Nothing writer cared about was objective (4-6)
I believe the answer is:
open-minded
'objective' is the definition.
(thesaurus)
'nothing writer cared about' is the wordplay.
'nothing' becomes 'O' (O resembles 0).
'writer' becomes 'pen' (something which writes).
'cared about' becomes 'minded' (I've seen this before. I am not sure about the 'about' bit.).
'o'+'pen'+'minded'='OPEN-MINDED'
'was' is the link.
Can you help me to learn more?
(Other definitions for open-minded that I've seen before include "Unprejudiced, receptive" , "Willing to consider new ideas" .)