Gödel’s Incompleteness Theorem is very important, and also not easy to understand. Looking through the article on Wikipedia, you will get a feeling of its full depth.
I didn’t know its meaning clearly until very recently. Some easily accessible explanations are very helpful.
“In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. “
It can be inferred
“You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements. “
It has many important implications:
All logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.
A computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths.
3. Human Being
You’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.
1. Follow Your Curiosity
“I have no special talent. I am only passionately curious.”
2. Perseverance is Priceless
“It’s not that I’m so smart; it’s just that I stay with problems longer.”
3. Focus on the Present
“Any man who can drive safely while kissing a pretty girl is simply not giving the kiss the attention it deserves.”
4. The Imagination is Powerful
“Imagination is everything. It is the preview of life’s coming attractions. Imagination is more important than knowledge.”
5. Make Mistakes
“A person who never made a mistake never tried anything new.”
6. Live in the Moment
“I never think of the future – it comes soon enough.”
7. Create Value
“Strive not to be a success, but rather to be of value.”
8. Don’t be repetitive
“Insanity: doing the same thing over and over again and expecting different results.”
9. Knowledge Comes From Experience
“Information is not knowledge. The only source of knowledge is experience.”
10. Learn the Rules and Then Play Better
“You have to learn the rules of the game. And then you have to play better than anyone else.”
From Paul Graham’s “Frighteningly Ambitious Startup Ideas”. As usual, it is a great essay where you could get inspired.
“Empirically, the way to do really big things seems to be to start with deceptively small things. Want to make the universal web site? Start by building a site for Harvard undergrads to stalk one another.
Neither Bill Gates nor Mark Zuckerberg knew at first how big their companies were going to get. **All they knew was that they were onto something.
Maybe it’s a bad idea to have really big ambitions initially, because the bigger your ambition, the longer it’s going to take, and the further you project into the future, the more likely you’ll get it wrong.
You’ll be better off if you operate like Columbus and just head in a general westerly direction. Don’t try to construct the future like a building, because your current blueprint is almost certainly mistaken. Start with something you know works, and when you expand, expand westward.”
Paul Buchheit’s reminiscence. very touching.
“The winter of 2004 felt so cold and rainy, the coldest I can remember it ever being here in California. The cold had a kind of depth that you can’t quite escape.
But on March 9th it was sunny and starting to get a little warm. Winter was over, and my brother had just died that morning.
We left the hospital and returned to his apartment in Menlo Park. It felt so wrong. He was gone, but his belongings were still there. Eventually we would have to pack them all into boxes, keeping some for ourselves, and donating the rest.
It does not feel good to pack up the remains of your brother’s life.”
Eight years passed.
“… Eight years since Steve died. I keep looking for meaning, but all I’ve found so far is that in order to be at peace with the present, we must be at peace with the past, because the present is a product of the past. Accept. Accept. Accept. Learn to love the present moment. What happened, happened.”
From travel writer Steven Crook：
Don’t stay in your apartment.
Don’t stay in your city.
Get to the east coast, get to the mountains.
Look inside that temple you walk past every day.
Wander down the little lanes and alleys.”