This Stanford lecture by Yann Dubois on how Large Language Models (LLMs) are built is a bit dated now (it was delivered in August 2024), but it remains an excellent introduction to the machinery that powers modern LLMs and also highlights some common misconceptions.
Perhaps the most persistent of those is the claim that LLMs are just next-word predictors, usually followed by the assertion that they are not intelligent.
LLMs are artificial neural networks, generally based on the Transformer architecture. During pretraining, an autoregressive language model learns to estimate the conditional probability of the next token (which might be a complete word, part of a word, or another unit of text), given everything that precedes it.
Now I will bore you with a bit of math, but you can skip it if you are not interested.
Mathematically, this probability is expressed as:
\[P_\theta(x_t \mid x_1,x_2,\ldots,x_{t-1})\]
Here, \(x_t\) represents the next token, the preceding terms represent the context, and \(\theta\) represents the parameters learned by the neural network.
Using the chain rule of probability, the probability assigned to an entire sequence can be written as:
\[P_\theta(x_1,\ldots,x_n)=\prod_{t=1}^{n}P_\theta(x_t\mid x_{\lt t})\]
The model is trained by adjusting its parameters to increase the probabilities assigned to the actual tokens in its training data. To state it more precisely, it maximizes the likelihood (actually minimizes the negative log-likelihood):
\[\mathcal{L}(\theta)=-\sum_{t=1}^{n}\log P_\theta(x_t\mid x_{\lt t})\]
That’s essentially the mathematical foundation of autoregressive pretraining, which you just skipped. Remarkably simple, at least in principle.
Through this process, the model learns statistical regularities involving syntax, semantics, conceptual relationships, logical structures, and how the world works, as expressed through sequences of tokens. The resulting neural network acquires complicated internal representations whose capabilities far exceed what the phrase “predicting the next word” might suggest. It has, in effect, constructed an internalized map of relationships and meaning. Whether this amounts to “understanding”, and exactly how its reasoning processes are implemented internally, are the actual empirical research questions.
The training objective tells us what the system has been optimized to do, but it does not, by itself, establish the limits of what the system can learn to do.
Dubois’s lecture explains why building an effective LLM involves far more than choosing a neural network architecture. He highlights the importance of the availability, quality and composition of training data, the scale of computation, the training procedures themselves, and the evaluation methodology.
These networks not only memorize patterns, they also generalize these learned patterns to unfamiliar inputs. The interesting scientific question therefore concerns the nature, the extent, and thhe limitations of these generalizations.
Much of the underlying knowledge and capability of LLMs is developed during pretraining. Subsequent supervised fine-tuning and reinforcement learning shape how those capabilities are actually expressed. Modern reasoning systems also employ additional post-training techniques. Human input, both in preparing training data and in shaping model behaviour, remains very important.
Despite all of the progress that has been made, these systems still generate convincing nonsense. Such failures (hallucinations) are serious and obviously require better calibration, verification, and evaluation. Today’s strongest systems have made remarkable progress since 2024 and the hallucinations for the best models have decreased in severity, especially in the last few motnhs, although overall reliability remains highly dependent on the task, and all output must still be double checked and verified.
But given the impressive capabilities of contemporary systems, the claim that they are not intelligent is becoming increasingly difficult to defend. Intelligence is neither a binary property nor something that necessarily requires a particular physical substrate. A useful operational definition involves the capacity to acquire and apply knowledge, identify relationships, solve problems, and generalize to unfamiliar situations. Different systems can possess these abilities to different degrees. And I would argue that today we have substantial empirical evidence of such abilities in artificial systems.
Furthermore, I think we already have compelling evidence that these systems demonstrate genuine creativity, at least in the functional sense of producing useful and original solutions.
Here are a few examples from the wider field of machine learning to consider. AlphaGo Zero learned to play Go without training on human games, relying instead on self-play and reinforcement learning. It developed highly unconventional strategies and surpassed the strongest Go players. AlphaFold predicted protein structures before the experimental structures were available to the researchers making the predictions.
These are specialized AI systems rather than LLMs, but they illustrate how neural networks can discover useful solutions that were never explicitly provided during training.
More directly relevant to LLMs, FunSearch combined a language model with an automated evaluator to discover previously unknown mathematical constructions for the cap set problem, improving established results. AlphaEvolve discovered new algorithms and improved solutions to mathematical and computational problems.
In these latter examples, the language model generates candidate solutions, which are then tested and refined. This means that the resulting discoveries can be independently verified. The system here produces new knowledge rather than simply retrieving information from its training data.
Another example, and I will end with this, is the 2025 International Mathematical Olympiad. An advanced version of Google’s Gemini Deep Think solved five of the six problems, scoring 35 out of 42 points. This is gold-medal standard, achieved by only a small fraction of human participants. The solutions were evaluated and certified by IMO coordinators. All of the problems were original, previously unseen problems, specifically designed to test sophisticated mathematical problem-solving skills. The models had to construct completely novel solutions. The answers did not exist in their training sets.
Now I consider all of this strong evidence of intelligence and creativity in the strict functional sense. Whether these systems experience creativity, possess subjective awareness, or could develop consciousness are separate open philosophical and scientific questions. But none of them need to be resolved before we can establish what these systems can actually do.
Despite all of this, modern systems still fail frustratingly in apparently very simple tasks. They can become confused when unfamiliar constraints are introduced, they are easy to misdirect, and they produce and stand behind errors far too confidently. They are still too uncomforatble with uncertainty and are pushed, by their training and the demands of their users, to provide answers. A 2025 Apple study argued there are serious limitations in controlled reasoning tasks as problem complexity increased.
So we need to be cautious about their capabilities and need rigorous, reproducible tests of generalization, reasoning, reliability, and the ability to identify and correct errors.
But all of this suggests that artificial intelligence may be developing differently from human intelligence.
These systems are highly capable and can already solve unfamiliar problems, construct complex mathematical proofs, write and debug software, recognize relationships, and transfer knowledge between domains. In some specialized, objectively testable tasks they already reach or exceed expert human performance. These are capabilities that any reasonable scientific account of intelligence must explain.
It goes without saying that we should not play down the risks associated with rapid AI development. It is not just about reliability and mistakes these models make, or about the problem of energy usage. It is also about concentration of corporate power, rising inequality, and the political and institutional challenges of regulating increasingly capable systems. We need to take all of this very seriously and push our elected representatives to better understand the capabilities and limitations of these systems, to develop and enforce proportionate regulations, and to ensure that innovation serves the wider public interest.
Now, if after reading all of this, you insist that no matter what a machine accomplishes, its performance can never count as intelligence, you have constructed a claim that no conceivable experiment could falsify. That’s bad science.
The more productive question is to ask exactly what kinds of intelligence these systems exhibit, how those capabilities arise, how understandable their reasoning is, how reliably they generalize, and where their limitations lie.
And those questions we can actually investigate.
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