WordPress Website





Artificial Intelligence has become an inseparable part of human life, the usage of AI has evolved over time surpassing complex computational use to a more intricate use in everyday lives. Google’s Med-PaLM 2 exhibited exemplary skill in answering questions of medical licensing exam[1], Tesla’s self-driving cars, chatbots used by online platforms to address grievances of the customers and so on. Being as inseparable, the question regarding the intelligence of a machine is bound to arise, however, Alan Turing a famous English mathematician, logician and computer scientist proposed a test to determine whether a machine can exhibit intelligence. He proposed that the Turing test is used to discover whether or not a machine can think intelligently like humans. [2]

In this essay, we are going to assess the Turing test, its mathematical objection and possibly extend its applicability.




Turing proposed “The Imitation Game” wherein three participants exist, the Interrogator, Computer and Human Responder. The premise is that the interrogator asks questions to both the computer and the human responder who are marked or represented by X and Y respectively or by any other alias. Both the participants answer the questions asked by the interrogator. If the machine can successfully mimic human actions through its answer and deceive the interrogator into believing that it is the human responder, the machine passes the Turing Test.

The test is quite simple and easy and works based on assessing the communication of the machine. The Turing Test faced many criticisms which were later incorporated as the “Objections to the Turing Test” some of which are the Theological Objection, Mathematical Objection, Lady Lovelace’s Objection, etc.[3]

Some of the objections were raised by Turing himself, one such objection is the “Mathematical Objection” which shall be the subject of our study for this essay.




The reason for the uniqueness of the mathematical objection is that it is one of the very few objections that speaks of a human’s incapacity to programme an “unanswerable” question, unlike the popular Lady Lovelace’s Objection wherein she claimed that a machine can only be capable of carrying out what is ordered by the human and not think by itself. The Mathematical Objection is essentially the combination of Godel’s Incompleteness Theorem, and Turing’s Halting Problem subjected to Lucas Penrose’s Argument. The basic premise of the Mathematical Objection is that there are some questions which are “unanswerable” and hence cannot be programmed into a machine.

  1. Godel’s incompleteness Theorem

Kurt Gödel presented two important discoveries in mathematical logic, known as Gödel’s incompleteness theorems, in 1931. These theorems demonstrate the intrinsic constraints of provability in formal axiomatic systems. The first incompleteness theorem states that no coherent set of axioms, whose theorems can be derived via an algorithmic technique, can prove all facts about natural number arithmetic.[4] Essentially, any such consistent formal system would always include claims about natural numbers that are true but cannot be proved inside it. Gödel’s proof makes use of a diagonal argument, which is the first of many similar theorems on the limits of formal systems. The incompleteness theorems apply to formal systems that are sophisticated enough to explain the fundamental arithmetic of natural numbers while being consistent and properly axiomatized.

The first incompleteness theorem carries significant implications for the foundations of mathematics. It shows that Hilbert’s program, which aimed to find a complete and consistent set of axioms for all mathematics, is unachievable. The theorem also implies that there are mathematical statements that cannot be proven or disproven within a given formal system.


  1. Turing’s Halting Problem

Alan Turing’s Halting Problem is a key notion in computer science that asks if it is feasible to predict whether a particular programme will ultimately stop or continue endlessly. Turing demonstrated in 1936 that there is no generic algorithm capable of solving this issue for all program-input pairings.

The Halting Problem is stated in terms of Turing machines, which are theoretical computational models capable of simulating almost any algorithm. A Turing machine is made up of an endless tape separated into cells, a read-write head that may travel along the tape, and a finite state control that governs the machine’s behaviour depending on the current state and symbol being read.[5] Turing’s demonstration of the Halting Problem’s undecidability is based on the idea of a self-referential programme, which is a programme that uses its own code as input to decide whether or not to stop. Turing demonstrated that if such a programme existed, it would result in a contradiction. Specifically, he created a programme that, when given its own code as input, would enter an endless loop if and only if the halting programme decided to stop. This indicates that the halting programme cannot accurately predict whether the self-referential programme will stop or not, contradicting the premise that such a programme exists.[6] The Halting Problem has important implications for the theory of computation and the design of programming languages. It shows that there are limits to what can be computed by algorithms, and that some problems are inherently unsolvable. It also highlights the importance of careful program design and testing, as it is not always possible to determine whether a program will halt simply by examining its code.


  1. Lucus Penrose’s Argument

Lucas Penrose’s argument on the Halting Problem is a criticism of Alan Turing’s work on the issue. Penrose, a mathematician and philosopher, claims that Turing’s argument of the Halting issue’s undecidability is inadequate, and that human intuition may be used to solve the issue. Penrose’s thesis is founded on the notion that human awareness and intuition do not have the same limits as algorithms and machines. He claims that human mathematicians can solve mathematical problems beyond the capabilities of even the most sophisticated computers. Penrose claims that this capacity stems from a fundamental distinction between human awareness and machine computation.[7] Penrose’s criticism of Turing’s Halting Problem is based on the premise that Turing’s demonstration is only valid for computers that follow a set of established rules. Penrose contends that such constraints do not apply to human awareness, and that people may instinctively understand solutions to problems that computers cannot solve.[8] Penrose’s argument has been met with criticism from the computer science community. Many argue that Penrose’s critique is based on a misunderstanding of Turing’s work and that human intuition is ultimately subject to the same limitations as machine computation.[9]

Despite the criticism, Penrose’s argument has sparked a debate about the nature of human consciousness and its relationship to machine computation. Penrose’s work raises important questions about the limits of machine intelligence and the potential for human intuition to solve problems that are beyond the reach of machines.



The application of the Turing Test so far has been used solely to assess the intelligence of a machine that mimics humans by way of giving deceptively similar answers, however, in today’s time, we have AI systems combined with robotics that have much more functionality and assessing intelligence solely based on express communication can not suffice.

The underlying premise of the Turing Test is to mimic a human’s response, the response however, does not jurisprudentially require to be in an expressly communicated form. What I propose here is that the Turing Test can be used to assess a machine through its decisions or actions. To take as an example, Tesla’s self-driving cars, which can reroute as per the real-time obstructions in the path. The action of rerouting or avoiding an obstruction naturally comes under the purview of the machine’s functioning but it can also be viewed as a form of intelligence by mimicking a human’s action of avoiding the obstruction.

In both ex vivo and in vivo bowel anastomosis, Axel Kriger’s Smart Tissue Autonomous Robot (STAR) matched and even outperformed human surgeons. It is an autonomous robot that only needs human consent for the plan but completes the procedure on its own. When used on phantom bowels, STAR performed better than human surgeons. Furthermore, STAR’s gut rebuilding improved the flow of viscous fluid, making it more laminar and smoother. After the surgeon has physically exposed the tissue margins, STAR develops a suture insertion plan based on the tissue’s thickness and structure. STAR asks the surgeon for the surgical plan necessary if the tissue deforms. This method is continued until the robot has completed the whole operation.[10] STAR performs functions that essentially require high levels of intelligence and has the ability to communicate with the surgeon for approval as well, the performance of this task that requires high precision and years of experience exhibits signs of human-like intelligence even if it might not deceive a human in terms of imitation in words or answers, it can very well mimic what a human action in such a situation. The author believes that restricting the application of Turing Test to mere mimicking of communication is counter-productive in assessing the current level of intelligence for AI systems in today’s era.







The Turing Test has been a groundbreaking concept in evaluating machine intelligence. However, as artificial intelligence continues to evolve and integrate into various domains beyond just communication and language, the limitations of the Turing Test become apparent. The test’s primary focus on assessing a machine’s ability to imitate human responses through conversation fails to capture the full extent of intelligence exhibited by modern AI systems, especially those combined with robotics and automation.

As exemplified by Tesla’s self-driving cars and the autonomous surgical robot STAR, these advanced AI systems demonstrate intelligence through their decision-making processes and precise execution of complex tasks, rather than solely through language-based interactions. Tesla’s vehicles intelligently reroute their paths based on real-time obstructions, mimicking human decision-making in navigating obstacles. STAR, on the other hand, performs intricate surgical procedures with a level of accuracy and precision that matches or even surpasses human surgeons, exhibiting a form of intelligence deeply rooted in action and execution. Restricting the evaluation of machine intelligence to the confines of the Turing Test’s conversational framework would be a disservice to the remarkable advancements in AI technology. It is imperative to expand the criteria for assessing machine intelligence to encompass not only language but also decision-making, task execution, and the ability to navigate complex real-world scenarios. By broadening the scope of the Turing Test to include these multifaceted dimensions of intelligence, we can better capture the true capabilities of modern AI systems and pave the way for more comprehensive and meaningful evaluations. As AI continues to permeate various aspects of our lives, adopting a more holistic approach to assessing machine intelligence will be crucial in understanding the profound implications and potential of these technologies.


[1] GOOGLE HEALTH, https://health.google/health-research/ (April 28, 2024).

[2] GEEKSFORGEEKS, https://www.geeksforgeeks.org/turing-test-artificial-intelligence/ (April 28, 2024).

[3] GRAHAM OPPY, DAVID DOWE, The Turing Test, Stanford Encyclopedia of Philosophy (April 29, 2024, 12:22 AM), https://plato.stanford.edu/entries/turing-test/#ArgExtSenPer.

[4] Id at 3.

[5] Karleigh Moore, Agnishom Chattopadhyay, Halting Problem, BRILLIANT (April 29, 2024 12:28 PM) https://brilliant.org/wiki/halting-problem/.

[6]MEDIUM, https://medium.com/@martalokhova/why-would-you-care-about-the-halting-problem-593cc27c943d (April 29, 2024).

[7] Jason Megill, The Lucas-Penrose Argument about Gödel’s Theorem, INTERNET ENCYCLOPEDIA OF PHILOSOPHY (April 29, 2024, 12:32 AM) https://iep.utm.edu/lp-argue/.

[8] Brian Tomasik, Replies to the Lucas-Penrose Argument, REDUCE SUFFERING  (April 29, 2024, 12:34  AM) https://reducing-suffering.org/replies-lucas-penrose-argument/.

[9]Id at 7.

[10]H. Saedi, J. D. Opfermann, Autonomous robotic laparoscopic surgery for intestinal anastomosis, SCIENCE (April 29, 2024, 12:38 AM) https://www.science.org/doi/full/10.1126/scirobotics.abj2908.

Leave a Reply

Your email address will not be published. Required fields are marked *