Geomancy and matrix algebra

Mathematical approaches to geomancy usually focus on its binary aspects. This is probably just Maslow’s hammer: our contermpoary lives are based around digital technology, and we see that technology in historical practices. Shoddy scholarship about historical transmission aside, I think this is less than ideal for a couple of reasons. First, historical geomancy was clearly understood in terms of modular arithmetic and parity. The same method for punctuation was used in sortes works like the Prenostica Socratis Basilei¹ to perform addition modulo 10.² Second, historical geomancers never bothered with arithmetic with figures that had less than or more than four rows.³ So, historical geomancy didn’t function like binary arithmetic.

Let me introduce a different mathematical approach, which I think is closer to the mark. First, instead of figures-as-numbers, let’s interpret them as arrays of numbers. Caput draconis, then, might be

It might not be incredibly clear why I chose to represent passive/active lines as -1 and 1, or why I chose to put them on the diagonal. However, there’s a good reason for this. Because of the way normal matrix multiplication works, this is actually equivalent to geomantic conjugation. Here is Caput draconis conjugated with Amissio, giving Puer:

There is already a nifty notation for a diagonal matrix, which is a matrix that has zero entries everywhere except the diagonal. Here’s the same conjugation as above:

We can also use the Hadamard product on column vectors and get something damn near equivalent to how conjugation is represented historically:

The question is: why would we?

The interpretation of historical mathematics

The simple fact of the matter is that historical geomancers did not think of geomancy as binary arithmetic. Neither did they think that it was a sort of matrix multiplication. In fact they had no knowledge of binary numerals, nor of matrices.

Most mathematicians are naive little-p platonists, that is, they believe that mathematical objects exist independently of us and outside of language. Or , at least, that’s how they talk: one uses verbs like ‘discover’ to describe mathematical achievements, not ‘invent’. And this is perfectly fine: I myself am a platonist, I suppose. However, this is not useful for historical research. If mathematical objects exist ‘out there’, they are timeless (Zalta uses the term ‘nonspatiotemporal‘). Yet as historians we are concerned with temporal and spatial objects.

Historians of mathematics are often too concerned with priority, even priority that had no historical influence.⁴ Importantly, mathematical objects are fungible, if not metaphysically, then in practice: the same ‘0’ I use is the same ‘0’ you use. Historical objects are not. Two witnesses of the same work are not the same, but have distinct biographies. Ideas are historical objects, too: their content matters, but their context matters too, maybe even more than the former.

It might be interesting to understand the mathematical underpinnings of a given historical practice, but we must remain steadfast methodological antiplatonists. For the purposes of historical research, pre-colonial Mayas, medieval Indians, and modern Westerners do not use the same ‘0’ but historically distinct ones with their own biographies, their own transmissions, their own lines of flight.

Footnotes

1. See Guardo, Albero Alonso. Prenostica Socratis Basilei: Étude, édition critique et traduction. Classiques Garnier, 2015. ↩︎
2. This is not strictly accurate — since this text predates the introduction of 0 to Europe, so it was held that $x \textup{ mod }x = x$. The same goes for prognostic spheres, which do not show 0 but do show their modulus. ↩︎
3. This again is not strictly accurate; a Byzantine manuscript (Harley MS 5596) seems to witness a simplified form of geomancy with three row figures. The 19th century Oraculum attributed to Napleon for some reason uses five row figures. My point still stands, though, as there doesn’t seem to be an arbitrary length of binary numbers used anywhere. ↩︎
4. For example, Ares, Juan, et al. “Who discovered the binary system and arithmetic? Did Leibniz plagiarize Caramuel?.” Science and Engineering Ethics 24.1 (2018): 173-188. Conspiracy theories about plagiarism notwithstanding. ↩︎