1. Sense and Denotation (Part 2) Caroline Bardini – Université Paris 7 April 23rd 2004

2. 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 ), 1892denotation 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 Denotation (the « object » the sign refers to) « A sign expresses its sense, stands for or designates its denotation. » Driving force : cognitive value of a statement

3. Frege’s examples The statement contains actual knowledge signs sense denotatio n « 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. « The denotation of ‘evening star’ would be the same as that of ‘morning star’, but not the sense. » 2. 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. A sign might have a sense but no denotation controversed « The expression ‘the least rapidly convergent series’ has a sense; but it is known to have no denotation, since for every given convergent series, another convergent, but less rapidly convergent series can be found. » 3.

4. Extending Frege’s ideas (1901) The expressions ‘Melbourne’ and ‘the australian capital’ have different senses but denote the same city. 4. 4’ and ‘8/2’ have the same denotation but express different ways of conceiving the same number. 5. More examples… Sense and denotation in mathematics educational research Denotation « within a universe » I - Duval Treatment / conversion II - Drouhard Interpretation / connotation III - ArzarelloAlgebraic sense/ contextualised sense

5. 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.

6. 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 Sense, denotation and Conversion Students know how to add decimal and fractions (= treatments), can’t shift from one representation to another (=conversion) Decimals and fractions: two different representation systems. 0,25+0,25 = 0,5 ¼ + ¼ = ½ Different senses, different treatment procedures Distinguishing sense from denotation is essential to conversion. Eg: if 0,25 and ¼ are not seen as refering to the same object, one cannot be thought as the substituent of the other. Therefore conversion cannot be conceived

7. 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 Representation Many algebraic difficulties described as deficiences in the way students master the invariance of denotation with respect to the sense. A symbolic expression denotes itself as a collection of signs One-to-one correspondance between sense, denotation and expression 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 Frege

8. 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. » 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. » representatio n sensedenotation objectivity

9. Algebraic sense and contextualised sense 1/2 (Arzarello) Algebraic sense: the way by which the expressions are given, through different rules. n(n+1) (x+5) = x n 2 +n x 2 +x+1 = 0 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 Interpretation (Drouhard) 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) n 2 +n) Ideography allows seasing the different contextualised senses by looking at the different algebraic senses.

10. 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 x 2 +x+1=0