Mind Tools: The Five Levels of Mathematical Reality

by Rudy Rucker

Explores postmodern mathematics, explains the five levels of reality, and discusses new realms such as gigaplexes, pixels, and Hibert space

Genres: Mathematics, Science, Nonfiction, Philosophy

328 pages, Paperback

First published January 1, 1987

About the author

Rudolf von Bitter Rucker is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk genre. He is best known for his Ware Tetralogy, the first two of which won Philip K. Dick awards. Presently, Rudy Rucker edits the science fiction webzine Flurb.

Reviews

[Tom Schulte · Rating 3 out of 5]

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. Into “Number”, after a discourse on Pythagorean metaphysics heading toward numerology, the book takes a turn into material that is worth keeping in print.

[See my full review on MAA Reviews.]

[Dan Ust · Rating 3 out of 5]

I read this in high school, so this is my recollection years later. I found it interesting to divide math into different approaches. I was already deeply interested in math. Anyhow, this is more a beginners intro to the conceptual issues in math: whets the appetite, but isn't the final word (of course!).

[Charles · Rating 4 out of 5]

A really excellent work in this field. I'm by no means a math whiz but this explained the topic well for those who aren't whizzes. If my grade school math teacher had told me just a few of these things I might have grown up loving math.

[Fred Cheyunski · Rating 4 out of 5]

Mathematical Thought and Origins of Related Visualization Explained - Rudy Rucker’s “Mind Tools” provides a very clear and step by step explanation of mathematical thought from basic concepts to their incorporation in computing. From the Introduction through the chapters the book covers the “five modes of thought” and proceeds accordingly. Namely, it addresses mathematics as (1) number, (2) space, (3) logic, (4) infinity and information.

I first learned about this book as a reference given related to data driven organization design (see my review of Rupert Morrison’s book with this title), graph database approaches to organizational modeling and the importance of visualization in human deliberations. Rucker’s explanation certainly does help one better understand the foundations for such activities and other uses of computers and visualization (for example see my review of David J Staley’s “Computers, Visualization, and History”).

Among my favorite parts included the author’s description of the analog and digital activities (right and left) aspects of the brain with the former providing the basis for recognizing and the latter manipulating the content of a scene in a photograph (page 9). Then, there is his discussion of the psychological roots of mathematical concepts including Carl Jung’s use of a tetrad or quarternity to examine brain processing and pattern (archetype) formation (pages 20-21). Figures with many simple number patterns (pages 52-62) offer ways of dealing with numbers. Rucker’s treatment of the different types of curves relates different means for working with space (e.g. see material on sine curves starting on page 142 or mention of life fractals like concept maps on page 180). The remarks on the limits of logic (such as on page 198) and the conceivability of patterns within information (page 300) are also revealing. Also, illuminating at different points and at the end of the book is the explanation of the mathematic foundation for computers and the ways graphical information are produced (for instance see page 307).

While the book is well laid out and proceeds systematically, it does take some attention and perseverance to begin to put it all together. Nevertheless, “Mind Tools” is worth the effort to further grasp mathematical thought, the analysis, and visualization it makes possible as well as their usefulness in our lives.

[TheQueensBooksII · Rating 3 out of 5]

Math nerds unite! This will thrill any math-o-phile. Me, not so much. I'm really good at geometry and I love algebra, but this book was way out of my league . . . and interest level. While I admire Prof. Rucker's obviously brilliant mind (although I couldn't tell you if the book were or were not fraught with errors as I barely could understand a word of it . . .), the subject didn't do much for me. From fractals to googols to caustic nephroids to polynomial quartic space . . . get the picture? And why did he write the book? From the last page (really, I don't think this will spoil anything):

"My purpose in writing Mind Tools has been to see what follows if one believes that everything is information. I have reached the following (debatable) (sic) conclusions:
1) The world can be resolved into digital bits, with each bit made of smaller bits.
2) These bits form a fractal pattern in fact-space.
3) The pattern behaves like a cellular automaton.
4) The pattern is inconceivably large in size and in dimensions.
5) Although the world started very simply, its computation is irreducibly complex.

(Not sure where he got the idea that "the world started very simply," but neither he nor I will ever be able to prove that our ideas about how the world started are true. Moving on . . . ) Prof. Rucker lost me on page 45 when he introduced logarithms, without explaining exactly what they are. I kept meaning to look it up so I could follow along, but I realized I really didn't care that much. So why did I read the book? It was recommended reading for an upcoming course I'm in. I can't wait to understand why someone would want to recommend this book to a lay audience, but perhaps it is just because it gives you another possible way to view the world.

The book didn't do much for me, but I know sharper minds will probably want to sleep with it under their pillows at night and dream about Godel's theorem . . . .

[David · Rating 5 out of 5]

Rucker rules! As a math noob, this has been very insightful and helpful, not because I'm going to be able to rush out and use any of the information here (like I've been able to with my recent incursions into linear algebra), but because it completely opened my eyes to the connections between various branches of "higher mathematics." Beyond that, it's been a taste of what it's like to actually THINK about mathematical concepts beyond what I have thought about before (which is basic computation and a tiny bit of number theory).

Bonus points for helping me finally start to grasp logarithms and what they might be _for_, which is a subject that has frustrated me to no end since the "Big O" notation (classifying the rate at which an algorithm will expend resources as you increase the number of things you shove into it) in computer science. And for Rucker's fearless use of ordinary conversational English and his humble and honest opinions.

Oh, and additional bonus points for being able to tie the five subjects together in a way that really did have lightbulbs going off in my head. I highly recommend this book to anyone who wants a taste of the interesting mathematical fields and how they relate!

Also, a little shocking fact: this book helped me realize that I actually do not enjoy mathematical logic. Which is bizarre since I'm a computer guy and computers are, at the circuit level, nothing but logic. Hmm.

[Richard Meese · Rating 5 out of 5]

Turing, Goedel, Cantor, a great treatment of infinity and computer science. Puts forward the idea that any thing can be coded, including human thought--which leads to an idea of the possibility of machine intelligence.

[Gus Lackner · Rating 5 out of 5]

Teaches one how to think like a mathematician. Skip the first chapter; the five star mathematical insights are far, far more valuable than the one star philosophical discussion.

[Adrian · Rating 5 out of 5]

Rudy Rucker links mathematics to reality and explains 5 ways we can look at it: in terms of number, space, logic, infinity and information. The concepts explained are rather simple to understand and I'm pretty sure everyone will find some things they didn't know before. Later in a book he argues that "reality as information" may be the most correct view and our universe can indeed be a computational process.
I suggest people whose interest touches corners of math read the book, otherwise you may get bored.

[Alterstuart · Rating 5 out of 5]

This one is moving me out of my mathematical comfort zone and keeping me reading.
edit: Now I'm done reading it, I can wholeheartedly recommend it for your kids. Grab it if you see it, this author has a good way with words.

[Whoof]

Overall good food for thought, touches on tons of topics (some better than others). The way he presents proofs and more formal stuff is hard to follow, or I just wasn't trying very hard.

Ends with a fun theory of what reality is

[Erik · Rating 5 out of 5]

Great survey of mathematics that reveals its importance as a pure philosophy for everyday use.

[Carrie · Rating 3 out of 5]

Very interesting. Does a good job of making connections that I never would have seen on my own.

[Christopher Gilmour · Rating 2 out of 5]

its a book for dull people

[Ami Iida · Rating 3 out of 5]

the story about Infinite ∞

I finished reading it just now. I knew this book
from complex system . I reference Infinite relative with complex system

[Aaron · Rating 1 out of 5]

wisdom unlimited bibliography