In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long ‘search’ for a purely mathematical incompleteness result in first-order Peano arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form.

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic (...) way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics. (shrink)

This paper investigates a first-order extension of GL called \. We outline briefly the history that led to \, its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \ is arithmetically complete. As expanded below, \ is a first-order modal (...) logic that along with its built-in ability to simulate general classical first-order provability―"\" simulating the the informal classical "\"―is also arithmetically complete in the Solovay sense. (shrink)

This work aim to analyse relations between Arithmetic and Geometry indicated by Piero della Francesca in his treatise Libellus de quinque corporibus regularibus. Piero della Francesca was a painter and scholar of perspective, geometry and arithmetic, in his time. He carried out investigations on pictorial, geometric and architectural issues. Of the treatises he wrote, only three are preserved, on perspective, Geometry and Arithmetic. The central document selected for this research was the manuscript Libellus de quinque corporibus regularibus, (...) deposited in the Vatican Apostolic Library in digitized copy. Throughout this work we have tried to understand the Libellus de quinque corporibus regularibus as a result of the work of an artist and scholar who proposed ways of relating Arithmetic and Geometry using both his practical knowledge and the works of authors of antiquity. (shrink)

Daniel Sutherland - Kant on Arithmetic, Algebra, and the Theory of Proportions - Journal of the History of Philosophy 44:4 Journal of the History of Philosophy 44.4 533-558 Muse Search Journals This Journal Contents Kant on Arithmetic, Algebra, and the Theory of Proportions Daniel Sutherland Kant's philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account (...) of his mature views on mathematical cognition, forcing readers to glean what they can from disparate contexts. Despite these hurdles, Kant's views have been of great interest to philosophers of mathematics. They have also been of interest to philosophers wishing to understand Kant's philosophy more generally, since Kant maintains that mathematical judgments are synthetic a priori and that the type of synthesis underlying mathematics is the same as that underlying the perception of objects . Our understanding of Kant's views has been greatly improved during the last four decades by work on Kant's philosophy of mathematics. Despite these advances, however, I think the recent work has largely neglected the importance of Kant's theory of magnitudes and the extent to which that theory is influenced by the Eudoxian theory of proportions presented in Euclid's Elements. As a consequence, an important feature of Kant's.. (shrink)

There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority (...) of this attack to the units view, with particular attention to pure units. Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not. (shrink)

Love, according to the poets, is something like a math problem. Whether it is two striving to become one or the triangulating effect of three, we find a venerable history of number-crunching in the literature of love, not least in ancient Israel's great poetic presentation of desire, the Song of Songs.

In this paper some of the history of the development of arithmetic in set theory is traced, particularly with reference to the problem of avoiding the assumption of an infinite set. Although the standard method of singling out a sequence of sets to be the natural numbers goes back to Zermelo, its development was more tortuous than is generally believed. We consider the development in the light of three desiderata for a solution and argue that they can probably (...) not all be satisfied simultaneously. (shrink)

This article explores the changing relationships between geometric and arithmetic ideas in medieval Europe mathematics, as reflected via the propositions of Book II of Euclid’s Elements. Of particular interest is the way in which some medieval treatises organically incorporated into the body of arithmetic results that were formulated in Book II and originally conceived in a purely geometric context. Eventually, in the Campanus version of the Elements these results were reincorporated into the arithmetic books of the Euclidean (...) treatise. Thus, while most of the Latin versions of the Elements had duly preserved the purely geometric spirit of Euclid’s original, the specific text that played the most prominent role in the initial passage of the Elements from manuscript to print—i.e., Campanus’ version—followed a different approach. On the one hand, Book II itself continued to appear there as a purely geometric text. On the other hand, the first ten results of Book II could now be seen also as possiblytranslatable into arithmetic, and in many cases even as inseparably associated with their arithmetic representation. (shrink)

This is a brief note about the history of the analysis of the collection of theories, RM3modn, in Meyer and Mortensen "Inconsistent Models for Relevant Arithmetics" Journal of Symbolic Logic 49 (1984).

Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts (...) James H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́. (shrink)

The paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument (...) against the possibility of logical deviancy. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: (...) Russell's logicism does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical (...) concepts. Numbers and arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

The Rise of Chance in Evolutionary Theory: A Pompous Parade of Arithmetic explores a pivotal conceptual moment in the history of evolutionary theory: the development of its extensive reliance on a wide array of concepts of chance. It tells the history of a methodological and conceptual development that reshaped our approach to natural selection over a century, ranging from Darwin’s earliest notebooks in the 1830s to the early years of the Modern Synthesis in the 1930s. Far from (...) being a “pompous parade of arithmetic,” as one early critic argued, evolution transformed during this period to make these conceptual and technical tools indispensable. This book charts the role of chance in evolutionary theory from its beginnings to the earliest days of modern evolutionary theory, making it an ideal resource for evolutionary biologists, historians, philosophers, and researchers in science studies or biological statistics. (shrink)

Building on previous scholarly work on the mathematical roots of assertoric syllogistic we submit that for Aristotle, the semantic value of the copula in universal affirmative propositions is the relation of divisibility on positive integers. The adequacy of this interpretation, labeled here ‘arithmetical dictum’, is assessed both theoretically and textually with respect to the existing interpretations, especially the so-called ‘mereological dictum’.

In a number of works, we have suggested that whereas the ordered field R of real numbers should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), the ordered field No of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In the present chapter, as part of a more general exposition of the absolute arithmetic continuum, we will outline some of the properties of the system of surreal numbers (...) that we believe lend credence to this mathematico-philosophical thesis. We will also provide an overview of No’s rich structure as a simplicity-hierarchical (or s-hierarchical) ordered field that recursively emerges from the interplay between its structure as an ordered field and its structure as a lexicographically ordered full binary tree. Finally, we will draw attention to how properties of the system of surreal numbers considered as an s-hierarchical ordered algebraic structure can be appealed to in conjunction with classical relations between ordered algebraic and geometric systems to help resolve, for the surreal case, one of the purported difficulties that lies at the heart of attempts to bridge the gap between the domains of number and of geometrical magnitude. In particular, we will explain how it is possible that despite the fact that every surreal number, considered as a member of an s-hierarchical ordered field, differs from every other in characteristic individual properties, the absolute (Euclidean) geometrical continuum, which is modeled by the Cartesian space over the ordered field No of surreal numbers, appears as an amorphous pulp of points that display little individuality. (shrink)

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent logicism is compatible with (...) intuitionism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:684 JOURNAL OF THE HISTORY OF PHILOSOPHY 33:4 OCTOBER 1995 "Private I.anguage" and the pivotal paper in the Stoic section, "The Conjunctive Model," bring out a third feature of Brunschwig's method. Many of his essays take their start from a small text or a relatively local problem, one which does not primafacie bear significantly on large philosophical issues. Yet in a rigorously conceived philosophical system, the whole is (...) often reflected in the details; where possible Brunschwig elicits the larger significance. He shows how Epicurean physics provides a normative model for Epicurean psychology without falling into the reductionist error; the Epicurean does extrapolate from physical theory to human experience, but in so doing he does not forget that human beings are not just atomic; they are atomic compounds. Reductionism does not follow from materialism, for Epicurus or for us. Similarly in "The Conjunctive Model" Brunschwig connects, in a manner which is speculative but nevertheless compelling, a important Stoic thesis in logic (that a conjunction is false if even one of its conjuncts is false) with a comprehensive understanding of Stoic ethics and physics, thereby giving definite content to the Stoic claims about the coherence of their system. The other Stoic papers focus on ontology and logic, showing how central these are to Stoicism. The titles speak for themselves: "Remarks on the Stoic Theory of the Proper Noun," "Remarks on the Classification of Simple Propositions in Hellenistic Logics," "The Stoic Theory of the Supreme Genus and Platonic Ontology," and "Did Diogenes of Babylon Invent the Ontological Argument?" (the answer is 'no'). One essay ("On a Stoic Way of Not Being") disguises by its title a crucial contribution to the philosophy of mind. The papers on Scepticism include "The 6oov ~t~ x~ ~.6"f.t 9 Formula in Sextus Empiricus," a definitive treatment of a verbal formula crucial to understanding the precise character of his sceptical stance, and "Sextus Empiricus on the xptxfl0tov," the only paper in the volume which has previously appeared in English. The papers "Once Again on Eusebius on Aristocles on Timon on Pyrrho" and "The Title of Timon's lndalmoi: From Odysseus to Pyrrho" deal with early Pyrrhonism, for which the evidence is so thin that no firm conclusions can be drawn. Yet even here Brunschwig's care and skill are exemplary, and his argument that we should look to Timon rather than Pyrrho as the creator of Pyrrhonism is as convincing as anything could be in this area. This is a collection which specialists must read and even nonspecialists will value for its elegant combination of rigor, intellectual honesty, and philosophical insight. BRAD INWOOD University of Toronto Michael J. B. Allen. Nuptial Arithmetic: Marsilio Ficino's Commentary on the Fatal Number in Book VIII of Plato's Republic. Berkeley: University of California Press, 1994. PP. x + 291. Cloth, $5o.oo. Marsilio Ficino was born in 1433 and died in 1499 at the age of 66. He would have been pleased if he could have known that. For three years before in 1496 (or slightly earlier) he declared in his Commentary on the Fatal Number that 6 was the perfect number and BOOK R~VIEWS 685 that contemplations of the sextile relations of the planets might lead to the return of the golden age of Saturn "some day," This is the fifth volume of Michael Allen's seminal studies of Ficino's Platonic commentaries; if his notable additional articles on further Ficino commentaries were assembled, as they sometime must be, it would be his sixth--not that he takes the same pleasure in numbers as his subject Ficino did. But 6 is the key to Ficino's unraveling of Plato's "fatal number" in Republic 546. To end the suspense, Ficino's solution, aided by Aristotle, Politics V. 1316, is (4 + 3 + 5)3or (6 x 2)3 or 1728. But 1728 what? There is no clue. The book is itself a valuable contribution to the history of philosophy. Allen reviews the history of this topos up to Brumbaugh and Rees. Although Ficino's entry was known and discussed in the sixteenth century, it has been lost from sight since, and Alien restores it importantly to... (shrink)

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to (...) Weak K\"onig's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. (shrink)

If true, this is one the the most important papers in the history of mathematics. the natural numbers are defined and one to one correspondence between the natural numbers is made precise. the paper deals with the very fundamentals of arithmetic and the logical principles differ quite markedly from those used by georg cantor.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:A History of American Music EducationSondra Wieland HoweA History of American Music Education, 3rd edition, by Michael L. Mark and Charles L. Gary. Lanham, MD: Rowman and Littlefield Education, 2007, 500 pp., $95.00 cloth, $44.95 paper.Mark and Gary's editions of A History of American Music Education are indispensable reading for every music education student, practicing professional music educator, and the general reader who is interested (...) in the development of music teaching in American public schools. Based on excellent secondary sources, this history presents an overview of music education from Colonial America, through the development of textbooks in the nineteenth century and the growth of professional organizations in the early twentieth century, to the expansion of the curriculum and the role of government in education since 1950.This third edition is written by Michael L. Mark, professor emeritus at Towson University in Maryland, and dedicated to the memory of his friend and coauthor, Charles L. Gary.1 Mark has written two other books that complement the history editions. Contemporary Music Education surveys various reform projects since the 1950s, trends in the philosophy of music education, government policies, and the development of new methods.2Music Education: Source Readings includes writings about music education from both the European and American perspectives.3 There are many recent articles and dissertations on the history of music education, but the only recent book is Terese Volk's Music, Education, and Multiculturalism, which describes school programs and materials emphasizing multiculturalism in the twentieth century from a historical perspective.4 Publications in musicology add to our knowledge of the history of music in American society, but musicologists rarely write about music in public schools. Publications on the history of education are helpful in understanding trends in music education, but these publications do not discuss music.The third edition of A History of American Music Education is expanded to 500 pages from the approximately 400 pages of the earlier editions, largely because of new formatting and [End Page 115] spacing that make the third edition easier to read. Each chapter has a reference list in APA style in contrast to the Chicago Manual style of the earlier editions, with endnotes plus a bibliography at the end of each chapter. The material in Mark's third edition is basically the same as the earlier editions, with some expansion of the history of music education after 1950.This book, based on secondary sources, is an overview of the history of music in American public schools, showing the response of education to societal needs during various historical periods. There is an emphasis on the role of national music organizations, especially MENC: The National Association for Music Education (MENC), in advocating for the importance of music for every child. The historical account includes the various justifications for teaching music and discussions of the meaning of music. There were times when music educators were interested in aesthetics, but there were also many times when they were mainly interested in a practical understanding of music and extrinsic motives for including music in the school curriculum.5The first section on "The Western Heritage" (chapters 1-3) briefly covers music in Jewish traditions through the Greek and Roman cultures because the insights of these ancient thinkers influenced all Western societies. The chapter on the Middle Ages through the Reformation shows the role of music as a practical subject, part of the quadrivium with arithmetic, geometry, and astronomy. As music became an integral part of the church, musicians developed methods of teaching note reading. Martin Luther promoted the singing of "good music" in schools and education in the vernacular for the middle classes. With the development of Italian conservatories and the patronage of music by the elite classes of the Renaissance, music training became specialized for musicians working in the church and courts. These various conflicting European opinions of the role of music in society influenced American education.The second section on "The New World" (chapters 4-5) emphasizes the use of music as a means of praising God. Spanish musicians established successful music education programs in Peru and Mexico to promote Christianity. The French brought Christianity to... (shrink)

Traces the origins and development of arithmetic, statistics, geometry, and calculus from the ancient civilizations to the present.

Unraveling all the mysteries of the khipu--the knotted string device used by the Inka to record both statistical data and narrative accounts of myths, histories, and genealogies--will require an understanding of how number values and relations may have been used to encode information on social, familial, and political relationships and structures. This is the problem Gary Urton tackles in his pathfinding study of the origin, meaning, and significance of numbers and the philosophical principles underlying the practice of arithmetic among (...) Quechua-speaking peoples of the Andes. Based on fieldwork in communities around Sucre, in south-central Bolivia, Urton argues that the origin and meaning of numbers were and are conceived of by Quechua-speaking peoples in ways similar to their ideas about, and formulations of, gender, age, and social relations. He also demonstrates that their practice of arithmetic is based on a well-articulated body of philosophical principles and values that reflects a continuous attempt to maintain balance, harmony, and equilibrium in the material, social, and moral spheres of community life. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.

Historians have long seen Sir William Petty’s ‘political arithmetic’ as an important contribution to the early social sciences, applying mathematics to the analysis of political and especially economic questions. A closer look at Petty’s political arithmetic manuscripts reveals, however, his political preoccupation with ‘transmuting the Irish into English’ by state manipulation of demography. Large-scale, coerced ‘counter-transplantations’ of ‘exchanges of women’ between England and Ireland would facilitate the ‘proportionable mixture’ and ultimately the ‘union’ of the two populations, stabilizing the (...) turbulent politics of late Stuart Ireland and securing English colonial government there. This paper explores Petty’s use of alchemical concepts in his transmutative political arithmetic, and argues that the idiom of alchemy was essential to Petty’s conception of politics and his solution to the problem of governing culturally distinct colonial populations. It makes a preliminary attempt to identify some of Petty’s likely sources among seventeenth-century proponents of corpuscularian, mechanical accounts of chemical change. Finally, it suggests that the influence of alchemy on the shape of early social-scientific thought may have been more significant than was previously realized.Keywords: Political arithmetic; Alchemy; William Petty; Ireland; Political economy. (shrink)

The volume is the first collection of essays that focuses on Gottlob Frege's Basic Laws of Arithmetic (1893/1903), highlighting both the technical and the philosophical richness of Frege's magnum opus. It brings together twenty-two renowned Frege scholars whose contributions discuss a wide range of topics arising from both volumes of Basic Laws of Arithmetic. The original chapters in this volume make vivid the importance and originality of Frege's masterpiece, not just for Frege scholars but for the study of (...) the history of logic, mathematics, and philosophy. (shrink)

Gottlob Frege’s Grundgesetze der Arithmetik (Basic Laws of Arithmetic, Vol. I/II; 1893/1903) is a modern classic. Since the 1930s it has belonged to an exclusive class of only eleven works in the history of symbolic logic, which contain the ‘first appearance of a new idea of fundamental importance’ [Church, 1936, p. 122], and its author is the only one whose other major works — Begriffsschrift (1879) and Die Grundlagen der Arithmetik (1884) — also belong to this distinguished group. (...) Together with the seminal paper ‘Über Sinn und Bedeutung’ these three monographs represent the essential contributions to one of the most fascinating foundational projects in mathematics, then and now. For decades Frege struggled with the question, ‘how are we to apprehend logical objects, in particular, the numbers? What justifies us in acknowledging numbers as objects?’ [Frege, 2013, ‘Afterword’, p. 265]. His favourite answer, for a long time, was that arithmetic is a complex branch of logic. But since there is no logicism without logic, Frege started in Begriffsschrift with the invention of a nearly perfect formalization of higher-order classical predicate logic. To discover how far we can get in arithmetic by analytic means alone, it was even necessary to clarify the semantic character of the concept of number. This was done in a marvellous way by Frege five years later, in Grundlagen. The next step was nothing less than the complete execution of his logicist project, a monumental undertaking; and it is well known that here Frege failed, due to Russell’s derivation of the antinomy in 1902. But, despite the contradiction, Grundgesetze contains Frege’s legacy to logicism, which became more and more attractive. (shrink)

James Gow's A Short History of Greek Mathematics provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts (...) I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. Particular attention is given to Pythagorus, Euclid, Archimedes, and Ptolemy, but a host of lesser-known thinkers receive deserved attention as well. (shrink)

Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory. The proof is constructive, using Tarski-style quantifier elimination and a four-part recursive comprehension principle for axiomatic consequence characterization. Presburger's proof for the completeness of first order arithmetic with identity and addition but without multiplication, in light of the restrictive formal metatheorems of Gödel, Church, and Rosser, takes the foundations of (...) class='Hi'>arithmetic in mathematical logic to the limits of completeness and decidability. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for (...) the applicability of arithmetic to matters of empirical fact?These are philosophical questions, part of a "wider puzzle of explaining the link between experience, language, thought, and the world" (18). But answers to them are heavily constrained by technical results in logic and mathematics, and most of Potter's protagonists are either mathematicians (Frege, Dedekind, Ramsey, Hilbert, Gödel) or mathematically trained philosophers (Kant, Russell, Carnap). (Only Wittgenstein fits neither category.) Potter, whose doctorate was in pure mathematics, is an able guide.It is not hard to think of answers to these questions taken singly; what is hard is answering both at the same time. Formalists can easily explain why arithmetic does not depend for its justification on empirical fact—it is just a game with symbols, not responsible to anything beyond its own rules—but they then have trouble explaining why the world of empirical fact is constrained by this game. Empiricists can explain the applicability of arithmetic in just the same way they explain the applicability of physics, but they then must then explain away the apparent a prioricity and necessity of arithmetic. Kant was the first to propose a plausible answer to both questions: arithmetic is both necessary and empirically applicable because it is grounded in the spatio-temporal form of any possible experience.What unites Frege, Dedekind, Russell, Ramsey, Hilbert, and Carnap is that they reject Kant's answer while continuing to seek an answer to his question: how can arithmetic's necessity be reconciled with its applicability? Potter helpfully classifies their projects by the surrogates they substitute for Kant's appeal to the form of spatio-temporal intuition: Frege and Dedekind appeal to thought, Russell to necessary features of the world, Wittgenstein, Ramsey, and Carnap to language, and Hilbert to the necessary structure of our experience of finitary combinations of objects. (Revealingly, the logicists are not grouped together in Potter's taxonomy.) A chapter (or two) is devoted to each of these thinkers, plus Kant and Gödel.This breadth of coverage does not come at the price of superficiality: Potter's chapters are (by and large) accurate, penetrating, and full of interesting insights. I cannot think of another book that offers such clear explanations of the many mathematical concepts and distinctions one must grasp in order to follow debates in the philosophy of arithmetic. The Russell chapters are enhanced by numerous quotations from archival material. The exposition of Hilbert's programme and the implications for it of Gödel's theorems is the best I have seen. The discussions of Wittgenstein's views on arithmetic and of Carnap's syntactical approach also shine. The chapters on Kant and Frege are somewhat weaker; the discussion of Kant, in particular, would have benefited from some engagement with Michael Friedman's views.But the real contribution of Potter's book is its integration of these nine thinkers into a single coherent story. No one will be surprised that Russell's motivations can only be understood in light of Frege's failures, or that many features of Wittgenstein's cryptic work become clearer when seen in light of the specific problems with Russell's system. But there are also some less obvious payoffs: for example, Hilbert's programme is usefully compared with Russell's regressive method (227-8), and Carnap and Russell are shown to face a common dilemma (269, 286).If there is a central theme to Potter's story, it is "how often we have had to reject an [End Page 455] account not for philosophical reasons but for technical ones" (278). Frege's program founders on Russell's paradox, Russell's... (shrink)

Brouwer and Husserl both aimed to give a philosophical account of mathematics. They met in 1928 when Husserl visited the Netherlands to deliver his Amsterdamer Vorträge. Soon after, Husserl expressed enthusiasm about this meeting in a letter to Heidegger, and he reports that they had long conversations which, for him, had been among the most interesting events in Amsterdam. However, nothing is known about the content of these conversations; and it is not clear whether or not there were any other (...) exchanges between them. The following is an attempt at some remarks on Husserl’s Philosophy of Arithmetic from an intuitionistic, by which I mean a Brouwerian, point of view. I will confine myself specifically to some intuitionistic musings on Husserl’s analysis of the concept of finite number, which he developed in his Habilitationsschrift of 1887, “On the Concept of Number,” and presented again in 1891 in the first four chapters of the Philosophy of Arithmetic. It will turn out that the intuitionistic account of number is very different from Husserl’s. For instance, they strongly disagree about the number 2. On the other hand, the roots of the projects of Husserl and Brouwer have much in common. Since this fact makes their disagreement pressing and lends it much of its significance, I will begin there. (shrink)

The aim of this essay is to develop a formal reconstruction of Greek arithmetic. The reconstruction is based on textual evidence which comes mainly from Euclid, but also from passages in the texts of Plato and Aristotle. Following Paul Pritchard’s investigation into the meaning of the Greek term arithmos, the reconstruction will be mereological rather than set-theoretical. It is shown that the reconstructed system gives rise to an arithmetic comparable in logical strength to Robinson arithmetic. Our reconstructed (...) Greek arithmetic is then compared to modern developments in the foundations of arithmetic. Finally, it is shown that our reconstruction can help to clarify some issues in Plato’s philosophy of arithmetic, especially the question if Plato’s views can be compared to 19th – 20th century developments in the logical construction of arithmetic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Humanist History of Mathematics?Regiomontanus's Padua Oration in ContextJames Steven ByrneIn the spring of 1464, the German astronomer, astrologer, and mathematician Johannes Müller (1436–76), known as Regiomontanus (a Latinization of the name of his hometown, Königsberg in Franconia), offered a course of lectures on the Arabic astronomer al-Farghani at the University of Padua. The only one of these to survive is his inaugural oration on the history (...) and utility of the mathematical arts.1 Regiomontanus tells his audience that the purpose of the oration is torelate first the origin of our arts, and among which nations they first began to be cultivated, in what way they were at last translated from various foreign tongues into Latin, which of our ancestors were famed in these disciplines, and to whom in our lifetimes recognition should be granted.2To this end, he offers a history of the quadrivial arts (arithmetic, geometry, music, and astronomy) and other important mathematical disciplines from [End Page 41] antiquity to his own time, praises their utility, and exhorts his audience to revive the languishing study of mathematics at Padua. Astrology, a discipline with which Regiomontanus himself was closely associated, is singled out for particular praise.Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenth-and-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled [End Page 42] after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various "algorisms" (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). Advanced students could read numerous more specific treatises by Arabic and Latin authors. For optics, the Perspectiva communis of John Peckham was the most common basic text, with texts... (shrink)

The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

The main goal of part 1 is to challenge the widely held view that Poincaré orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of (...) these readings. More fully, I argue that the justification Poincaré offers for the use of the group notion in geometry appears to extend to set-theoretic notions that would suffice to put arithmetic on a logical foundation, thus undermining his own case for the necessity of intuition in arithmetic. In part 2, I offer an interpretation of intuition’s role on which it justifies the use of group-theoretic, but not set-theoretic, notions. (shrink)