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Cognition and Culture : a symbiotic relationship Luis Moreno-Armella. Dartmouth, April 2007. We live in an artificial world, the world of culture . Biolog y. Cultur e. ?. Biolog y. Cultur e. Computa t ional cut. Human beings use an unique mode of computation : symbolic computation

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Cognition and Culture:

a symbiotic relationship

Luis Moreno-Armella

Dartmouth, April 2007




Computational cut

Human beings use an unique mode of computation:

symbolic computation

and keep an equilibrium between the analogue and

symbolic computational modes.


Voluntary memory

Cognitive transition

1.5 millions ago


Voluntary memory

Refinement of skills: cutting, throwing, manufacturing, TOOLS…


Voluntary memory

control of actions…

Explicit intentional act


Voluntary memory:

Distributed knowledge net, (community sense)



Mythic culture


Symbolic artifacts


Symbolic technology`




Externalization of memory…


The new symbolic capacity:

- Representation of quantity (bones)

- Envelopes and cuneiform writing

- Alphabetic writing (representing ideas --Greece)


External memory field

Symbolic technology

That works as a cognitive mirror


Writing became




Cognition and culture are the outcomes of

temporal processes.

Human cognitive capacity is the result of millions of years of evolution.

Human culture (even its earliest stages) is much more recent.




The study of human cognition has been too often

carried on as though humans had no culture, no

variability and no history.

Today, perhaps, things have changed on this respect.

Especially when it comes to the impact of culture on



Intentions of tools and actions were projected on

the tools and were crystallized in them

The first level of symbolization results from

crystallizing the intentionality and the

actions that emerge from that intentionality.


The genuine symbols might occur when instead

of projecting our intentionality onto the external

world, we project it insideourselves.

That is, when we use the symbols as crafted


Then, we enter the realm of metacognition.


Arithmetic: Ancient Counting Technologies

  • Evidence of the construction of one-to-one
  • correspondences between arbitrary collections of
  • concrete objects and a model set (a template)can
  • already be found in between 40,000 and 10,000 B.C.
  • Hunter-gatherers used bones with marks (tallies)
  • as reckoning devices.

In Mesopotamia, between 10000 B.C. and 8000, B.C.,

People used sets of clay bits as modeling sets.

However, this technique had severe limitations.

To deal with large collections, we would need increasingly

larger model sets with evident problems of manipulation

and maintenance.


The idea that emerged was to replace the

elements of the model set with clay pieces of

diverse shapes and sizes,

whose numerical value were conventional.

The counters that represented different amounts and sorts

of commodities—according to shape, size, and number—

were put into an envelope that was later sealed. To secure the

information contained in an envelope, the shapes of the counters

were printed on the outer side of the envelope.


The shape of the counter is impressed on the

outer side of the envelope.

The mark on the surface indicates the counter inside.

The mark on the surface keeps an indexical relation

with the counter inside as its referent.


Afterward, instead of impressing the counters against

the clay, due perhaps to the increasing complexity of the shapes involved, scribes began to draw on the clay the shapes of former counters.

But drawing a shape and

impressing a shape from a material object are

extremely different activities.

Drawing involves

a gesture-structure that goes deeper into

the intentionality that

crystallize the social co-action involved.


The contextual constraints of the diverse numerical

systems, constituted a conceptual barrier for the

mathematical evolution of the numerical systems.

Eventually, the collection of numerical–contextual

systems was replaced by the sexagesimal system:

a genuine numerical positional system.


There is still an obstacle to have a complete numerical system:

  • the presence of zero, that is of primordial importance
  • in a positional system to eliminate representational
  • ambiguities.
  • For instance, without zero, how can
  • we distinguish between 12 and 102?


Digital technologies…


The evolutionary transition from static to dynamic

inscriptions can be modeled through several stages of development, each of which can

still be evident in mathematics classrooms in the 21st Century.


Static Inert

In this state the inscriptions

is “hardened” or “fused” with

the media.

Early forms of writing

included calligraphy as art

form of writing since it

was very difficult to change the

writing once “fused” with the


In this sense it is inert.


Static / Kinesthetic

  • With the co-evolution of reusable media to inscribe upon,
  • we enter a second stage of use defined by erasability.
  • Chalk, pencils, for instance, allow a transparent use of writing and expression:
  • permanence is temporal, erased over time.

Static Computational

The intentional acts of a human are computationally refined. A simple example is a calculator where the notation system is processed within the media and presented as a static representation of the user’s input or interaction with the device.


Discrete Dynamic

As computational become less static, and user interactions become more fluid, the media within

which notations can be expressed becomes

more plastic and malleable.


Discrete Dynamic

As computational become less static, and user interactions become more fluid, the media within

which notations can be expressed becomes

more plastic and malleable.


The nature of mathematical symbols have evolved in recent years from static, inert inscriptions that users have little personal identification with but appropriate over time, to dynamic objects or diagrams that are






But when an element of a diagram is dragged, the resulting re-constructions are developed by

the environment NOT the user.

So what becomes important is that the environments feedback the intentions of the users.



when we work in a digital ecology, our semiotic becomes digital.

Executability is intrinsic to the new symbols and representations:

We are externalizing the memory AND the cognition!


If we conceive of mathematical objects through their digital instantiations, that is, by means of digital symbols,

we need to solve an epistemological problem: As formalization is relative to the medium in which it takes place, how do we develop a new methodology to prove that coheres with the

mathematical ecology of the new medium?



A digital theorem: the Hilbert filling curve

Given any screen resolution, there is a level in the recursive process that generates the curve, that fills that screen.