Mind Tools: The Five Levels of Mathematical Reality

Rucker’s book explores mathematics and reality from five points of view: number, space, logic, infinity and information. Rucker explores these concepts directly and simply. His idiosyncratic philosophy comes across as lacking in cohesion and reach when compared to works by Pickover, Hofstadter, etc. It bookends more formal explorations, however. Much of the central portions entitled “Space”, “Logic” and “Infinity And Information” are readable and entertaining expositions of topics from number theory, basic calculus, and analysis.

In my reading, something happened between considering a tetrahedron’s vertices labeled “Disease, Death, Loneliness, and Struggle” and the passage: “The symbol 0 seems egg-like, female, while 1 is spermlike and male. Can this really be an accident? …An egg is round, and a sperm is skinny. …Formally speaking, both zero and one are undefinable.” I began to wonder why Dover reprinted this peculiar work. In the section on “Number”, after a discourse on Pythagorean metaphysics heading toward numerology, the book takes a turn into material that is worth keeping in print.

In “Space” there is material on algebraic curves suitable to engage first year calculus students or math enthusiasts in the wonders and basics of this topic. Rucker takes the reader to a tour of the Hilbert space and a lengthy discussion of fractals with several black and white illustrations. I feel the strongest material in this book, in introducing higher concepts to the non-mathematician, is in “Logic” and covers Gödel, Chaitin’s Theorem, Turing, and the taxonomy of infinities.

Tom Schulte’s front room is wall-to-wall and floor-to-ceiling books of well-loved books, including Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter.