The morphology of the Pyrenees is characterized by the presence of high-elevation, low-relief surfaces. The origin of these Lower-Miocene surfaces is still debated. Two major interpretations have been proposed, both assuming that these surfaces are remnants of a single composite planation surface. The first interpretation proposes that this surface corresponds to a peneplain developed near sea level before the Late Miocene, subsequently uplifted and dissected. The present-day Pyrenees is therefore supposed to rise from the Late Miocene. In the second interpretation, the rise of the efficient base level of the chain induced the progressive inhibition of erosion and the smoothing of the relief before the Late Miocene, resulting in a highly elevated peneplain. According to this latter interpretation, the high elevation of the low-relief surfaces does not equate to post-orogenic uplift. We test these two interpretations by investigating, among other considerations, the relation between the elevation of the planation surface remnants and the deep structure of the chain. We find that (1) the isostatic compensation of the dissected Pyrenean planation surface by crustal thickening and (2) the absence of thinning of the lithosphere mantle below the chain favors the second interpretation.