Number Sense, Calculating Matter, Immanence

Elizabeth de Freitas

Adelphi University, NY defreitas@adelphi.edu

Guiding Questions

• What is the role of number and calculation in an immanent ontology?

• How might number and calculation factor into new materialist methods of inquiry?

• When is calculation an inventive practice that

doesn’t simply serve the control society?

An ethics for such onto-epistemology?

• Vicky Kirby asks: “How should we understand epistemology in such an instance where calculation is an ontologizing process of mutation?” (Kirby, 2011, p.41).

• How should we develop curriculum and instruction

if calculation is an activity of all matter? How do we cultivate children’s number sense as that which is folded into the material universe?

Number sense

• Animals of all kinds are said to develop some degree of number sense (Nieder & Dehaene, 2009).

• Tests of basic number sense typically demand counting and calculation of some kind, often using visual and symbolic quantities (Chinn, 2015).

• Cognitive research on “number neurons” and neural correlates of “computational thinking” aims to identify biological sources that might explain the emergence of number sense (Dehaene, 2011).

A calculating matter?

• Kirby (2011): “What arithmetic allows the body's internal differentiations to make sense, whereby what appears as "one" modality can translate to, or anticipate its difference from, another, and finesse discrepancies?” (p. 62).

• Carnality is “calculating and thinking material through and through” (p.63)

Outline of Paper: Two Sections

• One: Focus on current neuro-cognitive theories of number sense & the contemporary curricular shift towards computational number sense.

• Two: philosophical study of alternative ontologies of number, to show how infinitesimal calculation, and its monster-making tendencies, functions crucially in Deleuze and Guattari’s ontology.

Infinitesimal calculation

• A calculating common matter and a fractal image of life.

• They show how calculation can be machinic but non-axiomatic (operating through problematics)

• Fractal structures transform the concept of measure, introducing the notion of a recombinant fractal subject that breaks with the phenomenological subject.

Non-human number sense:Clever Hans

Rats tapping levers(Mechner 1958)

Number neurons

Image from Ansari (2012)

Dyscalculia

• Ladislav Kosc, formally introduced the term in the 1970s:• “A structural disorder of mathematical abilities which has its

origins in a genetic or congenital disorder in those parts of the brain that are the anatomical-physiological substrate of the maturation of the mathematical abilities adequate to age, without a simultaneous disorder of general mental functions.” (Kosc, 1974)

• Diagnostic and statistical Manual of Mental Disorders DSM 5 (2013): “patterns of difficulties characterized by problems with processing numerical information, learning arithmetic facts, and performing accurate or fluid calculation” (APA 2013, p. 67).

Diagnosing number sense

4 5

Diagnosing number sense

1 6

Diagnosing number sense

7 2

Diagnosing number sense

3 8

Diagnosing number sense

6 1

Diagnosing number sense

Diagnosing number sense

Diagnosing number sense

Diagnosing number sense

Attempts to include ordinality

• Lyons (2011; 2013): ordinality relates to “visuo-motor associations”– 2, 3, 4– 4, 3, 2– 2, 4, 3

0______________________________________100

Where is 20?

A limited concept of Number

• Focus on cardinality– Subitising– Sudden response time– Comparison of quantity

• What about ordinality?– calling forth (an)other, the new– Indexing the sequence– Intrinsically temporal– The engine of calculation

Rethinking number

The infinitesimal is an infinitely small interval.

The infinitesimal is “diabolical” because it undermines the atomism and fixity of individuals, and binds content with expression through infinite variation.

Evangelista Torricelli (1608-1647):Indivisibles: Every magnitude is composed of a string of “indivisibles” which

are the building blocks and which themselves cannot be divided.

Image from Alexander (2014)

The infinitesimal

• A changeling with one foot in the virtual and one foot in the actual.

• In A thousand Plateaus, Deleuze and Guattari (1987) will tap into this approach to the infinitesimal, describing it as that “intense matter” and “continuum of variation” that conjugates content and expression in “reciprocal presupposition” (D& G, 1987, p. 108-109).

What else might number be/do?

“The Numbering Number, in other words, autonomous, arithmetic, organization, implies neither a superior degree of abstraction nor very large quantities … These numbers appear as soon as one distributes something in space, instead of dividing up space or distributing space itself … The number is no longer a means of counting or measuring but of moving: it is the number itself that moves through space … The numbering number is rhythmic, not harmonic” (Deleuze & Guattari, 1987, p. 389-390).

What kinds of calculating machines operate in smooth space?

• Fractalization. Any space with a fractional dimension escapes conventional measures and is “the index of a properly directional space”. In other words, the dimensionality of a smooth space is determined by that which moves through it, rather than by some magnitude of containment. Dimension becomes degrees of freedom of movement. The Koch snowflake is a machine so complex in its folds that it seems to introduce a new kind of object between line and plane, a new dimension between 1 and 2.

Six iterations of the Hilbert space filling curve

Recombinant subjects

• The fold and the superfold• The individual and the dividual

• Two kinds of automaton – “the great spiritual automaton” and the “psychological automaton”

Mathematical monsters

• The Koch snowflake

With each iteration, smaller triangles are added to the previous edges, generating formulae for perimeter (P) and Area (A) after n iterations. As n approaches infinity, the perimeter expands without limit, but the area converges to a finite sum.

Deleuze’s ontology of immanence

• The “individual is a relation insofar as every relation is a measure, and insofar as every measure plunges into the infinite.” (Lectures on Spinoza)