I lean towards the idea that the mind is more than the brain, that the brain is just, as Huxley thought, a reducing valve for our Mind. Just the fact that when meditating, I can reach an experience of just being, of all my thoughts being stilled, of experiencing an absence of ego, however brief, yet still experience consciousness - the observer behind the thoughts, the metaprogram of which thoughts and ego are simply subroutines - for me, anyways, it gives weight to the idea that mind is separate from body. There is certainly a lot of anecdotal evidence to support that from historical Buddhist studies of consciousness.
Consciousness is one of the most difficult topics in both science and philosophy. Roger Penrose put forth a theory several years ago that Mind is a product of a series of collapses of the quantum wave function, that the brain is indeed a Quantum Computer. Too, there are growing numbers of scientists who are postulating that awareness is a fundamental property of the Universe, like Gravity - and that the greater the organizational complexity of matter, the greater the awareness. Awareness only achieves self-awareness at extremely high levels of complexity; to wit, the human mind. We are like crystallized focus points of Mind.
Too, in mathematics, Platonism is very much alive and well, though by no means universally held. Mathematical objects such as the Mandelbrot set are not invented or conceived so much as they are discovered.
In Philosophy, the concept of Qualia, or (to oversimplify), the qualitative nature of subjective experience, is held by many to be irreducible to physical or deterministic explanations. For if all that we are could be explained in reductionist terms, then mind-body dualism ala Descartes would be a non-issue; an illusion.
My personal opinion is that consciousness is probably non-reducible - but the jury is very much still out on this one, and will probably remain so for the foreseeable future.