“There is a secret world out there. A hidden parallel universe of beauty and elegance, intricately intertwined with ours.”
– Edward Frenkel
Vision
At PeakMath, our vision is to build a model of the mathematical landscape, which at least to some extent could convey the astonishing beauty, depth and interconnectedness of the ever-expanding world of mathematics. Such a model could potentially be a new medium of thought in the sense of Matuschak and Nielsen, allowing for new approaches to creativity in mathematical research, and for deeply meaningful dynamical learning paths all the way from K-12 to graduate level mathematics.
There are many different lines of thought converging at the idea of mathematics as a landscape. The purpose of these notes is to sketch a few of these lines, as background and as inspiration.
Reports from the frontier
Mathematicians speaking about their own research almost invariably describe their inner experience as the exploration of a landscape. If you for example listen to interviews at Math-life balance, or read the quotes in the book Mathematicians on Creativity (Borwein, Liljedahl, and Zhao), examples abound. Hardy “gazes at a distant range of mountains and notes down his observations” and for Cayley it’s a “tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood and flower”.
While the landscape is the main metaphor, individual mathematicians may use variations on the same theme. Bourbaki, known for a systematic but perhaps a bit rigid view of mathematics, doesn’t refer to the landscape, but to mathematics as a big city. Wiles, intensely focussed on a single “location” (Fermat’s Last Theorem), talks not about roaming in a landscape, but of stumbling around in a dark mansion. Hilbert, perhaps the last mathematician with a complete overview of human mathematical knowledge, speaks of mathematics as a garden, with beaten paths and hidden trails. And Langlands, explorer of new worlds, speaks of the “impression of standing before a virgin continent”.
Mathematical concept space
Every learner or researcher is navigating through mathematical concept space. Vygotsky proposed the zone of proximal development as the space where the learner sits on the edge between the known and the unknown, and suggests that this space is best navigated in dialogue with a “more knowledgeable other”. The biologist Stuart Kauffman promotes (as one of his four general laws) the principle of “the adjacent possible”, which when transferred to mathematics hints at the space just-beyond the known, that might be explored by a researcher in the near future.
A good model of the mathematical landscape would shine some light on the space of paths open to us in the landscape, guide the learner through the next-level threshold concepts and point towards the peaks. It would also display aspects of the rich web of functors and analogies connecting diverse parts of the landscape.
Beyond-formal aspects of mathematical thought
Mathematics in its idealised form is a formal structure, made of axioms, definitions, theorems and rigorous proofs. These building blocks can be formalised in a programming language like Lean and implemented at different levels of rigour in educational apps. But much of mathematical creativity and meaning-making relies on beyond-formal aspects of mathematical thought, like analogies, metaphors, images, heuristics and value statements. Integrating the formal and the beyond-formal is key to a deeply meaningful learning experience.
One way to understand the phenomenology of beyond-formal mathematical thought is through embodied cognition, sometimes referred to as 4E or even 7E cognition (Embodied, Embedded, Enactive, Extended, Emotional, Evolutionary and Exaptative).
Placing yourself mentally in the mathematical landscape activates some of the related neural functions, like exploration and mental mapping, the hunting and gathering of concepts, clues and connections, and the experience of pushing through the darkness to climb the next mathematical peak.
Mathematics as narrative
Mathematics is not a static body of knowledge, but is always emerging, expanding, re-forming. Madeline Muntersbjorn has suggested that mathematics is neither invented nor discovered, but cultivated and emergent.
This process of emerging takes place on the grand scale of human civilisation (“mathematical phylogeny”), but also on a smaller scale in the individual mind of every learner and explorer (“mathematical ontogeny”). One of the many insights found in the thesis of Ganesalingam is that in order to understand mathematical language, we must admit some notion of “mathematical time”.
When Terence Tao writes about “What is good mathematics”, his conclusion is that “the very best examples of good mathematics […] are part of a greater mathematical story”. A wider discussion of narrative perspectives, including the idea of archetypes, can be found in the book Circles Disturbed, with essays by (among many others) McLarty, Mazur, Harris, Gowers, Teissier and Corfield.
We would like to integrate notions of mathematical time and mathematical archetypes into the landscape, to make mathematical narratives come alive.
Existing maps of mathematics
When the Internet was young, Dave Rusin’s Mathematical Atlas was one of its ultimate attractions. The Atlas is now gone from the web, and only the Wayback version remains.
Since then, quite a few people have drawn their own maps of the mathematical landscape. Some of the most striking examples are Mathematopia (by Zhaorui Xu), Mathematistan (by Martin Kuppe), Land of Middle Math (by Franka Miriam Brückler), and the Leitfaden of A Singular Mathematical Promenade (by Étienne Ghys; see page 18 and 19 in the pdf).
There is something about all these maps that speaks to the soul.
Our journey
We are now taking the first steps towards the realisation of an immersive and interactive model of the mathematical landscape. The idea is to build a community around the landscape-building process, which may appeal to you if you seek an overview of mathematics as a whole and its many interconnections. And if you as a researcher, educator, game developer or visual artist see some room for improvement (and there will be plenty of such room!), then please do get in touch with us. We seek both collaboration and feedback.
While building the landscape, we will be on the lookout for clues that could lead to a new framework for L-functions and the immortal conjectures surrounding them. The RH Saga is our introductory course, an exploration of L-functions and the analogy between number fields and function fields. This is really the story of Weil’s Rosetta Stone, possibly the deepest of all mathematical analogies.
We hope you will find it interesting and meaningful to join us for this long journey, and if not now, then maybe at some point in the future.
Andreas Holmstrom, Oct 2023