When lecturing, saying the word “obvious” is a signal for the students to begin panicking and self-doubting. I wonder if some instructors do this intentionally.
Saying that something is ‘obvious’ can provide useful information to the listener of the form “If you think about this for a few minutes you’ll see why this is true; this stands in contrast with some of the things that I’m talking about today.” Or even “though you may not understand why this is true, for experts who are deeply immersed in this theory this part appears to be straightforward.”
I personal wish that textbooks more often highlighted the essential points over those theorems that follow from a standard method that the reader is probably familiar with.
But here I really have in mind graduate / research level math where there’s widespread understanding that a high percentage of the time people are unable to follow someone who believes his or her work to be intelligible and so who have a prior against such remarks being intended as a slight. It seems like a bad communication strategy for communicating with people who are not in such a niche.
There’s an excellent definition of how “obvious” should be used in mathematics: something is obvious if and only if “a proof immediately springs to mind”.
The problem remains that it’s not particularly helpful to know that a proof immediately springs to the lecturer’s or textbook author’s mind. And so, operating under the assumption that people are trying to communicate only relevant information, whenever I see ‘obvious’ or ‘easily seen’ in a mathematical text, I can’t help but read it as an obnoxious ‘you should know this already—unless you’re dumb or something’. I think that the best norm for using the word ‘obvious’ and its variations would be to not use it at all.
This might be a defensive mechanism. Being explicit and formal is very important in mathematics but when you choose which proofs to omit, you’re making a judgement about which kind of presentation will lead to the greatest level of understanding among the intended audience. This judgement is more about psychology rather than mathematics and is necessarily based on fuzzy intuitions. Maybe mathematicians are uncomfortable with that and call such omitted proofs ‘obvious’ to make criticism costly in terms of status.
No! How is this different than saying: There’s an excellent definition of how “obvious” should be used: something is obvious if and only if “an explanation immediately springs to mind”.
There a running joke in mathematics that saying that a statement is “obvious” means:
When lecturing, saying the word “obvious” is a signal for the students to begin panicking and self-doubting. I wonder if some instructors do this intentionally.
Saying that something is ‘obvious’ can provide useful information to the listener of the form “If you think about this for a few minutes you’ll see why this is true; this stands in contrast with some of the things that I’m talking about today.” Or even “though you may not understand why this is true, for experts who are deeply immersed in this theory this part appears to be straightforward.”
I personal wish that textbooks more often highlighted the essential points over those theorems that follow from a standard method that the reader is probably familiar with.
But here I really have in mind graduate / research level math where there’s widespread understanding that a high percentage of the time people are unable to follow someone who believes his or her work to be intelligible and so who have a prior against such remarks being intended as a slight. It seems like a bad communication strategy for communicating with people who are not in such a niche.
“Yes, gentlemen, it IS obvious!”
I always imagine that line being delivered by John Cleese (as in “he IS the Messiah!”).
There’s an excellent definition of how “obvious” should be used in mathematics: something is obvious if and only if “a proof immediately springs to mind”.
The problem remains that it’s not particularly helpful to know that a proof immediately springs to the lecturer’s or textbook author’s mind. And so, operating under the assumption that people are trying to communicate only relevant information, whenever I see ‘obvious’ or ‘easily seen’ in a mathematical text, I can’t help but read it as an obnoxious ‘you should know this already—unless you’re dumb or something’. I think that the best norm for using the word ‘obvious’ and its variations would be to not use it at all.
This might be a defensive mechanism. Being explicit and formal is very important in mathematics but when you choose which proofs to omit, you’re making a judgement about which kind of presentation will lead to the greatest level of understanding among the intended audience. This judgement is more about psychology rather than mathematics and is necessarily based on fuzzy intuitions. Maybe mathematicians are uncomfortable with that and call such omitted proofs ‘obvious’ to make criticism costly in terms of status.
No! How is this different than saying:
There’s an excellent definition of how “obvious” should be used: something is obvious if and only if “an explanation immediately springs to mind”.
Darn, you beat me to it.