Theorem Quotes - page 4
Neither the temporal succession nor the phantom of what ye might have chosen and didn't is itself Freedom. They are a lens. The picture is a symbol: but it's truer than any philosophical theorem (or, perhaps, than any mystic's vision) that claims to go behind it. For every attempt to see the shape of eternity except through the lens of Time destroys your knowledge of Freedom."
C. S. Lewis
I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you. I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have even forgotten how to integrate a good many things.
But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very well unlikely to be good economics: and I went more and more on the rules-(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in (4), burn (3). This last I do often.
Alfred Marshall
It always seems to me absurd to speak of a complete proof of a theorem being rigorously demonstrated. An incomplete proof is no proof and a mathematical truth not rigorously demonstrated is not demonstrated at all. I do not mean to deny that there are mathematical truths, morally certain, which defy and will probably to the end of time to defy proof, as, e.g. that every indecomposable integer polynomial function must represent an infinitude of primes. I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitation of our faculties in regard to time, which like space may be in its essence poly-dimensional, and that this and such sort of truths would become self-evident to a being whose mode of perception is according to superficially as distinguished from our own limitation to linearly extended time.
James Joseph Sylvester
The theorem tells you that maybe there must be something happening faster than light, although it pains me even to say that much. The theorem certainly implies that Einstein's concept of space and time, neatly divided up into separate regions by light velocity, is not tenable. But then, to say that there's something going faster than light is to say more than I know.
John Stewart Bell
Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does. Possibly philosophers would look on us mathematicians the same way as we look on the technicians, if they dared.
Paul Halmos
The Laundry is the British Government's secret agency for dealing with "magic.” The use of scare-quotes is deliberate; as Sir Arthur C. Clarke said, "Any sufficiently advanced technology is indistinguishable from magic,” so "magic” is what we deal with. Note that this does not involve potions, pentacles, prayers, eldritch chanting, dressing up in robes and pointy hats, or most (but not all) of the stuff associated with the term in the public mind. No, our magic is computational. The realm of pure mathematics is very real indeed, and the...things...that cast shadows on the walls of Plato's cave can sometimes be made to listen and pay attention if you point a loaded theorem at them. This is, however, a very dangerous process, because most of the shadow-casters are unclear on the distinction between pay attention and free buffet lunch here. My job-applied computational demonologist-comes with a very generous pension scheme, because most of us don't survive to claim it.
Charles Stross
Descartes ... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is ... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.
René Descartes