Shadows of the Mind: A Search for the Missing Science of Consciousness is a 1994 book by mathematical physicist Roger Penrose that serves as a followup to his 1989 book The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics.

Penrose hypothesizes that:

In 1931, the mathematician and logician Kurt Gödel proved his incompleteness theorems, showing that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. Further to that, for any consistent formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory. The essence of Penrose's argument is that while a formal proof system cannot, because of the theorem, prove its own incompleteness, Gödel-type results are provable by human mathematicians. He takes this disparity to mean that human mathematicians are not describable as formal proof systems and are not running an algorithm, so that the computational theory of mind is false, and computational approaches to artificial general intelligence are unfounded. (The argument was first given by Penrose in The Emperor's New Mind (1989) and is developed further in Shadows of The Mind. An earlier version of the argument was given by J. R. Lucas in 1959. For this reason, the argument is sometimes called the Penrose-Lucas argument.)

Penrose's theory of Objective Reduction predicts the relationship between quantum mechanics and general relativity. Penrose proposes that a quantum state remains in superposition until the difference in space-time curvature reaches a significant level. This idea is inspired by quantum gravity, because it uses both the physical constants ${\displaystyle \scriptstyle \hbar }$ and ${\displaystyle \scriptstyle G}$. It is an alternative to the Copenhagen interpretation, which posits that superposition fails under observation, and the many-worlds hypothesis, which states that each alternative outcome of a superposition becomes real in a separate world.

Penrose's idea is a type of objective collapse theory. In these theories the wavefunction is a physical wave, which undergoes wave function collapse as a physical process, with observers playing no special role. Penrose theorises that the wave function cannot be sustained in superposition beyond a certain energy difference between the quantum states. He gives an approximate value for this difference: a Planck mass worth of matter, which he calls the "'one-graviton' level". He then hypothesizes that this energy difference causes the wave function to collapse to a single state, with a probability based on its amplitude in the original wave function, a procedure taken from standard quantum mechanics.

When he wrote his first consciousness book, The Emperor's New Mind in 1989, Penrose lacked a detailed proposal for how such quantum processes could be implemented in the brain. Subsequently, Stuart Hameroff read The Emperor's New Mind and suggested to Penrose that microtubules within brain cells were suitable candidate sites for quantum processing and ultimately for consciousness. The Orch-OR theory arose from the cooperation of these two scientists and was developed in Penrose's second consciousness book Shadows of the Mind (1994).

Hameroff's contribution to the theory derived from studying brain cells (neurons). His interest centred on the cytoskeleton, which provides an internal supportive structure for neurons, and particularly on the microtubules, which are the important component of the cytoskeleton. As neuroscience has progressed, the role of the cytoskeleton and microtubules has assumed greater importance. In addition to providing a supportive structure for the cell, the known functions of the microtubules include transport of molecules, including neurotransmitter molecules bound for the synapses, and control of the cell's movement, growth and shape.

Penrose's views on the human thought process are not widely accepted in certain scientific circles (Drew McDermott, David Chalmers and others). According to Marvin Minsky, because people can construe false ideas to be factual, the process of thinking is not limited to formal logic. Further, AI programs can also conclude that false statements are true, so error is not unique to humans.