Loughborough University Wednesday, December 10th 2008

1. Loughborough University Wednesday, December 10th 2008 Semantic perspective in mathematics education A model-theoretic point of view Viviane DURAND-GUERRIER vdurand@univ-lyon1.fr Université de Lyon, Université Lyon 1, IUFM de Lyon & LEPS EA 4148 LIRDHIST http://lirdhist.univ-lyon1.fr

2. Preliminary We assume a logical perspective on semantics, that suits with the following definitions referring to Morris (1938) or Eco ( 1971) Semantics concerns relation between signs and objects they refer to. Syntax concerns the rules of integration of the signs in a given system Pragmatics concerns the relationship between subjects and signs : signs perceived according to their origin, the effects they produce, and the way they are used. According with Da Costa (1997), it is necessary to take in account these three aspects for a right understanding of logical mathematical field.

3. An example : addition of integers Semantics : the addition of two integers is defined as the cardinal of the union of two relevant discrete collections ; the result is independent of the nature of the involved object (with respect that mixing these objects will preserve their integrity) Syntax : Addition is defined as the iteration of the successor ; it does not necessitates reference to quantities. This provides algorithmic rules in a given system of numeration. Pragmatics : The articulation between both aspects is build by a forth and back between calculation (syntax) and effective counting (semantics).

4. Negation, between syntax and semantics

5. Negation (1) Semantics : the negation of a proposition exchanges the truth value; the negation of a property exchange those objects which own the property. 3 is an even number / 3 is not an even number To be an even number / not to be an even number Syntax : the negation of a given proposition follows precise rules with respect of the concerned language. In French : -for singular proposition, and universal propositions, we apply “ne ..pas” on the verb : 3 est un nombre pair/3 n’est pas un nombre pair Tous les nombres sont pairs/Tous les nombres ne sont pas pairs -for existential proposition, we use the quantifier “aucun” (no). Certains nombres sont pairs / aucun nombre n’est pair

6. Negation (2) Pragmatics : Some linguistic forms may lead to referential ambiguities “Tous les nombres ne sont pas pairs” (1) (not all numbers are even) is sometimes interpreted as “Aucun nombre n’est pair” (2) (No number is even),the contrary in Aristotle’ sense. This interpretation is reinforced by the possibility of changing “ ne sont pas pairs” (are not even) in “sont impairs” (are odd) in sentence (1), that gives “Tous les nombres sont impairs”, synonym of (2). Generally, but not always, the context permits to choose the right interpretation.

7. Concerning negation, it is likely that it is considered by most mathematics teachers as a rather simple notion, met and used early by young children, and build through practise of mathematics. However, negation, as soon as it operates on quantified statements, involved both syntactic and semantics criteria, that interoperate according to the specificity of the concerned language. This leads to actual difficulties for students in practising mathematics, in particular, confusion between negation and contrary that are largely underestimated in the teaching of mathematics, whatever the level.

9. In French a sentence such as “Tous les A ne sont pas B” (“All A are not B”) is ambiguous. According with the French linguistic norm, its meaning is: It is false that “all A are B”, eg. “not all A are B”, that means “some A is (are) not B”. However, it is often used to express that « No A is B » (the contrary, following Aristotle)

10. Exemple : Tous les diviseurs de 12 sont pairs (1) Tous les diviseurs de 12 ne sont pas pairs (2) Tous les diviseurs de 12 sont impairs (3) Selon la norme linguistique française comme (1) est Faux, (2) est Vrai Selon le principe de substitution (2) et (3) ont la même valeur de vérité Or (3) est Faux Ceci renforce l’interprétation « Aucun » Example All divisors of 12 are even (1) All divisors of 12 are not even (2) All divisors of 12 are odd (3) According with the French norm, as (1) is False, thus (2) is True According with the substitution principle, (2) and (3) have the same truth-value But (3) is False This reinforce the interpretation by “None” The French linguistic norm does not suit with logical syntax and does not respect the substitution principle

11. Effective difficulties for pupils and students An inquiry of the CI2U, a national commission of the IREM (Institute for Research on Mathematics Education) Questionnaire about absolute value-limits-logic Students at the very beginning of their first university year (340 answers analysed) Exercise : Give the mathematical negation of the following sentences • 1 - Toutes les boules contenues dans l’urne sont rouges.All the balls in the urn are red • 2 - Certains nombres entiers sont pairs.Some integers are even • 3 - Si un nombre entier est divisible par 4, alors il se termine par 4.If an integer is divisible by 4, then the last digit is 4

12. Toutes les boules contenues dans l’urne sont rouges All the balls in the urn are red • C0 : A sentence synonym of the affirmative answer 13% • C1 : A sentence synonym of “Il y a au moins un balle qui n’est pas rouge” (“There exits at least a ball in the urn that is not red”)38% • C2 : The ambiguous sentence: “Toutes les boules ne sont pas rouges”6% • C3 : A sentence synonym of “Aucune boule n’est rouge” (“No ball is red”)21% • Other answers : 10% • No answer :12%

13. Certains nombres entiers sont pairs. Some integers are even • C0 : The affirmative sentence “Tous les entiers sont pairs” (“All integer are even”) 5,5 • C2: The ambiguous sentence “Tous les entiers ne sont pas pairs” (All integers are not even)5,5% • C4: The sentence “Tous les entiers sont impairs”(“All integers are odd”)15,5% • C5: The sentence “Aucun entier n’est pair”(“No integer is even”)13% • C6: The sentence “Certains entiers ne sont pas pairs (sont impairs)” “Some integers are not even (are odd)”34% • Other answers11,5% • No answer : 15%

14. Si un nombre entier est divisible par 4, alors il se termine par 4 If an integer is divisible by 4, then the last digit is 4 For this item, 98 students (29%) did not gave an answer; only 34 students (10%) gave a correct answer, synonym of “There exists an integer dividable by 4 for which the last digit is not 4”. In 155 answers, there is an implication with various position for the negation;

15. In our population, a large number of students face difficulties to provide negation of quantified sentences, especially in the case of the sentence involving an implication. • As the population comes from various universities, we can suspect that many fresh students in France face these difficulties. • Due to the importance of negation, implication and quantification in the process of conceptualisation in mathematics, our results indicate that it is necessary to take in account these logical questions in the teaching of mathematics. • An international perspective could be to explore theses questions in various languages, and particularly in context where students learn mathematics in a foreign language, either in their own countries, or in a country where they are studying.

16. A model-theoretic point of view Some insights

17. Model theory is developed by Tarski in the continuity of his work on the semantic definition of truth for quantified logic. • A semantic definition of truth put in relation a formalized language with interpretative structures of this language.

18. A semantic version of propositional calculus (Wittgenstein, 1921) In Tractatus logico-philosophicus, Wittgenstein proposes a formalization of the notion of proposition. A proposition is a linguistic entity that is either true or false. The components of the system are “propositional variables”, that could be interpreted as propositions in some particular piece of discourse. There are two principles • First, the principle of bivalence, proposing that there are exactly two truth values in the system • Second, the principle of extension, which asserts that the truth-value of a complex sentence is entirely determined by the truth values of its elementary components. The truth value of a proposition expresses its agreement or its disagreement with the facts or the state of things that it pretends describing.

19. A tool and two fundamental notions Wittgenstein introduced the truth-tables, and consider all the possible distributions of truth values, in particular with two propositions (16 distributions among them those of the classical connectors). With the principle of extension, it is possible to determine the truth-table of any complex sentence. Among them, two play a particular role: • Tautology : a statement of the system true for any distribution of truth-value; so true for any interpretation in any piece of discourse. • Contradiction : a statement of the system false for any distribution of truth-value; so false for any interpretation in any piece of discourse. Tautologies such as  « S S’ » support the classical inference rules. Example : S : « p  (pq) » et S’: « q » ; « SS’ » is a tautology associated to Modus Ponens (P; and If P, then Q; hence Q)

20. The semantic conception of truth (Tarski, 1933, 1944) • The main problem is to give a definition of truth materially adequate and formally correct (Tarski, 1944, 1974, p.269)*. • In this study, I only look for grasping the intuitions expressed by the so named « classical » theory of truth, i.e. this conception that “truly”as the same signification as “in agreement with reality” (contrary with the “utilitariste” conception that “true” means useful under such or such aspect (Tarski, 1933, 1972, p. 160)*. * Our translation

21. A recursive definition of truth • Interpretation of a propositional function (a predicate) of a given formal language in a « domain of reality » by an open sentence. • Satisfaction of an open sentence by an object (an individual) of the discourse’s universe. • For all a, a satisfies the propositional function « x is white » if and only if (it is the case that) a is white • Definition of the truth of a complex sentence • Propositional connectors (Wittgenstein) • Quantifiers (in agreement with common sense)

22. Théorie des modèles (Tarski 1955) • A formalized language L, a syntax, well-formed statements (formulae) : F, G, H ….. • An interpretative structure  (a domain of reality, a mathematical theory). •  is a model of a formula F of L if and only if the interpretation of F in  is a true statement. • A formula H is a logical consequence of a formula Gif and only if any model of G is a model of H.

23. An example F : xy (S(x,y) S(y,x) x=y)  : set of ordered real numbers in which S is interpreted as the relationship  ‘to be inferior or equal’   The interpretation of F in  expresses that « the relationship ‘to be inferior or equal’ is anti symmetric ». This is true. Hence,  is a model of F ’ : set of ordered real numbers in which S is interpreted as the relationship  ‘to be equal’.   ’ is not a model of F

24. About logical consequences Logical consequences support validity, and hence classical mode of reasoning in mathematics (Quine, 1950) 1. G : p(x)  (p(x)  q(x)) H : q(x) H is a logical consequence of G (Modus Ponens) 2. F : p(x) G :  x p(x) H : x p(x) F is a logical consequence of G / G is not a logical consequence of F H is a logical consequence of F / F is not a logical consequence of H 3. F: xy p(x, y)xy q(x, y) G: xy (p(x,y)q(x,y)) F is a logical consequence of G / G is a not a logical consequence of F

25. Deduction theorem • Every theorem of a given deductive theory is satisfied by any model of the axiomatic system of this theory; moreover at every theorem one can associate a general logical statement logically provable that establishes that the considered theorem is satisfied in any model of this type. (Deduction theorem) • All the theorems proved from a given axiomatic system remain valid for any interpretation of the system.

27. Proof by interprétation • A proof that a given statement is not a logical consequence of the axioms of a theory consists in providing a model of the theory that is not a model of the formula associated with the statement in question. • Example : le fifth Euclid’s postulate and the non- Euclidean geometries.

29. Didactic perspectives

30. Objects and properties versus statements

31. Objects and properties versus statements (1) An example from Arsac & al. (1992) A general statement : n2-n+11 is a primary number for every n (false) A property of some objects (an open sentence) : n2-n+11 is a primary number True for integers from 1 to 10 False for every multiple of 11 ; False for 25 Exploring the statement in grade 7 (12-13), students work with objects (integers). Some of them declare the statement false as soon as they find 11 is a counter example Others state that the statement is neither true, nor false, or both true and false, or look for a domain where it is always true.

32. Objects and properties versus statements (2) • (40) Student : there is an exception, hence it is not always  • (41) Marie : That has been established. Except for this, it is always a prime number. What if we eliminated 11…? (…) • (63) Marie : Yes but 22 is twice 11; we can maybe try 33; I think that this will also be an exception. • (64) Marie : I think they have won, because 25 is also an exception. • (76) Marie: They are no longer exceptions because 22, 33, are all multiples.

33. Objects and properties versus statements (3) For Marie, the argument is not the number of counterexamples, but their relationship to the situation. She adds that to be sure of getting a true sentence, it is necessary to be under hundred. The authors report that asking the question anew some days later, several pupils declare the statement is false, citing the two counterexamples 11 and 25.

34. Objects and properties versus statements (4) • Our interpretation is that those students who do not want to declare that the statement is false as soon as a counterexample is found are not considering the closed statement. They are working with the open statement “n2-n+11 is a prime number”, in which they substitute numerical values for n. • The group of discussions reveals a disagreement between those students who consider the general statement and insist on the fact that “it is not always true, so it is false”, and those students who remain focussed on the particular cases they have used to make up their minds.

35. Objects and properties versus statements (5) This activity may lead students to make various true assertions: • the sentence is false; the sentence is not always true; • the sentence is true for all integers from one to ten; • the sentence is sometimes true, sometimes false; • the sentence might be true and might be false; • the sentence is true except for 11; the sentence is true except for the multiples of 11; • the sentence is false for every multiple of 11; • it is impossible to determine all the numbers for which the sentence is true (or false)

36. An invalid inference rule

37. An invalid inference rule (1) To prove • “Given two functions f and g defined in a subset A of the set of real numbers, and a an adherent element of A, if f(t) and g(t) have h and k respectively for limits as t tends to a remaining in A, then f+g has h+k for a limit in a”.

38. An invalid inference rule (2) A proof (Houzel, 1996) • “ By hypothesis, for all >0, there exists >0 such that t A and ta  imply • f(t) - hand g(t) - k ; thus we have f(t) + g(t) – (h + k)  f(t) – h + g(t) - k f(t) - hg(t) – k  ”

39. An invalid inference rule (3) The first assertion could be interpreted as the application of the invalid inference rule* : • “for all x, there exists y, such that F(x, y)”, • and • “for all x, there exists y, such that G(x, y)”, • hence • “for all x, there exists y such that F(x, y) and G(x, y)” * In some interpretations, it is possible that the two premises are true and the consequent is false

40. An invalid inference rule (4) The use of this invalid rule can be found in many situations, providing • aproof with a gap that can be easily completed, • an incorrect proof for a true statement, • An incorrect proof of a false statement. This may be encountered either in history (Abel, Cauchy, Liouville, Seidel) or in undergraduates or graduates students’ proofs (Durand-Guerrier &Arsac, 2005)

41. An invalid inference rule (5) This enlightens a very important difference between an expert and a novice in mathematics: an expert in a mathematical field knows when it is dangerous to slack off the rigorous application of rules of inference, while novices have to learn this at same time as they acquire the relevant mathematical knowledge. These two aspects of mathematics cannot be learned separately.

42. Conclusion

43. Adopting a semantic point of view, and being situated at a meta mathematic level, a model-theoretic point of view provides on the one hand a frame to analyse a priori the situations under both mathematical and didactical aspects, and on the other hand to analyse students’ activity, in particular by providing to the researcher a methodology to identify and study the place and the role of objects, besides statements, in the process of conceptualisation in mathematics, and to take in account the articulation between syntax and semantics, and truth and validity.

44. « La logique semble bien, contrairement à ce que pensait Wittgenstein, un indispensable moyen, non de « fonder » mais de comprendre l’activité mathématique. C’est-à-dire pour une part, explorer la relation de l’implicite à l’explicite d’une théorie.(…) Une part essentielle de l’analyse épistémologique est ainsi ouvertement prise en charge par l’analyse logique. (…) En même temps elle apparaît comme une épistémologie effective dans la mesure où la réflexion est orientée vers et investie dans l’agir.» (Sinaceur, H. 1991, Logique : mathématique ordinaire ou épistémologie effective ?, in Hommage à Jean Toussaint Desanti, TER) “Logic seems, opposite with what Wittgenstein thought, an indispensable mean, not of ‘founding’ but of understanding mathematical activity.That means for a part to explore the relation from implicit to explicit in a theory (…). An essential part of the epistemological analysis is so openly taken in account by logical analysis. (…). So it appears as an effective epistemology in the measure that the reflection is oriented and invested in action.* * Our translation

45. Some references • Durand-Guerrier, V. : 2003, Which notion of implication is the right one ? From logical considerations to a didactic perspective, Educational Studies in Mathematics53, 5-34. • Durand-Guerrier, V. Logic and mathematical reasoning from a didactical point of view. A model-theoretic approach. in electronic proceedings CERME 3 (Conference on European Research in Mathematic Education,, Bellaria, Italy, Februar 2003. http://www.lettredelapreuve.it/CERME3Papers/TG-Guerrier.pdf • Durand-Guerrier, V. & Arsac, G. : 2005, An epistemological and didactic study of a specific calculus reasoning rule, Educational Studies in Mathematics, 60/2, 149-172 • Durand-Guerrier, V. : 2008, Truth versus validity in proving in mathematics, Zentralblatt für Didaktik der Mathematik, 40/3, 373-384 • About logic, language and reasoning at the transition between French upper Secondary school and University.Negation, implication and quantification, ICME 11, Monterrey, 13-07-08, on line.