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>It's these contradictory intuitions that are simultaneously difficult and useful. You just need to pick the right one at the right time. I have heard it put t

>It's these contradictory intuitions that are simultaneously difficult and useful. You just need to pick the right one at the right time. I have heard it put that you actually need to learn to ignore your intuitions and follow the actual calculations to do higher-level math. Rather like Eliezer's "shut up and multiply" on Less Wrong. I have no personal opinion on that since I haven't learned any higher math, past first year calculus and discrete math, yet.

ced 16y ago
Many great mathematicians would disagree with that. I like this definition from a prof at Stanford: A proof is a nothing more than a clear, compelling argument. If it feels wrong, no matter how rigorous the math is, you still shouldn't blindly believe it. It's an opportunity to delve deeper, and either find a flaw in the proof (or your understanding thereof), or to educate your intuition.
- pjscott 16y ago
  Exactly: intuitions can be changed, and if they're leading you astray then they should be. It's very important to have a solid set of intuitions, especially in fields like higher math that aren't easily intuitive.
jerf 16y ago
As a rationalist might point out, you can not go into a problem unbiased. You must make some hypotheses before you can test them; your initial hypothesis space can not be the space of all possible hypotheses or you can not get anywhere. "Shut up and calculate" works for the final part of the problem, but you need your "intuition" to get to the part where you know what to shut up and calculate about.
jleader 16y ago
I once heard Richard Feynman give a talk where he mentioned that in grad school he would challenge the pure mathematicians to give him their hardest, most abstract problems, and he'd often solve them by thinking up a physical analogy about which he could reason intuitively. So in his case it wasn't a matter of ignoring his intuitions, but finding the right intuition, that actually corresponded correctly with the math.
- dwwoelfel 16y ago
  Are you sure you're remembering this right? In "The Pleasure of Finding Things Out" there is a story he tells about an ongoing challenge he had with the mathematicians. They would come to him with a theorem about topology, and he would tell them if it was true or false. That is much different from solving the theorems.
  - jleader 16y ago
    I assume you're right; my memory of a talk I heard almost 30 years ago is unlikely to be completely correct.

ced 16y ago

Many great mathematicians would disagree with that. I like this definition from a prof at Stanford: A proof is a nothing more than a clear, compelling argument. If it feels wrong, no matter how rigorous the math is, you still shouldn't blindly believe it. It's an opportunity to delve deeper, and either find a flaw in the proof (or your understanding thereof), or to educate your intuition.

pjscott 16y ago
Exactly: intuitions can be changed, and if they're leading you astray then they should be. It's very important to have a solid set of intuitions, especially in fields like higher math that aren't easily intuitive.

jerf 16y ago

As a rationalist might point out, you can not go into a problem unbiased. You must make some hypotheses before you can test them; your initial hypothesis space can not be the space of all possible hypotheses or you can not get anywhere. "Shut up and calculate" works for the final part of the problem, but you need your "intuition" to get to the part where you know what to shut up and calculate about.

jleader 16y ago

I once heard Richard Feynman give a talk where he mentioned that in grad school he would challenge the pure mathematicians to give him their hardest, most abstract problems, and he'd often solve them by thinking up a physical analogy about which he could reason intuitively. So in his case it wasn't a matter of ignoring his intuitions, but finding the right intuition, that actually corresponded correctly with the math.

dwwoelfel 16y ago
Are you sure you're remembering this right? In "The Pleasure of Finding Things Out" there is a story he tells about an ongoing challenge he had with the mathematicians. They would come to him with a theorem about topology, and he would tell them if it was true or false. That is much different from solving the theorems.
- jleader 16y ago
  I assume you're right; my memory of a talk I heard almost 30 years ago is unlikely to be completely correct.