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The set of all real numbers (ℝ) is uncountably infinite, yet all of the general theorems of math, all of the thoughts leading up to them, all of of the particles of the physical universe could be described by countably many numbers.
Does the fact that humans can grasp and correctly infer truths about an uncountably infinite structure like the real numbers imply that consciousness has a much larger “capacity” for information than the brain?
Does the uncountability of the "real" numbers imply that consciousness is much larger than the brain?
No. The 2 things are wholly unrelated. You are essentially requiring that the physical brain capacity required to contain the idea of something is related, in some sense, to a notion of the size of that something.
This is blatantly untrue for things that have physical size e.g. your brain doesn't need to be bigger than a horse to contain the idea of a horse.
Similarly, your brain doesn't need to contain the uncountable reals to contain the idea of uncountable reals.
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'.)
From a modern, non-Platonic, axiomatic point of view, mathematics does not collect facts, it studies the implications of groups of axioms. We obviously make up the axioms themselves. And our sense of which axioms-sets matter is derived from human intuition.
The idea of Euclidean space, for instance, is not a fact. If you consider the theory of relativity valid, we never encounter Euclidean space. The space we are in is never empty, and therefore never flat. But our intuition has evolved a very effective approximation to most of the space we encounter, and that approximation is Euclidean geometry.
The way we choose to model space involves infinite divisibility, but that does not mean that anything infinitely divisible actually exists. Chemistry seems to disagree, and physical theories like 'quantum foam' disagree even more profoundly. The easiest way for use to think of solids is as something infinitely divisible. But we know they are atomic. Even the underlying space may ultimately not be infinitely divisible.
So to think of the Real numbers as a factual thing that we are 'grasping' is not any more warranted than the idea that we live in Euclidean space. Imagination can create models that do not fit reality, but are useful. That does not mean that what we imagine is knowledge. Our capacity to hold onto our natural notions in spite of reality, and in spite of the fact that the idea requires things that are larger than reality can possibly be, does not mean that we really know, or even imagine anything truly more complex than reality itself.
To describe consciousness or conscious reality takes about 1000 billion numbers, namely coordinates and states of the neurons of a human brain. Nothing infinite or uncountable is required.
Further, the really real numbers that can be defined, such that two mathematicians know what they speak about, are not uncountable.
To believe in an uncountable set of "real numbers" requires to miss the fundamental mistake in the concept of countability or "finished infinity" (for closer information see chapters V and VI of https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf) and is certainly not a better indication for a large capacity of consciousness than the belief in Gods and Goddesses or in the flying spaghetti monster.
