Neural Networks:
Powerful Mathematics, Not Artificial Intelligence
The Mathematics Behind the Hype
Neural networks are often presented as a leap toward "artificial intelligence" but in reality, they solve a specific kind of mathematical problem, and one that is not easily addressed by classical methods.
At their core, neural networks generate nonlinear partial differential equations to describe a wide range of processes. These equations are notoriously difficult to solve exactly. During the Second World War, some of the greatest mathematical minds alive could spend months working out a single detailed solution.
The sheer urgency of wartime research led to the adoption of numerical methods, ways to generate approximate answers quickly. One such approach, the Monte Carlo method, focused on finding the most likely outcome through repeated random sampling.
This work laid the foundations for an entirely new branch of mathematics: stochastic calculus, the probabilistic counterpart to the differential calculus pioneered by Newton and others. Brought together as a formal theory by mathematician Kiyosi Itô in the immediate post-war years, stochastic calculus made it possible to describe complex events as the evolution of probabilities rather than precise deterministic outcomes.
Neural Networks as Probabilistic Tools
Today’s neural networks, whether powering a language model or an image generator, are direct descendants of these probabilistic approaches. Modern architectures such as ChatGPT have their roots in language translation.
The principle is straightforward:
For any given English sentence, there is a most probable French equivalent.
For any given question, there is a most probable answer.
By analysing vast amounts of paired input–output examples (whether sentences or question–answer pairs), neural networks learn to generate statistically likely outputs.
This is a remarkable feat of pattern recognition, but it is not intelligence. It is the application of probability to structured data, built on decades-old mathematical techniques.
Beyond the "AI" Label
The current wave of AI enthusiasm, often led by what might be called the "Tech Bro" narrative, rests on a fundamental misconception: that neural networks think. They don’t.
They are powerful, sometimes clunky, and often energy-intensive tools for solving certain mathematical problems. These tools have utility, but their capabilities are too often exaggerated into claims about emergent intelligence.
The mathematics required to model human-like reasoning or creativity, even crudely, does not currently exist. Artificial General Intelligence (AGI) remains a distant prospect, and it’s far from certain whether it will ever be achieved.
Why This Matters
Understanding what neural networks are really doing is not an exercise in pedantry, it’s essential for making sound decisions about where, when, and how to use them.
For SMEs, cultural organisations, and the public sector, this means:
• Recognising the limits of current AI tools
• Avoiding over-reliance on systems that cannot reason or create
• Investing in complementary human expertise where true problem-solving is required
Neural networks are a significant advance in applied mathematics. They deserve recognition for what they are, not for what marketing language claims them to be.
