The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle

Today we credit Pierre Wantzel with the first proof (1837) of the impossibility of doubling a cube and trisecting an arbitrary angle by ruler and compass. However two centuries earlier Descartes had put forward what probably counts as the first proof of these impossibilities. In this paper I analyze this proof, as well as the later related proof given by Montucla (1754) and the brief version of this proof published by Condorcet (1775) . I discuss the many novelties of these early arguments and highlight the problematic points addressed by Gauss (1801) and Wantzel. In particular I show that although Descartes developed many of the algebraic techniques used in later proofs he failed to provide an algebraic impossibility proof and resorted to a geometric argument. Montucla and Condorcet turned this proof into an algebraic one. I situate the analysis of the early proof of the impossibility of the two classical problems in the general context of early modern mathematics where mathematics was primarily viewed as a problem solving activity. Within such a paradigm of mathematics impossibility results arguably do not play the role of proper mathematical results, but rather the role of meta-results limiting the problem solving activity.     

Teaching Mathematics from an Applications Perspective

It is a widespread opinion among engineering faculty that undergraduates could be better prepared in mathematics when taking courses in their professional field of study. The lack of preparation in applying mathematical concepts may be due to the fact that examples from engineering disciplines are not widely used in mathematics courses. Most mathematics departments act as service departments to students majoring in various fields in addition to providing their own degree programs. As a result, it is not economically justifiable for them to custom tailor courses for customers from different disciplines. On the other hand, engineering has a distinct requirement of creatively applying mathematical concepts and principles to engineering problems studied in various courses. Some of the issues that must be addressed to ensure adequate preparation in the application of mathematics include: the mathematical competencies needed in engineering courses; which mathematics courses should cover such competencies; and what examples and problems related to students' major field should be developed and taught in these courses to enhance understanding and application of these concepts. The goal of this paper is not to resolve these issues, but rather to develop a conceptual framework for determining the answers using the Quality Function Deployment approach.     

Optimal Facility-Location

Dr. Christoph Witzgall, the honoree of this Symposium, can count among his many contributions to applied mathematics and mathematical operations research a body of widely-recognized work on the optimal location of facilities. The present paper offers to non-specialists a sketch of that field and its evolution, with emphasis on areas most closely related to Witzgall’s research at NBS/NIST.     

Productivity and mobility in academic research: evidence from mathematicians

Using an exhaustive database on academic publications in mathematics all over the world, we study the patterns of productivity by mathematicians over the period 1984–2006. We uncover some surprising facts, such as the weakness of age related decline in productivity and the relative symmetry of international movements, rejecting the presumption of a massive “brain drain” towards the US. We also analyze the determinants of success by top US departments. In conformity with recent studies in other fields, we find that selection effects are much stronger than local interaction effects: the best departments are most successful in hiring the most promising mathematicians, but not necessarily at stimulating positive externalities among them. Finally we analyze the impact of career choices by mathematicians: mobility almost always pays, but early specialization does not.     

The justification of mathematical statements

The uncompromising ethos of pure mathematics in the early post-war period was that any theorem should be provided with a proof which the reader could and should check. Two things have made this no longer realistic: (i) the appearance of increasingly long and complicated proofs and (ii) the involvement of computers. This paper discusses what compromises the mathematical community needs to make as a result.     

Scientific production: A statistical analysis of authors in mathematical logic

We show that scientific production can be described by two variables: rate of production (rate of publications) and career duration. For mathematical logicians, we show that the time pattern of production is random and Poisson distributed, contrary to the theory of cumulative advantage. We show that the exponential distribution provides excellent goodness-of-fit to rate of production and a reasonable fit to career duration. The good fits to these distributions can be explained naturally from the statistics of exceedances. Thus, more powerful statistical tests and a better theoretical foundation is obtained for rate of production and career duration than has been the case for Lotka's Law.     

Über die Entwicklung der Mathematik in Westeuropa zwischen 1100 und 1500

The twelfth century was a period of transmission and absorption of Arabic learning though it filtered outside of the Arabic world as early as the second half of the tenth century. In general, the lure of Spain began to act only in the twelfth century, and the active impulse toward the spread of Arabic mathematics came from beyond the Pyrenees and from men of diverse origins. The chief names are Adelard of Bath, Robert of Chester, Hermann of Carinthia and Gerard of Cremona. In this time the Latin world became acquainted with the Hindu numerals, the Arabic Algebra and Euclid'sElements. However, not only Spain, but also the Norman kingdom of southern Italy and Sicily occupies a position of peculiar importance, though the works of the translators did not become very influential. There were made direct translations from Greek into Latin. One had to wait a century more to obtain a translation from Greek into Latin of the chief Archimedean scientific and mathematical treatises by William of Moerbeke. In the thirteenth century Fibonacci and Jordanus Nemorarius stand at the threshold of European mathematics. Not only was Fibonacci the first to explain Arabic arithmetic, but his works, especially his later ones, contain many original ideas. Jordanus continued the Greco-Roman tradition rather than the Greco-Arabic one, but he did so with much independence. To Nicole Oresme (fourteenth century) was due a broadened view of proportionality, a geometric proof to determine the summation of convergent infinite series and the proof, evidently the first in the history of mathematics, that the harmonic series is divergent. The Configuration Doctrine was treated by Merton College authors and by Oresme. In the fifteenth century theDe triangulis omnimodis of Regiomontan, a systematic account of the methods for solving triangles, marked the rebirth of trigonometry.      

立磨液压系统的自动化控制技术的研究

数字技术的数学基础——离散数学、逻辑数学等，早在17、18世纪就已经出现。但是发展成为数字技术并付诸实用，随着半导体器件、集成器件和超大规模集成器件的出现，数字技术在液压系统中的应用迅速而又普遍地发展起来。     

Mathematical models of plant-soil interaction

In this paper, we set out to illustrate and discuss how mathematical modelling could and should be applied to aid our understanding of plants and, in particular, plant-soil interactions. Our aim is to persuade members of both the biological and mathematical communities of the need to collaborate in developing quantitative mechanistic models. We believe that such models will lead to a more profound understanding of the fundamental science of plants and may help us with managing real-world problems such as food shortages and global warming. We start the paper by reviewing mathematical models that have been developed to describe nutrient and water uptake by a single root. We discuss briefly the mathematical techniques involved in analysing these models and present some of the analytical results of these models. Then, we describe how the information gained from the single-root scale models can be translated to root system and field scales. We discuss the advantages and disadvantages of different mathematical approaches and make a case that mechanistic rather than phenomenological models will in the end be more trustworthy. We also discuss the need for a considerable amount of effort on the fundamental mathematics of upscaling and homogenization methods specialized for branched networks such as roots. Finally, we discuss different future avenues of research and how we believe these should be approached so that in the long term it will be possible to develop a valid, quantitative whole-plant model.     

Hilberts Logik. Von der Axiomatik zur Beweistheorie

This paper gives a survey of David Hilbert's (1862–1943) changing attitudes towards logic. The logical theory of the Göttingen mathematician is presented as intimately linked to his studies on the foundation of mathematics. Hilbert developed his logical theory in three stages: (1) in his early axiomatic programme until 1903 Hilbert proposed to use the traditional theory of logical inferences to prove the consistency of his set of axioms for arithmetic. (2) After the publication of the logical and set-theoretical paradoxes by Gottlob Frege and Bertrand Russell it was due to Hilbert and his closest collaborator Ernst Zermelo that mathematical logic became one of the topics taught in courses for Göttingen mathematics students. The axiomatization of logic and set-theory became part of the axiomatic programme, and they tried to create their own consistent logical calculi as tools for proving consistency of axiomatic systems. (3) In his struggle with intuitionism, represented by L. E. J. Brouwer and his advocate Hermann Weyl, Hilbert, assisted by Paul Bemays, created the distinction between proper mathematics and meta-mathematics, the latter using only finite means. He considerably revised the logical calculus of thePrincipia Mathematica of Alfred North Whitehead and Bertrand Russell by introducing the ε-axiom which should serve for avoiding infinite operations in logic.     

Rescher's Principle of Decreasing Marginal Returns of Scientific Research

In his book "Scientific Progress", Rescher (1978, German ed. 1982, French ed. 1993)developed a principle of decreasing marginal returns of scientific research, which is based, interalia, on a law of logarithmic returns and on Lotka's law in a certain interpretation. In the presentpaper, the historical precursors and the meaning of the principle are sketched out. It is reported onsome empirical case studies concerning the principle spread over the literature. New bibliometricdata are used about 19th-century mathematics and physics. They confirm Rescher's principleapart from the early phases of the disciplines, where a square root law seems to be moreapplicable. The implication of the principle that the returns of different quality levels grow theslower, the higher the level, is valid. However, the time-derivative ratio between (logarithmized)investment in terms of manpower and returns in terms of first-rate contributors seems not to belinear, but rather to fluctuate vividly, pointing to the cyclical nature of scientific progress. Withregard to Rescher's principle, in the light of bibliometric indicators no difference occurs betweena natural science like physics and a formal science like mathematics. From mathematical progressof the 19th century, constant or increasing returns in the form of new formulas, theorems andaxioms are observed, which leads to a complementary interpretation of the principle of decreasingmarginal returns as a principle of scientific "mass production".     

Vocational education as preparation for university engineeringmathematics

SARTOR 3 clearly states that A-levels and Advanced GNVQs are suitable preparation for accredited engineering degree courses. In this paper, a comparison is made between the mathematical competence of students with Advanced GNVQs, and those with A-levels. The basis of this comparison is the results of a mathematics diagnostic test taken during induction week at university. The test covers fundamental areas of mathematics which underpin the study of engineering in higher education. The results indicate that A-level students, of all grades, have better basic mathematical skills than those from the vocational route     

Johannes de Sacrobosco und die <Emphasis Type="Italic">Sphaera</Emphasis>-Tradition in der katholischen Zensur der Frühen Neuzeit

Johannes de Sacrobosco’s (c. 1195–c. 1256) On the Sphere, a short introduction into qualitative cosmology written in the thirteenth century, was the most widely used textbook on cosmology in the early-modern period, being reprinted, re-edited or commented over 320 times. While the reception and circulation of this work in the sixteenth and seventeenth century is well known, one fact has so far escaped the notice of scholars: Sphaera textbooks were subject to several acts of ecclesiastical censorship in the early modern period, even though the content of this work promoted a cosmology that opposed the allegedly heretical implications of Copernicanism. This paper investigates for the very first time the dynamics and motives behind Roman and Iberian censorship in relation to this cosmology treatise. Editions and commentaries published by Protestants were generally regarded as suspect, but rarely prohibited across-the-board. Instead, they were usually approved for scientific use after expurgations had removed problematic theological passages. However, the commentary (1550) authored by the Catholic Mauro da Firenze (1493–1556) was prohibited repeatedly and completely because it contained theologically dangerous ideas. The case studies presented in this paper shall shed light on the dynamics of knowledge within the Sphere tradition from a new perspective, that of the Catholic censorship of books. Moreover, a longitudinal study based on a specific genre of books provides insight into the ideology and practices of early modern catholic book censorship, whereby the well-known problematic relationship between science and religion in the pre-modern period is seen in the context of a confessionalisation of science.     

The random-fuzzy variables: a new approach to the expression of uncertainty in measurement

The good measurement practice requires that the measurement uncertainty is estimated and provided together with the measurement result. The practice today, which is reflected in the reference standard provided by the IEC-ISO "Guide to the expression of uncertainty in measurement," adopts a statistical approach for the expression and estimation of the uncertainty, since the probability theory is the most known and used mathematical tool to deal with distributions of values. However, the probability theory is not the only tool to deal with distributions of values and is not the most suitable one when the values do not distribute in a totally random way. In this case, a more general theory, the theory of the evidence, should be considered. This paper recalls the fundamentals of the theory of the evidence and frames the random-fuzzy variables within this theory, showing how they can usefully be employed to represent the result of a measurement together with its associated uncertainty. The mathematics is defined on the random-fuzzy variables, so that the uncertainty can be processed, and simple examples are given.     

A Survey of Portuguese Mathematics in the Nineteenth Century

The Portuguese University was briefly reformed in 1772, aiming to bring it to the level of its European counterparts; but this was soon cut short by the return to power of reactionary forces. As a consequence of this, and the political and social unrest that characterized the first half of the nineteenth century in Portugal, there was very little production of mathematics in this period. The military academies were the main centres of transmission of mathematical knowledge, and mathematical works were mostly published by the Lisbon Academy of Sciences. In the second half of the nineteenth century the country entered a period of stability. The education reform of 1836 and the Academy's new statutes of 1851 set in train a blossoming of mathematical activity, reflected in the restructuring of the military academies, or their transformation into Polytechnic Schools, of which the Polytechnic School of Lisbon is of particular importance. Mathematics research was further promoted from 1877 onwards by the publication of the first mathematics journal independent of the Academy, which aimed specifically at ending the isolation of Portuguese mathematics. In the final pages of this survey some data is given on the life and work of the two outstanding Portuguese mathematicians of the nineteenth century: Daniel Augusto da Silva (1814–1878) and Francisco Gomes Teixeira (1851–1933).     

Engineering mathematics in the UK: SARTOR-a timely opportunity forreform

SARTOR 3 provides a new framework for the formation of Engineers in the UK and within it the skilled application of a distinct body of knowledge based on mathematics is explicit. This paper presents a critical overview of the mathematical requirements and expected mathematical outcomes for engineering degrees. The issues associated with mathematical preparedness of students entering engineering degrees are examined. The requirements for both Chartered and Incorporated Engineers are identified and the rationale behind a set of proposals for their mathematical education is described. The paper outlines recommendations that will allow benchmarking of the mathematical aspects of degree programmes     

Mengenlehre—Vom Himmel Cantors zur Theoria prima inter pares

On the occasion of the 150th birthday of Georg Cantor (1845–1918), the founder of the theory of sets, the development of the logical foundations of this theory is described as a sequence of catastrophes and of trials to save it. Presently, most mathematicians agree that the set theory exactly defines the subject of mathematics, i.e., any subject is a mathematical one if it may be defined in the “language” (i.e., in the notions) of set theory. Hence the nature of formal definitions plays an important role within the logical foundations of mathematics. Its study is also helpful to answer the question of how it is possible that the set theory as a universal new ontology for the subject of mathematics (as people hoped around 1900) totally failed but nevertheless the language of set theory is successful in all the mathematical practice.     

An Engineering Design Curriculum for the Elementary Grades

The United States has historically excelled in the design of products, processes and new technologies. Capitalizing on this historical strength to teach applied mathematics and science has many positive implications on education. First, engineering design can be used as a vehicle for addressing deficiencies in mathematics and science education. Second, as achievement in mathematics and science is enhanced, a greater number of students at an earlier age will be exposed to technical career opportunities. Third, enhancing elementary and secondary curricula with engineering design can attract underrepresented populations, such as minorities and females, to engineering as a profession. This paper describes a new and innovative engineering design curriculum, under development in the Austin Independent School District (AISD) in Austin, TX. The philosophic goals upon which the curriculum is based include: integrating the design problem-solving process into elementary schools, demonstrating the relationship of technical concepts to daily life, availing teachers with instructional strategies for teaching applied (as opposed to purely theoretical) science and mathematics, and teaching teamwork skills that are so greatly needed in industry and everyday life. Based on these goals, kindergarten, first grade, and second grade engineering design lessons have been piloted in AISD, in conjunction with a University of Texas program for teacher enhancement and preparation.     

数理统计方法在分析测试中的应用

八十年代初,为判明误差,应用数学、试验设计、数据分析等被广泛应用在分析测试中,因Excel能运算一些的数学公式,使数理统计方法在监测中的应用更为便捷,为查明监测误差,并把所有误差减少到允许的范围内,为保证监测质量做出了贡献。