The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will (...) wish to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors' justification of the philosophical standpoint adopted at the outset of their work); the whole of Part 1 (in which the logical properties of propositions, propositional functions, classes and relations are established); section 6 of Part 2 (dealing with unit classes and couples); and Appendices A and B (which give further developments of the argument on the theory of deduction and truth functions). (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

mathematicians for over 60 years. Amazingly, the Argonne team's automated theorem-proving program EQP took only 8 days to find a proof of it. Unfortunately, the proof found by EQP is quite complex and difficult to follow. Some of the steps of the EQP proof require highly complex and unintuitive substitution strategies. As a result, it is nearly impossible to reconstruct or verify the computer proof of the Robbins conjecture entirely by hand. This is where the unique symbolic capabilities of (...) class='Hi'>Mathematica 3 come in handy. With the help of Mathematica, it is relatively easy to work out and explain each step of the dense EQP proof in detail. In this paper, I use Mathematica to provide a detailed, step-by-step reconstruction of the highly complex EQP proof of the Robbins conjecture. (shrink)

Motus quidem veros corporum singulorum cognofcere , & ab apparentibus actu diícriminare, difficillimum est ; propterca quod partes ípatij illius immobilis in quo corpora vere moventur, non incurrunt in sensus.

Foreword: Mathesis as a philosophy of the beauty of music -- The constructive relationships between music and mathematics : the Pythagorean conception of universal music and its spread in the worldview of later periods -- Semantic interpretation of the interaction between music and mathematics : mystic middle ages and the sacral baroque -- Constructive aspects of the interaction between music and other arts -- Musica mathematica in practice : aspects of analysis -- Constructive aspects of music composition -- Semantic (...) aspects of music composition -- The mathematized musical graph -- Implications of modern mathematical theories. (shrink)

This is the first study to deal with Greek mathematics from the viewpoint of cultural history. Mathematics, and especially the teaching of mathematics, did not proceed in isolation, but developed along lines parallel to the development of general literate culture.

On the occasion of the centenary of the publication of the first volume of Principia Mathematica, this article aims to shed some light on Whitehead’s philosophy of mathematics, to differentiate Whitehead’s version of logicism from Russell’s, and to show the unity of Whitehead’s thought. These aims rely on Whitehead’s basic insight that mathematics is the study of pattern.

Originally published in 1910, Principia Mathematica led to the development of mathematical logic and computers and thus to information sciences. It became a model for modern analytic philosophy and remains an important work. In the late 1960s the Bertrand Russell Archives at McMaster University in Canada obtained Russell's papers, letters and library. These archives contained the manuscripts for the new Introduction and three Appendices that Russell added to the second edition in 1925. Also included was another manuscript, 'The Hierarchy (...) of Propositions and Functions', which was divided up and re-used to create the final changes for the second edition. These documents provide fascinating insight, including Russell's attempts to work out the theorems in the flawed Appendix B, 'On Induction'. An extensive introduction describes the stages of the manuscript material on the way to print and analyzes the proposed changes in the context of the development of symbolic logic after 1910. (shrink)

Early in Principia Mathematica Russell presents an argument that "‘the author of Waverley’ means nothing", an argument that he calls a "definite proof". He generalizes it to claim that definite descriptions are incomplete symbols having meaning only in sentential context. This Principia "proof" went largely unnoticed until Russell reaffirmed a near-identical "proof" in his philosophical autobiography nearly 50 years later. The "proof" is important, not only because it grounds our understanding of incomplete symbols in the Principia programme, but also (...) because failure to understand it fully has been a source of much unjustified criticism of Russell to the effect that he was wedded to a naive theory of meaning and prone to carelessness and confusion in his philosophy of logic and language generally. In my paper, I (1) defend Russell’s "proof" against attacks from several sources over the last half century, (2) examine the implications of the "proof" for understanding Russell’s treatment of class symbols in Principia, and (3) see how the Principia notion of incomplete symbol was carried forward into Russell’s conception of philosophical analysis as it developed in his logical atomist period after 1910. (shrink)

In an article in "mind," 1971, pp. 282-283, the authors raised objections to a certain definition in "principia mathematica." in this paper we demonstrate that (a) their argument is faulty, (b) their suggested remedies are unsatisfactory, and (c) there is nothing wrong with the original definition.

This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia (...) Mathematica. (shrink)

Here the concern is with the logic underlying Whitehead and Russell's Principia Mathematica and with the relation of that logic to Russell's underlying metaphysics. The author emphasizes the fact that work is, strictly speaking, a theory of propositional functions, not of classes; sentences containing symbols for classes are defined by means of propositional functions. It is terms of the latter sort of entity that Russell's Paradox must be solved; the theory of types is, strictly speaking, a theory of the (...) stratification of propositional functions. (shrink)

This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full and then analyzed. The point (...) of the quotation is to make clear how murky, incompetent, and badly written the paper is. There is no way to determine from the article whether the editor or referees suggests improvements. (shrink)

La decouverte par le Dr. E. A. Fellmann de l'exemplaire des Principia Mathematica de Newton ayant appartenu a Leibniz et contenant des notes marginales manuscrites (datant vraisemblablement de 1689), apporte un temoignage direct de la premiere reaction de Leibniz devant l'oeuvre du savant anglais. La presente edition met le document original entre les mains du lecteur, en l'accompagnant des notes et commentaires objectifs indispensables (en allemand et en francais). Le rapport de Leibniz a Newton a souvent ete envisage dans (...) le cadre de la facheuse querelle de priorite dans l'invention du calcul differentiel et integral. Or Leibniz, prenant connaissance des Principia, n'y voit pas un traite purement mathematique: refusant d'enteriner la scission entre la mecanique et les lois de la pensee, il se soucie avant tout d'etablir une fondation metaphysique pour la dynamique. Les Marginalia ont l'originalite de nous presenter, non pas un Leibniz adversaire ou rival, mais un Leibniz lecteur de Newton, dispose a tirer le meilleur parti d'un systeme qui aurait pu, a bien des egards, lui sembler etranger, comme le confirmera la systemisation ulterieure du concept de vis visa. (shrink)