Language ⇄ Math
Both are rule-based symbol systems. Letters & words; numbers & operators. Grammar & algebra. Semantics that point to meaning-ideas in one case, quantities in the other. This page is a visual map of those parallels.
Two systems, one skeleton
Language & math both turn symbols into structure into meaning. Logic is the common skeleton we hang them on.
- Symbols: tokens in an alphabet
- Syntax: rules for valid form
- Semantics: how forms map to meaning
- Generativity: finite rules, infinite expressions
We can travel either direction: phrase → formula, or formula → description.
Symbols ↔ Tokens
Different costumes, same job: carry meaning under rules.
Vocabulary differs, structure rhymes.
Language parse (syntax tree)
Math expression tree
Semantics: one idea, two renderings
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Slide from a natural sentence to a formula. Colors track matching parts.
Precision ⇄ Ambiguity
Language spreads meaning across contexts (wider curve). Math pins it down (narrower curve). Choose a word to see the spread.
Love: many senses (romance, charity, delight…)
Translation: saying the same thing twice
- “five plus three equals eight”
- “if it rains then the street is wet”
- “not both A & B”
- 5 + 3 = 8
- r → w
- ¬(A ∧ B)
Once we fix the dictionary (symbols) & the grammar (syntax), meaning can travel cleanly across domains.
Logic: the shared backbone
Truth tables are the same no matter whether you describe them in words or symbols.