Arturo Herrero

New representations of thought

New representations of thought — written language, numerals, mathematical notation, data graphics — have been responsible for some of the most significant leaps in the progress of civilization, by expanding humanity’s collectively-thinkable territory.


A representation captures some aspect of a concept or phenomenon in a human-understandable form, thereby enabling a person to perceive and think about it.

For example, a piece of music can be represented as a verbal description or as a page of music notation (symbolic); a waveform plot or spectrograph (visual); an audio recording or click track (aural), the pattern of finger movements while playing the piano (tangible), or the experience of walking around an orchestra (spatial).

  • Before the 14th century, multiplication was considered a highly abstract concept, only for the mathematical elite. Once place-value Arabic numerals replaced Roman numerals, multiplication and division became mundane. It was this representation which made universal arithmetic literacy possible.

  • Before the 17th century, mathematical calculation was described in prose. The invention of algebraic notation made mathematical structure visible, and allowed for abstracting beyond numbers. This representation was the birth of modern mathematics.

  • Before the 19th century, data was presented in tables. Playfair invented the data plot. Without this form of representation, modern scientific discovery and communication would be inconceivable.

  • Faraday’s representation of magnetism as “lines of force” inspired Maxwell’s theory, which he wrote as twenty differential equations. Heaviside invented the language of vector analysis specifically to rewrite Maxwell’s equations in four simple lines. This representation was the birth of electrodynamics.

  • Dalton’s elements were a grab-bag, with no coherent framework or predictive power. Mendeleev found a way to represent the patterns of chemical properties with a “periodic table”. This representation enabled, for the first time, a theory of chemistry.

Why representations matter

These representations weren’t mere scientific “discoveries”. Each of them essentially enabled all subsequent scientific breakthroughs thereafter. A powerful new form of representation affects everything, forever.

Research agenda and former floor plan, Bret Victor.

June 09, 2019 | @ArturoHerrero