Sense and Denotation (Part 2)
Caroline Bardini – Université Paris 7
April 23rd 2004
Recall
“Every good mathematician is at least half a
philosopher, and every good philosopher is at
least half a mathematician.”
Friedrich Ludwig Gottlob Frege
(1848-1925)
Über Sinn und Bedeutung (On sense and denotation), 1892
Driving force : cognitive value of a statement reference
Sign /name: Any designation representing a proper name. It can consist of a
name, combination of words, letters or other signs.
Sense: The mode of presentation
of that which is designated
« A sign expresses its sense, stands
for or designates its denotation. »
Denotation (the « object » the sign refers to)
Frege’s examples
« Let a, b, c the lines connecting the vertices of a triangle with the midpoints of
the opposite sides. The point of intersection of a and b is then the same as the
point of intersection of b and c. So we have different designations for the same
point, and these names (‘point of intersection of a and b’, ‘point of intersection
of b and c’) likewise indicate the mode of presentation. »
1.
denotatio
n
signs
sense
The statement contains actual knowledge
2.
« The denotation of ‘evening star’ would be the same as that of ‘morning star’,
but not the sense. »
To the sign there corresponds a definite sense and to that in turn a definite denotation,
while to a given denotation (an object) there does not belong only a single sign.
3.
« The expression ‘the least rapidly convergent series’ has a sense;
but it is known to have no denotation, since for every given
A sign might have a sense but no denotation
convergent series, another convergent, but less rapidly convergent
series can be found. »
Extending Frege’s ideas
More examples…
4.
(1901) The expressions ‘Melbourne’ and ‘the australian capital’ have
different senses but denote the same city.
5.
4’ and ‘8/2’ have the same denotation but express different ways of
conceiving the same number.
Sense and denotation in mathematics educational research
I - Duval
Treatment / conversion
II - Drouhard
Interpretation / connotation
III - Arzarello
Algebraic sense/ contextualised sense
Denotation « within a universe »
Meaning and reference, ed. A. W. Moore, Oxford University
Press, 1993.
Translations from the Philosophical Writings of Gottlob Frege,
ed. P. Geach and M. Black, Oxford: Blackwell, 1952.
Frege’s theory of Sense and Reference. Its Origns and Scope. W.
Carl, Cambridge University Press, 1994.
Treatment and Conversion (Duval)
Treatment
Changing representations of an object within the same
semiotic system. Internal transformation.
Eg.: Calculating = substituting new expressions to given
expressions within the same s.system (i.e. writing numbers).
Conversion Transpose the representation of an information/ object/
situation into a different system. External transformation.
Eg.: Formulating a problem given in natural language into equations.
Sense, denotation and Treatment
0,25+0,25 = 0,5
¼+¼=½
Sense, denotation and Conversion
Decimals
and
fractions:
two
different
Eg: if 0,25
and
¼ are not
seen
as refering
representation
systems.
to the same object,
one cannot be thought
Different senses,
Students
as the substituent
know how
of the
to other.
add decimal
Thereforeand
different treatment fractions
conversion
(= cannot
treatments),
be conceived
can’t shift from
procedures
one representation to another (=conversion)
Distinguishing sense from denotation is
essential to conversion.
Interpretation and connotation (Drouhard)
Interpretation of an algebraic expression X within a given framework is every object
that ‘corresponds’ to the denotation of X within this framework.
3x +7
Graphical framework : line y = 3x+7
Arithmetical: the writing of a number congru
to 7 modulo 3
Math/
Extra-math
Connotation is the subjective perception someone has of an algebraic expression
Frege
Representation
Many
algebraiccorrespondance
difficulties described
as deficiences
the way students
master the
One-to-one
between
sense,indenotation
and expression
invariance of denotation with respect to the sense.
Pupils don’t see that algebraic transformation must preserve denotation, that
is the « value » of the square must remain the same throughout the operation.
Then proposing numerical values won’t help convincing them.
(a+b)2
A symbolic expression denotes itself as a collection of signs
Representation (Frege)
« Somedody observes the Moon through a telescope. I compare the moon
itself to the reference; it is the object of the observation, mediated by the real
image projected by the object glass in the interior of the telescope, and by the
retinal image of the observer. »
representatio
n
sense
« The optical image in the telescope is indeed one-sided and dependent upon
the standpoint of observation; but it is still objective, inasmuch as it can be
used by several observers. But each one would have his own retinal image. On
account on the different shapes of the observers’s eyes, even a geometrical
congruence could hardly be achieved, and an actual coincidence would be out
of the question. »
representation
sense
objectivity
denotation
Algebraic sense and contextualised sense 1/2 (Arzarello)
Algebraic sense: the way by which the expressions are given, through
different rules.
Interpretation (Drouhard)
n(n+1)
n2+n
(x+5) = x
=0
x2+x+1
Contextualised sense: Formulas express different thoughts, with respect
to the different contexts they are used.
Elementary number theory : product of 2 consecutive nbrs
n(n+1)
Geometry: Area of a rectangle of (integer) sides n, n+1
The formula mimics in its own form and shape the main relationships
among the different objects involved.
Ideography allows for changes of sense by suitable manipulations
on shape of formula (eg. : n(n+1)
n2 +n)
Ideography allows seasing the different contextualised senses by looking
at the different algebraic senses.
Algebraic sense and contextualised sense 2/2 (Arzarello)
Senses may change without a corresponding change either in the formula or
in the denoted object.
n(n+1) can be seen as area or product of numbers
Algebraic transformation can produce different expressions holding
different algebraic senses, but with the same denotation.
It is not always true that two expressions having the same denotation can be
mutually reduced by means of algebraic transformations.
(x+5)2 = x and x2+x+1=0