The French fashion is in part one promoting particular philosophy (9)
'one promoting particular philosophy' is the definition.
'the french fashion is in' is the wordplay.
'the french' becomes 'plat' (I can't explain this - if you can you should give a lot more credence to this answer).
'fashion' becomes 'ton' (I've seen this before).
'in' is an insertion indicator.
'plat' placed around 'tonis' is 'PLATONIST'.
'part' is the link.
This may not be right. It may be part of another bit of the clue.
(Other definitions for platonist that I've seen before include "Philosopher", "Student of philosophy".)