A Quote by Leopold Kronecker
Analysis does not owe its really significant successes of the last century to any mysterious use of sqrt(-1), but to the quite natural circumstances that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line.
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It is generally believed that our science is empirical and that we draw our concepts and our mathematical constructs from the empirical data. If this were the whole truth, we should, when entering into a new field, introduce only such quantities as can directly be observed, and formulate natural laws only by means of these quantities.
In less than a century we experienced great movement. The youth movement! The labor movement! The civil rights movement! The peace movement! The solidarity movement! The women's movement! The disability movement! The disarmament movement! The gay rights movement! The environmental movement! Movement! Transformation! Is there any reason to believe we are done?
It follows that the word probability, in its mathematical acceptance, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it will vary. Probability is the expectation founded upon partial knowledge.
In the last analysis, our only freedom is the freedom to discipline ourselves.
The free trade movement in the middle of the last century represents the first conscious recognition of these new circumstances and of the necessity to adapt to them.
The individual man, in introspecting the fact of his own consciousness, also discovers the primordial natural fact of his freedom: his freedom to choose, his freedom to use or not use his reason about any given subject. In short, the natural fact of his "free will." He also discovers the natural fact of his mind's command over his body and its actions: that is, of his natural ownership over his self.
The greatest blessing of our democracy is freedom. But in the last analysis, our only freedom is the freedom to discipline ourselves.
Our faith in freedom does not rest on the foreseeable results in particular circumstances, but on the belief that it will, on balance, release more forces for the good than for the bad ... Freedom granted only when it is known beforehand that its effects will be beneficial is not freedom.
And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities...?
New York was no mere city. It was instead an infinitely romantic notion, the mysterious nexus of all love and money and power, the shining and perishable dream itself. To think of 'living' there was to reduce the miraculous to the mundane; one does not 'live' at Xanadu.
The importance of our being free to do a particular thing has nothing to do with the question of whether we or the majority are ever likely to make use of that particular possibility. To grant no more freedom than all can exercise would be to misconceive its function completely. The freedom that will be used by only one man in a million may be more important to society and more beneficial to the majority than any freedom that we all use.
It seems sensible to discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron... Instead it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only relations between observable quantities occur.
As a survival-happy species, our successes are calculated in the number of years we have extended our lives, with the reduction of suffering being only incidental to this aim. To stay alive under almost any circumstances is a sickness with us. Nothing could be more unhealthy than to “watch one’s health” as a means of stalling death. The lengths we will go as procrastinators of that last gasp only demonstrate a morbid dread of that event. By contrast, our fear of suffering is deficient.
Existence is movement. Action is movement. Existence is defined by the rhythm of forces in natural balance. (...) It is our appreciation for dance that allows us to see clearly the rhythms of nature and to take natural rhythm to a plane of well-organised art and culture.
In mathematical analysis we call x the undetermined part of line a: the rest we don't call y, as we do in common life, but a-x. Hence mathematical language has great advantages over the common language.
The freedom that will be used by only one man in a million may be more important to society and more beneficial to the majority than any freedom that we all use.
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