"All quantitative determinations are in the hands of mathematics, and it at once follows from this that all speculation which is heedless of mathematics, which does not enter into partnership with it, which does not seek its aid in distinguishing between the manifold modifications that must of necessity arise by a change of quantitative determinations, is either an empty play of thoughts, or at most a fruitless effort. In the field of speculation many things grow which do not start from mathematics nor give it any care, and I am far from asserting that all that thus grow are useless weeds, among them may be many noble plants, but without mathematics none will develop to complete maturity." (Johann F Herbart, Werke, 1890)
"If philosophers understood mathematics, they would know that indefinite speech, which permits each one to think what he pleases and produces a constantly increasing difference of opinion, is utterly unable, in spite of all fine words and even in spite of the magnitude of the objects which are under contemplation, to maintain a balance against a science which instructs and advances through every word which it utters and which at the same time wins for itself endless astonishment, not through its survey of immense spaces, but through the exhibition of the most prodigious human ingenuity which surpasses all power of description." (Johann F Herbart, Werke, 1890)
"Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skillful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures."(Johann F Herbart, 1890)
"Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself." (Johann F Herbart, Werke, 1890)
"Mathematics renders its best service through the immediate furthering of rigorous thought and the spirit of invention." (Johann F Herbart, "Mathematischer Lehrplan fur Realschulen", 1890)
"Mathematics, the priestess of definiteness and clearness." (Johann F Herbart, Werke, 1890)
"Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"The great science [mathematics] occupies itself at least just as much with the power of imagination as with the power of logical conclusion." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"The real finisher of our education is philosophy, but it is the office of mathematics to ward off the dangers of philosophy." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"There is perhaps nothing which so occupies, as it were, the middle position of mathematics, as trigonometry." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"The idea that aptitude for mathematics is rarer than aptitude for other subjects is merely an illusion which is caused by belated or neglected beginners." (Johann F Herbart, "Umriss pädagogischer Vorlesungen", 1902)
"The study of mathematics - from ordinary reckoning up to the higher processes - must be connected with knowledge of nature, and at the same time with experience, that it may enter the pupil's circle of thought." (Johann F Herbart, "Letters and Lectures on Education", 1908)
"Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself." (Johann F Herbart, Werke, 1890)
"Mathematics renders its best service through the immediate furthering of rigorous thought and the spirit of invention." (Johann F Herbart, "Mathematischer Lehrplan fur Realschulen", 1890)
"Mathematics, the priestess of definiteness and clearness." (Johann F Herbart, Werke, 1890)
"Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"The great science [mathematics] occupies itself at least just as much with the power of imagination as with the power of logical conclusion." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"The real finisher of our education is philosophy, but it is the office of mathematics to ward off the dangers of philosophy." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"There is perhaps nothing which so occupies, as it were, the middle position of mathematics, as trigonometry." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)
"The idea that aptitude for mathematics is rarer than aptitude for other subjects is merely an illusion which is caused by belated or neglected beginners." (Johann F Herbart, "Umriss pädagogischer Vorlesungen", 1902)
"The study of mathematics - from ordinary reckoning up to the higher processes - must be connected with knowledge of nature, and at the same time with experience, that it may enter the pupil's circle of thought." (Johann F Herbart, "Letters and Lectures on Education", 1908)
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