Henry Carter Hull

An introduction to the fourth dimension that explains the fundamentals of hyperspace and higher-dimensional geometry.

In 1962 John Hubley came to Harvard University as the first teacher of animation in the new Visual Arts Center. It was his idea to make a film based on Edwin Abbott's famous novel about life in a two-dimensional world, Flatland.

Now turn to logic puzzles, trying to distinguish between knights who only make true statements, and knaves who only tell falsehoods. Start with simple cases. Then introduce tricky "if–then" statements. Next, what if the knight or knave is insane and thus has false beliefs? This makes things trickier! Finally, add a third category: normal people who are sometimes truthful, sometimes not.

Lewis Carroll, author of Alice's Adventures in Wonderland, wrote a book of logic puzzles for children. Take a crack at some of these fun exercises, which Carroll designed to illustrate the principles of Aristotelian logic. See what you can conclude from such categorical statements as "all wasps are unfriendly, and all puppies are friendly." Carroll's syllogisms get progressively more elaborate.

Finish your study of logic with puzzles where you must draw conclusions based on what other people can infer from information they are given. Your first example is the "muddy children" puzzle, in which children with muddy faces must conclude with logical certainty (without looking in a mirror, feeling their faces, or being told) that they have muddy faces. Such puzzles are unusually subtle.

Discusses the unity among Johann Sebastian Bach's multifaceted compositions, graphic artist M.C. Escher's paradoxical drawings, and the mathematical concepts of Kurt Godel, exploring the Strange Loop phenomenon that occurs in all their work, in which moving upwards or downwards through the levels of a hierarchical system leads right back to the beginning.

Learn about biconditional statements of the form, "p if and only if q." Then tackle the "Hardest Logic Puzzle Ever," devised by philosopher George Boolos. You have three yes/no questions to identify three gods: the god who always answers truthfully, the god who always lies, and the god who randomly mixes true and false answers. One big problem: They answer in a language you don't speak.

This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition of mathematic fundamentals. Rather than a system of theorems with completely developed proofs or examples of applications, readers will encounter a coherent presentation of mathematical ideas that begins with the natural numbers and basic laws of arithmetic...