From afar a group of Stone Age hunters are observing your behaviour. You seem very familiar to them, although a bit quirky. They notice you prioritize objectives such as staying warm and having access to water. You also have harnessed a fire to keep yourself warm, although they did mock you as it took you a while to get it going. However, they were puzzled by the fact that you were burning your catch of the day on the fire also. They had never considered this act before, but they could not comprehend its importance and simply regarded it as one of your quirks. They continued to look on as you slept which further convinced them that you may be one of them and a friendly creature to approach.

Suddenly, a gust of wind whipped through the fire, sending sparks crackling high into the air. The wind caught, what seemed to the hunters, a thin piece of bark. They cast their eyes on the bark and noticed the odd shapes. The individual symbols seemed chaotic, but they were neatly ordered on the bark. They could not understand what had been transcribed on the bark. This discovery only morphed their confusion into fear. Was this creature one of us? Does it experience pain, thirst, or hunger? Can we trust it?

I write this story to draw an analogy between the endeavour to develop artificial intelligence systems that are smarter than humans. Artificial general intelligence is a system with the capacity to excel at many tasks humans can do and outperform humans on many other tasks. Such a system will inevitably have many consequences, and this is the main focus of AI safety research. The human in the story corresponds to an artificial general intelligence, and we humans are represented by the Stone Age hunters. The fact that the human has many goals aligned with the hunters, such as staying warm and having a water supply, is equivalent to the instrumental goal hypothesis which states that generally intelligent systems will have to pursue a set of instrumental goals, such as collecting resources, to achieve its objectives. The cooking of the rabbit is analogous to an AI system capitalising on current technologies in a way that we cannot imagine due to our limited intellect. Such a system will operate on a different conceptual level to us and produce outputs that we cannot comprehend, represented by the jotted notes on the piece of bark. The fear then experienced by the hunters is akin to the fear of AI research that such a powerful system may have further goals that are not aligned with human goals, leading to an existential threat to humanity.

AI safety research begins to ask questions as to whether we can trust, or ensure that such an intelligent system with have goals compatible with ours. If we let it operate in society, would it work with us or suppress us to achieve its goals? Developing solutions to this problem is the work of alignment research, and is important so that we can utilise the many benefits of this technology without having concerns about its potential harms.

Let us suppose that there is a single notion of quantifiable intelligence for any individual and it relates to the structure of their brain. These assumptions may be false, but we will discuss this later. As our ability to prove the brain of a living individual is limited, we are left to perform empirical investigations to obtain external signals that are representative of the internal structure. For example, IQ tests are ubiquitously used as a proxy for intelligence. These tests correlate performance on a range of mental tasks to an individual's intelligence level. For the most part, they match our internal sense of intelligence, however, it is still susceptible to spurious correlations and adversarial manipulation. Meaning it is missing some fundamental aspect of the notion. The IQ test was developed by conducting investigations to identify correlations with intelligence and then exploiting these correlations in the test's design.

It is important to note at this stage that an underlying assumption being made here is that intelligence is an inherently human attribute. Who is to say that human intelligence is the only manifestation of intelligence in the fundamental sense? How are we to know that other systems, organic and non-organic, are not exhibiting some other form of intelligence? Currently, we are developing artificial systems that mimic human intelligence, so it is reasonable to assume that the intelligence they behold has much overlap with our own. However, it is still not clear that this is the only form of intelligence that may arise. For example, one often says that the weather has a mind of its own. That's interesting, does this mind operate the same intelligence as our own?

By definition, we are not capable of comprehending these alternative forms of intelligence. Although it is reasonable to assume that we could spot when it is present. For example, we humans live in a three-dimensional world. We are aware that a fourth dimension could exist even if we cannot comprehend it. Who is to say that some alien organism is not restricted to three dimensions, but has access to this fourth dimension? Then from our perspective, this intelligent alien may seem docile, however, in the extra fourth dimension it may be executing actions beyond our imagination. As we train more powerful artificial systems, we begin to traverse a treacherous border of capabilities. It may appear on the surface that these systems still lack high levels of intelligence, however, it may just be operating in a domain beyond our comprehension. Exhibiting a form of intelligence we are blind to, which may have the capacity to exploit and inflict power onto humans. There is much debate as to whether such an intelligence agent would be power-seeking and pose a risk to humanity, spanning topics including human values, morality, and consciousness.

It has been popular to assess the capabilities of these artificial systems by letting them complete standardised tests and comparing their results to human performance. Thus we get the IQ, and other standard intelligence measures, of many of the cutting-edge models and use them as a metric to compare them. Recall, that these tests were developed by performing tests on humans, who is to say that we can extrapolate these results as a proxy for artificial intelligence? Indeed, it has been the case that these models are performing well on these tests, but we are still yet to say that any of these models are generally intelligent. On the one hand, they may possess a narrow intelligence that allows them to excel at a small set of tasks that correlate with these standardised tests. On the other hand, they lack the general intelligence required to complete tasks such as long-term planning.

Consequently, we should be asking questions about the way we are testing intelligence currently in our society, as it seems that high performance on these tests can be achieved without possessing all the qualities we would associate with an intelligent individual. Therefore, when a model achieves a score of 140 on an IQ test, what comparison can we make to a human individual with an IQ of 140? It is clear the model has some form of reasoning, or memorisation, capability, but it is not clear how this compares with human intelligence. We are missing a fundamental component of what it means to be intelligent.

In this vein, we question the assumption that intelligence is solely a function of the brain. Consider a modern human placed within a hunter-gatherer society. Would it be reasonable to assume that this human would be the smartest individual in this society? On the surface, we would probably say yes. If we put aside their increased amount of acquired knowledge, by being brought up in a substantially more developed world, we could assume that their mental skills such as problem-solving and reasoning are more advanced. However, we can attribute this to the structured curriculum the individual went through. It bears no resemblance to the different architectural properties of the brain, and we know this because, over the relatively short period between these societies, evolution has not had enough time to alter the brain structure. It is not the case therefore that intelligence is purely a function of the brain. It is also not the case that intelligence is an attribute of a single human. Intelligence may be an emergent phenomenon arising from a large collection of people. A large cohort can organise themselves to resonate with their intelligence and enhance the collective intellectual capacity that permeates the individual level.

Consider language, a construct we have developed to communicate our thoughts and ideas. My hypothesis is that language and thought have evolved simultaneously in a positive feedback loop. Humans have mechanisms for generating sound which over time we learn to use to encode patterns and gain the attention of other humans. From an evolutionary perspective, this was beneficial as it could be used strategically to reduce one's likelihood of death. With continued experience in operating our vocal mechanisms, we learned how to encode more sophisticated information, which then motivated the ability to construct more sophisticated sounds, and so on. Eventually, we developed a construct that could encode information that allowed one to organise their thoughts and increase their capacity to think abstractly. It is because of this that we have become a more intelligent species, despite there being no advances in our brain architecture. This is by no means a claim that all our intelligence is attributed to the environment, as other animal species have not developed sophisticated forms of longs. However, our intelligence is due in large part to the society we are embedded within, and perhaps the general intelligence we are trying to artificially create is an emergent phenomenon of this.

Still using the notion of evolution, we can claim that humans possess the lower form of intelligence possible for establishing a civilisation. Major advances in technology, nutrition, philosophy, well-being, and the natural sciences can all be attributed to work carried out in the recent past. The rate of societal change and improvement has been growing exponentially. Humanity has developed into an established civilisation within a preposterously minuscule amount of time compared to the grand scheme of life on Earth. In comparison, breakthroughs on the front of evolution have been essentially non-existent. Why was it then that complex societies were formed earlier? Well, perhaps it is because they only formed once humans reached the minimum intellectual capacity required to operate such societies. Hence, if we assume that intelligence levels exist on a scale, from this observation we may deduce that our point on the scale is far below the end of the scale, supposing that such an end exists. We may also conclude that super-intelligent artificial intelligence is possible. However, it is not clear that organisms that exist above our positive on the scale reap all the benefits we may think come with that greater intellectual prowess. Perhaps, the human intelligence level sits in a Goldilocks zone where we have sufficient capacity to develop societies but are not advanced enough to induce our failure modes.

The Freedom Fable

Uriel Abulof, Lisa Cameron, Ankhi Mukherjee

This debate was centred around the topic of freedom. What does it mean to have freedom? Is freedom necessarily a good thing? How can we achieve personal and collective freedom?

Uriel Abulof repeatedly emphasised the importance of distinguishing freedom from liberty. Freedom is having the ability to make a choice, whereas liberty is not being controlled by an external power. Often we conflate freedom and liberty, however, carefully distinguishing these is important.

With Lisa's experience as a politician and visiting other cultures, she notes the importance of incorporating the cultural context into freedom. Freedom can manifest in different ways for different cultures. Cultures view different things as contributing to one's freedom. When people in different cultures demand freedom, they are probably advocating for different opportunities. Therefore, when assessing the extent to which one has freedom, it is important to view freedom in the context of the culture.

Ankhi Mukherjee identified the use of freedom as a literary device for persuasion and manipulation. The promise of freedom can rally large swathes of people to act towards a unified cause. Often these promises are misguided and blown to unreal proportions. It seems to be in human nature to strive toward freedom, whatever that may mean.

This raised the question as to whether total freedom is necessarily a good thing. There is an argument to be made that there must be some restrictions to facilitate personal progression. For instance, without the constraint of having to pay for your food, you may not be motivated to work to earn money and pay for the food. Therefore, economic markets can be seen as an example where applying some constraints encourages progress. Without these markets, one wouldn't be able to work toward developing a living. There would be no way for an individual to progress. This can be perhaps viewed as a restriction in liberty leading to an increase in freedom.

Returning to liberty, we can amplify this idea by taking it to its extreme. If everyone were required to have the same liberty, then there would be no progress as progress would require some organising structure that will implicitly restrict some liberty. On the other hand, if liberty were centralised to an individual, then clearly the rest of society would be suffering.

Nothing and Everything

Graham Priest

In this talk, Graham Priest confronts nothing. He introduces the paradox surrounding this concept and indicates its fundamental nature.

The nothing paradox is the notion that to speak of nothing is to speak of something and so nothing must be something. One can make some progress in resolving this paradox by distinguishing the different uses of the word nothing. As a quantifier, there is no paradox. For example, nothing can travel faster than the speed of light. In this sense, nothing refers to the absence of things, in particular, it is describing the notion of nothingness. However, when used as a noun phrase, nothing is self-referential and thus the paradox emerges. Therefore, Priest goes on to start defining what nothing is.

Nothing is the fusion of no things. That is nothing is the antithesis of everything. This construction is rather abstract, and indeed in a subsequent part of the talk Priest notes this. This abstractness is not relevant from the perspective of logic, which is the one taken by Priest, however, from a practical perspective it causes issues. For example, in this definition, there is no notion of compatibility in the fusion between things. In the real world the fusion of non-sensical things is still likely to be referred to as nothing, whereas under this definition they constitute a thing and thus are not nothing. However, from the practical perspective, the fusion of all things seems to be a well-defined object, it is everything. Therefore, it seems that for the fusion of things to be sensical, the components must be connected through some meaning. Consequently, from the practical perspective, we must endow things with meaning to be able to coherently discuss the fusion of things. Of course, these meanings are non-existent in the language of logic, indeed logic is precisely designed to ignore concepts such as meaning. A notion of nothingness that is more suited to the practical perspective can still be constructed on logical foundations, but semantics for the logic must also be considered.

Going along with Priest's logical construction of nothing, it follows that objects are dependent on being distinct from nothing. Priest illustrates this as a plane, where the plane represents nothing and protrusions from the plane represent things. This raises the question as to whether nothing is a relative concept. By shifting the ground plane can we neutralise the protrusion of certain things such that they are no longer distinct from the notion of nothing? On a logical level, this may not be possible, but from a practical perspective, this may indeed be the case. Raising the level from which to compare objects to determine things we could potentially prevent the fusion of non-sensical objects from being considered things.

Priest's summary of the notion of nothing is that nothing is a paradoxical object. It is both something and no thing. It is the ground of all things. Reality is composed of beings distinct from this ground, so realiti’s foundation is inherently paradoxical.

The Consciousness Test

Yoshua Bengio, Sabine Hossenfelder, Nick Lane, Hilary Lawson

In this debate, the panellists were discussing the utility of Turing's test to determine whether a system is conscious.

A partial consensus from this debate was that to discern consciousness we need to look at what happens within the system and not just on its inputs and outputs. Therefore, something like the Turin test would be an inadequate test for consciousness.

Consequently, much of the debate was on trying to understand what we mean by consciousness such that we know what to look for within the internals of these systems. Although the debate mainly considered ideas in an abstract setting, often it was contextualised around machine learning models, and in particular neural networks, since this technology is currently the most viable candidate for a form of alternative consciousness. It was noted however that neural networks may not be the right level at which to consider alternative forms of consciousness. Indeed Hilary Lawson repeatedly emphasised the observation that we only consider questions of consciousness for some neural networks, say large language models. More specialised neural networks are not even considered as exhibiting any form of consciousness. There is still a lot we do not know about our form of consciousness, and so arriving at definitive answers to these questions is probably still out of reach.

Lawson made the case that an essential component of consciousness is self-awareness. Lawson notes that experience is dependent on observing something, and consciousness is the observer observing themselves. Assuming this idea, tests for consciousness seem feasible as one can formulate this with the mathematical language of self-similarity, which helps explain mathematical objects such as fractals.

Throughout the debate, a constant distinction between intelligence and consciousness was made. One must not conflate these ideas as they are distinct.

Nick Lane's view was that any test for consciousness must focus on the feelings of the system. It is the feelings that help drive one's experiences and thus consciousness. For example, many argue that under anaesthesia humans are no longer conscious, and it is the role of anaesthesia to make sure that we feel nothing during painful procedures.

Yoshua Bengio argued that artificial systems will be able to exhibit consciousness. For him it is not the matter, say our brains, that drives consciousness, but rather the connection between the matter. It is the computation and information transfer that drive our conscious experience. That is not to say that computation alone provides consciousness, otherwise we would be attributing consciousness to every neural network. Instead, it is the content of the information transferred within these computations that is important. It is probably the case that the information represented in the computations of large language models is sufficient for inducing these experiences.

The contradiction between Bengio and Lawson boiled down to the nature of reality. Lawson disregards the materialistic view of the world. He believes that there is more to the universe beyond what our equations and theories provide. These theories merely provide a model that works well to describe our experiences. Therefore, he is sceptical of the possibility of neural networks manifesting conscious experiences. Bengio on the other hand is confident that consciousness is a derivable phenomenon from this materialistic viewpoint. His arguments are agnostic to the true nature of reality and instead rely on the relationships between the constituent components.

The Language Wars

Babette Babich, Sandra Laugier, Graham Priest

Language is incredibly dynamic and personal. The meaning of words is a function of their use, the culture they are used in and the personal experience of the one speaking them. Therefore, the meaning of words and phrases evolves. It is no surprise then that conflicts have risen due to the misinterpretation of language. However, often this misinterpretation may be a disguise for a disagreement about power. Often these conflicts arise as individuals are trying to inflict their power with their words. Thus power incompatibilities arise as a misinterpretation of language. For instance, a leader may be misconstruing the meaning of territory to justify the invasion of a neighbouring piece of land. Much of this disagreement may be induced intentionally. One just disguises their power struggles as language problems to detach themselves from them.

At a societal level, the nature of language is a representation of reality. It identifies the prominent ideas of the day. It can demonstrate the mood of society.

At an individual level, language can act as a mechanism for agency. One can use language to motivate their actions, set goals, and construct plans to complete their objectives.

Due to the potential ambiguities in language, and its ability to adopt multiple meanings. There is the challenge of effectively discerning meaning when in conversation. Resolving this clarity is essential to avoiding miscommunication and conflict.

An illustrative example presented by Priest involves American and British individuals. They are discussing the components of a peace agreement after a conflict. The British are putting ideas on the table to be discussed and incorporated into the agreement, whereas the Americans are tabling ideas such that they are not discussed and left out of the agreement. This discrepancy in the meaning of the phrase tabling an idea led to endless disagreements between the parties that could have been avoided.

Quantum and the Unknowable Universe

Sabine Hossenfelder, Roger Penrose, Slavoj Zizek

There was not much I gauged from this debate apart from some interesting points.

• Roger Penrose constructed an interesting thought experiment regarding superposition. Suppose there is a planet far away, on the order of light years, hosting some complex dynamical system, such as a weather pattern, which is in a superposition. Suppose we send a probe to that planet, capture its state, and transmit the information back to Earth. At what stage does the system collapse due to being observed? When the state is captured or when we observe the data from the transmission? Penrose uses this experiment to consider the case that superpositions have lifetimes, and the collapse we observe is the death of a superposition according to this lifetime.

• Penrose also postulated that consciousness arises as the non-computability of the universe, and our malfunctioning of processing information present in the universe. On the one hand, I have some reservations with this idea as it just seems as though we a lumping consciousness into something we do not know, therefore, side-stepping the problem of trying to figure out what it is. On the other hand, I see some validity in this idea, as it is often when things go wrong that we become most aware of our surroundings. For example, many of us have become accustomed to there always being WIFI. It is only when we lose signal that we realise how dependable we are on it.

A World of Order and Chaos

Marcus Du Sautoy, Nick Lane, Marika Taylor

In this debate, the panellists considered chaos and entropy. In particular, whether the second law of thermodynamics is true. The second law of thermodynamics states that the entropy of a system increases over time. Therefore, the reason for considering the validity of this statement is the observation that humans seem to operate in a system of decreasing entropy. The panellists are quick to point out that this observed paradox is because we are observing a closed system. On a larger scale, one needs to account for the earth and the sun to realise that humans are not violating the second law of thermodynamics. However, since the second law of thermodynamics is only a side-effect of our current fundamental theories, the question remains as to whether the law is a fundamental aspect of reality. It could be the case that our current theories are not entirely accurate and therefore the law may not hold. Empirically the law has been repeatedly validated and so scientists are fairly confident in its validity. Despite this, some physicists still consider the possibility that the law does not hold. This leads to some exotic theories of reality. Showing the validity of these exotic theories in explaining reality may start to put the second law into question.

When discussing the second law, it is important to understand the difference between open and closed systems since the second law only holds for closed systems. To do this one must also precisely define what sort of entropy they are considering. For example, one can have informational entropy or chemical entropy. Having decided on this one then needs to track all the sources and sinks of entropy to determine when a system is open is closed. A system is likely to be closed with regard to informational entropy but not closed with regard to chemical entropy.

After this discussion of the second law, the debate started to consider the notion of order and chaos. Chaos has been studied extensively within mathematics, culminating in a chaos theory. Indeed, Marcus Du Satouy envisions mathematics as the rigorous search for order.

From a physics standpoint, Marika Taylor notes that order and chaos are objective phenomena, they are independent from observers and can be precisely defined. In particular, order and chaos can exist at different levels. Perhaps at a certain level, we can precisely describe the events occurring, and thus we have order. However, as we scale these events it may be harder to keep track of all these events and thus we enter into chaos.

This idea of scale is also relevant to the discussion on the second law. When one starts considering things at the quantum level, it is difficult to keep track of all the sources and sinks of entropy. Therefore, at this scale, it is harder to validate the second law. It is precisely at this scale that physicists are developing theories that do not implicitly predict that this law holds. It may be the case that the second law holds at the macroscopic level, but not at the quantum level. It may be the case that due to these quantum effects, with certainty we can only say that the universe as a whole is a closed system.

Marika Taylor makes the interesting observation that gravity is the only attractive force. Where the other fundamental forces cancel each other out at scale, gravity creates localised pockets of order due to its attractive nature. Humans are situated in such a pocket, in the form of our solar system. This allows us to concentrate energy and use it to increase order, seemingly violating the second law.

The second law of thermodynamics is inherently temporal. It provides a direction to time, namely forward in time is the direction which observes entropy increasing. Marcus Du Satouy observed that mathematics is seemingly independent of time. A prime number is a prime number whenever you consider it. Therefore, could we say that mathematics violates the second law of thermodynamics since its entropy is constant throughout time?

The Journey In Search of a Destination

Babette Babic, Sandra Laugier, Frank Tallis, Jonathan Webber

The panel reached the consensus that there is no ultimate goal or meaning to life, and thus we should not be searching for one. If we are continually searching for some fundamental meaning out in the universe we will become lost since one does not exist. However, the panellists note that someone with a purpose is more resilient, and thus there is utility in finding a meaning. We should be constructing our meanings and striving to achieve those. To find our meaning we have to first find our true selves.

This process need not be explicit. Indeed it is often the case that someone who shows signs of having a purpose can not articulate it. Instead, we should always be seeking ways to do better, and our true meaning will emerge. It is important to note that happiness is not the correct metric by which to measure this progress, suffering is an inherent part of reality and is crucial for illuminating our identity. Moreover, one must note that goals and meaning can be temporal, therefore the search for purpose is continually evolving.

Babette Babic poignantly captured the idea that there is no ultimate meaning with the following articulation. Life is a gift, we have no right to it and therefore we must enjoy it as a gift. Its finiteness gives meaning and reconciles the fact we have no right to it.

Sartre vs Baldwin: The Unknown Other

Marie-Else Bragg, Joanna Kavenna, Jonathan Webber

The tension between Sartre and Baldwin lies in our ability to obtain a true understanding of the other. Sartre says that we are all alone with our subjectivity and the other is inaccessible. Baldwin on the other hand argues that to be alive is to be socially connected with the other. Specifically, Baldwin believes that ideas are more real than the people.

Both think that we are influenced by the structure of the outside world but driven from within. Therefore, we are a product of the path we have taken through society. Sartre believes that this limits our ability to fully relate to others as the traversal along the path is a deeply personal experience. Baldwin on the other hand thinks that this is evidence that we can relate to the other, as this path is embedded in society and can be accessed.

Baldwin in his works captures the unique connections we make beyond language to make this case that the other is accessible. He notes that there is an experience beyond language that exists in society that is accessible to all. Language therefore has an inherently social meaning, and therefore it is conceivable that the other can be understood. Baldwin articulated the power of language beyond its literal use with the observation that it can sometimes impose rigorous rationality to the extent that it makes us blind to certain things. By finely detailing language to describe an object we can often become blind to its true meaning and purpose. This meaning is detected in the absence of language and thus exists in a realm beyond language.

On the other hand, Sartre claims that this experience is more than just language and meaning. It is structured by the goals you are pursuing and your internal emotions. These are all intimately influenced by your background but are driven by your internal subjectivity. In particular, Sartre concluded that you can get an outline of someone else's experience but you cannot feel it without tracing the same path as them, which is an infeasible challenge.

Philosophy of the Senses

Barry C Smith

Platonic ideas have largely been refuted throughout scientific development. The idea that the world is constructed from elements in the shape of five platonic solids has long been departed from, with the modern-day equivalent being the standard model of physics. However, some platonic ideas have remained despite there being increasing evidence refuting them. One such example is the notion that there are only five senses and that these senses operate independently of one another. It turns out this is far from reality. Elucidating this was the aim of Barry Smith's talk.

Smith first observes that there are many more senses than five, it is somewhere in the thirties. These additional senses include ones which allow us to balance, feel acceleration or monitor our internal bodily processes.

Secondly, Smith argues that we should not be speaking of sense independently of one another. The senses on their own are rather limited, it is their combination that fosters rich experiences. Indeed, we often describe sensory experiences using terminology associated to the other sense. For instance, one may describe the colour bright green as sour, or one may describe a deep humming sound as dark. Smith pointed to several studies that demonstrated a manipulation of one sense by controlling another. A canonical example is demonstrating the differences in taste of something by altering the way it smells. Demonstrating our dependence on each of our senses to form our conscious experience. Moreover, this highlights how often we think we are using one sense to act, whereas, in reality, we are relying on a different sense. Our sense of taste is heavily limited, and when we are tasting food we are incorporating a lot of our sense of smell. Understanding in more detail these connections can help us better understand how we form our conscious experiences.

Smith further emphasised the subjective nature of our senses by highlighting how they are inherently related to our culture. For example, individuals in the West may describe the smell of vanilla as sweet as it is often associated with sweet dishes such as ice cream. However, in Asia, the smell of vanilla has savoury connotations as it is often used in more savoury dishes. This pushes the notion that sense are far from a platonic ideal. They are deeply related to culture and inherently dependent on each other.

When we, humans, are solving, relatively, challenging tasks, we like to think that we are reasoning through the steps that we need to complete to solve the task. It is difficult to gauge how we deduce those steps, are we replaying the steps we observed someone else make when they tackled a similar task or are we generating these steps using our intuition?

Nowadays, we are all intimately connected through social media and transport and so we are heavily influenced by a host of different individuals. Thus it is plausible that the majority of our perceived reasoning is copied from others. However, there is still so much variety and individuality in the world that it is likely that the challenges we encounter differ in nuanced ways from those of others. Therefore, it is probably still the case that we are also generating novel patterns of reasoning. Indeed, humans did not evolve reasoning capabilities from observing an alien species. Ultimately, we started without this ability and have slowly learned it, and thus at least some non-negligible amount of mental processing ought to be novel generation.

Large language models, on the other hand, are trained on data that contains countless instances of reasoning. In deployment, we observe large language models performing similar patterns of reasoning. However, there remains the question as to whether they are just contextualising previously observed patterns of reasoning, or whether they are forming their novel patterns of reasoning.

We can provide some thoughts on this question by considering the storage capacity of these large language models and the memory necessary to memorise algorithmic patterns.

Estimation

Model Capacity

Current large language models have many parameters, with upper-bound estimates on those of the OpenAI models around 1 trillion parameters. However, it is probably less than this, and it has been shown, on open-source models, that their capabilities are largely unaffected through some compression techniques that remove up to 90% of the model’s parameters. Therefore, for our estimates, we will use an upper-bound on, effective, parameter count of 200 billion. Supposing that these parameters are stored in float32, it follows that the effective storage capacities of these models is around 800 billion bytes.

Algorithmic Storage Requirements

Now we concern ourselves with estimating the amount of memory required to memorise a reasoning trajectory, where a reasoning trajectory refers to a framework that can be contextualised to solve an encountered task. For example, consider the challenge of opening a door. We all know the rough steps, or reasoning trajectory, required to complete this action, however, when we encounter a door in practice we have to contextualise these steps such that they are applicable. We will suppose that this reasoning trajectory is formulated at a conscious level of granularity, as otherwise, one could continually reduce the sequence of steps into finer and finer details.

1. Approach the door to within an arms-length distance.

2. Elongate and grab onto the handle with your hand.

3. Unlike the door handle if necessary.

4. Manipulate the handle to release the door catch.

5. Contract your arm in a way as to pivot the door around its hinges.

6. Proceed through the open doorway.

7. Manoeuvre your arm and body such to reposition the door in the closed position.

8. Release your hand from the handle in a way to re-engage the catch.

9. Lock the door handle if required.

10. Proceed away from the door.

OpenAI's new o1 model explicitly undergoes chain-of-thought reasoning to construct a similar set of steps that it can implement to complete a task prompted by the user, which in this setting acts as the context. However, understandably, we do not have access to the raw set of steps the model uses as this could be quickly reverse-engineered by a competitor or adversary.

Opening a door is a rather simple reasoning strategy. It is probably reasonable to assume then that most of our general reasoning strategies can be summarised with around 10-100 lines of pseudo-code. Humans can certainly execute longer reasoning strategies, however, these are probably an application of several general reasoning strategies, and we probably have memorised relatively few of these. Therefore, we do not consider them in this estimation.

Now supposing that each line has around 50 extended ASCII characters, it follows that, on average, a human reasoning trajectory would require around 500 to 5000 bytes of memory to store. Of course, here we are assuming that these trajectories are stored in something akin to natural language, at least in terms of memory requirements.

Consequences

Assuming that these large language models use all of their storage to memorise these reasoning trajectories, it follows that large language models can memorise around 160 million to 1.6 billion reasoning trajectories. As we have mentioned before, these reasoning trajectories need to be contextualised to be useful. Hence, it cannot be the case that all of a large language model's storage is used for memorising these trajectories, it must use some, probably a significant amount, of it to store knowledge.

Suppose that humans also store these reasoning trajectories as these lines of natural language. Then given that the human brain has a capacity of around 2.5 million billion bytes, at a minimum 1.6 billion reasoning trajectories would occupy around 0.032% of a human's memory. Along with the fact that each human does not have an identical set of reasoning trajectories to other humans, it is likely that to effectively interact with users large language models would have to memorise significantly more reasoning trajectories than 1.6 billion. There is even room to accommodate the fact that the model does not have to memorise all these trajectories as some are inherently physical.

It is perhaps more likely then that these models have a mechanism for generating reasoning patterns in a way that exhausts the vast space of human trajectories more efficiently than simply memorising ones they have observed in the training data. Of course, these methods of generating reasoning trajectories are flawed, leading to some poor behaviours. It is the work of those developing these models to find ways to ensure that the methods the model learns to generate these reasoning trajectories are as robust as those we use to generate our own.

In this spirit, I have, with the assistance of ChatGPT, created and deployed a couple of websites to tackle problems I have faced.

FindYourPhD

Throughout my journey of finding a PhD supervisor, I have scrolled endlessly through faculty pages to discover principal investigators with research interests that align with my own. Due to the specificity of my desired goals for a PhD, this was a long process. Naturally, I was thinking of ways to make this process more efficient. A relatively simple idea I had was to create a centralised location where supervisors could advertise PhD positions they were offering, and where students could quickly discover opportunities relevant to their interests. FindYourPhD is a platform where supervisors can upload their positions, and students can navigate a table of positions based on their interests. The website then offers individual supervisor pages where students can learn more about the advertised opportunities and understand how to apply.

The challenge I am having with this website is the lack of data it presents. I am experiencing a chicken and egg problem, as students will only be drawn to the platform if positions are advertised, however, supervisors will only be inclined to post their positions if there is an audience who will receive them.

I am not prepared to scrape web pages to populate this table for multiple reasons.

1. I wouldn't be able to obtain permission from supervisors to display their positions. It does not feel right to advertise positions without the consent of the supervisor.

2. The positions displayed on web pages are often not up to date. Therefore, the table of positions wouldn't be up-to-date, which is an issue I am trying to solve.

3. My areas of interest are narrow and do not reflect those of the broader student population. I wouldn't want to skew the scope of the platform.

MOAT

In any field, it can be challenging to stay up-to-date with the latest research. Even with a complete list of recently published papers, it can be difficult to decide which papers to read with the limited time availability you have. Utilising ideas of

relativism, I developed MOAT search which analyses recently published articles to identify pairs of articles with unique connections. The intention is that by exploring the papers in pairs, one can draw connections between the ideas and extract meaningful information that may not be present when reading papers individually.

MOAT is an acronym for Measure of all things, referencing the famous statement,

made by the ancient Greek philosopher Protagoras. Protagoras was a Sophist who pioneered the idea of relativism.

I am fascinated by the idea of exploring the connections between topics. In the context of machine learning, I am exploring various ways of connecting neural network robustness and generalisation. More generally, there have been numerous instances when the connections between ideas have been incredibly fruitful, such as the Langlands program in mathematics. On the other hand, one of the greatest mysteries in modern-day physics is understanding the connection between quantum mechanics and general relativity.

It would be interesting to expand the scope of MOAT and create similar pairings for news articles. There are always two sides to every story, so digesting articles that take different perspectives on the same topic can help mitigate biases.

To identify the pairs to display I am using text embedding models. In particular, I am consulting the Massive Text Embedding Benchmark to identify the appropriate models for the task. My current limitation is on the size of these models and the size of their embedding spaces. As my approach to discovering the pairs requires multiple different models, this quickly leads to memory issues on the web hosting services. Therefore, I have had to optimize and restrict the application of the embedding models, meaning the results are not as optimal as they could be. However, even with this bottleneck, my qualitative experience of the results is positive.

The transition to adopting rational thinking can be illustrated in a phenomenon known as the child's thinking issue. More specifically, we observe that when a child is born they cannot think rationally. Their understanding of the world passes through a series of phases, each providing a different perspective on the world. In the first couple of years of infancy, a child is in their sensory-motor phase. They are not aware of their surroundings and struggle to differentiate foreign objects from themselves. It is only the things they touch that are noticed and registered. From here the child enters the pre-operational phase, which is relevant until the age of around six. Here the child has a better understanding of their surroundings, with a sense of themselves and the environment. However, they still cannot do concrete operations. For instance, a child in this phase may understand zero as an object, say as the digit, however, they struggle to connect this to the concept of nothing. The concrete operation phase then lasts until the age of around eleven. During this phase, the child gains an understanding of concepts. They can grasp mental constructs. The formal operation phase then follows, where the child is able to use basic concepts to inform their actions. What is apparent in this evolution of reasoning is that the child is progressively more able to abstract and grasp mental concepts and use them to inform their actions. The transitions from primitive to rational thinking can be seen as a refinement of this process, where a more sophisticated representation of mental concepts is achieved, facilitating principally informed actions.

Interesting evidence for the transition from primitive to rational thinking can be observed in how the early literature of Western philosophy was formalised. Homer's preliminary work on Western philosophy was formalised in the form of poems. Poetry is a connector between abstract mental imagery and structured language. Through its rhythm and structure, poetry can capture meaning beyond the written words. In these primitive stages of Western philosophy, the methods of rational thinking were not developed, and so early scholars resorted to poetry to try and fruitfully express their ideas. Indeed, with the advent of rational thinking, we utilise written language in the form of logic, mathematics and critical analyses to express our ideas. However, poetry is still effective for what it lacks in rigour and specificity it makes up for in providing a mood. Providing a mood behind philosophical arguments is critical for eliciting support and generating a following for the ideas. Poetry prevails as a medium for unifying individuals under a common idea, for it taps into a human connection that exists beyond the words of language.

This is not to say that the primitive mode of thinking is not useful, or inferior to the rational mode of thinking. The primitive mode of thinking has been established as a set of cognitive abilities necessary for survival. However, to flourish we must adopt a rational mode of thinking.

In this era of pre-rational thinking, Western philosophers held beliefs. They were limited in their capacity to support their claims with sound arguments. Consequently, many philosophers in this era were labelled as mystics. A feature of this mode of thinking is the contemplation of metaphysical ideas. An ability to escape the realm of reality and question what is beyond our perception. For example, early Western philosophers contemplated the existence of entities beyond the constraints of time. Are there objects that exist eternally not because they persist through time, but because they exist outside of it? Hercaltius entertained the notion that such an eternal object exists. Despite this Heraclitus fundamentally believed that everything changes, which is reflected in his belief that everything was fire. On the other hand, Parmenides was adamant that nothing changes, and our perceptions deceive us. His argument for suggesting that nothing changes is that the existence of something in the present moment implies its existence in the immediate past, and therefore its existence in the eternal past. Therefore, the thing must have maintained a constant meaning and thus everything must remain constant. Stuart Russell pushes back on this argument in his book ‘History of Western Philosophy’. Specifically, he uses the example of words and their meanings to demonstrate how things must change over time. It may seem at first that the meaning behind words remains constant throughout time, however, their perpetual change is concealed by the fact that these changes cause no effect on the truthfulness or falsehood within the phrases they are used.

Clearly these thoughts entertained by Heraclitus and Parmenides cannot be explored through rational thinking. However, their contemplation can motivate rational investigations into our reality. It is often the case that exercises in primitive thinking develop into a field of rational thinking. Alchemy motivated the study of chemistry, metaphysics prompted the development of physics.

After this period of primitive, indulgent thinking Russell notes a decline in the vigour of philosophy. The philosophy after this period lacked an emphasis on man and its comparison to the universe.

• Scepticism focuses on how we know rather than how we can come to know.

• Socrates emphasised our ethics.

• Plato rejects the world and instead emphasises the self-created world of thought.

• Aristotle held a belief in the absolute pursuit of science.

The more metaphysical contemplations entertained by mystic philosophers would not be revived until the Renaissance. So although rational modes of thinking enhanced our capacity to pursue scientific agendas, expand our technological capacity, more effectively manage our ethics, and develop cohesive societies. It is too restrictive to entertain our thoughts that take us beyond our current reality. Progress requires a balance of rigorously pursuing technological advancements and a primitive exploration of what lies beyond our senses.

The development of philosophies is intrinsically linked to the environment, as its purpose is to solve issues besieging the residents. In particular, a philosophy’s focus is on confronting geographical, historical, national, linguistic, religious, and societal challenges. These factors intimately embed a characteristic that permeates those who adopt its teaching. Therefore, understanding the scene during which a philosophy is developed is crucial for comprehending its ideologies.

Ancient Greece was situated around the Aegean Sea and occupied a much larger region than modern Greece does today. Ancient Greece stretched across the Mediterranean to include regions such as Sicily and meandered eastward as West Asians migrated to the region. Its centrality meant it connected a diverse range of cultures. Its accessibility from land and air facilitated the exchange of ideas across borders. The culmination of this diversity enlightened inhabitants and motivated individual contemplation. Inevitably a unifying philosophy was necessary to facilitate the exploration and transmission of ideas.

In the eighth century before Christ, the unification began with the Homer era. With his epics, Homer introduced a formal language for written communication. The Greek used by Homer allowed individuals to speak and write, providing a common language for the communication of ideas. Eventually, the Greek used by Homer would evolve into Classical Greek, the language used by Plato to introduce the idea of rational thinking. This language would extend across the region, and be adopted by many including the Roman Empire. Classical Greek would later be used to write the Bible.

Christianity as a religion was not present in the early period of the Ancient Greek civilisation. Instead early Ancient Greek society preached to religions founded on mythology rather than scriptures. Just as any religion attempts to answer fundamental questions regarding the universe, Ancient Greek religions provided explanations as to where the universe and mankind came from. Its answers took the form of myths portrayed through romantic language. For instance, the Dionysus religion worships Dionysus the God of wine, who is regarded as the manifestation of the spirit. Other Ancient Greek religions diverge from many modern religions as they are polytheistic rather than monotheistic. For instance, the Olympus religion worships twelve Gods headed by Zeus.

Without the anchoring of Ancient Greek religions to a single scripture or a single God, the resulting philosophy is less restrictive. It is not required to conform to the specific portal of humanity provided by a person or book. Thus it can be dynamic and adapt. It can combine different components of the religion to form perspectives that are most applicable to certain situations. A cornerstone feature of Ancient Greek philosophy is its capacity to evolve and be applicable at different points in history.

After the era of Homer, Ancient Greek society accelerated as iron catalysed its productivity and manufacturing. The society entered into a more technologically advanced phase, which ultimately led to the division into slave owners and slaves. In particular, thinking philosophically was only available to the slave owners since it required intellectual freedom to explore one's curiosity. As a slave, one is restricted and exhausted to the point where cognition is preoccupied with survival. This potentially casts a limitation on Western philosophy, which often overlooks concepts such as suffering.

However, the main triumph of Ancient Greek philosophy was its capacity to organise large collections of individuals. Just as slave owners wanted to control, and optimise their population of slaves, groups of settlers were interested in building houses and forming city-states, or polis. Throughout Ancient Greece there were collections of poleis, and the challenge of coherently organising them was tackled by the philosophers. Eventually, Ancient Greece would be united with over two hundred poleis. With Athens being the acropolis, the central region that each polis saw was worthy of defence.

It seems as though the geographic component is the most influential in the construction of a philosophy. For it dictates the organisation of individuals, who in turn contribute to the nation, language, society and religion.

# Birth, Development and Decline

Philosophies have a birth, a development and then an eventual decline. The identification of these phases is associated with the ideological development of the philosophy. Understanding the core components of each phase elucidates the fundamental nature of the philosophy and its applications.

Greek philosophy saw its birth around the sixth century before Christ, with the construction of schools by individuals such as Heraclitus and Pythagoras. Each of these schools proposed a rather independent philosophy and there was little cohesion between their ideas and practices. Due to the infancy of the subject, the preserved work from this era is limited. There is little evidence from which to reconstruct the ideas contemplated by these schools. Moreover, at this stage philosophical language was under-development, and so it seems unlikely that many manuscripts were developed in the first place. The scholars of this time had fragmentary ideas and their explorations were not systematic. They lacked rational arguments, and due to their geographical separation, they were heavily influenced by local characteristics. Despite this, we can obtain a vague sense of what each of these schools promoted. For instance, Thales indulged in scientific activities. He travelled across the region to places such as Egypt and Babylon and eventually concluded that the source of all things was water. On the other hand, Miletus, a student of Thales, concluded that the source of all things is unbounded. Therefore, we notice that in these primitive stages, there are large disparities of thought. Thales attributes the source of all things to something physical, whereas Miletus appeals to something more abstract. Pythagoras meanwhile pioneered mathematical thinking.

The early stages of Greek philosophy started to mature during the fifth and fourth centuries before Christ, in what is known as the classical period. Prominent figures of this period include Socrates, Plato and Aristotle. This period was ignited by the sophistic movement, which saw intellectuals from the Greek polis congregate in the Athens polis. On the one hand, this congregation of intellectuals facilitated the exchange and development of ideas. On the other hand, it acted as a mechanism to recruit students and enabled scholars to pass their knowledge on to future generations through teaching. It is from here that the educational process became more active. Students could learn and form their ideas by listening to different scholars. It provided avenues for individuals to pursue their intellectual curiosity. Having the majority of philosophical enquiry conducted within a geographical constraint meant ideas could evolve rapidly. A particularly important evolution of ideas came from the sequence of teacher-student relationships involving Socrates, Plato and Aristotle. Socrates is renowned for his emphasis on the virtue of knowledge, and that individuals should know themselves. Plato established the first comprehensive schools on the subject. Then Aristotle advanced work on metaphysics and logic.

The turmoil inflicted by Alexander the Great's invasion of the East marked the end of the Classical Greek polis and initiated the Hellenistic age. During this age, notable schools of philosophy included the Epicurean, Stoicism and Scepticism. The Epicurean school was concerned with the teachings of Epicurus which included atomic philosophy theory. Moreover, it was one of the first schools that allowed children and women to participate, which broke the perception that philosophical endeavours could only be conducted by slave owners. The school of Stoicism was notable for its involvement in the philosophy of the Roman Empire. The school of Scepticism required one to think deeply about the world and be a suspect of everything it offers.

The culmination of the development of Greek philosophy was that of the logos. Logos is a rational form of thinking that is grounded on theoretical arguments. It grounds the logical methods established by the philosophers at a societal level.

It is important to note that despite much of the teachings of Greek philosophy can be traced back to its birth in the sixth century before Christ, the scholars of this time were influenced by barbarians who made the pilgrimage to Greece. For instance, barbarians from Persia and India eventually resided in the Greek polis. A lot of the early work of Greek philosophy was speared headed by scholars from Ionia in southern Italy. Consequently, we notice that the birth of Greek philosophy and its subsequent Classical period were ignited by the spatial collapse of individuals. As people aggregate, the local characteristics in their ideas merge. As this congregation becomes more pronounced the exchange of ideas is more fluid and scholars can start developing from the ideas of those before them, rather than starting from scratch. The decline of Greek philosophy can then be measured by the gradual evaporation of local characteristics in its theory. With this evaporation, the philosophy becomes applicable internationally. The local characteristics of Greek philosophy were originally those of Ionia Italy, and then those of Athens. With Alexander the Great's invasion the characteristic became diffuse as the theory spread across the Mediterranean with the expansion of the Roman Empire and then into the Islamic states during the fall of the Roman Empire. Despite this, the spirit of Greek philosophy remained, that is of rational and theoretical thinking, even though it started to lose its name.

Greek philosophy globalised the world by fruitfully merging local and world characteristics. It is important to note however that the Chinese culture continued to develop rather independently of the spread of Greek philosophy. It is only recently that Greek philosophy has acted as a source of wisdom that has renewed Chinese culture.

Game theory is a rich theory that has considered the evolution of many different types of scenarios. A famous example includes the prisoner dilemma. These scenarios are largely theoretical and do not often reflect the true nature of humans. Largely because humans are not rational agents, and humans do not operate in the heavily simplified worlds in which these theoretical games are played. Nevertheless, these games can be contextualised to provide useful heuristics on how groups of humans can interact. However, we cannot apply this same translation to interacting AI systems for they have many different properties with add nuance to the dynamics.

• AI systems are largely rational. They perform their actions based on the observations and are not influenced by emotion.

• AI systems are not as affected by external factors. A human’s mood may change based on external factors which influence the decisions they make. Whereas the internal state of an AI is largely independent of the external world, and thus their decisions are more robust.

• AI systems still lack a general sense of common reasoning which we assume humans have. For instance, if there is an esoteric solution to a game, we do not expect this to manifest in human dynamics since it would violate an individual’s common sense. However, we cannot at present ensure that AI systems will not follow these esoteric solutions. This problem falls under the scope of AI alignment, as we can think of our morals as a sense of common reasoning.

These differences mean that on the one hand, the interactions between AI systems are more predictable, as their decisions largely persist over time and are not as influenced by external factors. However, our inability to understand what a model is doing means that it is more challenging to contextualise these theoretical games.

Some other differences affect the style of the interactions between AI systems.

• The source code for an AI system is likely to be open-sourced and available to other AI systems. Therefore, one AI system could simulate the other, and get a sense of the likely output of the other system.

• The increased amounts of computing resources available to these systems means that their interactions could become more complex. For example, they can become longer and be carried out over shorter time frames.

Since every party involved is aware that these AI systems can simulate, memorise, and hold more complex interactions, the dynamics of these interactions will be vastly different to those we are currently familiar with.

One may argue that these complexities will not emerge since we will develop perfectly rational agents, and thus there would be no need to contextualise these theoretical games. The interactions will just play out in the way that game theory tells us. However, it is not in our interest to develop perfectly rational agents. The Traveller's Dilemma was a game formulated by Kaushik Basu. Individuals Alice and Bob have had a set of their antiques damaged whilst travelling with an airline. They seek compensation from the airline, however, the airline is not capable of valuing these antiques as they are relatively obscure set. Therefore, to ensure that Alice and Bob do not inflate the price of the antiques they get each of them to secretly write down the value of the antiques on pieces of card. The airline will then reimburse the pair with the lowest value written on the cards, penalise the individual who wrote down the higher value and reward the individual who wrote down the lower value. For concreteness, we can suppose that the airline is willing to reimburse any value between £2-£100, and the airline penalises/rewards the individual who gives the higher/lower value by £2. If consider one perspective, say Alices, then we quickly see that the rational choice is to write down £2. Indeed, at first, she thinks she should write down £100, but then assuming that Bob will do the same she is then inclined to write down £99 as then Bob will be penalised £2 and she will be rewarded £2 giving her a total of £101. However, she then realised Bob would consider the same arguments and thus she must further reduce her value to £98. Continuing in this way we end up with Alice reasoning to write down the value of £2.

This scenario shows how a rational agent can descend to unreasonable solutions. Here we considered a low-states scenario, however, it is not difficult to imagine scenarios where the stakes are not so mundane.

Similarly, we would not want a system that tries to please everyone. A system designed to please everyone would be increasingly fractious and wouldn't be able to make any decisions since you cannot please everyone.

These nuances in the construction and implementation of capable AI systems mean we need to rethink how we study their interactions. Optimizing certain characteristics may lead to unintended consequences. AI systems will not operate in the same way as humans. Our traditional notions of how to successfully operate can no longer be applied.

• Some argue that this scenario is feasible, and is close to being initiated with current machine learning models.

• Others agree that such a scenario is feasible, however, they note that current machine learning models are not capable of triggering such an explosion.

• Then some argue that intelligence does not work in this way. More specifically, other external factors affect the intelligence of a system, and thus the explosion is extrinsically limited.

Fundamentally, each of these perspectives is founded upon the articulation one has of intelligence and its interplay with the environment. For example, thinking that intelligence is unbounded and an inherent property of the individual system may lead one to concur with the first perspective. Since the environment cannot limit the systems’ intelligence. On the other hand, if one views intelligence as a property determined by the individual system and its interaction with the environment, then one is probably inclined to adopt the third perspective

Although current machine learning models possess lots of knowledge, it is still unclear whether they are intelligent. They have been empirically shown to have some reasoning capabilities, however, they are still susceptible to making trivial mistakes. It is possible to improve these systems through algorithmic advancements. Recently, Ryan Greenblatt from Redwood Research enhanced an LLM’s capacity to solve ARC-AGI by fine-tuning the method by which the LLM solves the problems. Therefore, one could argue that current models are more intelligent than humans and it is just that they cannot extract their latent knowledge effectively. On the other hand, one could argue that this inability to effectively utilise their latent knowledge precisely means that the models are not intelligent. Indeed intelligence is the capacity to utilise knowledge appropriately by inferring information from your environment. Consequently, it seems as though the intelligence of a system is inherently linked to the connection between its knowledge and the environment. Currently, LLMs have vast amounts of knowledge but lack efficient mechanisms, and perhaps even the skill, to incorporate this knowledge into the environment. Humans on the other hand have evolved symbiotically with the environment and thus their capacity to interact with the environment is greater. Therefore, despite humans having less knowledge than LLMs, our ability to utilise the knowledge we have appropriately with the environment is greater and thus we would regard ourselves as more intelligent. Maybe if we were placed in an alternative environment an LLM would be regarded as more intelligent than humans. For instance, if our environment was just an IDE where we were tasked with writing boilerplate code, then GPT4 would be more intelligent than most humans.

Thus, we could be confronted with a scenario where existing systems suddenly become more intelligent as they move into a different environment. For example, existing LLMs could, unintentionally, be incredibly proficient at leveraging a particular tool that is not currently invented. At the point of construction the LLM may not seem intelligent as it does not have access to this tool, however, as the tool becomes developed and made accessible to the LLM, we may see a vast improvement in the LLM’s intelligence as it can use the tool. For example, consider biologists before and after the microscope.

With this, we see that intelligence is directly related to the environment since intelligence is measured by interactions with the environment. However, knowledge acquisition also plays a role in intelligence, and the capability of a system to acquire knowledge is linked to its environment. Indeed, a biologist with a microscope has more access to their environment and can thus acquire more knowledge and become more intelligent. LLMs acquire knowledge through training, which is a notably energy and time-intensive process. We could theoretically train LLMs more efficiently in a more forgiving environment, such as outer space where the low temperatures mean that less energy is required for cooling which leaves more energy devoted to knowledge acquisition. This is analogous to how in the solar system life has only prospered on Earth which is situated in a Goldilocks zone, providing the ideal conditions for life to emerge.

We now return to the question as to whether an intelligence explosion is plausible. With our present construction of intelligence, we see that this question is related to the constraints imposed by knowledge and the environment. In particular, the constraints imposed by these can be linked to energy. Acquiring knowledge and interacting with the environment requires energy. Energy is in abundance in our current environment however it is not easily extracted. It seems as though human intelligence has managed to extract just about as much energy as we can from the environment, pending the implementation of fusion energy, and thus we can assume that humans will not trigger an intelligence explosion. Of course, we will still become more intelligent, we just will not see an increasing exponential rate of intelligence. Hence, a system that triggers an intelligence explosion will need to utilise an innovative source of energy, say a Dyson sphere.

There is evidence that this energy-centric approach as to whether an intelligence explosion can occur is viable. Throughout history, as humans have been able to extract greater amounts of energy from the environment, humans have become more intelligent. Fire meant less energy expended for digestion, then hunter-gatherers adopted farming, farmers industrialised their practices and then electricity powered the digital age. It seems as though the next step in this journey is the widespread adoption of fusion energy. If there is anything beyond that is yet to be seen. Maybe GPT-5 will discover it first...

A concept activation vector attempts to identify a concept with a vector. It simply takes a collection of data representing a concept, and a collection of data not representing the concept and learns to differentiate the activations of the network between the sets of data with a linear classifier. The trained linear classifier provides a vector, which can be thought of as pointing in the direction of the target concept. Note that under this construction we are supposing that concepts are delineated linearly by the network. This is quite a strong assumption that does not hold in all cases, particularly for rather esoteric concepts. However, it is observed that a network linearly encodes a large proportion of general concepts.

We can investigate how well a concept activation vector summarises a concept by studying a neural network trained to classify hand-written digits. More specifically, we first train an autoencoder model to reconstruct the images. This way we obtain a representation of the 28x28 images in a lower-dimensional latent space, in this case, we set the latent space to have 16 dimensions. Moreover, in the lower-dimensional latent space, the model is incentivised to linearly encode the concepts which increases the effectiveness of concept extraction. We then obtain our concept activation vectors by training the linear classifiers on the activations of the different classes of images in this latent space. Consequently, we obtain 10 different concept activation vectors, each one summarising a particular hand-written digit. There are different ways one can understand how well a concept activation vector summarises a concept. A simple approach would be to study the accuracy of the linear classifier.

For our case, we will explore the influence of the concept activation on downstream tasks, which is also the approach taken by the original paper. Namely, we train a neural network to classify the reconstructed images of the autoencoder, and then observe how the concept activation vector influences the prediction of the classifier. In all cases, we see that perturbing the latent space activation of an image by the corresponding concept activation vectors leads the classifier to more strongly classify the image with that label. In more detail, we can approximate the gradient with which the concept activation vector influences the target logit of the classifier. As noted, this gradient was always found to be positive.

Instead of influencing an image’s latent space activation with its corresponding activation vector, we can also influence it by the concept activation vector of a different concept.

We can continue perturbing the latent space activation to the point where the classification model reclassifies the image with the label of the other concept. The necessary amplitude of perturbation required to change the output of the classifier can indicate how well the concept activation vector summarises that concept and the relationship of the concept to the original label of the image.

However, there is a mild form of self-improvement for which there is already evidence of a system almost performing. This form of self-improvement involves the system generating its data, on which it continues to train itself to improve its capabilities. The intuition here is that a sufficiently capable system can generate higher quality data than that it was trained on. Although this seems as though we are having our cake and eating it, this dynamic is reminiscent of a student-teacher dynamic. In this scenario, the student is the AI system and the teacher is a human expert. The teacher trains the students using their expertise, and eventually, the student can conduct their explorations and progress on the knowledge provided by the teacher. Throughout history, this dynamic has progressively expanded the scope of human knowledge and expertise, and thus it is plausible that an AI could bootstrap its capabilities through data generation.

Indeed, AlphaGo was initially trained using human examples of good moves in the game of Go. It was then left to play against itself and learn how to improve its capabilities. Eventually, it beat the leading human Go player. An attribute of the success of the resulting system was that its style of play was divergent from the well-established strategies found by human players. For instance, in one of the games the system played a move that contradicted common notions. The move would become known as move 37, as it turned out to be decisive for the outcome. Subsequently, AlphaGo has rejuvenated the scope of approaches taken to play the game of Go.

However, the AlphaGo system still had some flaws. A group of researchers discovered some adversarial examples for which the AlphaGo system failed, and hence they beat the system. This is perhaps to be expected, for the data generation process is likely to make the capabilities of the system rather brittle, for it only reinforces behaviour along a certain direction, and the blind spots of the system remain.

Simple Models

We can test this data generation form of self-improvement with a simple model. We consider a generative model trained to encode and decode images from the MNIST dataset, which is just a collection of hand-written digits.

We supply the model with an initial set of data, let it train on this data, generate a new set of data using the trained model, let the model continue training itself with this new dataset and repeat.

To relate this to the scenario of the intelligence explosion, we slightly corrupt the initial batch of data to reflect that human data may not be the ground truth. However, we evaluate the modelling during training using the ground truth, to simulate the higher intelligence level the model is striving for.

In our experiments, we consider an autoencoder model and a variational autoencoder. The corruption of the initial dataset comes in the form of random noise of varying amplitudes or horizontal lines of varying lengths added to the edges of the digits. For each model, we conduct a control experiment where no noise is added to the initial images.

In each experiment, the model is trained on 20 batches of data, 19 of which are self-generated.

For the autoencoder, our control experiment shows that the autoencoder continues to improve as it trains on its generated data. However, for the variational autoencoder, the loss immediately increases when it starts to train on its generated data. It is only after a few iterations that its loss decreases again, however, in our experiments the loss never arrives back at its original value. This perhaps reflects that the latent space of a variational autoencoder is regularised to resemble a probability distribution, therefore, as the images it generates lie on a different manifold to the original dataset the previously learned latent space is no longer applicable. Consequently, loss rises, and it takes the model many training runs to alter its latent space distribution to represent the new manifold on which its training data lies.

As we increase the amplitude of random noise corruption we see that this effect for the variational autoencoder still holds. Similarly, we see this effect when adding lines to the digits. However, in any case, we see that the model can recover and continue to improve towards the ground truth.

With the autoencoder, we do not see this initial loss in performance as the model continues to train itself on its own generated data. Instead, it continues to improve towards the ground truth. This is perhaps explained by the fact that the autoencoder has no regularisation on its latent space, and thus it is easier to adapt its latent representation to new inputs.

Human existence is the deeply personal pursuit of meaning through action and responsibility.

One Paragraph Summary

The depth of our unconsciousness is the centre of our human existence. Consciousness stems from this centre and is a means to find meaning through action and responsibilities that transcend any comprehension. The relationship between the unconscious and conscious is profoundly personal, and hence meaning cannot be inflicted externally but must be found. Pursuing meaning must be an endeavour directed at something beyond the being themselves. It is only then that true value and fulfilment will emerge.

Rehumanising Psychotherapy

Viktor Frankl has a uniquely religious and inherently human perspective on consciousness and the meaning of life. Likely influenced in large parts by spending three years in concentration camps during the height of World War Two.

The Adlerian School of Psychology is a practice based on encouragement. The role of the psychotherapist is to encourage the patient to overcome their inferiority feelings. Similarly, the Freudian psychoanalyst takes an objective approach to the practice. It is a reductionist approach that considers values as nothing more than a reaction formation and a defence mechanism.

The Freudian School of Psychology was developed by Freud who, unsurprisingly, saw the unconscious as an instinct rather than a spirit. We will see that Viktor Frankl adopts the view that the unconscious also holds a spiritual component. In particular, the Freudian approach to psychology treats man as being ruled and governed by a homeostatic principle, that is a principle that is reactionary and externally driven.

Viktor Frankl argues that this has led to an objectification of neurotic patients. Leading to the proliferation of the neurotic triad, namely depression, addiction and aggression. According to Frankl this engineering approach that treats man as a machine is detrimental. Frankl supports this with various studies showing the increased prevalence of existential vacuum and nöogenic neurosis. The former is a sense of meaninglessness and emptiness, and the latter is a frustration for the lack of will for meaning.

Frankl is deeply troubled by these studies for he holds the view that human existence is a deeply personal pursuit for meaning. Frankl supports these views by citing a study that shows the feeling of existential vacuum and nöogenic neurosis are particularly prominent among suicide attempters. Moreover, the studies highlight that apart from these neuroses, the suicide attempters were largely successful in other parts of life, such as their education, career and finances.

Frankl dismisses the Freudian approach to psychiatry with the following prompting questions.

• If meaning and values are nothing but a defence mechanism, then is life worth living?

• If a man is just seeking pleasure by getting rid of tensions caused by his needs, then why worry?

• If a man is a victim of outer and inner influences, and his behaviours are nothing but reflexes, then who is justified in demanding that man improve?

Therefore, with "Man's Search For Ultimate Meaning", Frankl argues that we should start humanizing psychiatry - the deneuroticisation of humanity requires the rehumanisation of psychotherapy. To do so, Frankl proposes Logotherapy which is the education of the individual on their responsibility. It leaves it to the patient to decide what is meaningful to them, rather than trying to force them down a particular path. More specifically, it focuses the patient on discovering their will to meaning, how to identify the meaning in suffering, and instilling a freedom of will.

Human Existence and Consciousness

Frankl's logotherapy conducts an existential analysis, which is where a human is interpreted as being responsible. Specifically, man has to answer to life by answering life, that is responding by being responsible. Man can only exist authentically when he is not driven but responsible. This existence is rooted in the unconscious depths of man, where there is a contention between existence and spiritual facticity. This contrasts the Freudian perspective that only identifies the existence facticity. The existence of man being embedded in the unconscious necessarily means that man cannot reflect, but must rather act. It is only through action that man can reveal the motivations of the unconscious. The more comprehensive a man's meaning becomes, the less comprehensible it becomes. Moreover, the determination of what experiences become conscious or remain unconscious is itself unconscious. In this sense consciousness is irrational, and it is inscrutable. This alludes to a transcendental component of existence, which Frankl will later elaborate on by identifying the theological components of our existence.

Frankl specifies this peculiarity of the mind by describing what is disclosed to consciousness as something that is, and describing what is revealed as something that ought to be. Consequently, consciousness can be attributed with an intuition for it anticipates what is not yet but is to be. Frankl suggests that this instinct comes in different forms. The vital instinct is disclosing what is required, whereas the ethical instinct enables man to discover their unique requirement of a situation. Frankl personifies the ethical instinct by comparing it to love. The feeling of love discloses a solution that is personal to the man's relationship with another being. Attempting to manipulate these primal instincts is detrimental to the creative process. Encouraging de-reflection helps liberate the creative process. The role of the therapist should help the patient convert unconscious potential into conscious action, which is then stored back in the unconscious as a habit. This process encourages the ethical instinct to instantiate a unique and liberating solution to an individual’s inhibitions.

Man most readily observes their spiritual unconsciousness through dreams. Dreams are an expression of the spiritual unconscious.

Freedom has a "from what" and a "to what" aspect. In Frankl's perspective, the "from what" is a being's drive, whereas the "to what" is the being responsibility, which is his consciousness. Therefore, consciousness is a willingness for freedom from a transcendent origin. To understand being free the existential quality of human reality is sufficient. However, to understand being responsible requires the transcendent quality of consciousness. It is in this vein that the irreligious man is limited. He treats consciousness as a psychological facticity. He goes no further, so as to not lose the firm ground underneath his feet. From his perspective consciousness is the "to what" he is responsible, whereas the religious know it is the penultimate step with the ultimate summit behind its fog. It is necessary to appeal to the religions to notice the transcendental and obscurity of consciousness, such that one can traverse to the ultimate summit. It is a hazardous endeavour for which religion acts as a comforting guide. In turn, despite the self having the function of acting on drives and instincts, the self cannot be traced back to any of them since they are motivated behind the façade of consciousness.

Having understood the religious components of the conscious, Frankl then goes onto to investigate unconscious religiousness, which he identifies as the latent relation of man to the transcendent. That is the relationship between the immanent self and the transcendent being. Frankl describes this hidden relation as directed to an unconscious God. Despite this description, Frankl notes that he is not attributing a divinity to the unconscious self. What Frankl is trying to make clear is that no knowledge can come to know itself without rising above itself. Moreover, man's religiousness is profoundly personal.

In linking religiousness to the unconscious, it is necessarily the case, in Frankl's perspective, that genuine religiousness has the character of deciding rather than being driven. The spiritual unconscious is an existential agent rather than an instinctual factor. It stems from a personal centre, and thus if repressed or manipulated the internal angel with turn demonic. Furthermore, due to the inherent obscurity of the unconscious, the religiousness of an individual must unfold in its own time. Indeed, since the process of unconscious material becoming conscious has a therapeutic effect, it follows that the unfolding of religiousness in man provides man with psychotherapy. Frankl argues that this unfolding provides the most psychotherapy man can imagine, however, he notes that religiousness demands more of him. The main psychotherapeutic aspect of the unfolding of religiousness is the dignification of man, which in turn leads to his freedom. Here Frankl uses the term dignity as appreciating the value of something in itself rather than its personal value.

With this perspective of existence and consciousness Frankl arrives at the following conclusions.

• Consciousness is a human achievement that other animals do not possess. Animals do not fulfil drives and instincts in search of meaning, and thus they cannot experience consciousness. Frankl describes the behaviour of animals as anticipatory anxiety. They are responding to the changing environment instinctually rather than spiritually.

• Man is concerned with gratifying needs and satisfying drives and instincts. Man's existence is directed to something or someone other than themselves. In particular, the more he forgets himself the more human he becomes. Man does not seek pleasure, instead, pleasure is the side effect of living out the transcendence.

• Lack of meaning and purpose foreshadows emotional maladjustment. Thus Frankl can explain the rise in the feeling of existential vacuum in modern society by observing how our societies have changed over time. In particular, in the modern era, man is no longer told what he must do by his drives and instincts. Moreover, man is no longer guided on what he should be through tradition and values.

• Since achieving meaning and fulfilment is necessary for existence, it follows that goodness can be defined as that which fosters meaning and the fulfilment of a being. Consciousness is the means to discover meaning. Due to its inscrutability, we cannot be certain of its correctness, however, all man can do is stick to it. It is only in this way man resists the existential vacuum.

• Frankl then discusses the meaning present in suffering and the importance of identifying this meaning. Suffering without meaning is despair and the acceptance of unnecessary suffering is masochism. In the unavoidable suffering, however, there is meaning

• Human existence, and thus survival, is dependent on having a direction toward meaning. For mankind, there is hope when we are united by a common will to a common meaning.

Wisdom is knowledge and the knowledge of its limits.

Intelligence is an abstract phenomenon. Generally, we envision it as the ability to reason and plan effectively through novel scenarios. Jean Piaget succinctly described intelligence as "what you use when you do not know what to do". Note that these descriptions of intelligence are not anthropocentric. Meaning that apriori it is not clear whether humans are intelligent systems. From our perspectives, it seems as though we are, however, it could be the case that through our infant years, we amass sufficiently many experiences that throughout our later life we can operate in our environment by replaying those experiences with different values to fit the context. Then due to our inability to remember our infancy, we have the perception of encountering novel tasks, whereas unconsciously we are just replaying past experiences. Therefore, knowing whether a system is intelligent is fundamentally linked to identifying novel.

From our definition of intelligence, we observe that it is not a binary feature. It is probably safe to assume that humans exist on the spectrum of intelligence. As our definitions are imprecise it is difficult to test for intelligence. For instance, the IQ test attempts to measure intelligence through a series of comprehension and pattern recognition tasks. However, one can revise to reduce the novelty of the scenarios encountered on the test, making it a flawed test for intelligence. Moreover, it excludes less canonical forms of novelty that account for emotional intelligence.

With our non-anthropocentric notion of intelligence, we can ponder the intelligence of artificial systems, such as machine learning models. More importantly, we can consider artificial systems with a greater intelligence than our own. It is not clear whether developing such a system is possible, and whether it takes the form of current machine learning techniques. It is certainly the case that current state-of-the-art machine learning models exhibit a strong set of abilities that significantly impact society. Whether those abilities constitute a generally intelligent system is a great source of current debate. Regardless as to whether these systems are intelligent according to our definition, it is still important to understand the consequences they will have on our economy, society and world order. To ponder these questions, we will slightly abuse our use of the word intelligence and refer to artificial general intelligence as a system that can automate tasks that humans can competently perform. We proceed with the understanding that the word intelligence is used loosely, current machine learning models are not an artificial general intelligence and it is not clear that current machine learning techniques will lead to an artificial general intelligence.

As an aside, we have made no connection between intelligence and consciousness. Whether consciousness is a necessary condition for intelligence is another source of debate. An interesting approach to test for consciousness suggested in Lex Fridman's podcast with Roman Yampolskiy was to supply models with optical illusions and observe their reactions. Models that demonstrate a reaction to the illusions may have a subjective visual experience. Of course, this does not demonstrate that the system is conscious, but it may be an indication that something peculiar is happening within the internals of the system. To be a robust test we must ensure that the supplied optical illusions elicit unique experiences not present in the training data.

Progress to AGI

A lot of effort is being made to achieve AGI as it would revolutionise industries such as healthcare and education. The potential benefits of these systems are so profound that many argue it would be morally wrong not to develop such systems. However, many are also aware of the potential risks and harms that such a system would have. The current approach to achieving AGI is by scaling up large language models. More specifically, we are training large language models with larger architectures on more data. This is motivated by current scaling indicating a reliable increase in performance. Therefore, assuming we have an idea of the performance required to achieve AGI, and assuming these scaling laws hold, we can arrive at rough predictions as to when AGI will arrive. Indeed, this is precisely the analysis conducted by Leopold Aschenbrenner, who also explores the consequences AGI will have on society.

I do agree that the current progress of this technology warrants attention as there are risks. For instance, due to their scalability and reproducibility, they can contribute immensely to information warfare. Consequently, any entity possessing such technology will have a considerable advantage over its adversaries. Furthermore, it may be difficult for a single entity to fully control the impact of their application of the technology, leading to unintended consequences that can affect vast amounts of people.

At one stage, Leopold uses the token thought rate to demonstrate how capabilities may scale. However, I do not think this metric encapsulates a model's capacity to think. When I am thinking I believe there is a lot more unconscious processing going on than just the tokens I am reciting in my head. Hence, by simply giving models more tokens to predict, I do not think we will get the sorts of increases in capabilities that Leopold predicts. There have been countless instances in recorded history where breakthroughs were made when people were not consciously thinking about a problem. Take Albert Einstein, who made many of his discoveries whilst working at a patent office. Therefore, an LLM’s incapacity to focus more intently on harder problems will require more than just letting it spend more tokens thinking about the particular problem.

Leopold's prediction for the future is based on the current scaling laws. In particular, Leopold predicts that AGI will arise as we scale up the training clusters. However, this is predicated on the assumption that energy and data supply will increase to facilitate such scaling. Although Leopold addresses these issues I believe that the increases he predicts will not be as swift as he imagines. Bureaucratic friction and limited public focus will mean that developing such infrastructure will be a slow process.

Throughout the essays, the capabilities of current systems is evaluated against benchmarks which I think are inherently flawed and not a good metric to determine whether these models are capable of replacing human workers. Most benchmarks get the model to answer questions about a series of topics. Due to the sheer scale of the models, these questions ultimately boil down to memorisation and are not indicative of whether the model can reason. Some of the models perform well on these benchmarks, beating humans with PhDs. From this Leopold immediately makes the conclusion that these models have an intelligence equal to that of individuals with PhDs. However, obtaining a PhD involves a lot more than answering the questions in the benchmarks. A paper from Google DeepMind shows that LLMs struggle with trivial counting and searching problems, despite performing well on standard benchmarks.

Will We Get AGI?

Using the definition that an AGI can automate most human tasks, it will be clear once we achieve AGI. However, using our more abstract definition of intelligence it is unclear when we will achieve AGI, as in practice it is difficult to ensure novelty and test for ability. Many benchmarks try to assess levels of intelligence, however, with the scale of current models, the benchmarks are likely part of the training data. The ARC benchmark is designed to be robust against memorisation and provide novel scenarios to reason through. Currently, the best models achieve around 35% on this benchmark, where it is assumed that humans can achieve around 85%. Indicating that at present these LLMs lack a form of intelligence.

In summary, I think the scaling of LLMs will lead to impactful technologies. In the past scale has provided these systems with some emergent capabilities. For instance, induction heads emerged as transformers were scaled from one to two layers. I am sceptical that scaling systems in their current form will lead to intelligent systems in the broad sense of the word. Moreover, I am sceptical that scaling will happen as dramatically as Leopold suggests. However, these systems will be knowledgeable and contribute to many human tasks. Therefore, we ought to be cautious about their deployment. To achieve full AGI I anticipate that current machine learning architectures will have to be augmented. The rate of

Modern Western philosophy is rooted in the works of the ancient Greeks who pioneered the method of rational and logical thinking. Developing a methodology to explore abstract concepts and facilitate the progression of societies to the point we observe today. Although it may seem obvious to us now how to form logical arguments, there was a time when this thinking mode was beyond the scope of many. Indeed, the great Western philosophers such as Plato and Aristotle are arguably some of the first humans to adopt this style of thinking. To develop this rationality they had to separate themselves from more primitive forms. As Ancient Greek civilisations had religious practices at the core of their culture, the typical mode of thinking was detached from one’s own consciousness. Priests would speak the word of God, they would not derive their own interpretation. This primitive thinking mode involved a lot of direct and specific language, there would be comprehensive descriptions of objects that lacked analysis, and the concept of the self was replaced by collective references. One of the first Ancient Greek philosophers who started the transition from this primitive mode of thinking to more rational thinking was Heraclitus. Living in the fifth century before Christ, Heraclitus pioneered alternative modes of thinking. One of his most iconic lines of thought led him to determine that fire was the source of all things. Moreover, he argued that the universe is the same for all, a rather absurd thought at the time due to the significance of religion in the culture. Heraclitus frequently appeals to the analogy to formalise his thinking. He used his imagination to build a universe of relations. His obsession with fire perhaps comes from its malleability to analogy. Therefore, we see that the first Western philosophy capitalised on relations to make sense of the world and thus it is no surprise that these ideas permeate our modern thinking. A reason why this contrastive thinking was attractive for early philosophers is up for debate. However, one could possibly make an evolutionary argument. Knowing good from bad is important for survival as we must avoid the bad things and capture the good things when they arise. It would be detrimental if we conflated these notions. Hence, having strong abilities to develop relations seems like a necessary skill for survival.

The Impact on Our Political Opinions

Frank Furedi said in an interview that we tend to form our views as the opposite of those held by people we disagree with. Clinging onto our evolutionary argument, we could say this is because we see those who disagree with us as the enemy, and thus to differentiate them we must form opposing views. Furedi claims that this is fuelling the current culture wars, where we seem to be ever more divided with the possibility of reconciliation becoming less plausible. As this style of thinking may be attributed to our primitive brain, it may be the case that this behaviour is largely unintentional by the individual, but rather an instinctive response. Consequently, it is not clear whether the large political changes we are observing are in response to this instinctive behaviour, or represent a reasoned shift in perspective. One would hope that everyone is forming their beliefs through rational thinking rather than through spite.

These opposing forces shaping the political landscape have meant that the main political ideologies are not static entities, but evolve. In The Network State by Balaji Srinivasan, several examples are provided that show the dynamic position of the right and the left ideologies. In some instances, they have represented the same views, albeit at different points in time. Despite this, we note that there has always been a right and a left. They never seem to converge. They may be closer to each other at different points in history, but as a civilisation, we are never inclined to consider them as one entity. This is perhaps due to our reliance on relational thinking. If there were not a right and a left, then we would not be able to make sense of their ideologies as there would be nothing to compare them to. How often do you categorise your political views on one side by understanding how your views are not aligned with those on the other side?

Intelligence and Emotion

Developing concepts through relational arguments is perhaps one of the key factors of our intelligence. To learn something it is not sufficient to know when you are right or when you are wrong, but you must understand how right and how wrong you are. Therefore, having the capacity to develop detailed connections leads to more effective learning. Moreover, our emotions are heavily governed by contrast. For example, a sunny day after a rainy day is more satisfying than a sunny day after consecutive sunny days. In my opinion, we cannot experience such happiness and satisfaction without experiencing their negative counterparts. Therefore, we should often seek challenges and discomfort to exemplify and encounter positive emotions.

The Power of Two

It is not clear why we seek to attribute meaning to things through relations, and specifically why these relations are between two objects. We can certainly derive meaning from three objects. A simple answer would just be that keeping track of two objects is easier than keeping track of any higher number of objects, but why is it the case that the comparison between two objects is sufficient to capture the necessary meaning? For example, why can we broadly characterise the political opinions of society as being right or left? Now it may be just that I am asking these questions in the wrong order. Perhaps as we are only capable of keeping track of two objects, we have developed a society such that sufficient meaning can be derived from the comparison between two objects. However, looking from a less anthropomorphic perspective we see that when dealing with more than two objects things get significantly more complicated. Consider, the dynamics between objects that exert a gravitational force on one another. With two objects the dynamics are well understood and can be predicted. However, as soon as we add a third object we arrive at the three-body problem, where the resulting dynamics of the system is notoriously difficult to simulate. Similarly, the double pendulum is an example of a chaotic system. Here the chaos arises due to the interaction between the force of gravity, the centripetal force of the first pendulum, and the centripetal force of the second pendulum. From these examples, it seems as though there is a natural tendency for the interaction between two objects to be meaningful and stable, but with the interaction between more objects being chaotic. Therefore, we may argue that our relational mode of thinking is a natural consequence of the structure of the universe.

It turns out the task of predicting the next word in the sequence is sufficient for the model to be incentivised to learn basic forms of reasoning. For instance, on some small language models, it has been shown that the model learned to detect and utilise skip trigrams to inform their prediction of the next word in the sequence [1].

Therefore, LLMs are potentially a useful tool to study concepts such as memory, which are intricately related to language. Similarly, we can study the structure of language, as LLMs are designed to generate statistically significant representations of the connections between words.

Memory

We are still unsure how memories are encoded in the human brain. There are many peculiar observed phenomena regarding memory that it is difficult to construct a unifying theory. For instance, many people do not have strong recollections of memories from their infancy. It is apparent that the formation of memories is dependent on many different factors.

From a user experience perspective, LLMs can be made more useful by instilling them with memories such that they can recall previous interactions to form more appropriate answers. As we are unsure how memories are stored in human brains, this task is not so straight forward. I do not think just allocating some computer memory to store previous interactions is the way to give LLMs memory, in a way that resembles humans or leads to realistic conversations. The reasons for my scepticism here are due to the following unanswered questions that emerge from such an implementation.

1. How long should we store memories for? The answer to this cannot be a fixed rule, say that the potency of memories decays at some pre-determined rate, as I can recall memories from my childhood and forget what I did last week. Therefore, it is going to be a function of may different factors which will hard to precisely model.

2. When space is limited which memories do we remove to make room for new memories?

3. For any given interaction a LLM only has access to the text. However, the user interacting with the LLM has an experience that is beyond just the text. Therefore, we need to understand whether just storing the text of the interaction is sufficient for it to be useful in future interactions. Similarly, we need to understand how much of the textual information from an interaction we store. Are we required to store every word of the interaction, or is only recording a few words sufficient?

4. Human memories are not perfect and are often subjected to the individual’s prior beliefs and experiences. This makes the recollection of memories a highly subjective and personal experience. Thus when we converse with someone who recalls a memory we are engaged by the nuances in their recollection. What personality should we provide the LLM such that when it forms and recalls its memories it does so in a personable and engaging manner?

5. Human memories are not stored in their entirety due to memory constraints. We attend to different components of an interaction and connect it to previous memories. What parts of interactions should we make the LLM attend to, and how to we incorporate a given interaction into existing memories?

6. Human memories come with triggers such that we can recall them. We are not able to search our brains for a particular memory, indeed sometimes we seem to just stumble across memories and we are not really sure on why we recalled them. What triggers should we apply to LLM memories such that they are recalled in a realistic manner?

Currently, for many applications of LLMs the size of the context length is sufficient to store the necessary context to allow for an appropriate interaction between the LLM and the user. However, as we aim for these systems to become more personalised to the individual, the context length may not be sufficient. Moreover, as this technology should eventually be accessible to many through handheld devices, thus we need to be able to store memories efficiently.

[2] constructs a memory module that augments an existing LLM with memory storage and memory retrieval capabilities. The paper emphasises the detail and specificity of the memories stored in its module. It employs the Ebbinghaus curve to vary the strength of memories to reflect the fact that we slowly forget memories and that memories we have some recollection of are easier to relearn. I think this implementation is insufficient for endowing LLMs with the capability to have memories the way humans do. Indeed, I do not recall the timestamps and specific details of interactions I have with others, which is what MemoryBank claims to do. One may argue that as computers have the capacity to store such information then they should do so. However, I would argue that the lack of specificity in my memories is a feature, not a bug. It means that I have to store it with an abstract representation so that I can reconstruct the details upon recall. In storing the intricate details I would be overfitting to the situation and would not be able to extract the general principles of the interaction. Humans have evolved an intricate attention pattern so that we only consciously register around 20 bits of the 11 million or so bits of information we are exposed to. If we tried to attend to all 11 million bits our lives would be chaotic and we would be rife with anxiety in trying to process all the information.

In contrast to MemoryBank, I think that memories should be stored implicitly in the weights of the model, just as we store our memories in our neurons. After all, it seems as those humans forms memories more strongly in response to learning. If something is surprising to you store that event more vividly in your memory. Therefore, if we instead allow the model to continually learn and update its weights, then we may be able to mimic the memory-forming structure of humans in the weights of the model.

Language

LLMs may be an interesting object to investigate from the perspective of language theory. After all, they are trained by studying large corpora of text, to develop connections that then lead to text generation that complies with syntactic rules. By interpreting its inner-computations we may be able to arrive at a theory of language, not of human psychology. We would only be able to understand the structure of our language, but we cannot make any inference about the human condition from this. Therefore, we would be studying language from the perspective that it is external to the human condition, much like mathematics is thought to be a fundamental aspect of nature.

In this study, we may be able to arrive at canonical interpretations of written texts. Although the canonical interpretation may not be the interpretation intended by the author, the canonical interpretation would be the one dictated by the structure of language. Therefore, as the form of language changes over time, it is probably the case that the canonical interpretation of a piece of text may change over time. The advantage of LLM systems is that we can train them on specific corpora of text to understand the interpretation of a piece of text that is offered by the structure of language present in a particular time period. The LLM is not influenced by texts it is not trained on, hence, we can observe how the meaning of language changes over time.

Summary

LLMs have a remarkable capacity to generate coherent text that conforms to grammatical conventions. Therefore, as we are now developing sophisticated methods for interpreting these models, we can use them to help investigate phenomena such as memory. Moreover, their scale means that they can offer global insights into the structure of language. However, we ought to be careful that we do not over-anthropomorphise these systems as their architecture and development do not resemble the human brain or human learning strategies. Indeed Moravec's paradox is evidence that we cannot immediately make the links between humans and LLMs.

References

[1] A Mathematical Framework for Transformer Circuits. https://transformer-circuits.pub/2021/framework/index.html

[2] Zhong, W., Guo, L., Gao, Q., Ye, H., & Wang, Y. (2024). MemoryBank: Enhancing Large Language Models with Long-Term Memory. _Proceedings of the AAAI Conference on Artificial Intelligence_, _38_(17), 19724-19731. https://doi.org/10.1609/aaai.v38i17.29946

In mathematics, the study of dynamical systems investigates the evolution of a state under a deterministic update function. For instance, suppose you were given a number between zero and one and imagine it as a sequence of digits

Then an example of a dynamical system would at time step n to identify the nth digit. For example, the first few steps of the dynamical system corresponding to

would be 1,5,3,9,0,2,1... We can reference this particular dynamical system as E10, as it considers numbers expanded in base 10. One could equally choose a number between zero and one, consider its expansion in base 2, and then identify its digits. In any case, these dynamical systems exhibit what is known as a sensitive dependence on initial conditions. Loosely speaking, this means that a small perturbation in the initial number you are given will lead to significantly different dynamics. For example, if we were given random numbers identical up to the tenth decimal point, that is they differ by less than 0.0000000001, then up to the tenth iteration of E10 the dynamics would be the same. However, after the tenth iteration, the dynamics would diverge.

The dynamical system E10 is a discrete dynamical system as it evolves in steps. Continuous dynamical systems evolve along infinitesimally small steps. A prominent example of a continuous dynamical system is the Lorentz system,

Despite the Lorentz system also exhibiting sensitive dependence on initial conditions, under certain parameter values it exhibits an attractive behaviour. Its dynamics are concentrated on a near pre-determined structure. Although the exact manner in which the system evolves over the structure is sensitively dependent on the initial conditions, the global dynamics of the system are largely predictable.

The above are examples of deterministic dynamical systems, as the dynamics of the systems is completely determined by the set of initial conditions. This is because the evolution functions are deterministic functions. If instead one used a random evolution function then one would obtain a random dynamical system. For example, a random dynamical system may involve taking the previous state value, performing a deterministic manipulation of the value and adding random noise to the output.

Dynamical systems are ubiquitously used to model and investigate real-world processes. Of course, this requires an understanding of the governing equations of the dynamics of the process you intend to model. This a strong requirement that cannot always be met in practice, hence, the development of random dynamical systems. The randomness component is intended to model the complexity of real-world processes for which we have no set of governing equations. It is a fundamental question of physics as to whether natural processes are inherently random, and thus cannot be modelled by any deterministic dynamical system. Although quantum mechanics is probabilistic, it is still not clear whether quantum mechanics is the governing theory of the universe. It may be the case that we have developed a random theory to compensate for our lack of fundamental knowledge of the universe.

In any case, the theory of random dynamical systems is plentiful and has facilitated the modelling of many complex systems. In particular, one can make the argument that human civilisation is a random dynamical system. In which case we ask, is the dynamics of human civilisation configured such that its dynamics are constrained to an attractive surface?

If the answer is affirmative then there is a case to be made that the course of our history is pre-determined. An affirmative answer does not get rid of our free will as the exact trajectory over the attracting surface is still susceptible to perturbation, as we saw with the Lorentz system. What it does mean is that events in history are inevitable, it is just a matter of time as to when our civilisation will cross that region of the attracting surface.

History

The Rest of History Podcast pondered whether if any mildly conservative patriotic German leader would have been led down the same path as Hitler in a pre-World War Two era. It questions as to whether the forces of the random dynamical system overcome the desire of its individual components, making certain events inevitable. Perhaps, the exact events of history are not determined, but the global events are. For example, the exact trigger of World War Two is not pre-determined, but there was always going to be an eventual trigger due to the political tensions and geopolitical climate of the time.

Throughout history, we have often seen the same narrative, where a small set of individuals slowly amass into a gathering projecting an ideology. Eventually gaining sufficient power to enforce change and impose order. At a critical mass, the individuals can no longer effectively control the perception and implementation of their ideology. Leading to a sudden collapse of power, instilling chaos similar to that from which the embers of the original ideology emerged. The exact events governing this cycle vary due to a host of factors, however, the general narrative of the cycle remains largely intact.

There is something eerily Markovian about this cycle. Over successive generations that past is slowly forgotten, the lessons learned from previous tragedies dissipate, and society is left to commit the same mistakes. Consider the present climate, where rhetoric surrounding nuclear war is murmured in political discourse. After the Cold War, such murmurings have been strongly muted for the public were still recovering from the close calls. However, a large proportion of the society that vividly remembers those calls has moved on, leaving a naïve youth who have not experienced the trauma that such murmurings inflicted.

Therefore, in the Network State Balaji Srinivasan consistently argues for the importance of studying history. Balaji visualises this historical cycle as a cyclical staircase. That is, despite progressing along one dimension, say the technological dimension, the large-scale events of history are recurring. By studying history we ensure that we do not forget the lessons learned from past events, mitigating opportunities for mistakes to be repeated. However, we must be aware that we are not returning to the same point in history, we have evolved along several dimensions. Just as a staircase looks like a circle when viewed along one dimension, we need to carefully view history to make explicit the dimensions along which we are evolving. Understanding this will help us predict how the recurring events of history will contextualise in the modern era.

Indeed we have, from a quantitative perspective, better managed this cycle. In Better Angels of Our Nature, Steven Pinker demonstrates that across practically all metrics, we are better off than any previous society. More specifically, when metrics are averaged globally across time they all exhibit the same declining pattern. Of course, there are still local hotspots where violence spikes and points of civil instability. However, the severity of the localised regions of instability is also declining. The perceived severity of these local points in history is all relative to the current global climate. That is not to say that the suffering experienced in these localised regions is not a devastating human tragedy, it is just that in the larger context of human civilisation such suffering is becoming more seldom. Before we have mentioned that the Markovian nature of history occurs on the orders of generations, Steven Pinker argues that our memory of historical events in our own lifetime can be shrouded by misconceptions.

Steven Pinker's analysis shows that in general terms society is progressing in a positive direction. This is evidence to the fact that major historical cycles are prevalent, as their repetition offers opportunities for us to learn and better deal with the challenges when they reemerge. Consequently, their effects are less detrimental to society, facilitating our progress toward a more prosperous existence. Therefore, it is important we take the advice of Balaji and study history to maintain the lessons learned by previous generations, it is only by doing this that we can continue to progress as a civilisations. It is my worry that we begin to forget the lessons of the past and start suffering on a scale that inhibits societal progress.

Before trying to understand whether free will exists, we should understand what we mean by free will. It cannot simply be the ability of an individual to perform any action they desire as they are limited by physical constraints. Indeed, searching on Google for 'what is free will' we see that free will is "the power of acting without constraint of necessity or fate; the ability to act at one's discretion".

However, this still holds some ambiguity as it is not clear whether the perception of having the ability to do something and not do it, is the same as possessing the ability to do something. For example, consider a time before Einstein discovered that nothing can travel faster than the speed of light. Now some individuals living in this time may have the desire to travel at a speed faster than the speed of light, but choose not to as they cannot build a vehicle to do so as they are busy doing other things. Therefore, can one say that this individual chose not to travel faster than the speed of light? From the perspective of the individual, they made this choice, however, from the perspective of physics there was no choice to be made as nothing can travel faster than the speed of light.

Despite this, we will, for discussion, suppose that free will is the ability to act on our desires which we will assume will be executable desires.

Determinism and Free Will

Determinism is the idea that the future state of the universe can, in principle, be computed from the current state of the universe using the laws of physics. That is, the laws of physics are not random but deterministic. Currently, our understanding of the quantum world is inherently probabilistic, and thus this may lead us to believe that the universe is not deterministic but random. However, our probabilistic understanding of the quantum world may just be an abstraction that we have made to make sense of the quantum world. It is entirely plausible that at a more fundamental level, we discover these quantum processes are governed by deterministic laws.

For example, consider this stream of data.

By Eouw0o83hf - Own work, CC BY-SA 4.0.

It may appear random at first, but once you understand that it was generated by the rule 90 cellular automaton you realise that it is a completely deterministic process.

Where determinism is the antithesis of randomness, we should understand that determinism is not the antithesis of free will. It is entirely plausible that we live in a deterministic universe and also possess free will. Randomness cannot be free will since it is governed by the law of probability. Random systems have long-term dynamics that can be predicted. For example, if one were to repeatedly flip a coin, then in the long run one would expect to obtain the same, or very similar, number of heads and tails. Therefore, even though the coin flip is supposedly random, we would not say that the coin chooses to be heads or tails. Hence, if the universe were driven by a random process, it does not follow that the things within that universe possess free will. Although, from the perspective of the things inside the universe it may seem that they have free will, where in reality their actions are dictated by a universal equivalent of flipping a coin.

The distinction between free will and determinism arises as they operate at different scales. Determinism refers to the laws of physics governing the fundamental units of the universe. Whereas free will is an emergent phenomenon at the level of organisms.

The Perception of Free Will

Despite having a desire to do something we may be inadvertently constrained by social pressures to not carry out these desires. This social pressure may be implicit and be developed by our time operating in social environments. However, the pressure may be explicit, we may forced not to do something as evolution has made us reluctant to be subject to the social ramifications. In which case, have we chosen not to do something or has society forced us not to do so? After all, if society was structured or functioned differently we may even be encouraged to execute our desires. Better Angels of Our Nature, by Steven Pinker, identifies many instances of this. For example, there was a time when many gathered at the Coliseum to witness deathly battles. An individual in this scenario may be inclined to clap and cheer at the death of a man. However, in modern society, such behaviour would be greatly condemned and thus the modern individual will not be inclined to celebrate the death of another man.

The Physics of Free Will

One can take a physics-based approach to free will to try and circumvent ambiguities relating to perception.

In a podcast with Lex Fridman, Lee Cronin utilises the fundamentality of time to explore the concept of free will. More specifically, Lee Cronin explains that to have free will time must be fundamental. That is, the process by which events unfold is a fundamental aspect of our universe. For if time were not fundamental, then it would be driven by an external force that would rid you of your free will. Lee Cronin then goes on to make an interesting argument for why time must be fundamental. However, note that this does not then imply that we have free will, as it is conceivable that we do not have free will and time is fundamental. Lee Cronin's argument for why time must be fundamental is the idea that the universe is not large enough to contain its future. That is, the structure of the universe at present cannot contain all the necessary information to describe the future states of the universe. Note how this is not the same as the universe being deterministic. Indeed, the universe can be deterministic and not be large enough to contain all the information regarding the future. This links to the ideas of Stephen Wolfram regarding computational irreducible, who explains that although some systems are driven by explicit rules, evolving the system to a future state necessarily requires computing all intermediary states.

Lee Cronin makes this idea explicit by referring to the law of excluded middle. The law of excluded middle states that every statement is true or false. For events that have occurred in the past, we can apply the law of excluded middle as we can say with certainty whether an event has occurred or has not occurred. However, we cannot apply the law to events in the future as there is uncertainty, from our perspective, as to whether they will occur. Therefore, from our perspective, the universe is too large to contain itself as we cannot apply the law of the excluded middle to events in the future.

Similarly, Sean Carrol argues that since we cannot predict the future, the universe is too complex to contain itself and therefore it must be driven forward through our decisions. Sean Carroll distinguishes varieties of free will. Libertarian free will posits that our decisions cannot be explained by the laws of physics. That is, free will is incompatible with a deterministic universe. On the other hand, there is compatibilist free will which understands that we are made of atoms that follow the laws of physics, however, we have an emergent capacity to make our own decisions. This resonates with the distinction between determinism and free will made previously. Indeed Sean Carroll is on the side of compatibilist free will and argues that since the universe is too large to contain itself, we will never have the capacity to determine the future using the laws of physics.

For me it is unclear whether the universe being too large to contain itself is an inherent property of the universe, or reflects our incapacity to ponder the universe and its entirety. Observe how we are consistently developing technology to access more intricate details of our universe. In 2023 the Nobel Prize in Physics was awarded to a group of scientists who developed methods to generate light pulses at the attosecond level. Such methods allow us to take measurements of the universe at time intervals of one-millionth of a millionth of a millionth of a second. If progress in these methods continues, then what is to say that we cannot eventually reach a point where we know all there is to know about the universe and thus predict its future? Answering such questions will not help answer the question of free will, it merely eliminates libertarian free will, whereas compatibilist free will remains plausible since compatibilist free will is an emergent property.

The Conscious and Unconscious Mind

The compatibilist view of free will distinguishes between the atomic and human scale. It understands that at the atomic scale, everything is driven by the, not necessarily deterministic, laws of physics and thus free will is not present, however, at the human scale free will emerges. Thus we are left questioning whether free will is a uniquely human phenomenon. Is there a Goldilocks scale at which free will can exist? Indeed, at the scale of planets and the cosmos, there does not appear to be any evidence of free will.

Here we explore this human-centric view of free will and determine whether investigations of the human mind indicate that there exists a phenomena akin to free will.

In an Andrew Huberman podcast, Dr Paul Conti emphasises that the human mind is structured like an iceberg. With the unconscious mind consuming most of the brain's functionality, and the conscious mind only accounting for a minor portion of the activity. This seems to suggest that the majority of our decision-making is driven by processes in the unconscious mind. However, from a personal perspective, it seems as though I am consciously making decisions all the time. Indeed, I accept some of my actions occur unconsciously, such as darting my hand away from a hot surface I accidentally touched. However, I feel as though I consciously chose to write this blog post about free will, rather than being unconsciously driven to do so. It may be the case that unconsciously I make decisions all the time, however, when my unconscious mind is undecided it appeals to the conscious made to make the final call.

The question now is whether this conscious decision-making experience is free will or just the perception of free will. Is my unconscious brain giving me these choices to make, or has it already decided on the action I am to take? On the one hand, it may just seem wasteful for the brain to expend energy on providing already-answered questions for conscious consideration. On the other hand, the unconscious brain may do this to elicit the conscious experience in the organism it inhabits. The conscious experience is beneficial to the organisms for it provides an added dimension to the sensory information it is perceiving. It helps the organism to arrive at a more complete picture of the world which can then help the decision-making process. Thus, the conscious experience may just be an evolutionary advantageous property of organisms that is ignited by the unconscious mind through constructing a perception of free will.

Is there any way that we can alter the unconscious mind to then implicitly drive our decision-making processes? If so, then it is plausible that we have free will, otherwise, our unconscious mind drives all our decision making and we are just passengers. Robert Sapolsky argues that we are all moulded by our experiences and the environment. Our actions are merely a collective consequence of these experiences, and thus we have no free will. He supports these claims with experimental evidence that shows our unconscious brain to engage before the conscious brain registers a stimulus. Therefore, our response to a stimulus is dictated by the reaction of the unconscious brain. However, just as compatibilist free will recognises there is a spatial scale at which free will emerges, is there also a temporal scale at which free will emerges? It may be the case that immediate reactions to stimuli are initially recognised in the unconscious brain, but what about reactions over longer time scales?

The Implications of Free Will

Why do we need to determine if we have free will or not? Well, it can help settle some important moral conundrums. For example, if we do not have free will then it seems immoral to punish someone for crimes they committed, as ultimately they did not choose to commit the crime. Instead, we should offer criminals an environment to change and rehabilitate their unconscious minds such that their reactions are not criminal.

In any case, I feel as if I have free will. Even if it is an illusion displayed by my unconscious mind or an emergent phenomenon from the physical universe, I still choose to live my life as if I have free will as it fills me with purpose and determination.

Generative machine learning is the application of machine learning to construct novel outputs from an input. For example, ChatGPT is a generative machine learning model as it generates a text-based response from user input.

Generative machine learning has gained increasing amounts of public interest since the release of ChatGPT and has been at the centre of public machine learning discourse. However, it is important to make clear that generative machine learning does not represent the field of machine learning in its entirety.

Current Progress

Just recently there have been major advances in the field of generative machine learning.

• OpenAI released demonstrations from their SORA model which is a text-to-video model leveraging advances in diffusion model architectures. On the surface, the model seems to demonstrate a remarkable improvement in video generation from previous models. More specifically, the model is more physically coherent and thus generates more realistic videos.

• Google released their Gemini 1.5 Pro model which offers improved context-length capabilities over their Gemini 1.0 models. In particular, the Gemini 1.5 Pro model can accurately digest over 700,000 words. Note that 700,000 words are about seven times the number of words in the average novel.

These models show the rapid improvement that is occurring in the field of generative machine learning. Innovations are being made in the training, architecture and implementation of the models. All of this has provided many benefits not just to the field of machine learning, but also to the sciences and society as a whole. Indeed, due to the generality of these models, they can applied in various domains.

However, there is the risk that too much focus is being placed on generative machine learning. Machine learning as a field is more than just these large generative models. Currently, it feels as if we are counting too much on the success of generative AI which, despite all of the attention it is receiving, still has major flaws that limit its capacity to be useful in practice.

Advantages

Due to the vast amounts of data that they are trained on, these generative models elicit seemingly novel responses. Thus, they can be used as a great source of inspiration and idea generation. Even just having an on-demand conversation with a human-like agent can help individuals probe their thoughts. The outputs of these models can provide a foundation from which individuals can work. This shouldn't be surprising as often these models are trained in precisely this manner. They are handed a stream of data and tasked with predicting the subsequent data in the stream.

Furthermore, these models have access to a vast amount of information. Indeed the corpus of data they are trained on includes a wide range of material. Although the models still need to be able to effectively retrieve this information, this breadth of knowledge means that they are applicable in many domains. This is contrary to previous machine learning practices that were specialised to a specific application. On the one hand, the specialisation improves the efficiency of developing the machine learning models, however, too much focus on a specific task may make the model fragile. One could argue that increasing generality in machine learning models may augment a model’s ability to perform a specialised task.

Consequently, these large generative models can effectively act as search engines. Perplexity AI is working on improving the information retrieval mechanisms of these models such that they can efficiently be used as a search engine. Unlike search engines, these models can receive as input larger and more intricate prompts. Making the searching experience more personalised and efficient, as the user can effectively navigate using natural language.

Limitations

Due to the commercial implications of this technology, many organisations training these large models shroud their data sources and architectures in secrecy. This can impact the equitable distribution of the technology, the alignment with the desires of society and the accountability for negative impacts.

For example, the just-released Gemini 1.5 model has been demonstrated to possess strong biases in an attempt to be politically correct and satisfy supposed social standards. This has received pushback from the community as it is clear that Google overshot the current social standards. It is thought that the alignment of the Gemini model was spearheaded by a few who thought they were acting in the best interests of society. It is concerning that despite Google being a large corporation, a model with heavily polarised views was deployed.

Furthermore, it is unclear whether the data being used to train the models has been obtained legitimately. Through reverse engineering, it has been identified in some cases that copyrighted material has been used to train these models. This has raised legal and ethical questions regarding whether human artists should be attributed for their work that is being used to augment the outputs of these generative models.

Current generative machine learning systems have static learned representations. They cannot update their beliefs or understanding in the presence of new information. Once these models have been trained, contradictory representations are difficult to fix. Although methods in mechanistic interpretability can tune certain parameters in small models to remedy wrong representations, it is unclear whether such techniques will be successful in these incredibly large models that may potentially have on the order of a trillion parameters. Moreover, it is unclear how these methods would affect other representations of the model. They may introduce more inconsistencies than they fix. This static representation of the world is not ideal, as it means that the model will always be out-of-date. The world is constantly changing, and so for a generative model to be effective in the real world it must adapt, just as humans update their actions and beliefs based on their experiences.

A lot of the attention surrounding generative models, such as large language models including ChatGPT is centred on statistics that demonstrate its performance is better than humans at some tasks. To arrive at these statistics the developing companies test their models on benchmarks, which are a series of tests designed to identify a particular characteristic of these models. However, many of these benchmarks have flaws. For example, AI Explained has investigated the MMLU benchmark and found that many of the tests within this benchmark are wrong. More specifically, the benchmark is designed to test the knowledge of these models and thus comprises many multiple-choice questions. It was found that many of the questions were ambiguous, gave the wrong solution, or the correct solution wasn’t even an option. Despite these flaws, it should be apparent that testing the knowledge of a model through a series of multiple-choice is not optimal in determining whether the model has an astute knowledge of a subject. Indeed it may work to test the model as an encyclopedia for a subject, but it will not work to determine whether the model could act as an expert for the subject. Furthermore, as information regarding these benchmarks is scattered across the internet, it is entirely plausible that they are contained within the large data pools used to train these generative models. Therefore, using the benchmark to test the trained language model is similar to a student taking a test with the solutions on their desk.

Generative models, such as large language models, are trained to detect statistical patterns in data, and then use these patterns to generate outputs. Consequently, it is difficult to instil a notion of certainty into these models. For example, these models often struggle to deal with simple mathematical expressions. Furthermore, the performance of generative models is going to be inherently biased toward patterns in the data for which statistical correlations are strongest. In other words, if a model is trained on real-world data, it is going to perform best on concepts that are popular in the real world as these concepts are present in more of the data. This is the main reason why models work so well in English, but struggle to perform in less common languages.

For instance, a large language model may know lots of information regarding a movie actor, including the names of their family members. However, when the model is queried on the family members directly they demonstrate very little knowledge. Indeed, the model may not recall that the family member is related to the famous movie actor. From a human perspective, we know with certainty that the movie actor is the son of their mother, however, a large language model lacks this sort of reasoning as its reasoning is derived statistically.

Future Directions of Progress

To try and eliminate some of the limitations outlined above I think it will be essential for these models to leverage multi-modal datasets to develop coherent representations of the world. However, to do so will require the development of bespoke techniques to handle these different formats of information. For instance, convolutional neural networks are optimised for image processing and graph neural networks are optimised for processing graphs. Recently, there have been numerous AI hubs that are intended to work on this very problem.

However, it is also important that in conjunction with this, the quality of data ought to be improved. As the common saying goes, garbage in garbage out. That is, using sub-optimal data will lead to sub-optimal representations in the model. Just as a student struggles to learn from poor educational resources, we cannot expect a model to achieve high-level performance if the data it is supplied with is poor. This also raises the question as to whether a model can achieve higher than human-level performance without synthetic data. Data curated by humans is inherently limited by the abilities of humans. For models to surpass human performance it is essential that they can beyond human data and bootstrap on synthetically generated data.

Another potentially fruitful direction of progress may be to optimise the strategies by which models are made to interact with each other. Just as humans form companies and societies to promote development and innovation. Models should be organised collectively to distribute tasks and share resources. There have been recent advances in the architectural designs of these large generative models that are principled on the idea that a collection of smaller models can effectively collaborate to outperform a single larger model.

Applications

The critical feature of this technology compared to other technologies is its capacity to be distributed. ChatGPT has over 100 million users and is widely available to all demographics, facilitating the spread of information and providing many with sources of intelligence.

Consequently, this technology can evolve and improve on rapid timescales. More specifically, they can be adapted, or fine-tuned, for specific applications such as health care. For example, these models can be adapted to offer general advice and treatment for common illnesses and refer individuals to specialists when required. Helping to alleviate the burden on health services and allow healthcare professionals to focus their attention on those requiring critical care. Moreover, as these systems have a much larger memory capacity than any individual, they can cross reference symptoms across a large database and personalise their services by utilising the medical history, and potentially even genetic data of the patient.

Similarly, these models can be finetuned to act as personal tutors for school children across the globe. Due to the generality of these models, they can operate in most languages and subjects as well as adapt their output to the proficiency of the student. The replicability of these systems means it is possible to individualised support that one teacher in a class of thirty cannot provide. In turn, this increases the accessibility of education as these models can be deployed in an offline fashion and thus access remote corners of the world where the environmental and economic situations are tough.

In the near term, I do not expect these large generative models will replace humans, they are simply not reliable or accurate enough. However, they will augment human roles and improve the efficiency with which humans can carry out their tasks.

Moreover, I do not believe that these systems could be used as an instrument by which to methodically investigate human psychology. They could be used to analyse society as a whole since they are trained to compress human knowledge and thus will inevitably extract societal-level patterns. However, that is not to say that the process by which they generate their outputs is driven by the same mechanisms driving individual human psychology.

Ultimate Goal

Ideally, there will be a time when we each have personalised AI systems that communicate with other personal AI systems. Leading to an ecosystem of cooperative AI systems tailored to individual needs that augment the desires of individuals. An individual should have autonomy over their AI assistants and it should be largely decentralised from any top-down control. Such personalisation of technology has happened before, and I think it is a matter of time before the same happens with AI. For example, telephone boxes shrunk into the pockets of individuals and the printing press facilitated the decentralised sources of knowledge. I think to realise the full potential of AI systems should be implemented at finer scales.

AI Risks

From the current limitations of generative models, I do not think there is a risk of generative models taking over and controlling the world by setting their own goals and making plans to achieve those goals. What’s more, I do not think that such detrimental capabilities will emerge by scaling up current approaches. The greater threat is from humans anthropomorphising the outputs of the models, and being manipulated by their outputs. Or, equivalently using the models to manipulate others.

Having an ecosystem of personalised AI agents that learn in an online fashion will reduce this risk as misaligned traits will be suppressed by the majority, just as the human species has been successful in suppressing the motivations of malevolent individuals.

I am optimistic that the advances in machine learning, as a whole, will be hugely beneficial for society. However, the current limitations of generative models highlight that focus should also be maintained on other machine-learning techniques and we should rest too much weight on the idea that generative models can realise the full potential of AI.

Games

Collaboration and interactions are an inherent part of society. Our actions and the actions of others shape our everyday lives. Whether we are aware of it or not, the actions of others have significant effects on our actions. Indeed, the actions of others can make us act rather irrationally.

We can investigate the dynamics of interactions and determine the optimal means by which to interact by formalising interactions as games. An optimal strategy for a game will be one that either benefits an individual over an adversary or mutually benefits all participants equally. As with all attempts at modelling real-world systems, there are limitations. More specifically, a provably optimal strategy for a game may not extend to an optimal strategy for the game re-contextualised in the real world. This is because human psychology is demonstrably selfish, and thus globally equal rewards may not be viewed as such from the perspective of an individual. For example, the victim of an altercation between individuals often exacerbates the harm inflicted upon them by the perpetrator. Whereas the perpetrator is likely to downplay the harm they caused and offer reasonings for their actions. Indeed, an experiment was conducted where individuals shocked each other to the extent that the other person shocked them. Unsurprisingly, the participants shocked the other with increasingly greater power, eventually far exceeding the strength of the initial shock.

Zero-Sum Games

A zero-sum game is one where the gain of a group of participants is at the expense of the others. More specifically, the gain of the group is equal to the losses of the other. In this sense, the amount of reward in the game stays constant. For example, tic-tac-toe is a zero-sum game. A player wins tic-tac-toe only if the other player loses. Similarly, chess is a zero-sum game. However, the stock market is not a zero-sum game. In this game, a trader is trying to increase their capital by investing in prosperous stocks. As stocks can increase without other stocks decreasing by the same amount the game is not zero-sum.

Zero-sum games are relatively easy to investigate as often the factors influencing the reward of the game are internal, which means that they can be tracked. In tic-tac-toe and chess, the factors affecting the reward of each player are the available moves on the board, which can be tracked. However, in the stock market, many external factors can influence the price of a stock, which makes it inherently harder to analyse. Moreover, these external factors mean that the game is not zero-sum as rewards can filter into the game.

Positive-Sum Games

A positive sum game is where the participants can potentially gain a reward without the other participants being negatively affected. Indeed, the stock market is an example of a positive sum game. Positive sum games are more ubiquitous in the real world since many external factors can contribute to one’s perception of reward. Sometimes it is the case that a zero-sum game contextualised into the real world becomes a positive-sum game. For example, suppose you are playing tic-tac-toe against a child who is still trying to grasp the strategies of the game. Despite being able to win against the child you may be incentivized to let them win. The pleasure they'll experience by beating you will surpass the pleasure you'll experience if you win. Moreover, their display of pleasure at beating you may cause you to feel some pleasure. Hence, this game of tic-tac-toe is no longer a zero-sum game. Indeed, human psychology has introduced an external reward system that breaks the conservation of reward present in tic-tac-toe.

The Prisoner's Dilemma

The prisoner’s dilemma is a positive sum game and is set up as follows. Two prisoners are held in custody separately and interrogated by the detaining officers. If one prisoner testifies against the other, whilst the other remains loyal then the defecting prisoner is set free whilst the other is imprisoned for 10 years. If both prisoners remain loyal to the other, then they are each sentenced to 6 months. If both prisons testify against each other then they each receive a sentence of 6 years.

The reason the prisoner’s dilemma is a positive-sum game is that if both remain loyal to each other then they both receive a reduced sentence, that is they are rewarded.

The subtly of the prisoner dilemma is that despite both prisoners being able to reap rewards if they remain loyal, it turns out the prisoners will defect. From the point of view of an individual prisoner, it is better to defect rather than cooperate. If they defect then they will go free if the other remains loyal, and they will only be sentenced to 6 years, rather than 10 years, if the other also defects.

The Iterated Prisoner's Dilemma

The iterated prisoner's dilemma is an augmentation of the prisoner's dilemma where each prisoner interacts repeatedly to accumulate rewards. This variation of the game is more resemblant to interactions between individuals in the real world, and can thus be used as a framework to study human interaction.

Tit-for-Tat

Tit for tat is a general strategy for confronting iterated prisoner dilemma-style situations. The tit-for-tat strategy says that on the first move, you should always cooperate with the other player, and then after that only cooperate if they cooperate otherwise you should defect. The intuition behind this strategy is to set up a friendly correspondence with the other player. With your first move of cooperation, they are less incentivised to defect based on their individualistic perspective. However, the generosity of the tit-for-tat strategy runs out after the first move. After the first move, the other individual must earn your cooperation. Thus, the tit-for-tat strategy avoids one’s kindness being taken for granted.

Moreover, the pattern of the tit-for-tat strategy is easy to identify. Thus, the opposing player can quickly understand that their generosity will be reciprocated and thus are incentivised to be generous.

Generous Tit-for-Tat

The tit-for-tat strategy has empirically been shown in simulations to be a relatively effective strategy for confronting these types of games. However, there is a flaw in the tit-for-tat strategy that arises when it is applied in the real world. More specifically, if two players are employing the tit-for-tat strategy then once one player defects both players are caught in a cycle of defecting. This is not ideal, as in the real world there are many factors which may cause a player to defect without them necessarily having the spite to do so.

The generous tit-for-tat strategy aims to rectify this issue. With the generous tit-for-tat strategy, the player sometimes gives the other player the benefit of the doubt and cooperates when the other has previously defected. This helps to break the cycle of defecting and provides an opportunity for the players to set up a collaborative relationship.

Contrite Tit-for-Tat

There is also a flaw to the generous tit-for-tat strategy if there are players who will always defect. If a player always defects, then a generous tit-for-tat player will frequently give that player the benefit of the doubt and not be reciprocated for their generosity. Unfortunately, in the real world, overly generous players can be exploited, however, the majority of the time your generosity will be acknowledged and rewarded, even if not immediately.

The contrite tit-for-tat strategy is more selective in where it allocates generosity. More specifically, a contrite tit-for-tat player will remember their decisions, and only be cooperative with other players if the mutual defection of a previous interaction was caused by an error or misunderstanding on their behalf. However, if the mutual defection was caused by the other player, then a contrite tit-for-tat player will forever defect against that player.

Consequently, interacting players who are both employing a tit-for-tat strategy can reconcile any previous differences and settle into a collaborative relationship.

Is the World Zero-Sum?

I think a lot of the contention in the world is linked to the puzzle as to whether the world as a system is zero-sum or positive-sum. At a physical level, there are conservation laws, such as the conservation of energy, momentum and so forth. Similarly, in the mathematics of optimisation, there are theorems colloquially known as no free lunch theorems, which indicate that over a uniform space of possibilities, no optimisation algorithm outperforms random guessing [2]. From this, it is common to use the phrase no free lunch colloquially to indicate that any gain is counterbalanced by a drawback.

Hence, at a societal level, there has been a consensus that has leaned toward the world being zero-sum. However, these no free lunch theorems apply to the rational worlds of mathematics and physics. We have seen that when games are contextualised into the real world they can potentially lose their conservative properties.

Those who think of the world as zero-sum may hold the view that billionaires have only amassed their wealth by capitalising on the detriment of others, rather than accumulating their wealth in a positive sum manner. Similarly, the debate around artificial intelligence systems' dangers can be grounded in this conundrum. On the one hand, those cautious of powerful AI systems think that the rise of these systems will result in the demise of humans. On the other hand, those accelerating AI capabilities research are focused on the benefits of AI systems which they believe will outweigh the risks. It is certainly the case that the world is not entirely zero-sum as otherwise there would be no room for progress and we would still be stuck in the stone ages. However, it is also the case that many gather rewards by exploiting others for their rewards.

In the case of the prisoner's dilemma, we saw that there was an opportunity for the prisoners to capitalise on the positive-sum nature of the game, however, their incentives motivated them otherwise. In the setting of the prisoner's dilemma, this individualistic strategy is not optimal, however, for games set in the real world it may be the case that not always taking the positive sum reward is a good thing. As in the real world, positive rewards are inherently subjective. One person's positive reward may not be so positive for another. Moreover, human psychology can be contradictory, and optimise for short-term gratification, therefore seeking to always obtain the positive reward may not be all it seems to be.

For example, reinforcement learning is a process that trains a machine learning model to complete a certain task by rewarding behaviour that resembles the task, and discouraging behaviour that deviates from the task. To quantify this the practitioner constructs a reward function that takes in the action of the model and gives it a reward based on how good its action was. In simple settings, where there are few external factors, this may work fine as a reward function may be constructed that precisely details the desired task. However, in a more complex environment, determining a useful reward function becomes difficult as there are multiple external factors to consider.

With the reward function set, a machine learning model relentlessly optimises their behaviour to seek positive rewards. However, as we noted the reward function may not be able to fully capture complex tasks, thus over-optimising the reward function may lead the machine learning to exhibit behaviour that deviates from the intended behaviour. In some cases, these deviations may be significant and cause inhibit the model being used for the desired tasks. For example, in a paper by DeepMind researchers tried to teach a virtual robot to pick up a ball by rewarding actions that progressively got it closer to picking up the ball. After some time the robot seemed to be learning the action of how to pick up the ball. However, it turned out the robot was only making the correct gestures in the space in front of the ball. Indeed, from the perspective of the reward system, it seemed as though the robot was completing the intended task, however, in reality, it was not [3].

The real world has zero-sum and positive-sum aspects. One must strive for rewards to foster the benefits that innovations bring. However, one must operate with a sense of caution so that potential consequences do not suffocate the benefits. Moreover, one must maintain an awareness to ensure that the rewards they are striving for are indeed positive, and not a proxy reward signal.

Bibliography

[1] The Better Angels of Our Nature - Steven Pinker

[2] Wikipedia contributors. (2024, January 15). No free lunch theorem. In Wikipedia, The Free Encyclopedia. Retrieved 05:44, February 17, 2024, from https://en.wikipedia.org/w/index.php?title=No_free_lunch_theorem&oldid=1195771929

[3] https://deepmind.google/discover/blog/specification-gaming-the-flip-side-of-ai-ingenuity/

• For our purposes, a space can just be thought of as a collection of points. For example, a line is just a collection of adjacent points, a circle is a collection of points that are at a certain distance from a specified point. It may be useful to think of a space as a lump of Play-Doh, with a particular moulding and separation of the Play-Doh giving a particular space.

• With the Play-Doh analogy, it becomes clearer what we mean by the shape of a space. The Play-Doh may be moulded into a spherical shape, or elongated into a thin strip. Moreover, it may be broken apart into smaller balls, which collectively still make up a space, which can then also be squashed together. Algebraic topology says that spaces have the same shape when one space can be moulded into the other without breaking or glueing the space.

A classical example of spaces with the same shape is the doughnut and the coffee mug. Indeed, given a piece of Play-Doh in the form of a doughnut one can mould it into a coffee mug. From the everyday usage of the term shape, it may seem wrong to say that a doughnut and a coffee mug have the same shape. However, taking a global perspective we see that both spaces are just objects with a hole in them. Algebraic topology takes this more global perspective so that it can formalise the concept of shape to more complex spaces without worrying about minor details. In particular, it makes it easier to reason about the shape of spaces that do not necessarily exist in three dimensions.

Ok, so algebraic topology says that a doughnut has the same shape as a mug, but a different shape to a sphere. How is this useful? The benefit is to facilitate the formalisation of shape to enable the exploration of higher dimensional spaces where our Play-Doh analogy starts to break down.

But why do we need to know the shape of higher dimensional spaces? Suppose you are analysing a set of data points with multiple attributes. Then this point cloud of points lives in a high dimensional space. Namely, there is a dimension for each attribute of the data points. Many data analysis tasks are intrinsically connected to understanding the shape of the collection of data points. For example, clusters of data points in the high dimensional space may correspond to a subset of data points with a particular property, and thus understanding the shape of the data corresponds to performing a classification task.

Note how in the case of clusters of data points, we are not interested in the exact ordering or location of the individual points. We are just concerned with the global shape of all the points. Therefore, the algebraic topology perspective on shape seems more reasonable. Indeed, if the data points were measured differently, the exact ordering of the points may change but the global structures will remain present. Moreover, using the colloquial interpretation of shape would require the tracking of the locations of each of the individual points in the data set, which could be vast.

Although mathematical endeavours need not be motivated by practical applications, indeed the pleasure of exploring abstract mathematics should be the main source of motivation, it is clear that algebraic topology, in conjunction with being an incredibly elegant subject also has practical applications.

Loops

To extend our Play-Doh analogy to higher-dimensional and more complex spaces we use loops. More specifically, suppose you have a two-dimensional disc, such as a frisbee, and imagine placing a loop on this space with a piece of string. Without tearing the string you can deform and shrink the loop. In particular, you could contract your loop, by shortening the string, down to a small point. Doing such transformations does not change the shape of your loop.

Now take your disc and puncture a hole in it. Again suppose you have a piece of string and make a loop on your punctured disc. In this case, if your loop wrapped around the hole then you could still deform and shrink the string, however, there will be a limit to how much you can shrink the string. In particular, eventually, your loop will wrap around the hole and you will no longer be able to shrink the string without tearing it or leaving the disc. However, if your loop does not wrap around the hole, then as before you can shrink it to a point.

Therefore, we see that the disc with no hole only has only one type of loop, namely the ones that can be shrunk to a point. Whereas, the disc with a hole has two types of loop, namely the ones that can be shrunk to a point and ones that eventually wrap around the hole. We conclude that the shapes of the disc and the punctured disc are different. Indeed this is consistent with our analogy that a Play-Doh disc cannot be moulded into a Play-Doh disc with a hole without tearing the hole into the playdoh.

Algebraic topology formalises this usage of loops as a way to investigate the shape of more complicated spaces.

Loops on a Circle

Let us explore in more detail the types of loops on a punctured disc. For simplicity, we can just assume that our space is a circle of points, as one can mould a punctured disc so that it is just a circle of points. So now we are thinking of loops as wrapping around a circle. Indeed a loop must start at a point and go all the way around the circle and finish at the same point.

In particular, we make the following observation, if a loop goes around the circle twice clockwise, then It cannot be deformed into a loop that goes around the circle once clockwise without tearing it. Therefore, these loops must be different in the context of algebraic topology. Furthermore, if I have a loop that goes twice clockwise and then once anti-clockwise, this is the same as the loop that goes just once clockwise.

This suggests that we can just identify loops with a number n. If n is positive then this says the loops go around the circle n-times clockwise. If n is negative, then the loop goes around the circle n-times anti-clockwise. Moreover, we can think about adding loops together. For example, a $n$ loop and a $m$ loop combine to give a $n+m$ loop. Indeed, if have the 2 loop and adjoin the -3 loop, I get a loop that first goes two times clockwise and then three times anti-clockwise, namely the -1 loop. Therefore, the set of types of loops on a circle can be associated with the integers.

Glueing Spaces Together

Ok, so algebraic topology provides us with a tool to investigate more complex spaces, that potentially live in dimensions higher than we can visualise. However, as we cannot visualise these spaces, how are we able to define them?

Well as algebraic topology differentiates spaces by whether or not they can moulded into one another without tearing or gluing, it makes sense to construct spaces by tearing and gluing existing spaces. Indeed it is common to take relatively simple spaces, that we can investigate easily, and combine them to form more complex spaces.

For example, imagine you have a strip of paper, where one set of opposite edges is longer than the other set of opposite edges. The shape of this piece of paper is the same as that of the disc as one can imagine moulding the strip into a disc. Take this strip of paper, add a half turn to one of the short edges and then glue the short ends together.

What you are left with is called a Mobius strip, and it has the peculiar property of having only one side. Indeed, trace your finger around the strip and you'll notice that you traverse the entire strip and return to where you started.

One can go a step further and imagine glueing two Mobius strips together along this edge. Doing so gives rise to a four-dimensional space known as the Klein bottle.

In turns out that many spaces can be constructed in this way. For example, a circle is just a line that has its edges glued together. The surface of a three-dimensional sphere, like a blown-up balloon, can be thought of as a two-dimensional disk whose boundary is attached to a single point.

A doughnut can be considered taking a strip and glueing both sets of opposite edges together.

Algebraic topology formalises this process of glueing and therefore facilitates the study of high-dimensional complex spaces by extrapolating the investigations done on simpler spaces.

Understanding Neural Networks Using Topology

As previously mentioned, a set of data points can be thought of as having a shape. More specifically, the data points can be thought of as a set of samples from an underlying manifold of a high dimensional space. The shape of the manifold is likely, and so the role of the neural network is to simplify the shape of the manifold to facilitate inference [1]. With this perspective [1] attempts to demystify the black-box nature of neural networks by tracking the shape of the data throughout the network. Roughly speaking [1] looks at the different dimensional holes present in the input data as it progresses through the neural network.

Tracking the holes present in a set of data as it progresses through a network provides a quantitative insight into the inner workings of a neural network, moreover, it provides a metric to establish a neural network’s capacity to deal with data. Suppose a network is being trained for a binary classification task. If the input data has many holes of different dimensions, the data points relating to the two classes are likely intertwined in a complex way. Therefore, to perform well at the classification task, the network is incentivised to disentangle the data and eliminate these holes. Hence, as data is propagated through a well-trained network one should observe the reduction in holes present in the data.

With this insight, [1] can elucidate the underlying processes of a neural network, and understand the correlation between the architectural properties of the network to its capacity to perform a certain task. The conclusions made by [1] through these investigations are the following.

1. Smooth activation functions, such as the hyperbolic tangent function, have a lower capacity for simplifying the shape of input data compared to nonsmooth activation functions such as ReLU.

2. The majority of the shape simplification a neural network performs occurs in deeper layers. Indeed, as the layers of a neural network increase, the capacity of a neural network to disentangle input data increases.

Bibliography

[1] Naitzat, Gregory, Andrey Zhitnikov, and Lek-Heng Lim. ‘Topology of Deep Neural Networks’. arXiv, 13 April 2020. https://doi.org/10.48550/arXiv.2004.06093.