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Brains, Machines, and Mathematics
Traces the relationship between the development of computing machines and our knowledge of brain functioning, and introduces corresponding mathematical models designed to describe this relationship. Begins with a historical overview tracing the rise of cybernetics to the current interchange of ideas between AI and brain theory. Subsequent chapters introduce neural sets and finite automata, the crucial cybernetic concepts of feedback and realization, pattern recognition networks, "semi-neural" learning networks, capabilities of Turing machines and automata which construct as well as compute. The final chapter presents two accessible proofs of Gödel's Incompleteness Theorem.
Hardcover
First published January 1, 1976
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Displaying 1 - 2 of 2 reviews
December 18, 2008
A lost gem among the wonderful flood of books merging biology, mathematics, and computers following Norbert Weiner's "Cybernetics" in the 1960s. The only reason I discovered it is because the author is a professor at my university.
I suspect that the very reason I gave this book 4 stars will be the reason that many will give it 2 or avoid it all together - its heavy on mathematical notation. Assuming you've got some set theory and a hearty constitution for exploring forests of equations, this is a stellar little handbook covering automata, neural nets, perceptrons, communication theory, and error-correction - the heart of the early attempts at mathematical models of the brain. Computational neuroscience has moved into many other areas today but this is still a great place to start if you're interested in the subject matter.
The clincher for me (and the fourth star above) was Arbib's clear and concise presentation of recursive logics and Godel's Incompleteness Theorem. Perhaps other books have filled this role since, but I've never found such an accessible presentation of it. I was literally trembling page-by-page as I entered the world of recursively enumerable sets and arithmetical logics somewhere over the airspace of Dallas, TX, emerging in a daze of sheer amazement at the airport.
I credit this book with making me comfortable with a great deal set theory and functional notation that has recently come in handy tackling modern algebra and the works of Rene Thom.
My biggest criticism is that this book does not convey the excitement of the author at all. The book reads exactly like a textbook. Profound and interesting conclusions are followed with sentences like, "The reader may be interested in learning more among the literature [see ref. 1 and 2]." Also, while I appreciated the mathematical rigor, I believe well-placed analogy and colloquialisms would have served this book well. I suppose the style was a product of both the era and the author's country of origin (Australia). In any case, this book still christened one of the more memorable plane rides I've ever had.
I suspect that the very reason I gave this book 4 stars will be the reason that many will give it 2 or avoid it all together - its heavy on mathematical notation. Assuming you've got some set theory and a hearty constitution for exploring forests of equations, this is a stellar little handbook covering automata, neural nets, perceptrons, communication theory, and error-correction - the heart of the early attempts at mathematical models of the brain. Computational neuroscience has moved into many other areas today but this is still a great place to start if you're interested in the subject matter.
The clincher for me (and the fourth star above) was Arbib's clear and concise presentation of recursive logics and Godel's Incompleteness Theorem. Perhaps other books have filled this role since, but I've never found such an accessible presentation of it. I was literally trembling page-by-page as I entered the world of recursively enumerable sets and arithmetical logics somewhere over the airspace of Dallas, TX, emerging in a daze of sheer amazement at the airport.
I credit this book with making me comfortable with a great deal set theory and functional notation that has recently come in handy tackling modern algebra and the works of Rene Thom.
My biggest criticism is that this book does not convey the excitement of the author at all. The book reads exactly like a textbook. Profound and interesting conclusions are followed with sentences like, "The reader may be interested in learning more among the literature [see ref. 1 and 2]." Also, while I appreciated the mathematical rigor, I believe well-placed analogy and colloquialisms would have served this book well. I suppose the style was a product of both the era and the author's country of origin (Australia). In any case, this book still christened one of the more memorable plane rides I've ever had.
September 28, 2025
Will not pretend to have followed all of the proofs, but the prose and diagrams in this book are excellent. Hyper relevant to 2025, despite being written over 50 years ago. Beautiful kernel connecting three branches of computation.
On the proofs and notation: I feel (perhaps biased by my time) that some systems and problems are helped and some are hurt by compact mathematical notation. Set theory and neural networks are elegantly compressable. Turing machines not so much.
On the proofs and notation: I feel (perhaps biased by my time) that some systems and problems are helped and some are hurt by compact mathematical notation. Set theory and neural networks are elegantly compressable. Turing machines not so much.
Displaying 1 - 2 of 2 reviews