Limits of counting

and the linguistic roots of the

concept of number

Gennaro Chierchia

Harvard University

Siena,30 ottobre 08

An ongoing debate

i. how are our linguistic abilities

related to our intelligence?

ii. how much of language is natural

and how much is conventional?

A pole of the debate:

• Intelligence:

- ability to extract statistical regularities from

our environment

- flexible problem solving capacity

• Language:

- the way intelligence copes with the problem of

communicating

• Languages:

- different modes of coping

- different histories/cultures

In other words…

Intelligence:

- unspecialized

- not language-like

- it ‘produces’ language

The other pole of the debate

Language:

- not a product of intelligence

- highly structured computational device

- autonomous from extra-linguistic abilities

- independent of the pragmatics of

communication

Some more specific questions

• How do we count?

• How do number systems develop?

• How do we individuate?

Even though our abilities to count and to

individuate have extralinguistic roots,

grammar ‘reinvents’ them with novel

characteristics.

Number marking and the mass-count

distinction

Singular vs. plural marking

a. That chair (is nice)

b. Those chairs (are nice)

– Tables, boys, spoons,…

c. That blood (is rh positive)

d. * Those bloods (are rh positive)

– water, bread, gold, rice,…

Articles

Indefinite

a. a chair, a boy

b. *a bread, a gold

Definite

e. the water on the floor

maximality

f. the people in this room

g. the person in this room uniqueness

Numbers

a. three chairs

b. John’s chairs are at least twenty

c. * three bloods

d. * John’s blood is at least two

e. John’s blood (the blood John donated) is

at least two grams

Quantifiers (the D-system)

I. Count Ds

every boy; * every blood

II. Mass Ds

much water; little wine, *much chair

III. Mixed Ds

all water all boys

molta acqua molti ragazzi

Measures

a. That rice is 2 kilos

b. Those **apple**s are 2 kilos

c. That water is two liters

d. That gold is 20 karats

e. The water in the pool is 20 degrees

liters, kilos, degrees, karats, …

A restriction on measure phrases

(R. Schwarzschild)

f. I ate two kilos of rice (/**apple**s)

g. I drank two liters of water

h. * I drank 20 degrees of water

i. * I bought 20 karats of gold

j. I never drink 20 degree water

k. I bought a lot of 20 karat gold

What’s the difference between liters and degrees?

In Italiano

• f. Quel whisky è due litri

• g. Quel whisky è ben trenta gradi

• h. Ho bevuto 2 litri di whisky

• i. * Ho bevuto 30 gradi di whisky

• l. Ho bevuto whisky a 40 gradi

• m. * Ho bevuto whisky a 2 litri

Main generalizations

Mass Count

• Plural no yes

• Numbers no yes

• Indef. art no yes

• Def art. yes yes

• Complex distribution of Ds and measure phrases

Extralinguistic roots

• a. substances b. objects

→ mass

→ count

- Objects have boundaries, move as wholes along continuous

paths, retain their identity upon colliding with each other

- Substances don’t

• Three month old infants distinguish between

objects and substances. (Spelke, Carey)

Rhesus monkeys do too

(Hauser)

Problems: ‘Fake’ mass nouns

• a. Furniture, foliage, footwear, luggage…

mobilia, fogliame, argenteria,…

• b. That furniture is all is left from the fire

• c. * Those three furnitures is all is left

Nota bene: fake mass ≠ collective

• d. group, bunch, committee,…

• i. Who has more toothpaste?

• ii. Who has more chairs?

• iii. Who has more furniture?

[Barner and Snedeker, Cognition]

Crosslinguistic variation

Interlinguistic: Germanic vs. Romance

Capello Hair

Munizione Ammunition

Calzatura Footwear

Servitù Servant

Mobile Furniture

Intralinguistic:

a. mobili vs. mobilia

b. coins vs. change

Interim conclusions

• The mass count distinction cannot be assimilated to a

pre/extralinguistic ontological/conceptual one

• Is it pure morphosyntax?

– There is no morpheme specialized for ‘mass’

(cf. gender morphemes)

– ‘counting water(s)’ feels like a semantic mistake

– Nouns of substances are universally mass

Semantic Modelling

a+b+c

a+b a+c b+c

a b c

• a is a dog, b is a dog, c is a dog [singularities /‘atoms’]

• a+b are dogs, etc. [pluralities]

• Dogs are not on the wedge of extinction

• I cani non sono estinti; i dinosauri sì.

• a+b+c is the dog-kind

‘dogs’

→

a+b+c

a+b a+ c b+c

‘dog’ → [a b c]

Domains of Discourse: objects

a+b+c+d ….

a+b+c a+b+d b+c+d ….

a+b a+ c b+c a + d …

a b c d e…

• singularities

• pluralities

• kinds

Substances

a

[a1 a2]

b1+b2

a1

b3+b4

a2

water or rice can be partinioned in many ways

Domains of discourse

a+b+c

d

a+b b+c a+c

[d1 d2] [d1 d2 d3] …

a b c

• Some kinds (dogs, chairs) have standard minimal

samples (‘objects’)

• Others (rice, water) don’t (‘substances’)

Mapping syntax into meaning

dogs

a+b+c

water

[ d ]

a+b

b+c a+c

dog

[d1 d2] [d1 d2 d3]

[ a b c ]

• Nouns divide Domains into qualitatively homogenous regions

sets of atoms

• N

kinds

• Substances must be encoded as kinds

Counting

• How many things are there?

– There’s this table; then there is this table’s leg,…

• How many groups of people are there?

– There’s John and Bill; then there’s John and Bill

and Paul,...

• How many quantities of water are there?

To count we need a homogenous, disjoint set.

Three ways to count

I. atoms

• We can count through ‘atomic’ sets:

chairs, people, events,…

• In this case we simply count the atoms

(smallest samples of a kind)

• The dogs are three:

a+b+c

a+b b+c a+c

[ a b d ]

Three ways to count:

II. Context dependent nouns

two quantities of **apple**s

a+b+c+d+e

quantity of

**apple**s

a+b+c d+e a+c b+d+e etc.

**apple**

a b c d e

Three ways to count:

III. Measure Phrases

two pounds of **apple**s

a+b+c+d

pound of

**apple**s

**apple**

a+b c+d a+c b+d a+d b+c

a b c d

Numbers as measure phrases

Two hundred **apple**s

hundred **apple**s:

{a 1 +…+ a 100 a 2 +…+ a 101 a 3 +…+ a 103 …}

Two disjoint members of this set

How numbers come about

• Simple numbers:

one, two,…ten, hundred, thousand, million

• Multiplicative numbers:

three hundred, three thousand, thirty, …

• Additive numbers

hundred and ten,

twenty three drei und zwanzing

fourteen (vs. forty)

Munduruku: an eight number system

Pug

one

xip xip

two

Ebapug three Eba = arms

Ebadipdip

four

two + two

Pug pogbi

five

one hand

Dipdip pogbi

ten

Ebapug pogbi fifteen

Ebadipdip pogbi twenty

The birth of arythmetic

Numbers are created using:

• A base (fingers, bodyparts,…)

• coordination (= addition)

- give me a thousand (candies) and two (candies)

• complementation /recursion (= multiplication)

- give me two ( cats)

- give me two ( hundred ( cats))

- give me two ( kilos of (**apple**s/bread))

Summing up:

three kinds of count nouns

• ‘atomic’ nouns: **apple**

[a b c d e]

• context dependent nouns: quantity of **apple**s

[ a+c+d b+e ]

• measure nouns: kilo of **apple**s

[ a+b c+d]

• plurals: ‘closure’ under sum formation

Mass nouns as kind/totality denoting

• Canonical mass nouns: water

[ w ]

( w1 w2 w3) (w3 w4)

• Fake mass nouns: furniture

[ a + b + c]

Consequences

Two reasons why mass nouns are unsuited for

counting

– ‘formal’: to count we need a set; kinds are not sets

– ‘substantive’: mass kinds don’t individuate

univocally a set of atoms which can be counted

• To count with a mass noun we need to go from

the kind to a countable set

• liter of water = {x: x ≤ water & x measures 1l}

A different way of mapping syntactic categories

into semantic types

a+b+c

d

a+b b+c a+c

[d1 d2] [d1 d2 d3]

a b c

• N kinds

– water d

–man a+b+c

• All nouns will need a classifier or a measure phrase

for counting

Classifier languages: Chinese

• a. Count classifiers

i. San ge ren ii. San ben shu

three CL people three CL book

• b. Neutral classifiers

iii. San bang (de) rou iv. Yi da zhang zhi

three pounds meat one big CL paper

Language vs

extralinguistic conceptual systems

• Semantic domains: two sorted join semilattice (with a

univocally atomic portion and a non univocally

atomic one) + a reasoning facility = a logic

• Mapping syntax into semantics:

–N kinds

–N sets (of ‘atoms’) N kinds

• Mapping semantic structures onto other conceptual

systems

– Count nouns ↔‘objects’

– Mass nouns ↔‘substances’

A missing protagonist.

liters vs. degrees

• I drank twenty liters of scotch

• * I drank twenty degrees of scotch

⇒ liter(x) < liter(y)

• x < y

=/=> degree(x) < degree(y)

• liters are monotone with respect to water

• liter of water: take all one liter sized water amounts

and partition them into non overlapping sets

• preserving the semantic property of the head:

n members of ‘liter of water’ measure n liters

The architecture of denotational structures is…

• not reduceable to

– extralinguistic conceptual domains: Substances vs Objects

– the pragmatics of communication

• C’è del sangue sulla tua giacca

• Ci sono dei sangui sulla tua giacca

• Ci sono delle quantità di sangue sulla tua giacca

• universal

• parametrized

(i.e. associated with ‘variable’ mapping principles)

– there is semantic variation: man and ren have different denotations

– limited to ‘logical packaging’: [a b c] vs. a+b+c

a set vs. a kind (or a class-as-many vs a class-as-one)

In short….

• Denotational Structures have the same

architecture as (narrow) syntax

• They constitute a reasoning facility on

syntactic structures

• As any specific computational device,

Denotational Structures can be selectively

impaired

The strange case of F.A. (73)

[Semenza et al.(1995)]

Isolated deficit affecting the mass-count distinction

Correction tasks:

There is table in this room

There is a sand in the shoe

Production task:

Car/road

I saw a car on the road

Butter/bread

I spread a butter on the bread

• F.A. has not lost her marbles

• She has lost her ‘kinds’

-N kinds

-bread a+b+c bread [a b c]

Nature of the explanation

Formal model of a cognitive capacity

• Language specific

• Realized in the brain

• Possibly philogenetically related to

other cognitive capacities