Imagine that you’re running a design studio in ancient Rome. The royal tax collector visits you and asks for a vision of the future of accounting systems. With the benefit of hindsight, we can spot a disastrous conceptual limitation: they’re using Roman numerals to do accounting! Trivial arithmetic operations are difficult and error-prone. There are surely many other problems with their practices, but could any design studio’s process have facilitated the invention of Hindu-Arabic numerals?

That number system represents both a profound work of design and also a profound mathematical insight. Place value is a great exemplar: the meaning of a numeral changes according to its position, but each position’s numerals use the same visual representations (digits) and operations (when adding 22 + 33, you add 2 + 3 in two different places). This self-similarity is critical to number system’s power. It expresses both deep mathematical ideas (commutativity, associativity, distributivity) and deep ideas in design (abstract representation, parsimony, spatiality).

To invent that number system, you’d need extraordinary proficiency in both design and mathematics, capable of an Insight through making feedback loop which uses the evolving number system not just to improve your own mathematical understanding, but to have original, world-class mathematical insights; and which uses those insights to improve the evolving number system.

Consider the studio design process. You’d interview everyone involved thoroughly to understand their process. You’d expand/rip the brief—trying to really access what’s meant by “accounting systems,” or even questioning why the accounting needs to happen in the first place. You’d bring the accountants and the royal mathematicians into your design process with rapid-fire idea generation and sketching sessions. You’d regularly return to the studio to synthesize contributors’ sketches and your observations into something new.

This practice would surely create improvements to the accountants’ systems. But it wouldn’t yield anything like the mathematical insight needed to arrive at Hindu-Arabic numerals, which would appear to require years (or generations?) of expert development by a mathematician–designer.

Matuschak, A., & Nielsen, M. (2019 0). *How can we develop transformative tools for thought?* https://numinous.productions/ttft