Mathematics 
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Mathematics
Eustacio Barcatan Elementary Shool

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accordance achievement activity actually admit Analysis applied basis Calculus called claim completely component concepts consciousness constitute constructed continued creation critical curiosity determination discovery distinct doctrine domain doubt educated essential estimate example exist experiment express external fact faculties fields final finite flourish follow former foundations Function fundamental genius geometry hand higher hope human hundred ideas imagination implied importance infinite interests kind knows learning less light magnitude mathe mathematician mathematics matics matter means measurement merely method mind movement namely nature never nevertheless notion objects observation once operations opinions philosopher physical principles problems processes propositions pure question realm reason regarding relation require scientific sense sensuous serve significance single space speak spirit student Symbolic Logic symbols Theory things thought thousand tion transformations true universe valid variety whole
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Page 19  ... one of the greatest discoveries of our age" — is the fact that the basis of logic is the basis of mathematics also. Thus the two great components of the critical movement, though distinct in origin and following separate paths, are found to converge at last in the thesis: Symbolic Logic is Mathematics, Mathematics is Symbolic Logic, the twain are one. Is it really so? Does the identity exist in fact? Is it true that so simple a unifying foundation for what has hitherto been supposed two distinct...
Page 44  Not in the ground of need, not in bent and painful toil, but in the deepcentred playinstinct of the world, in the joyous mood of the eternal Being, which is always young, Science has her origin and root; and her spirit, which is the spirit of genius in moments of elevation, is but a sublimated form of play, the austere and lofty analogue of the kitten playing with the entangled skein or of the eaglet sporting with the mountain winds.
Page 12  Group, thus simply exemplified, though it had barely emerged into consciousness a hundred years ago, has meanwhile become a concept of fundamental importance and prodigious fertility, not only affording the basis of an imposing doctrine — the Theory of Groups— but therewith serving also as a bond of union, a kind of connective tissue, or rather as an immense cerebrospinal system, uniting together a large number of widely dissimilar doctrines as organs of a single body.
Page 26  ... physicist, the geologist, or other student of objective science looks abroad in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth into the universe of logic in quest of the things that are there; exploring the heights and depths for facts — ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless...
Page 42  Again, it is in the mathematical doctrine of Invariance, the realm wherein are sought and found configurations and types of being that, amid the swirl and stress of countless hosts of transformations, remain immutable, and the spirit dwells in contemplation of the serene and eternal reign of the subtile...
Page 11  Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint, — an enchanted realm where thought is double and flows throughout in parallel streams.
Page 10  ... the penetrative power of radium emanations, are none of them done by direct repeated application of a unit or by any direct method whatever. They are all of them accomplished by one form or another of indirection. It was perception of this fact that led the famous philosopher and respectable mathematician, Auguste Comte, to define mathematics as "the science of indirect measurement.
Page 29  Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a palette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering.
Page 25  For, phrase it as you will, there is a world that is peopled with ideas, ensembles, propositions, relations, and implications, in endless variety and multiplicity, in structure ranging from the very simple to the endlessly intricate and complicate. That world is not the product but the object, not the creature but the quarry of thought, the entities composing it — propositions, for example, — being no more identical with thinking them than wine is identical with the drinking of it. Mind or no...
Page 10  ... capable of increase and decrease and measurement." Measurability was the essential thing. That definition of the science was a very natural one, for magnitude did appear to be a singularly fundamental notion, not only inviting but demanding consideration at every stage and turn of life. The necessity of finding out how many and how much was the mother of counting and measurement, and mathematics, first from necessity and then from pure curiosity and joy, so occupied itself with these things that...