# Drawing Is Computing: KidLisp as an Interactive Medium for Graphical Linear Algebra **A position paper and collaboration proposal** Jeffrey (prompt.ac / @jeffrey) · Aesthetic Computer · 2026-06-11 --- ## Abstract Graphical Linear Algebra (GLA) reframes linear algebra as a diagrammatic language: linear maps and relations become *string diagrams* — generators wired together, composed in sequence and in parallel — and proofs become visual rewrites. Its pedagogy and its research artifacts, however, are overwhelmingly *static*: PDFs, slides, chalk. This paper argues that Aesthetic Computer (AC) and its embedded language KidLisp are a natural *interactive* substrate for GLA. I sketch three concrete bridges — a diagram renderer, an executable term language for props, and an animated signal-flow playground — note honestly where the tree/graph mismatch makes the problem interesting rather than trivial, and close with a proposal for collaboration with the GLA / Applied Category Theory (ACT) community. --- ## 1. What Graphical Linear Algebra is GLA is the research program of Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi (with Robin Piedeleu and others), popularised through Sobociński's long-running blog *graphicallinearalgebra.net*. Its core move is to stop writing linear algebra in matrices and symbols and start *drawing* it. A diagram is built from a tiny set of **generators** — addition, the constant zero, copy, discard, and multiplication-by-a-scalar — drawn as boxes with wires. Two composition operators glue them: - **sequential** (`;`, `∘`): connect the outputs of one diagram to the inputs of the next, left to right; - **parallel** (`⊗`): stack two diagrams side by side. The algebra of these diagrams is captured by the symmetric monoidal theory of **Interacting Hopf Algebras (IHA)**, a complete presentation of the *prop of linear relations* — relations that are also linear subspaces. "Complete" is the punchline: two diagrams denote the same linear relation **if and only if** one can be rewritten into the other by the IHA equations. You no longer *calculate*; you *redraw*. A headline application is **signal flow graphs** — the dataflow diagrams engineers already use for linear dynamical systems and circuits — which GLA gives a fully compositional semantics. The key conceptual inversion: in GLA the picture is not an *illustration* of the mathematics. The picture **is** the mathematics. ## 2. What Aesthetic Computer and KidLisp are Aesthetic Computer is a mobile-first runtime and social network for creative computing: a musical-instrument-like interface where users discover memorisable paths and run published "pieces" — small interactive programs addressed by URL. **KidLisp** is its embedded Lisp dialect for generative art: S-expressions over ~118 graphics, timing, and math primitives, hot-reloaded live in the browser, every piece shareable by a short link. Two properties of AC matter for what follows: 1. **It is live and shareable by default.** A piece is a URL. There is no build step between an idea and a running, linkable artifact. 2. **It treats spatial/visual interfaces as carriers of real computational meaning** — the same bet GLA makes about diagrams, aimed at a different domain. ## 3. The bridges ### 3.1 A diagram renderer (the easy, immediately useful one) A string diagram is a planar wiring of boxes: sequential composition runs left-to-right, parallel composition stacks vertically. KidLisp already draws boxes, lines, and text. A renderer that lays out GLA generators and their wires is a short piece — the companion artifact `gla.lisp` in this repository is a first, hand-placed instance: two inputs feeding an **add** node, whose sum feeds a **copy** node that fans out to two outputs, with animated dots showing the signal propagate. It runs in the browser today. The value here is pedagogical and immediate: GLA's teaching materials are static; an embeddable, linkable, *animated* diagram is not. ### 3.2 An executable term language for props (the interesting one) The naive pitch — "Lisp is already a diagram language" — undersells the problem, and saying so honestly is the whole point. **Lisp S-expressions are trees. String diagrams are graphs.** A tree has exactly one root and no sharing; an expression `(f (g x) (h x))` *names* `x` twice but the term structure is still a tree with variables doing the sharing. GLA's power lives precisely in the structure trees cannot express directly: - **copy / fan-out** — one wire becoming two — is sharing made first-class, *without* a variable to name it; - **discard** — throwing a wire away — is structural weakening; - **feedback loops** — a signal flow graph's defining feature — are cycles, which a term tree simply does not have. So "render Lisp as diagrams" is not the task. The real task is to design a **textual syntax for a prop** — a term language whose denotation is a morphism in a symmetric monoidal category, with explicit `copy`, `discard`, sequential `∘`, and parallel `⊗` combinators, plus a trace / feedback operator. This is a known and respected problem (the *internal language of monoidal categories*; related to the algebra of Frobenius/Hopf structures and to hypergraph categories). A KidLisp-flavored surface syntax that **evaluates** to a concrete linear relation — and renders the corresponding diagram as it goes — would be a genuine contribution: an executable, inspectable bridge between the textual and the diagrammatic, with a live runtime attached. Concretely, a generator set might read: ```lisp (seq ; sequential composition ∘ (par in-A in-B) ; parallel composition ⊗ add ; ● two wires in, one out copy ; △ one wire in, two out (par out-1 out-2)) ``` where evaluation produces both the matrix/relation it denotes *and* the drawn diagram — two views of one object, exactly GLA's thesis made interactive. ### 3.3 A signal-flow playground (the demonstrable one) Signal flow graphs are GLA's flagship application and are *intrinsically animated*: values stream down wires, scalars amplify, feedback accumulates. AC is built for exactly this — real-time, frame-driven, interactive. A piece where you wire generators on a canvas and **watch the linear system run** — adjust a scalar and see the trajectory change — is something the ACT community largely does not have. It is the most compelling demo of all three because it shows the diagram *computing*, not just sitting. ## 4. Why this is worth a paper, a tool, or a thesis The honest framing of the opportunity: - **For pedagogy:** GLA wants to be approachable — that is why the blog exists. Interactive, linkable diagrams lower the entry cost further than any PDF can. - **For research:** §3.2 is a real open-flavored problem at the intersection of programming-language design and category theory — a term language and live evaluator for props/linear relations. - **For the field's culture:** ACT prizes *working* compositional tools. A browser-native, shareable GLA playground is a contribution in a form the community values and currently lacks. A fair caveat, stated plainly: a PhD in these groups is **formal category theory** — props, monoidal categories, completeness proofs, papers. The tool-building is an excellent entry credential and complement, not a substitute for the abstract mathematics the degree demands. The right posture is to lead with the artifact and learn the formalism, not the reverse. ## 5. The people and the path The GLA authors are the centre of gravity of Applied Category Theory: - **Paweł Sobociński** — Professor, *TalTech* (Tallinn, Estonia); leads the Laboratory for Compositional Systems and Methods; supervises PhD students and coordinates a doctoral programme. Author of the GLA blog and an outreach-minded researcher. **The most natural first contact.** - **Fabio Zanasi** — Professor, *UCL* (London); co-author of the foundational GLA papers. - **Filippo Bonchi** — *University of Pisa*; the third co-author. Two timely venues, both chaired by Sobociński in Tallinn: - **ACT 2026** (Applied Category Theory), 6–10 July 2026; - **DIAGRAMS 2026**, 24–28 August 2026 — arguably the better fit for an interactive-diagrams contribution. Estonian doctoral positions are **salaried**, sidestepping the tuition-debt model of the US/UK. ### Proposed first move Not "may I do a PhD," but: **ship the smallest working demo and point at it.** A KidLisp piece that renders and animates a GLA string diagram (now in hand) plus a short note to Sobociński — "here is what I build; I'd like to build it for category theory; DIAGRAMS 2026?" — tests the collaboration, funding, and thesis paths simultaneously, on the strength of a running artifact rather than a cold ask. Practitioners in this field notice tools. ## 6. Kindred voices: Richard Southwell and the diagrammatic turn The case for an *interactive* diagrammatic mathematics is not mine alone. In his 2026 talk *Speculations about the future of mathematics*, the category-theory educator **Richard Southwell** — author of *Categories and Toposes: Visualized and Explained* — argues that mathematics is shifting "from this sort of algebraic, encoded style of maths to something more like" drawing diagrams. His framing arrives, independently, at the exact metaphor underpinning this paper and my whistlegraph practice: > "I liken [algebra] to **sheet music** — as opposed to [the diagram], > which is the **actual music**. This is what our visual cortex can > process." — Richard Southwell His worked example *is* the companion piece `gla.lisp`: "a number flows in from the left; the black node **copies** it; the white node **adds** — put a 3 in, get a 3 and a 3, add them, get 6." He closes with a plea that is, almost verbatim, the mission of Aesthetic Computer: that people who understand graphical methods should "spend more time trying to make it easier for the rest of us." This is the audience for an AC-native GLA playground, and Southwell — an independent educator actively seeking collaborators who make the subject visual — is plausibly a lower-friction first contact than a cold email to a department. **A calibration, stated honestly.** Southwell's talk pairs GLA with a second, far more speculative program: Norman Wildberger's *box arithmetic*, which takes multisets (drawn as nested boxes, with "anti-boxes" for negatives) as a finitist foundation that rejects infinities and the real numbers. The two are **not** the same tier. GLA is rigorous, peer-reviewed, and complete (the IHA theorem); box arithmetic is heterodox, ultrafinitist, and — by Southwell's own admission — "the wild west," not even proven consistent. For a serious collaboration or thesis pitch, build on GLA. Box arithmetic is better treated as provocation and art-fuel. Notably, its nested-box structure is *operadic* — rooted **trees** — which places it on the opposite side of the same tree-vs-graph divide drawn in §3.2: GLA is the graph side, box arithmetic the tree side, and AC's score practice straddles both. Full transcript of the talk is archived alongside this study as `studies/southwell-future-of-math-transcript.md`. ## 7. Status and next steps - [x] `gla.lisp` — animated signal-flow string diagram, live in AC. - [ ] Generalise from hand-placed coordinates to a small layout pass (a list of generators → auto-wired diagram). - [ ] Prototype the §3.2 term language: `seq` / `par` / `add` / `copy` / `discard` / `scale` evaluating to both a relation and a drawing. - [ ] Add interaction: drag generators, edit scalars, watch the signal flow respond (§3.3). - [ ] Draft a one-page note + demo links for Sobociński; target DIAGRAMS 2026. ## References - Bonchi, Sobociński, Zanasi — *Interacting Hopf Algebras* / *A Categorical Semantics of Signal Flow Graphs*. - Sobociński — *Graphical Linear Algebra*, graphicallinearalgebra.net. - Bonchi, Piedeleu, Sobociński, Zanasi — *Graphical Affine Algebra* / *Graphical Resource Algebra*. - Sobociński — *Laboratory for Compositional Systems and Methods*, TalTech (compose.ee/pawel). - Richard Southwell — *Speculations about the future of mathematics* (2026), youtube.com/watch?v=l619jN5wkJA; *Categories and Toposes: Visualized and Explained*. - Norman Wildberger — *box arithmetic* lectures (multiset foundations; ultrafinitist).