The Unreasonable Effectiveness of Mathematics in the Natural Sciences

First, read about this topic on wikipedia.

Wigner points out that mathematics explains the natural phenomenon too
well. That this needs explanation.
Hamming (20 yrs later) attributes this to human observational bias.
Tegmark (17 yrs after Hamming) posits that the universe itself might be a
mathematical structure.

The originals are listed in the reference section, and are well worth
reading.

I am a Strange Loop

A book by Douglas Hofstadter about the relationship of the brain with consciousness, and the loop that forms when a symbolic processor must reason about the symbol that represents itself.

There is also a reasonable Douglas Hofstadter video, with Daniel Dennett named “Vicitm of the Brain”

Physics and Five Problems in the Philosophy of Mind

by Stuart Kauffman (available on arXiv)

1. how does mind act on matter?
2. If mind does not act on matter is mind a mere epiphenomenon?
3. What might be the source of free will?
4. What might be the source of a responsible free will?
5. Why might it have been selectively advantageous to evolve consciousness?
6. What is consciousness?

The Free Will Theorem

First read about through slashdot.
Mathematicians John Conway (inventor of the Game of Life) and Simon Kochen of Princeton University have proven that if human experimenters
demonstrate ‘free will’ in choosing what measurements to take on a
particle, then the axioms of quantum mechanics require that the free will
property be available to the particles measured, or to the universe as a
whole. Conway is giving a series of lectures on the ‘Free Will Theorem’
and its ramifications over the next month at Princeton. A followup article
strengthening the theory (PDF) was published last month in Notices of the
AMS.”

arXiv paper
ams notices

The EPR experiment

For a good explanation of the EPR experiment, and how it demonstrates that there is no local hidden variable theory.
Bell’s theorem is given a good explanation on LessWrong

Quantum Mechanics

Also, whenever you get into a discussion about the abilities of quantum
phenomena to do weird things, remember that there are some interesting limitations: In particular, the no cloning theorem states that it is not possible to copy a quantum state, because any copying process ends up destroying the original. This is actually a feature when it comes to secure communications.

Gödel’s Incompleteness Theorems

For some fundamental limitations on symbolic logic it really pays to be
aware of Gödel’s Incompleteness Theorems