When Bohr first meets Oppenheimer, he tells him algebra is more than just reading but about hearing the music. What did he mean by that?
"The important thing isn't can you read music, it's can you hear it. Can you hear the music, Robert?"
By this, Bohr means that there's something more important than being able to do the calculations. One must have insight into what these calculations mean, intuit it, have inspiration.
He means that just like anyone being able to play the notes when they're able to read sheet music, anyone can do algebraic calculations, given enough study. But not everyone has the inspiration to come to new understandings. But Oppenheimer, as is implied often during the movie, can.
[...] and those who were seen dancing were thought to be insane by those who could not hear the music.
Can you hear the music? probably relates to the above. Its origin and paternity are totally unknown/unresolved as far as I know.1 In this case, Bohr sees far more than the mere perception of signs and numbers. He actually forsees much farther where this will lead them into.
Metaphorically speaking, because there are links between music and emotions2, you can probably also understand that quote as "(do you hear) the music celebrating for the (future) success?".
I read from a man called Brian Caudill -- BS Psychology a way to interpret it:
The easiest way to interpret a phrase is to break it down into its most basic parts.
“and those who were seen” = people
“dancing”= doing something (action)
“were thought to be” = others believed (reaction)
“insane” = not thinking correctly
“by those” = other people
“who did not hear” = did not have access to
“the music” = the cause
people doing something (action), others believed (reaction), not thinking correctly, other people, did not have access to; the cause.
it means that:
When people are doing something which others do not know the cause, then others are likely to see them as not thinking correctly.
When people do not know the reason for your actions they will not understand your reaction.
1. discussed in many places, many sources are investigated at Quote Investigator ®
There are well-known parallelisms between mathematics and music.
Mainly, they are both languages bound to logic and formal syntactic rules possibly more strict than the rules of the natural language.
An important difference, though, is how people come to learn and appreciate them. You can appreciate music just listening to it, without knowing its formal rules and in fact not even being able to read a score.
Algebra, or mathematics, works the other way around. First you struggle--sometimes painfully--with the rules and the drills. Only later and only a few arrive to really understand what is going on and to grow a feeling of what is right and wrong without having to resort to an actual formal proof.
I've read Bohr's line in the following sense: do you understand algebra to the point that it flows in you and you have acquired such instinct?
P.S.: I am an Algebra university teacher and yes, I do hear its music.