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# The Conscious-Paper Paradox

Steffen Grünewälder

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# The Conscious-Paper Paradox

Steffen Grünewälder

Department of Mathematics

University of York

**Abstract.** In a similar vein to Searle's Chinese Room thought experiment, we introduce a paradox which concerns the implementation of consciousness on a Turing machine. In distinction to Searle's argument, our thought experiment centers on 'isomorphisms' between a computer's memory, a pen-and-paper copy of said memory, and an ink spot. Our line of reasoning provides a type of rejection test for consciousness which can be used to guide the search for conscious systems.

In the 20th century widely celebrated thought experiments concerning consciousness of computers have been devised. With the rise of Large Language Models, these once purely theoretical thought experiments have become the center of heated arguments about concrete AI systems. The most well-known of these thought experiments is *Turing's test* for intelligent behavior of machines [11]: given solely the transcripts of a conversation between a machine and a person, an evaluator has to identify who the machine is. The machine passes Turing's test if the evaluator cannot reliably distinguish between the two. It will come as little surprise to anyone who has interacted with a Large Language Model that these models pass various forms of the Turing test and that it is becoming increasingly harder to distinguish between a human and a machine in a conversation (e.g. [6]). Turing designed his test with the intention to identify machines that demonstrate intelligent behavior. While passing the Turing test can be seen as a necessary requirement for a machine to possess a certain type of intelligence, it is less clear that passing the Turing test is sufficient to establish the intelligence of a machine. Using another famous thought experiment, *the Chinese room argument*, Searle argues persuasively against the sufficiency of the Turing test for deciding if a machine is intelligent or conscious [10]. The thought experiment is the following: a person who does not understand Chinese is isolated in a room and is given a book that contains detailed instructions to draft responses to any Chinese text that might be handed to that person. A second person, who understands Chinese, then writes a note, passes this note to the room, and receives a response from the person in the isolated room without ever directly interacting with that person. The person in the isolated room writes the response by meticulously following the instructions in the book without understanding the original note or the response they are writing. In this thought experiment, it is impossible for the person outside the room to know if the person inside the room understands Chinese or not. In the context of the Turing test, Searle's argument states that it is possible for a machine to imitate intelligent behavior without being intelligent or conscious.

Searle attempted to extend his Chinese room argument beyond the Turing test to derive a general statement about the insufficiency of computers for the understanding of language. But here the argument is less persuasive. At the heart of this extended reasoning lies the idea that computers are purely *syntactical* while humans are *semantical* in their understanding [3]. Searle proceeds to argue that, therefore, only biological processes are sufficient for a semantical understanding. The first problem with his reasoning is that there is no forceful argument for why computers cannot have semantical understanding; the Chinese room argument, from which this line of reasoning originated, simply states that we cannot judge understanding based solely on inter-
actions with a system. The second problem is that it is unclear which insufficiencies of computers biological processes are meant to overcome and how they allow for semantical understanding.

We develop a different argument to address the question of understanding of a system and of consciousness. In contrast to Searle's Chinese room argument, we focus on the state of a computer (stored in its memory) and sequences of such states. One challenge in the study of consciousness is to define what exactly consciousness is. We will not attempt to provide such a definition and we will simply assume that the reader themself is conscious and therefore does not object to the *assumption that conscious systems exist*. *A second assumption* we will need is that there are different ‘states’ of consciousness, for instance, spending a day at the beach is not quite the same experience as visiting the dentist.

The central idea underlying our argument has its roots in topology and algebra: in topology, homeomorphic transformations are used to identify geometric objects with equivalent topological properties; roughly speaking, from a topological point of view, it is not possible to distinguish between a tea cup and a doughnut because there is a homeomorphism that turns a tea cup into a doughnut. In algebra, the related concept of an isomorphism is used to identify indistinguishable objects. We will argue that the state of a computer memory is isomorphic (by a slight abuse of terminology) to random ink spots on paper. This then implies, among other things, that a computer memory is insufficient to represent multiple states of consciousness.

**Related Literature.** The literature on the subject of consciousness is naturally vast and spans many disciplines ranging from Philosophy to Neuroscience and Artificial Intelligence (AI). For our work, the links to AI are particularly important. A recent paper [2] provides a substantial review of this area. The authors focus, in particular, on what they call ‘indicator properties’ of conscious systems. These indicator properties are extracted from prominent theories of consciousness and are used by the authors to evaluate current AI systems – roughly speaking, the more of these indicator properties are present in an AI system the more likely it is to exhibit consciousness. The key theories that are analyzed are the Recurrent Processing Theory [7], Global Workspace Theory [1; 4] and Higher-Order Thought Theories [9]. There are some similarities to our approach. In particular, the Higher-Order Thought Theories speculate about a ‘metacognitive monitoring mechanism’, which is fundamental for conscious systems. Through a different line of reasoning, we arrive at the conclusion that a ‘self-interpreting’ property is required for conscious systems. A fundamental difference between these approaches and our approach is the assumption of ‘computational functionalism’ that is used in [2]. This assumption states that consciousness is a purely computational effect which is orthogonal to our line of reasoning.

**Isomorphy between memory states and ink spots.** Consider an implementation of consciousness on a computer – or, more formally, on a Turing machine. For simplicity, let us assume that the machine is in a state that represents the consciousness experiencing joy. Now imagine that we write this state of the machine on paper. By doing so, do we create a consciousness that experiences joy? Can sheets of paper filled with zeros and ones feel joy? What if, instead of zeros and ones, we use pluses and minuses or any other encoding – will we have created the same consciousness? In fact, does the paper ‘know’ the difference between zeros and ones and random scribbles or a spot of ink? Figure 1 is a visualization of this question.

We might also wonder how the paper can ‘know’ that zeros and ones that are adjacent on the sheet are in fact adjacent? To have any hope of creating a consciousness by writing on a paper there would need to be, at the very least, some measurable physical interaction between the dif-
<figure><img src="image_3.png" /><figcaption>Figure 1. A transformation of binary code in computer memory to scribbles on paper. Can a paper distinguish between binary code and arbitrary scribbles? Is a paper able to do handwriting recognition and understand that substitution of +/− for 1/0 does not change the state?</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2. The figure visualizes two different states of the consciousness, represented in computer memory as an array of zeros and ones. If the paper cannot interpret the writing then we can transform both states into the same ink spot. In other words, to the paper, they will be the same state of consciousness.</figcaption></figure>

ferent symbols on the paper. For example, if we take scissors and cut out each individual letter or number that is written on the piece of paper, place the pieces apart from each other, would we break the system and destroy the consciousness?

**Insufficiency of memory to represent different states of consciousness.** There is obviously no communication between the symbols on the paper and the idea that a sheet of paper could feel is absurd. Arguing against a system being conscious solely on the grounds of absurdity does not appear to be a very robust approach. This is especially true given the subtlety of the question of consciousness. It seems therefore useful to pivot slightly and to place another idea at the center of the argument. To that end, let us go back to the assumption that the consciousness has at least two different states and let us assume that there is no meaningful physical interaction between the ink on the paper and the paper cannot 'distinguish' between what is written on it. Copying the two different states from the computer memory to a sheet of paper, we can transform each of the states into the same ink spot without the paper 'noticing' the difference. But then both states are, in fact, the same state and we can represent at most one state of consciousness using pen-and-paper. Figure 2 visualizes this idea.

**Sequence of states.** It seems clear that there is no inherent difference between ink spots and meaningful sequences of symbols. The difference only becomes apparent when the papers are read and interpreted. Since an algorithm that modifies and updates the state of the memory is interpreting the state that it is modifying, it seems possible that the algorithm instills meaning into the various symbols and allows a consciousness to emerge. There are two natural ways in which an algorithm might help in this regard. A deterministic algorithm ties the different memory states together and provides a certain coherence across the different memory states. In fact, in the absence of outside inputs, the whole sequence of memory states can be reconstructed based on
<figure><img src="image_5.png" /><figcaption>Figure 3. The top row visualizes the output of an algorithm that generates a *coherent* sequence of memory states. Copying this sequence of states onto a sheet of paper gives us a coherent order of symbols. However, for the paper itself this is indistinguishable from an ink spot.</figcaption></figure>

knowledge of the first memory state and the algorithm. The top row in Figure 3 visualizes such a sequence of memory states where the state transitions are performed by some algorithm. We can now copy the states sequentially onto a sheet of paper thereby creating a sheet of paper with coherence across, say, different rows. But this coherence is not ‘visible’ to the paper and the carefully generated sequence of symbols is not any different from an ink spot on the paper.

The second way in which an algorithm might allow a consciousness to emerge is through the state transitions themselves. Here, we like to propose a variant of Searle’s Chinese room thought experiment that is inspired by our isomorphism approach. Consider a sheet of paper with the different memory states written in different rows as on the right side of Figure 3. Now, consider a photocopier that copies one row after another onto a new sheet of paper (Fig. 4). Here, the photocopier performs the state transitions, but, similarly to a sheet of paper, is completely oblivious to what it copies. In fact, neither the photocopier nor the paper would observe a difference between the states and a sequence of ink spots. Finally, there remains the question of whether the flipping of a memory entry from zero to one can be done in a fundamentally different way from our photocopier example to allow for a consciousness to emerge. The consciousness would then arise not from the memory content or the state transitions themselves, but from an additional process.

<figure><img src="image_6.png" /><figcaption>Figure 4. The figure shows a variant of Searle's Chinese room thought experiment in which the state transitions are performed by a photocopier. To the photocopier and the paper it copies to, there is not difference between a sequence of memory states and a sequence of ink spots.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5. The top row shows the processing of sensor inputs by a computer. The bottom row shows the same process but implemented by a photocopier and sheets of paper. To the photocopier and the paper there is no difference if the first state was read out from a sensor or is just a random scribble on a paper, or an ink spot for that matter.</figcaption></figure>

**Sensor input.** As with the state of the memory itself, we have the problem that neither the symbols nor their ordering have inherent meaning or structure. It seems useful to consider how, in other contexts, the memory states and transitions between states become meaningful. Let us consider an algorithm that adjusts the temperature in a room based on the readings of a thermometer. We can run the algorithm that computes the control signals for the thermostat using a pen and paper. The scribbles now become meaningful as the signals that are computed by the algorithm are transferred to the heating system which ‘interprets’ them and translates them into physical observations. We can imagine that we trace back from these physical observations to the different symbols on the paper by running the algorithm in reverse, thereby giving meaning to the various scribbles on the paper. The problem with this approach is that the physical observations help us, as outside observers, interpret what is written in the memory or on the paper, but, from the point of view of the paper itself, there is no difference between random scribbles and sensor readings that are propagated by the algorithm. In fact, we can reproduce the entire sequence of state transitions and sensor inputs by using a photocopier and sheets of paper (see Figure 5). In this setup, the sensor readings have no special meaning and are indistinguishable from an ink spot. We can also imagine the system *sending signals* to the heating system, which in turn influences future sensor inputs. The coherence introduced in this way is no different from that produced by state transitions, and both the paper and the photocopier will be indifferent to them.

**Probabilistic programs.** The pen-and-paper approach does not allow us to incorporate stochasticity directly. However, we can use the same approach as with sensor inputs: provide a pre-generated list of random numbers and copy them onto the sheet of paper. The paper itself will be indifferent to how these numbers were generated.

**Non-isomorphic self-interpreting systems.** Since we know that conscious systems exist, the central question is: what are we missing? Are there computational models beyond Turing machines which allow us to implement a consciousness, or is there a loophole in our argument that
can be exploited to create a consciousness? To circumvent the complications with the pen-and-paper approach we need, at the very least, a system that is *not isomorphic to written text on paper*. In other words, a system that cannot be represented and executed using pen-and-paper. Furthermore, the system has to have some form of *self-interpreting* property.

The former question is potentially easier to address: one of the main problems with the pen-and-paper approach is that different scribbles are disjoint objects that are not inherently connected. Can we replace this with a system that does not treat these as disjoint objects? To take some inspiration (in the most naive way possible) from neuroscience, we could postulate that *electromagnetic waves* might be a key object for creating consciousness and not just a random byproduct of neural activity. While it is not obvious how such waves can create a conscious interpreter, they at least address the problem of disjoint scribbles and ink patterns by binding them together into a single object – we will use electromagnetic waves as a running example to discuss further ideas below as it is useful to have a concrete approach in mind.

Another potential way to circumvent the paradox is to exploit physical properties like *quantum entanglement*. For example, Deutsch [5] argues that quantum Turing machines are fundamentally different from standard Turing machines and probabilistic Turing machines. His argument hinges on Bell's inequality, which sets limits on correlations that can be observed in classical physical systems. Quantum systems can surpass this limit thanks to quantum entanglement and one can wonder if this provides a way to develop a system that is fundamentally different from a Turing machine and our pen-and-paper approach – the pen-and-paper approach allows us to run a Turing machine, if we assume that we have infinite amount of paper and we use a pencil and eraser to track the changes on the memory tape.

**Approximations and simulations.** A natural question that arises is whether we can approximate or simulate a system capable of implementing a consciousness. For example, we could aim for a *perfect* simulation of the physical world within our computer. Simulating electromagnetic waves should be easy enough. But if this is indeed the key to building a consciousness then we will ultimately face an insurmountable problem with this approach: the result would still be just another program with which we try to create a consciousness and this program could again be run by using pen-and-paper.

It is surprising that such a barrier should exist. For example, differential equations that model physical systems can be approximated arbitrarily well with standard hardware. If we consider again the electromagnetic wave then there are partial differential equations (PDEs) that we can solve to understand the behavior of such a wave to a very high precision. Similarly, quantum Turing machines differ from probabilistic Turing machines but they can be simulated by probabilistic Turing machines. Such a simulation is inefficient: the computational complexity of a probabilistic Turing machine can be many orders of magnitude greater than that of a quantum Turing machine. However, in terms of computability – that is, the set of functions each machine can compute – there is no fundamental difference between them. In fact, the *Church-Turing thesis* postulates that *universal Turing machines* are sufficient as computational devices and that there is no benefit in using quantum Turing machines or other computational devices – provided that we disregard questions of computational efficiency.

There is the possibility that the Church-Turing thesis is false, for example, because there are physical processes that cannot be simulated by a universal Turing machine. However, if we assume that the Church-Turing thesis is correct, then we will be able to simulate a conscious system
using pen-and-paper. The question is then if there is a significant difference between the sim-
ulation and the real system. From a purely operational point of view, a solution of a PDE that
describes with high precision the behavior of a physical system might be sufficient. For exam-
ple, we can reproduce the readings of a thermometer in a heated room by solving a PDE. To any
system that only has access to the physical world through the thermometer, the simulated mea-
surements would be indistinguishable from genuine physical measurements.

Let us go back to our running example and assume that an electromagnetic wave is used to
tie together our various visual, acoustic and other sensory signals. Why would evolution favor the
use of an electromagnetic wave and a conscious system to navigate the world, and not a simula-
tion of it? One reason might be the efficiency of computation. It might just be much faster and
less resource demanding to implement an electromagnetic wave than to track a solution of a com-
plex PDE. In the context of quantum computing, we have similar effects in that certain quantum
algorithms like Shor's algorithm or Grover's algorithm are believed to be by magnitudes more
efficient than their classical Turing machine counterparts. In fact, Richard Feynman suggested in
the 80s that there are essential difficulties in simulating quantum mechanical systems efficiently
on classical Turing machines [8:p.7].

**Benefits and the big reveal.** While we focused on the question of *how* a conscious system can be created, there is the equally curious question of *why* conscious systems evolved at all. If we stay with our running example of an electromagnetic wave and assume that this is the key to a conscious system then we might wonder why an evolutionary process would go to such lengths to produce an information-processing system. A partial answer might be that there is a natural path to such systems: it is conceivable that in simpler organisms, neurons were just primitive information-processing units that coincidentally produced electromagnetic waves. These waves might initially not have served any purpose, but through fine-tuning, the neurons might have evolved to create waves with particular characteristics and such waves might now form a central piece of the information processing system. But what could these electromagnetic waves provide beyond what well-established techniques like deep neural networks cannot? While we have no answer to this question, it seems clear that there must be a significant advantage to such systems. Furthermore, finding such an advantage in an approach that addresses the interpretability problem could be a telltale sign that we are closing in on creating a conscious system.

**References.**

[1] Bernard J. Baars. *A Cognitive Theory of Consciousness*. Cambridge University Press, 1988.

[2] Patrick Butlin, Robert Long, Eric Elmoznino, Yoshua Bengio, Jonathan Birch, Axel Con-
stant, George Deane, Stephen M. Fleming, Chris Frith, Xu Ji, Ryota Kanai, Colin Klein,
Grace Lindsay, Matthias Michel, Liad Mudrik, Megan A. K. Peters, Eric Schwitzgebel,
Jonathan Simon, and Rufin VanRullen. Consciousness in artificial intelligence: Insights
from the science of consciousness. arXiv, 2023.

[3] David Cole. The Chinese Room Argument. In Edward N. Zalta and Uri Nodelman, editors, *The Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University, Winter 2024 edition, 2024.

[4] Stanislas Dehaene and Lionel Naccache. Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework. *Cognition*, 79(1-2):1–37, 2001.
[5] David Deutsch. Quantum theory, the Church-Turing principle and the universal quantum computer. *Proc. R. Soc. Lond. A*, 1985.

[6] Cameron R. Jones and Benjamin K. Bergen. Large Language Models pass the Turing test. *ArXiv*, 2025.

[7] Victor A. F. Lamme. Towards a true neural stance on consciousness. *Trends in Cognitive Sciences*, 10(11):494–501, 2006.

[8] Michael A. Nielsen and Isaac L. Chuang. *Quantum Computation and Quantum Information: 10th Anniversary Edition*. Cambridge University Press, 2010.

[9] David M. Rosenthal. *Consciousness and Mind*. Oxford University Press, 2005.

[10] John Searle. Minds, brains and programs. *Behavioral and Brain Sciences*, 1980.

[11] Alan M. Turing. I.— Computing machinery and intelligence. *Mind*, 1950.