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Updated: May 10, 2025
It says that whether a given formula, derived mathematically from one that was first read from nature, still expresses some fact of nature, cannot be decided by pure mathematical logic, but only by testing it against truly observable phenomena.
Mountains and hills shall be tunnelled, rivers shall be bridged, houses shall be levelled; a road shall be cut through forests; and, when I have finished my march, the course over which I have passed shall be a mathematically straight line. Thus will I show to the world that, when a prince desires to travel, it is not necessary for him to go out of his way on account of obstacles."
"An analogue computer is a machine that ... that sets up conditions mathematically similar to the conditions in some problem and then lets all the operations proceed while it draws a graph a prediction of how the real conditions would turn out.
Professor Zollner, for example, says: "Science can make no use of the substance of intellectual revelations, but must be guided by observed facts and by the conclusions logically and mathematically uniting them" a passage which is quoted with approval by Professor Reichel, and would seem to be endorsed by the silence concerning the religious side of the question which is observed by most of our great scientific supporters.
I inquired of some of the boys how much 811 pennies put into dollars and cents would amount to, but all were so visibly embarrassed that I, remembering my own mathematically tortured childhood, desisted before the schoolmaster could hear.
He entirely failed to see that before dealing with physical astronomy, it must be dealt with mathematically. "It is well to remark," as Mr. Ellis says, "that none of Newton's astronomical discoveries could have been made if astronomers had not continued to render themselves liable to Bacon's censure."
Then we discover, singularly enough, that property may indeed manifest itself accidentally; but that, as an institution and principle, it is mathematically impossible. So that the axiom of the school ab actu ad posse valet consecutio: from the actual to the possible the inference is good is given the lie as far as property is concerned.
One day, while staring with a weary look at the walls of my cell, I suddenly began to feel how irresistibly thick the stone was, how strong the cement which kept it together, how skilfully and mathematically this severe fortress was constructed. It is true, my first sensation was extremely painful; it was, perhaps, a horror of hopelessness.
Such variation is called "chance variation," not because it takes place by "chance" in the properly accepted sense of the term, but because it is so tremendously varied is evidently due to such complicated and little-understood circumstances that it can best be studied mathematically, using statistical applications of the "theory of probabilities."
I presume that the bank there is managed honestly, although I neither know nor care whether it is. But this at least is certain the cagnotte gains 3 per cent. on every spin. Mathematically, a man is bound to lose the capital he invests in every thirty throws when his luck is neither good nor bad. In the long run his luck will leave him with a balanced book minus the cagnotte.
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