During my days as a college undergrad, I’ve always felt the same whenever I looked at a derivation (or any piece of text). A board filled with equations, assumptions and some textual punctuation. I tell myself, “Is this the best we can come up with?” While others found comfort in sequentiality and linear comprehension, I most often could wrap up what-is-to-understand in a simple graphic. A nonlinear map charting primitive concepts and their relationships.
To this day, humanity’s favorite mode of recording knowledge is formulaic in nature. The graphical mode serves as an auxiliary and was never given the weight it deserves. To me, it is an underrated mode of reasoning and expression. An uncharted territory where even the grammar is still at its early stages.
For my knowledge-representationalist mind, even more fundamental than knowledge itself, representation is everything. What is knowledge if not represented? Nothing. We need to perceive it somewhere either realized (in nature) or represented (by another intellect).
Realizing how important the notion of representation is to all knowledge endeavors, my problematic became about finding visual means of expressing knowledge that are as powerful as, old and still persisting formulaic ways.
It was a normal afternoon when my mind found the leading thread.; a flash of imagery under the title of the craft of theory and that of engineering are one and the same.
It came to me twofold.
Engineering is all about machines. They are the product and ingredients of the process. Bigger devices involves the wiring of smaller ones. And the most primitive kind of machines converts energy.
Through a clever composition of primitives one can effect a conversion–chain that produces a meaningful action. Such composition can even be bundled into a machine whose sole purpose is to perform the said action.
Theory, when distilled to its purest components, is mainly a body of constraints. Mathematical statements and even formulated in-text laws, when looked up close, express relationships between variables and freedoms proper to the phenomenon under study.
Constraints can be combined to express newer ones and it is always possible to find a set of primitives capable of generating all of the theory’s content. Whether we are dealing with quantitative or qualitative descriptions, it is all about the relations — all about constraints.
From there the parallels draw as follow.
Both machines and constraints, (1) can be built through combination of smaller realizations (unless we are dealing with primitives), (2) have interfaces and (3) are about the conversion of something. The first parallel is clear enough. The second attains the same level of clarity when we forward the fact that, interfaces imply an I(nput)/O(utput) aspect, which is found in both. The last parallel is complete by stating that, constraints convert information.
Now that we have a mapping, we can take one last dig to get exactly where I want you to be.
From a mathematical point of view, a constraint — or an equation — is a tensor combination, a relation among variables and an equality between two containers of information. We can arrange an equation so that we have a single tensor on one side of the equality. Doing so, allows us to view the other side as a tensor composition which defines the isolated one. The situation is without doubt similar to how a machine is constructed through composition of similars, as if they are standing on the other side of equation.
Bottom line, tensors are abstract machines.
It is no exaggeration to say, all of it is just engineering. The distinction is made only at the level of application, abstract or concrete. All possible implications of the previous statement beside, I want to focus on the representational aspect.
For “concrete” engineering, circuit representation is an important technique. It shows the underlying composition of a device as a network of interconnected components. Such graphic serves both a descriptive and prescriptive purpose. The ingredients of such visual formulation are lines and vertices. They stand for wiring and components respectively.
Per the established parallel and late considerations, we affirm the possibility of “abstract” circuits — a diagrammatic representation for tensors.
Just as machines have interfaces, tensors have indexes which allow us to interact with their various components. Reusing the previous recipe, vertices stand for a tensor — where the legs identify with the indices — and lines are then used for index contractions within the same (or between different) tensor(s).
Tensor calculus is well established in its formulaic state. But, what is known of the kind insights and shortcuts we might uncover by representing information visually?
Personally, I’d say very little.