The search for intuitive mathematics

"I now view the effort to find a fundamental theory of physics as in many ways another challenge in language design — perhaps even the ultimate such challenge. Designing a computational language is to create a bridge between two domains: the abstract world of what is possible to do, and the “mental” world of what people understand and are interested in doing. The challenge is [...] to give people a way to describe these [abstractions]." -- Stephen Wolfram

Wolfram's quote could be viewed through the lens of Seymour Papert. Educators often use physical analogies and embodiments to describe complex mathematical and physical phenomena. On the other hand, practitioners (such as scientists) use symbolic representations for faster exploration. This fundamental gap between education and practice was pointed out by Seymour Papert (in his book Mindstorms, Chapter 8) when he was designing the LOGO programming language.

This gap is also evident in our math education trajectory. Standard mathematics curricula start off with picture based math or manipulable objects (e.g., Montessori Sensorial materials), grounded in intuitive and embodied interactions such as categorizing shapes, counting, grouping, and drawing. We then move on to learn symbolic abstraction, manipulation, and eventually Computer Algebra Systems in order to explore with abstractions faster.

In practice, many students lose the intuitive underpinnings of those abstractions and resort to memorizing rules of symbol manipulation which become cognitively arbitrary and severed from their intuitive grounding in embodied interactions. The current mathematical tools, workflow, and computational languages also favor symbol-based math and programming as one advances to later stages.

My PhD dissertation creates a novel framework for Computer Algebra in which sketching and embodied interactions are the main activities to generate and work with symbolic abstractions. The framework consists of three unique brushes (iconic element, list, and function) and two interactions (fused representations and abstraction layers).

It closes the gap between the two ends of the math activities spectrum, retaining embodied interactions and yet allowing a learner/practitioner to compose and explore algebraic expressions. Interactive sketching enables one to create personalized artifacts that they most relate to, and do algebra with layers of such metaphors.

Humans invent, understand, and represent math by layers of physical and abstract metaphors. The framework embraces this cognitive theory, and favors bottom-up construction of mathematical expressions that are grounded in iconic metaphors, as opposed to the current top-down symbolic abstractions that require us to be power users of symbols and think using abstractions based on symbol manipulation. The framework is implemented as a sketch interface called Noyon.

What is Embodied Mathematics?

Embodied Mathematics can mean a lot of things really. However, my work heavily uses the "layers of metaphors" concept popularized by George Lakoff and Rafael Núñez. The embodied mathematics theory posits that our innate mathematical capabilities are limited to counting, categorizing and grouping. More specifically,

(a) categorizing and grouping objects or their iconic representations,

(b) subitizing (i.e., differentiate and count a few objects), and

(c) small amounts of addition and subtraction, when presented in terms of adding or removing objects from a group.

The authors proposed that there are four grounding metaphors (or the "4G"s as the authors call them) that help us understand and extend to higher level arithmetic and math operators, which are based on our innate capabilities and require little instruction.

The 4Gs are,

  1. Object Collection: categorizing and grouping objects,

  2. Object Construction: understanding parts and whole of an object, and composing different objects from different parts,

  3. Measuring Stick: denoting an amount/cardinality by the dimension of an object, and addition and subtraction as increasing/decreasing the length, and

  4. Motion along a Path: similar to the \textit{Measuring Stick}, but using the travelled length from the path origin as the unit. Their book further describes how advanced mathematical concepts are created based on these metaphors.

Based on the grounding metaphors, there are linking metaphors that then establish connections to even higher level concepts (metaphors such as a number as a point on a line, geometry figures as algebraic equations). In fact, the authors demonstrate that abstract mathematics consists of layers and layers of such metaphors that were established over many centuries. It should also be noted that a lot of the traditional abstract mathematics that we learn are linking metaphors, which require explicit instructions and long training.

Okay. So what's it got to do with my work exactly?

I favor the above theory on how mathematics is constructed in our mind, and I use it as a design guide in my work.

  • My aim is to leverage the basic grounding metaphors to design interactions for computer algebra and associated algorithms.

  • It is possible that there are other grounding metaphors that underlie abstractions that are fundamentally different from arithmetic and basic algebra. My secondary goal is to study and look for such metaphors.

    • These basic metaphors and constructs could be found from many sources. For example, history of mathematics, psychological studies, Brain MRI studies, etc. I aim to collaborate with appropriate scholars to find a basic set of metaphors underlying different fields of math, programming, and sciences.

Less is more. Or is it?

My usual aim is to find the minimum number of primitives, brushes, and metaphors that capture different fields of mathematics. I assume sketching is the main activity by which embodied math can be done on top of the world (in mixed reality).

For example, my work in standard algebra demonstrated that it is possible to construct and manipulate algebraic expressions using three brushes. Can the same be said about PDEs, graphs, and other fields? My current investigation involves PDEs and graphs, and capturing the classes of problems most commonly tackled in these fields. These classes then inform the design of the brushes. For example, so far, I and my students discovered that most graph representations used in books and papers can be captured using three kinds of brushes that lets us paint three different kinds of relationships between nodes. For PDEs, currently my investigation points towards six different brushes. Currently, I am developing sketch interfaces and VR/AR interfaces around these brushes.

We often tend to think that elegance means less. Less is beautiful. Less is more. My work is one among many works in this philosophy that demonstrates so. However, is less always more? The physicist Sabine Hossenfelder puts this debate into perspective. In her book "Lost in Math: How Beauty Leads Physics Astray," she says: "The laws of physics don't care about what humans find beautiful."

Less is not always more. The power of symbols and algebra-based manipulation, even if they are long complicated expressions computed by a machine, is undeniable. The physicist Leonard Susskind put it aptly. "Space itself may be more than three dimensions, but you and I evolved in a world of three dimensions, our neural architecture is appropriate for three dimensions. I can't visualize a five dimensional space, but I can add more letters to x, y, and z, and manipulate them using algebra."

The moonshot goal of any embodied math-based design of an interface, if I may suggest, would be to figure out the balance between symbolic manipulation (or symbol-based description) and embodied interactions and primitives. I am currently writing a paper on what this might look like for algebra, graphs, and PDEs.