Mind Tools: The Five Levels of Mathematical Reality

Rating:
Summary: Basically good introduction to mathematical concepts
Review: In this book Rudy Rucker provides a novel way of classifying mathematical thinking - as number, space, logic, infinity, and information. He uses many standard examples and some more unusual ones such as classifying numbers as small, medium, large and inconceivable. It provides a good introduction for the general reader of mathematics, especially on the mathematical frontier, with such concepts as transfinite numbers, Goedel's incompleteness theorem, and cellular automata theory. It does have some errors, such as calling "Every sex act is sacred", "Every sex act is evil" imply "Some evil acts are sacred"; is a valid logical argument; not so. Consider this interpretation: "Every irrational integer is irrational", "Every irrational integer is an integer"; hence "Some integers are irrational". But in general I would recommend this book to the general reader.

Rating:
Summary: Ends with erroneous conclusions
Review: On the last page of this book, after bouncing around and occasionally relying on cutesy dialogue, the author makes five conclusions. Each appears below, with reviewer comments immediately after

1. The world can be resolved into digital bits, with each bit made of smaller bits.

By definition, a digital bit is irreducible, so the sentence makes no sense.

2. These bits form a fractal pattern in fact-space.

Fractals are noted for having the appearance of complexity, even though they are defined by very simple rules. All information indicates that the rules that define the universe are more complex than the human mind can comprehend.

3. The pattern behaves like a cellular automaton.

A cellular automaton is defined by rules where life and death are determined by the status of neighbors. By their definition, facts do not live or die on the basis of their neighbors.

4. The pattern is inconceivably large in size and dimensions.

No argument here.

5. Although the world started very simply, its computation is irreducibly complex.

The author is the only person that I have ever encountered who considers the world to have had simple origins. The second part is a direct contradiction to statements 1, 2, and 3, particularly 1.

Another distressing situation occurs when the author coins the word numberskulls. An obvious allusion to numskulls and used to refer to people who reduce things to numbers, it is a very poor joke. While this may appeal to those afraid of numbers, it ignores the fact the modern world has a numeric definition.
If you are looking for a book dealing with the world and how it functions, reach past this one and grab "The Emperor's New Mind" by Roger Penrose (Oxford University Press, 1989). It costs a little more, but is well worth it.

Published in Journal of Recreational Mathematics, reprinted with permission.


Rating:
Summary: inspiring
Review: Page 32 gives a chart which shows the evolution of the strands of mathematics from ancient times until the present. This makes the book.

Rating:
Summary: Mind (Expanding) Tools
Review: Really nice survey of important ideas underneath the application of mathematics to real world analysis and understanding. Actually started a company based on one of his "someone should write a progam that ..." statements.

Rating:
Summary: Mind (Expanding) Tools
Review: Really nice survey of important ideas underneath the application of mathematics to real world analysis and understanding. Actually started a company based on one of his "someone should write a progam that ..." statements.