Our Vision

From Words, to Pictures, to Equations, to Programs

One defining feature of humanity is our innate use of language to describe the world around us. Indeed, the arc of human history and progress can be framed in terms of the many profound transformations in the nature of the languages that we have used to model our physical reality.

Pre-Socratic philosophers like Heraclitus and Democritus used natural language (words and sentences) to produce qualitative descriptions of how they believed the world worked, sparking a tradition of natural philosophy carried forwards by Epicurus and Aristotle, and lasting until at least the French Revolution. A parallel tradition of the use of geometrical language (points, lines, shapes, etc.) to describe natural phenomena in optics, astronomy, and mechanics began with Archimedes in Greece, was continued by Ptolemy in Egypt, greatly elevated by al-Haytham in Iraq, and culminated in the staggering achievements of Kepler and Galileo in Western Europe.

Almost contemporaneously, the revolutionary ideas of Descartes connected the geometrical language of points and lines to the analytical language of symbols and equations, ushering in the era of analysis: Newton and Leibniz, building off ideas presaged by Archimedes himself, developed an alien new language of derivatives and integrals, thus enabling the systematic study not just of static points and lines, but of dynamic curves and trajectories. Natural philosophy, which had remained essentially qualitative since the pre-Socratics, now became quantitative for the very first time.

In some sense, this language of analysis (subsequently tightened up by Euler, Cauchy, Weierstrass, and many other mathematicians) is what gave rise to the modern science of physics, as distinct from earlier natural philosophy. Straightforward extensions of the language that Newton and Leibniz first pioneered for celestial mechanics would later give rise to the 19th and 20th century theories of hydrodynamics, thermodynamics, electromagnetism, and general relativity. Ultimately, analysis gave humanity a systematic language for investigating continuous mathematical structures, and it turns out that many foundational aspects of nature, from the energy and momentum of a distant star to the geometrical structure of space and time themselves, can be accurately modeled as continuous mathematical structures.

In the early 20th century, Turing and Church, in attempting to generalize earlier results of Gödel, inadvertently developed a fourth type of linguistic description: the computational. If analysis gave humanity a systematic tool for studying the continuous, then computation gave us a systematic tool for studying the discrete. The computational paradigm quickly brought conceptual clarity to many longstanding philosophical issues going back at least to Democritus: Church and Turing finally formalized what it is we really mean by a “procedure” or a “process” (resolving questions first posed mathematically by Hilbert); Shannon, Kolmogorov, and Chaitin finally formalized what it is we really mean by “information” or “entropy” (resolving paradoxes first posed by Maxwell); and Zuse, Fredkin, and Wolfram investigated what alternative foundations for physics might look like if they were to be based on discrete computational rules, as opposed to continuous analytical ones (resolving speculations first posed by Democritus and Epicurus).

Almost simultaneously with Turing and Church, physicists such as Planck were coincidentally discovering that many facets of nature (such as the energy levels of hydrogen) also appear to be fundamentally discrete. Thanks to this fortuitous co-evolution of the computational and the quantum, many of our most fundamental models of reality, from black hole event horizons to condensed matter systems, are now discrete, computational, and information-theoretic in nature: an abrupt transition from the earlier Newtonian paradigm.

When posed in these terms, this transition from natural language, to geometrical language, to analytical language, to computational language, seems almost inevitable. Humanity went from writing words, to drawing pictures, to solving equations, to writing programs. It is tempting (not least as a computational scientist) to conclude that this development was necessarily associated with a corresponding increase in the power, sophistication, and fundamentality of our scientific models. Yet the advent of artificial general intelligence forces us to interrogate that assumption more closely, and to confront the richer and more complex interactions between the various languages that humanity has used for describing the universe.

Artificial Intelligence as a Pre-Socratic Paradigm

From a purely historical perspective, it is clear that artificial intelligence has developed as an extension of the computational paradigm. Indeed, Turing famously predicted its arrival as a natural consequence of his own formalism, almost immediately after proposing it. Yet, upon closer observation, it becomes clear that artificial intelligence is not, in fact, another stage in this progression of linguistic paradigms, but rather a loop back to an earlier (pre-quantitative) era.

With the advent of artificial general intelligence, we have sharply transitioned back from the era of equations and programs to the era of words and sentences. Heraclitus and Democritus would not immediately have understood the concept of a differential operator or a CFD simulation, but they would immediately have understood the concept of a system prompt or an agent harness. Skills, prompts, and agents may well be built upon post-Turing infrastructure, but they remain fundamentally pre-Socratic ideas.

In some sense, many of the key advancements in the reliability of artificial intelligence over the past couple of years may be thought of in terms of the progressive incorporation of humanity’s various “higher” linguistic concepts (the geometrical, the analytical, and the computational) into the ultimately pre-Socratic substrate of modern language models. By endowing agents with computational language, thus allowing them to write programs, we gave them a systematic means of checking their work, just as we did. By endowing computational languages with geometrical and analytical features (for instance, by using formal proof assistant languages like Lean or Rocq), we gave agents, in turn, the ability to draw pictures and solve equations, just as we did. And at each turn the agents get smarter and more powerful, and proceed to develop better and more accurate models of reality, just as we did. The development of modern agent infrastructure is an accelerated recapitulation of the development of human intellectual culture, albeit in a somewhat scrambled historical order.

Connecting artificial intelligence to the computational world gave us coding agents for software engineering: one of the earliest and most striking demonstrations of the automation of human knowledge work. Connecting them, furthermore, to the geometrical and analytical worlds gave us autoformalization agents for pure mathematics: one of the earliest and most striking demonstrations of the automation of human genius.

So to take the obvious limit of this process, what will happen when agents are given simultaneous access to the mathematical, computational, and physical worlds on approximately equal terms, via a unified formal language that synthesizes the pre-Socratic, Archimedean, Newtonian, and Turingean paradigms into a single coherent description of reality?

This is what we’re building at Lanyon.

The Lanyon Philosophy: Science as Product vs. Science as Process

On the surface, our near-term objective at Lanyon is straightforward: to extend the grand accomplishments of artificial intelligence in software development and pure mathematics, bringing them to bear upon applied mathematics, computational physics, and ultimately upon the entirety of engineering and the physical sciences. Just as recursive self-improvement in programming and mathematics required the existence of formal tools (compilers, type-checkers, static analyzers, proof assistants, etc.) to act as verifiers within reinforcement learning loops, so too will such tools be required for optimizing capabilities in engineering, numerics, and the natural sciences going forward. This remains a fundamental aspect of the Lanyon R&D roadmap: to build a type system for the physical universe.

Yet there is much more to our mission than that. The advent of artificial general intelligence forces us to distinguish between two cultures of science, in a way that we have previously been able to avoid throughout much of our history. The first is science as product, based around an external objective, such as “cure cancer” or “solve fusion.” The second is science as process, based around an internal objective, such as “learn more about proteins” or “understand turbulence.” We built compasses long before we understood terrestrial magnetism, steam engines long before we understood thermodynamics, and light bulbs long before we understood quantum mechanics. Science as product has almost never been bottlenecked by science as process. They remain, in essence, separate pursuits.

Much of the attention around the application of artificial intelligence to science thus far has hinged on the view of science as product. Explainability, tractability, and alignment with human cultural and ideological norms have largely been viewed as secondary concerns to the primary criteria of “is it true?” and “does it work?” For many of the civilization-defining challenges that our species will face in the coming decade, such as developing clean and renewable energy, exploring deep space, and expanding critical infrastructure to developing parts of the world, this arguably seems like the correct sequence of priorities. Solve the existential problems first, satiate human intellectual curiosity later.

At Lanyon we do not accept such compromises as inevitable. We believe that it is possible to adopt a radically pro-human attitude towards science as product – tackling humanity’s greatest scientific challenges in an energetically results-oriented fashion – whilst simultaneously retaining a radically pro-human attitude towards science as process – building incrementally upon existing human intellectual and cultural artifacts in an explainable and comprehensible manner. We view artificial intelligence not as a technology for circumventing human creativity, curiosity, and understanding, but for augmenting and democratizing them.

This conviction underlies the very design of our core product. Rather than affording artificial intelligence a free reign to invent its own non-human abstractions, we constrain it to assemble its abstractions out of curated, human-understandable primitives. Likewise, we do not permit artificial intelligence to write mathematical proofs or computational algorithms directly, but rather to do so using symbolic pipelines that constrain it to use only human-understandable proof tactics or algorithmic constructs. Indeed, we view the design and implementation of the Lanyon domain-specific language as an attempt to crystallize and formalize the millennia of accumulated mathematical, scientific, and cultural wisdom of our species, and to disseminate that wisdom through the most powerful communication technology of our time: agentic artificial intelligence.

Words gave us the ability to express our scientific intuitions. Mathematics gave us the ability to constrain them. Computation gave us the ability to express them at scale. And now agentic artificial intelligence gives us the ability to disseminate them to all.

The constraint of human understandability may seem like an overly burdensome restriction, but we firmly believe that it remains the only scalable and reliable way to ensure that artificial general intelligence continues to serve the best scientific and technological interests of our species. And, in any case, the grammatical constraints of language, the axiomatic constraints of mathematics, and the syntactical constraints of computation likely seemed like overly burdensome restrictions in their times too. At Lanyon we are simply taking the lessons that our civilization has learned, and applying them to this new frontier of science and technology to the very best of our ability.