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

Hady Ba "Numerical Cognition, Linguistic Relativism & the Ontology of Numbers" ECAP 7 Milan 2011

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.


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”


Linguistic Relativity & Numerical Cognition

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



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



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 [email protected]

mailto:[email protected]

numerical cognition, linguistic relativity and the ontology of numbers

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