Arithmetic mean

“ And the second devoted her effort to the portrayal of arithmetic, and she strove hard to represent clearly how it wisely enumerates the days and the hours of time, and the water of the sea drop by drop, and then all the sand, and the stars one by one, knowing well how to tell the truth, and how many leaves there are in the woods: such is the skill of arithmetic that numbers have never deceived her, nor will she ever be in error when she wishes to apply her sense to them. ”

“ Arithmetic is the art or science of computing by numbers. It is national, political, military, and commercial. It is of the highest importance to the community; because it pre-eminently teaches us to take care of Number I. Our ministers succeed according to their knowledge of the science of numbers. Witness the skilful management of majorities of the lower house.
He who understands the true art of Addition, Subtraction, Multiplication, and Division, as here laid down, will not be considered a mere cipher in [4] the world ”

“ So in all the affairs of life, the arithmetical part of the business is the dominant one. How many and how much have we? How many and how much do we want? How constantly does noble Arithmetic of the finite lose itself in base Avarice of the Infinite, and in blind imagination of it! ”

“ But mere counting does not imply either the possession or the use of anything that can be really called the mathematical faculty, the exercise of which in any broad sense has only been possible since the introduction of the decimal notation. The Greeks, the Romans, the Egyptians, the Jews, and the Chinese had all such cumbrous systems, that anything like a science of arithmetic, beyond very simple operations, was impossible ”

“ When we enter into the spirit of the methods of arithmetic, we perceive that they all flow clearly and simply from the very principles of numeration, from some precise definitions, and from certain ideas of relations between numbers, which all minds easily perceive, and which they even possessed in advance, before their teacher made them recognize them and taught them to class them in a methodical and fruitful order. ”

“ The principle is obvious enough and would probably be admitted in theory by any intelligent teacher, but in practice we are still wedded to conventional usages which arose long before the psychology of arithmetic was studied. For example, we inherit the convention of studying addition of integers thoroughly, and then subtraction, and then multiplication, and then division, and many of us follow it though nobody has ever given a proof that this is the best order for arithmetical learning. ”

“ In every subject the blackboard is freely used, and in many has become indispensable. The old-fashioned mental arithmetic has given place to the so-called "practical" arithmetic, a name which seems to be a misnomer, since the student of it is, for the rest of his life, committed to the use of pencil and paper for any mathematical computation higher than the multiplication table. ”

“ All we need to say is that in any system for the calculation of real world quantities that employs logically true and non-falsifiable statements of arithmetic these statements can be replaced by a statement or statements that are not non-falsifiable statements of arithmetic. ”

“ It is possible, as one knows from experience, for a boy to have mastered all the arithmetical rules and tables with which his master has supplied him, and to have all his measures and weights at his fingers' ends, and yet to be so destitute of the arithmetical sense as to give without a moment's misgiving an entirely nonsensical answer to a simple arithmetical problem,—to say, for example, as I have known half a class of boys say, that a room is five shillings and sixpence wide. Such a boy, though his head may be stuffed with arithmetical information, has no knowledge of arithmetic. ”

“ Arithmetic especially is only an instrument, a tool, the theory of which we certainly ought to know, but the practice of which it is above all important most thoroughly to possess. The methods of analysis and of mechanics, invariably lead to solutions whose applications require reduction into numbers by arithmetical calculations. ”

“ While the putting of arithmetic after geometry and algebra may excite the greater surprise, it is practically the most defensible part of the whole inversion. The most important processes of advanced arithmetic are only explainable on algebraic or geometric grounds. ”

“ On the contrary, the premisses of arithmetic can be put in other forms, and, furthermore, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them. Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science. It is no more than an interesting subdivision in a general theory. ”

“ They should not include any elaborate practice with very large numbers, or decimals beyond thousandths, or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year.
When we have an adequate sociology of arithmetic, stating accurately just who should use each arithmetical ability and how often, we shall be able to define the task of the elementary school in this respect. ”

“ Efforts to arrive at a much more accurate definition of the term arithmetic are apt to meet with disappointment. On the one hand, we meet with contradictory classifications among works of the highest authority. The great mathematical encyclopedia which is being published almost simultaneously in German and French includes determinants under arithmetic, while the International catalogue of scientific literature places determinants under algebra. ”

“ When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers. ”

“ But mathematics uses these facts to deduce more and more complicated theorems, while philosophy seeks, by analysis, to go behind these facts to others, simpler, more fundamental, and inherently more fitted to form the premisses of the science of arithmetic. The question, “What is a number?” is the pre-eminent philosophic question in this subject, but it is one which the mathematician as such need not ask, provided he knows enough of the properties of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher’s question. ”

“ In seeking to determine with precision in what determinations of values properly consist, we easily recognize that they are nothing else but veritable transformations of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by analysis. In this point of view, the calculus of values might be simply conceived as an appendix, and a particular application of the calculus of functions, so that arithmetic would disappear, so to say, as a distinct section in the whole body of abstract mathematics. ”

“ Given that tally systems are contingent with regard to number, the logically true system of arithmetic is not only useful but essential. This might not be immediately apparent because it might seem that tally systems can be identified as systems of counting by the fact that the totals they produce stand in a one-to-one relation with objects in the real world. ”

“ It should be employed widely in accounting and the treatment of commercial problems, and would be except for outworn conventions. It is a leading contribution of algebra to business and industrial life. Arithmetic can make it nearly as well. It saves more time in the case of drills on reducing fractions to higher and lower terms alone than is required to learn its meaning and use. To rewrite a quantitative problem [Pg 78] as an equation and then make the easy selection of the necessary technique to solve the equation is one of the most universally useful intellectual devices known to man. ”

“ Why not instead adopt the easier course of simply trying to translate each of the applications of mathematics?" In order to give the empirical/realist account we need not say that every statement of arithmetic is induced from observations of real world quantities. Nor need we say that the system of arithmetic is open to falsification, but is not in fact never falsified by the observation of real world quantities. ”

“ All such careful arithmetical computations add clearness and definiteness to historical and geographical ideas. The natural sciences have been so little systematically taught in our common schools, that we are scarcely able to realize what connection may be made between them and arithmetic. ”

“ If one were inclined to adopt the common definition that arithmetic is the science of the relations existing between numbers, one would be perplexed by the fact that the theory of groups of finite order is classed with arithmetic in the encyclopedia mentioned above, while it might be difficult to name any other mathematical subject which makes less direct use of numbers than this theory does. ”

“ Since the abilities which together constitute arithmetic ability are thus specialized, the individual who is the best of a thousand of his age or grade in respect to, say, adding integers, may occupy different stations, perhaps from 1st to 600th, in multiplying with integers, placing the decimal point in division with decimals, solving novel problems, copying figures, etc., etc. ”

“ Arithmetic was not practised methodically by the setting of sums to be worked out by the pupils, but consisted chiefly in demonstrating the more obvious properties of numbers, such as odd, even, prime, perfect, etc., together with many fanciful absurdities. ”

“ The transformation of functions belongs strictly to algebra. Teachers who love arithmetic, and who have themselves success in working out the so-called numerical conundrums, defend with much earnestness the current practice which uses so much time for arithmetic. They see in it a valuable training for ingenuity and logical analysis, and believe that the industry which discovers arithmetical ways of transforming the functions given in such problems into plain numerical operations of adding, subtracting, multiplying, or dividing is well bestowed. ”

“ IN this age the knowledge of arithmetic is so widespread that it is difficult to conceive that it is comparatively young. The three R's, Reading, 'Riting and 'Rithmetic, seems to our mind almost axiomatic and fundamental in regard to education. It is, therefore, very interesting to read in histories of arithmetic that the earliest known book which contains a systematic exposition of the decimal system of numeration was written in the seventh century, and that our familiar arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, can not be traced to an earlier century than the seventh. ”

“ Results led to the conclusion that arithmetical ability tends to be represented in two main divisions: (1) the power to compute with ease and readiness, and (2) the power to solve problems by arithmetic, which involves the application of a higher degree of ability than is required in computation.
Arithmetical tests of various kinds correlate more closely than do arithmetical tests with non-arithmetical tests. “Hence we are compelled to interpret this relationship as evidence distinctly in favor of Burt’s suggestion, that there is an essential unity in arithmetical ability.” ”

“ All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses, in the manner we have described.
We may, if we please, call the proposition, "Three is two and one," a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. ”

“ Months, if not years, are required to store away the perplexities, incongruities, and inconsistencies of the existing weights and measures, and then memory must often fail in reproducing them. The mystery of compound arithmetic is essential in the calculations they require. All this is done away by the decimal progression, so that the first four rules of arithmetic are ample for the pupil.
Looking closely at the metric system, we must confess its simplicity and symmetry. Like every creation of science, it is according to rule. Master the rule and you master the system. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”