Rethinking geometrical exactness

A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’s geometry and EPG.     

The mathematical technique in Fermat's deduction of the law of refraction

This article attempts to explain Fermat's not quite obvious calculations connected with his deduction of the law of refraction in Analysis ad refractiones (1662), and to describe the development which led to these calculations. In 1657 Fermat tried to deduce a law of refraction based on the principle that light follows the quickest path between two given points. He did not succeed because he found that the calculations were too long and tedious. The calculations are indeed complicated, but if Fermat, in 1657, had been willing to accept Descartes' law of refraction he would probably also have seen that it solved his problem. However, Fermat was of the opinion that Descartes' law was wrong and, therefore, he did not expect that solution. Only in 1662, when he succeeded in reducing the calculations substantially, did he realize that they led to the sine law of Descartes.     

Descartes and the geometrization of thought: The methodological background of Descartes' géométrie

After the relationship between geometry and algebra exemplified in his distinction between geometrical and mechanical lines is examined, the basis for Descartes' limited approach to analytical geometry is discussed in connection with his reflections on method. It is argued that his epistemology, which required that conceptual thinking be accompanied by a construction supplied by the imagination, in conjunction with the significant role he attributed to mnemonic devices, helps to clarify the methodological background for Descartes' distinctive approach to geometry.     

Descartes and the birth of analytic geometry

The traditional thesis that analytic geometry evolved from the concepts of axes of reference, co-ordinates, and loci, is rejected. The origins of this science are re-defined in terms of Egyptian, Greek, Babylonian, and Arabic influences merging in Vieta's Isagoge in artem analyticam (1591) and culminating in a work of his pupil Ghetaldi published posthumously in 1630. Descartes' Vera mathesis, conceived over a decade earlier, served to revive and strengthen the important link with logic and thereby to extend the field of application of this analytic method to the corporeal and moral worlds.     

What Descartes knew of mathematics in 1628

The aim of this paper is to give an account of Descartes’ mathematical achievements in 1628–1629 using, as far as is possible, only contemporary documents, and in particular Beeckman’s Journal for October 1628. In the first part of the paper, I study the content of these documents, bringing to light the mathematical weaknesses they display. In the second part, I argue for the significance of these documents by comparing them with other independent sources, such as Descartes’ Regulae ad directionem ingenii. Finally, I outline the main consequences of this study for understanding the mathematical development of Descartes before and after 1629.     

Princess Elizabeth of Bohemia and Descartes’ letters (1650–1665)

After Descartes’ death in 1650, Princess Elizabeth generously shared with others several letters she had received from the philosopher, which contained philosophically as well as mathematically exciting material. In this article I place the transmission of these copies in context, revealing that Elizabeth steadily became an intellectually inspiring figure, attracting international attention. In the 1650s she stayed at Heidelberg where she discussed Cartesian philosophy with professors and students alike, including the professor of philosophy and mathematics Johann von Leuneschlos. In the mid-1660s, an initiative was taken from the English side of the Channel (Pell, More) to obtain Descartes’ mathematical letters to Elizabeth that had not yet been published. One letter of Elizabeth herself on this very subject has been preserved. The letter, addressed to Theodore Haak, will be published here for the first time. It is of special interest, because the princess supplies a general outline of her solution to the mathematical problem Descartes gave her to solve in 1643. It substantiates the hypothesis regarding Elizabeth’s solution earlier proposed by Henk Bos.     

Huygens' foundations of probability

It is generally accepted that Huygens based probability on expectation. The term “expectation,” however, stems from Van Schooten's Latin translation of Huygens' treatise. A literal translation of Huygens' Dutch text shows more clearly what Huygens actually meant and how he proceeded.     

The ovals in the Excerpta Mathematica and the origins of Descartes’ method of normals

The only occurrence of Descartes’ method of normals before La Géométrie (1637) is to be found in the Excerpta Mathematica. These mathematical fragments, published posthumously among others works in 1701, and dated by Tannery before 1629, deal with curves used in dioptrics which Descartes called ovals. I study in detail two of the texts on ovals together with the related texts in La Géométrie in order to shed light on the geometrical origins of Descartes’ method of normals.     

Quelques theoremes de dualite combinatoire

The purpose of this paper is to prevent some new and unified proofs of a number of known results in combinatorial programming: generalized versions of Minty's lemma, of Ford and Fulkerson's and Hoffman's theorems the length-width inequality and the strong complementary slackness theorem. All these results are derived from a “main duality theorem” which can be deduced from the duality theorem of linear programming or from more general results of convex analysis.     

The origin of “Zorn's Lemma”

The paper investigates the claim that “Zorn's Lemma” is not named after its first discoverer, by carefully tracing the origins of several related maximal principles and of the name “Zorn's Lemma.” Previously unpublished information supplied by Zorn is included.     

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.     

Integration in the umbral calculus

The adjointness between “multiplication” and “derivation” is a recurrent theme in Rota's umbral calculus. The subject of this paper is the companion adjointness between “division” and “integration.”     

Matroid designs of prime power index

A class of BIBD's is constructed with parameters generalizing those of the finite projective geometries. These designs are used to construct matroids in which the hyperplanes are equicardinal and the complement of every hyperplane has prime power cardinality. These so-called matroid designs of prime power index “almost” have the property that the flats of any given rank are equicardinal.     

Gray codes in graphs of subsets

Let G(n,k) be a graph whose vertices are the k-element subsets of an n-set represented as n-tuples of “O's” and “1's” with k “1's”. Two such subsets are adjacent if one can be obtained from the other by switching a “O” and a “1” which are in adjacent positions, where the first and nth positions are also considered adjacent. The problem of finding hamiltonian cycles in G(n,k) is discussed. This may be considered a problem of finding “Gray codes” of the k-element subsets of an n-set. It is shown that no such cycle exists if n and k are both even or if k=2 and n?7 and that such a cycle does exist in all other cases where k?5.     

Envy-free rights assignments and self-oriented preferences

Very recently a new solution to Sen's “Impossibility of a Paretian liberal” has been suggested where the focus is on the rights assignments per se (Austen-Smith, 1979). It was shown that the concept of fairness, when applied to rights, admits the existence of social decision functions which satisfy Sen's original conditions. Unfortunately this result collapses when individuals have rights over more than one pair of alternatives.In order to obtain possibility results for this more general case the present paper proposes to restrict individuals' preference orderings. It is proved that envy-free collective choice rules exist if individual preferences are self-oriented and if, in addition, people attach primary importance to their own private sphere alternatives. These restrictions are quite severe, but they may be justified if one values the absence of envy in rights allocations very highly.     

A confinement problem for a linearly elastic Koiter's shell

In this Note, we propose a natural two-dimensional model of “Koiter's type” for a general linearly elastic shell confined in a half space. This model is governed by a set of variational inequalities posed over a non-empty closed and convex subset of the function space used for modeling the corresponding “unconstrained” Koiter's model. To study the limit behavior of the proposed model as the thickness of the shell, regarded as a small parameter, approaches zero, we perform a rigorous asymptotic analysis, distinguishing the cases where the shell is either an elliptic membrane shell, a generalized membrane shell of the first kind, or a flexural shell. Moreover, in the case where the shell is an elliptic membrane shell, we show that the limit model obtained via the asymptotic analysis of our proposed two-dimensional Koiter's model coincides with the limit model obtained via a rigorous asymptotic analysis of the corresponding three-dimensional “constrained” model.     

Sequential games as stochastic processes

This paper examines the stochastic processes generated by sequential games that involve repeated play of a specific game. Such sequential games are viewed as adaptive decision-making processes over time wherein each player updates his “state” after every play. This revision may involve one's strategy or one's prior distribution on the competitor's strategies. It is shown that results from the theory of discrete time Markov processes can be applied to gain insight into the asymptotic behavior of the game. This is illustrated with a duopoly game in economics.     

Sandwich semigroups of binary relations

This paper introduces new semigroups of binary relations that arose naturally from investigating the transfer of information between automata and semigroups associated with automata. In particular we introduce a new multiplication on binary relations by means of an arbitrary but fixed “sandwich” relation. R.J. Plemmons and M. West have characterized Green's relations in the usual semigroup of binary relations, and we use these to investigate Green's relations in our semigroups. We give algorithms for constructing idempotents and regular elements in these new semigroups.     

Zermelo's discovery of the “Russell Paradox”

In his 1908 paper on the Well-Ordering Theorem, Zermelo claimed to have found “Russell's Paradox” independently of Russell. Here we present a short note, written by E. Husserl in 1902, which contains a detailed exposition of Zermelo's original version of the paradox. We add some comments concerning the date of Zermelo's discovery, the circumstances which caused Husserl to write down Zermelo's argument, and the argument itself.