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Also from the same book: "I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up example

Also from the same book: "I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) – disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, ‘False!’"

kqr 9y ago
Not from the book but personal experience: This is also useful to understand preconditions by reduction. I.e. if you want to understand a theorem, it can sometimes be useful to start by figuring out the reason behind the preconditions. "Why does this apply only to balls that have hair?" Simply go, "What would the theorem imply if I start with a smooth ball instead?" This practise can also lead to generalisations. Oftentimes starting with a smooth ball will make you go "What? That't can't be possible." But sometimes, starting with a smooth ball leads you to, "Huh, that's really, really weird. But it's not a contradiction in and of itself. I could use that result in another context!"
j2kun 9y ago
What kind of second-rate mathematician tries to prove a theorem without writing down examples? That's Proving Theorems 101.
- ColinWright 9y ago
  I have no idea why you were being down-voted, and I wonder how many of the down-voters are actually mathematicians. If they were responding to your tone they need to know that it's incredibly mild compared with what one could say. I upvoted you, but clearly didn't undo all the downvotes.
- Koshkin 9y ago
  Right. The very first step in trying to prove a theorem is to try and disprove it (by looking for a counter-example).
  - jwdunne 9y ago
    Which are a form of example. You don't store those counter examples in your head - you write them down. If they don't turn out to be counter examples, what are they?

kqr 9y ago

Not from the book but personal experience: This is also useful to understand preconditions by reduction. I.e. if you want to understand a theorem, it can sometimes be useful to start by figuring out the reason behind the preconditions. "Why does this apply only to balls that have hair?" Simply go, "What would the theorem imply if I start with a smooth ball instead?" This practise can also lead to generalisations. Oftentimes starting with a smooth ball will make you go "What? That't can't be possible." But sometimes, starting with a smooth ball leads you to, "Huh, that's really, really weird. But it's not a contradiction in and of itself. I could use that result in another context!"

j2kun 9y ago

What kind of second-rate mathematician tries to prove a theorem without writing down examples? That's Proving Theorems 101.

ColinWright 9y ago
I have no idea why you were being down-voted, and I wonder how many of the down-voters are actually mathematicians. If they were responding to your tone they need to know that it's incredibly mild compared with what one could say. I upvoted you, but clearly didn't undo all the downvotes.
Koshkin 9y ago
Right. The very first step in trying to prove a theorem is to try and disprove it (by looking for a counter-example).
- jwdunne 9y ago
  Which are a form of example. You don't store those counter examples in your head - you write them down. If they don't turn out to be counter examples, what are they?