“Fundamentally” and all of its near-synonyms: “really”, “essentially”, “at bottom”, “actually”, etc.
Usually, these mean “not”. (“How was that party you went to last night?” “Oh, it was all right really.”) (“Yes, I kidnapped you and chained you in my basement, but fundamentally, underneath it all, I’m essentially a nice guy.”)
On a related note, I often find myself starting a sentence with “The fundamental issue”—and when I catch myself and ask if what I’m talking about is the single issue that in fact underlies all others, and answer myself “no”—then I revise the sentence so something line “One important issue”… Here the lullaby is in two parts, a) everything is less important than this thing and b) there is only this one thing to care about. It’s rarely the case that either is true, let alone both.
A joke along these lines has the math professor claiming that the proof of some statement is trivial. They pause for a moment, think, then leave the classroom. Half an hour later, they come back and say, “Yes, it was trivial.”
I heard about a professor (I think physics) who was always telling his students that various propositions were “simple”, despite the fact that the students always struggled to show them. Eventually, the students went to the TA (the one I heard the story from), who told the professor.
So, the next class the professor said, “I have heard that the students do not want me to say ‘simple’. I will no longer do so. Now, this proposition is straightforward...”
At the Princeton graduate school, the physics department and the math department shared a common lounge, and every day at four o’clock we would have tea. It was a way of relaxing in the afternoon, in addition to imitating an English college. People would sit around playing Go, or discussing theorems. In those days topology was the big thing.
I still remember a guy sitting on the couch, thinking very hard, and another guy standing in front of him, saying, “And therefore such-and-such is true.”
“Why is that?” the guy on the couch asks.
“It’s trivial! It’s trivial!” the standing guy says, and he rapidly reels off a series of logical steps: “First you assume thus-and-so, then we have Kerchoff’s this-and-that; then there’s Waffenstoffer’s Theorem, and we substitute this and construct that. Now you put the vector which goes around here and then thus-and-so...” The guy on the couch is struggling to understand all this stuff, which goes on at high speed for about fifteen minutes!
Finally the standing guy comes out the other end, and the guy on the couch says, “Yeah, yeah. It’s trivial.”
We physicists were laughing, trying to figure them out. We decided that “trivial” means “proved.” So we joked with the mathematicians: “We have a new theorem—that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial.”
The mathematicians didn’t like that theorem, and I teased them about it. I said there are never any surprises -- that the mathematicians only prove things that are obvious.
Most of the time I’ve run into the word “obviously” is in the middle of a proof in some textbook, and my understanding of the word in that context is that it means “the justification of this claim is trivial to see, and spelling it out would be too tedious/would disrupt the flow of the proof.”
I assert that it (“obviously” in math) is most often used correctly, but that people spend more time experiencing it used incorrectly—because they spend more time thinking about it when it is not obvious.
Indeed! See also this classic essay by Jerry Weinberg on Lullaby Words. “Just” is one of them, can you think of others before reading the essay? ;)
“Fundamentally” and all of its near-synonyms: “really”, “essentially”, “at bottom”, “actually”, etc.
Usually, these mean “not”. (“How was that party you went to last night?” “Oh, it was all right really.”) (“Yes, I kidnapped you and chained you in my basement, but fundamentally, underneath it all, I’m essentially a nice guy.”)
Good one.
On a related note, I often find myself starting a sentence with “The fundamental issue”—and when I catch myself and ask if what I’m talking about is the single issue that in fact underlies all others, and answer myself “no”—then I revise the sentence so something line “One important issue”… Here the lullaby is in two parts, a) everything is less important than this thing and b) there is only this one thing to care about. It’s rarely the case that either is true, let alone both.
In mathematics, “obvious” is one of those words. It tends to mean “something I don’t know how to justify.”
A joke along these lines has the math professor claiming that the proof of some statement is trivial. They pause for a moment, think, then leave the classroom. Half an hour later, they come back and say, “Yes, it was trivial.”
I heard about a professor (I think physics) who was always telling his students that various propositions were “simple”, despite the fact that the students always struggled to show them. Eventually, the students went to the TA (the one I heard the story from), who told the professor.
So, the next class the professor said, “I have heard that the students do not want me to say ‘simple’. I will no longer do so. Now, this proposition is straightforward...”
-- Surely you’re joking, Mr. Feynman!
Most of the time I’ve run into the word “obviously” is in the middle of a proof in some textbook, and my understanding of the word in that context is that it means “the justification of this claim is trivial to see, and spelling it out would be too tedious/would disrupt the flow of the proof.”
I thought the mathematical terms went something like this:
Trivial: Any statement that has been proven
Obviously correct: A trivial statement whose proof is too lengthy to include in context
Obviously incorrect: A trivial statement whose proof relies on an axiom the writer dislikes
Left as an exercise for the reader: A trivial statement whose proof is both lengthy and very difficult
Interesting: Unproven, despite many attempts
Well, that’s what it’s supposed to mean. One of my professors (who often waxed sarcastic during lectures) described it as a very dangerous word...
Do you really assert that it is more often used incorrectly (that the fact is not actually obvious)?
I assert that it (“obviously” in math) is most often used correctly, but that people spend more time experiencing it used incorrectly—because they spend more time thinking about it when it is not obvious.
No, I guess not.
A list of common proof techniques. ;)
Voted up because that’s an excellent link.
http://c2.com/cgi/wiki?AlarmBellPhrase
I failed to predict any of the words, but I didn’t think about it for very long.