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The Role of Cognitive Representations in Formation of Mathematical Concepts

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Irina Starikova

on 28 August 2016

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Transcript of The Role of Cognitive Representations in Formation of Mathematical Concepts

A received view:

"Groups are typical
algebraic
objects. Traditional approach to groups is a
combinatorial
approach."
Case study from current mathematics: Geometric Group Theory

Mathematical representations:
Representation of one mathematical object by another.
Groups -> generated groups
Graphs of groups
Metric spaces

Cognitive representations:

Notation
Formulas
Tables
Diagrams
Mathematical vs. Cognitive Representations
Algebraic notation:
Letters:
a,b,r,f
Signs: (,),<,>, *
Words:
abbab
Sentences:
ab=ba=e
manipulation with symbols
Groups can be represented by graphs:

Geometric approach:
Visualise!
Graphs are still
algebraic
objects!!!!
So what?
Z(1): 1 2 3 4 5 6 7 8 9 10 11 ...



Zooming out!
Word metric is boring.
Introduce metric on Cayley graphs?
Groups as metric spaces: Hyperbolic groups
Visualising
+ zooming in and out
the visualisations
Results so far
The Role of Cognitive Representations in Formation of Mathematical Concepts

Hyperbolic triangles
Groups can be studied as
geometric
objects !(hyperbolic metric spaces)
Crossing borders and linking domains
Making new concepts available (unavailable in combinatorial context)
Solving problems
Starting a new program!
R____________________________


Metric, hyperbolicity, triangle
can be applied to algebraic groups !
OK, we can use groups to study geometry:
Symmetries of geometric objects
Fundamental groups - F. Klein
A group G is a
set
of elements with a binary operation, which satisfies the group
axioms
:
closure: a • b, is also in G
associativity: (a • b) • c = a • (b • c)
identity: e • a = a • e = a
inverse: a • b = b • a = e
"Groups can be represented as
geometric
objects and studied by
geometric
methods!"
Focus
:
pre-formal philosophy
examples from current mathematics
the role of
cognitive
representations
vs.
Combinatorial group theory
Why this case study:
sharp contrasts
advanced mathematics
accessible
Contrast in two dimensions:
1. Combinatorial/algebraic and geometric approaches
2. Mathematical and cognitive representations

Mikhail Gromov, "Hyperbolic groups."
Essays in group theory
, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
Algebraic vs. geometric

Algebra/Combinatorics


discrete
sets
out of space
distribution of elements and operations
countable and finite

Geometry

continues
figures
in space
measure
spatial relations
curvature
lists (rows and columns)
tables
matrix-like displays
+ diagrams
Mathematical
representations/objects
Cognitive representations
A group

A
generated
group

A Cayley graph
New!
'delta
-hyperbolicity'
'hyperbolic
group'
'triangle'
'quasi-isometry'
By the way,
e.g. adjacency matrix
wouldn't help!
Concluding Message
:
Changes in cognitive representations can play a significant epistemic role by making new concepts and new resources available.
Thank you!
Triangles?! - Yes!
Angles - ?
"0-slimness"
Large-scale geometry
Classification of groups
Tarski monster group
Isomorphism problem (are the two presented groups isomorphic?)
Word problem (are two arbitrary words equal?)
Tarski monster group
Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by A. Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 10^75. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
metric space
generated group

__________________
sets
groups
generated groups
presented groups
lines
figures
symmetries
transformations of geometric figures
spaces
Combinatorics/Algebra
Geometry
Cognitive representation
Mathematical
representation
Mathematically
point on a plane can be represented by an ordered pair of numbers.
Cognitively
it can be represented by a dot.
Claim
:
Mathematical representations play a significant epistemic role. But sometimes it is the
cognitive
representations which play a significant epistemic role: they lead us to
new resources
and concepts
.
A group? What is it?
A metric space
New!

Quasi-isometry
equivalence
New instruments:
Cayley
graphs,
quasi-
isometry and
delta
-hyperbolicity
Abel Prize "for his
revolutionary
contributions to geometry".
Group elements can be represented by a multiplication table
{..., -2, -1, 0, 1, 2, 3, ...}
{...,-1+
1
, 0+
1
, 1+
1
, ...}
Starikova, I. “From Practice to New Concepts:
Geometric Properties of Groups”,
Philosophia Scientiæ
, 16 (1), 2012, pp.129-151.
Review
:
contrasted mathematical-cognitive representations in an combinatorial and geometric approaches
observed how they function in the case study
now to conclusions

http://www.ihes.fr/~gromov/index.html


Irina Starikova
University of San Paulo
Supported by CAPES

Cognitive representations in algebra are symbolic and based on
How to see geometry in groups? By changing representation.
Let's look at applications:
natural sciences (chemistry) - mathematics
mathematics (graphs) - computer
mathematics - mathematics (combinatorics/algebra - geometry)

How the contact is possible?
Graphs are good mediators:
allow for intuitive grasp (chemical structure)
computable
beautiful
Possibilities:
graphs (visualised)
computers (limits; data storage, classification of finite simple groups)
more elaborate mathematics, nice properties (cognitive)
A graph can be represented by an
adjacency

matrix


https://www.math.ubc.ca/~anstee/math223/223Petersen.pdf

Problem: What is the role of cognitive representations in mathematics?
"
hidden arguments
" J. Norton
"picture-
proofs
" J.Brown

Is a justificatory role
the only
important epistemic role ?

Concept formation?
Communication between domains?
More questions:
Mathematical objects -what are they?
Foundations of mathematics?
Mathematical educational strategies?
Graph Theory?
Cayley graph
Structure of the talk:
Problem
Example
Discussion
Representations
:
mathematical - abstract
cognitive - e.g. diagrams, formulas
Full transcript