Two-Year College Mathematics Journal最新文献_第9页

Inventor's Paradox 发明家的悖论

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027074

W. Page, M. Siu

引用次数: 1

Integration by Geometric Insight-A Student's Approach 几何洞察力的整合——一个学生的方法

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027075

A. D. Holley

引用次数: 0

A Space-Filling Torus 一个空间填充环面

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027071

Dan Wheeler, D. Sklar

引用次数: 0

The Universal Domination of Geometry. 几何学的普遍统治。

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027068

J. Dieudonne

{"title":"The Universal Domination of Geometry.","authors":"J. Dieudonne","doi":"10.2307/3027068","DOIUrl":"https://doi.org/10.2307/3027068","url":null,"abstract":"Contrary to popular belief, the idea of \"pure\" geometry, divorced from algebra, is quite recent (early nineteenth century) and was completely foreign to the Greeks. They had no algebra in the modern sense, but their geometry was really an \"algebro-geometry,\" a complex mixture of purely geometric arguments, and computations with ratios of segments. For instance, most properties of conics are proved by Apollonius by using what we now would call the cartesian equation of the curve with respect to two axes consisting of a tangent to the conic and the corresponding diameter. On the other hand, the solution of quadratic equations is presented as a geometric construction of areas. Finally the method of solving an algebraic problem by finding the intersection of two curves is one of the oldest in Greek mathematics: the solution given by Maenechmus to the problem of \"duplicating the cube\" (i.e., finding a ratio x/y such that (x/y)3 = a/b, a given ratio) was to intersect the two curves xy = ab, ay = x2, and this introduced the conics in mathematics. As soon as the invention of coordinates enormously enlarged the scope of geometry, both Descartes and Newton emphasized the possibility of using this new tool to solve far more complicated equations by intersection of curves. Thus, again contrary to what most people think, the method of coordinates worked both ways, linking algebra and geometry rather than replacing one by the other. This possibility of interpreting algebraic problems in geometric terms had a great appeal for mathematicians ever since the end of the seventeenth century. Unfortunately it was limited to problems dealing with 2 or 3 independent variables, but with the development of Mechanics and Astronomy, problems with an arbitrary number of variables (the \"degrees of freedom\") became more and more common; furthermore, the passage from n variables to n + 1 variables usually did not modify the algebraic treatment in an appreciable way. The temptation to use in such problems a language inspired by geometry thus became irresistible by the middle of the nineteenth century; after 1870 it was generally agreed that one could use in mathematics a conventional language, derived from ordinary geometry, without of course claiming any more that it corresponded to an underlying physical reality. Instead of speaking of a system of n numbers, one would say it was a \"point in n-dimensional space;\" the set of such systems satisfying a linear equation would be called a \"hyperplane,\" etc. The success of this idea has been amazing, and has proliferated during the last century in an unexpected variety of ways. In the remainder of this paper I would like to emphasize some of the highlights of this evolution.","PeriodicalId":448750,"journal":{"name":"Two-Year College Mathematics Journal","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117205116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 12

Shouldn't We Teach Geometry?. 我们不应该教几何吗?

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027069

B. Grünbaum

{"title":"Shouldn't We Teach Geometry?.","authors":"B. Grünbaum","doi":"10.2307/3027069","DOIUrl":"https://doi.org/10.2307/3027069","url":null,"abstract":"The point of view stressed by Dieudonne sees geometry in almost all currently active and important branches of mathematics. Naturally, from such a point of view, we teach geometry whenever we teach mathematics. This view goes far back-was it not customary to call every mathematician \"geometer\"?-but somehow, I cannot find much solace or joy in this view. It seems to me that to call all the activities described by Dieudonne by the name \"geometry\" is rather analogous to-and as justified as-calling the activities of a French chef \"agriculture\". Indeed, in both cases much love, thought and effort is given to specially chosen and suitable objects (meats, vegetables, theorems) while converting them into delicacies for the true connoisseurs. And in both cases, many of the \"raw materials\" must be wasted if the only destiny deemed worthwhile for them is the transformation into gourmet dishes, or highly abstract mathematics. Dropping the simile, it seems to me that the situation of geometry in colleges and universities is much less satisfactory than the opinions of my colleagues might have led you to believe. To justify this assertion, I should probably first give a brief explanation of what I mean by \"geometry\". A good point of departure is provided by the illustration in Figure 1, which I found in a card store a short time ago. I hope that it conveys to you the same idea it conveys to me-that geometry is a special way of replacing objects of the real world by \"simpler\", \"idealized\" \"figures and shapes\", and then investigating the mutual relations of these. This, I believe, is what geometry-\"intuitive geometry\"-is, and should be. Geometry arose in such manner and-as I hope to elaborate shortly-this is how meaningful and interesting questions continue to arise. In Greek times this process of abstraction led to the points, segments, circles, triangles and all the other objects of Euclidean geometry, and in the course of its development gave the ancients the opportunity to create the deductive system we know as Euclidean geometry.(l) Unfortunately, in later times this great achievement of the ancients was perverted to a religion. And even the Age of Reason did not dispel the superstition of the divine or a priori root of Euclidean geometry until about 150 years ago for the most \"radical\" among mathematicians, -and not at all, to this day, for most mathematicians and laymen alike. Very few people are conscious of the fact that all geometry -and Euclidean geometry in particular-is a product of our thinking and represents just one of the ways in which we try to communicate about our surroundings and understand certain aspects of reality.","PeriodicalId":448750,"journal":{"name":"Two-Year College Mathematics Journal","volume":"341 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123119019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 8

Structure vs. Substance: The Fall and Rise of Geometry. 结构与物质:几何学的兴衰。

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027070

R. Osserman

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引用次数: 6

Geometry Is Alive and Well and Living in Paris under an Assumed Name 几何活得很好，化名住在巴黎

Two-Year College Mathematics Journal Pub Date : 1981-09-01 DOI: 10.2307/3027067

D. Albers

引用次数: 1

An Interview with Stan Ulam 采访Stan Ulam

Two-Year College Mathematics Journal Pub Date : 1981-06-01 DOI: 10.2307/3027387

Anthony Barcellos, S. Ulam

引用次数: 1

Corrections to an Earlier Capsule 对早期胶囊的修正

Two-Year College Mathematics Journal Pub Date : 1981-06-01 DOI: 10.2307/3027391

R. Johnsonbaugh, E. F. Schuster

引用次数: 0

On Dice-Sum Frequencies 骰子和频率

Two-Year College Mathematics Journal Pub Date : 1981-06-01 DOI: 10.2307/3027395

V. N. Murty

引用次数: 2