Which Mathematical Logic is the Logic of Mathematics?

The main tool of the arithmetization and logization of analysis in the history of nineteenth century mathematics was an informal logic of quantifiers in the guise of the ??epsilon?Cdelta?? technique. Mathematicians slowly worked out the problems encountered in using it, but logicians from Frege on did not understand it let?alone formalize it, and instead used an unnecessarily poor logic of quantifiers, viz. the traditional, first-order logic. This logic does not e.g. allow the definition and study of mathematicians?? uniformity concepts important in analysis. Mathematicians?? stronger logic was rediscovered around 1990 as the form of independence-friendly logic which hence is not a new logic nor a further development of ordinary first-order logic but a richer version of it.     

“Untiring Labor Overcomes All!” The History of the Dutch Mathematical Society in Comparison to Its Various Counterparts in Europe

The Netherlands, like some other European countries, witnessed the emergence of several amateur mathematical societies in a “philomathy” atmosphere during the 18th and early 19th century. One of them, the Amsterdam Mathematical Society “Untiring Labor Overcomes All” (nowadays known as the Wiskundig Genootschap), during the early 19th century became a national institution which embodied almost the entire Dutch mathematical community. It would fulfil its role as a national mathematical Society even before the 1860s, when pure mathematics became the subject of professional research and mathematical Societies of “professionals” were founded all over Europe. This article points out the Dutch social climate (the gap between the social classes was not as enormous as elsewhere in Europe, and engineering courses were part of the mathematics curriculum at the university) and changes within the Society itself, thus describing how it was possible for the Wiskundig Genootschap to become a link between two kinds of societies (“amateurs” vs “professionals”) which should be clearly distinguished. Copyright 2001 Academic Press.Zoals in vele Europese landen ontstonden er ook in Nederland gedurende de 18de en vroege 19de eeuw wiskundige genootschappen van “liefhebbers.” Een van deze amateurgezelschappen, het Amsterdams Wiskundig Genootschap onder de zinspreuk “onvermoeide arbeid komt alles te boven,” groeide gedurende de eerste decennia van de 19de eeuw uit tot een nationaal genootschap dat de Nederlandse wiskundige gemeenschap vertegenwoordigde. Het genootschap vervulde haar rol van nationaal genootschap zodoende reeds lang voordat in de jaren 1860 onder invloed van professionalisering van (wiskundig) onderzoek dergelijke genootschappen van “professionals” in de andere Europese landen opkwamen. In dit artikel wordt gewezen op het sociale klimaat (er was nauwelijks sprake van een kloof tussen de sociale klassen zoals elders in Europa, en ingenieurswiskunde maakte deel uit van het universitair curriculum) en veranderingen binnen het genootschap zelf, waarmee beschreven wordt hoe het Wiskundig Genootschap een verband vormt tussen twee soorten van genootschappen (“amateurs” vs “professionals”) die nadrukkelijk van elkaar onderscheiden dienen te worden. Copyright 2001 Academic Press.MSC 1991 subject classifications: 01A50, 01A55.     

The Mathematical Basis of Population Genetics∗

In this paper the topic known to biologists as population genetics is presented in terms of the Cartesian product. The gene, being a set of an ordered pair of elements (alleles), provides an example of a lattice in which the axes represent the composition of the gene in respective parents, and shows as co‐ordinates the possible genetic composition of individuals of the next generation with respect to that gene. When the axes are made to represent, respectively, the set of male and female parents of a population as a whole then it is the next generation as a whole that is represented by the co‐ordinates.

Population genetics is also shown to provide an example of the binomial expansion (p+q)ⁿ in which n is 2, while on those occasions when the gene can be composed of any two of a number of elements (alleles) it expands to (p+q + r+ ... + w)² with p,q,r.....w, each representing the frequencies of an allele of the gene in the population.

It is upon these mathematical models that the study of population genetics depends and the incidence of alleles in any population is established.

     

The Spin–Orbit Resonances of the Solar System: A Mathematical Treatment Matching Physical Data

In the mathematical framework of a restricted, slightly dissipative spin–orbit model, we prove the existence of periodic orbits for astronomical parameter values corresponding to all satellites of the Solar System observed in exact spin–orbit resonance.     

From National to International Society: The London Mathematical Society, 1867–1900

The London Mathematical Society had been founded in 1865 as little more than a college club. Thanks to the support of prominent members from the British mathematical community, it had quickly grown in size and stature during its first two years; yet, while firmly established at home, it had still to secure an academic reputation overseas. This paper, a sequel to [35], examines the principal developments that occurred in the period from 1867 until the turn of the century, during which time the Society would consolidate its position as a prestigious learned body in both the national and international mathematical arenas.Copyright 1998 Academic Press.The London Mathematical Society a été fondée en 1865 et a débuté en tant que club de collège. Grâce au soutien de membres éminents de la communauté mathématique britannique, le nombre d'adhérents et la stature de cette Société se sont rapidement accrus pendant ses deux premières années; cependant, bien que résolument établie en Grande Bretagne elle devait encore établir sa renommée à l'étranger. Cet exposé qui fait suite à [35], examine les développements importants qui se sont produits entre 1867 et la fin du dix-neuvième siècle, une période pendant laquelle la Société a consolidé sa position de Société prestigieuse dans les arènes nationales et internationales.Copyright 1998 Academic Press.Als die London Mathematical Society 1865 gegründet wurde, war sie kaum mehr als ein Universitätsverein. Weil prominente Mathematiker aus ganz Großbritannien sie unterstützten, nahm sie in den ersten zwei Jahren schnell an Größe und Gestalt zu; obwohl sie sich im eigenen Lande beständig entwickelte, mußte sie sich im Ausland zunächst noch um einen akademischen Ruf bemühen. In dieser Arbeit werden (im Anschluß an [35]), die wesentlichen Entwicklungen in der Zeit von 1867 bis zur Jahrhundertwende untersucht, also in der Zeit, in der die “Society” ihre Stellung als anerkannte gelehrte Vereinigung sowohl auf nationaler als auch auf internationaler Ebene gefestigt hat.Copyright 1998 Academic Press.MSC 1991 subject classification: 01A55     

Cardinality of the Set of Delta–Closed Classes of Functions of Multi–Valued Logic

A closure under composition operation and weak version of inversion operation is considered on the set of functions of k–valued logic. The cardinality of the set of all such closed classes is calculated.     

Mathematical properties of a discontinuous Cournot–Stackelberg model

The object of this work is to perform the global analysis of a recent duopoly model which couples the two points of view of Cournot and Stackelberg [17], [18]. The Cournot model is assumed with isoelastic demand function and unit costs. The coupling leads to discontinuous reaction functions, whose bifurcations, mainly border collision bifurcations, are investigated as well as the global structure of the basins of attraction. In particular, new properties are shown, associated with the introduction of horizontal branches, which differ significantly when the constant value is zero or positive and small. The good behavior of the model with positive constant is proved, leading to stable cycles of any period.     

The Balassa–Samuelson effect in Romania – The role of regulated prices

Due to the Balassa–Samuelson effect, the candidate countries to the European Monetary Union face a possible conflict between the effects of tradable productivity increase on inflation and on the appreciation of the real exchange rate on one hand, and the Maastricht criteria regarding inflation and exchange rate stability on the other hand. A restrictive monetary policy could succeed in reaching inflation criteria, but only with the cost of slowing down real convergence. This paper deals with the Balassa–Samuelson effect in Romania. The main conclusion of the paper is that in the period 1998–2006 the average annual rate of inflation generated by the Balassa–Samuelson effect in Romania was on average 0.6% in the case of the classical model. Due to the existence of government-regulated prices (mostly non-tradable goods) accounting for as much as 21% of the CPI basket in Romania – the price of non-tradables has increased by less than in the case those prices were market prices. If we include these regulated prices in non-tradable, which is named by us extended model, the impact of Balassa–Samuelson on inflation could have been on average 2.46%. Therefore, the Balassa–Samuelson effect is expected to result into higher inflation with future price liberalization for non-tradables.