A word previously well known to mathematicians but probably not to the general public, algorithm had begun to seep into the general awareness during the early years of the computer age. As the computer age mutated into the information age, algorithm became one of the buzz words, echoing around the world and seeming to transmute from a piece of vocabulary into a sentient being. Social media became littered with talk of sexist algorithms, racists algorithms, blind algorithms… With the supposed rise of AI, the much vaunted and eagerly sort after, but at the same time feared, artificial intelligence, talk turned to the search for the elusive intelligent algorithm. In little more than the seventy years since the Second World War the word algorithm has come to occupy a dominant position in much of the public discourse.
But what exactly does the word algorithm mean? Where did it come from? What is an algorithm? The word algorithm has an almost thousand-year history and over the centuries its meaning has mutated and evolved and the computer algorithms of today are not the same as the algebraic algorithms of medieval mathematics. Jeffrey M Binder, who describes himself as a programmer, historian, and writer, has written a book, Language and the Rise of the Algorithm,[1] which tracks those mutations and the evolution of the current meaning of the word algorithm over the time since it first appeared in the early thirteenth century.
I will start my review by saying that Binder’s book is excellent and if you have any interest in the topic at all then you should definitely read it. It certainly has the potential to become a classic in the tangled field of the histories of mathematics, language, logic, and computer science.
As Binder points out early in the introduction algebra was introduced into Europe by the Latin translation of Muḥammad ibn Mūsā al-Khwārizmī’s al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah in the twelfth century; its title gave us the word algebra and the mangled transliteration of his name into Latin the term algorithm. Because algebra is the practice of doing mathematics with symbols a large part of Binder’s book is a review over the centuries of how algebra was perceived, understood, and accepted or not. Part of that perception involved the question whether symbolic algebra was a language, so the book also traces the thoughts on the nature of language over the same period.
Central to Binder’s narrative is his systematic debunking of the commonly held belief that the computer age was heralded by Leibniz with his calculating machine and his attempts to create a calculus ratiocinator to resolve differences of opinion, expressed in the famous quote:
The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.
To a certain extent the book is divided into three sections pre-Leibniz, Leibniz, post-Leibniz. A finer division is presented in the fact that the book takes the reader chronologically through the history of symbolic mathematics, and the evolution of symbolic logic out of it from the sixteenth century down to the present. Throughout this journey Binder shows how the various actors used and or defined the concept of the algorithm and how the term took on differently meanings in different contexts. He shows how the term algorithm, that today non-experts seem to consider has always meant roughly the same, has actually been a linguistic chameleon taking on many different meanings over the last eight hundred years.
Binder packs far too much detailed information and analysis into each chapter of his book for me to attempt a detailed chapter by chapter review. To do so I would probably end up writing a review as long as the book itself. I can’t see anything of real relevance that Binder has left out of his account despite the fact that his book is hardly more that two hundred pages long. I will, however, give brief outlines of the five chapters and coda.
The opening chapter is a whirlwind tour of the introduction of both the Hindu-Arabic number system and algebra in the medieval and early modern periods, starting with Brahmagupta in the sixth century and ending with Descartes’ unification of algebra in the seventeenth century. The Hindu-Arabic number system because as Binder correctly points out algorithm, usually then spelt algorism, was the name for the rules governing the use of this new arithmetic. Despite its brevity this tour is excellently done.
The second chapter starts in the seventeenth century, enter Leibniz and his attempt to create a universal symbolic language that translates natural language. This chapter looks not only at Leibniz’s thoughts on language, both symbolic and natural, but at this of other protagonists of the so-called scientific revolution, in particular John Wilkins but also George Delgarno, John Ray, Descartes, Locke et al. It also covers the discussion amongst the mathematicians of the use of symbolism in the newly created calculus of Leibniz and Newton.
Moving into the eighteenth century, the third chapter centres on the thoughts on language, algebra, and symbolism of Marie Jean Antoine Nicolas Caritat, Marquis de Condercet. As in the previous chapter there is a list of significant contributors to the debates on these topics such as Locke, Euler, Jean Le Rond d’Alembert, and the Abbé de Condillac. A central question that is discussed by these participants, is algebra a language? Once more Binder covers a complex of thoughts and ideas briefly but comprehensively and clearly. There follows an account of the thoughts of the English mathematician, Francis Maseres, who firmly rejected the modern, continental thoughts on the relationship between algebra and language. I found particularly interesting that at this late date Maseres still had problems with the acceptance of negative numbers. For me surprisingly, this view was shared by his associate the political radical, Willian Frend, after all Frend’s eldest daughter Sophia Elizabeth married Augustus De Morgan. This is followed by a highly informative essay on an English ally of Condorcet, Charles Mohon, third Earl of Stanhope, the creator of the Stanhope Demonstrator, a logic machine. The chapter closes with rumination on Immanuel Kant and how he fits or doesn’t into these ongoing debates.
Throughout these chapters Binder draws the readers attention to the varying and various ways that the proponents in the diverse debates used and defined both implicitly and explicitly the term algorithm.
In terms of symbolic systems, language, and logic, the nineteenth century saw the dawning of a new age that Binder takes us through in his fourth chapter. At the centre of this new perception is George Boole and his algebraic logic. Boole divorces his logical symbols from natural language. The symbols are no longer defined in terms of an interpretation in natural language but instead through the rules of the system for their use. They don’t not have a single fixed linguistic interpretation but can be used to stand for many different things. Binder’s presentation of Boole’s logic and his motivations for creating it is excellent. Although Boole separated logic and natural language Binder points out that this development ran parallel to the new theories on the genesis of language developed by the Romantics.
The introduction to Boole is followed by an essay on the calculating wonder Zerah Colburn and the question as to whether the methods he used could be generalised as algorithms for others to learn. This is followed by the work on symbolic mathematics produced by the Cambridge Analytical Society, in particular the algebra of George Peacock, leading into a wide ranging examination of the thoughts of many other nineteenth century thinkers including Józef Wroński, John Venn, William Wordsworth, Maria Edgeworth, Mary Everest Boole, Samuel Taylor Coleridge, Augustus De Morgan, William Stanley Jevons, Ernst Schröder, Gottlob Frege, and others through which Binder weaves the thoughts and concepts of Boole.
Chapter five takes us into the twentieth century and finally into the age of the computer. Before the arrival of the computer, we have the meta-logical/meta-mathematical theories of Church, Post, and Turing setting the formal limits on what can and what cannot be calculated or computed. It is nice to see Post getting the credit that is due to him for once, he so often gets overlooked. Of course, Gödel gets a look in as do Andrey Markov, and Stephen Kleene. The latter two with varying definitions of the algorithm. There is a long discourse on the meta-logical and philosophical debates in mathematics and logic kicked off by Whitehead and Russell, with their Principia, involving Carnap, Turing, Wittgenstein, Church et all.
Near the end of this section is my favourite paragraph in the whole book, because of my perpetual war with the “Alan Turing invented the computer blockheads.”
That one must think of computing machines in terms of either of these models is not self-evident. Babbage had already imagined a programmable computer a century before Church and Turing, and the designers of some early computers, such as Konrad Zuse (1936–38) and the IBM Mark I (1939–43), were initially unaware of their work. John Von Neumann, often held up as the designer of the standard computer architecture, knew Turing personally and was deeply familiar with his paper on the decision problem, but it is not clear that Turing’s imaginary machines had any strong influence on his plan. Yet Church and Turing did eventually become common reference points for the discipline of computer science. The most important thing they provided was less a paradigm for the design of actual machines that a theoretical framework for reasoning mathematically about what came to be called “algorithm.” [2]
The computer has arrived and with it computer programming languages, at the beginning COBOL, FORTRAN, and ALGOL. We meet Grace Hopper, one of the later creators of COBOL, and her invention of the compiler–a program that automatically translates a human-readable sequence of instruction into a machine-executable form[3]. This presaged a minor program language war in the early days that Binder outlines. Some wanted to make programming languages more symbolic, mathematical, and strictly formal to avoid the pitfalls made obvious by the meta-logical results of Church, Turing et al. Other aware of the potential market for computer use by non-mathematicians and non-logicians wanted to write programs in more normal languages to make them accessible to these potential users. Binder points to the emergence of the Apple computer as a vindication for the user friendly party in this dispute.
Binder now considers the story of ALGOL, usually explained as algorithmic language although the original name was International Algebraic Language, which he sees “as a major factor in securing the widespread adoption of algorithm as a general term for computational procedures.”[4] The aim of ALGOL was it seems to produce a universal language for describing algorithms. The aims and failures of the ALGOL program are discussed in quite a lot of detail in comparison to other approaches to programming, leading into a wider discussion of approaches to creating computer algorithms.
Binder opens his introduction with the following paragraph:
In May 2020, as much of the world focused on the COVID-19 pandemic and as racial justice protests took place across the United States, a technical development sparked excitement and fear in narrow circles. A computer program called GPT-3, developed by the OpenAI company, produced some of the best computer-generated imitations of human writing yet seen: fake news articles that were, according to the authors, able to fool human readers nearly half the time, and poems in the style of Wallace Stevens.[5]
His journey through the history of the algorithm ends with a twenty-one-page coda, The Age of Arbitrariness, which deals with the newly emerging age of machine learning and the associated change in the meaning of algorithm.
If classical algorithms are divided from human understanding, they are also divided from data.
[…]
Machine learning (ML) changes this. The “algorithms” are no longer designed by engineers but instead tuned by machines based on large amounts of data.[6]
Binder closes out his book ruminating on this difference.
Binder has obviously invested an enormous amount of research in his book, a fact that is reflected in the 991 endnotes from just 225 pages of text, most of which refer to the 35-page bibliography of books and papers he consulted. The book also has a good index.
I have only two very minor negative comments on this excellent book. At one point Binder refers to “abacus and counting board” as if they were two separate things. The counting board is an abacus, Binder obviously doesn’t know that the wire and bead frame abacus that people now think of when they read the word abacus didn’t enter Europe until the early eighteenth century well after the use of the counting board had ceased to be used in everyday calculating. My other problem is that he twice refers to Martin Davis as a popular science writer when referencing one of Davis’ popular books, The Universal Computer: The Road from Leibniz to Turing (3rded. CRC Press, 2018). Given the subject of his own research, Binder really should know that Martin Davis was one of the 20th centuries leading meta-mathematicians/meta-logicians!
In his journey from al-Khwārizmī to GPT-3 Binder covers an incredible amount of complex material in a comparatively small number of pages. However, his writing is never cluttered or in anyway incomprehensible, it is always clear, lucid, and easy to follow and somewhat surprisingly, given to topic, actually a pleasure to read. As I said above, it certainly has the potential to become a classic in the tangled field of the histories of mathematics, language, logic, and computer science and I very much think it deserves to do so.
[1] Jeffrey M. Binder, Language and the Rise of the Algorithm, University of Chicago Press, Chicago and London, 2022
[2] Binder p. 176
[3] Binder p. 177
[4] Binder p. 179
[5] Binder p. 1
[6] Binder p. 205