MacTutor
Proof is the idol before whom the pure mathematician tortures himself.
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about `and'.
We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own.
It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.
To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies.
Human life is proverbially uncertain; few things are more certain than the solvency of a life-insurance company.
Shuffling is the only thing which Nature cannot undo.
I believe there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe and the same number of electrons.
There once was a brainy baboon,
Who always breathed down a bassoon,
For he said, "It appears
That in billions of years
I shall certainly hit on a tune".
Schrödinger's wave-mechanics is not a physical theory, but a dodge -- and a very good dodge too.
For the truth of the conclusions of physical science, observation is the supreme Court of Appeal. It does not follow that every item which we confidently accept as physical knowledge has actually been certified by the Court; our confidence is that it would be certified by the Court if it were submitted. But it does follow that every item of physical knowledge is of a form which might be submitted to the Court. It must be such that we can specify (although it may be impracticable to carry out) an observational procedure which would decide whether it is true or not. Clearly a statement cannot be tested by observation unless it is an assertion about the results of observation. Every item of physical knowledge must therefore be an assertion of what has been or would be the result of carrying out a specified observational procedure.
Let us suppose that an ichthyologist is exploring the life of the ocean. He casts a net into the water and brings up a fishy assortment. Surveying his catch, he proceeds in the usual manner of a scientist to systematise what it reveals. He arrives at two generalisations: No sea-creature is less than two inches long. (2) All sea-creatures have gills. These are both true of his catch, and he assumes tentatively that they will remain true however often he repeats it.
In applying this analogy, the catch stands for the body of knowledge which constitutes physical science, and the net for the sensory and intellectual equipment which we use in obtaining it. The casting of the net corresponds to observation; for knowledge which has not been or could not be obtained by observation is not admitted into physical science.
An onlooker may object that the first generalisation is wrong. "There are plenty of sea-creatures under two inches long, only your net is not adapted to catch them." The icthyologist dismisses this objection contemptuously. "Anything uncatchable by my net is ipso facto outside the scope of icthyological knowledge. In short, "what my net can't catch isn't fish." Or -- to translate the analogy -- "If you are not simply guessing, you are claiming a knowledge of the physical universe discovered in some other way than by the methods of physical science, and admittedly unverifiable by such methods. You are a metaphysician. Bah!"
The quest of the absolute leads into the four-dimensional world.
The solution goes on famously; but just as we have got rid of all the other unknowns, behold! V disappears as well, and we are left with the indisputable but irritating conclusion:
0 = 0
This is a favourite device that mathematical equations resort to, when we propound stupid questions.
The mathematics is not there till we put it there.
It is also a good rule not to put overmuch confidence in the observational results that are put forward until they are confirmed by theory.
If your theory is found to be against the second law of theromodynamics, I give you no hope; there is nothing for it but to collapse in deepest humiliation.