Where the formal language Analogy Breaks

Calling mathematics a formal language 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.

Software architecture also lives by abstraction: a good model compresses many cases while preserving the invariants that matter.

What the metaphor gets right

Software architecture also lives by abstraction: a good model compresses many cases while preserving the invariants that matter.

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 mathematically correct model can still encode the wrong physical assumptions. Certainty inside a formal system is not the same as empirical certainty about the world.

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

Mathematics constructs formal structures from definitions, relations and inference. In science its power comes from mapping aspects of physical systems onto structures from which consequences can be derived.

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

1. What mechanism does this comparison help me see?
2. Which engineering assumption does the natural system not share?
3. 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

Greek mathematical traditions tied proof, ratio and geometry to philosophical questions; the historical Pythagoras is particularly hard to reconstruct because no securely attested writings survive.

Mathematics is where symbolic reasoning most dramatically escapes immediate experience and then returns with predictions about reality.

Do mathematical structures describe reality because we selected useful structures, because reality is mathematical, or because the distinction is incomplete?

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

• Stanford Encyclopedia of Philosophy — Pythagoras
• Stanford Encyclopedia of Philosophy — Pythagoreanism

These links are starting points for the scientific and historical ideas. The systems interpretation, analogies and conclusions here are my own.