BLACK-SCHOLES, FEYNMAN DIAGRAMS, AND INTERFACES AS ENGINES
Why the most powerful representations reshape the domains they describe
In An Engine, Not a Camera, Donald MacKenzie argues that the Black-Scholes formula did not describe the options market as it already existed but rather constituted the market it purported to model. Before the formula's publication in 1973, options prices deviated substantially from the values the model predicted. After traders began using it as the standard framework for pricing, prices converged toward its outputs, not because the model had discovered some hidden truth about market structure but because the model itself reshaped pricing behavior, trading infrastructure, and the institutional fabric of financial markets until reality came to resemble what the formula assumed. MacKenzie calls this performativity, the capacity of a representation to bring about the conditions it describes rather than merely recording what was already there.
The phenomenon extends well beyond finance. In every domain where representations mediate human thought and action, there exists a question that is rarely asked but always consequential. Does the representation merely record what already exists, or does it make new operations possible that were inconceivable before the representation was available? The history of intellectual progress suggests that the most powerful representations belong to the second category, reshaping the cognitive landscape they claim to map.
The GUI as Institutional Knowledge
The conventional understanding of a graphical user interface is that it primarily reduces the friction between intention and execution, replacing typed commands with clickable elements and memorized syntax with discoverable menus. While this understanding is not incorrect, it is incomplete because it treats the interface as a thin veneer over pre-existing functionality.
When a product team constructs a software interface, they are not simply attaching faster input mechanisms to features that already exist. Instead, they are making a series of decisions about which operations matter, in what order they should be performed, what the user needs to see at each stage, and what should remain hidden to avoid overwhelming cognitive bandwidth. In doing so, they are encoding domain knowledge, the accumulated understanding of how work actually happens in a particular context, into the structure of the tool itself.
That is, the interface does not merely present the system's capabilities but organizes them according to a theory of the work, one that reflects how practitioners actually think, what sequences of action lead to reliable outcomes, and where human judgment remains indispensable. The interface, in this sense, is an argument about the structure of the work, rendered in pixels and interaction patterns, and its quality depends on both visual polish and the faithfulness of the domain model it embodies.
The Notation That Made Arithmetic Possible
The Hindu-Arabic numeral system, which reached Europe through Fibonacci's Liber Abaci in 1202, offers perhaps the purest illustration. Roman numerals represent quantity perfectly well. XLVII denotes forty-seven with complete fidelity, and for purposes of labeling quantities or recording transactions, the system is entirely adequate.
What becomes visible only when one attempts multiplication is that the notation does not merely make certain operations difficult but renders them effectively impossible to perform as fluent mental acts. Try multiplying XLVII by MCMXII, and what becomes apparent is not that the calculation is hard but that the representational system provides no conceptual handles for decomposing the problem into manageable steps. The Roman system lacks positional structure, and without positional structure, there is no place for carrying, no mechanism for the columnar decomposition that makes long multiplication tractable as a cognitive operation.
The Hindu-Arabic system did not label the same quantities more conveniently. It brought into existence an entire domain of cognitive operations, long division, decimal expansion, algebraic manipulation, that could not be performed, or even coherently imagined, within the representational framework it replaced. The notation was an engine that generated computational capacity the Roman system structurally could not support.
Seeing What Cannot Be Calculated
Richard Feynman introduced his diagrams at the Pocono Conference in 1948 to address a specific problem in quantum electrodynamics. Calculating the probability of particle interactions required summing over an enormous number of terms in a perturbative expansion, and before the diagrams, these calculations consumed pages of dense integral calculus. Freeman Dyson later proved that Feynman's pictorial method was mathematically equivalent to the conventional formalism, but equivalence in expressive power is not equivalence in cognitive effect.
The diagrams made certain physical processes seeable. A photon exchange between two electrons, previously buried inside a series expansion, became a line connecting two vertices. Higher-order corrections, previously requiring extraordinary algebraic stamina to enumerate, became variations on a visual pattern. What had been an exercise in symbolic endurance became an exercise in spatial perception.
The diagrams did not increase the formal power of the theory, as every calculation they enabled could, in principle, have been performed without them. What they did was restructure the cognitive relationship between the physicist and the formalism, making certain questions askable and certain patterns recognizable that had previously been buried beneath the surface of the notation.
Representations as Affordance Generators
As I have explored elsewhere, representations that carve reality at its joints do not merely describe structure but make it available for thought and action in ways it was not available before. Mendeleev's periodic table predicted undiscovered elements by revealing gaps in its pattern. Bricklin's spreadsheet grid made scenario analysis cognitively tractable by organizing quantitative relationships spatially. What MacKenzie adds to this picture is the recognition that the most powerful representations do not stop at making the existing world more legible but actively reshape the world by enabling operations that alter the domain they represent.
The question this poses for anyone building software is not what information to display but what cognitive operations to enable. An interface that faithfully mirrors the state of a system is a camera. An interface that restructures how its user perceives, reasons about, and acts upon that system is an engine. The history of notations, formalizations, and interfaces suggests that the engines are what matter.