that it is within the spirit of finitism, and explains how to represent sets, functions and elementary calculus in strict constructivism. Appendix A proves that the essentials of Bishop and Bridges' book Constructive Analysis can be developed within strict constructivism. Appendix (...) B further develops, within strict constructivism, the essentials of the functional analysis applied in quantum mechanics, including the spectral theorem, Stone's theorem, and the self-adjointness of some common quantum mechanical operators. Some comparisons with other related work, in particular, a comparison with S. Simpson's partial realization of Hilbert's program, and a discussion of the relevance of M. B. Pour-El and J. I. Richards' negative results in recursive analysis are given in Appendix C. ;Chapter 2 explores the possible philosophical implications of these technical results. It first suggests a fictionalistic account for the ontology of pure mathematics. This leaves a puzzle about how truths about fictional mathematical entities are applicable to science. The chapter then explains that for those applications of mathematics that can be reduced to applications of strict constructivism, fictional entities can be eliminated in the applications and the puzzle of applicability can be resolved. Therefore, if strict constructivism were essentially sufficient for all scientific applications, the applicability of mathematics of mathematics in science would be accountable. The chapter then argues that the reduction of mathematics to strict constructivism also reduces the epistemological question about mathematics to that about elementary arithmetic. The dissertation ends with a suggestion that a proper epistemological basis for arithmetic is perhaps a mixture of Mill's empiricism and the Kantian views. (shrink)

The volume contains twenty essays devoted to the philosophy of mathematics and the history of logic. They have been divided into four parts: general philosophical problems of mathematics, Hilbert's program vs. the incompleteness phenomenon, philosophy of mathematics in Poland, mathematical logic in Poland. Among considered problems are: epistemology of mathematics, the meaning of the axiomatic method, existence of mathematical objects, distinction between proof and truth, undefinability of truth, Goedel's theorems and computer science, philosophy (...) of mathematics in Polish mathematical and logical schools, beginnings of mathematical logic in Poland, contribution of Polish logicians to recursion theory. (shrink)

I discuss here the pragmatic problem in the philosophy of mathematics, that is, the applicability of mathematics, particularly in empirical science, in its many variants. My point of depart is that all sciences are formal, descriptions of formal-structural properties instantiated in their domain of interest regardless of their material specificity. It is, then, possible and methodologically justified as far as science is concerned to substitute scientific domains proper by whatever domains —mathematical domains in particular— whose formal structures (...) bear relevant formal similarities with them. I also discuss the consequences to the ontology of mathematics and empirical science of this structuralist approach. (shrink)

A way to argue that something plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in (...) this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it. (shrink)

This very short introduction will first outline how ontological investigations and questions of practice go together. The second section will bring in the next pole of this entire book, history of mathematics. How do ontology, practice, and history go together? Is this a forced marriage or one born in true love? That is, do these three belong together in some very basic way? One chapter in the section argues that the philosophy of mathematical practice intersects with the (...) history of mathematics essentially. If the first section of this introduction is persuasive that will also show that the history of mathematics intersects with ontological questions as well, specifically, that historical questions are very much about the ontological commitments of past mathematicians. The last section will take up what might seem a very different theme, namely, how these things matter to mathematics education. This section is connected closely to the second section since mathematics education has taken an interest in the role of history of mathematics in mathematics education, and much insight can be gained by seeing the issues in relation to the ontology-practice-history triad. The question of education is also important since teachers must consider what it is exactly that they are teaching, or, to adapt the title of Reuben Hersh’s well-known book, “What is mathematics really about?” And the “aboutness” of mathematics is really the ontological question. (shrink)

Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to (...) the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical factions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the opposite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. In his textbook on the philosophy of mathematics (Shapiro, 2000), Stewart Shapiro characterizes this attitude as ‘philosophy last, if at all.’. (shrink)

In this contribution I reply to Heather Wiltse and Róisín Lally’s commentaries. Both stress the importance of not only looking at gaps, but also accounting for relation and connection—an endeavor which was less visible in my original piece but which I wholly support, and on which I elaborate more here, making use of the example of ‘algorithmic mediation.’ In the process, I attempt to clear up some possible misunderstandings with regard to my initial formulations, as concerns object-oriented ontology’s stance on (...) relation, the question of whether postphenomenology already has a ‘transcendental’ account, the variety of possible connotations associated with the term ‘transcendental,’ and my proposals for looking specifically at what happens in between ‘small’ (‘technology with a small “t”’) and ‘big’ (‘Technology with a capital “T”’) perspectives. It turns out that my commentators and I are pretty much on a similar track—though we may differ in the first instance as to the conceptual instruments we would choose to deploy in that, nonetheless shared, project. (shrink)

Utilizing Einstein’s comparison of General Relativity and Descartes’ physics, this investigation explores the alleged conventionalism that pervades the ontology of substantival and relationist conceptions of spacetime. Although previously discussed, namely by Rynasiewicz and Hoefer, it will be argued that the close similarities between General Relativity and Cartesian physics have not been adequately treated in the literature—and that the disclosure of these similarities bolsters the case for a conventionalist interpretation of spacetime ontology.

In the last two centuries, our world would have been a safer place if philosophers such as Rousseau, Marx, and Nietzsche had not given intellectual encouragement to the radical ideologies of Jacobins, Stalinists, and fascists. Maybe the world would have been better off, from the standpoint of sound practice, if philosophers had engaged in only modest, decent theory, as did John Stuart Mill. Yet, as Ronald Beiner contends, the point of theory is not to think safe thoughts; the point is (...) to open intellectual horizons. In Philosophy in a Time of Lost Spirit, Beiner reflects on the dualism of theory and practice. The purpose of the theorist is not to offer sensible guidance on the conduct of social life but to test the boundaries of our vision of social order. Whereas the liberal citizen should embody the practical virtues of prudence and moderation, the theorist should be radical, probing, and immoderate. Looking back at the liberal-communitarian debate of the 1980s, Beiner recognizes that the antidote to our spiritless times lies neither in the embrace of community over individualism nor of individualism over community: both individual and community need to be submitted to radical questioning. It is by exposing ourselves to the challenge of fearless thinking encountered at the philosophical extremities that we are most likely to understand our own world at a deeper level. In this collection of essays and reviews, Ronald Beiner helps us to think critically about the thought-worlds of our foremost contemporary thinkers, including Hannah Arendt, Allan Bloom, Michel Foucault, Hans-Georg Gadamer, J¦rgen Habermas, Will Kymlicka, Christopher Lasch, Richard Rorty, Judith Shklar, Leo Strauss, Charles Taylor, and Michael Walzer. (shrink)

In this paper I explore how the discussion about the existence of laws in biology, more specifically laws about species taxa, bears on the issue of whether species are kinds or individuals. One of the main arguments offered in favor of the view that species are individuals is that it explains the lack of laws about species taxa, since laws cannot refer to individuals. My aim in this paper is to question the premise that there are no laws about species (...) axa and consequently to show that the proposed argument fails. I will argue that even if there are no strict scientific laws about species taxa, still, scientifically interesting, law-like generalizations are made which are used for explaining phenomena and predicting properties of species members. The existence of these law-like generalizations, in turn, suggests that species are, at least prima facie, best conceived of as kinds. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important (...) alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative (...) foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)

This thesis presents and criticizes Hilary Putnam's argument that mathematics is as empirical as science, in particular the argument that the switch from Euclidean geometry to Riemannian geometry as the approporiate geometry for physical space constituted an instance of revising mathematics as a result of observation. The thesis explains Putnam's views on mathematics as following from his theory of meaning and reference for natural kind terms. It is argued that Putnam's account of reference is unsuitable for mathematical (...) terms and that an "if-thenist" approach to mathematical knowledge is more fruitful. (shrink)

Reductionism in the Philosophy of Science develops a novel account of reduction in science and applies it to the relationship between classical and molecular genetics. However, rather than addressing the epistemological issues that have been essential to the reductionism debate in philosophy of biology, the discussion primarily pursues ontological questions, as they are known, about reducing the mental to the physical. For Sachse construes reductionism as a purely philosophical endeavor and defends the possibility of reduction in (...) principle, which may not be relevant to understanding reductionist reasoning and explanation occurring in scientific practice, as discussed by philosophers of science. Likewise, the conceptual framework used stems more from metaphysics and philosophy of mind than philosophy of science. Sachse's aim is twofold. First, he argues for the special sciences' being reducible to physics, by deriving the in principle possibility of epistemological reduction from ontological reduction. Second, he attempts to simultaneously make room for the legitimacy of the special sciences, effecting a conservative reduction rather than an elimination of the special sciences. (shrink)

_The Universal_ proposes a radically new philosophical system that moves from ontology to ethics. Drawing on the work of De Beauvoir, Sartre, and Le Doeuff, among others, and addressing a range of topics from the Asian sex trade to late capitalism, quantum gravity, and Merleau-Ponty's views on cinema, Dorothea Olkowski stretches the mathematical, political, epistemological, and aesthetic limits of continental philosophy and introduces a new perspective on political structures. Straddling a course between formalism and conventionalism, Olkowski develops the concept (...) of an ontological unconscious that arises from our "sensible" relation to the world-the information we absorb and emit that affects our encounters with the environment and others. In this "realm of the senses," or the field of vulnerability defined by our experience with pleasure and pain, Olkowski is able to rethink the space-time relations put forth by Irigaray's notion of the "interval," Bergson's "recollection," Merleau-Ponty's idea of the "flesh," and Deleuze's "plane of immanence." This aesthetic sense is shared by all humankind and nonhuman entities in the organic and inorganic world. The sensible universal can be applied to categories of pure and practical reason; experiential binaries of male-female and subject-object; and issues of autonomy, moral laws, and the regulation of perception. (shrink)

Since the beginning of the twentieth century, prominent authors including Jean Piaget have drawn attention to Edmond Goblot’s account of mathematical thought experiments. But his contribution to today’s debate has been neglected so far. The main goal of this article is to reconstruct and discuss Goblot’s account of logical operations (the term he used for thought experiments in mathematics) and its interpretation by Piaget against the theoretical background of two open questions in today’s debate: (1) the relationship (...) between empirical and mathematical thought experiments and (2) the question of whether mathematical thought experiments can play a justificatory function in proofs. The main corollary of this analysis is that Piaget’s interpretation is seriously flawed and insufficiently appreciative of important theses of Goblot’s account. First, Goblot can be easily defended against Piaget’s main criticism, and second, Goblot developed ideas about mathematical thought experiments that still deserve attention. (shrink)

There are several forms of the Ontological Argument, but it is more or less fair to say that all hang on the contention that the notion of a perfect being entails the existence of that being, since existence is involved in perfection. My first interest is in the word "perfect." The word, I think, is usually vague but it seems to me that, in the context of the proof, it has a meaning which turns out to be much more (...) pedestrian than we would suspect. The argument, if it is valid, must assert that any perfect being would have to have every possible characteristic. It alleges that existence is a characteristic and it is, of course, over this that most debate has centered. It seems clear, however, that "perfect," as employed in the argument, merely means what we would call "complete" in less august transactions. In some forms of the proof, indeed, this is made somewhat clearer since the expression "greatest possible" being is substituted for "perfect." Ordinarily, no one would be tempted to pass lightly from the notion of "that which is complete" to the notion of "the deity." Both "perfect" and "greatest possible" appear to have moral connotations and "complete" usually does not. The slide, however, is made easier by the recurrent notion that evil is not a positive quality but merely the absence of another positive quality, usually goodness. If this view is held, it certainly does follow that whatever is "complete" must be "perfect" in the moral sense as well as in the other usual senses. It is not, here, my concern to pass judgment on this notion. I am, at the moment, merely trying to trace possible sources of confusion in the argument. (shrink)