This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form of the symbolic language of algebra. Thus the (...) paper develops further the method of reconstruction which the author introduced for the analysis of the development of geometry. (shrink)

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, (...) arithmetic would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, (...) arithmetic would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

In the early 1930s W. O. Kermack and W. H. McCrea published three papers in which they attempted to prove a result of E. T. Whittaker on the solution of differential equations. In modern parlance, their key idea consisted in using quantized contact transformations over an algebra of differential operators. Although their papers do not seem to have had any impact, either then or at any later time, the same ideas were independently developed in the 1960–1980s in the framework (...) of the theory of modules over rings of microdifferential operators. In this paper we describe the results of Kermack and McCrea and discuss possible reasons why such promising papers had no impact on the mathematics of the twentieth century. (shrink)

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the early (...) development of algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that (...) are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory. This book is Open Access under a CC BY licence. (shrink)

The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character (...) of algebra. (shrink)

What was the impact of Lavoisier's new elementary chemical analysis on the conception and practice of chemistry in the vegetable kingdom at the end of the eighteenth century? I examine how this elementary analysis relates both to more traditional plant analysis and to philosophical and mathematical concepts of analysis current in the Enlightenment. Thus I explore the relationship between algebra, Condillac's philosophy and Lavoisier's chemical system, as well as comparing Lavoisier's analytical approach to those of (...) his predecessors, such as Baumé and Bucquet. With reference to the aims of vegetable analysis, I show how the dominance of elementary analysis devalued a tradition that sought to isolate immediate principles , marginalizing the chemical practices of many doctors and pharmacists in the context of the new chemistry in France. (shrink)

The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with (...) the related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)

Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyse entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter's ability (...) to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard non-relativistic quantum theory and classical relativistic field theory. (shrink)

The book is a collection of the author¿s selected works in the philosophy and history of logic and mathematics. Papers in Part I include both general surveys of contemporary philosophy of mathematics as well as studies devoted to specialized topics, like Cantor's philosophy of set theory, the Church thesis and its epistemological status, the history of the philosophical background of the concept of number, the structuralist epistemology of mathematics and the phenomenological philosophy of mathematics. Part II contains essays (...) in the history of logic and mathematics. They address such issues as the philosophical background of the development of symbolism in mathematical logic, Giuseppe Peano and his role in the creation of contemporary logical symbolism, Emil L. Post's works in mathematical logic and recursion theory, the formalist school in the foundations of mathematics and the algebra of logic in England in the 19th century. The history of mathematics and logic in Poland is also considered. This volume is of interest to historians and philosophers of science and mathematics as well as to logicians and mathematicians interested in the philosophy and history of their fields. (shrink)

Recently the interest in Bohm realist interpretation of quantum mechanics has grown. The important advantage of this approach lies in the possibility to introduce non-locality ab initio, and not as an "unexpected host". In this book the authors give a detailed analysis of quantum potential, the non-locality term and its role in quantum cosmology and information. The different approaches to the quantum potential are analysed, starting from the original attempt to introduce a realism of particles trajectories (influenced by de (...) Broglie's pilot wave) to the recent dynamic interpretation provided by Goldstein, Durr, Tumulka and Zanghì, and the geometrodynamic picture, with suggestion about quantum gravity. Finally we focus on the algebraic reading of Hiley and Birkbeck school, that analyse the meaning of the non-local structure of the world, bringing important consequences for the space, time and information concepts. (shrink)

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of (...) Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883?1884 in connection with the composition of Peirce's ?On the Algebra of Logic: A Contribution to the Philosophy of Notation? that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional. (shrink)

In a paper read before the last Congress for the Unity of Science, Dr. Milton Singer distinguishes three main phases in the recent history of logic. The achievement he considers most characteristic of the first period is the development of the class calculus or so-called Boolean algebra. It begins with the work of Boole and DeMorgan and culminates in Schroeder's Algebra of Logic. In a minimum formulation, the results of this first stage can be summed up as, (...) first, simplification and generalization of the traditional syllogistic machinery by means of a rather efficient new symbolism, and, second, adoption of the axiomatic method. To restate the last point: Boolean algebra was, since Leibniz, the first explicit attempt to construct what we call a formal system or calculus. Not realized at that time, however, or at least not adequately expressed, was the importance of the more recent distinction between scientific calculi on the one hand and purely linguistic or logical calculi on the other. As a consequence of this lack of explicit insight, the class calculus appeared to be on a par with scientific calculi, say, for instance, axiomatized geometry, Euclidean or otherwise. Its analytical character, even if it was seen, could not be formulated. (shrink)

De Morgan's Formal Logic, which was published on virtually the same day in 1847 as Boole's The Mathematical Analysis of Logic, contains a logic of complex terms (LCT) which has been sadly neglected. It is surprising to find that LCT contains almost a full theory of Boolean algebra. This paper will: (1) provide some background to LCT; (2) outline its main features; (3) point out some gaps in it; (4) compare it with Boole's algebra; (5) show that (...) it is a lattice-theoretical formulation of Boolean algebra; (6) discuss some issues of historical priority; and (7) conclude with the puzzle of LCT's lack of influence. (shrink)

In Plato's Parmenides we find on the one hand that the One is denied every property , and on the other hand that the One is attributed every property . In the course of the history of Platonism , these assertions - probably meant by Plato as ontological statements of an entirely formal nature - were repeatedly made the starting points of metaphysical speculations. In the Mystical Theology of the Pseudo-Dionysius they became principles of Christian mysticism and negative theology. (...) I shall show that the two assertions can each be interpreted within the ontological framework of ancient and medieval logic in such a manner that it becomes true, and I shall make plausible that they were understood, with regard to their logical core, by pagan and Christian Platonic metaphysicians just as is indicated by that interpretation. The mentioned ontological framework is basically the Boolean algebra of first-order properties. The main points of the interpretation are on the one hand the identification of the One with the maximal element of the algebra of properties, and on the other hand two alternative intuitively prominent mereological definitions of ontic predication. (shrink)

François Viète is considered the father both of modern algebra and of modern cryptanalysis. The paper outlines Viète’s major contributions in these two mathematical fields and argues that, despite an obvious parallel between them, there is an essential difference. Viète’s ‘new algebra’ relies on his reform of the classical method of analysis and synthesis, in particular on a new conception of analysis and the introduction of a new formalism. The procedures he suggests to decrypt coded messages (...) are particular forms of analysis based on the use of formal methods. However, Viète’s algebraic analysis is not an analysis in the same sense as his cryptanalysis is. In Aristotelian terms, the first is a form of 'analysis,' while the second is a form of 'diaresis.' While the first is a top-down argument from the point of view of the human subject, since it is an argument going from what is not actual to what is actual for such a subject, the second one is a bottom-up argument from this same point of view, since it starts from what is first for us and proceed towards what is first by nature.Keywords: Analysis; Cryptanalysis; Algebra; Aristotle; Viète. (shrink)

It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental (...) propositions of his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries. (shrink)

J. S. Silverberg, The Most Obscure and Inconvenient Tables ever Constructed.- D. J. Melville, Commercializing Arithmetic: The Case of Edward Hatton.- C. Baltus, Leading to Poncelet: A Story of Collinear Points.- R. Godard, Cauchy, Le Verrier et Jacobi sur le problème algébrique des valeurs propres et les inégalités séculaires des mouvements des planètes.- A. Ackerberg-Hastings, Mathematics in Astronomy at Harvard College Before 1839 as a Case Study for Teaching Historical Writing in Mathematics Courses.- J. J. Tattersall, S. L. McMurran, "Lectures (...) for Women" and the Founding of Newnham College, Cambridge.- D. Waszek, Are Euclid's Diagrams "Representations"? On an Argument by Ken Manders.- B. Buldt, Abstraction by Embedding and Constraint-Based Design.- W. Meyer, The Birth of Undergraduate Modern Algebra in the United States.- P. Liu, History as a Source of Mathematical Narrative in Developing Students' Interpretations of Mathematics.- F. Kamareddine, J. P. Seldin, Thoughts on Using the History of Mathematics to Teach the Foundations of Mathematical Analysis. (shrink)

Abraham Robinson has twice been the initiator of trends in the foundations of mathematics which have later become recognized as profound and important, although they were generally ignored at first: metamathematical problems of algebra and non-standard analysis. This book considers the second topic; before we begin analysis—especially a treatment of the classical theorems of calculus—we need basic results from logic, model theory in particular. Robinson then sketches non-standard arithmetic and proceeds to develop the usual properties and relations (...) of differentiability and integrability of functions, convergence; he then examines general topology in this new context and fortified with these results develops "modern" functional analysis in some detail: measure theory and functions of a real variable, functions of a complex variable, normed and linear spaces, topological groups and some harmonic analysis. The last two chapters cover selected topics in analysis drawn principally from applied mathematics and the history of the calculus with special reference to infinitesimals. Robinson views the development of non-standard mathematics as the renaissance of ideas about infinitesimals as serious topics of mathematical study. Robinson primarily has confined himself to analysis and not attempted to cover non-standard algebraic systems; several important results of Alling concerning non-standard arithmetic and rings of continuous functions appeared too late to be included. Of course, the real test of non-standard analysis will come in allowing us to prove new theorems about standard analysis, but although Robinson does not attempt to do this here—although he has obtained results in this direction—other mathematicians have already begun to do so: the future of this new branch of mathematics already looks secure.—P. J. M. (shrink)

The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are (...) typically allowed only with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ?given?, that is, known or ?data?, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ?unknown? objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky. (shrink)

In the framework of the De Regula Aliza, Cardano paid much attention to the so-called splittings for the family of equations $$x^3 = a_1x + a_0$$ x3=a1x+a0 ; my previous article deals at length with them and, especially, with their role in the Ars Magna in relation to the solution methods for cubic equations. Significantly, the method of the splittings in the De Regula Aliza helps to account for how Cardano dealt with equations, which cannot be inferred from his other (...) algebraic treatises. In the present paper, this topic is further developed, the focus now being directed to the origins of the splittings. First, we investigate Cardano’s research in the Ars Magna Arithmeticae on the shapes for irrational solutions of cubic equations with rational coefficients and on the general shapes for the solutions of any cubic equation. It turns out that these inquiries pre-exist Cardano’s research on substitutions and cubic formulae, which will later be the privileged methods for dealing with cubic equations; at an earlier time, Cardano had hoped to gather information on the general case by exploiting analogies with the particular case of irrational solutions. Accordingly, the Ars Magna Arithmeticae is revealed to be truly a treatise on the shapes of solutions of cubic equations. Afterwards, we consider the temporary patch given by Cardano in the Ars Magna to overcome the problem entailed by the casus irreducibilis as it emerges once the complete picture of the solution methods for all families of cubic equations has been outlined. When Cardano had to face the difficulty that appears if one deals with cubic equations using the brand-new methods of substitutions and cubic formulae, he reverted back to the well-known inquiries on the shape of solutions. In this way, the relation between the splittings and the older inquiries on the shape of solutions comes to light; furthermore, this enables the splittings to be dated 1542 or later. The last section of the present paper then expounds the passages from the Aliza that allow us to trace back the origins of the substitution $$x = y + z$$ x=y+z, which is fundamental not only to the method of the splittings but also to the discovery of the cubic formulae. In this way, an insight into Cardano’s way of dealing with equations using quadratic irrational numbers and other selected kinds of binomials and trinomials will be provided; moreover, this will display the role of the analysis of the shapes of solutions in the framework of Cardano’s algebraic works. (shrink)

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In (...) order to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Originality of Descartes’s Conception of Analysis as DiscoveryBenoît TimmermansAccording to Descartes, his Meditations employ the method of analysis. This method of proof, says Descartes, “shows the true way by means of which the thing in question was discovered methodically and as it were a priori.” 1 Such a definition of analysis poses a problem that seems to have attracted little attention among commentators until now, (...) namely, why Descartes considers analysis a method of discovery whereas the Scholastic tradition asserts just the contrary, that synthesis allows the discovery of new things, whereas analysis is a method for judging findings or verifying them a posteriori. What is more, the definition that Descartes gives of his analysis contains another divergence, this time brought up by the commentators, 2 for he curiously considers analysis an a priori movement and not an a posteriori movement as dictated by tradition.These issues are all the more important as the analytical method acquired particular prestige in the seventeenth century. This prestige concerned not only philosophy (Descartes, Leibniz, Locke, etc.) but science as well. Thus algebra lost some of its rough edges and gradually took on the name of “analysis.” 3 Descartes created a type of geometry that was soon dubbed “analytic,” and Leibniz developed his analysis infinitorum or differential and integral calculus. If the scientific revolution and the new philosophies that were associated with it attached so much importance to a method that had already been known in ancient Greece, was it not due to a change of the meaning of analysis or at least a rearranging of some of its functions? This is the question that we propose to [End Page 433] tackle while trying to determine to what extent Cartesian analysis breaks with the forms of analysis that preceded it. 4To do this we shall first examine the teachings of Scholastic philosophy concerning the connections between analysis and discovery in general and then study the logical possibilities of discovery through analysis that draw their inspiration from Aristotle’s Analytics and that were developed extensively by the School of Padua in the sixteenth century. Finally, we shall attempt to determine whether the mathematical definition of analysis derived from the works of Euclid and Pappus gives grounds for the correlation with discovery.The Link Between Analysis and Discovery According to the ScholasticsOne might well think that Descartes was completely unaware of the contradiction between his definition of analysis and the teachings of the Scholastics. Had not he always exhibited a true disdain for the latter? However, in September 1640, that is, four months before writing his text on analysis, Descartes began girding for the Jesuit Fathers’ objections to his Meditations. This involved looking for an “abstract of the whole of scholastic philosophy, this would save the time it would take to read their huge tomes.” 5 He finally found what he was looking for in the shops of Leiden, namely, Eustachius a Sancto Paulo’s Summa Philosophiae. There is reason to believe that Descartes read this opus with interest, for two months later he referred to it as “the best book of its kind ever made.” 6 Later he said that he would definitely have chosen the Summa Philosophiae as the basis for refuting the School’s doctrine if he had not given up such a project. 7 Now, Eustachius a Sancto Paulo’s Summa Philosophiae is categorical on the subject of analysis versus synthesis. Analysis of a definition “is perfectly compatible with the doctrines for teaching and learning; as for the other disciplines of discovery, the other method, that is the composite or synthetic method, must be used.” 8According to the Scholastic doctrine as stated by Eustachius a Sancto Paulo, analysis thus plays a part in the teaching of well-known things and in all processes of appraisal, assessment, or judgment. This is easy to explain if one considers that, etymologically, analysis breaks links (ana-luein), goes backward, or climbs back up from the consequences to the principles. Starting from an observation, proposition, or question, it wends its way back to the simplest, [End Page 434] surest items of knowledge that shed light on and verify said starting... (shrink)

The narrative about the nineteenth century favored by many philosophers of mathematics strongly influenced by either logic or algebra, is that geometric intuition led real and complex analysis astray until Cauchy and Kronecker in one sense and Dedekind in another guided mathematicians out of the labyrinth through the arithmetization of analysis. Yet the use of geometry in most cases in nineteenth century mathematics was not misleading and was often key to important developments. Thus the geometrization of complex (...) numbers was essential to their acceptance and to the development of complex analysis; geometry provided the canonical examples that led to the formulation of group theory; and geometry, transformed by Riemann, lay at the heart of topology, which in turn transformed much of modern mathematics. Using complex numbers as my case study, I argue that the best way to teach students mathematics is through a repertoire of modes of representation, which is also the best way to make mathematical discoveries. (shrink)

The history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, (...) although logic is evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart). In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided: 1. What were the reasons for the philosophers' lack of interest in formal logic? 2. What were the reasons for the mathematicians' interest in logic? 3. What did "logic reform" mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic? 4. Was mathematical logic regarded as art, as science or as both? (shrink)

The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an (...) class='Hi'>analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal κ > א1, then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive. (shrink)

The concept of scientific history / Isaiah Berlin -- The limits of scientific history / W.H. Walsh -- The objectivity of history / J.A. Passmore -- Explanation in science and in history / C.G. Hempel -- The Popper-Hempel theory reconsidered / Alan Donagan -- The autonomy of historical understanding / Louis O. Mink -- Historical continuity and causal analysis / Michael Oakeshott -- Causal judgment in history and in the law / H.L.A. Hart and (...) A.M. Honoré -- Causes, connections and conditions in history / Michael Scriven -- The historical individual / A.C. Danto -- Methodological individualisms : definition and reduction / May Brodbeck -- Societal laws / Maurice Mandelbaum -- Determinism in history / Ernest Nagel. (shrink)

There is a paradox lying at the heart of the study of heredity. To understand the ways in which features are passed down from one generation to the next, we have to dig deeper and deeper into the ultimate nature of things - from organisms, to genes, to molecules. And yet as we do this, increasingly we find we are out of focus with our subjects. What has any of this to do with the living, breathing organisms with which we (...) started? Organisms are living. Molecules are not. How do we relate one to the other? In Genetic Analysis, one of the most important empirical scientists in the field in the twentieth century attempts, through a study of history and drawing on his own vast experience as a practitioner, to face this paradox head-on. His book offers a deep and innovative understanding of our ways of thinking about heredity. (shrink)

The “centre-periphery” relationship historically structured scientific exchanges between metropolis and province, between the fount of empire and its outposts. But the exchange, if regarded merely as a one-way flow of scientific information, ignores both the politics of knowledge and the nature of its appropriation. Arguably, imperial structures do not entirely determine scientific practices and the exchange of knowledge. Several factors neutralise the over-determining influence of politics—and possibly also the normative values of science—on scientific practice.In examining these four examples of Indian (...) scientists in encounters with their peers at the centre, exceptional scientists are seen in a social context where the epistemology of science supposedly describes its practice. Imperialism imposes practices and patronage, which moderate the exchange of scientific knowledge. But, at Level Two, the politics of knowledge and the patterns of patronage within it mediate exchanges between the centre and the periphery.The first step in reconfiguring exchanges between centre and periphery —in this case, between Europe and India during the period 1850 to 1930— is to recognise the relation between the acquisition of resources and the maintenance of legitimacy and identity.67 Political life is not confined to the core of political institutions.68 Second, in examining science as practised in the colonies, it is necessary to see stages of scientific institutions, whose development structures the exchange.From the encounter of Ramchandra and De Morgan, it is evident that the centre-periphery framework should be separated from the models of transmission embedded within it. The notion of “translation” helps to suggest that scientists bring personal motives and meanings to each encounter. Ramchandra, for example, sought a novel method of teaching Indians calculus, while De Morgan's interest lay in finding a place for algebra in a liberal education.The hierarchy inherent in the centre-periphery framework compels the conclusion that, at Level Two, the autodidact outside the institutions of science must have his work presented to scientists at the centre by authoritative figures from the centre. This is not mainly a question of imperialism, but rather of patronage. The peripheral scientist could not be granted direct entry into the collegial circle until his efforts at the periphery could be translated into the language and concerns of the central community. Ramanujan's enigmatic formulas were translated into the language of analysis by Hardy, which enabled the creation of a field to which Hardy was committed. Scientists from the periphery who were already part of the circle by virtue of their training, were not necessarily subject to the same degree of attestation as other scientists from the periphery. P.C. Ray, with his DSc from Edinburgh, and his position at Calcutta University, had less difficulty in winning the trust of colleagues at the centre, even when he returned to India. On the contrary, remaining at the periphery, he moved from a context of patronage to a sphere of competition. In addition, Ray's collegiality, even at Level Two, was more comprehensive, and connected him with Level One.Eventually, the professional Indian science graduate found collegiality within the international community of scientists. Saha's self-imposed progressive nationalism constrained his identification with the centre and made him a potential competitor instead. Once having achieved eminence in the world of science, C.V. Raman and Saha shifted their work to journals of physics published in India in order to further the cause of physics research in their own country.69To go beyond the limitations of the centre-periphery model, it is necessary not merely to examine exchanges between scientists functioning in a “shared epistemological universe”,70 but also to recognise the part played by institutions, the experience of colonialism, and the forms of patronage characterising both colonialism and science. Put another way, although there is relative epistemological autonomy within the disciplinary research communities of science, the interplay between knowledge and power structures this exchange.The scientific links between colonial India and Britain at the turn of the century were mediated by structures which prefigured change. Does structure determine all? If it does, we are left with an Orientalist reconstruction of the docile native, and a passive cultural medium into which science percolates. But this neglects the role of scientists in creating new structures within which they worked. A middle position—one more sensitive to the exigencies of colonial scientific life—would be one where the participants are seen not as the dupes of “structure nor the potentates of action”, but as occupying a ground between the two.71. (shrink)

The thematization of writing has often been seen as Derrida's personal contribution to modern philosophy, but it is significant that in his earliest extended discussions of it, he presented it as a sign of the times. This chapter focuses on Derrida's discussion of writing in the first part of Of Grammatology and provides an analysis of its stakes by bringing it into conversation with Althusser's new theory of reading. Althusser was the most powerful figure in the philosophy department. Derrida's (...) first publication, the “Introduction” was a contribution to the history and philosophy of science, a fact recognized by the Société des Amis de Jean Cavaillès, which awarded it their book prize in 1964. Writing, especially mathematical writing, for instance in algebra, seemed to represent best the formal systems of mathematics. (shrink)

Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new (...) notion of realizability, namely herbrandized modified realizability. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic. (shrink)

For the Introduction of a Conceptual Perspective in Mathematics: Dedekind, Noether, van der Waerden. „She [Noether] then appeared as the creator of a new direction in algebra and became the leader, the most consistent and brilliant representative, of a particular mathematical doctrine – of all that is characterized by the term ‚Begriffliche Mathematik‘.“2 The aim of this paper is to illuminate this “new direction”, which can be characterized as a conceptual [begriffliche] perspective in mathematics, and to comprehend its roots (...) and trace its establishment. Field, ring, ideal, the core concepts of this new direction in mathematical images of knowledge, were conceptualized by Richard Dedekind (1831–1916) within the scope of his number theory research and associated with an understanding of a formation of concepts as a “free creation of the human spirit”3. They thus stand for an abstract perspective of mathematics in their entirety, described as ‘modern algebra’ in the 1920s and 1930s, leading to an understanding of mathematics as structural sciences. The establishment of this approach to mathematics, which is based on “general mathematical concepts” [allgemein‐mathematische Begriffe]4, was the success of a cultural movement whose most important protagonists included Emmy Noether (1882–1935) and her pupil Bartel L. van der Waerden (1903–1996). With the use of the term ‘conceptual’, a perspective is taken in the analysis which allows for developing connections between the thinking of Dedekind, the “working and conceptual methods” [Arbeits‐ und Auffassungsmethoden]5 of Noether as well as the methodological approach, represented through the thought space of the Noether School as presented under the term “conceptual world” [Begriffswelt]6 in the Moderne Algebra of van der Waerden. This essay thus makes a contribution to the history of the introduction of a structural perspective in mathematics, a perspective that is inseparable from the mathematical impact of Noether, her reception of the work of Dedekind and the creative strength of the Noether School. (shrink)

This paper assesses the philosophical heritage of Jacob Klein’s thought through an analysis of the key tenets of his Greek Mathematical Thought and theOrigin of Algebra. Threads of Klein’s thought are distinguished and subsequently singled out (phenomenological, epistemological, and anti-ontological; historical, ontological, and critical), and the peculiar way in which Klein’s project brings together ontology and history of mathematics is investigated. Plato’s theoretical logistic and Klein’s understanding thereof are questioned—especially the claim that the Platonic distinction between practical (...) and theoretical logistic is historically neglected because of Plato’s understanding of the manner of being of mathematical objects—in order to advance the claim that Klein ontologically overdetermines the history of mathematics in a manner that ends up limiting some of his most brilliant analyses of the Greek conception of “numbers” and the philosophical meaning of the notion of “multiplicity.”. (shrink)

This paper assesses the philosophical heritage of Jacob Klein’s thought through an analysis of the key tenets of his Greek Mathematical Thought and theOrigin of Algebra. Threads of Klein’s thought are distinguished and subsequently singled out (phenomenological, epistemological, and anti-ontological; historical, ontological, and critical), and the peculiar way in which Klein’s project brings together ontology and history of mathematics is investigated. Plato’s theoretical logistic and Klein’s understanding thereof are questioned—especially the claim that the Platonic distinction between practical (...) and theoretical logistic is historically neglected because of Plato’s understanding of the manner of being of mathematical objects—in order to advance the claim that Klein ontologically overdetermines the history of mathematics in a manner that ends up limiting some of his most brilliant analyses of the Greek conception of “numbers” and the philosophical meaning of the notion of “multiplicity.”. (shrink)

The views of some historians and philosophers of history as to the possibility of fruitful historical generalization seem at odds with the underlying methodology of the other social sciences. A formal model of the world historical process is here presented within which this apparent contradiction is seen to be resolvable in terms of modern theories of probability and stochastic processes. This is done by giving rigorous form to procedures and statements in the social sciences. A formal treatment of the (...) dependence of an investigation in one discipline on previous studies both in that area and in other social and natural sciences then follows naturally. (shrink)

For most of its history, mathematics was fairly constructive: • Euclidean geometry was based on geometric construction. • Algebra sought explicit solutions to equations. Analysis, probability, etc. were focused on calculations. Nineteenth century developments in analysis challenged this view. A sequence (an) in a metric space is said Cauchy if for every ε > 0, there is an m such that for every n, n ≥ m, d (a n , a n ) < ε.

Adriaan van Roomen published an outline of what he called a Mathesis Universalis in 1597. This earned him a well-deserved place in the history of early modern ideas about a universal mathematics which was intended to encompass both geometry and arithmetic and to provide general rules valid for operations involving numbers, geometrical magnitudes, and all other quantities amenable to measurement and calculation. ‘Mathesis Universalis’ (MU) became the most common (though not the only) term for mathematical theories developed with that (...) aim. At some time around 1600 van Roomen composed a new version of his MU, considerably different from the earlier one. This second version was never effectively published and it has not been discussed in detail in the secondary literature before. The text has, however, survived and the two versions are presented and compared in the present article. Sections 1–6 are about the first version of van Roomen’s MU the occasion of its publication (a controversy about Archimedes’ treatise on the circle, Sect. 2), its conceptual context (Sect. 3), its structure (with an overview of its definitions, axioms, and theorems) and its dependence on Clavius’ use of numbers in dealing with both rational and irrational ratios (Sect. 4), the geometrical interpretation of arithmetical operations multiplication and division (Sect. 5), and an analysis of its content in modern terms. In his second version of a MU van Roomen took algebra into account, inspired by Viète’s early treatises; he planned to publish it as part of a new edition of Al-Khwarizmi’s treatise on algebra (Sect. 7). Section 8 describes the conceptual background and the difficulties involved in the merging of algebra and geometry; Sect. 9 summarizes and analyzes the definitions, axioms and theorems of the second version, noting the differences with the first version and tracing the influence of Viète. Section 10 deals with the influence of van Roomen on later discussions of MU, and briefly sketches Descartes’ ideas about MU as expressed in the latter’s Regulae. (shrink)

In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré’s works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one (...) question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré’s works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré’s stated motivations for topology and the Poincaré conjecture. (shrink)

Thought experiments are ubiquitous in science and especially prominent in domains in which experimental and observational evidence is scarce. One such domain is the causal analysis of singular events in history. A long‐standing tradition that goes back to Max Weber addresses the issue by means of ‘what‐if’ counterfactuals. In this paper I give a descriptive account of this widely used method and argue that historians following it examine difference makers rather than causes in the philosopher’s sense. While difference (...) making is neither necessary nor sufficient for causation, to establish difference makers is more consistent with the historians’ more ultimate purposes. †To contact the author, please write to: Department of Philosophy, Erasmus University, 3000 DR Rotterdam, The Netherlands; e‐mail: [email protected]. (shrink)

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...) obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. -/- I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. -/- My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like "If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. (shrink)