Nowadays, with mass industrialisation and growing digitalisation, it is almost impossible to ignore the discourse around the digital computer processes that underpin modern society. However, the term “algorithm” is quite often either misused or simply used without much knowledge of the history and theory behind algorithms.
Background
It is commonly conceived that algorithms were brought about when computers were built. Nothing could be further from the truth. Precise, mechanical procedures for performing arithmetic, which may well be regarded as algorithms, have existed from the time of the ancient Babylonians, who not only performed long division but approximated the square root of two long before Pythagoras was actually born. Hence, one could say that algorithms were invented at the same time as mathematics and that any mathematical process, at least a basic one, inherently involves a computational step. However, in order to formalise such a procedure, one needs the key notion of a variable and an algebraic abstraction, which was not commonly used by the Babylonians and Pythagoreans.
Al-Khwarizmi
The missing piece came about with the publication of The Concise Book of Calculation by Restoration and Balancing, or more commonly referred to in Arabic as Al-Jabr wal-Muqabaleh, now known as “Algebra”, by Muhammad ibn Musa Al-Khwarizmi. Little is known about the life and origins of al-Khwarizmi, other than that he was presumably born in the Khwarazm region, then part of Greater Iran, in present-day Uzbekistan and Turkmenistan.
As an influential scholar of the House of Wisdom in Baghdad during the Golden Age, al-Khwarizmi introduced the first systematic approach for solving a simple single-variable quadratic equation. Al-Khwarizmi introduced a geometric approach that manipulated the sum of squares to find the root of the polynomial. Since the notion of negative numbers was not conceived of at the time, the method only produced the positive real roots of the equation.
What is relevant from this method is the fact that it is the first effective, step-by-step procedure for solving a problem involving arbitrary variables. This is what distinguishes an algorithm from any written solution to a particular problem; an algorithm is meant to be a procedure for solving any instance of a general problem. This is how al-Khwarizmi, in addition to founding the field of algebra, accidentally laid the foundations for computing.
Algorism
Al-Khwarizmi’s books, along with notes on Hindu and Arabic numeral systems, in which he introduced calculations in base 10, were first translated into Latin under the title of Algoritmi de numero Indorum–Al-Khwarizmi on the Hindu Art of Reckoning. The circulation of this work around the West led to the common usage of the term ”algorism”, used to refer to any computation of arithmetic using numeral tables, and hence the profession of algorists in the 16th and 17th centuries came into being.
It is worth noting that the emergence of procedural and operational computation as a profession eventually led to algorithms becoming practically rich and rigorous. The importance of clarity, effectiveness, and efficiency of algorithms was first noticed at this point, and hence al-Khwarizmi’s purely algebraic method became a formal one that could be applied in a variety of settings. Eventually, this technique would simply be called an algorithm for centuries to come.
Algorithms & Computation
The methodology of algorithms continued to flourish in both applied and pure areas of mathematics and science. It went on to allow us to carry out powerful calculations and solve previously intractable problems. For a while, it seemed as if algorithms were, in an abstract sense, the ultimate solution to determining and deciding the general concept of truth. So intriguing was this idea that the great German mathematician David Hilbert famously said, “Wir müssen wissen. Wir werden wissen,” translated as: “We must know. We shall know.” It was believed by Hilbert’s school that essentially all propositions in mathematics could be formalised and decided algorithmically.
Hilbert’s dreams were utterly crushed by Kurt Gödel and subsequently by Alan Turing and Alonzo Church, who proved that there are limits to algorithmic procedures. The beautiful irony is that in proving the limitations of algorithms, they founded the whole enterprise of computer science upon which modern digital technology is entirely dependent. Turing and Church’s work went on to inspire generations of mathematicians and computer scientists in extending and applying the foundations of computing to create new enterprises in science and technology.
AI & Quantum
What does an algorithm mean for us today? One is instantly reminded of Artificial Intelligence (AI); seemingly intelligent agents working by the very same principles by which algorists performed their calculations. It is truly remarkable to see how far foundations that were once proven by Turing to be limited have come to create entirely new forms of technology. The inescapable conclusion is that the concept of what an algorithm is and what its limits are is constantly being challenged and altered.
With newer technologies depending on AI for automatic decision-making, it very often feels as if the world is becoming subject to algorithms. Can we allow the world’s events to be solely judged by algorithms? Have we, in exploiting the immense power of algorithms, lost control of our own creation? It has never been as crucial to ponder such questions about the nature of algorithms and their ethical implications.
If algorithms were developed dynamically throughout history, then how can they change in the future? With David Deutsch’s formulation of quantum computing, it is natural to ask what the next chapter in this story will be and whether quantum computing might introduce a new notion of algorithms and their limits. From a quest to solve simple quadratic equations geometrically to the long-desired dream of formalising mathematics, all the way to building autonomous machines, al-Khwarizmi’s vision keeps reappearing with more elegant forms.
