Does the uncountability of the "real" numbers imply that consciousness is much larger than the brain?
I'm not sure this is a direct answer, but some reflections and arrows on the problem. Quickly, though, one resource in particular I would recommend taking a look at that feels on point is Albert Lautman's Mathematics, Ideas and the Physical Real.
One thing I'm thinking about here: the reals are more "profoundly" infinite than the rationals -- it is curious how the respective jargons line up here in a certain way (and that the subtle difference between these primordial orders of infinition, and whether there is anything 'between' them, should be of some concern to 'everyone' today, philosophers and mathematicians.)
There is a subtle capaciousness and possibility of an infinite precision in the "real" which is inherently super-physical -- not realizable as true properties of materialities within the cosmos, so far as we understand -- i.e., that there is a fundamental quantization of the energy fields making up the universe; and hence a 'continuum many' anything is right out, except on paper, in thought experiments -- in a strange middle modality of existence which is rigorous and logical, but social and interpersonal as well -- the 'lawfully countable' being of syntax which is caught up in the dialectical development of mathematical armamentarium.
(Although maybe there's a way to get an arbitrary amount of information in a condensed region through infinite photon density? Hypercomputation for instance seems to demand something like this.)
The 'real' point I came here trying to say, however, was something like this: thought moves at an infinite speed when creating philosophical concepts -- a concept only coheres through a particular combination of its component elements; these 'objects' which may have nothing to do with each other as a group, and seem collectively perhaps to 'do' little with the concept, but nevertheless they function as part of a diagrammatic assemblage to create or recreate the concept.
But there are auxiliary functions of the concept as well: cognitive operations which work to discover and 'smooth' a plane of consistency for operations that are syntactical, social, legal, political, economic, psychic, etc.; machines to encode or unfold.
The concept works, but is only 'perceptible' or 'cognizable' provided you attain this pure speed necessary to undergo the concept as event, in such a way that you write (or become) the creation or reconstitution of the concept along with an appropriate plane of consistency (writing, jargon.)
Following Laruelle, perhaps we can even say the concept is in a certain way even perhaps a kind of transcendental computer -- a combinatorial or categoreal 'engine' of objective intensities and movements, coordinating specific functions in material assemblages -- but only capable of being grasped "as-such" by an effectively 'unwritable' operation (of Consciousness but we could also say Love, since it is becoming-everyone/the whole world, being everywhere-at-once, etc.)
At any rate you can read more about this line of thinking about what philosophy is actually up to in What is Philosophy? (especially this point about thinking only creating or reconstructing the concept when moving 'at infinite speed'.)