How To Understand Things

🌈 Abstract

The article discusses the importance of deeply understanding concepts, rather than just memorizing algorithms or procedures to pass exams. It highlights the habits and qualities of highly intelligent people, such as:

• Continuously exploring different proofs or approaches to a problem, even after solving it
• Having the "will to think" and the drive to truly understand something, rather than just accepting answers
• Developing physical intuition and concrete visualizations to complement abstract knowledge
• Being unafraid to ask "dumb" questions to clarify their understanding
• Taking the time to ponder and think deeply about a topic before diving into external sources

The article also cautions against the tendency of educational systems to prioritize speed and passing exams over true understanding, and provides advice on how to develop a deeper comprehension of complex topics.

🙋 Q&A

[01] The smartest person the author knew

1. What habit did the smartest person the author knew have?

• After solving a problem or proving a theorem, the smartest person would continue thinking about the problem and try to find different proofs or approaches, sometimes spending hours on a problem they had already solved.

2. How did the author's approach differ from the smartest person's?

• The author had the tendency to stop once they had reached the end of a proof, since they had "gotten the answer", whereas the smartest person would explore multiple proofs and deepen their understanding.

3. What did the author conclude about intelligence?

• The author concluded that intelligence is as much about virtues like honesty, integrity, and bravery, as it is about "raw intellect".

4. What is the key difference between "hardware" and "software" traits of intelligence?

• "Hardware" traits like processing speed and working memory can vary greatly among intelligent people, but the "software" traits, such as the unwillingness to accept unsatisfactory answers, are common among the smartest people and can be learned through effort.

[02] The quality of "not stopping at an unsatisfactory answer"

1. What are the components of this quality?

• Energy: Thinking hard takes effort, and it's easier to just stop at an answer that seems to make sense.
• Intrinsic motivation: Deeply understanding something requires a lot of effort, so most people don't do it.
• Honesty/integrity: The ability to avoid lying to oneself and constantly question whether one truly understands something.

2. How does writing help in developing this quality?

• Writing forces clarity and makes it harder to fool oneself that one understands something, as opposed to just thinking about it.

[03] Faraday's approach to understanding

1. What was Faraday's approach to understanding scientific concepts?

• Faraday believed in experimentally demonstrating things himself, rather than just reading or hearing about them. He would repeat and extend the experiments of others to establish his own understanding.

2. How did Faraday's physical intuition lead to important discoveries?

• Faraday's focus on physical experiments, rather than abstract mathematics, allowed him to develop a deep, intuitive understanding that led to crucial discoveries in science.

3. How does the ability to generate concrete examples or visualizations relate to deep understanding?

• Generating visual representations or concrete examples can help develop a stronger grasp of concepts, compared to just understanding them at an abstract, mathematical level.

[04] The author's experience with learning calculus

1. What was the author's initial struggle with the "dy/dx" notation in calculus?

• The author was confused by the "dy/dx" notation, as it looked like a fraction but was not actually division, and the proofs they were taught did not make logical sense.

2. What was the author's experience with the pressure to learn the operations rather than deeply understand calculus?

• The author was frustrated by the inadequate proofs but felt pressured to just learn the operations to pass exams, as the class needed to move on to the next topic. This tendency of educational systems to prioritize speed over understanding was seen as "killing the will to understanding" in people.

3. What is the author's advice for understanding complex topics?

• The author advises going slow, spending time pondering the questions yourself before reading, and developing a mental framework to better retain and integrate the information learned from external sources.

[05] Helpful questions to ask when trying to understand something

1. What are some of the mantra-like questions the author suggests asking?

• "But what exactly is X? What is it?"
• "Why must X be true? Why does this have to be the case? What is the single, fundamental reason?"
• "Do I really believe that this is true, deep down? Would I bet a large amount of money on it with a friend?"

[06] Parables about understanding

1. What is the key insight from the parable of Agassiz and the fish?

• The parable suggests that true understanding comes from direct observation and experience, rather than just regurgitating information from textbooks or other sources.

2. What is the key insight from the parable about the student who couldn't write about Bozeman?

• The parable shows that understanding is not a binary "yes/no" but has layers of depth, and that true understanding comes from direct observation and reasoning, rather than just repeating what has been heard before.

3. What is the main advice the author derives from these parables?

• The advice is to prioritize direct experience and first-hand data gathering, rather than relying solely on secondary sources or popular narratives, in order to develop a deeper, more robust understanding of a topic.