The Universal Computer: The Road from Leibniz to Turing

by Martin D. Davis

Computers are everywhere today -- at work, in the bank, in artist's studios, sometimes even in our pockets -- yet they remain to many of us objects of irreducible mystery. How can today's computers perform such a bewildering variety of tasks if computing is just glorified arithmetic? The answer, as Martin Davis lucidly illustrates, lies in the fact that computers are essentially engines of logic. Their hardware and software embody concepts developed over centuries by logicians such as Leibniz, Boole, and Godel, culminating in the amazing insights of Alan Turing. The Universal Computer traces the development of these concepts by exploring with captivating detail the lives and work of the geniuses who first formulated them. Readers will come away with a revelatory understanding of how and why computers work and how the algorithms within them came to be.

Genres: Computer Science, Science, Mathematics, Nonfiction, History, Computers, Technology

257 pages, Hardcover

First published October 1, 2000

About the author

Martin David Davis (born 1928) is Professor Emeritus at New York University's Computer Science Department.

Reviews

[Andrew Nguyen · Rating 4 out of 5]

Originally assigned as an optional read in a theory of computation class 5 years ago, I finally got around to reading this (Dr. Lutz, please revise my grade to an A). I'm going to heavily caveat this review because I'm a big fan of both history and computing, a pretty specific niche. Further, a reader probably won't enjoy this book without a little formal training in mathematics.

This book is a whirlwind tour of mathematical ideas and people that led to the birth of the digital computer. Starting with Gottfried Leibniz and his dream of a language to describe abstract ideas and ending with Alan Turing's universal computing machine, Davis does a really nice job in drawing the line from abstract logic to formal mathematical systems to the computers we see today.

I really enjoyed the biographical portions of this book. Academic holy wars are part and parcel of any revolutionary idea, and each mathematician has a worthy foe: institutionalized ideas, vengeful colleagues or himself. Unlike in a Michael Lewis biography, these men aren't deified ad nauseam. Rather, Davis lays out their insecurities, horrifying political views and academic failures very plainly. Even the most brilliant mathematicians have failures of foresight or imagination in the realm of their greatest power. These failures really emphasize the difficulty of the journey from abstract thought to physical computer and gives the book a fun story arc.

Besides just biographical details, each chapter describes the ideas pioneered by each mathematician. Davis tries to explain these ideas in the most elementary ways possible and this succeeds for the most part. I personally enjoyed the description of diagonalization proofs, and the Turing Machine examples made me nostalgic for computer science curriculum. I followed very closely until the chapter on Godel's completeness theorem. This theorem is so subtle and gets referenced so often, I really wished that section would have given more explicit examples instead of oblique references to Principa Mathematica or Peano's Arithmetic.

Overall, I really enjoyed this read. Definitely recommended if you're interested in math, computers and history.

[Chethan R · Rating 5 out of 5]

Most books on the theory of computation start off with automatons like DFAs, NFAs, Pushdown Automatons, and Turing machines without really talking about the reason for these models of computation. The history and the reason behind the way computers are the way they're today is often omitted.

This book is an essential prerequisite for anyone studying theory of computation. From the vision of Leibniz to reduce all computation to a set of axioms and build upon them to Hilbert who pushed for building Arithmetic on such a formal system to Godel who proved that such a system is incomplete to Turing who designed the Turing machine on which all computers today are based, this book is a good blend of the ideas of computation for the last three centuries blended with the peculiarities of these mathematicians' personalities, it's a must-read for anyone interested in what computers are, why they are so and what all can computers compute!

[Maurizio Codogno · Rating 5 out of 5]

Il 2012 è stato il centenario della nascita di Alan Turing, e nell'occasione sono state pubblicate molte opere su di lui. Questo libro in realtà è del 2000, ma Martin Davis l'ha aggiornato in alcune parti, compreso un rapido accenno al programma di computer Watson che nel 2011 vinse una gara del quiz televisivo "Jeopardy", proprio per l'occasione.

L'approccio di Davis non è esattamente biografico, anche se il testo contiene varie brevi biografie da quella di Leibniz a quella di Turing; il filo conduttore è il raccontare la storia di come l'umanità sia arrivata a definire il concetto di computazione in modo indipendente dalla mente umana: si parte appunto dall'intuizione un po' naif di Leibniz, che sperava che con un "calculemus!" si potessero appianare le controversie, alla prima formulazione specifica di Boole, al tentativo di Frege fatto crollare da Russell di sistematizzare tutta l'aritmetica, alla grande struttura immaginata da Hilbert di cui Gödel minó definitivamente le fondamenta, alle applicazioni pratiche delle teorie di Turing sulla computazione che sono alla base dei nostri attuali computer. (Davis è uno dei piú accaniti sostenitori della preminenza di Turing rispetto a von Neumann a riguardo della nascita dell'informatica, per la cronaca).

Il libro contiene un po' di formule di logica matematica, che però non sono troppo ostiche da seguire anche senza una formazione specifica: non preoccupatevi, non ci sono dimostrazioni. La lettura resta molto scorrevole e chiara, e ci sono moltissime note a fne testo per chi volesse approfondire gli argomenti: insomma é altamente consigliabile non solo agli appassionati ma anche ai semplici curiosi. L'unica pecca che trovo è che Davis non perde occasione di farci sapere che in un modo o nell'altro ha avuto a che fare con tutti i matematici dello scorso secolo...

[Alex · Rating 3 out of 5]

Since picking up "Logicomix", I realized I have a few other books on the mathematical foundations of computing. While Logicomix disappointed, "Engines of Logic" certainly had to meaty math goods I was looking for. If nothing else, it was a good pointer to topics that warrant further investigation from a variety of other sources.

I can't imagine anyone but a computer-dork like me would find this interesting. Kinda like Rush…for books. But if you're into this sort of thing, it's worth picking up.

[Paromita · Rating 3 out of 5]

Good book but does try to cover a lot - could have been slightly longer perhaps. But the bigger issue for me was the detailed mathematical logic notation in certain chapters without sufficient explanation, the starting examples felt reasonable but then a couple of steps were skipped so the transition to more formal notation felt abrupt.

[Steve · Rating 4 out of 5]

Martin Davis, a notable logician who work for (and with) very notable mathematicians and scientists, writes about the relationship amongst math, logic, and computation.

He surveys the lives and achievements of thinkers from Leibniz and Babbage to von Neumann and Turing and discusses what these ideas mean for modern computing.

The Universal Computer is a rather quick read, with the biographical content being particularly brisk, and there are points where some readers may like more detail, but this can be viewed as an accomplishment considering the density of the topics covered, including Boolean Algebra, set theory and diagonal method, algebraic invariants, Godel's undecidable propositions, and Turing machines. Also, the author includes a lengthy notes section explaining the beginnings of the finer points of these matters and references for working through the actual mathematics and technology discussed.

[Griffin Strain · Rating 4 out of 5]

I was tasked to read this discussion of the evolution of logical thought and mathematics for my final class in my computer science degree, introduction to computational theory. Overall, I was very impressed with the work and the ability of the author to work the line between presentation of mathematical theory and historical background on the important figures in the creation of computable thought. On several occasions, it is evident that the author has strong opinions on certain facts and makes those opinions known, especially with some of the discussion about who should receive the credit for first putting the idea of a computing machines into practice. However, overall, I think the work reads as a fairly comprehensive view at a good number of mathematical ideas that came together to form computational theory. I would definitely recommend this book to anyone interesting in the origin of computing or computational logic.

[Adam Casto · Rating 5 out of 5]

An excellent overview of the history of mathematics as it pertains to the development of the concept of the modern computer. It can be a little difficult to follow at times as it chronologically jumps around between references and anecdotes. However with a bit of attention, it works to weave a wonderful picture of how a machine many of us take for granted these days came to be.

[Julia · Rating 4 out of 5]

Wir müssen wissen.
Wir werden wissen.

[Chinchilla_clouds · Rating 3 out of 5]

Ένα βιβλίο και για (εμάς) τους μη γνώστες. Καλογραμμένο (όπως και η μετάφρασή του) και προσιτό, περιέχει βιογραφικά στοιχεία για τον εκάστοτε επιστήμονα, κάτι που το κάνει πολύ συμπαθητικό και ξεκουράζει κατά την ανάγνωση, αφήνοντας την αίσθηση πως διηγείται ιστορίες, με έναν πιο ακαδημαϊκό τρόπο.

[Aadesh · Rating 4 out of 5]

Great interesting book and finished it in one go. Had read bit and pieces of all the awesome people mentioned in this book. This book provided details in their life and their contributions for the advancement of mathematics, computation and human knowledge.

[John Doherty · Rating 4 out of 5]

Pretty good but I would recommend anyone interested in this subject first read Charles Petzold's book "The Annotated Turing". It goes into much more depth. If I hadn't read the Petzold book first I think I would have been confused by this one.

[John Doe · Rating 5 out of 5]

Its a must read for any person works on computers.

There are many people already commented on the contents of this book. Here I want to talk about how and why I picked up this book, or how the hell this book interests me in the first place.

So I was learning and reading how to write programs and getting familiar with several programing languages, then what intricate me most is how similar all those so called programing languages are and in a way they all follow certain patterns, like they all have these very basic concepts: class, objects, structures, logic operations and most importantly control flow structure aka the iteration and recursion loop. But how are all these related to programing? What is programing exactly? there are so many sources saying that its kind of a smart way to solve problems. Then how about algorithms? Isn't Algorithm the way to solve problems? Like step by step find answer to a specific problem? well, I guess everyone is right, but I still couldn't understand what the hell are all those things, and most importantly why bother??

I bet lots of people have the same confusion as me.
And the key to understand these two basic questions is .....eeemmm... this book. lol
I am not joking.

Here comes my personal version of understanding. Centuries ago, people who were searching for truth, god's existence, laws of nature,wanted to find a way to deductively/mathematically/logically/systematically unfold *All The Truth*, aka Leibniz's crazy idea... then comes with the continual efforts of great mathematicians, logicians and philosophers, finally we have computable machines, encodings which use math logic operations to abstract lots of information of human knowledge into machine language: 0s and 1s. Boom we have computer that almost fufills Leibniz's dream. Yet, its a litter different from the original hope. That's machine mostly can just operate under defined instructions(leave machine learning and neural network alone). And the biggest advantage of machine is that it can exhaustively do the same thing over and over again, aka loops.

Finally I have an idea what the hell is programing and algorithms. That is to build an abstraction such that the problem in our human knowledge could be expressed/understood in a simulated/repetitive/iterative/recursive way such that it could be whole heartily embraced by machines 0s and 1s..

To put it in a rude and unprecise way, when we say we are programing, we are indeed trying to turn things into loops. Nonetheless, according to mathematicians world view, everything is based on addition, the acceleration brought by computers binary operation makes our world looping effortlessly. Ins't it the case???

[Giovanni Rasera · Rating 3 out of 5]

Io innamorato dell’informatica.
Ma il libro è scritto male.
Da rileggere.

[Ushan · Rating 4 out of 5]

This short book consists of mini-biographies of Leibnitz, Boole, Cantor, Hilbert, Frege, Gödel and Turing. Davis is a co-discoverer of the Davis-Putnam algorithm, and he knew personally some of the people he mentions; other than his short reminiscences, there is little in this book that a reader of Neal Stephenson does not already know. There is of course much more to the story; the P=?NP problem was first formulated in Gödel's letter to von Neumann; Davis mentions a biographer of Leibnitz who was executed by the Nazis for anti-Nazi activities, but not the great German logician who starved to death in a Czech prison because he was a Nazi.

[Kerem · Rating 5 out of 5]

Probably one of the most fascinating "history" books I've ever read. Starting with Leibniz' far ahead vision, the book goes through the history of "logic" that resulted in today's computers (and whatever will come next of them). The exciting lives of Boole, Frege, Cantor, Hilbert and Gobel, how they reached the major milestones in this history, and their failures and problems are presented almost in a novel fashion (rather than heavy maths). The book's final is a lengthy tribute to Turing, who was again well ahead of his time, and got frustrated and suffered a lot that he well deserves such a tribute. Truly exciting book with some of the key logic and maths behind, and strongly recommend.

[Jim Mccormick · Rating 5 out of 5]

Outstanding review of key personalities behind the development of logic. Interesting summary of the design of the earliest computers. Very reasonable perspective based on first-hand experience of mid-century computer developments. Objective presentation of divergent personal views of many of these great minds.

[Manuel · Rating 3 out of 5]

"Il calcolatore universale" percorre la storia della logica matematica che è alla base della teoria della computabilità e quindi alla base dei moderni calcolatori elettronici.

Il libro parte da Leibniz, in particolare dal suo sogno, ossia quello di riuscire a ridurre un qualsiasi problema matematico a un insieme di simboli che, dopo essere elaborati, forniscano la soluzione desiderata. Nonostante sia conosciuto per lo più per il suo fondamentare apporto al calcolo infinitesimale, riuscì a creare una logica rudimentale basata su operatori e relative proposizioni. Si passa poi a Boole, che riuscì a tradurre frasi scritte in un linguaggio comune in un insieme di espressioni algebriche. Dopo una breve parentesi su Cantor, si passa alla seconda metà su Hilbert, Gödel, per mostrare l'idea di metamatematica e il famoso teorema di indecidibilità.

La parte più concreta del libro si ha con l'ultima su Turing, si mostra il funzionamento della macchina di Turing, che è l'astrazione matematica del funzionamento di un calcolatore, e le vicende riguardanti la decriptazione delle macchine enigma durante la guerra.
Infine nell'ultimo capitolo si ha qualche cenno sui primi calcolatori costruiti negli anni 40/50 (Colossus/ACE/Eniac/Edvac/Mark 1) e sull'architettura di Von Neuman.

È un libro "matematico", quindi ci sono diversi concetti anche complessi intervallati da eventi biografici. Anche se capire i singoli elementi è difficile, nel complesso fornisce un percorso convincente sull'idea logica dietro ai moderni calcolatori.
Io cercavo un approfondimento riguardante la macchina di Turing e ne sono rimasto abbastanza soddisfatto.
3.5 ⭐

[Andrea Samorini · Rating 5 out of 5]

pag.233 _______________________________

Turing proseguiva chiedendosi «in che misura sia possibile, in linea di principio, per una macchina calcolatrice simulare attività umane», e questa domanda lo portava a considerare la possibilità di una macchina programmata per imparare, e alla quale fosse permesso di commettere errori. «Invece di avere una situazione in cui la macchina a volte non dà risposte, potremmo aggiustare le cose in modo che essa dia ogni tanto risposte sbagliate. Anche il matematico umano prende qualche cantonata quando sperimenta nuove tecniche... In altre parole, se ci si aspetta che la macchina sia infallibile, allora essa non può essere anche intelligente. Ci sono diversi teoremi che lo affermano quasi esattamente, ma che non dicono nulla su quanta intelligenza può essere esibita da una macchina che non abbia pretese di infallibilità» (riferimento indiretto al teorema di incompletezza di Gödel, sul quale avremo ancora qualcosa da dire nel prossimo capitolo). Turing invocava «fair play nei confronti delle macchine» osservando che «ogni matematico umano è sempre sottoposto a un addestramento prolungato». E concludeva: «... perché dovremmo aspettarci di più da una macchina?». Il gioco degli scacchi - disse - poteva essere un esercizio adatto, per cominciare. All'epoca, nessuna apparecchiatura di questo tipo era stata realizzata! Il pubblico, stupefatto, rimase in silenzio.

[Dan Cohen · Rating 4 out of 5]

A great book on the history of the mathematical concepts that relate to the foundations of computing and computability. It covers Leibnitz, Boole, Frege, Cantor, Hilbert. Gödel, Turing, von Neumann and others, and then finishes with an account of the early development of computing hardware.

I found it surprising that the author managed to do justice to such a varied range of subjects in a relatively short book. He even managed to convey a flavour of things as obscure as Cantor's transfinite set theory and Gödel's incompleteness theorems - this is an excellent introduction to these subjects as well as to Turing's work on computability.

Although the scope of the book was perfect for me (I was introduced to most of these areas many years ago as a student and have had a life-long interest in the foundations and history of IT), I did think it a bit odd that the author chose to cover some of these areas, given the supposed purpose of the book to describe the development of the concepts of logic that led to the development of modern IT. It's not really necessary to cover transfinite set theory for this, or the philosophy of the foundations of mathematics, or Gödel's work. A reader looking for more on pragmatic aspects and less mathematical history might be disappointed, but I was delighted.

[Ville · Rating 4 out of 5]

Kiinnostava kirja kaikille, joiden opintoihin on joskus sisältynyt yliopistotason matematiikkaa. Kirja käy läpi eri matemaattikoiden ja loogikoiden elämiä sekä matemaattisia pohdintoja ja saavutuksia 1600-luvulta 1900-luvulle asti. Kaikki (kirjan potentiaalisen yleisön edustajat) ovat kuulleet esimerkiksi Boolen algebrasta, mutta mikä tuo Boole oikeastaan oli miehiään? Entä äärettömyyksiä tutkinut Georg Cantor? Kurt Gödel? Alan Turing?

Aivan kirjan lopussa pohditaan vielä muutaman sivun verran, tietävätkö tietokoneet oikeastaan yhtään mitään, ja mitä itse asiassa on älykkyys tai tietoisuus. Näitä selitetään kiinalaisen huoneen ajatuskokeella sekä Turingin testillä. Kirja on vuodelta 2000, ja onkin hieman hupaisaa, että siinä kerrotaan vielä viisivuotiaan lapsenkin käyttävän kieltä sujuvammin kuin paraskaan tietokoneohjelma tekee, ja että Turingin testin läpäisevän ohjelman valmistuminen on vielä "kaukana tulevaisuudessa". Näin ChatGPT:n aikakaudella voitaneen jo todeta, että "kaukana tulevaisuudessa" tarkoitti noin 22 vuotta.

[Jordan Zhu · Rating 5 out of 5]

The breathtaking rapid pace of change in computing makes it easy to overlook the pioneers who began it all. Written by Martin Davis, respected logician and researcher in the theory of computation, The Universal Computer explores the abstract logical concepts underlying modern computers and their physical realization. In this groundbreaking book, you can benefit greatly from reading Martin Davis's fine ramble through the history of logic and the erudite, gripping, and humane illustrating of its pioneers.

Martin Davis has a career of more than six decades devoted to the important interface between logic and computer science. His first-hand experience, of teaching and programming on the room-sized behemoths, and lucid descriptions about logic, have encouraged my own investigations in the future.

[jailbird · Rating 3 out of 5]

the book used to be called "the universal computer". i'm glad that this has been changed, because i think that's a misleading title. i was captivated by the hints of the origins of computation in the earlier chapters, but the book is more focused on mathematics. the historical content is excellently written and immersive, however i'm a little bit disappointed by the tendency to hold up an assortment of computational idols. i was very disappointed by the dismissal of poland's involvement in the development of the bombe. but it's understandable that this book is written with more of a western bias.

and yes the mathematical portions are very complicated. i would not recommend for the average reader

[Novromeda · Rating 3 out of 5]

L’ho trovato un saggio interessante, anche se mi aspettavo qualcosa di leggermente diverso. Ho comprato questo libro per caso, perché mi sembrava giusto approfondire la storia della logica e dell’informatica, ma credevo l’autore avesse un approccio più leggero e concettuale sull’argomento mentre, di fatto, sono presenti molti calcoli complessi e ragionamenti ostici che non sono per chi vuole semplicemente avere un’infarinatura generale.
Nonostante ciò lo consiglierei a tutti coloro i quali hanno curiosità particolare e voglia di approfondire la storia della matematica che ha portato alla creazione del primo calcolatore.
Inoltre è presente un breve sommario della storia di Turing, forse il più grande matematico che sia mai esistito.

[Garrett · Rating 3 out of 5]

I'd be lying if I said I understood all of it. Even this pop-science retelling of the history of mathematics went way over my head sometimes. Still, it was fun to see the line of thought from Leibniz to Turing and beyond. It's funny, sometimes you just kinda assume that math has always been the way it is, but it turns out that new ideas are being discovered all the time (the idea itself of mathematical "discoveries" kind of blows my mind).

I wish there could have been more explanation of how computers actually work, but it was still interesting to meet a lot of the people who had an influence on them.

[Bravo27 · Rating 4 out of 5]

Questolibroripercorreilpercorsocheilpensierofilosofico-matematicohaseguitoperarrivareaimodernicalcolatori.Lacifradelraccontononèmoltoformale,anzisisoffermamoltosullatoumanodeipersonaggiseguitieperchivolesseapprofondireillatomatematico-scientificotrovaampispuntinellabibliografiaapprontata.Homoltoapprezzatolafranchezzaconcuil'autoreprendeposizioninelledisputeancoraapertespiegandomoltobenelevarieragioniedichirandoespressamenteleproprieopinioni.

[Roberto Paredes · Rating 4 out of 5]

I really enjoyed this one. It has some math and logic concepts that are easy to digest, along with their historical context.

The Universal Computer will show you who are the fathers of modern computing: how their lives where; their science, ideas, and how each one put a piece on the puzzle that was finally solved by Alan Turing.

[Schureed · Rating 4 out of 5]

The book is about the stories along the history of computer. Most of the content is actually about the personal life and opinions instead of the rigorous contribution of these mathematicians. Fun to read overall to see how science is developed in a tortuous way, but not too ideal for understanding CS materials.

[Jeannie · Rating 4 out of 5]

Read the translated version. It was a really insightful book, introducing the development of logic applied to computing. A little redundant on the life stories of mathematicians, but overall worth a read.

[Rodrigo Sampaio · Rating 2 out of 5]

I did not finish this book. The Kindle edition was faulty with wrong/missing symbols throughout, and the author’s technical explanations were also quite hard to follow / alienating, at least for someone without technical training or, maybe, sufficient intelligence :)