From Numbers to Rings: The Early History of Ring Theory

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From Numbers to Rings: The Early History of Ring Theory

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Reform of the construction of the number system with reference to Gottlob Frege

Due to missing ontological commitments Frege rejected Hilbert’s Fundamentals of Geometry as well as the construction of the system of real numbers by Dedekind and Cantor. Almost all of school mathematics is ontologically committed. Therefore, H.-G. Steiner considered Frege’s viewpoint of mathematics fundamentals, refined by Tarski’s semantics, as suitable for math education. Frege committed numbers ontologically by using measurement to define numbers. He invented the concept of quantitative domain (Größengebiet), which – it is now known by reconstruction of that concept by the New-Fregean Movement – agrees with the concept of quantity domain (Größenbereich) as established in the German reform of the application-oriented construction of the system of real numbers. Concepts of quantity (ratio-scale) and interval-scale in comparative measurement theory – going beyond Frege – show the way how the negative numbers can be ontologically committed and the operations of addition and multiplication can be included. In this work it is shown how Frege’s viewpoint of mathematics fundamentals, as propagated by H.-G. Steiner, can be better implemented in the current construction of the system of real numbers in school.     

From the History of Dispersion Relations in the Chew–Low Model

We formulate a model problem in the dispersion theory of strong interactions at low energies. The problem is based on the crossing symmetry and elastic unitarity of the S-matrix. We point out the role of the extensive studies on dispersion relations performed in Dubna in the 1960s.     

The number of lines in Frege proofs with substitution

We prove that for sufficiently large , there are tautologies of size that require proofs containing lines in axiomatic systems of propositional logic based on the rules of substitution and detachment. Received October 19, 1995     

The polynomial bounds of proof complexity in Frege systems

The concept of a determinative set of variables for a propositional formula was introduced by one of the authors, which made it possible to distinguish the set of hard-determinable formulas. The proof complexity of a formula of this sort has exponential lower bounds in some proof systems of classical propositional calculus (cut-free sequent system, resolution system, analytic tableaux, cutting planes, and bounded Frege systems). In this paper we prove that the property of hard-determinability is insufficient for obtaining a superpolynomial lower bound of proof lines (sizes) in Frege systems: an example of a sequence of hard-determinable formulas is given whose proof complexities are polynomially bounded in every Frege system.     

Frege,Dedekind, and the Modern Epistemology of Arithmetic

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.     

由历史看数学的应用研究

本文从历史的角度 ,研究了数学应用的情况 .     

On the Transmission of Riemann's Ideas to Portugal

Henrique Manual de Figueiredo is a secondary figure in the history of science in Portugal. His name is not recorded in any of the main biographical encyclopedias on his country, in international compilations such as the detailed work of May (Bibliography and Research Manual of the History of Mathematics, Toronto: Univ. of Toronto Press, 1973), or in studies on the history of science and culture in his country. However, he deserves to be remembered as a unique pioneer in the transmission to Portugal of Riemann's work, in particular of Riemann surfaces and the theory of algebraic curves. Although trained within the French tradition, and on friendly terms with French scientists till the end of his life, Figueiredo as a young man turned in the direction of the mathematical ideas then being developed in Germany. His life and work are also interesting from the point of view of the study of the transmission of science to and within peripheral countries and of their choice of foreign models. They suggest that, far from being a slow process of regular diffusion, the transmission of mathematical ideas from leading to peripheral mathematical communities is a complex process with selective sharp advances. Figueiredo was a respected mathematician within the structures of his own country, a professor at the University of Coimbra, who held several official positions in his country and represented it at one of the first international encounters involving science and technology in which peripheral countries took an active participation: the Paris Universal Exhibition of 1900Copyright 1999 Academic Press.Henrique Manuel de Figueiredo é uma figura de segundo plano na história da ciência em Portugal. Não se encontra qualquer referência ao seu nome, quer nas principais enciclopédias biográficas do seu paı́s, quer em publicações internacionais, tais como o minucioso trabalho de May (Bibliography and Research Manual of the History of Mathematics, Toronto: Univ. of Toronto Press, 1973), nem mesmo em estudos sobre a história da ciência e cultura do seu paı́s. Contudo, ele merece ser recordado como pioneiro na divulgação, em Portugal, do trabalho de Riemann, em particular superficies de Riemann e teoria das curvas algébraicas. Apesar da sua formação na escola francesca e de ter mantido laços de amizade com cientistas franceses, durante toda a sua vida, Figueiredo enquanto jovem deixou-se influenciar pelas ideias matemáticas então desenvolvidas na Alemanha. Na sua vida e obra tiveram também um papel importante a divulgação de ciência em paı́ses periféricos e o contributo para a escolha de modelos estrangeiros. Parece que a transmissão das ideias matemáticas dos centros principais para as comunidades matematicas periféricas, longe de ser um processo lento e regular, foi antes um processo complexo com progressos altamente irregulares. Figueiredo foi um matemático conceituado nas estruturas do seu próprio paı́s, era Professor na Universidade de Coimbra, ocupou vários cargos oficiais no seu paı́s e representou-o num dos primeiros encontros internacionais de ciência e tecnologia, a Exposição Universal de Paris, em 1900, na qual paı́ses periféricos tiveram uma participação activa.Copyright 1999 Academic Press.MSC 1991 subject classifications: 01A55; 01A60; 01A70.     

神经网络优选组合预测模型在电力负荷预测中的应用

针对以往的组合预测模型中，最优权重不能保证非负性的问题，引入了神经网络优选组合预测模型。实例验证表明，此模型具有很强的自适应性和较高的预测精度。     

大学数学教学中数学思想的培养

在大学数学教学中引入数学历史人物的生平、研究问题以及研究方法的介绍对培养大学生正确的数学思想有着重要的意义。     

Coherent structure visiometrics: From the soliton to HEC

In this paper I give a brief personal review of the history of the early soliton days and how they led to thevisiometrics and reduced modeling paradigm that has been a part of my approach to nonlinear science in the last three decades. I illustrate it with HEC (Hybrid Eiliptic-Contour): a fast, minimal, asymptotically motivated model for unforced, 2-dimensional incompressible weakly dissipative turbulence (U2DIT).The work on reduced modeling of two-dimensional turbulence is on-going with David G. Dritschel and Hongbing Yao (incompressible); and Jaideep Ray, Thomas Scheidegger and Ravi Samtaney (compressible).