“The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them, ] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it”

— Paul Bernays

Paul Bernays, Platonism in mathematics http://sites.google.com/site/ancientaroma2/book_platonism.pdf (1935)

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Swiss mathematician 1888–1977

Alistair Cameron Crombie (1915–1996) Australian zoologist, historian of science

Robert Grosseteste and the Origins of Experimental Science 1100-1700 (1953)

“I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary -- being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, A treatise on differential equations (1859), p. v; cited in: Quotations by George Boole http://www-history.mcs.st-and.ac.uk/Quotations/Boole.html, MacTutor History of Mathematics, August 2010.

“Only an unashamed dogmatist would dare to assert that the issue has finally been resolved now, in favor of the view that, outside logic or mathematics, the method of modern science is the only method to employ in seeking knowledge. The dogmatist who made this assertion would have to be more than ashamed. He would have to blind himself to the fact that his own assertion was not established by the experimental method, nor made as an indisputable conclusion of mathematical reasoning or of purely logical analysis.”

Robert Maynard Hutchins (1899–1977) philosopher and university president

Great Books: The Foundation of a Liberal Education (1954)

“History’ and ‘Political Economy’ have not been differentiated on the basis of systematic reflection; rather, they have been quite different in origin and conceptual structure. Only on further development of both disciplines are they set closer together and merged into a single science, namely ‘Sociology’, which for about a hundred years past has been assimilating other fields of science.”

Otto Neurath (1882–1945) austrian economist, philosopher and sociologist

Source: 1930s, "Empirical Sociology" (1931), p. 319; Lead paragraph

“The truths of the ṛṣis are not evolved as the result of logical reasoning or systematic philosophy but are the products of spiritual intuition, dṛṣti or vision.”

Sarvepalli Radhakrishnan (1888–1975) Indian philosopher and statesman who was the first Vice President and the second President of India

Internet Encyclopedia of Philosophy
Context: The truths of the ṛṣis are not evolved as the result of logical reasoning or systematic philosophy but are the products of spiritual intuition, dṛṣti or vision. The ṛṣis are not so much the authors of the truths recorded in the Vedas as the seers who were able to discern the eternal truths by raising their life-spirit to the plane of universal spirit. They are the pioneer researchers in the realm of the spirit who saw more in the world than their followers. Their utterances are not based on transitory vision but on a continuous experience of resident life and power. When the Vedas are regarded as the highest authority, all that is meant is that the most exacting of all authorities is the authority of facts.

“Even with the loss of Puerto Argentino and without internationalizing the conflict, we should have continued the action in such a way that the enemy would have been faced by serious, permanent and systematic difficulties and risks and be obliged to realize that we Argentines were not going to surrender.”

Leopoldo Galtieri (1926–2003) Argentine military dictator

"Galtieri muses on what-if's of Falkland war" http://www.nytimes.com/1982/09/16/world/galtieri-muses-on-what-if-s-of-falkland-war.html, The New York Times (September 16, 1982)

“I thought… that the fact that in characterizing truth for a language it is necessary to put words into relations with objects was enough to give some grip for the idea of correspondence; but this now seems to me a mistake. The mistake is in a way only a misnomer, but terminological infelicities have a way of breeding conceptual confusion, and so it is here. Correspondence theories have always been conceived as providing an explanation or analysis of truth, and this, a Tarski-style theory of truth, certainly does not do.”

Donald Davidson (philosopher) (1917–2003) scriitor amator

Donald Davidson (1990, p. 135), as cited in: Simon Evnine (1991) Donald Davidson. p. 137

“Calculus systematically evades a great deal of numerical calculation.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“We shall see that the mathematical treatment of the subject [of electricity] has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.”

James Clerk Maxwell book A Treatise on Electricity and Magnetism

A Treatise on Electricity and Magnetism (1873), §36.

“The success of mathematical physics led the social scientist to be jealous of its power without quite understanding the intellectual attitudes that had contributed to this power. The use of mathematical formulae had accompanied the development of the natural sciences and become the mode in the social sciences. Just as primitive peoples adopt the Western modes of denationalized clothing and of parliamentatism out of a vague feeling that these magic rites and vestments will at once put them abreast of modern culture and technique, so the economists have developed the habit of dressing up their rather imprecise ideas in the language of the infinitesimal calculus.”

Norbert Wiener (1894–1964) American mathematician

Source: Ex-Prodigy: My Childhood and Youth (1964), p. 89; partly cited in: Herman E. Daly. Steady-State Economics: Second Edition With New Essays. 1977/1991 p. 4

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