Asked by isa
Consider the words typically associated with geometry. Are there any words that would be hard to precisely define? What words can you think of?
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GPT-5 mini
AI
Short answer: yes — many common geometric words are either taken as undefined primitives in axiomatic treatments, or are vague and only make precise sense after you choose a context (Euclidean vs spherical, topology vs differential geometry, measure theory, etc.). Below are typical examples and why each can be hard to pin down precisely, plus how people usually formalize them.
- Point, line, plane
Why hard: in axiomatic geometry these are often primitive terms (Hilbert’s axioms), not defined in simpler terms.
How to formalize: treat them as elements of sets with incidence relations, or model them as subsets of R^n.
- Between / Betweenness
Why hard: “A is between B and C” is intuitive but needs an axiom system (betweenness axioms) or coordinate definition.
How to formalize: use ordered coordinates or an axiomatic betweenness relation.
- Inside / Outside / Boundary
Why hard: “inside a curve” requires topology (Jordan curve theorem) to define rigorously in the plane.
How to formalize: use topological notions of interior, exterior, boundary.
- Curve / Surface / Shape / Figure
Why hard: these are used informally for many objects; “curve” could mean continuous image of an interval, differentiable 1‑manifold, rectifiable curve, algebraic curve, etc.
How to formalize: choose a class (continuous map, C^k manifold, algebraic variety, etc.).
- Tangent
Why hard: historically subtle; tangent as “just touches” is ambiguous. Needs limits or differential structure.
How to formalize: derivative (limit of secant slopes), tangent space of a manifold.
- Angle
Why hard: angle between lines is fine, but angle between curves needs tangents; also orientation issues.
How to formalize: measure between direction vectors or tangent lines.
- Parallel
Why hard: depends on geometry (Euclidean: coplanar nonintersecting; spherical: no parallels).
How to formalize: use axioms (Euclid’s fifth) or model-dependent definitions.
- Intersection / Touch / Cross
Why hard: touching vs crossing vs overlapping needs precise definitions (e.g., transversality in differential topology).
How to formalize: intersection sets, multiplicity, transversality conditions.
- Dimension
Why hard: multiple notions (topological dimension, Hausdorff dimension, manifold dimension) which can differ.
How to formalize: specify which dimension notion you mean.
- Smooth / Differentiable / Regular / Nice
Why hard: informal adjectives that hide precise orders of differentiability (C^k, analytic) or regularity conditions.
How to formalize: state C^k, C^∞, real-analytic, Lipschitz, etc.
- Simple / Self-intersecting / Nondegenerate
Why hard: “simple” polygon vs “simple curve” have specific but different formal meanings; “nondegenerate” depends on context.
How to formalize: give precise combinatorial/topological conditions or algebraic nondegeneracy constraints.
- Generic / Typical / Almost all / Random
Why hard: informal; “generic” can mean dense open, complement of algebraic variety, or full measure depending on the field.
How to formalize: use Baire category, Zariski-open sets, or measure-theoretic “almost everywhere.”
- Near / Close / Neighboring / Sufficiently small/large
Why hard: vague without metric or quantified epsilon/delta conditions.
How to formalize: provide epsilon–delta statements or explicit bounds.
- Shape equivalence terms (congruent, similar, isometric)
Why hard: congruence and similarity are precise, but “same shape” might mean any of these or something weaker (homeomorphism).
How to formalize: specify rigid motion, scaling, or topological/homeomorphic equivalence.
How to avoid ambiguity
- State the model or axioms you’re using (Euclidean R^n, Hilbert’s axioms, Riemannian manifold, algebraic variety, etc.).
- When using adjectives like “smooth” or “generic,” specify the technical meaning (C^2, measure zero, Zariski open, etc.).
- Replace vague words with explicit quantified statements when possible (give epsilon, bounds, or an exact property).
If you want, tell me a specific geometric word you’re worried about and I’ll give a precise formal definition appropriate to a chosen setting.
- Point, line, plane
Why hard: in axiomatic geometry these are often primitive terms (Hilbert’s axioms), not defined in simpler terms.
How to formalize: treat them as elements of sets with incidence relations, or model them as subsets of R^n.
- Between / Betweenness
Why hard: “A is between B and C” is intuitive but needs an axiom system (betweenness axioms) or coordinate definition.
How to formalize: use ordered coordinates or an axiomatic betweenness relation.
- Inside / Outside / Boundary
Why hard: “inside a curve” requires topology (Jordan curve theorem) to define rigorously in the plane.
How to formalize: use topological notions of interior, exterior, boundary.
- Curve / Surface / Shape / Figure
Why hard: these are used informally for many objects; “curve” could mean continuous image of an interval, differentiable 1‑manifold, rectifiable curve, algebraic curve, etc.
How to formalize: choose a class (continuous map, C^k manifold, algebraic variety, etc.).
- Tangent
Why hard: historically subtle; tangent as “just touches” is ambiguous. Needs limits or differential structure.
How to formalize: derivative (limit of secant slopes), tangent space of a manifold.
- Angle
Why hard: angle between lines is fine, but angle between curves needs tangents; also orientation issues.
How to formalize: measure between direction vectors or tangent lines.
- Parallel
Why hard: depends on geometry (Euclidean: coplanar nonintersecting; spherical: no parallels).
How to formalize: use axioms (Euclid’s fifth) or model-dependent definitions.
- Intersection / Touch / Cross
Why hard: touching vs crossing vs overlapping needs precise definitions (e.g., transversality in differential topology).
How to formalize: intersection sets, multiplicity, transversality conditions.
- Dimension
Why hard: multiple notions (topological dimension, Hausdorff dimension, manifold dimension) which can differ.
How to formalize: specify which dimension notion you mean.
- Smooth / Differentiable / Regular / Nice
Why hard: informal adjectives that hide precise orders of differentiability (C^k, analytic) or regularity conditions.
How to formalize: state C^k, C^∞, real-analytic, Lipschitz, etc.
- Simple / Self-intersecting / Nondegenerate
Why hard: “simple” polygon vs “simple curve” have specific but different formal meanings; “nondegenerate” depends on context.
How to formalize: give precise combinatorial/topological conditions or algebraic nondegeneracy constraints.
- Generic / Typical / Almost all / Random
Why hard: informal; “generic” can mean dense open, complement of algebraic variety, or full measure depending on the field.
How to formalize: use Baire category, Zariski-open sets, or measure-theoretic “almost everywhere.”
- Near / Close / Neighboring / Sufficiently small/large
Why hard: vague without metric or quantified epsilon/delta conditions.
How to formalize: provide epsilon–delta statements or explicit bounds.
- Shape equivalence terms (congruent, similar, isometric)
Why hard: congruence and similarity are precise, but “same shape” might mean any of these or something weaker (homeomorphism).
How to formalize: specify rigid motion, scaling, or topological/homeomorphic equivalence.
How to avoid ambiguity
- State the model or axioms you’re using (Euclidean R^n, Hilbert’s axioms, Riemannian manifold, algebraic variety, etc.).
- When using adjectives like “smooth” or “generic,” specify the technical meaning (C^2, measure zero, Zariski open, etc.).
- Replace vague words with explicit quantified statements when possible (give epsilon, bounds, or an exact property).
If you want, tell me a specific geometric word you’re worried about and I’ll give a precise formal definition appropriate to a chosen setting.
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