Pascal\u27s Arithmetical Triangle

In almost all books on College Algebra, the Pascal Triangle is placed in such a position as to show the binomial coefficients in a horizontal line. This position is appropriate when the only use of the triangle is to help students remember these coefficients; but Pascal had a wider purpose in mind. Pascal\u27s Construction. Pascal first sets up a lattice-work consisting of equal squares, assigning to each square a definite positive integer which he determines as follows. 1. The number in each square of the 1st (top) row, and also in each square of the 1st (left-hand) column is to be unity. 2. To each other square is assigned an integer determined by the recurrence relation which he states thus: The number in any other square is equal to the sum of the two numbers immediately to the left, and immediately above it

Bernoulli Numbers

The first appearance of the set of rational numbers of which I am speaking was in James Bernoulli\u27s Ars Conjectandi. This work was published in 1713, eight years after the death of its author. In the 2nd chapter, Bernoulli deals with permutations and combinations

The Address of the President: Science and Progress

Science and human welfare have always risen and fallen together. We find this generalization borne out in all recorded history, and even farther back in the dim horizons of prehistory. From the first attempts at counting and measuring, observation of the precision of the periodic recurrence of the seasons, forging of metal tools and weapons, domestication of animals, planting and harvesting of crops, we see that the slow advance of the status of man in his world has been accompanied or even preceded by the advance in science. In the earlier stages, of course, we find art, culture, and the more spiritual phases of life largely rudimentary in their development, and confined at that to a very small fraction of the human race, the elite who were only enabled to maintain their relatively high type of life at the expense of the forced labor of millions. It is a commonplace of history that the abolition of slavery has never been accomplished by purely moral or humanitarian or even religious influences, but in so far as slavery has disappeared, it has been under the impulse of developing industrialism and technology, making free labor more efficient and hence in the long run more economical than slave labor. And even though today we must recognize that far too many human beings are still what may be called wage slaves we can none the less see that the hope for betterment of these conditions, the hope for further progress in other words, lies first of all in the continued advance of science, in its more complete understanding and conquest of the fields it already occupies, and in its extension into ever new and broader realms

Early History of the Iowa Mathematics Section

On a certain evening in November, 1915, a group of six mathematics teachers had supper together in a small restaurant in downtown Des Moines. As I remember the group, it consisted of Professors Smith, Baker, and Reilly of the State University, Professor Neff of Drake, Professor Wester of State Teachers, and myself. Since we had all been attending the discussions in the Iowa Education Association, we very naturally fell to conversing about the problem of mathematics teaching at the university and college level. We all felt that the programs of the State Teachers Association, as it was then named, left a good deal to be desired when it came to the problems peculiar to the college teacher. When someone, probably Professor A. G. Smith, suggested that what was needed was a new organization which would concern itself with the subject matter of the college field, we all agreed enthusiastically. We finally parted after agreeing that each one in his own sphere would endeavor to rouse sufficient interest in such a new organization that in the near future we might hope to see our desire turned into reality. I am sorry to say that of this group but two remain* ; and I regret very much that Professor Neff is unable to be at this meeting to give his own recollections of that informal gathering which proved so pleasant and profitable to us all. *Within the state of Iowa, I have been informed that Professor Wester is at present living on the Pacific Coast

Primordialists and Constructionists: a typology of theories of religion

This article adopts categories from nationalism theory to classify theories of religion. Primordialist explanations are grounded in evolutionary psychology and emphasize the innate human demand for religion. Primordialists predict that religion does not decline in the modern era but will endure in perpetuity. Constructionist theories argue that religious demand is a human construct. Modernity initially energizes religion, but subsequently undermines it. Unpacking these ideal types is necessary in order to describe actual theorists of religion. Three distinctions within primordialism and constructionism are relevant. Namely those distinguishing: a) materialist from symbolist forms of constructionism; b) theories of origins from those pertaining to the reproduction of religion; and c) within reproduction, between theories of religious persistence and secularization. This typology helps to make sense of theories of religion by classifying them on the basis of their causal mechanisms, chronology and effects. In so doing, it opens up new sightlines for theory and research

Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure