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Any progress in the theory of partial differential equations must also bring about a progress in Mechanics.
--
"Vorlesungen über Dynamik"

 

Carl Gustav Jacob Jacobi

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It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.

 

Paul Dirac

 

After you have practiced for a while, you will realize that it is not possible to make rapid, extraordinary progress. Even though you try very hard, the progress you make is always little by little. It is not like going out in a shower in which you know when you get wet. In a fog, you do not know you are getting wet, but as you keep walking you get wet little by little. If your mind has ideas of progress, you may say, "Oh, this pace is terrible!" But actually it is not. When you get wet in a fog it is very difficult to dry yourself. So there is no need to worry about progress.

 

Shunryu Suzuki

 

Progress? There's no such thing as progress. There's only change. You dig a hole in the ground, you build up a city, and you fight a war, and you call it progress?

 

Charles Manson

 

I believe it safe to say that all progress must lead, not to further progress, but finally to the negation of progress, a return to the point of departure.

 

Eugene Delacroix

 

The ultimate objective test of free will would seem to be: Can one predict the behavior of the organism? If one can, then it clearly doesn't have free will but is predetermined. On the other hand, if one cannot predict the behavior, one could take that as an operational definition that the organism has free will ... The real reason why we cannot predict human behavior is that it is just too difficult. We already know the basic physical laws that govern the activity of the brain, and they are comparatively simple. But it is just too hard to solve the equations when there are more than a few particles involved ... So although we know the fundamental equations that govern the brain, we are quite unable to use them to predict human behavior. This situation arises in science whenever we deal with the macroscopic system, because the number of particles is always too large for there to be any chance of solving the fundamental equations. What we do instead is use effective theories. These are approximations in which the very large number of particles are replaced by a few quantities. An example is fluid mechanics ... I want to suggest that the concept of free will and moral responsibility for our actions are really an effective theory in the sense of fluid mechanics. It may be that everything we do is determined by some grand unified theory. If that theory has determined that we shall die by hanging, then we shall not drown. But you would have to be awfully sure that you were destined for the gallows to put to sea in a small boat during a storm. I have noticed that even people who claim everything is predetermined and that we can do nothing to change it, look before they cross the road. ... One cannot base one's conduct on the idea that everything is determined, because one does not know what has been determined. Instead, one has to adopt the effective theory that one has free will and that one is responsible for one's actions. This theory is not very good at predicting human behavior, but we adopt it because there is no chance of solving the equations arising from the fundamental laws. There is also a Darwinian reason that we believe in free will: A society in which the individual feels responsible for his or her actions is more likely to work together and survive to spread its values.

 

Stephen Hawking