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with the old one until they believe that the new system works.
Except---what we did in a marvelous project at IBM---for constantly eating your own cooking. We were
constantly running the latest version of our software, constantly compiling our optimizing compiler
through itself. By constantly using it ourselves, we got instantaneous feedback on the performance and
on the design, which was constantly evolving. In my opinion that's the only way to develop a large piece
of software: a totalitarian top-down approach cannot work. You have to design it as you go, not at the
very beginning, before you write a single line of code.
And I feel that my experience in the real world debugging software, yields a valuable philosophical
lesson: Experimentation is the only way to "prove" that software is correct. Traditional mathematical
proofs are only possible in toy worlds, not in the real world. The real world is too complicated. A
physicist would say it like this: Pure math can only deal with the hydrogen atom. One proton, one
electron, that's it! The quasi-empirical view of math may be controversial in the math community, but
it is old hat in the software business. Programmers already have a quasi-empirical attitude to proof.
Even though software is pure mind-stuff, not physical, programmers behave like physicists, not
mathematicians, when it comes to debugging.
On the other hand, a way to minimize the debugging problem is to try at all costs to keep software
intellectually manageable, as illustrated by my discussion of LISP. In our IBM project, I did this by
re-writing all my code from scratch each time that I advanced in my understanding of the problem. I
refused to use ad hoc code and tried instead to base everything as much as possible on clean,
systematic mathematical algorithms. My goal was crystalline clarity. In my opinion, what counts most
in a design is conceptual integrity, being faithful to an idea, not confusing the issue!
The previous paragraph is very pro-math. However, my design evolved, as I said, based on computer
experiments, and the experimental method is used by physicists, not by mathematicians. It's too bad
that mathematicians feel that way about experimentation!
Remember, I think that math is not that different from physics, for we must be willing to add new
axioms:
Physics: laws->Computer->universe Math: axioms->Computer->theorems
To get more out, put more in!
Another lesson I'd like you to take from this book is that everything is connected, all important ideas
are---and that fundamental questions go back millennia and are never resolved. For example, the
tension between the continuous and the discrete, or the tension between the world of ideas (math!)
and the real world (physics, biology). You can find all this discussed in ancient Greece. And I suspect
we could even trace it back to ancient Sumer, if more remained of Sumerian math than the scrap paper
jottings on the clay tablets that are all we have, jottings that give hints of surprisingly sophisticated
methods and of a love for calculation that seems to far outstrip any possible practical application.
[Did Sumer inherit its mathematics from an even older civilization---one more advanced than the ancient Greeks---that was destroyed by
the glaciers, or when the glaciers suddenly melted, or by some other natural catastrophe? There is no way for such sophisticated
computational techniques to appear out of nowhere, without antecedents.]
On Creativity
Having emphasized and re-emphasized the importance of creativity, it would be nice if I had a theory
about it. Nope, but I do have some experience being creative. So let me try to share that with you.
The message of Gödel incompleteness, as I've said again and again in this book, is that a static fixed
FAS cannot work. You have to add new information, new axioms, new concepts. Math is constantly
evolving. The problem with current metamath is that it deals only with---it refutes---static FAS's. So
where do new math ideas come from? Can we have a theory about that? A dynamic rather than static
view of math, a dynamic rather than a static metamath, a kind of dynamic FAS perhaps?
Since I don't have that theory, I think an anecdotal approach might be best. This book is full of amazing
case studies of new, unexpected math ideas that reduced the complicated to the obvious. And I've come
up with a few of these ideas myself. How does it feel to do that?
Well, you can't find them if you don't look for them, if you don't really believe in them.
Is there some way to train for it, like a sport?! No, I don't think so! You have to be seized by a demon,
and our society doesn't want too many people to be like that!
Let me describe what it feels like right now while I'm writing this book.
First of all, the ideas that I'm discussing seem very concrete, real and tangible to me. Sometimes they
even feel more real than the people around me. They certainly feel more real than newspapers,
shopping malls and TV programs---those always give me a tremendous feeling of unreality! In fact, I
only really feel alive when I'm working on a new idea, when I'm making love to a woman (which is also
working on a new idea, the child we might conceive), or when I'm going up a mountain! It's intense,
very intense.
When I'm working on a new idea I push everything else away. I stop swimming in the morning, I don't
pay the bills, I cancel my doctor appointments. As I said, everything else becomes unreal! And I don't
have to force myself to do this.
On the contrary, it's pure sensuality, pure pleasure. I put beautiful new ideas in the same category with
beautiful women and beautiful art. To me it's like an amazing ethnic cuisine I've never tasted before.
I'm not depriving myself of anything, I'm not an ascetic. I don't look like an ascetic, do I?
And you can't force yourself to do it, anymore than a man can force himself to make love to a woman he
doesn't want.
The good moments are very, very good! Sometimes when I'm writing this I don't know where the ideas
come from. I think that it can't be me, that I'm just a channel for ideas that want to be expressed. ---
But I have been concentrating on these questions for a long time. --- I feel inspired, energized by the
ideas. People may think that something's wrong with me, but I'm okay, I'm more than okay. It's pure
enthusiasm! That's "God within" in Greek. Intellectual elation---like summiting on a high peak!
And I'm a great believer in the subconscious, in sleeping on it, in going to bed at 3am or 5am after
working all night, and then getting up the next morning full of new ideas, ideas that come to you in
waves while you're taking a bath, or having coffee. Or swimming laps. So mornings are very important
to me, and I prefer to spend them at home. Routine typing and e-mail, I do in my office, not at home.
And when I get too tired to stay in the office, then I print out the final version of the chapter I'm
working on, bring it home---where there is no computer---and lie in bed for hours reading it, thinking
about it, making corrections, adding stuff.
Sometimes the best time is lying in bed in the dark with my eyes closed, in a half dreamy, half awake
state that seems to make it easier for new ideas, or new combinations of ideas, to emerge. I think of the
subconscious as a chemical soup that's constantly making new combinations, and interesting
combinations of ideas stick together, and eventually percolate up into full consciousness. --- That's not
too different from a biological population in which individuals fall in love and combine to produce new
individuals. --- My guess is that all this activity takes place at a molecular level---like DNA and
information storage in the immune system---not at the cellular level. That's why the brain is so [ Pobierz całość w formacie PDF ] - zanotowane.pl
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