Novembre/Décembre 1998


Intuition and Proof:
Reflecting on Fischbein's paper

Maria Alessandra Mariotti

Dipartimento di Matematica
Università di Pisa - Italia


Some years ago, in an article entitled "Intuition and Proof", E. Fischbein (1982) presented the results of a research project concerning the theme of proof within the general framework of his reflections on intuitive reasoning. In the spirit of that discussion, I will develop some ideas on the same theme.

A basic legacy that is left us by Efraim Fischbein, is his original approach to educational problems centred on the complex notion of intuition. The synthesis of this approach is contained in his book "Intuition in Science and Mathematics" (1987), where a "theory of intuition" is sketched and offered to the community of researchers as a useful tool for the interpretation of educational phenomena.

In the same way that it is impossible to conceive of a theory emptied of intuitive meaning, so one cannot conceive of mathematics deprived of its theoretical organisation: axioms, definitions and theorems constitute mathematics as much as its ideas and models. But theory and intuition may be distant and conflicting poles difficult to reconcile. Yet, sometimes, contradictory conceptions merge into new compromise conceptions, a classic example of which being is the notion of infinity (Fischbein et al., 1979): the dynamic representation of infinity can be considered as a compromise between the finite structure of intellectual schema and the formal infinity itself (Fischbein 1987, p. 205). But not always compromises are successful, and rather than compromises harmonisation may have to be sought.
   The need for harmonising intuition and mathematical notions constitutes a basic issue of education and a contribution to this difficult task comes from those studies which focus on conflicts and discrepancies and seek to identify their origins .

Empirical and formal approach

Theorems constitute the basic chunks of mathematical knowledge, as organised in a specific theory, and they can be considered a particular product of the process of knowledge acquisition. Besides the direct acquisition of information, mostly related to factual evidence attained through experience, human culture has developed a complex way of obtaining information and knowledge, which is not direct, but mediated through means such as language, logic and reasoning. As a consequence of this mediation, the structural unity between cognition and adaptive reactions is broken: "Knowledge through reasoning, becomes a relatively autonomous kind of activity, not directly subordinated to the adaptive constraints of the behaviour of human beings" (ibid., p.15). In particular, a crucial differentiation occurs between empirical verification and logical deduction with the result that their relationship becomes problematic.

A comparison between truth evaluation in terms of factual verification and logical validity in terms of deductive inferences leads one to consider the effect of a factual confirmation on the validity of a statement. Of course there are different attitudes which can be described with reference to an empirical approach, and to a theoretical approach: despite the fact that a formal proof confers a general validity to a mathematical statement, further checks seem to be desirable in order to confirm that validity (Fischbein, 1982).
   Thus, the discrepancy between empirical verification -- typical of common behaviour -- and deductive reasoning - typical of theoretical behaviour, is a source of difficulties, an obstacle to an appreciation of the sense of proof.
   In school practice, it is very common to confuse these two points of view which can confuse students, who see 'examples' as playing a basic role in stating axioms and "discovering" theorems, only to find them forbidden when they are asked to prove a statement where one or few examples are not acceptable as a "proof". And what can be said about the role of counterexamples where one single example can invalidate a theorem.
   Actually, the relationship, crucial for mathematics, between empirical truth and logical validity, is a complex and delicate relationship which must be developed through out education.
   The sense of proof is far from common sense. Although in history it is possible to find mathematicians who have felt uneasy with a theorem despite the acceptance of its proof - Cantor is one of the most famous - it is generally the case that a feeling of general validity, is what a mathematician attains whenever a theorem is proven; but that feeling is new and 'strange' with respect to the natural attitude of mind.

Intuition and theory

Looking more carefully at the relationship between the intuitive and the theoretical approach leads one to consider the problem of proof more globally where the unity among statement, proof and theory must be recognised (see the notion of theorem introduced in Mariotti et al. 1997). An analysis of the relationships between theorems (statement, proof and theory) and intuition can be undertaken in to two opposite directions.

  • On the one hand, a statement expresses the implicit relationships between the principles assumed in the theory, and the thesis of the theorem, under the conditions stated by the hypotheses. Making these relationships which are implicit at the intuitive level (Fischbein, 1987, p. 50) explicit, constitutes the first step towards the construction of an argumentation, which, in the framework of a theory, can become a proof.
  • On the other hand, a theorem represents a piece of knowledge and as such must be appropriated by the learner; in other words, in order that it can be used in productive reasoning, a theorem should acquire the status of an intuition. But this can only occur if the unity or fusion between statement and proof, previously artificially separated, is restored. Statement and proof must condense into an intuitive knowledge (Fischbein, 1982). In other words the unity between statement and proof claims not to be broken: the process of analysis which led to the proof, has to be recomposed into a single chunk to acquire that immediacy which makes it productive.

To summarise, as far as theorems are concerned, intuition is differently involved at the level of both the statement and its proof:

- the truth of a statement;
- the structure of the proof: the necessity of a logic as relationship between (the logical articulation of) the single steps of the proof;
- the validity (generality) of the statement as a necessity imposed by the proof.

The articulation between the first and the second level represents a crucial point in the elaboration of a proof: uncertainty may trigger the exploration of motivations and start a process of argumentation.
   The second level is the junction between the first and the third level; in fact, grasping the logical structure of a proof corresponds to inserting the statement within a coherent framework of intuitions, that can guarantee its evidence, necessity and complete acceptability. It will reach the status of "cognitive belief" (Fischbein, 1982, p. 11). Finally, it allows a theorem in its unity of statement and proof, to condense into a new intuition and to become a productive intellectual instrument.

" ... The logical form of necessity which characterises the strictly deductive concatenation of a mathematical proof can be joined by an internal structural form of necessity which is characteristic of an intuitive acceptance." (Fischbein 1982, p. 15)

Interesting to remark that the description of a similar process can be found in Descartes:

Hoc enim fit interdum per tam longum consequentiarum contextum, ut, cum ad illas devenimus, non facile recodermur totius itineris, quod nos eo usque perduxit; ideoque memoriae infirmitati continuo quodam cogitationis motu succurrendum esse dicimus. [...] Quamobrem illas continuo quodam imaginationis motu singula intuentis simul et ad alia transeuntis aliquoties percurram, donec a prima ad ultimam tam celeriter transire didicerim, ut fere nullas memoriae partes relinquendo; rem totam simul videar intueri. (Descartes, Regula VII)

Implication at the didactic level

The sense of proof may definitely contrast with the common behaviour towards the acceptability of a statement based on factual verification. School practice seems to neglect or at least undervalue the difficulties related to the discrepancy between a practical and a theoretical behaviour, that explains most of the failures of traditional teaching.

Traditionally at school, students learn theorems which others have produced and only very late in their school life, by imitating the products that they learnt, they might be required to produce a theorem. But confining school practice to repeating proofs that others have produced, and doing this moreover for statements that are self-evident and do not appear to need any justification, is likely to be useless if students are to construct the complex relationship between the intuitive and theoretical attitude.
   Students may not develop a correct mental attitude towards theorems -- they may follow their common sense and ask for supplementary examples to corroborate their confidence, so as accept the possibility of exceptions. These results as reported by Fischbein (1982) have been confirmed more than once.

Besides the possible discrepancies between the theoretical and the intuitive approach, intuition can constitute an obstacle: when the immediacy of a statement inhibits the process of analysis of the implicit links and thus the construction of the analytic structure that constitutes a proof. In this case, it becomes impossible to understand the meaning of proof because the self-evidence, immediacy and the feeling of certitude that characterises intuitive statements can inhibit any kind of argumentation, i.e. the elaboration of the analytic structure, "step by step", which constitutes a proof. The process is blocked and so is the path to proof.

A suggestion immediately follows, which is that pupils' introduction to theorems would benefit by facing situations where there are no self-evident solutions.
   The basic point concerns the process of production of theorems, so in this case a comparison with the common practice in the mathematicians community is illuminating. A mathematician has direct experience of producing theorems and can always profit from this experience when he/she relates to any theorem, in contrast to students who do not have such opportunities.

Recent results (Boero et al., 1996, Bartolini, in press) confirm that open-ended problems are very suitable in the early approach to theorems. Open-ended situations may generate a feeling of uncertainty which calls for indirect means for getting knowledge, in particular, those problems which require the production of a conjecture. Moreover, the process of production of the conjecture is essential for the introduction of pupils to argumentation. But, engaging in an argumentation is not enough (Balacheff, 1987; Duval, 1992-93); the unity among statement, proof and theory shall not be broken, requiring the construction of the complex relationship between stated principles and consequences (Mariotti et al. 1997). The preservation of this unity maintains the link with the intuitive level, the basic condition for the autonomous production of theorems, and the productive use of theorems in mathematical reasoning.

Here again a traditional school practice must be criticised. When experience is confined to "ready-made" theorems (formulated and proven by others) the link between a theorem and its intuitive counterpart can be underestimated and finally neglected. Of course, from the point of view of formal logic, any theorem is completely independent of its interpretation, so that it can loose any link with intuition. But this cannot be the perspective of education.

Generally speaking, the main point here is how to overcome conflicts and construct a correct relation between intuition and theoretical attitude, i.e. a complementarity between different forms of knowledge, the intuitive and the formal, so distant may be, with the aim of making them two aspects of the same mental behaviour.

Fischbein taught us to look carefully at conflicts, incongruent phenomena, in order to detect deep reasons, which can indicate how to overcome the obstacles. Mathematical education aims to harmonise intuition and theory, but keeping in mind the possible obstacles: there is nothing more dangerous for mathematics learning than neglecting the deep discrepancies between spontaneous thinking, sometimes common sense, and mathematical thinking.
   Maybe the case of proof, is exemplary, although it is not the only one -- ; in fact, definitions present similar problems (Mariotti & Fischbein, 1997). Actually, proving is an activity characteristic of doing mathematics, but also an activity which differentiates substantially mathematics from common thinking and real life practice.


Reactions? Remarks?

The reactions to the contribution of Maria Alessandra Mariotti will be
published in the January/February 99 Proof Newsletter

© M. A. Mariotti 1998

Edited by Celia Hoyles




Balacheff, N. (1987) Processus de preuve et situations de validation, Ed.St. Math.18, 147-76
Bartolini Bussi M., Boni M., Ferri F. & Garuti R. (in press), Early Approach To Theoretical Thinking: Gears In Primary School. Ed. St. Math.
Boero, P., Garuti, R. & Mariotti, M.A.(1996) Some dynamic mental processes underlying producing and proving conjectures, Proc. of PME-XX, Valencia
Descartes, R. (1701/1964). Rules for the direction of the mind. In R. Descartes, Philosophical essays (pp.145-236; L. Lafleur, Trans.). Indianápolis: Bobbs-Merrill.
Duval, R. (1992-93) Argumenter, demontrer, expliquer: continuité ou rupture cognitive?, Petit x , n° 31, 37-61.
Fischbein, E. (1982) Intuition and proof; For the learning of mathemarics 3 (2), Nov., 8-24.
Fischbein, E. (1983) Intution and analitical thinking in Mathematics Education, Z.D.M.2, 68-74.
Fischbein, E. (1987) Intuition in science and mathematics, Dordrecht: Kluwer
Fischbein, E., Tirosh, D. & Melamed, U. (1979) Intution of infinity, Ed.St.Math.10, 3-40.
Garuti, R.; Boero, P.; Lemut, E. & Mariotti, M.A. (1996) Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems, Proc. of PME-XX, Valencia
Mariotti M.A. & E. Fischbein, (1997) Defining in classroom activities,Ed.St.Math., 34, 219-248
Mariotti M.A., Bartolini Bussi, M., Boero P., Ferri F., & Garuti R. (1997) Approaching geometry theorems in contexts: from history and epistemology to cognition, Proc. of PME-XXI, Lathi, pp. I- 180-95.

Nota bene

"For this [admitting of truths which are not immediate consequences of first and self-evident principles] may sometimes be accomplished through such a long chain of inferences that when we have arrived at the conclusions we do not easily remember the whole procedure which led us to them; and thus we say that we must come to the assistance of our weak memory by means of a certain continuous process of thought…. Because of this, I have learned to consider each of these steps by a certain continuous process of the imagination, thinking of one step and at the same time passing on to others. Thus I go from first to last so quickly that by entrusting almost no parts of the process to the memory, I seem to grasp the whole series at once." [BACK]


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