Introducing the Main Characters

2019 ◽  
pp. 14-34
Author(s):  
Ohad Nachtomy
Keyword(s):  
Infinite Series ◽  
Infinite Number ◽  
Potential Infinity ◽  
The Law

This first chapter introduces the central concepts and distinctions that Leibniz uses in articulating his view of infinity. In other words, the author introduces the main players in this book. These include: Leibniz’s rejection of infinite number; his distinction between infinite being and infinite number; degrees of infinity; the distinction between actual and potential infinity; indivisibility; his syncategorematic approach to infinite terms; his distinction between infinite number and infinite series; the law of the series; and the distinction between primitive force and derivative force. The chapter’s aim is to present at the outset some of the terminology and concepts used in the book in order to present Leibniz’s approach to infinity—that is, to clarify the major resources needed in order to present his complex views. At the same time, this serves as a sketch of (what the author takes to be) Leibniz’s approach to infinity.

scholarly journals On the Reduction of Ferrers' Associated Legendre Function

1929 ◽  
Vol 1 (4) ◽  
pp. 241-243
Author(s):  
Hrishikesh Sircar
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Legendre Function ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Reduced Order ◽  
Zonal Harmonics ◽  
Integral Degree ◽  
Associated Legendre Function

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

scholarly journals Some remarks on the paper by B. G. Marsden “An attempt to reconcile the dynamical and radar determinations of the Astronomical Unit”

1965 ◽  
Vol 21 ◽  
pp. 237-239
Author(s):  
Eugene Rabe
Keyword(s):  
Least Squares ◽  
Infinite Series ◽  
Infinite Number ◽  
Least Squares Method ◽  
Fundamental Principle ◽  
Astronomical Unit ◽  
Number Of Solutions ◽  
Least Squares Solution ◽  
Minimum Value

While Marsden's solution C leaves residuals with the relatively small [vv] of 13.73, it should be realized that this representation of the observations of Eros does not satisfy the fundamental principle of the least squares method, in so far as the associated value of [vv] is not a minimum with respect to small arbitrary deviations from solution C. As a matter of fact, there is an infinite number of “solutions” with [vv] between the 13.73 of Marsden's solution C and the 8.66 of his solution A, each of these being associated with a certain arbitrarily prescribed value of the mass of Mars and with a related mass of Earth + Moon. Of this infinite series of solutions, only solution A is a least squares solution in the true sense, with a minimum value of [vv]. This can be seen and verified as follows.

Geometrical ray theory: Edge‐diffracted rays and their traveltimes (second‐order approximation of the traveltimes)

Geophysics ◽  
1994 ◽  
Vol 59 (1) ◽  
pp. 148-155 ◽  
Author(s):  
Jianguo Sun
Keyword(s):  
Infinite Number ◽  
Order Approximation ◽  
Second Order ◽  
Transformation Matrix ◽  
Ray Theory ◽  
Second Order Approximation ◽  
Edge Diffraction ◽  
Global Properties ◽  
The Law ◽  
Reflecting Surface

The general laws presenting global properties of reflected and transmitted rays are extended to the edge‐diffracted rays by the use of the law of edge diffraction. Similar to reflected and transmitted rays, edge‐diffracted rays and their traveltimes are described by the same four 2 × 2 matrices that are submatrices of the symplectic ray transformation matrix. If the incident rays have no line and point focus in the reflector having edges, one cone of diffracted rays corresponds just to one diffraction point. Otherwise, there may be an infinite number of cones of diffracted rays that are from one and the same diffraction point, and there may be no diffracted ray if the incident rays have a point focus in the reflecting surface with edges. For some cases, edge diffractions are the same as those from point diffractors and all diffracted rays are focused on the point focus of the reflected rays in the earth’s surface.

scholarly journals A Version of Hagen's Proof of the “Law of Error.”

1919 ◽  
Vol 38 ◽  
pp. 24-26
Author(s):  
R. F. Muirhead
Keyword(s):  
Least Squares ◽  
Infinite Number ◽  
Method Of Least Squares ◽  
Maximum Value ◽  
The Law ◽  
Image Position ◽  
8Th Edition

Hagen's proof (1837), as described in the 8th edition of Mansfield Merriman's “Method of Least Squares,” is based on the assumption that the error may be supposed to consist of the algebraic sum of an infinite number of infinitesimal errors of equal amount ε, each one of which is equally likely to be positive or negative. Thus if 2m is the number of the infinitesimal errors, the probability of the error x ≡ 2p ε occurring isand the maximum value of P occurs when p = 0, and is.

Wholes, Parts, and Infinite Collections

Philosophy ◽  
1992 ◽  
Vol 67 (261) ◽  
pp. 367-379 ◽  
Author(s):  
P. O. Johnson
Keyword(s):  
Infinite Series ◽  
Infinite Number ◽  
Common Sense ◽  
The Other ◽  
Bertrand Russell ◽  
Popular View ◽  
The Common ◽  
The One ◽  
Infinite Set

In his book, The Principles of Mathematics, the young Bertrand Russell abandoned the common-sense notion that the whole must be greater than its part, and argued that wholes and their parts can be similar, e.g. where both are infinite series, the one being a sub-series of the other. He also rejected the popular view that the idea of an infinite number is self-contradictory, and that an infinite set or collection is an impossibility. In this paper, I intend to re-examine Russell's wisdom in doing both these things, and see if it might not have made more sense, and caused his enterprise fewer problems, if he had simply stuck to our commonplace ideas. To this end, I shall also be considering his treatment of certain paradoxes that he claims can only be resolved by the abandonment of the above notions, as well as certain others which his theories appear to have generated.

Constitutional Theory as a Problem of Constitutional Law: On the Constitutional Court's Total Revision of Austrian Constitutional Law

1998 ◽  
Vol 32 (4) ◽  
pp. 567-590
Author(s):  
Alexander Somek
Keyword(s):  
Infinite Number ◽  
Constitutional Law ◽  
Constitutional Theory ◽  
Legal Reason ◽  
The Law

La. I clear it thus out of Sir Edw. Coke 1 Inst. Sect. 138. that this [Legal Reason] is to be understood of an artificial perfection of Reason gotten by long Study, Observation and Experience, and not of every Mans natural Reason; for Nemo nascitur Artifex. This Legal Reason is summa Ratio; and therefore if all the Reason that is to be dispersed into so many several heads were united into one, yet could he not make such a Law as the Law of England is, because by so many successions of Ages it hath been fined and refined by an infinite number of Grave and Learned Men.Ph. […] I grant you that the knowledge of the Law is an Art, but not that any Art of one Man, or of many how wise soever they be, or the work of one and more Artificers, how perfect soever it be, is Law. It is not Wisdom, but Authority that makes a Law. […] That the Law hath been fined by Grave and Learned Men, meaning the Professors of the Law is manifestly untrue, for all the Laws of England have been made by the Kings of England, consulting with the Nobility and Commons in Parliament, of which not one of twenty was a Learned Lawyer.

scholarly journals PHILOSOPHICAL AND RELIGIOUS VIEWS OF FLORENSKY AND THE IDEA OF ACTUAL INFINITY

2019 ◽  
pp. 108-113
Author(s):  
Dmitry N. Radul ◽  
Keyword(s):  
20Th Century ◽  
Western Europe ◽  
Early 20Th Century ◽  
Actual Infinity ◽  
European Culture ◽  
Potential Infinity ◽  
History Of ◽  
The Law ◽  
The One

The article briefly observes the history of the idea of the actual infinity in European culture until the beginning of the 20th century. Special attention is paid to the role of Cantor set theory in reviving interest in the idea of actual infinity in Western Europe and Russia. The influence of the Cantor’s philosophy of religion on the Western European theology of the late 19th century - early 20th century is given. The influence of Cantor’s ideas on the formation of Florensky’s views is described. A detailed analysis of the application of the idea of actual infinity in the book “The Pillar and the Statement of Truth” is given. Florensky describes the understanding of the connection of Kant’s antinomical of reason and the idea of a potential infinity. The potential infinity is considered by Florensky as a source of imperfection and sinfulness. Special attention is paid to the understanding of truth as actual infinity. The introduction of the actual infinity allows Florensky to remove the one-sidedness of the law of identity and the law of sufficient basis in the Supreme unity...

On the expansion in a series of the attraction of a spheroid

1833 ◽  
Vol 2 ◽  
pp. 162-163
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Joint Effect ◽  
The Other ◽  
Original Function ◽  
Simple Mode ◽  
Rectangular Coordinates ◽  
Rotatory Motion

Mr. Ivory’s principal object in this paper appears to be the removal of some difficulties in the demonstration of the method of developing the attractions of spheroids in an infinite series, as employed by Laplace in the Mécanique Céleste . It is natural to think, he observes, that the theory of the figure of the planets would be placed on a firmer basis if it were deduced directly from the general principles of the case, than when it is made to depend on a nice and somewhat uncertain point of analysis; and he conjectures that the theory will probably be found to hinge on this proposition,—that a spheroid, whether homogeneous or heterogeneous, cannot be in equilibrium by means of a rotatory motion about an axis, and the joint effect of the attraction of its own particles and of the other bodies of the system, unless its radius be a function of three rectangular coordinates; for if this proposition were clearly and rigorously demonstrated, the analysis of Laplace, on changing the ground on which it is built, would require little or no alteration in other respects. Without, however, attempting to demonstrate this proposition in all its extent, the author has substituted a more direct and simple mode of argument than that of Laplace, which is perfectly conclusive with respect to all the cases to which the theorem in question can possibly require to be applied. He has shown that by immediately transforming a given expression into a function of three rectangular coordinates, we obtain the same development as is deduced in the Mécanique Céleste , by a more general and complicated mode of reasoning, which seems to be so far objectionable, as it tends to introduce a variety of quantities into the series which do not alter its total value, since they destroy each other, but which may possibly interfere with the accuracy of its application to particular cases, in which it may be employed as a symbolical representation: for example, when any finite number of terms is assumed as affording an approximate value; since, if the expression developed has not been reduced to the form of a function of three rectangular coordinates, the development may contain an infinite number of terms, which are introduced by the operation without being essential to its final result. He takes for the example of such a case the equation of a spheroid, prominent between the equator and the poles, somewhat resembling the figure which was once attributed to Saturn; and he shows that its development in the form required will contain an infinite number of quantities arising from the expansion of a radical, which are not to be found in the original function.

What is the purpose of cognition?

2020 ◽  
Vol 43 ◽  
Author(s):  
Aba Szollosi ◽  
Ben R. Newell
Keyword(s):  
Infinite Number ◽  
Human Cognition ◽  
The Way

Abstract The purpose of human cognition depends on the problem people try to solve. Defining the purpose is difficult, because people seem capable of representing problems in an infinite number of ways. The way in which the function of cognition develops needs to be central to our theories.

Ethical Challenges: Less About Moral Wrongdoing and More About Communication Breakdown

2015 ◽  
Vol 20 (3) ◽  
pp. 72-84 ◽  
Author(s):  
Paula Leslie ◽  
Mary Casper
Keyword(s):  
Ethical Challenges ◽  
Speech Language Pathology ◽  
Thickened Liquids ◽  
Communication Breakdown ◽  
Professional Duty ◽  
Language Pathology ◽  
The Law ◽  
Personal Case

“My patient refuses thickened liquids, should I discharge them from my caseload?” A version of this question appears at least weekly on the American Speech-Language-Hearing Association's Community pages. People talk of respecting the patient's right to be non-compliant with speech-language pathology recommendations. We challenge use of the word “respect” and calling a patient “non-compliant” in the same sentence: does use of the latter term preclude the former? In this article we will share our reflections on why we are interested in these so called “ethical challenges” from a personal case level to what our professional duty requires of us. Our proposal is that the problems that we encounter are less to do with ethical or moral puzzles and usually due to inadequate communication. We will outline resources that clinicians may use to support their work from what seems to be a straightforward case to those that are mired in complexity. And we will tackle fears and facts regarding litigation and the law.

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