Abstract
From the physico-mathematical view point, the imitation game between man and machine, proposed by Turing in his 1950 paper for the journal “Mind”, is a game between a discrete and a continuous system. Turing stresses several times the laplacian nature of his discrete-state machine, yet he tries to show the undetectability of a functional imitation, by his machine, of a system (the brain) that, in his words, is not a discrete-state machine, as it is sensitive to limit conditions. We shortly compare this tentative imitation with Turings mathematical modelling of morphogenesis (his 1952 paper, focusing on continuous systems, as he calls non-linear dynamics, which are sensitive to initial conditions). On the grounds of recent knowledge about dynamical systems, we show the detectability of a Turing Machine from many dynamical processes. Turings hinted distinction between imitation and modelling is developed, jointly to a discussion on the repeatability of computational processes in relation to physical systems.
Invited conference, Colloquium on Cognition, Meaning and Complexity, Roma, June 2002 (version française dans Intellectica, n. 35, 2002/2, pp. 131–162, suivi par une réponse aux articles de commentaires, pp. 199–216). Reprinted also in “Parsing the Turing Test” Epstein et al. (eds.), Springer, 2008.
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Notes
- 1.
The term “Turing machine” is traceable to A. Church, review of (Turing 1936) in Journal of Symbolic Logic, 2, 42–43, 1937. The expression employed by Turing to designate his machine is “logical computing machine”.
- 2.
Crucial technical aspect of Gödel’s proof, 1931: it allows the encoding of the formal-deductive meta theory of Arithmetic in Arithmetic itself (see Gödel et al. 1989).
- 3.
A mathematical description of a forced pendulum can be found in Lighthill (1986).
- 4.
A system is deterministic, if we know to (or think we can) write a finite number of equations or rules of inference that will determine its evolution. In classical physics, determinism is inherent to the construction of scientific objectivity: the possibility to “determine” a system by a finite number of equations or of rules is intrinsic to its theoretical approach. Within this classical framework, Poincaré has demonstrated that equational determinism does not imply the predictability of the physical system. But we will come back to this, during an intermission.
- 5.
This reader, while the others read the §.2, could consult the following page http://www$\sim$cse.ucsc.edu/-charlie/3body/ for about ten extraordinary examples of mechanical iteration of perfectly regular orbits, for 3, 6, …, 19, 99 bodies (crossed 8s, fantastical flowers … absolutely no chaos). Once found, the exact initial conditions that generate these periodical orbits, thanks to very difficult mathematics, the machine, at each click of the observer, starts over with the exact same trajectories, as perfect as unreal. Unreal, because these orbits are critical: the gravitational field of a small comet at 10 billion kilometers would topple these “planets” far away from their periodical trajectories. Some of these images give rise to laughter (and the admiration for the mathematicians who worked on them), so much are they physically absurd: even in physics, some sense of humor can help us distinguish between real world and virtual reality.
- 6.
- 7.
In these two last cases of programmable ergodicity, it is the global knowledge of the past which says nothing about the future (the series have the appearance of globally random sequences – they can concentrate for a long time near certain values, change suddenly of attraction zone, topple a group of values very far, with no apparent regularities), but, locally, we perfectly know the next step – we have explicitly described (programmed) the laws of determination, conversely to dice and Lottery. It is the similar geometry of trajectories that allow to call ergodic all these series, physical or programmable: they show no visible regularities.
- 8.
Thom’s and Prigogine’s points of view have enormously enriched our knowledge and, despite important differences, they are mathematically and physically compatible: the analysis in (Petitot 1990) shows it quite well. Unfortunately, the trap of ontologizing Platonism gives rise to inescapable quarrels, because it leads to confound the mathematical construction of scientific objectivity that constitutes itself between us and the world, with preexisting ontologies. An objectivity constituted between us and this reality which canalizes and causes friction upon our organisative propositions, propositions that are in no way arbitrary because they are the result of our action in this world and they are embedded in our cognitive practices and structures (Longo 2003a,b). In effect, the mathematical concepts require a conceptor who draws them on the phenomenal veil starting upon regularities that impose themselves upon his/her cognitive structure (those he/she “manages to see”); the mathematical explicitation of these regularities are part of the very process of the construction of mathematical knowledge and objectivity. To put it in husserlian terms, Platonism reduces and confounds transcendental constitution and transcendence. How much damage has this understandable reaction, in foundational reflections, of numerous great mathematicians (Gödel, Thom, Connes … caused by the dominating formalist philosophies, which are technically difficult, but conceptually poor (those of foundations in meaningless logico-formal calculations, see next intermission). For example, in the quarrel about determinism, we even get to a dualistic separation that gives a different ontological status to fluctuation, a material cause, than to the global mathematical structure (the equations of a dynamic), efficient or formal cause, in the aristotelian terminology so dear to Thom. This latter would be the “in-itself” or the platonic idea and would precede the phenomenal appearance (Petitot 1990). The revitalization of Aristotle’s fine causal analysis is very interesting (but one must not forget the “final cause”, see (Stewart 2002)); there is, however, no need of an ontological (platonician) distinction among these four different causes. To the contrary, their interplay and temporal and conceptual simultaneity, within physical and biological phenomena, with their ‘teleonomy’, is the scientific challenge of today.
- 9.
“[The game] is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart from the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. […] We now ask the question, ‘What will happen when a machine takes the part of A in this game?’ Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, ‘Can machines think?”’ (Turing 1950).
- 10.
“A man provided with paper, pencil, and rubber, and subject to a strict discipline, is in effect a universal machine!… LCMs (logical computing machines, see note 1) can do anything that could be described as ‘rule of thumb’ or ‘purely mechanical’ ” (Turing 1948). And Wittgenstein continues: “Turing’s ‘Machines’. These machines are humans who calculate.” (Wittgenstein 1980). “No insight or ingenuity on the part of the human being carrying out the computation”: the LCM is the breaking down of formal thought into the simplest mechanical gesture, but as a human abstraction, upon a finite sequence of meaningless signs, outside of the world.
- 11.
Turing refers to the brain as, at least, a dynamical physical system. To stay within his image, take a turbulent system that is at the same time very stable and very unstable, very ordinate and very inordinate; insert it sandwich style between different levels of organization that regulate it and that it integrates. You will then have a very pale physical image of a biological entity. Among these entities, quite material, soulless and without software distinct from the hardware (the modern dualism of the cognitivism of the formal rule and of the program), you will also find bodies with nervous systems that integrate and regulate them (as networks of exchange and communication), within which they integrate themselves (as organs) and by which they are regulated (by hormonal cascades. for example). These systems organize the action of the body by keeping it in a state that is physically critical, yet extended (it subsists in time and following relatively spaced out rails (see Bailly, Longo, 2010); within the limits of this state, we can find both stability and instability, variance and invariance, integration and differentiation, see Bailly, Longo 2003b). And all this in a dynamic ecosystem and in the changing history of a community of bodies-brains that interact by gestures and language (ulterior levels of organization, external to, but generated by the biological objects, this time).
- 12.
May it be said between us that the winning strategy proposed above for a dynamical system also applies to a man (or a woman): ask a thousand questions that require a few lines of answers each, to the human and to the machine, via a teleprinter as Turing would want. Ask the same questions the next day: you will not obtain the same responses from the human, only a continuity of meaning. In this case, the random mechanical genesis of variants is more of an attempt to trick than a mathematical counter-strategy like those of which we speak above, because there is the vexed question of meaning as well as the dynamic stability of the biological object’s identity, which would show the difference. But that goes beyond the modest ambitions of this article: here we are only talking about digital machines and Physics.
- 13.
- 14.
But why change the name given by Turing to the imitation game between a machine and a man/woman? The slip of scientific vision, implicit in this change of name, is very well underlined by Lassègue (1998). But would have these authors failed to grasp the profound and dramatic irony of this improbable game in which to make a computer participate: to play the difference between man and woman? Would have they ignored the evolution and the mathematical stakes of Turing’s scientific project, at the same time as the tragedy of the “game” lived by this man of genius who first projectedhimself into a machine (human computer), then condemned for his homosexuality and soon to commit suicide; would they have so badly understood his mathematics as much as ignored his suffering between being and imitation: man/woman/machine?
- 15.
This issue of well explicating the hypotheses must be a feature of the Greats (Laplace, Frege, Hilbert, Turing, …): probably because they understand the novelty of the original conceptual framework they are proposing. If not, one may find, even quite recently, people who say they have “demonstrated” Church Thesis; small implicit hypothesis: the Universe, with all of its sub-systems, is an enormous laplacian machine. But, Church Thesis is an implication, which goes from all informal definition, that of potentially mechanizable deductive calculus à la Hilbert, to specific formal systems (Church, Turing, …). As an implication, today one could say that it is certainly within the limits of truth, in Thom’s sense: “the limit of the true is not the false, but the insignificant” (see for a modem appreciation (Aceto et al. 2003)). Quite obviously the ultimate goal of these “proofs” is to talk of the brain, finite sub-systems of the Universe (for a brief history of Church’s Thesis – Church–Turing’s, more specifically – and of its physical and cognitive caricatures (see Copeland 2002).
- 16.
“The model simplifies, the metaphor complicates” (Nouvel 2002); it adds information, it refers to a (another) impregnating conceptual framework, a universe of methods and of knowledge that we transform onto the first one. “When a model functions as metaphor, the model becomes an object of seduction for thought if we then use it as a suggestion for the solution of a philosophical question, we will manage, abetted by this confusion, to make this metaphor appear as a ‘philosophical consequence”’ of mathematical modelling (Nouvel 2002).
- 17.
Web page: Giuseppe Longo, ENS (the papers quoted below are downloadable).
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Longo, G. (2010). Randomness, Determinism and Programs in Turing’s Test. In: Carsetti, A. (eds) Causality, Meaningful Complexity and Embodied Cognition. Theory and Decision Library A:, vol 46. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3529-5_5
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