Occasions Quotes - page 4
P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert's sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory.... In the absence of a precise notion of what it means to be evident, either in the intuitive or in the abstract realm, we have no strict proof of Bernays' assertion; practically speaking, however, there can be no doubt that it is correct...
Paul Bernays
Constitution,-the apex of all its intelligences and mighty instincts and dumb longings: it is I? William Conqueror's big gifts, and Edward's and Elizabeth's; Oliver's lightning soul, noble as Sinai and the thunders of the Lord: these are mine, I begin to perceive,-to a certain extent. These heroisms have I,-though rather shy of exhibiting them. These; and something withal of the huge beaver-faculty of our Arkwrights, Brindleys; touches too of the phoenix-melodies and sunny heroisms of our Shakspeares, of our Singers, Sages and inspired Thinkers all this is in me, I will hope,-though rather shy of exhibiting it on common occasions.
Thomas Carlyle
My philosophy isn't only not conducive to misanthropy, as it might appear to a superficial reader, and as many have accused me. It essentially rules out misanthropy, it tends toward healing, to dissolving discontent and hatred. Not knee-jerk hatred but the deep-dyed hatred that unreflective people who would deny being misanthropes so cordially bear (habitually or on select occasions) toward their own kind in response to hurts they receive-as we all do, justly or not-from others. My philosophy holds nature guilty of everything, it acquits mankind completely and directs our hate, or at least our lamentations, to its matrix, to the true origin of the afflictions living creatures suffer, etc.
Giacomo Leopardi