[WWL+23]: Numeral Frequency and Roundness

[Sig88] [WWL+23]

Corpus study showing number frequency is predicted by: (a) log magnitude, and (b) graded roundness via Sigurd/Jansen & Pollmann k-ness properties.

Key finding: each k-ness property has an independent positive effect on frequency, with 10-ness being the strongest predictor and multipleOf5 the weakest.

Register Effect

Informational texts (Wikipedia) show stronger roundness effects than non-informational texts (fiction, conversation), suggesting roundness interacts with communicative goals.

β coefficients from the negative binomial regression (§4.2, [WWL+23])

structure WoodinEtAl2024.RoundnessCoefficient : Type

Regression coefficient for each k-ness property on log frequency.

Higher β = stronger positive effect on numeral frequency in corpora. All coefficients are positive: each property independently increases frequency.

• property : String
• β : ℚ

@[implicit_reducible]

instance WoodinEtAl2024.instReprRoundnessCoefficient : Repr RoundnessCoefficient

Equations

• WoodinEtAl2024.instReprRoundnessCoefficient = { reprPrec := WoodinEtAl2024.instReprRoundnessCoefficient.repr }

def WoodinEtAl2024.instReprRoundnessCoefficient.repr : RoundnessCoefficient → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def WoodinEtAl2024.β_tenness : RoundnessCoefficient

β coefficients ordered by magnitude (§4.2).

Equations

• WoodinEtAl2024.β_tenness = { property := "10-ness", β := 446 / 100 }

def WoodinEtAl2024.β_2_5ness : RoundnessCoefficient

Equations

• WoodinEtAl2024.β_2_5ness = { property := "2.5-ness", β := 384 / 100 }

def WoodinEtAl2024.β_5ness : RoundnessCoefficient

Equations

• WoodinEtAl2024.β_5ness = { property := "5-ness", β := 339 / 100 }

def WoodinEtAl2024.β_2ness : RoundnessCoefficient

Equations

• WoodinEtAl2024.β_2ness = { property := "2-ness", β := 274 / 100 }

def WoodinEtAl2024.β_mult10 : RoundnessCoefficient

Equations

• WoodinEtAl2024.β_mult10 = { property := "multipleOf10", β := 245 / 100 }

def WoodinEtAl2024.β_mult5 : RoundnessCoefficient

Equations

• WoodinEtAl2024.β_mult5 = { property := "multipleOf5", β := 6 / 100 }

def WoodinEtAl2024.roundnessHierarchy : List RoundnessCoefficient

The 6 coefficients in descending order.

Equations

• WoodinEtAl2024.roundnessHierarchy = [WoodinEtAl2024.β_tenness, WoodinEtAl2024.β_2_5ness, WoodinEtAl2024.β_5ness, WoodinEtAl2024.β_2ness, WoodinEtAl2024.β_mult10, WoodinEtAl2024.β_mult5]

theorem WoodinEtAl2024.hierarchy_ordering : β_tenness.β > β_2_5ness.β ∧ β_2_5ness.β > β_5ness.β ∧ β_5ness.β > β_2ness.β ∧ β_2ness.β > β_mult10.β ∧ β_mult10.β > β_mult5.β

Frequency-weighted roundness score

def WoodinEtAl2024.weightedRoundnessScore (n : ℕ) : ℚ

Frequency-weighted roundness score using Woodin et al.'s β coefficients.

Unlike the unweighted roundnessScore (which counts properties equally), this weights each property by its empirical frequency effect.

Equations

• One or more equations did not get rendered due to their size.

theorem WoodinEtAl2024.weighted_100_gt_50 : weightedRoundnessScore 100 > weightedRoundnessScore 50

theorem WoodinEtAl2024.weighted_50_gt_110 : weightedRoundnessScore 50 > weightedRoundnessScore 110

theorem WoodinEtAl2024.weighted_7_eq_zero : weightedRoundnessScore 7 = 0

theorem WoodinEtAl2024.weighted_50_pos : weightedRoundnessScore 50 > 0

weightedRoundnessScore 50 > 0: 50 has multipleOf5, multipleOf10, 2.5-ness, 5-ness, and 10-ness (50 = 5 × 10¹), so its weighted score is the strictly positive sum of those β coefficients.

theorem WoodinEtAl2024.weighted_50_gt_7 : weightedRoundnessScore 50 > weightedRoundnessScore 7

theorem WoodinEtAl2024.roundness_prior_monotone : weightedRoundnessScore 100 > weightedRoundnessScore 50 ∧ weightedRoundnessScore 50 > weightedRoundnessScore 7

RSA utterance prior from corpus frequency. Rounder numerals have higher prior weight, so weightedRoundnessScore doubles as an empirically-grounded RSA utterance prior: rounder numerals are more likely to be chosen, all else equal. The strict-monotonicity chain 100 > 50 > 7 realises this on representative cases.

Register effect data

inductive WoodinEtAl2024.Register : Type

Register type from corpus analysis.

Informational registers (Wikipedia) show stronger roundness effects than non-informational registers (fiction, conversation).

• informational : Register
• nonInformational : Register

@[implicit_reducible]

instance WoodinEtAl2024.instReprRegister : Repr Register

Equations

• WoodinEtAl2024.instReprRegister = { reprPrec := WoodinEtAl2024.instReprRegister.repr }

def WoodinEtAl2024.instReprRegister.repr : Register → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance WoodinEtAl2024.instDecidableEqRegister : DecidableEq Register

Equations

• WoodinEtAl2024.instDecidableEqRegister x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

structure WoodinEtAl2024.RegisterEffectDatum : Type

Register effect datum: roundness β is larger in informational texts.

• register : Register
• roundnessEffectMagnitude : String
• notes : String

def WoodinEtAl2024.instReprRegisterEffectDatum.repr : RegisterEffectDatum → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance WoodinEtAl2024.instReprRegisterEffectDatum : Repr RegisterEffectDatum

Equations

• WoodinEtAl2024.instReprRegisterEffectDatum = { reprPrec := WoodinEtAl2024.instReprRegisterEffectDatum.repr }

def WoodinEtAl2024.informationalEffect : RegisterEffectDatum

Equations

• One or more equations did not get rendered due to their size.

def WoodinEtAl2024.nonInformationalEffect : RegisterEffectDatum

Equations

• One or more equations did not get rendered due to their size.