Beyond Turing
This book carries in its title the name of an English mathematician, and it owes him an explanation. Alan Turing is remembered by the general public for two reasons: the machine that at Bletchley Park helped break the Enigma codes, shortening the war, and the question that in 1950 opened the debate on machine intelligence: "Can machines think?" But before either of these, in 1936, a twenty-four-year-old Turing had done something more profound: he had defined, with mathematical precision, what it means to compute. And in the very gesture with which he founded computer science, he had traced its boundary: there exist problems, perfectly defined, that no machine - none, ever, however powerful - will be able to solve.
This is why the chapter stands at the beginning of the book, and not in an appendix. The "incalculable remainder" that gives this work its title was not born as a romantic protest against technology, nor as a claim of faith against calculation. It was born inside mathematics, at the hands of its giants. Before ever being a hope of philosophy or a certainty of the faiths, the limit of the computable is a theorem. The man who proved it was no humanist quarrelling with machines: he was the man who had just invented them.
1. The Turing machine: what it means to compute
In 1936 the problem on the mathematicians' table had been posed by David Hilbert and bore a solemn German name, Entscheidungsproblem, the decision problem: does there exist a mechanical procedure capable of establishing, for any mathematical statement, whether it is provable? To answer, Turing first had to answer a question that no one had formalised: what exactly is a "mechanical procedure"? His answer is the machine that today bears his name: an unlimited tape divided into cells, a head that reads and writes one symbol at a time, a finite set of states and rules. Nothing else. A deliberately meagre idealisation, almost humbling in its simplicity - and for that very reason universal.
For the stroke of genius is the second step: Turing proved that there exists a universal machine, a single machine capable of simulating any other, provided its description is supplied on the tape. That description we now call a program, and that universal machine we carry in our pockets. The smartphone, the hospital server that stores medical records, the corporate information system that processes orders, the platform that trains large language models on big data: in their mathematical essence, they are all the same machine of 1936, merely made faster. The thesis formulated in those years by Alonzo Church and by Turing himself - never refuted in ninety years of computer science - says precisely this: everything that is computable in the intuitive sense is computable by a Turing machine. It is one of the most powerful ideas in the history of thought, and every chapter of this book pays it homage: the pipelines, the sensors, the profiles, the models encountered in these pages are variations, admirable ones, on that single theme.
But an exact definition has a price that the merchants of wonders prefer to pass over in silence: what is defined exactly has exact boundaries. To define computation is to be able to prove what computation cannot do. And it is here that the story turns vertiginous.
2. The halting problem: the boundary proved
Let us return to Hilbert's question: does there exist a mechanical procedure that decides every mathematical question? Turing's answer was to construct a question that no machine can decide. The most celebrated takes the form of a practical paradox every programmer knows: given any program and any input, is there an algorithm that establishes in advance whether that program will terminate or run forever? This is the halting problem. And Turing's proof - a limpid diagonal argument, by contradiction - establishes that such an algorithm cannot exist. Not "has not yet been found": it cannot exist, with the same necessity by which there is no largest prime number. If it existed, one could build a program that consults it about itself and does the opposite of what the oracle predicts: the prediction bites its own tail, and the oracle collapses.
It is worth breathing the air of this result, because it is more radical than it seems. It does not say that the halting problem is hard, costly, beyond the reach of today's computers. It says that it is undecidable: no increase in power, no future processor, no larger model will ever bring it inside the fence of the computable. The limit is not technological; it is logical. Nor is it an isolated case. A theorem of 1953, due to Henry Gordon Rice, generalises the blow: every non-trivial property of program behaviour is undecidable. Do you want an algorithm that certifies, in general, whether another program is correct? Whether it will never violate a rule? Whether it will always do what it promises? It does not exist, and it never will. Let the reader who has read the chapters on governance and on human oversight find here the deep root of what is asserted there: human supervision of automated systems is not merely a legal precaution written into the regulations - it is what remains necessary when mathematics itself proves that total automatic control is impossible. Article 14 of the AI Act has, without citing it, a foundation that comes from 1936.
There is, moreover, an even more vertiginous way of looking at the same boundary, and it rests on a simple count. The possible programs are infinite, but with a countable infinity: they can be listed, one after another, like the natural numbers. The possible problems - the functions to be computed - are infinite with a larger infinity, that of the continuum, which no list can exhaust: it is the diagonal argument that Georg Cantor had discovered half a century earlier. The consequence is disconcerting and inescapable: the computable functions are a countable island in an uncountable ocean. In a mathematically precise sense, almost everything that can be defined cannot be computed. The incalculable is not the exception at the margins of computation: it is the rule. It is the computable that is the rare miracle.
3. Gödel: truth exceeds proof
Five years before Turing, a young logician from Brno had already opened a crack in the same edifice, from the other side. In 1931 Kurt Gödel proved the incompleteness theorems: every consistent formal system rich enough to contain arithmetic contains true statements that the system itself cannot prove; and no such system can prove its own consistency from within. Hilbert's dream - a mathematics wholly mechanisable, capable of certifying itself - died twice in the same decade: Gödel showed that truth exceeds every single system of proof, Turing that decision exceeds every machine. They are two faces of the same boundary, and the history of logic has welded them together: Turing's result can be read as the computational form of incompleteness.
On what these theorems mean for the human mind one must be honest, because honesty is the method of this book. An illustrious line of thought - from the philosopher John Lucas in 1961 to the mathematical physicist Roger Penrose - has argued that incompleteness proves the mind is not a machine: man "sees" the truth of the statement that the formal system cannot prove, and therefore exceeds every formal system. It is a fascinating argument, and a controversial one: rigorous objections have shown that the conclusion does not follow of necessity, because nothing guarantees that the human mind is consistent in the sense required by the theorem, nor that it knows its own formal specification, if it has one. This book has no need to force Gödel beyond what Gödel proved. It is content with what the theorems establish beyond controversy: first, every formal system has an outside - a territory of truth that is structurally inaccessible to it; second, the gesture with which Gödel and Turing saw and proved these boundaries was not itself the execution of a known algorithm, but an act of understanding that did what no procedure of their time provided for: it looked at the system from outside. Anyone who wishes to reduce thought entirely to computation must at least explain how computation was able to prove, about itself, what no computation decides.
And here the reader recognises the lexicon of the whole work: the existence of an outside. The signal and the symptom, the data and the subject, the profile and the customer: every chapter has encountered a plane that the system formalises and a plane that exceeds it. The mathematics of the twentieth century is not the metaphor of this structure: it is its founding case, the one in which the excess was proved with the rigour of a theorem.
4. The imitation game: what the Turing test measures
In 1950, in a philosophy journal, Turing finally posed the question that made him famous - can machines think? - and with an engineer's instinct immediately declared it ill-posed: too vague to be decided. He proposed replacing it with an operational experiment, the imitation game: if, conversing in writing with a hidden interlocutor, a human examiner cannot tell the machine from the man, we shall say that the machine has passed the test. It is a methodologically masterful move, and it must be read for what it is: Turing did not define thought; he defined a criterion of behavioural indistinguishability. The test measures the quality of the imitation, by construction. On whether perfect imitation coincides with the thing imitated, the test - honestly - is silent.
For decades the question remained academic, because no machine came close. No longer: large language models converse with a fluency that would have astonished Turing himself, and recent experimental studies document that, in short conversations, examiners mistake them for humans at remarkable rates (Jones and Bergen 2024). The decisive point, for this book, is to understand what this success proves - and what it does not. It proves that linguistic behaviour is, to a large extent, computable: that the surface of conversation can be learned by a Turing machine trained on a sufficient portion of human language. It does not prove that behind the surface someone has appeared. As early as 1966 Joseph Weizenbaum had observed, with his simple program ELIZA, how little it takes for man to attribute understanding to what executes rules: and he was so troubled by it that he devoted the rest of his life to warning us. The error does not lie in the machine, which does what it is: it lies in the inference of the observer, who deduces being from functioning. The Prologue of this book has a name for this illegitimate passage, the category error, and the Epilogue unfolds it in full. Here it is enough to fix the point with the rigour of the chapter: the Turing test is a measure of performance, not a certificate of existence. Passing it says much about the imitation and nothing, by construction, about the imitator.
5. The mathematical remainder: why "beyond" Turing
One can now say with precision what the title of this book means. "Beyond Turing" is not against Turing - it is, literally, what lies on the far side of the boundary that Turing himself traced. No one had a better claim to trace it: the father of the universal machine is also the father of its first impossibility. And there is a moving nobility in this double primacy, which Turing's own life embodies: the man who mechanised calculation was also the man in whose hands calculation served to save lives, at Bletchley, in silence; a man whose greatness his own time failed to safeguard, and of whom history has asked forgiveness. The machines he invented have changed the world. The theorems he proved say that the world is not exhausted by them.
The technical reader will note that this chapter has spoken of logical limits, not practical ones - and that between the two lies a further band of territory: problems computable in principle but intractable in practice, where computation times explode beyond every physical resource. That is the province of complexity theory, and it reinforces the picture: even within the island of the computable, what is truly within the machines' reach is a still smaller region. But the heart of the chapter is the outer boundary, because it is there that the book's thesis finds its hardest foundation. When the pages that follow say that the symptom exceeds the signal, that the subject exceeds the data, that guilt exceeds the score, let the reader know that this structure - a formalisable inside, an irreducible outside - is no nostalgia of the philosophers: it is the very form that mathematics discovered by looking at itself.
And here lies the conclusion, which is at once a historical fact and a vertigo. The limit of the machine was not discovered by its enemies: it was proved by its fathers, in its own language. Before the first computer was born, humanity already knew - with mathematical certainty - what no computer would ever be able to do. The island of the computable is countable; the ocean that surrounds it is not. And on that island, for ninety years, we have been building wonders - this book traverses them all - without the ocean having retreated by a single metre. The incalculable remainder, before dwelling in the face of man, in his freedom and his hope, dwells in the theorems. Faith safeguards it, philosophy questions it: mathematics, quite simply, has proved it.
References cited in the text: Turing A.M. (1936), On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society; Church A. (1936), An Unsolvable Problem of Elementary Number Theory, American Journal of Mathematics; Gödel K. (1931), Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik; Rice H.G. (1953), Classes of Recursively Enumerable Sets and Their Decision Problems, Transactions of the American Mathematical Society; Cantor G. (1891), Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung; Turing A.M. (1950), Computing Machinery and Intelligence, Mind; Weizenbaum J. (1966), ELIZA - A Computer Program for the Study of Natural Language Communication between Man and Machine, Communications of the ACM; Weizenbaum J. (1976), Computer Power and Human Reason; Jones C., Bergen B. (2024), People Cannot Distinguish GPT-4 from a Human in a Turing Test, arXiv; Lucas J.R. (1961), Minds, Machines and Gödel, Philosophy; Penrose R. (1989), The Emperor's New Mind; Regulation (EU) 2024/1689, Article 14.