Perspectives
Where the coordinate system Analogy Breaks
A coordinate system is deliberately chosen. Physical spacetime and biological form impose constraints that may not fit everyday Euclidean…
Calling geometry a coordinate system is useful for about five minutes. The comparison can expose structure that ordinary language hides. The problem begins when the comparison stops behaving like a tool and starts behaving like a claim about what the thing literally is.
Architects make similar choices when selecting schemas, reference frames and abstractions: representations can simplify reasoning without becoming the thing represented.
What the metaphor gets right
Architects make similar choices when selecting schemas, reference frames and abstractions: representations can simplify reasoning without becoming the thing represented.
This is why engineering metaphors are so attractive. They compress a complicated relationship into something we can manipulate mentally. In the best case, the metaphor produces predictions and sharper questions.
What it smuggles in
A coordinate system is deliberately chosen. Physical spacetime and biological form impose constraints that may not fit everyday Euclidean intuition.
Engineering words often carry hidden assumptions: a designer, a specification, modular ownership, explicit goals, a stable machine underneath the program. Those assumptions must be tested rather than inherited.
Return to the physical story
Geometry describes relations of position, distance, angle, curvature and transformation. A coordinate system makes some relationships convenient to express, but it does not create the underlying geometry.
Once the mechanism is back in view, the analogy can be kept on a short leash. We can state which relation is genuinely similar and which difference would make the analogy generate a wrong prediction.
My three tests for a metaphor
- What mechanism does this comparison help me see?
- Which engineering assumption does the natural system not share?
- What would I predict incorrectly if I took the comparison literally?
The third test is the most important. An analogy that cannot fail is decoration, not explanation.
Beyond the borrowed language
Euclidean geometry built a powerful deductive system; non-Euclidean geometries later showed that apparently obvious spatial assumptions can be alternatives rather than necessities.
Geometry is a discipline of invariant relationships: descriptions can change while structure persists.
Which properties belong to reality, and which belong to the coordinate system we chose?
A good metaphor should eventually make itself less necessary. It gets us close enough to the real system that we can see where the scaffolding ends.
Reading trail
These links are starting points for the scientific and historical ideas. The systems interpretation, analogies and conclusions here are my own.