This paper rejoins the debate surrounding Thomas Tymockzko’s paper on the surveyability of proof, first published in the Journal of Philosophy, and makes the claim that by attending to certain broad features of modern conceptions of proof we may understand ways in which the debate surrounding the surveyability of proof has heretofore remained unduly circumscribed. Motivated by these historical reflections, I suggest a distinction between local and global surveyability which I believe has the promise to open (...) up significant new advances in the philosophy of mathematics. (shrink)

The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non-constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can be (...) defined as a semiotic meta-code that depends on the underlying mode of signification (semiosis), the selected code and the underlying semiotic space and determines the individual mode of integration (selection, combination, blending) into a narrative structure (proof). Finally, we examine certain historical types of styles of mathematical proofs, to elucidate our viewpoint. (shrink)

In this paper, we suggest the broader concept of proof-event, introduced by Joseph Goguen, as a fundamental methodological tool for studying proofs in history of mathematics. In this framework, proof is understood not as a purely syntactic object, but as a social process that involves at least two agents; this highlights the communicational aspect of proving. We claim that historians of mathematics essentially study proof-events in their research, since the mathematical proofs they face in the extant sources (...) involve many informal components, often not completely formalizable, and convey some kind of semantic content calling for understanding and verification. We illustrate the application of this methodological approach in some outstanding historical cases, paying particular attention to the process of proof interpretation that makes a proof-event alive. Finally, we suggest a classification of proof-events, according to the conditions imposed upon problem-solving. This enables us to speak about broad classes of proof-events in history of mathematics that share a common characteristic. (shrink)

John von Neumann's proof that quantum mechanics is logically incompatible with hidden varibales has been the object of extensive study both by physicists and by historians. The latter have concentrated mainly on the way the proof was interpreted, accepted and rejected between 1932, when it was published, and 1966, when J.S. Bell published the first explicit identification of the mistake it involved. What is proposed in this paper is an investigation into the origins of the proof rather (...) than the aftermath. In the first section, a brief overview of the his personal life and his proof is given to set the scene. There follows a discussion on the merits of using here the historical method employed elsewhere by Andrew Warwick. It will be argued that a study of the origins of von Neumann's proof shows how there is an interaction between the following factors: the broad issues within a specific culture, the learning process of the theoretical physicist concerned, and the conceptual techniques available. In our case, the ‘conceptual technology’ employed by von Neumann is identified as the method of axiomatisation. (shrink)

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and (...) argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning. (shrink)

This book provides a detailed, updated exposition and defense of five of the historically most important (but in recent years largely neglected) philosophical proofs of God’s existence: the Aristotelian, the Neo-Platonic, the Augustinian, the Thomistic, and the Rationalist. It also offers a thorough treatment of each of the key divine attributes—unity, simplicity, eternity, omnipotence, omniscience, perfect goodness, and so forth—showing that they must be possessed by the God whose existence is demonstrated by the proofs. Finally, it answers at length all (...) of the objections that have been leveled against these proofs. This work provides as ambitious and complete a defense of traditional natural theology as is currently in print. Its aim is to vindicate the view of the greatest philosophers of the past— thinkers like Aristotle, Plotinus, Augustine, Aquinas, Leibniz, and many others— that the existence of God can be established with certainty by way of purely rational arguments. It thereby serves as a refutation both of atheism and of the fideism that gives aid and comfort to atheism. (shrink)

Proofs of God in Early Modern Europe offers a fascinating window into early modern efforts to prove God’s existence. Assembled here are twenty-two key texts, many translated into English for the first time, which illustrate the variety of arguments that philosophers of the seventeenth and eighteenth centuries offered for God. These selections feature traditional proofs—such as various ontological, cosmological, and design arguments—but also introduce more exotic proofs, such as the argument from eternal truths, the argument from universal aseity, and the (...) argument ex consensu gentium. Drawn from the work of eighteen philosophers, this book includes both canonical figures (such as Descartes, Spinoza, Newton, Leibniz, Locke, and Berkeley) and noncanonical thinkers (such as Norris, Fontenelle, Voltaire, Wolff, Du Châtelet, and Maupertuis). -/- Lloyd Strickland provides fresh translations of all selections not originally written in English and updates the spelling and grammar of those that were. Each selection is prefaced by a lengthy headnote, giving a biographical account of its author, an analysis of the main argument(s), and important details about the historical context. Strickland’s introductory essay provides further context, focusing on the various reasons that led so many thinkers of early modernity to develop proofs of God’s existence. -/- Proofs of God is perfect for both students and scholars of early modern philosophy and philosophy of religion. (shrink)

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum (...) of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

Since Plato's time, philosophers have concentrated on developing moral theories to guide our actions. They have said we ought to act to maximize happiness; we ought to act to fulfill human potential; etc. But all of them have largely ignored a key question: Regardless of which acts are morally obligatory, can moral obligation as such be proven? Early in his book, Allen clarifies what sort of demonstration or justification can suffice as a proof that we are subject to moral (...) obligation. He analyzes some twentieth-century ethical theories which initially appear to serve as such a demonstration. Next, he examines at length the theory of contemporary English philosopher R.M. Hare. And finally, he reworks Hare's ideas into a complete proof that we are bound by moral obligation. Philosophers should value this book because it brings to light and defines a neglected but critical problem, and develops an innovative, thought-provoking solution. Serious students, too, will find it helpful because it provides a clearly written historical study of a central theme in twentieth-century ethics. (shrink)

Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy [1991]; Shin [1994]; and Giaquinto [1994]) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.

An anthology of the most important historical sources, classical and modern, on the subjects of presumptions and burdens of proof In the last fifty years, the study of argumentation has become one of the most exciting intellectual crossroads in the modern academy. Two of the most central concepts of argumentation theory are presumptions and burdens of proof. Their functions have been explicitly recognized in legal theory since the middle ages, but their pervasive presence in all forms of (...) argumentation and in inquiries beyond the law--including politics, science, religion, philosophy, and interpersonal communication--have been the object of study since the nineteenth century. However, the documents and essays central to any discussion of presumptions and burdens of proof as devices of argumentation are scattered across a variety of remote sources in rhetoric, law, and philosophy. Presumptions and Burdens of Proof: An Anthology of Argumentation and the Law brings together for the first time key texts relating to the history of the theory of presumptions along with contemporary studies that identify and give insight into the issues facing students and scholars today. The collection's first half contains historical sources and begins with excerpts from Aristotle's Topics and goes on to include the locus classicus chapter from Bishop Whately's crucial Elements of Rhetoric as well as later reactions to Whately's views. The second half of the collection contains contemporary essays by contributors from the fields of law, philosophy, rhetoric, and argumentation and communication theory. These essays explore contemporary understandings of presumptions and burdens of proof and their role in numerous contexts today. This anthology is the definitive resource on the subject of these crucial rhetorical modes and will be a vital resource to all scholars of communication and rhetoric, as well as legal scholars and practicing jurists. (shrink)

In this section, we apply the notions obtained above to a famous historical example of a false proof. Our goal is to demonstrate that this proof shows a sufficient degree of distinctiveness for a formalization in a Naproche-like system and hence that automatic checking could indeed have contributed in this case to the development of mathematics. This example further demonstrates that even incomplete distinctivication can be sufficient for automatic checking and that actual mistakes may occur already in (...) the margin between the degree of distinctiveness necessary for formalization and complete distinctiveness. (shrink)

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn’t produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz’s drafts on logic contained sufficient ingredients to prove them by an algebraic method –which we call the Leibniz-Cayley (LC) system– without having to make use of the more expressive and complex machinery of first-order (...) quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method –which we call the McColl-Ladd (ML) system– employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz’s historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic. (shrink)

The aim of this paper is to offer a conceptual analysis of Weinberg's proof of the spin-statistics theorem by comparing it with Pauli's original proof and with the subsequent textbook tradition, which typically resorts to the dichotomy positive energy for half-integral spin particles/microcausality for integral-spin particles. In contrast to this tradition, Weinberg's proof does not directly invoke the positivity of the energy, but derives the theorem from the single relativistic requirement of microcausality. This seemingly innocuous difference marks (...) an important change in the conceptual basis of quantum physics. Its historical, theoretical, and conceptual roots are here reconstructed. The link between Weinberg's proof and Pauli's original is highlighted: Weinberg's proof turns out to do justice to Pauli's anti-Dirac lines of thought. The work of Furry and Oppenheimer is also surveyed as a “third way” between the textbook tradition established by Pauli and Weinberg's approach. (shrink)

In this paper, I will argue from a historical perspective that Gentzen’s 1935 consistency proof of 1st order Peano Arithmetic PA principally aimed to give a finitist interpretation of implication and this aspect of the 1935 proof emerged as the attempt to cope with the non-finiteness in BHK-interpretation of implication. My argument consists of two parts. First, I will explain that the fundamental idea of the 1935 proof is to show the soundness of PA on some (...) finitist interpretation and Gentzen had attempted to use BHK-interpretation as such an interpretation. It is claimed that he eventually thought that there is a ‘circularity’ which is due to the non-finiteness of BHK-interpretation of implication, so it cannot be used as a finitist interpretation. Secondly, I will argue that the principal aim of the 1935 proof is to give his own finitist interpretation of implication which avoids such a ‘circularity’. Finally, this paper’s conclusion and an open question are presented. (shrink)

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of Constructions. The book (...) also covers the essence of proof checking and proof development, and the use of dependent type theory to formalize mathematics. The only prerequisites are a good knowledge of undergraduate algebra and analysis. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarize themselves with the material. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Proof and Persuasion in Black Athena:: The Case of K. O. MüllerJosine H. BlokNon tali auxilio.Virgil, Aeneid II, 521When in 1824 the German classical scholar Karl Otfried Müller (1797–1840) set down to write a review of Champollion’s first Letter to M. Dacier (1822), he was profoundly interested. 1 For several years he had been working on Egypt, and as he told his parents in 1820, “I have come (...) to love Egyptian antiquity so much, that, if I were not constricted by the schedule of my classes, I would have set myself to the decipherment of the hieroglyphs, which I would not deem impossible by the clues I have found.” 2 During the same years (1820–25) that he wrote his early books on ancient Greece, he [End Page 705] reviewed sixteen studies on Egypt alone. 3 In his review of the Letter, then, he wanted to draw “the attention of [the] readers to this, certainly not unfounded, discovery,” 4 though he feared the author was inclined to jump to conclusions. But in August 1824 he reviewed Champollion’s more complete publication of his findings of the same year and, recalling his previous anxiety, commented:Now, however, the reading of the present work has totally convinced me, that the usage of hieroglyphs to indicate sounds is as ancient as this writing system itself. [... This discovery should also mean that...] the history of Egyptian religion and state will be reconstructed and expanded. 5From that moment he sided unequivocally with Champollion, politely but clearly reminding German colleagues who still stumbled on with decipherments of their own, that this would not do. 6With these facts in mind, Martin Bernal’s rendering in his Black Athena, volume I, is bound to surprise: “[U]nlike Humboldt, Niebuhr and Bunsen [Müller] disregarded the sensational scholarly developments between 1815 and 1830. There is no indication that he paid any attention to Champollion’s decipherment” (316). The contradiction between Bernal’s statements and the sources on Müller’s life and work turns out not to be an incidental error but part of a larger pattern. To understand this pattern and its objectives, let me first summarize Bernal’s argument and my own.In his challenging book Bernal argues that the ancient Greek world was founded on the colonization of Greece by Phoenicians and Egyptians in the second millennium bc. 7 The material which is to prove his thesis is offered in volume II, though volume I includes a summary. In this first volume Bernal observes that from late antiquity until the eighteenth century, awareness of the Afroasiatic roots of ancient civilization had been retained as the heritage of ancient Greece itself. Hence Bernal labels this perception underlying Western classical scholarship of the early modern period the “Ancient Model.” By or soon after the 1820s, however, it was replaced by the “Aryan Model,” which took Greek civilization to be partly autochthonous and partly [End Page 706] shaped by invasions coming from the North. To reveal that the fall of the Ancient Model was not the result of internal developments in classical scholarship but of externalist influences, notably the belief in progress, the defense of Christianity, Romanticism, and most importantly racism, is the core theme of volume I. Bernal sustains his argument by contending that historical source criticism was not really or entirely an internal development but created to serve the external ends, and by presenting Müller as the embodiment of Romanticist racism, who was responsible for overthrowing the Ancient Model and inaugurating its Aryan successor. Hence, by thus designating Müller as the axis around which the overall turn of classical scholarship revolved, he wants to prove that the fall of the Ancient Model was unjustified in terms of scholarship but only occurred due to dishonorable ideologies.Through a limited number of representative issues I want to argue that Bernal’s rendering of Müller and the context of his work is untenable in the light of the source material and that Bernal’s explanation of the fall of the Ancient Model is untenable as well. My aim is not just to clear Müller’s name... (shrink)

The Hegel Lectures Series Series Editor: Peter C. Hodgson Hegel's lectures have had as great a historical impact as the works he himself published. Important elements of his system are elaborated only in the lectures, especially those given in Berlin during the last decade of his life. The original editors conflated materials from different sources and dates, obscuring the development and logic of Hegel's thought. The Hegel Lectures series is based on a selection of extant and recently discovered transcripts (...) and manuscripts. Lectures from specific years are reconstructed so that the structure of Hegel's argument can be followed. Each volume presents an accurate new translation accompanied by an editorial introduction and annotations on the text, which make possible the identification of Hegel's many allusions and sources. Lectures on the Proofs of the Existence of God Hegel lectured on the proofs of the existence of God as a separate topic in 1829. He also discussed the proofs in the context of his lectures on the philosophy of religion (1821-31), where the different types of proofs were considered mostly in relation to specific religions. The text that he prepared for his lectures in 1829 was a fully formulated manuscript and appears to have been the first draft of a work that he intended to publish and for which he signed a contract shortly before his death in 1831. The 16 lectures include an introduction to the problem of the proofs and a detailed discussion of the cosmological proof. Philipp Marheineke published these lectures in 1832 as an appendix to the lectures on the philosophy of religion, together with an earlier manuscript fragment on the cosmological proof and the treatment of the teleological and ontological proofs as found in the 1831 philosophy of religion lectures. Hegel's 1829 lectures on the proofs are of particular importance because they represent what he actually wrote as distinct from auditors' transcriptions of oral lectures. Moreover, they come late in his career and offer his final and most seasoned thinking on a topic of obvious significance to him, that of the reality status of God and ways of knowing God. These materials show how Hegel conceived the connection between the cosmological, teleological, and ontological proofs. All of this material has been newly translated by Peter C. Hodgson from the German critical editions by Walter Jaeschke. This edition includes an editorial introduction, annotations on the text, and a glossary and bibliography. (shrink)

This chapter contains sections titled: Hegel's Discussion of the Proofs On “Proof” and “Existence” The Proofs, Religious Elevation, and the Communion of Spirit The Multiplicity of Proofs and the One God The Cosmological Proof The Teleological Proof The Ontological Proof The Dialectic of the Proofs and the Speculative Reversal Hegel's Proofs Today.

Is science getting at the truth? The sceptics – those who spread doubt about science – often employ a simple argument: scientists were sure in the past, and then they ended up being wrong. Such sceptics draw on dramatic quotes from eminent scientists such as Lord Kelvin, who reportedly stated at the turn of the 20th century “There is nothing new to be discovered in physics now,” shortly before physics was dramatically transformed. They ask: given the history of science, wouldn’t (...) it be naïve to think that current scientific theories reveal ‘the truth’, and will never be discarded in favour of other theories? Through a combination of historical investigation and philosophical-sociological analysis, Identifying Future-Proof Science defends science against such potentially dangerous scepticism. It is argued that we can confidently identify many scientific claims that are future-proof: they will last forever, so long as science continues. How do we identify future-proof claims? This appears to be a new question for science scholars, and not an unimportant one. It is argued that the best way to identify future-proof science is to avoid any attempt to analyse the relevant first-order scientific evidence, instead focusing purely on second-order evidence. Specifically, a scientific claim is future-proof when the relevant scientific community is large, international, and diverse, and at least 95% of that community would describe the claim as a ‘scientific fact’. In the entire history of science, no claim meeting these criteria has ever been overturned, despite enormous opportunity. (shrink)

In this paper I criticize the current standards for the acceptability of expert testimony in current US legislation. The standards have been the subject of much academic literature after the Frye and Daubert cases. I expose what I call the Paradox of Proof, and argue that the historical and current standards have sidestepped the problem of determining who is an expert and who is not in a court of law. I then investigate the problem of recognizing expertise from (...) the layperson’s standpoint, and suggest what courses of action the future research ought to take on the problem of identifying expertise. (shrink)

Public and professional debates have come to rely heavily on a special type of reasoning: the argument-from-ignorance, in which conclusions depend on the _lack_ of compelling information. "I win my argument," says the skillful advocate, "unless you can prove that I am wrong." This extraordinary gambit has been largely ignored in modern rhetorical and philosophical studies. Yet its broad force can be demonstrated by analogy with the modern legal system, where courts have long manipulated burdens of proof with skill (...) and subtlety. This legal, philosophical, and rhetorical study by Richard H. Gaskins provides the first systematic treatment of arguments-from-ignorance across a wide range of modern discourse—from constitutional law, scientific inquiry, and moral philosophy to organizational behavior, computer operation, and personal interaction. Gaskins reviews the historic shifts in constitutional proof burdens that have shaped public debate on fundamental rights and, by analogy, on the fundamental status of intellectual and cultural authority. He shows how similar shifts have dominated polemical battles between scientific and ethical modes of authority, affecting both academic and popular discussion. Finally, he discovers the philosophical roots of default reasoning strategies in the arguments of Kant and nineteenth-century Kantian schools. Concluding that shifting proof burdens are inescapable in a world of scientific and moral uncertainty, Gaskins emphasizes the common strategic ground shared by dogmatic and skeptical reasoning. Using Hegelian strategies, he describes a more pluralistic temper that can move critical thinking beyond polemics and strengthen our capacities for common discourse. (shrink)

Introduction to the Scientific Proof of the Natural Moral Law This paper proves that Aquinas has a means of demonstrating and deriving both moral goodness and the natural moral law from human nature alone. Aquinas scientifically proves the existence of the natural moral law as the natural rule of human operations from human nature alone. The distinction between moral goodness and transcendental goodness is affirmed. This provides the intellectual tools to refute the G.E. Moore (Principles of Ethics) attack against (...) the natural law as committing a "naturalistic fallacy". This article proves that instead Moore commits the fallacy of equivocation between moral goodness and transcendental goodness in his very assertion of a "naturalistic fallacy" by the proponents of the natural moral law. In the process the new deontological/kantian theory of natural law as articulated by John Finnis, Robert George, and Germain Grisez is false historically and philosophically. Ethical naturalism is affirmed as a result. (shrink)

If one of Gentzen’s consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert’s program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen’s second proof can be finitistically justified. In particular, the focus is on Takeuti’s purportedly finitistically acceptable proof of the well ordering of ordinal notations in Cantor normal form.The paper begins with a historically informed discussion of finitism and its (...) limits, before introducing Gentzen and Takeuti’s respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti’s proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti’s proof, and therefore Gentzen’s proof, conforms to. (shrink)

hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as (...) part of Hilberts Programme. According to Hilberts Programme one could provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics. (shrink)

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse (...) Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. (shrink)

While the ontological arguments of Anselm and Descartes continue to be the source of controversy among philosophers and theologians in the West, scant attention has been paid to the ontological argument first formulated by Ibn Sina (370/980 - 429/1037), and thereafter reformulated by various Muslim philosophers throughout the centuries up to the present day. Here several versions of the argument will be presented in historical sequence, and some of the most important recent discussions of the argument by contemporary Muslim (...) philosophers will be mentioned. Some reflections on the argument and the discussions of it will then be presented along with comments of a comparative nature regarding contemporary Islamic philosophy and Western philosophy of religion. (shrink)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book (...) opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

The Legacy of Aristotelian Enthymeme provides a historical-logical analysis of Aristotle's rhetorical syllogism, the enthymeme, through its Medieval and Renaissance interpretations. Bringing together notions of credibility and proof, an international team of scholars highlight the fierce debates around this form of argumentation during two key periods for Aristotle's beliefs.Reflecting on medieval and humanist thinkers, philosophers, poets and theologians, this volume joins up dialectical and rhetorical argumentation as key to the enthymeme's interpretation and shows how the enthymeme was the (...) source of a major interpretive conflict. As a method for achieving the standards for proof and credibility that persist across diverse fields of study today including the law, politics, medicine and morality, this book takes in Latin and Persian interpretations of the enthymeme and casts contemporary argumentation in a new historical light. (shrink)

A solid grasp of the main themes and arguments of the seventeenth century philosopher Rene Descartes is an essential tool towards understanding modern thought, and a necessary entree to the work of the empiricists and Immanuel Kant, and to the study of contemporary epistemology and philosophy of mind. Clear and accessible, this book serves as an introduction to Descartes's ideas for undergraduates and as a sophisticated companion to his Meditations for more advanced readers. After a thorough discussion of the main (...) themes and arguments of the Meditations, the historical background of the work, and its critical reception, the author offers his own reflections on Cartesian doubt, the cogito, the causal and ontological proofs of God's existence, the Cartesian circle, Cartesian dualism, and Descartes' views of the material world. The commentary includes and cross-references the full text of Meditations I, II, and V, and most of Meditations III and VI, employing John Cottingham's highly-praised translation. (shrink)

A survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937–1939, and later testimony of Goodstein and Kreisel; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g., Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of Tarski's (...) semantical work, both for our understanding of Wittgenstein's remarks on Gödel, and our understanding of Gödel's theorem itself. (shrink)

Historical epistemology is a form of intellectual history focused on “the history of categories that structure our thought, pattern our arguments and proofs, and certify our standards for explanation”. Under this umbrella, historians have been studying the changing meanings of “objectivity,” “impartiality,” “curiosity,” and other virtues believed to be conducive to good scholarship. While endorsing this historicization of virtues and their corresponding vices, the present article argues that the meaning and relative importance of these virtues and vices can only (...) be determined if their mutual dependencies are taken into account. Drawing on a detailed case study—a controversy that erupted among nineteenth-century orientalists over the publication of R. P. A. Dozy'sDe Israëlieten te Mekka —the paper shows that nineteenth-century orientalists were careful to examine the degree to which Dozy practiced the virtues they considered most important, the extent to which these virtues were kept in balance by other ones, the extent to which these virtues were balanced by other scholars’ virtues, and the extent to which they were expected to be balanced by future scholars’ work. Consequently, this article argues that historical epistemology might want to abandon its single-virtue focus in order to allow balances, hierarchies, and other dependency relations between virtues and vices to move to the center of attention. (shrink)

This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew (...) how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof. (shrink)

An exploration of mathematical style through 99 different proofs of the same theorem This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and (...) practice of mathematics. Inspired by the experiments of the Paris-based writing group known as the Oulipo—whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp—Ording explores new ways to examine the aesthetic possibilities of mathematical activity. 99 Variations on a Proof is a mathematical take on Queneau’s Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor. Readers will gain not only a bird’s-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape. (shrink)

The most fundamental subject and aim of the Islamic belief system is the subject of maʿrifatullah (knowing Allah). Studies on this subject are mostly called ithbāt al-wājib (the demonstration of God) in the literature. They are considered the most valuable work for kalām, philosophy and mysticism schools. Kalām schools started to use this conceptualization intensively after Fakhr al-Dīn al-Rāzī, mainly under the influence of Ibn Sīnā. Sūfis, on the other hand, most participated in these studies based on the theory of (...) the unity of existence. In kalām, all the principles of belief are ultimately grounded to explain and strengthen faith in Allah. For this reason, belief in Allah is referred to as aṣl al-uṣūl (the origin of all principles). Throughout history, scholars of kalām discussed the creedal issues of Islam in the Muslim society while it tried to answer the creedal issues abroad by the expanding Islamic geography with studies on ithbāt al-wājib. In this framework, the problems of prophecy and the hereafter are included in kalām. Since the beginning of the history of kalām, it has used the theory of atomism to a large extent, especially in defending the existence of Allah against materialists. The systematisation of Islamic philosophy and mysticism in time, the kalām cosmological argument, the argument from contingency of philosophers and the argument from kashf and ilhām of Sūfis strengthened this area from three different branches. On the other hand, the interaction of these schools within themselves and the diversification of the interlocutors of studies on ithbāt al-wājib brought these arguments closer together. The convergence in terms of terminology, subject and methodology, especially after Shahrestānī and Rāzī, revealed the tradition of philosophical theology. In this process, theologians began to use both the concepts of temporality (hudūth) and contingency (imkān) to use ontological, cosmological and teleological proofs. Dawānī, one of the eminent figures of these studies in the last periods of Ottomans, is the subject of our research, due to his skills in utilising these proofs of ithbāt al-wājib and his influence in the studies after him. His review of the matter can be observed in his effort in his works where he deals with the so-called deficiencies and errors of his predecessors regarding ithbāt al-wājib. For this reason, many annotations and commentaries were made for his treatises and ideas until the 19th century. Especially, the advance of materialism in the West in this century showed great influence on many scholars and intellectuals in the Ottoman Empire seems to be a parallel historical experience. Therefore, it has become necessary to utilise the arguments for the existence of Allah by supporting each individual argument together. In this regard, in our study, to see the process, we also discussed the views of Aḥmad Nūrī, one of the scholars in the last period Ottoman, who dealt with the common arguments about ithbāt al-wājib produced by the three primary schools of Islamic thought: kalām, Islamic philosophy and mysticism. (shrink)

The paper expresses appreciation for Williams's fine study, which restores Hegel's lectures on the philosophy of religion and on the proofs of the existence of God to a central place in his system, and rejects the anti-metaphysical reading of Hegel that is regnant today. The paper attempts to show how the proofs are co-constitutive and self-supporting. It demonstrates the importance to Hegel of both the concrete historical "this" and the community of faith. It ends with reflections on Hegel's lectures (...) on the philosophy of world history, which offer an interpretation as to how God is efficaciously present in history without violating the fabric of history. (shrink)

This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders the proof of the so–called fixed point theorem transparent. We also point out various historical details and make some observations on circularity and some comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motive for arithmetization and what it should accomplish are made obvious. We suggest this as (...) a way to teach the incompleteness results to students that have had a basic course in logic, which is more efficient than the standard textbooks. For the sake of self–containment Cantor’s original diagonalization is included. A broader and more technical perspective on diagonalization is given in [Gaifman 2005]. Motivated partly by didactic considerations, the present paper presents things somewhat differently. It also includes various points concerning natural language and circularity that appear only here. (shrink)

The paper covers a topic that sits between theology and philosophy of language and is based on completely unpublished material. The bulk of the paper consists in the critical edition and annotated translation of a section of an unpublished and anonymous commentary on the _Ratnatrayaparīkṣā_ by Śrīkaṇṭha. This section describes the transition of the indeterminate knowledge to the determined one according to the early Śaiva Siddhānta perspective. The introduction contains parts that are more “philological” or “historical” and others that (...) are more “philosophical”. The paper contains one appendix on the significant variant readings of the _mūla_ text. (shrink)

From Kant’s first published work to recent articles in the physics literature, philosophers and physicists have long sought an answer to the question, why does space have three dimensions. In this paper, I will flesh out Kant’s claim with a brief detour through Gauss’ law. I then describe Büchel’s version of the common argument that stable orbits are possible only if space is three-dimensional. After examining objections by Russell and van Fraassen, I develop three original criticisms of my own. These (...) criticisms are relevant to both historical and contemporary proofs of the dimensionality of space (in particular, a recent one by Burgbacher, F. Lämmerzahl, C., and Macias). In general I argue that modern “proofs” of the dimensionality of space have gone off track. (shrink)

We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.

Ever since Greek antiquity "disembodied knowledge" has often been taken as synonymous with "objective truth." Yet we also have very specific mental images of the kinds of bodies that house great minds--the ascetic philosopher versus the hearty surgeon, for example. Does truth have anything to do with the belly? What difference does it make to the pursuit of knowledge whether Einstein rode a bicycle, Russell was randy, or Darwin flatulent? Bringing body and knowledge into such intimate contact is occasionally seen (...) as funny, sometimes as enraging, and more often just as pointless. Vividly written and well illustrated, Science Incarnate offers concrete historical answers to such skeptical questions about the relationships between body, mind, and knowledge. Focusing on the seventeenth century to the present, Science Incarnate explores how intellectuals sought to establish the value and authority of their ideas through public displays of their private ways of life. Patterns of eating, sleeping, exercising, being ill, and having (or avoiding) sex, as well as the marks of gender and bodily form, were proof of the presence or absence of intellectual virtue, integrity, skill, and authority. Intellectuals examined in detail include René Descartes, Isaac Newton, Charles Darwin, and Ada Lovelace. Science Incarnate is at once very funny and deeply serious, addressing issues of crucial importance to present-day discussions about the nature of knowledge and how it is produced. It incorporates much that will interest cultural and social historians, historians of science and medicine, philosophers, sociologists, and anthropologists. (shrink)

Nyāya, which is one of the orthodox Brahmanical schools in India, accepts the authority of both the Vedic scriptures and God as its composer. Nyāya has specialized in logic and argumentation from ancient times while at the same time gradually strengthening its theistic tendency. Nyāya polemicist, Udayana, is famous for his contribution to the rational proof of the existence of God. In this paper, I will consider a tiny part of his proof of the existence of God given (...) in his theistic monograph, the Nyāyakusumāñjali, and in particular a topic called prāmāṇyavāda, or the Theory of Validity, from the viewpoint of the historical development of this argument. In this topic, it is argued how we justify our cognition and undertake actions. Nyāya polemicists preceding Udayana argued this topic in order to justify the Vedic scriptures and to encourage people to perform the Vedic rituals. And this topic has had little relationship with theological context. However, Udayana seems to link this argument to his theory of theology in an implicit way. I will suppose in his argument the implicit linkage between the prāmāṇyavāda and the assumption of an omniscient being. (shrink)

If one of Gentzen’s consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert’s program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen’s second proof can be finitistically justified. In particular, the focus is on Takeuti’s purportedly finitistically acceptable proof of the well ordering of ordinal notations in Cantor normal form.The paper begins with a historically informed discussion of finitism and its (...) limits, before introducing Gentzen and Takeuti’s respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti’s proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti’s proof, and therefore Gentzen’s proof, conforms to. (shrink)

It should be evident from the foregoing discussion that one man's natural selection is not necessarily the same as another man's. Why should this be so? How can two theories, which both Matthew and Darwin believed to be nearly identical, be so dissimilar? Apparently, neither Matthew nor Darwin understood the other's theory. Each man's viewpoint was colored by his own intellectual background and philosophical assumptions, and each read these into the other's ideas. The words sounded the same, so they assumed (...) the concepts must als be the same.123As Ghiselin has pointed out, historians attempting to evaluate Darwin's predecessors have been similarly blinded by a preoccupation with words, without regard to their proper context.124 In the case of Matthew, the practice of quoting only brief passages from the appendix to Naval Timber and Arboriculture, without relating them to the rest of his work, has suggested a greater resemblance to Darwin's theory than actually exists.It is clear, both from the use which Matthew made of his ideas and from the philosophical roots of his natural world view, that he could not have arrived at the concept of natural selection by the same thought process which Darwin employed. His discussion of natural selection is presented not as an argument, but as an axiom. No theory is proposed, no evidence marshaled to support it. Natural selection is stated as a fact, a Law of Nature, unquestioned, and presumably, unquestionable.Despite his clamor for recognition as the discoverer of natural selection, Matthew recognized and acknowledged this very fundamental difference between Darwin and himself. In a letter to the Gardener's Chronicle of May 12, 1860, he wrote:To me the conception of this law of Nature came intuitively as a self-evident fact, almost without an effort of concentrated thought. Mr. Darwin here seems to have more merit in the discovery than I have had—to me it did not appear a discovery. He seems to have worked it out by inductive reason, slowly and with due caution to have made his way synthetically from fact to fact onwards; while with me it was by a general glance at the scheme of Nature that I estimated this select production of species as an a priori recognisable fact—an axiom, requiring only to be pointed out to be admitted by unprejudiced minds of sufficient grasp.125In the same letter, Matthew maintained that his ideas had not been accepted because “the age was not ripe for such ideas.”126 Nor, he said, was the present age. He considered the inability of most of Darwin's critics to grasp his theory to be “incurable.” Yet he did not argue that natural selection should be accepted because of the evidence, but rather, that it should be accepted on faith:Belief here requires a certain grasp of mind. No direct proof of phenomena embracing so long a period of time is within the compass of short-lived man. To attempt to satisfy a school of ultra skeptics, who have a wonderfully limited power of perception of means to ends... would be labour in vain.... They could not be brought to conceive the purpose of a handsaw though they saw its action, if the whole individual building it assisted to construct were not presented complete before their eyes... Like a child looking upon the motion of a wheel in an engine they would only perceive and admire... without noticing its agency in... affecting the purposed end.127Here, then, is the final irony. In a passage urging acceptance of Darwin's theory, a theory which was to banish design and purpose from the natural world, we find echoes of Paley and of Providence.Loren Eiseley has lamented the fact that Matthew “did not bring his views into the open, because the amount of ground he was able to cover in a few paragraphs suggests that he might have been able to sustain a longer treatise.”128 Now that the intellectual and historical context of Matthew's ideas are known, this statement is no longer tenable. Matthew was not a scientist, and his books were not written as biological treatises. His discussions of natural selection were not attempts to “cover ground” in advancing a particular scientific theory, but were simply reflections of his own assumptions about the natural world.Furthermore, despite Matthew's acceptance of evolution and natural selection, his biological thought was basically conservative on points where Darwin's was radical. Where Matthew saw a series of stable worlds interrupted by violent upheavals, Darwin saw a continuous process of change in an ever-fluctuating world. Where Matthew conceived of species in terms of Aristotelian classes and essences, Darwin revolutionized our concept of species by treating them as populations. Where Matthew saw a world of design and beauty functioning according to natural laws laid down by benevolent Providence, Darwin abolished design and Providence from nature and ushered in a world which cycles ever onward according to laws of chance and probability.It is not even particularly useful to point to Matthew as evidence that evolution was “in the air” prior to 1859.129 His ideas did not represent the first wave of a coming revolution, but were the product of his own personal philosophical outlook, as expressed in the context of the biological thought of the 1830's. Matthew is important in the history of ideas, not simply because he accepted the concept of evolution or thought of something resembling natural selection, but because he did so without overthrowing, in his own mind, any of the basic philosophical assumptions which had underlain biological science since Aristotle. In recognizing Matthew's failure to do so, we are in a position to appreciate more fully the significance of the Darwinian Revolution. (shrink)

AS AGAINST HUME’S VIEW THAT "A MIRACLE CAN NEVER BE PROVED SO AS TO BE THE FOUNDATION OF A SYSTEM OF RELIGION" I ARGUE THAT THE POSSIBILITY OF MIRACLES CAN BE DEFENDED ON PHILOSOPHICAL GROUNDS, THAT THERE IS HISTORICAL EVIDENCE FOR THE OCCURRENCE OF CERTAIN MIRACLES AND THAT SUCH MIRACLES CAN IN FACT GIVE GROUNDS FOR THE PREFERENCE OF ONE SYSTEM OF RELIGIOUS BELIEF OVER ANOTHER.

The essay collection "Kant on Proofs for God's Existence" provides a highly needed, comprehensive analysis of the radical turns of Kant's views on proofs for God's existence.— In the "Theory of Heavens" (1755), Kant intends to harmonize the Newtonian laws of motion with a physico-theological argument for the existence of God. But only a few years later, in the "Ground of Proof" essay (1763), Kant defends an ontological ('possibility' or 'modal') argument on the basis of its logical exactitude while (...) he praises the physico-theological argument for its beauty and appeal to the common sense. In the first "Critique" (1781/7), Kant replaces traditional constitutive ontological, cosmological, and physico-theological proofs with his own regulative theoretical and moral-practical religious arguments. He continues to defend a moral argument in the second "Critique" (1788). But in the third "Critique" (1790), Kant reintroduces a physico-theological besides an ethicotheological argument in order to unify the critical system of philosophy. Kant develops further moral arguments and arguments from evil in the "Theodicy" essay (1791) and the "Religion" (1793/4), and still searches for the right kind of proof for God's existence in the "Opus postumum" (1796–1804).—Part one of this volume is dedicated to an analysis of Kant's proofs for God's existence in their historical order that explains which proofs Kant favors or rejects in various periods of his thought. Part two contains a systematic classification of main kinds of proof for God's existence in Kant that outlines the argumentative structure of particular kinds of proof and discusses Kant's potential reasons for their variations and modifications. The essay collection speaks to Kant specialists, philosophers, and theologians, but introduces the topic to non-academic readers also. -/- Contributers: Uygar Abaci (Pennsylvania State University), Craig Bacon (University of South Carolina), Andrew Chignell (Princeton University), Eckart Förster (Humboldt University, Johns Hopkins University), Ina Goy (Beijing Normal University) Paul Guyer (Brown University), Graham Oppy (Monash University), Lara Ostaric (Temple University), Stephen Palmquist (Independent Scholar), Lawrence Pasternack (Oklahoma State University), Konstantin Pollok (University of Mainz), Oliver Sensen (University of Tulane), Allen Wood (Bloomington University, Stanford University). (shrink)

_ Source: _Volume 35, Issue 1, pp 66 - 74 Context and argumentative style of Grotius’s De veritate are that of Reformation controversialist theology and of humanist historical notions of truth. Controversialism, however, no longer operated from shared principles, and the textual criticism of humanist scholarship implied looking at the book of revelation as an historical document, in a double sense: a product of history, and historical narratives. To what intellectual juggling this leads Grotius, is evident in (...) his considering the historical proofs of the resurrection of Christ, the role therein of witnesses and of pagan historical support. Justifying Christ’s resurrection with pagan theories of metempsychosis, however, was another step towards a rational justification of Christianity. His defence of a minimalist doctrinal content was not yet deist or neological, as his reliance on miracles and resurrection demonstrate. One thus might locate this text at the ‘false dawn’ of the Enlightenment, as a comparison with Edward Lord Herbert of Cherbury’s homonymous book, abstract and general enough to become a really Deist manifesto, finally shows. (shrink)