Professor Peter Greinrod.
Artificial intelligence (AI) is already changing the way we see the world and the way the world sees us. But so far, most AI systems fall into a few well-known categories. Some analytical engines operate in data-rich environments. Examples include pattern and object recognition, supervised classification, anomaly detection, control systems, and prediction across images, videos, sensor streams, and high-throughput machines. Others are generative, such as large-scale language models, media synthesis tools, and conversational agents.
Recent years have seen the rise of “agent” AI, systems that can coordinate many components, tools, and subprocesses in pursuit of a set goal. AI can now perform many high-frequency, “gruntal” tasks, increasing bandwidth and allowing users to spend more time on “what really matters.”
Although these approaches have produced impressive results, they also share critical weaknesses of opacity and implicit bias. Whether we’re discussing neural networks trained on millions of images or language models that orchestrate external actions and procedures, the internal logic of these systems is often difficult to examine, explain, and formally trust. As AI moves from recommendations to decision support to partial autonomy, this opacity becomes a serious concern.
If we want AI systems to exhibit the creativity, abstraction, and imagination that are uniquely human, we will need new mathematical frameworks.
The assignment is now well rehearsed. Is the data adequate and representative? What biases are embedded in training sets and calibration procedures? Can a system be made fair? Who gets to define “fairness”? What subjectivity and blind spots are allowed (so far) within individual applications? How vulnerable are they to malicious manipulation? What can we do about illusions?
These are not peripheral questions. These are laying the foundation for what it means for AI systems to function reliably in the real world.
It is precisely at this fundamental level that mathematics can and must play a leading role. Mathematics is too often treated as something that can be “added” to AI, as a tool for interpretation, evaluation, or error analysis. This is a big misunderstanding. Mathematical structures are not accessories to intelligent systems. It’s their footing. Without it, we are left with only heuristics, pragmatics, and empiricism, powerful but weak, effective but difficult to justify when things go wrong.
Additionally, mathematics provides deeper concepts and abstractions that can be the catalyst for next-generation AI. Mathematics offers something very distinctive. It is a language that expresses strictly provable results, logic, reliability, and performance limits, and some degree of guarantee of behavior and (foreseeable and unexpected) performance.
Mathematics provides principled ways to reason about data, uncertainty, and evidence. Through probability, geometry, and topology, it helps you understand the structure and shape of data space, why certain representations work, where the boundaries for decision-making lie, and how small perturbations can lead to large changes in outcomes. Reveal the convergence, stability, and failure modes of learning algorithms through optimization, numerical analysis, and dynamic systems. Through information theory, we clarify what can and cannot be inferred from finite data.
Many of today’s most innovative technologies are based on mathematical concepts that once seemed abstract or esoteric.
Equally important is the mathematics behind explanatory and exploratory AI. This allows us to not only build systems that perform well, but also to ask why they perform the way they do. Explainability, interpretability, and robustness are not purely engineering add-ons. These are mathematical properties that can be analyzed, proven, and stress tested. Examples of AI spoofing, vulnerabilities to adversarial attacks, and hallucinations require understanding how and why these occur, and defining and justifying appropriate mitigations. The same is true for operational bias issues arising from conditioning data sets and methodologies, as data drifts between calibration and operation. This is the difference between ex post explanations and models that can be interpreted by design.
There is also a positive side. As interest in neuromorphic and brain-inspired computing grows, mathematics becomes even more central. If we want AI systems to exhibit uniquely human forms of creativity, abstraction, and imagination, we will need a new mathematical framework that draws on fields such as category theory, stochastic processes, non-classical logic, and the mathematics of learning and adaptation. These are not incremental adjustments to the existing architecture. They are conceptual changes.
This is the intellectual space in which the Erlangen AI Hub operates.
Our goal is not just to make current AI methods safer or more efficient, but to strengthen their foundations. By incorporating powerful abstract ideas from across mathematics directly into real-world AI challenges, we aim to build more reliable, controllable, and transparent systems.
Importantly, this work is practice-based and responsive to national priorities. Our partners include the BBC, Ofcom and Capgemini, as well as many large and small businesses across industries of all sizes and sectors, policy experts and regulators, funders and national strategic decision makers. The goal is not mathematics per se, but mathematics that yields actionable insights, mathematics that changes what AI can responsibly do.
In the next phase of AI, mathematics will be both an innovator and a disruptor. It will take us beyond systems that are merely correlated, to systems that reason, adapt, and justify their actions within known limits.
This approach is very important for the UK. If the UK’s sovereign AI initiative simply replicates the trajectory of the US, China, India or the European Union, it will be difficult to distinguish itself. Scale is not our only comparative advantage. Intellectual leadership is possible. By developing truly new concepts, methods, and guarantees for AI, rooted in deep mathematics, we can lead rather than follow.
There is precedent for this. Many of today’s most innovative technologies are based on mathematical concepts that once seemed abstract or esoteric. Public-key cryptography, post-quantum cryptography, compressed sensing, and modern control theory all began as mathematical insights before becoming industrial necessities. AI is no exception.
In the next phase of AI, mathematics will be both an innovator and a disruptor. It will take us beyond systems that are merely correlated, to systems that reason, adapt, and justify their actions within known limits. It helps replace blind trust with guaranteed trust. And you’ll be able to unleash your creativity, not only in generating content, but also in solving truly new problems with rigor and accountability.
If society wants reliable AI, mathematics must be at the heart of it. It is not a constraint on progress. It is a condition that makes progress sustainable. The Erlangen AI Hub is an asset to Oxford and its academic, commercial and institutional collaborators. And the UK must thrive within a global and competitive community.
