Customer Reviews

Logic Machines and Diagrams

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This is an interesting, if brief look back at historical computing mechanisms. It is interesting for two reasons. First, that it focuses exclusively on tools for logical reasoning, and not on numerical tools like a slide rule or abacus. Second, it was written at the dawn of the electronic computing era, before even one-at-a-time transistors were used, and long before the earliest integrated circuit.

It briefly describes Lull's rotating disks, a little like the child's toy that can put a lion's head on a turtle's body. Lull used dozens of variants (up to 14 disks, in one case) to explore combinations of theological and moral ideas, but not for reaching conclusions from basic premises. Gardner goes on to describe set-inclusion diagrams pioneered by Euler and popularized by Venn, Pierce's arcane "existential diagrams", and a few other pencil and paper notations. He describes a few clever tools based on different cards with windows cut into them, so that the alignment of windows in overlapping cards represents joining of logical statements. He also mentions mechanical, electrical, and relay-based gadgets, as well as one based on rotating, counter-rotating, and jammed gears.

For some reason, he chose not to mention a once-popular tool based on stacks of cards, with holes punched or notches cut in the edges. Stack the cards, slide a pin through the aligned holes representing (for example) male subjects, and shake loose all the cards with notches that don't snag the pin. Repeat the process on the male or female stack of cards, using other holes representing different traits, and the desired subset of subjects (if any) remain. Although used for filing, this tool could easily have handled dozens of logical variables, not just the four or five seen in other mechanisms.

The second facet of this book gives it a quaint feeling. Gardner ends the book with a set of predictions about the future of logic machines. He dismisses programmable computers as unwieldy and inefficient. Still, he hopes for inference engines to be sped up electronically, perhaps using analog computation. Other predictions sound equally odd - some because they did not come to pass, others (like H. G. Wells' World Brain) because they did in some way.

This book is an interesting bit of modern history, and a worthwhile collection of visual tools for representing logical concepts. If nothing else, it gives insight into how complex problems of four logical variables were once thought to be - much simpler than the computer circuits and even logic puzzles of today.

//wiredweird

The advent of the digital computer has rendered logic diagrams and manipulatives into historical footnotes. Nevertheless, they are significant in the historical development of logic as an area of mathematics. In this book Gardner describes the various ways syllogisms and propositions have been represented in diagrams, but he also spends a great deal of time explaining the personalities.
Ramon Lull was quite a character. Early in life he was a hedonist, but due to a religious vision, was converted into somewhat of a religious fanatic. While some groups considered him divinely inspired and he has been beatified, the Dominicans considered him a madman and his prolific writings contain a great deal that could be considered heretical. He even wrote a book containing dialogues in which he and a priest argue as to which one has had the most preposterous life. On three occasions, he made pilgrimages to Moslem cities in Africa in order to engage in religious debates with Islamic leaders and point out the errors inherent in Islam. The first two times he managed to escape with his life, but the last time, at the age of 83, he was stoned to death by a mob. In many ways, it is thought of as his way of committing suicide.
His logic diagrams were a set of circular manipulatives that could be rotated to create different combinations of symbols. Various symbols represented God, the soul, sins, virtues and other elements of religious belief. While they are of interest, after reading of his rather eventful life, they seem anticlimactic.
The diagrams of John Venn, network diagrams that represent the propositional calculus, tables with parts that slide to evaluate propositions, computing using binary values and even spinning wheels are all used to perform logical computations. The number of different ways and materials that can be used to evaluate propositions was surprising. I found the use of spinning wheels to be of greatest interest. If the wheel spins in one direction it is true and if it is spinning in the other direction, it is false. A proposition is then true if the spins interact to allow another wheel to spin the in the positive direction.
This is a book that is now of historical interest only. However, I think that I may introduce the spinning wheel example in class the next time I teach discrete math. It is easy to understand and demonstrates another way in which logic can be applied to describe the world.