Numerical Cognition, linguistic relativity and the ontology of numbers

ECAP 7 — Workshop on Numerical Cognition and Mathematical
Ontology
Milan September 3, 2011

“Numerical cognition, linguistic relativity and
the ontology of numbers”

Mouhamadou El Hady BA
Institut Jean Nicod ENS-EHESS Paris

Introduction:

– Cardinal view of numbers: numbers are second order concepts applying to
collections of objects (Frege, Russell)
– Ordinal view of numbers: numbers are individuated by their position in a
sequence (Dedekind)
– Which conception is better?
– Should we accept Platonism about numbers like most mathematicians?
– Having a look at the performance of one-two many language speakers
could be helpful to address these questions.

Hady Ba "Numerical Cognition, Linguistic Relativism & the Ontology of Numbers" ECAP 7 M

Overview of the Talk
Numerical cognition, linguistic relativity and the ontology of numbers

Stating the obvious
One-two many languages
Linguistic relativity
• Sapir about SWH
• Whorf about SWH
• Taking stock
Linguistic Relativity & Numerical Cognition
• Peter Gordon testing Pirahas (Science 2004 vol 306)
• Pica & alii testing Mundurukus (Science 2004 vol 306)
Linguistic determinism and Numerical Ontology
• What God really gave us
• What about Platonism?
• Platonism and the analogy with language


Stating the obvious

Science is not equivalent to voodoo!


Some amazonian languages have words for only a handful of distinct

numbers

Members of these communities use numbers in a somewhat fuzzy way

They have problem with exact calculations involving even small figures

Case in point Pirahas & mundurukus



Piraha's counting system
Two number words (for one and two)

Not always used for exact quantities (actually “roughly one”
like in the english “a couple of”)

Never used recursively, nor productively to engender greater
numbers

Another word meaning “many”



Piraha's counting system (P. Gordon, Everett)



Munduruku's counting system (Pica & alii)
Number words for 1 through 5

Lack of use of the word for “5” by younger children

No use of the numerals in counting sequence nor with
precision but they can do so when asked

Never used recursively to produce greater numbers

Presence of determiners like “many”, “some” used for
quantification


Munduruku's counting system (Pica & alii)



Questions
Does such a numerical system retroact on the cognitive abilities of
one-two many language speakers?

Can they help us assess Linguistic relativity?

Does it teach us anything about the nature of numbers?


Linguistic Relativity
Weak version:
The language we speak determine our thinking and how we perceive objective
features of the world.
– Pretty trivial

Strong version (aka Sapir-Whorf Hypothesis):
The influence of language is so pervasive that people from different linguistic
backgrounds can have conflicting and incommensurable conceptualization of the
same objective reality.
– Highly controversial



Sapir-Whorf hypothesis:
Sapir is supposed to be a reluctant proponent of SWH “It would be naïve to
imagine that any analysis of experience is dependent on pattern expressed in
language” (Sapir & Swadesh)

Cognitive scientists cf. Pullum (1991), Pinker (1994): SWH is a urban legend
promoted by Whorf and stemming from an ignorance of the nature of our
language faculty (cf. Whorf on Eskimos, Sapir : “language is no part of
biology”)



Sapir about SWH :
Thought itself is a product of language :
“It is, indeed, in the highest degree likely that language is an instrument originally
put to uses lower than the conceptual plane and that thought arises as a refined
interpretation of its content. The product grows, in other words, with the instrument,
and thought may be no more conceivable, in its genesis and daily practice, without
speech than is mathematical reasoning practicable without the lever of an appropriate
mathematical symbolism. No one believes that even the most difficult mathematical
proposition is inherently dependent on an arbitrary set of symbols, but it is
impossible to suppose that the human mind is capable of arriving at or holding such a
proposition without the symbolism.” (1949: 15)



Sapir about SWH :
Language is an encompassing formal system :
“Language (…) is also a self-contained, creative symbolic organization,
which, not only refers to experience largely acquired without its help but
actually defines experience for us by reason of its formal completeness (…) .
In this respect, language is very much like a mathematical system, which,
also, records experience, in the true sense of the word, only in its crudest
beginnings but, as time goes on, becomes elaborated into a self-contained
conceptual system which previsages all possible experience in accordance
with certain accepted formal limitations. ” (1931: 578)



Sapir about SWH :
Language defines experience and induces
incommensurability:
“Such categories as number, gender, case (…) are, of course, derivative of
experience at last analysis, but, once abstracted from experience, they are
systematically elaborated in language and are not so much discovered in
experience as imposed upon it because of the tyrannical hold that linguistic
form has upon our orientation in the world. Inasmuch as languages differ
very widely in their systematization of fundamental concepts, they tend to be
only loosely equivalent to each other as symbolic devices and are, as a matter
of fact, incommensurable (…)” (1931: 578)


Whorf about SWH :
Linguistic determinism :
“We dissect nature along the lines laid down by our native languages. The
categories and types that we isolate from the world of phenomena we do not
find because they stare every observer in the face; on the contrary, the world
is presented in a kaleidoscopic flux of impressions which has to be organized
by our minds -and this means largely by the linguistic systems in our minds.
We cut nature up, organize it into concepts, and ascribe significances as we
do, largely because we are parties to an agreement to organize it in this way.
(…) The agreement is, of course, an implicit and unstated one but its terms
are absolutely obligatory; we cannot talk at all except by subscribing to the
organization and classification of data which the agreement decrees..”
(1940:213-4)



Whorf about SWH :
A new Principle of Relativity :
“We are thus introduced to a new principle of relativity, which holds that all observers
are not led by the same physical evidence to the same picture of the universe, unless
their linguistic backgrounds are similar, or can in some way be calibrated.” (1940:214)

“From this proceeds what I have called the “linguistics relativity principle,” which
means, in informal terms, that users of markedly different grammars are pointed by
their grammars toward different types of observations and different evaluations of
externally similar acts of observation, and hence are not equivalent as observers but
must arrive at somewhat different views of the world..” (1940-bis:222)


Whorf about SWH :
Even Science is Relative :
“From each such unformulated and naïve world view, an explicit scientific world view
may arise by a higher specialization of the same basic grammatical patterns that
fathered the naïve and implicit view. Thus the world view of modern science arises
by higher specialization of the basic grammar of the Western Indo-European
languages.” (1940-bis:222)

“What surprises most is to find that various grand generalizations of the Western
world, such as time, velocity, and matter, are not essential to the construction of a
consistent picture of the universe. (…) Hopi may be called a timeless language. It
recognizes psychological time(…) Among the peculiar proprieties of Hopi time are
that it varies with each observer, does not permit simultaneity, and has zero
dimensions.” (1940:216)

Whorf about SWH :
But Relative doesn't mean gratuitous:
“Science of course was not caused by this grammar; it was simply colored by it. It
appeared in this group of languages because of a train of historical events that
stimulated commerce, measurement, manufacture, and technical invention in a
quarter of the world where these languages were dominant” (1940-bis:222)

Even a timeless language could develop physics
“How would physics constructed along these lines work, with no T (time) in its
equations? Perfectly as far as I can see” (1940-217)
No “velocity” other concepts like “intensity” Objective/Subjective



Taking stock :
Language is a creative formal system
Producing the inescapable categories under which we
conceptualize the world
So different languages give rise to different cognitive and
conceptual frameworks
And different “sciences”



Consequence :

If SWH is true, one-two many language speakers should have
different cognitive capacities with regard to number manipulation
than English or French speakers.

So, what about Munduruku and Pirahas speakers?


Peter Gordon testing Pirahas (Science 2004 vol 306)

Matching task: Putting tokens so as to match the number of objects of the
reference group
Relatively good accuracy for two/three items but considerable deterioration
beyond that, up to 8-10



Time constraints affect performance: correlation between
set size and accuracy beginning at set size 3
Performance for set size above 3 is poor but not random
Ability to estimate and compare larger sets is preserved



===> Pirahas can't manipulate exact quantities above 3 and the only plausible
explanation is that having a one-two many counting system affects their numerical
cognition.
===> Vindication of Sapir Whorf's linguistic determinism?
“Such categories as number (…) are not so much discovered in experience as
imposed upon it because of the tyrannical hold that linguistic form has upon our
orientation in the world.” Sapir (1931: 578)
“(...) the nature of the language is the factor that limits free plasticity and rigidifies
channels of development in the more autocratic way. This is because a language is a
system, not just an assemblage of norms.” Whorf (1939)
=======>Not everyone agrees though!



Pica & alii testing Mundurukus (Science 2004 vol 306)

Even though Munduruku's numbers range from 1 to 5, only the first two
are used accurately to describe a set. E.g.: 4 units can as well be described
as 3 than as 5
Almost as good as French subjects in approximating tasks (comparing
disproportionate sets 20/80, approximate operation with large numbers,
distance effect... )
Systematic failure at exact arithmetic calculation involving numbers
above 4



Pica & alii testing Mundurukus (Science 2004 vol 306)

Pica & coll. still won't accept SWH arguing that it is not the absence of
exact number lexicon that is responsible for the lack of arithmetic
competence but the lack of a “procedure for fast apprehension of exact
numbers beyond 3 or 4”.
This is an ad hoc hypothesis as far as we can see. Such a procedure would
have been linguistic
We'd rather agree with Gordon, Sapir & Whorf that having a language
encoding a lexicon for precise numbers is essential for the ability to
understand and manipulate exact numbers.
Consequences for the ontology of numbers?


Linguistic determinism and Numerical
Ontology

What God really gave us
Leopold Kronecker:
“God gave us the integers; the rest is the work of Man”
Under linguistic determinism, SWH style:

What's universal is a cognitive faculty devoted to estimating and
comparing quantities

To go from that faculty to the production of integers and exact
calculation, we need a natural language encoding exact numbers

Recursivity is not sufficient (cf. Everett on Pirahas)
=====> even the integers are not God given


Ontology

What about Platonism?
Our constructivist approach is compatible with Platonism but...
We could use Occam's razor :

Ordinality comes first: without a linguistically encoded sequence,
no arithmetics nor mathematics (Sorry Valeria Giardino!)

Cardinality and operations upon numbers are constructed via
language

then there is no need to give numbers ontological reality


Ontology

Platonism and the analogy with language
Universal Grammar + Environment =====> Language


But no need for platonism about Language
Cognitive endowment + Appropriate language=====> Numerical system

No need either for platonism about numbers or mathematical
entities


Many thanks for your attention and feel free to reach
me at mehba@ehess.fr


Numerical Cognition, linguistic relativity and the ontology of numbers

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