Mathematics
Comment
Stakeholder Type
GESDA
"Cellulose Planets" by Elise Ansart, ETH Zurich
Photo: "Cellulose Planets" by Elise Ansart, ETH Zurich

Topic

Mathematics

Anticipation Committee Chair:

Minhyong Kim

Director / Sir Edmund Whittaker Professor

International Centre on Mathematical Sciences in Edinburgh

Anticipation Committee:

Carina Curto, Eric Maskin, Hyejin Youn, Karthik Ramani, Nima Arkani-Hamad, Rashid Sumaila, Sarah Teichmann, Terence Tao
Topic Brief

Mathematics

Unlike any other discipline, mathematics is deeply rooted in almost every human culture. This has been the case for as long as written records have existed. The practices of counting, measuring, arranging, quantifying and comparing objects in the physical world around us is so useful that we begin to teach them to our children at a very young age.

Unlike any other discipline, mathematics is deeply rooted in almost every human culture. This has been the case for as long as written records have existed. The practices of counting, measuring, arranging, quantifying and comparing objects in the physical world around us is so useful that we begin to teach them to our children at a very young age.

The modern academic study of mathematics is a world away from these basic activities, but remains an essential part of the functioning of human civilisation. In the 21st century, mathematical fields such as linear algebra, differential calculus and statistical analysis underpin a wide swathe of our everyday activities, from online shopping to medical diagnosis.

The field has always been aware of its shortcomings: mathematics has a long tradition of raising awareness of unsolved challenges. At the beginning of the 20th century, for instance, the mathematician David Hilbert outlined 23 significant challenges, 20 of which now have full or partial solutions. At the turn of this century, the Clay Institute offered a $1 million prize for the solution of any of seven problems, only one of which has been solved so far.

Mathematicians are aware of plenty of other less-celebrated gaps that also need to be filled, both for the “completeness” of mathematics and because those gaps can contain tools useful for the advancement of humanity’s interests. It has always been extremely difficult to predict the pace of progress and breakthrough in mathematics, and to know which as yet undeveloped mathematical tools will be useful to academics in other disciplines of the natural and social sciences. However, it is clear from past experience that future developments in a variety of mathematical disciplines will accelerate progress in science and technology, facilitate greater societal stability and push forward the frontiers of medicine.

Achieving these advances will require increased collaboration between mathematicians and academics from other disciplines, and improvements in the equity of access to advanced mathematics education.

KEY TAKEAWAYS

Humans have been counting, measuring and comparing aspects of the physical world, as well as defining abstract mathematical concepts and their interactions, for many tens of thousands of years. While the uses of mathematics have broadened, some of the central original applications remain, albeit with significantly more complexity. Modern mathematics is essential to the study of Nature, whether that is for understanding our planet’s past and possible futures, or broader cosmological considerations. Taking this further will require new mathematical tools. The wealth of data available to us in the 21st century has facilitated the development of Machines that can assist with the complexity of modern mathematical calculation and logical inference. With judicious development, such AI could soon become a useful tool for mathematicians. Whether in supply chains, medical technologies, transport logistics or public administration, the central role of mathematics in human Society is set to continue. Suitably applied, mathematical models can assist with the development of better futures for human groups of all sizes, though challenges remain. There are also challenges in applying mathematics to the study of biological Life, which necessarily involves dealing with complex components and processes that are often difficult to formalise as mathematical entities and procedures. Nonetheless, progress is being made, raising the possibility that the application of mathematics can improve future human health and our scientific understanding of life.

Topic:

Anticipation Potential

Mathematics

Sub-Fields:

Nature
Machines
Society
Life
Mathematics is poised to fundamentally reshape scientific and technological frontiers, with experts anticipating significant breakthroughs across its various domains. Areas where mathematics intersects with Machines, particularly AI and robotics, are seen as holding the most transformative potential, with impactful advances already being deployed. While the application of fundamental mathematics to describe the mechanics of Life also presents considerable transformative opportunities, there is a higher degree of uncertainty surrounding these future breakthroughs. Developments in mathematics related to Nature and Society are likewise expected to undergo substantial scientific and technological shifts, albeit over a longer time frame. Across these diverse sub-topics, coordinated international action is frequently highlighted as crucial for exploiting anticipated opportunities.
SubField

Nature

Our understanding of the natural world, in terms of both Earth systems and deeper physical realities, depends to a large extent on mathematical modelling. Although this has been valuable in a variety of applications, phenomena such as climate change have vastly increased the complexity of the systems that researchers would like to model. For instance, the ocean-climate-human system probably involves numerous feedbacks that are currently beyond mathematical models, as is the economic quantification of “ecosystem services”.1 This will require innovations in solving mathematical challenges such as finding solutions to complex and chaotic equations of fluid flow, and their integration with models that accurately reflect human behaviour and trends in other species’s movements and characteristics at a number of scales. Using higher-resolution data and more powerful computing resources to build and explore models will also improve our understanding.2 Programmes that ensure more equitable access to data and computational resources will bring much-needed depth and strength to these efforts.

Future Horizons:

×××

5-yearhorizon

Research models effects of policy on humans

Enhanced mathematics for studying feedback loops provides a way to model human effects such as the benefits of actions resulting from the Paris agreement. This stimulates further research and greater confidence in policy-making. Mathematicians provide conceptual tools for quantising gravity where space, time and quantum mechanics are secondary, perhaps emergent, phenomena and not central pillars of the model. Geometric renderings of physical properties of fundamental systems offer insights into ways to simplify representations of particle and other interactions.

10-yearhorizon

Mathematicians quantify value of ecosystem services

Innovations in mathematics allow researchers to quantify the value of ecosystem services such as carbon sequestration. Bioeconomic models are enhanced by real-time data from satellite monitoring and other sources.

25-yearhorizon

AI informs bio-abundance predictions

AI enables mathematicians to reliably predict the abundance of ocean species.
The same is true in fundamental physics and cosmology, where insights about both the birth and the deep-future fate of the universe strongly suggest a need for new mathematical ideas and conceptualisations.3 In universe-modelling, previous generations of mathematicians have had predictive success, but more recently, mathematics has responded to discoveries by physicists. However, both groups now need to think about possible new physics and mathematics that will transcend the ideas of quantum mechanics and space-time that dominated the 20th century. These new ideas will be simple, rather than complex, baroque constructions. Insights into novel geometric structures and their properties are proving promising avenues for advances in fundamental physics,4 and the hope – based on past trends – is that such advances will trickle down to other disciplines and fields, seeding progress across the sciences.

Nature - Anticipation Scores

SubField

Machines

Recent progress in the development of AI and machine learning (ML) has led to numerous scientific, medical and even societal innovations that are set to improve the human experience. However, a range of significant challenges remain.5 Mathematicians are still seeking a clear theoretical understanding of exactly how and why AI and ML work,6 for example, with some models showing a mysterious ability to solve mathematical problems through guesswork in a manner that has yet to be understood. For this and other reasons, it is likely that AI will be most useful in its performance on tasks that are extremely challenging for human intelligence, and AI and human approaches to mathematics could ultimately be complementary rather than competitive.7

Future Horizons:

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5-yearhorizon

Benchmarks help improve AI performance in mathematics

A rolling set of benchmarks for measuring AI progress in solving partial differential equations is established. AI assists formal verification routines to correct errors in proofs. Improved access to data assists ML modelling of feedback between ocean, human and climate systems. AI integrates symbolic and physical models for problem-solving.

10-yearhorizon

AI assists with proofs

AI fed small lemmas is able to formally prove modular parts of large formal proofs. Projects that data-dump iterations of flawed proofs enable AI to learn the process of human theorem-proving. Success here births a database of mathematical activity that complements mathematical achievement and provides AI with training data for achieving mathematical progress in the same style as human mathematicians. AI assists a number of projects centred on climate change, such as quantifying the value of ecosystem services (carbon sequestration, for example), intergenerational discounting, dynamic ocean modelling that manages climate-related displacement of biomass and integrating bioeconomic models into mainstream economic thought. AI-driven robotics interacts with the physical world and learns to mathematically encode physics through experience.

25-yearhorizon

AI ubiquitous in mathematics research

AI is integrated into the workflow of most mathematicians’ research as a kind of digital assistant for accomplishing routine tasks. Mathematical innovation allows AI-driven robotics systems to operate safely and intelligently in human environments.

A complication comes from the tendency for AIs doing mathematics to “hallucinate” mathematical truths in ways that make their output unsuitable for use in formal proof. As yet, the field has yet to agree on what constitutes a good set of benchmarks for AI performance in a number of mathematical fields,8,9 making it extremely difficult to measure progress. Nonetheless, it is expected that theoretical advances in AI and ML will make their mathematics increasingly reliable and useful for pressing issues such as modelling the interactions of ocean, human and climate systems, as well as in basic science.10

Generative AI and statistical ML are being used by applied mathematicians and assisting with the design of physical systems. However, building AI that understands the real world means building “embodied” AI that works via the maths-based rules behind real-world physical processes, integrating symbolic and physical models. This will also produce more robust, efficient and explainable intelligence. Artificial general intelligence will not be achieved without new mathematical insights and architectures that unify and streamline the variety of paths currently being taken.

Machines - Anticipation Scores

SubField

Society

Mathematics is central to much of the work carried out in the social sciences. Calculus underpins our understanding of economic systems; algebra and game theory define the forces at work when we model social interactions such as elections and responses to policy changes such as inter-state relations and healthcare provision; a deep grasp of statistics is vital to the task of anticipating future societal needs and ensuring that resource management is achieved effectively and efficiently. However, individual human behaviour resists mathematical modelling11 because of its complicated nature, including susceptibility to priming and framing effects that exert considerable influence on short timescales.

Future Horizons:

×××

5-yearhorizon

Data analysis supports new urban initiatives

Major cities begin to experiment with the application of AI-powered data analysis and network theory to design new urban initiatives. Mathematicians assist in efforts to improve voting systems to create better-functioning democracies.

10-yearhorizon

Models predict citizen response to policy

Experimental economics helps predict how citizens will respond, and how their welfare will be affected, when government policy changes.

25-yearhorizon

Decision-making forecasts improve

AI and network theory enable reasonably accurate forecasting of human decision-making, based on patterns of previous behaviour.

Mathematical tools may nonetheless be able to extract rules and descriptions for human behaviour in the aggregate.12 There are, for instance, mathematical relationships between quality of societal infrastructure, population size, crime statistics and income distributions. In addition, understanding the kinds of network structures that exist in online and other communities, or organisational structures in particular disciplines, can help describe and model human behaviour and characteristics. Such phenomenological modelling can reveal the core dynamics that drive large-scale transformations in complex systems where first-principles models are impossible. These insights, if gained with sufficient mathematical rigour, can be applied to help shape human behaviour, not just to describe it. Data analysis carried out on societal systems facilitates the exposure of systemic risks13 or hidden biases, such as might be found in legal, governmental or corporate decision-making. Mathematically-derived insights can also provide ways to go beyond simple market dynamics, facilitating new, urgently required hybrid markets such as those needed for sustainable development and healthcare.14 Understanding of social-network structures can help with robust communication in an information-saturated world, enabling the exchange of ideas beyond the originator’s bubble or to avoid echo-chamber effects.15

One impediment to progress is not lack of mathematical tools but a lack of data on human behaviour and decision-making. Historical records are too sparse to create precise economic models, and there is a dearth of controlled experiments generating useful, cleanly interpretable data. Mathematically driven data science can be expected to help fill the gap, maybe incorporating new advances in the understanding of human behaviour such as that provided by neuroscience.16

Society - Anticipation Scores

SubField

Life

Whether for medical technologies such as diagnostic imaging, surgical intervention, prevention of infectious disease outbreaks and cancer treatment, or for biological applications in neuroscience, genomics and improving agricultural practices, mathematics plays a central role in our study of living systems.17

Future Horizons:

×××

5-yearhorizon

Mathematicians flow into biology

Collaborations enable mathematicians to formalise a number of key problems facing biology, encouraging a flow of mathematicians into the field.

10-yearhorizon

Neuroscience benefits from theory-formation

Theory-formulation takes over from data analysis as the prime role of mathematicians working in neuroscience.

25-yearhorizon

Mapping initiatives facilitate clinical research

Success in a number of mapping initiatives, from the Human Cell Atlas to the topology of neuronal networks, facilitates clinical research that improves the outcome of cancer, dementia and other treatments.

One example is the Human Cell Atlas,18 which aims to make a cellular and molecular-level resolution three-dimensional map of the human body. This involves the integration of vast amounts of data, inclusion of annotation at relevant resolutions and incorporating the facility to interrogate the underlying data. The mid-term aim is to build a foundation model of the human body at cellular and molecular resolution.19

In neuroscience, there is a lack of mathematical models to describe high-dimensional, non-linear complex systems.20 If the mathematical tools for describing the dynamics of such systems were to be developed, there would probably be applications in many other areas, such as economics.

For these projects and others, which promise a revolution in the ability to understand and engineer the mechanisms of life, lack of data is not always the main problem, though data for a comprehensive Human Cell Atlas would be transformational for biology and medicine. Instead, there is a need to formalise the mathematical requirements of biology21 so that mathematicians can efficiently and effectively work with biologists to make progress — there are progress opportunities here for both fields. There is also a need to understand how deep or complex the models of various systems need to be in order for progress to be made — not all biological systems require the same level of granularity for useful analysis.22

Life - Anticipation Scores

References

Citations

5.3.1 Nature

  1. A. M. M. Sequeira et al.. Ecosystem services “on the move” as a nature-based solution for financing the Global Biodiversity Framework https://doi.org/10.1038/s44183-024-00073-7
  2. V. Eyring et al.. Pushing the frontiers in climate modelling and analysis with machine learning https://doi.org/10.1038/s41558-024-02095-y.
  3. R. Loll et al.. Quantum Gravity in 30 Questions https://arXiv.org:2206.06762
  4. N. Arkani-Hamed and J. Trnka. The Amplituhedron https://arXiv.org:1312.2007

5.3.2 Machines

  1. C. Dessimoz and P. D. Thomas. AI and the democratization of knowledge https://doi.org/10.1038/s41597-024-03099-1
  2. W. Douglas Heaven. Large language models are amazing but nobody knows why https://www.technologyreview.com/2024/03/04/1089403/large-language-models-amazing-but-nobody-knows-why
  3. C. Drösser. AI Will Become Mathematicians’ ‘Co-Pilot’: interview with Terence Tao https://www.scientificamerican.com/article/ai-will-become-mathematicians-co-pilot/
  4. M. Eriksson et al.. Can We Trust AI Benchmarks? An Interdisciplinary Review of Current Issues in AI Evaluation https://arXiv.org:2502.06559
  5. N. McGreivy and A. Hakim. Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations https://doi.org/10.1038/s42256-024-00897-5
  6. President’s Council of Advisors on Science and Technology. Supercharging Research: Harnessing Artificial Intelligence to Meet Global Challenges https://bidenwhitehouse.archives.gov/wp-content/uploads/2024/04/AI-Report_Letter-ExSumm-29APRIL2024_SEND.pdf

5.3.3 Society

  1. S. Hochrainer-Stigler et al.. Measuring, modelling, and managing systemic risk: the missing aspect of human agency https://doi.org/10.1080/13669877.2019.1646312
  2. V. Chuqiao Yang et al.. Regulatory Functions from Cells to Society https://arXiv.org:2409.02884
  3. K. Lucas et al.. Systemic Risks: Theory and Mathematical Modeling https://doi.org/10.1002/adts.201800051
  4. E. Maskin. Mechanism design for pandemics https://doi.org/10.1007/s10058-021-00270-7
  5. S. Bhattacharya et al.. Unveiling Scaling Laws in the Regulatory Functions of Reddit https://arXiv.org:2407.12063
  6. D. Peixoto et al.. Decoding and perturbing decision states in real time https://doi.org/10.1038/s41586-020-03181-9

5.3.4 Life

  1. H. Peter Fischer. Mathematical modeling of complex biological systems: from parts lists to understanding systems behavior https://pmc.ncbi.nlm.nih.gov/articles/PMC3860444/
  2. O. Rozenblatt-Rosen et al.. The Human Cell Atlas: from vision to reality https://doi.org/10.1038/550451a
  3. J. E. Rood et al.. The Human Cell Atlas from a cell census to a unified foundation model https://www.nature.com/articles/s41586-024-08338-4
  4. K. Morrison et al.. Diversity of emergent dynamics in competitive threshold-linear networks https://doi.org/10.1137/22M1541666
  5. S. T. Vittadello and Michael P.H. Stumpf. Open problems in mathematical biology https://doi.org/10.1016/j.mbs.2022.108926
  6. K. A. White et al.. Charting a New Frontier Integrating Mathematical Modeling in Complex Biological Systems from Molecules to Ecosystems https://doi.org/10.1093/icb/icab165