In closing, I can only apologize for not having given any positive account of either mathematical truth or mathematical necessity. I can only say that I have not given such an account because I think that the search for such an account is a fundamental mistake. It is not that there is nothing special about mathematics; it is that, in my opinion, the investigation of mathematics must presuppose and not seek to account for the truth of mathematics. But this is the beginning of another paper and not the end of this one.
Hilary PutnamRelated topics
account beginning closing end fundamental given having investigation mathematics mistake necessity nothing opinion paper positive say search seek special think truthRelated quotes
Anyone who compares the bibliography with the reprinted papers will notice that mathematics and physics have not been given equal treatment. There is a strong bias in favor of mathematics. The bias arises from the fact that mathematical papers, provided that are correct and not trivial, have permanent value, whereas most papers in physics journals are ephemeral. For this reason, it is customary to publish complete collected works of mathematicians but only selected works of physicists.
Freeman Dyson
The world and the mind are states of being. The supreme is not a state. It pervades all states, but it is not a state of something else. It is entirely uncaused, independent, complete in itself, beyond time and space, mind and matter. (...) it leaves no traces. There is nothing to recognize it by. It must be seen directly, by giving up all search for signs and approaches. When all names and forms have been given up, the real is with you. You need not seek it. Plurality and diversity are the play of the mind only. Reality is one.
Nisargadatta Maharaj
The following work is not a republication of a former treatise by the Author, entitled, "The Mathematical Analysis of Logic." Its earlier portion is indeed devoted to the same object, and it begins by establishing the same system of fundamental laws, but its methods are more general, and its range of applications far wider. It exhibits the results, matured by some years of study and reflection, of a principle of investigation relating to the intellectual operations, the previous exposition of which was written within a few weeks after its idea had been conceived.
George Boole
All this business of a labour to accomplish, before I can end, of words to say, a truth to recover, in order to say it, before I can end, of an imposed task, once known, long neglected, finally forgotten, to perform, before I can be done with speaking, done with listening, I invented it all, in the hope it would console me, help me to go on, allow me to think of myself as somewhere on a road, moving, between a beginning and an end, gaining ground, losing ground, getting lost, but somehow in the long run making headway.
Samuel Beckett
America could carry on a two years' war by the confiscation of the property of disaffected persons, and be made happy by their expulsion. Say not that this is revenge, call it rather the soft resentment of a suffering people, who, having no object in view but the good of all, have staked their own all upon a seemingly doubtful event. Yet it is folly to argue against determined hardness; eloquence may strike the ear, and the language of sorrow draw forth the tear of compassion, but nothing can reach the heart that is steeled with prejudice.
Thomas Paine
