Charles Peirce's place in philosophy

Peirce's publications on the method of scientific investigation (as distinct from his work in formal logic and mathematics) are his most important and valuable contributions to philosophy. His views on this subject are superior in clarity and cogency to his voluminous writings on metaphysics and cosmology. He subscribed to a fallibilistic conception of knowledge that is poles apart from a wholesale skepticism; his formulations of the conditions for meaningful discourse and of the pragmatic maxim, though not free from difficulties, have been fruitful sources of much subsequent philosophical and scientific analyses; and his classification of and discussions of types of argument or reasoning employed in scientific inquiry continue to be valuable and insightful clarifications of this important subject. In contrast to his account of scientific method, Peirce's evolutionary theory of ultimate reality, though marked by originality and ingenious speculation, has little merit as a contribution to genuine knowledge.     

Properly efficient and efficient solutions for vector maximization problems in euclidean space

Recently Benson proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone K is any nontrivial, closed convex cone. We give an equivalent definition of his notion of proper efficiency. Our definition, by means of perturbation of the cone K, seems to offer another justification of Benson's choice above Borwein's extension of Geoffrion's concept. Our result enables one to prove some other theorems concerning properly efficient and efficient points. Among these is a connectedness result.     

Euclid and the fundamental theorem of arithmetic

Slight changes or benevolent interpretations of certain theorems and proofs in Euclid's Elements make his demonstration of the fundamental theorem of arithmetic satisfactory for square-free numbers, but Euclid's methods cannot be adapted to prove the uniqueness for numbers containing square factors.     

On the finiteness theorem of Siegel and Mahler concerning diophantine equations

In this paper we present a new proof, involving so-called nonstandard arguments, of Siegel's classical theorem on diophantine equations: Any irreducible algebraic equation f(x,y) = 0 of genus g > 0 admits only finitely many integral solutions. We also include Mahler's generalization of this theorem, namely the following: Instead of solutions in integers, we are considering solutions in rationals, but with the provision that their denominators should be divisible only by such primes which belong to a given finite set. Then again, the above equation admits only finitely many such solutions. From general nonstandard theory, we need the definition and the existence of enlargements of an algebraic number field. The idea of proof is to compare the natural arithmetic in such an enlargement, with the functional arithmetic in the function field defined by the above equation.     

Mathematical methodology in the thought of charles S,Peirce

It is fitting that the celebration of Peirce's New Elements of Mathematics should be taking place in New York City, where Peirce was often to be found attending mathematical meetings at Columbia University and where he consulted the resources of the old Astor Library for the production of many of his writings. This paper considers Peirce—a lifelong student of logic—as he examined scientific and mathematical methodology on all levels, in ages past as well as in the then-contemporary literature. Peirce hoped to create an exact philosophy by applying the ideas of modern mathematical exactitude. He developed a semiotic pattern of mathematical procedure with which to test validity in all areas of investigation.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

Gregory's converging double sequence: A new look at the controversy between huygens and gregory over the “analytical” quadrature of the circle

As is well known, upon publication of his Vera circuli et hyperbolae quadratura (Padua 1667), James Gregory became involved in a bitter controversy with Christiaan Huygens over the truth of one of his major propositions. It stated that the area of a sector of a central conic cannot be expressed “analytically” in terms of the areas of an inscribed triangle and a circumscribed quadrilateral. Huygens objected to Gregory's method of proof, and expressed doubts as to its validity. As Gregory's iterative limiting process, employing an infinite double sequence, uses a combination of geometric and harmonic means, one may apply to it methods developed by the young Gauss for dealing with a similar process based on the combination of arithmetic and geometric means. This yields both the Leibnizian series forπ/4 and the product found by Vie`te for2/π, and thus serves to illuminate the structure of Gregory's procedure and the nature of Huygens' criticism.     

A proof of the existence of Batanin's initial operad

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.     

Carolyn Eisele's place in Peirce studies

Carolyn Eisele's unique, ongoing career as a scholar is sketched, and the importance of her contributions to Peirce Studies and other fields is emphasized. The essay concludes with a series of suggestions about how to interpret Peirce's works based on themes related to the pioneering efforts of Dr. Eisele.     

Śrîpati: An eleventh-century Indian mathematician

?rîpati (fl. a.d. 1039–1056) is best known for his writings on astronomy, arithmetic, mensuration, and algebra. This article discusses ?rîpati's arithmetic, the Ga$\text{n}\text{?}$itatilaka, as well as the arithmetical and algebraic chapters of the Siddhânta?ekhara. In addition to discussing the kinds of problems considered by ?rîpati and the techniques he used to solve them, the article considers the sources upon which ?rîpati drew. A glossary of Indian treatises and technical terms is provided.     

Neo-Kantian foundations of geometry in the German Romantic period

While mathematics received relatively little attention in the idealistic systems of most of the German Romantics, it served as the foundation in the thought of the Neo-Kantian philosopher/mathematician Jakob Friedrich Fries (1773–1843). It fell to Fries to work out in detail the implications of Kant's declaration that all mathematical knowledge was synthetic a priori. In the process Fries called for a new science of the philosophy of mathematics, which he worked out in greatest detail in his Mathematische Naturphilosophie of 1822. In this work he analyzed the foundations of geometry with an eye to clearing up the historical controversy over Euclid's theory of parallels. Contrary to what might be expected, Fries' Kantian perspective provoked rather than inhibited a reexamination of Euclid's axioms. Fries' attempt to make explicit through axioms what was being implicitly assumed by Euclid while at the same time wishing to eliminate unnecessary axioms belies the claim that there was no concern to improve Euclid prior to the discovery of non-Euclidean geometry. Fries' work therefore serves as an important historical example of the difficulties facing those who wanted to provide geometry with a logically secure foundation in the era prior to the published work of Gauss, Bolyai, and others.     

A subspace theorem approach to integral points on curvesPoints entiers sur les courbes et théorème des sous-espaces

We present a proof of Siegel's theorem on integral points on affine curves, through the Schmidt subspace theorem, rather than Roth's theorem. This approach allows one to work only on curves, avoiding the embedding into Jacobians and the subsequent use of tools from the arithmetic of Abelian varieties. To cite this article: P. Corvaja, U. Zannier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 267–271.     

The Method of Scientific Discovery in Peirce’s Philosophy: Deduction, Induction, and Abduction

In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, passing then to the distinction between mathematics and logic. In the end, we propose a sketch of a comparison between Peirce and Whitehead concerning the two thinkers’ view of mathematics, hoping that this could point to further inquiries.     

Social welfare and the gini coefficient revisited

The question of conflict between the rankings of income distributions with the same mean by the Gini coefficient and by individualistic social welfare functions is re-examined. The negative result of Newbery (1970) is extended. However, positive results are obtainable which reverse Newbery's conclusion by admitting into the individual's utility index a measure of his position in the income distribution, or of his deprivation with respect to others' incomes.     

Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries

Computing in hypernumber arithmetics is discussed, and specifically that of M-algebra, which includes the operations of complex, quaternion, and Cayley numbers (octaves or octonions) as subsets of itself. It is shown that modern quantum gravitation theory requires minimally the 16-dimensional space of the author's M-arithmetic (announced in Appl. Math. and Comput., 1976, p. 211 f. and 1978, p. 45 f.), which has 4320 units (including positive and negative) rather than the mere 2 units of ordinary or “real” arithmetic, the 4 units of complex arithmetic, the 24 units of quaternion arithmetic, or the 240 units of octonion or octave arithmetic. Thus computer programming is the natural tool for computations in advanced quantum physics. It turns out that more than three kinds of i-type hypernumbers and more than three kinds of the \Ge-type are needed to ensure the necessary nilpotent and noncommutative algebra required in unified field theory. It is also shown that “more than three” here means “at least seven”; and it turns out that a 16-dimensional arithmetic is needed for such computations. The following paper contextualizes, characterizes, and specifies that arithmetic as the apex of a hierarchy susceptible of clear geometric definition. And the hypernumbers needed in quantized unified field theory are specified.     

Empirical bounds for ruin probabilities

We consider the classical model for an insurance business where the claims occur according to a Poisson process and where the distribution for the cost of each claim fulfills Cramér's tail-condition. Under these conditions Lundberg's constant R is of fundamental importance for ruin calculations.We derive estimates of R, based on an observation of the insurance business and investigate the statistical properties of those estimates. We further derive bounds and confidence intervals for ruin probabilities.     

Legendre's Theorem and Quadratic Reciprocity

As Gauss noted already, his Quadratic Reciprocity Law cannot be deduced from Legendre's Theorem without the existence of primes in arithmetic progressions. Here the deduction is made, with Dirichlet's Theorem replaced by the more elementary result of Selberg, which states that every non-square is a quadratic residue to half the primes.