Logic Machines, Diagrams and Boolean Algebra

by Martin Gardner

Logic Machines, Diagrams, and Boolean Algebra, by Martin Gardner, Dover Pubs, 1968, 13.5x21.5cm, 157pp + ads.

Genres: Science, Mathematics

157 pages, Paperback

First published January 1, 1958

About the author

Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature (especially the writings of Lewis Carroll), philosophy, scientific skepticism, and religion. He wrote the Mathematical Games column in Scientific American from 1956 to 1981, and published over 70 books.

Reviews

[Douglas Summers-Stay · Rating 5 out of 5]

It is nearly impossible to find out information from secondary sources about logic machines and diagrams. You can find scanned versions of the primary sources a lot of times, but the older ones (like Ramon Llull) are in Latin, and even the 18th and 19th century sources are often verbose and opaque. Martin Gardner, on the other hand, even by 1958, was already a master of clear and simple explanations. He goes through several historical examples of logic machines and diagrams-- diagrams being a kind of step halfway between a mathematical notation and a machine, where the solution of a logical problem becomes visually apparent as soon as it is diagrammed. His own invented method of diagrams are fairly clear, but would only later evolve into the method I like best.
The book is a link between several of the things I talked about in Machinamenta (Llull's disks, Jevons' logical piano) and the vector space reasoning method I've been working with in my research. The latter is mentioned in the appendix, but the best discussion of it is actually here. Some authors get so into biographical details that they neglect the technical aspects, but while Gardner does throw in some story for color, he keeps the focus on how the methods work, where it belongs.
The book was written near enough to the invention of the (digital, electronic) computer that it was still very much up for debate quite how such things would be built. Part of what the author was doing, I suspect, was thinking about how the different methods could be captured in silicon.

[Jerry · Rating 4 out of 5]

Once again in the history of science a subject of seemingly academic interest only, pursued entirely for its own sake, suddenly turns out to have enormous practical value. As we move into the new industrial age of electronic automation, there is every reason to believe that the dull, detailed work of the symbolic logicians will assume increasing practical importance in the designing of efficient circuits for the more complicated automata.

This is a 1958 survey of the history of “logic machines and diagrams”. That is, methods and devices for solving logic puzzles of the sort:

If B is true, then A is false.
Either B or C or both are true.
A is true.

I only vaguely remember these from high school math classes. They always seemed far too restricted for use in real-world problems, which is why they’ve remained “a recreational task” of mathematicians and amateur puzzlers.

It’s written in the semi-popular style combined with a deep dive into the especially interesting that Martin Gardner would become known for. Much of the later Gardner is here; he’s already quoting Carroll, Baum, and Chesterton, for example, though only sparingly (except for Carroll, who created his own logic diagram/machine hybrid).

By this point he’d already published In the Name of Science (later Fads and Fallacies in the Name of Science) and Mathematics, Magic, and Mystery. The latter bridged his previous career as a magic writer with his more recent career as a math writer. He’d started his “Mathematical Games” column for Scientific American a year earlier in 1957.

Because of when it was written, Gardner talks very little of digital computers and not at all about modern programming languages, except to predict that they’d play a major role in logic machines, and vice versa. He’s focused entirely on weird mechanical logic machines, paper-and-pencil diagrams that mimic them, and fascinating electronic (wired) logic machines.

He distinguishes between methods and machines that mimic the logic they’re solving, and those that do not. The former are easier to understand, and, thus, often more useful for real-world problem solving; the latter can be more innovative but also incomprehensible to the point of uselessness.

He starts with a machine that was almost but not quite a logic machine, and was famous in its time, the ars magna of Ramon Lull, which as described seems to have been very similar to those plot creation devices that are occasionally popular. He then covers the early history of logic diagrams, such as Venn diagrams.

As he moves into the “modern” era, there are more photographs of the wonderful devices, now wired, for solving logic problems. He ends with a couple of smart analogies to then-modern science fiction stories, such as Simak’s “Limiting Factor”.

I’m intrigued by what Gardner would have covered if he’d written the book only two years later, after the invention of logic gates. He was clearly aware of research in the area: he mentioned the research of Claude Shannon and others on how electronic circuits could mimic symbolic logic.

Giving orders to a giant brain, telling it to perform certain steps under certain circumstances, is more a logical than arithmetical matter, and the new electronic computers are being constructed with more and more attention paid to special circuits designed specifically to handle the logical aspects of the computer’s work.

I have a much better understanding of propositional calculus, symbolic logic, and Venn diagrams than I did going in, but that’s not really the point of the book. It’s meant to be a fascinating overview of several fascinating advancements in a then-obscure field that tended to fascinating amateurs like Gardner. It meets that goal very well.

[Lost Planet Airman · Rating 3 out of 5]

A part of my Martin Gardner, first-to-last retrospective. A good review of logic and logic machines from a mathematical perspective, but 99.44% overcome by the integrated circuit and the digital computer.

[Federico Kereki · Rating 2 out of 5]

The "modern" parts about current computers are, logically, outdated, but the historical parts are good.

[Sue Dounim · Rating 5 out of 5]

I would suggest to people mainly interested in the history of computing to actually start with the last chapter or two. Gardner's speculations on the future of electromechanical logic machines, aided by his deep familiarity with the science fiction literature of that time (1958) make really absorbing and fascinating reading.
The main content of the book was equally interesting to me. In several places, he cites contemporary inventors (and critics) of logic problem solving techniques and devices as, sometimes apologetically, observing that such and such a device or technique is interesting or clever, but ultimately useless as complicated propositional logic puzzles are almost completely impractical.
Yet here we are in the 21st century, where 1s and 0s (true and false) order the entire world...hmmm...