Mathematics is the best example we have of the eternal, universal forms suggested by ‘Socrates’ in the dialogues of Plato. Penrose affirms this understanding in his discussions of General Relativity (the Large) and Quantum Mechanics (the Small), then considers some of the obstacles to a Unified Grand Theory, and suggests how science might investigate the connection between the physical and mental realms. The passage from the axiomatic through the problematic to the speculative makes this a fascinating read in the philosophy of science.

Einstein’s discovery of the principles of General Relativity―not by observation, but through pure mathematical reasoning―indicates the congruence between mathematics and the physical world. Alas. After Einstein, we can describe the structure of space-time with extraordinary accuracy, but we cannot explain how that structure came to be. The cosmological uniformity that we can see now could only have evolved from an absurdly precise initial state, and that precision, says Penrose, must have something to do with the union of quantum mechanics and general relativity―except that quantum mechanics and general relativity are incompatible theories as they currently stand.

Quantum mechanics (also derived from ‘unreasonably effective’ mathematics) can explain the stability of atoms, chemical forces, the reliability of inheritance through DNA, lasers, superconductors, etc, etc, but cannot be magnified to the classical level without changing some fundamental rules (collapse of the wavefunction!!). Penrose reviews the fun bits of quantum theory (wave-particle duality, spin, non-local effects) before alighting on the paradox (‘the measurement problem’) which indicates that the theory is incomplete, wrong, or…something else. Rejecting the conventional ‘many-worlds’ response to Schrödinger’s live/dead Cat, Penrose suggests that the solution may have something to do with our understanding of Perception. Wow.

In the third chapter, “Physics and the Mind,” Penrose takes up the question of mind/matter duality. His position is that the physical action of the brain evokes awareness, but this physical action cannot be simulated computationally. The brain is not a computer. (We understand natural numbers not because we have derived them from some set of computational rules, but because we have been able to make ‘contact’ with the Platonic world of mathematics.) Penrose speculates that the neural networks of the brain may exhibit large-scale quantum coherent activity and thus provide clues to a new theory of quantum gravity, which could serve to bridge the gap between quantum mechanics and general relativity.

In the last section of the book, peers of Penrose offer their critiques of his ideas, reiterating for the reader key concepts and considerations. This is the best kind of book on the pursuit of knowledge, one that recognizes the gaps in our understanding while emphasizing the curiosity and wonder inherent in the endeavor. Bravo. ( )

Incoherent ( )

POINTS OF INTEREST

ROGER PENROSE: “So you cannot say that the impossibility is at any specific place in the picture [see an Impossible Triangle] – the impossibility is a feature of the whole structure. Nevertheless, there are precise mathematical ways in which you can talk about such things. This can be done in terms of breaking it apart, glueing it together and extracting certain abstract mathematical ideas from the detailed total pattern of glueings. The notion of cohomology is the appropriate notion in this case.” “Let us come back to the question of what is going on. Gödel’s argument concerns particular statements about numbers. What Gödel tells us is that no system of computational rules can characterize the properties of the natural numbers. Despite the fact that there is no computational way of characterising the natural numbers, any child knows what they are.” “Thus, I am saying that, more generally, mathematical understanding is not a computational thing, but something quite different, depending upon our ability to be aware of things.” “Light cones represent the most important structures in space-time. In particular, they represent the limits of causal influence. The history of a particle in space-time is represented by a line travelling up the space-time diagram, and this line has to lie within the light cone. This is just another way of saying that a material particle cannot travel faster than the speed of light. No signal can travel from inside to outside the future light cone and so the light cone does indeed represent the limits of causality.” “Some people find it hard to conceive of this [Platonic] world as existing on its own. They may prefer to think of mathematical concepts merely as idealisations of our physical world – and, on this view, the mathematical world would be thought of as emerging from the world of physical objects. Now this is not how I think of mathematics, nor, I believe, is it how most mathematicians or mathematical physicists think about the world. They think about it in a rather different way, as a structure precisely governed according to timeless mathematical laws. Thus, they prefer to think of the physical world, more appropriately, as emerging out of the (‘timeless’) world of mathematics.”

STEPHEN HAWKING: “Personally, I get uneasy when people, especially theoretical physicists, talk about consciousness. Consciousness is not a quality that one can measure from the outside. If a little green man were to appear on our door step tomorrow, we do not have a way of telling if he was conscious and self-ware or just a robot. I prefer to talk about intelligence which is a quality that can be measured from the outside.” “Basically, he’s a Platonists believing that there’s a unique world of ideas that describes a unique physical reality. I, on the other hand, am a positivist who believes that physical theories are just mathematical models we construct, and that it is meaningless to ask if they correspond to reality, just whether they predict observations.”

NANCY CARTWRIGHT: “One thing that is clear is that Roger’s proposal – first to posit macroscopic quantum coherence across the microtubules of the cytoskeleton and then to look for the special non-computational features of consciousness in a new kind of quantum-classical interaction – is not a detailed programme.” ( )

What is the relationship of the "intangible" mind or consciousness and physical reality, namely, the brain? This is the question that Penrose addresses, and along the way deals with minor matters such as the Big Bang and Big Crunch, the relationship between classical and quantum physics (especially gravitation). All done with very little actual mathematics and in less than 200 pages! Clearly addressed to the general reader, I'm still glad that this is the third time I've been through these issues. And the general reader must "trust" Penrose that he got his mathematics right. His primary theses: (1) mathematics describes the physical world in a marvelously precise way, and mathematics resides in a world of ideas, i.e., he is a Platonist; (2) consciousness may be the emergent property of the quantum "entangling" of the thousands of neurons in the brain, specifically in the microtubules. "Action at a distance" is one of those very confusing and difficult to comprehend concepts of quantum mechanics, but Penrose insists the possibility needs to be investigated, with an entirely new approach to physics! I have always admired Penrose as a writer, taking very, very difficult subjects and making them more accessible to the general reader. I have been of the opinion that if someone cannot state things clearly, their own thinking is muddy as well. An excellent read for those interested in how physicists currently (well, as of 12 years ago) view the world around us.

Incoherent ( )