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High Up (On Vagueness and Mathematics)

Imagine walking down Olympus Mons — the largest volcano in the solar system, on Mars. Its slopes are so gradual that you might walk for hours barely noticing the descent. At the top, you are clearly “high up.” At the bottom, you are clearly not. But somewhere in between, the description falters. You become unsure whether “high up” applies or not. That fuzzy middle zone is what philosophers call vagueness.

We can make this precise — or try to. Map the mountain onto a number line: 0 at the summit, 1 at the base. The set of points where you count as “high up” has a boundary. Call it sup — the least upper bound, the last point before “high up” runs out. The set of points where you count as “not high up” also has a boundary: inf, the greatest lower bound, the first point where “not high up” begins.

Now suppose — reasonably — that sup itself counts as “high up” (since every point below it does), and that inf counts as “not high up.” What happens when we ask how sup and inf relate to each other?

There are only three possibilities, and each one produces a contradiction:

  • If sup = inf, then that single point is both “high up” and “not high up.”
  • If sup > inf, then the real number line — which is dense, meaning there’s always another number between any two — guarantees a point z sitting between them. That point would be both “high up” (since it’s below sup) and “not high up” (since it’s above inf).
  • If inf > sup, the same logic applies in reverse.

The standard response is to blame the vague word. “High up” is imprecise — a folk term, not a technical one. Strip it away and mathematics, supposedly, is safe.

But stripping the vague term doesn’t solve the problem. It moves it.

Consider the wave theory of light. Its mathematical core — the equation governing refraction:

sin(α) / sin(β) = μ

— looks clean and precise. But the philosopher Mary Hesse pointed out that the equation, on its own, is ambiguous: it can be interpreted in multiple, entirely different ways. The symbols don’t come pre-labeled. Perhaps α and β aren’t angles of light at all — perhaps they’re the angles between the Pole Star and two planets at midnight. The mathematics would fit. Which interpretation is correct? The equation doesn’t say. Meaning doesn’t live in the symbols alone.

Vagueness and ambiguity are usually treated as distinct problems. Ambiguity means a word or expression has more than one possible meaning. Vagueness means a word has unclear edges — cases where it’s genuinely uncertain whether it applies. But consider: what if a word were both at once?

Thai has a word, krup, that technically means “yes” but functions more like a polite acknowledgment — because outright agreement can feel presumptuous, as if you’re confirming what the listener already knows. It occupies a middle space between assertion and non-assertion.

Now invent a word: snook. It means “tall” in some contexts and “not tall” in others. When applied to someone of borderline height — someone exactly at the edge of where “tall” is uncertain — is snook ambiguous, vague, or somehow both? Is there a vagueness between vagueness and ambiguity? If so, what does that do to the apparent clarity of mathematical symbols?

Even pure mathematics — mathematics with no interest in mountains or light — is soaked in vagueness. The discipline’s foundational concepts carry it: continuity, completeness, integral, limit. These are not casual approximations. They are the load-bearing terms of analysis, the branch of mathematics that underlies calculus.

And they are vague. Any careful textbook in real analysis will show you functions that slip through the formal definition of continuity — technically satisfying the definition while still behaving in ways the definition was meant to exclude. The definition doesn’t quite capture the intuition. The intuition doesn’t quite surrender to the definition. The gap between them is not a failure waiting to be fixed. It is where thinking happens.

Without words like “continuity” and “completeness” — words that mean something intuitively before they mean something formally, and that keep some of that intuitive life even afterward — mathematics would be unlearnable. Students would have nothing to grab onto. The vagueness isn’t what mathematics tolerates in spite of itself. It’s part of what mathematics thinks with.

The fantasy of a perfectly precise formal world, unsullied by the messiness of natural language, is just that — a fantasy. Vagueness goes all the way down.