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 (...) class='Hi'>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)

Poincaré’s conventionalism has been interpreted in many writings as a philosophical position emerged by reflection on certain scientific problems, such as the applicability of geometry to physical space or the status of certain scientific principles. In this paper I would like to consider conventionalism as a philosophical position that emerged from Poincaré’s scientific practice. But not so much from dealing with scientific problems, as from the use of two specific methodologies proper to modern mathematics and the modern natural sciences: methodological (...) structuralism and the hypothetical-deductive method—thus, as a philosophical position which emerged from a way (or rather, two ways) ofdoingscience. With this approach, I try to deepen the analysis of connections between Poincaré’s scientific practice and his philosophy. (shrink)

Moritz Pasch (1843ber neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved. Pasch also presented in these lectures the main tenets of his philosophy of mathematics, which he continued to elaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivist methodology with a radically empiricist epistemology for mathematics. By taking into consideration publications from the entire span of Paschs (...) highly original, but virtually unknown, philosophy of mathematics is presented. (shrink)

We set out a doctrine about truth for the statements of mathematics?a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics?and argue that this doctrine, which we shall call ?mathematical relativism?, withstands objections better than do other non-realist accounts.

In the article, questions from the field of philosophy of mathematics are studied. The author is driven by the need to achieve a balance between the philosophy of science and the history of science in formation of concepts of the science development. In this regard, the author justifies the reliance on the methodology of implicit knowledge, combined with the epistemology principle of criticism in studying the development of mathematics as the most expedient and effective. The author expresses the necessity of (...) criticizing the methodology of counterexamples typical for mathematical empiricism declared by I. Lakatos. In the article, an explanation of the concept of implicit knowledge, as well as the examples from mathematics are given. The author believes that application of the concept of implicit knowledge in philosophical and mathematical terms allows us to reveal the unique specificity of mathematical science deeper. The author concludes that it is necessary to recognize the fundamental importance of the methodology of implicit knowledge for the epistemology, as well as for the development of philosophy and methodology of science in general. (shrink)

Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...) are (1) the (quasi-)empirical character of mathematics and (2) the rejection of axiomatic deductivism as the basis of mathematical knowledge. In this paper Lakatos' later views on the quasi-empirical nature of mathematical theories and methodology are examined and specific attention is paid to what this view implies about the nature of mathematical argumentation and its relation to the empirical sciences. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

According to the deductivist view of mathematics which became the rule during the nineteenth century, formal proofs working with symbolic formulas replaced the intuitive knowledge that used to be gained by the step-by-step construction of geometric fgures and diagrams. Twentieth century epistemological refection on symbolic formulas and formal proofs, however, took them to be diagrams respectively exhibiting formal relations and transformations. The claim was also made that, for such diagrams to be proofs, temporal displays of transformations—and of other speech acts—were (...) required. By returning to the signifcant elements of contemporary theories of mathematical proofs, we will here show how these proofs came to be a matter for semiotics and pragmatics as much as for formal logic. Once this is done, we will provide a few arguments against the standard objections to the ability of diagrams to present temporal order or perform speech acts. (shrink)

Despite his well‐known deductivism, in his early (unpublished) writings, Popper held an inductivist position. Up to 1929 epistemology entered Popper's reflections only as far as the problem was that of the justification of the scientific character of these fields of research. However, in that year, while surveying the history of non‐Euclidean geometries, Popper explicitly discussed the cognitive status of geometry without referring to psycho‐pedagogical aspects, thus turning from cognitive psychology to the logic and methodology of science. As a consequence (...) of his reflections on the problematic relationship between geometrical‐mathematical constructions and physical reality Popper was able to get over a too direct notion of such a relationship, cast doubts on inductive inference and started conceiving in a new (strictly non‐inductivist) manner the relationship between theoretical and observational propositions. (shrink)

Illuminates legal reasoning -- and its justification At least since plato and Aristotle, thinkers have pondered the relationship between philosophical arguments and the "sophistical" arguments offered by the Sophists -- who were the first professional lawyers. Judges wield substantial political power, and the justifications they offer for their decisions are a vital means by which citizens can assess the legitimacy of how that power is exercised. However, to evaluate judicial justifications requires close attention to the method of reasoning behind decisions. (...) This new collection illuminates and explains the political and moral importance in justifying the exercise of judicial power. Explores enduring questions Focusing on work over the past century, the essays address important recurring questions, such as: When a judge or a lawyer reasons to a conclusion about what is legally required in a given case, must he also ask what is morally required? To what extent do a judge's personal, political, or moral biases affect his legal reasoning? What is the impact of such biases? Can all such biases be avoided? Is legal reasoning similar to reasoning in mathematics, logic, and linguistics, the physical sciences, the social sciences, or literature and history? Do formal logical modes of argument play any roles in legal reasoning? Solid coverage, well organized The articles were chosen to present some of the most influential works on the topic, as well as less familiar works that are thought provoking and informative. Each volume also offers a representative range of theoretical approaches to its topic, contains an introduction that locates the subject within the larger framework oftheories of legal reasoning and jurisprudence, and includes bibliographical notes on further readings. Many 19th century legal theorists argued that deduction is the central mode of legal reasoning, and that legal argument is like a deductive proof in mathematics or logic. They were attracted to this "deductivist" concept because it suggested that legal reasoning could be politically and morally neutral. This volume brings together some of the most thought-provoking articles on both sides of the debate. It also contains several leading articles that explain the role of probabilistic judgments and presumptions in various types of legal arguments, including the laws of evidence and criminal procedure. The collection is a solid introduction to the basic modes of logical argument in legal reasoning. (shrink)

In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether second-order logic can count as logic in some proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not have (...) the "epistemic innocence" which would be needed for, e.g., foundational programmes in mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense. (shrink)

As in the case of great books in all branches of philosophy, Pierre Duhem's Le Théorie Physique , first published in 1906, can be looked to as the progenitor of many different and even conflicting currents in subsequent philosophy of science. On a superficial reading, it seems to be an expression of what later came to be called deductivist and instrumentalist analyses of scientific theory. Duhem's very definition of physical theory, put forward early in the book, is the quintessence of (...) instrumentalism: A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws [p. 19]. (shrink)

I describe some interpretive strategies used by physicists in the development of quantum electrodynamics in the 1930s and 1940s, using Wimsatt's account of how to reason with false models as a guide. I call these “interpretive” strategies because they were used not just to derive empirical predictions, but also to derive information about the world besides the aforementioned predictions. These strategies were regarded as mathematically unrigorous, yet they were crucial to the development of a better theory of quantum electrodynamics. I (...) argue that these strategies are not easily assimilated into conventional axiomatic, deductivist views of what theories tell us about the world. Furthermore, it is unclear if these strategies are necessarily less reliable than strategies based solely on mathematically rigorous inferences. I suggest that these less than fully rigorous strategies are worth considering as general strategies for working with theories in physics. (shrink)

Kevin Hoover explores retroduction as a means of theorizing in macro-economics, rather than exclusive reliance on deductivism or inductivism. Retroduction is discussed as involving conceptualization with respect to empirical evidence as a means of identifying causal mechanisms as the basis for theory in the form of mathematical models. It is put forward as a preferred alternative to more fundamental reform of macro-economics, whose justification Hoover dismisses as being ‘ideological’. Yet ideology refers to a position on metaphysics such as (...) Hoover himself sets out in terms of open systems, which would significantly extend the scope for retroduction. (shrink)

The paper advocates an epistemological interpretation of the Peano School axiomatics. The construction of axiom systems is presented as a cognitive enterprise unveiling the internal dynamics, evolution, and architecture of axiomatic systems as well as connections to applications. This approach reveals that the study of the relation between axioms and theorems not only serves to reduce a theory to a minimum number of principles and increase the certainty or justification of the latter, but also to study alternative settings of a (...) given mathematical theory. The paper explains why the Peanian conception of axiomatics differs from axiomatic deductivism, if-thenism and scientific deductivism, and analyzes Peano’s conception of rigor. The paper also postulates that Peano had a set-theoretic but also an algorithmic conception of function, and prioritized functions over relations. Two seemingly contradictory aspects of Peano’s axiomatics, global and local reasoning, are illustrated through the examples of rigor and simplicity. Finally, the paper claims that Peano’s axiomatics was a dynamic mathematical enterprise rather than an a priori axiomatization grounded on logical principles, which explains the difference with respect to Frege’s project. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Is Bioethics Applied Ethics?Robert M. VeatchBioethics is often referred to as a kind of applied ethics. The term applied ethics can be controversial if it is taken to imply that ethical theory from philosophy or religious ethics has to be the starting point for ethical analysis of some practical field such as medicine or law or politics. The term can be understood as requiring some premises from an ethical (...) theory that are then "applied" to some problem area, as a "top down" way of doing the work of moral reflection in a way that axioms in geometry are applied to present proofs of mathematical claim.The publication of an important work in the history of medical ethics, A History of Medical Ethics, provides an opportunity to reflect more carefully on what applied ethics means. That massive multi-authored volume edited by Robert Baker and Laurence McCullough (2007) is scheduled for publication this fall. With the permission of the Press, we have asked Baker and McCullough to prepare an article-length version of one of the central essays in that work. Their paper challenges the idea that ethical theory should be "applied" to medicine or other fields, putting forth an alternative of the "appropriation" of an ethical theory by the formulation of concepts, principles, moral rules, or other specifications that may turn out to be unique or at least modified for the particular field in question.In this issue of the Kennedy Institute of Ethics Journal, we publish the Baker-McCullough article together with five critical commentaries. We include two medical historians who have a sustained commitment to studying the history of ethics in medicine, two systematic ethicists who have an unusual level of commitment to studying ethics historically, and a physician with unusual sensitivity to the history of medical ethics. Each comments on the Baker-McCullough thesis and offers an alternative view of the link between ethical theory and the field of biomedical ethics.Daniel Fox triggered some of the initial reflection on the history of medical ethics in the 1970s. His work is discussed by Baker and McCullough. Fox is sympathetic with the Baker-McCullough appropriation thesis, viewing their position as complementary to his own, although he [End Page 1] urges more attention to the political aspects of medicine, professional dominance, and professional control of medical resources.Ulf Schmidt, a specialist on the historical aspects of research during the Nazi era, focuses on Baker and McCullough's use of that historical example of the appropriation of medical ethics.The two ethicists who comment both have had many years experience with historical aspects of medical ethics, and both worked on the historically important Belmont Report of the National Commission for the Protection of Human Subjects. Both question whether Baker and McCullough overstate the problem of the relation between application and appropriation. Albert Jonsen, as a member of the National Commission, concentrates on the Belmont Report, showing how he sees the report appropriating philosophical ethics.Tom Beauchamp, who was the primary staff person responsible for the report, provides a somewhat different read on the report and particularly on the role of Kantian ethics in the report's discussion of autonomy. Beauchamp accuses Baker and McCullough of having too narrow an understanding of the meaning of the term "applied," claiming that no people doing medical ethics today would simply apply ethical theory in a deductivist manner.Finally, Jeremy Sugarman, the physician ethicist, is, in a somewhat surprising turn, the cautious defender of the application that Baker and McCullough challenge. Although he ends up sympathetic with the appropriation that implies going beyond straight-forward top-down application, he outlines the value of ethical theory for the practical analysis of ethical problems in fields such as medicine.ReferenceBaker, Robert, and McCullough, Laurence, eds. 2007. A History of Medical Ethics. New York: Cambridge University Press, in press. [End Page 2]Google ScholarCopyright © 2007 Johns Hopkins University Press... (shrink)

As in the case of great books in all branches of philosophy, Pierre Duhem's Le Théorie Physique, first published in 1906, can be looked to as the progenitor of many different and even conflicting currents in subsequent philosophy of science. On a superficial reading, it seems to be an expression of what later came to be called deductivist and instrumentalist analyses of scientific theory. Duhem's very definition of physical theory, put forward early in the book, is the quintessence of instrumentalism:A (...) physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws [p. 19]. (shrink)

The present paper elaborates a deductivist account of natural language argu-ment in the context of pragma-dialectics. It reviews earlier debates, criticizes some standard misconceptions in the literature, and argues that the identification and analysis of deductive argument schemes can be the basis of a compelling theory of argumentative discourse.

Deductivism has been variously presented as an evaluative thesis and as an interpretive one. I argue that deductivism fails as a universal evaluative thesis, and as such that its value as an interpretive thesis must be supported on other grounds. As a reconstructive strategy, deductivism is justified only on the grounds that an arguer is, or ought to be, aiming at the deductive standard of evidence. As such, the reconstruction of an argument as deductive must be supported (...) by contextual and situational factors including facts about the arguer. Further, the plausibility of deductivism as a normative thesis is not tied to its plausibility as a descriptive or interpretive thesis. (shrink)

This essay proposes and defends a general thesis concerning the nature of fallacies of reasoning. These in distinctive ways are all said to be deductively invalid. More importantly, the most accurate, complete and charitable reconstructions of these species and specimens of the informal fallacies are instructive with respect to the individual character of each distinct informal fallacy. Reconstructions of the fallacies as deductive invalidities are possible in every case, if deductivism is true, which means that in every case they (...) should be formalizable in an expressively comprehensive formal symbolic deductive logic. The general thesis is illustrated by a detailed examination of Walter Burleigh's paradox in his c. 1323 work, De Puritate Artis Logicae Tractatus Longior (Longer Treatise on the Purity of Logic), as a challenge to the deductive validity of hypothetical syllogism. The paradox has the form, ‹If I call you a swine, then I call you an animal; if I call you an animal, then I speak truly; therefore, if I call you a swine, then I speak truly'. Several solutions to the problem are considered, and the inference is exposed as an instance of the common deductive fallacy of equivocation. (shrink)

This is a concise book that introduces students to the basics of logical thinking and important mathematical structures that are critical for a solid understanding of logical formalisms themselves as well as for building the necessary background to tackle other fields that are based on these logical principles. Despite its compact and small size, it includes many solved problems and quite a few end-of-section exercises that will help readers consolidate their understanding of the material. This textbook is essential reading (...) for anyone interested in the logical foundations of Informatics, Computer Science, Data Science, Artificial Intelligence, and other related areas. Written with undergraduate students in these disciplines in mind, this book can very well serve the needs of interested and curious readers who wish to get a grasp of the logical principles upon which these fields are built. This book does not require readers to possess math skills beyond those learned in high school. (shrink)

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part (...) II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their (...) own. The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

"Deductivism" is the thesis that all logic is deductive. Stove lays out the arguments for the existence of non-deductive logic.

In Gemes (1990) I examined certain formal versions of hypothetico-deductivism (H-D) showing that they have the unacceptable consequence that "Abe is a white raven" confirms "All ravens are black"! In Gemes (1992) I developed a new notion of content that could save H-D from this bizarre consequence. In this paper, I argue that more traditional formulations of H-D also need recourse to this new notion of content. I present a new account of the vexing notion of the natural axiomatization (...) of a theory. The notion is used to construct a form of H-D that allows for the type of selective confirmation without which Glymour (1980a,b) claims H-D is hopeless. (shrink)

Since Christensen refuted the Bootstrap theory of confirmation in 1990, there have been some trials to improve the Hypothetico-Deductive theory of confirmation. After some trials, Gemes (1998) declared that his revised version completely overcame the difficulties of Hypothetico-Deductivism without generating any new difficulties. In this paper, I will assert that Gemes's revised version encounters some new difficulties, so it cannot be a true alternative to the Bootstrap theory of confirmation and to classical Hypothetico-Deductivism. Also I will assert that, (...) in principle, such new difficulties cannot be overcome by any trials dependent only on formal logic. (shrink)

: Any precise version of H-D needs to handle various problems, most notably, the problem of selective confirmation: Precise formulations of H-D should not have the consequence that where S confirms T, for any T', S confirms T&T'. It is the perceived failure of H-D to solve such problems that has lead John Earman to recently conclude that H-D is "very nearly a dead horse". This suggests the following state of play: H-D is an intuitively plausible idea that breaks down (...) in the attempt to give it a precise formulation. Indeed I think that fairly captures the view among specialists in the field of confirmation theory. Here I argue that the truth about H-D is largely the reverse: H-D can be given a precise formulation that avoids the longstanding technical problems, however, it relies on a fundamentally unsound philosophical intuition. The bulk of this paper involves reviewing the problems affecting previous attempts at giving precise formulations of H-D and displaying some recent versions that can handle these problems. It then briefly explains why the basic intuition behind H-D is itself unsound, namely, because H-D involves a tacit assumption of inductive scepticism. Finally, the historical relation between H-D and the positivists' quest for a criterion of empirical significance will be reconsidered with the surprising result that having glossed H-D as fundamentally unsound it is concluded that a sound version of the criterion of empirical significance is now available. The demarcation criterion, the positivists' philosopher's stone that serves to separate claims with empirical significance from claims lacking empirical significance having finally been found, it is argued that we should regard empirical significance as just one among a variety of virtues and not follow the positivists in taking it to be a sin qua non for all meaningful statements. (shrink)

The non-justificationist deductivism (or critical rationalism) of Karl Popper constitutes the only approach to human knowledge, including of course the natural and social sciences, that is capable of overcoming all the failings, and the plain contradictions, of the traditional doctrine of inductivism and of its modern incarnation, Bayesianism.

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Alleged counter-examples deployed in Park [Erkenntnis 60: 229–240] against the account of selective hypothetico-deductive confirmation offered in Gemes [Erkenntnis 49: 1–20] are shown to be ineffective. Furthermore, the reservations expressed in Gemes [ibid] and [Philosophy of Science 62: 477–487] about hypothetico-deductivism are retracted and replaced with the conclusion that H-D is a viable account of confirmation that captures much of the practice of working scientists. However, because it cannot capture cases of inference to the best explanation and cases of (...) the observational confirmation of statistical hypotheses, it is concluded that H-D cannot supply a complete theory of confirmation. (shrink)

It has been a traditional claim that Newtonian inductivism sharply contradicts Cartesian deductivism, and that Newton's rejection of the method of hypothesis is intended as a criticism of the Cartesian scientific methodology. There have been some sharp attacks against the received view that Descartes aimed at the construction of a purely a priori science, but despite this two beliefs still dominate even recent interpretations of Descartes' work. The first is the belief that a significant part of Descartes' natural philosophy (...) was meant to be a priori, and the second is the belief that the Cartesian empirical scientific method was a version of the method of hypothesis. Also, new light has been shed on Newton's famed inductivism and his rejection of the method of hypothesis, a light which shows first, that the traditional inductive interpretation of Newton's method falls significantly short of capturing the richness of this method, and second, that hypotheses play an important role in shaping the conceptual framework in which Newton's work was developed. Despite these recent interpretations, however, an accurate account of Newton's experimental method and an examination of the exact scope and significance of Newton's critique and rejection of the method of hypothesis are still needed. ;The central thesis of this dissertation is that a study of the methodological work and actual scientific practice of Descartes and Newton reveals profound similarities that can be traced back to their employment of the same scientific method of discovery, namely the method of analysis. To that end, I have undertaken a comparative study of the methodological and scientific work of both thinkers. I argue that the official Cartesian method is not the method of hypothesis, and that Descartes did not think any of the parts of his science to be purely a priori. I also argue that the content of Newtonian methodology exceeds that of simple induction, and I try to reconstruct this method as accurately as possible. The conclusion is reached that both Newton and Descartes employed the same scientific method, and that the differences in the fruits delivered by this method are due to the differences between the conceptual frameworks within which Descartes' and Newton's thoughts were developed. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability (...) of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)

The contrast between the two approaches alluded to in the title has gained a certain prominence in our own day. With the knowledge of hindsight it will be of interest therefore to study its incidence in an earlier period, in the writings of Whewell and Mill, Which may thus yield added significance for a later generation. Right at the start there is a difficulty. Not all inductivists agree on their principles, or their interpretation of the logic of scientific reasoning, and (...) the same is true of deductivism, and the differences can therefore be discussed only in connection with individual writers. When this is done, we do find moreover some considerable variations in the treatment of their respective doctrines by the members of each of our two schools. To define these differences, we need a finer structure of elements of classification, indicative of criteria for the acceptance of scientific hypotheses. (shrink)

John Fox argued that deductivism must be supplemented with ‘epistemic syllogisms’, non-deductive arguments whose vindication is trivial if deductivism is correct. I resist this attempt to surpass deductivism.

Albert Lautman (1908-1944) was a French philosopher of mathematics whose work played a crucial role in the history of contemporary French philosophy. His ideas have had an enormous influence on key contemporary thinkers including Gilles Deleuze and Alain Badiou, for whom he is a major touchstone in the development of their own engagements with mathematics. Mathematics, Ideas and the Physical Real presents the first English translation of Lautman's published works between 1933 and his death in 1944. Rather than being preoccupied (...) with the relation of mathematics to logic or with the problems of foundation, which have dominated philosophical reflection on mathematics, Lautman undertakes to develop an understanding of the broader structure of mathematics and its evolution. The two powerful ideas that are constants throughout his work, and which have dominated subsequent developments in mathematics, are the concept of mathematical structure and the idea of the essential unity underlying the apparent multiplicity of mathematical disciplines. This collection of his major writings offers readers a much-needed insight into his influence on the development of mathematics and philosophy. (shrink)

In “Deductivism as an Interpretative Strategy: A Reply to Groarke’s Defense of Reconstructive Deductivism,” David Godden distinguished two notions of deductivism. On the one hand, as an interpretative thesis, deductivism is the view that all-natural language argumentation must be interpreted as being deductive. On the other hand, as an evaluative thesis, deductivism is the view that for a conclusion to follow, it has to follow of necessity from the premises—or, in other words, that being a (...) good inference implies being deductive. The main goal of this paper is to show that evaluative deductivism is wrong. (shrink)

We use recently developed approaches in argumentation theory in order to revamp the hypothetico-deductive model of confirmation, thus alleviating the well-known paradoxes the H-D account faces. More specifically, we introduce the concept of dialectic confirmation on the background of the so-called theory of dialectical structures (Betz 2010, 2012b). Dialectic confirmation generalises hypothetico-deductive confirmation and mitigates the raven paradox, the grue paradox, the tacking paradox, the paradox from conceptual difference, and the problem of surprising evidence.