Welcome to online math help,
I agree that it seems a little odd that we don't bother to define
'opposite' precisely; but the comparison to 'open' or 'bounded' is not
really fair, since in those cases a simple word is being given a
complicated definition, while 'opposite' is a simple word being given
the simplest possible definition. more definitions on online math tutors.
Also, the definition probably
doesn't enter into any proofs in such a way as to require clarity; you
only need the definition to see what someone is talking about. That's
important, of course, examples on math forum, but there is a lot of ordinary language that
mathematicians use that doesn't need definition just because we use
them enough to know we are using them the same way. Often it's only in
talking to kids that we realize we don't know how to define a word.
I agree that it seems a little odd that we don't bother to define
'opposite' precisely; but the comparison to 'open' or 'bounded' is not
really fair, since in those cases a simple word is being given a
complicated definition, while 'opposite' is a simple word being given
the simplest possible definition. more definitions on online math tutors.
Also, the definition probably
doesn't enter into any proofs in such a way as to require clarity; you
only need the definition to see what someone is talking about. That's
important, of course, examples on math forum, but there is a lot of ordinary language that
mathematicians use that doesn't need definition just because we use
them enough to know we are using them the same way. Often it's only in
talking to kids that we realize we don't know how to define a word.
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