--- title: "Definitions as Design Decisions" description: "A definition is chosen rather than discovered, and the choice decides which theorems become provable. Comparing two candidate definitions of one idea is the fastest way to see what a definition costs." canonical: https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d source: https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Definitions as Design Decisions A definition is chosen rather than discovered, and the choice decides which theorems become provable. Comparing two candidate definitions of one idea is the fastest way to see what a definition costs. Subject: Mathematics · Area: Mathematical Thinking · Ages 22 to 24 Page: https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d ## Ready when they can - Compare two definitions of one idea and say what each one makes easy - Show that a theorem fails under a slightly weaker definition - Explain a definition's edge cases and why they are included or ruled out ## Lesson: Definitions are choices with a price A definition is chosen rather than discovered, and the choice decides which theorems become provable. The fastest way to see the price is to compare two candidate definitions of one idea. One definition says a function is a rule; another says it is a set of input-output pairs. Under the pairs view a function literally is its graph, so deciding whether two formulas define the same function becomes a mechanical check of pair sets. **Example.** Compare f(x) equals x squared over x with g(x) equals x. The formulas agree wherever both are defined, but f is undefined at 0 while g accepts 0. Different domains mean different functions. On the integers from negative 2 to 2 they agree at 4 points and differ only at 0. Thomae's function sharpens the moral: it is 0 at irrationals and 1 over q at p over q in lowest terms, hence continuous at every irrational yet wild at rationals. Edge cases expose the design. The empty product is set to 1 so that product formulas keep working when the index set runs dry. The value 0 factorial equals 1 is adopted for the same reason: it keeps n factorial equals n times (n minus 1) factorial true at n equals 1, and no theorem forces it. The subspace definition says nonempty because closure alone would let the empty set pass vacuously. With the indicator f of the rationals, f(0) plus f(root 2) plus f(1 over 2) is 1 plus 0 plus 1, giving 2. **Tip.** Test whether a hypothesis is load-bearing by dropping one word and watching the theorem fail. Pointwise continuity lets the allowed gap depend on the location, while uniform continuity demands one tolerance for every pair at once. That global quantification is exactly what breaks for x squared on the whole line. If removing a clause breaks a theorem, that clause carries real weight. **Recap.** Compare candidate definitions, probe their edge cases, and drop clauses to find which words do the work. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Sigma-Algebras and Measurable Sets](https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk) - [Topological Spaces: Continuity Without Distance](https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI) - [Writing a Proof Someone Else Can Check](https://lightmysky.com/learn/mathematics/writing-a-proof-someone-else-can-check-mt_nZ-zMzj18N) ## Opens up - [Counterexamples and Testing a Conjecture](https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM)