Our starvation is a physical manifestation of our inner hunger. Our infections are manifestations of our inner disease. Our wars are manifestations of our divisions within ourselves.
It is for this reason that people in America are depressed. When you remove the physical manifestations of suffering, the suffering does not go away; its spiritual nature simply becomes more obvious. Then we call it depression.
If you want to end world hunger, learn to feel full and happy. If you want to cure cancer, learn to be in harmony with yourself. If you want to fight pollution, learn to think pure and beautiful thoughts.
You can’t remove other people’s pain. But people can’t learn to remove their own pain if they don’t have an example to follow.
That poor man that you refused to help? That suffering animal that you neglected and let die? You are him. You are it. The world is your body; end your suffering and you end all suffering. Save yourself and you save the world.
Things I have found to be helpful for doing math:
2. Don’t guess and check. This can take subtle forms. E.g., “I have no real idea how to prove this theorem, but I think this method might work, even though I haven’t really thought it through; but I’ll try it, since I don’t have any other ideas.”
3. Start by understanding the question. If you really understand the question, usually the answer will be obvious. So you could just spend 95% of your thinking time trying to understand the question, and then the time you spend looking for an answer approaches zero, because it will just come to you.
4. If you have a hard question, try rephrasing it. Find a theorem that’s equivalent to the theorem you’re trying to prove, or find a structure that’s equivalent to the structure you’re studying. Hell, invent a new field of math if you need to (viz. Evariste Galois).
5. Math is about showing that things are the same when they obviously aren’t. Deep similarities behind obvious distinctness. 3 * 7 is the same as 21, a circle is homeomorphic to a square, addition and multiplication both form abelian groups. Usually when I solve a hard problem, 90% of the solution consists of ekeing out some deep similarity that seemingly has little to do with the hard problem, and then the remaining 10% is an easy solution to the hard problem that employs the deep similarity.
My eventual graduate thesis will be on the inconsistency of mathematics, and its implications. Topics will include:
- A proof of the inconsistency of all theories of mathematics in first-order logic.
- The equivalence of truth and provability, via Tarski’s truth schema.
- The equivalence of all first-order theories of mathematics, via the truth schema.
- The consequences of naïve set theory:
- Properties of the set of all sets, including the combinatoric indescribability of its cardinality.
- Infinitely deep sets, and a proof of the continuum hypothesis.
- The existence of various large cardinals.
- The existence of indefinable sets.
- Whatever I figure out about the “singularity point” of mathematics: the location of the border between consistency and inconsistency in the hierarchy of increasingly strong theories.
- My philosophy of mathematics, including mathematical nondualism: the view that every statement is ultimately true and false. As well as the view that formal proof does not solely dictate what propositions we are to accept.