Mulmuley’s geometric complexity theory is still where I would start. It’s based on continuum mathematics, but extending it to boolean objects is the ultimate goal. A statement of P!=NP in GCT language can be seen as Conjecture 7.10 here. (Also downloadable from Mulmuley’s homepage, see “Geometric complexity theory I”.)
Mulmuley’s geometric complexity theory is still where I would start. It’s based on continuum mathematics, but extending it to boolean objects is the ultimate goal. A statement of P!=NP in GCT language can be seen as Conjecture 7.10 here. (Also downloadable from Mulmuley’s homepage, see “Geometric complexity theory I”.)