Classical science was based upon the belief that it is possible to formulate both the position and velocity at one time of any given particle. It followed that knowledge of the position and velocity of a given number of particles would enable the future behavior of the whole collection to be accurately predicted. The principle of Heisenberg is that given the determination of position, its velocity can be stated only as of a certain order of probability, while if its velocity is determined the correlative factor of position can be stated only as of a certain order of probability. Both cannot be determined at once, from which it follows necessarily that the future of the whole collection cannot possibly be foretold except in terms of some order of probability.
John DeweyRelated topics
certain classical correlative factor future given knowledge number once order position possible probability science velocity while HeisenbergRelated quotes
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
This morning I received a letter from de Bellio. He writes that he does not believe scientific research into the nature of color and light can help the artist, neither can anatomy nor the laws of optics. He wants to discuss these questions with me and find out my views. Now everything depends on how this knowledge is to be used. But surely it is clear that we could not pursue our studies of light with much assurance if we did not have as a guide the discoveries of Chevreul and other scientists. I would not have distinguished between local color and light if science had not given us the hint; the same holds true for complementary colors, contrasting colors, etc. 'Yes', he will tell me: 'but these have always been taken into account, look at Monet' It is at this point that the question becomes serious!
Camille Pissarro
The free being with absolute freedom proposes to itself certain ends. It wills because it wills, and the willing of an object is itself the last ground of such willing. Thus we have previously determined a free being, and any other determination would destroy the conception of an Ego, or of a free being. Now, if it could be so arranged that the willing of an unlawful end would necessarily - in virtue of an always effective law - result in the very reverse of that end, then the unlawful will would always ANNIHILATE ITSELF. A person could not will that end for the very reason because he did will it; his unlawful will would become the ground of its own annihilation, as the will is indeed always its own last ground.
Johann Gottlieb Fichte
It is not possible, I think, to rise from the perusal of the arguments of Clark and Spinoza without a deep conviction of the futility of all endeavors to establish, entirely à priori, the existence of an Infinite Being, His attributes, and His relation to the universe. The fundamental principle of all such speculations, viz. that whatever we can clearly conceive, must exist, fails to accomplish its end, even when its truth is admitted. For how shall the finite comprehend the infinite? Yet must the possibility of such conception be granted, and in something more than the sense of a mere withdrawal of the limits of phænomal existence, before any solid ground can be established for the knowledge, à priori, of things infinite and eternal.
George Boole
While it is true that we are paying out far more money and maintaining a much stronger Military Establishment than ever before, because of the conditions stated, we have been able to pursue a moderate course. Our people have had all the war, all the taxation, and all the military service that they want. They have therefore wished to emphasize their attachment to our ancient policy of peace. They have insisted upon economy. They have supported the principle of limitation of armaments. They have been able to do this because of their position and their strength in numbers and in resources. We have a tremendous natural power which supplements our arms. We are conscious that no other nation harbors any design to put us in jeopardy.
Calvin Coolidge