November 21st update: the final version of Gila Hanna paper “
Reflections on proof as explanation” will no longer include the comment which justified this post. However, taking into account this comment was important and the following clarification of the misleading diagram is necessary. I thank Gila for the quality of our exchange and for giving me the opportunity of this clarification.
For about 30 years, I have used the Venn diagram reproduced here, without noticing how seriously it could be misleading once separated from its context. I realized that when reading recently Gila Hanna “
Reflections on proof as explanation” (
Hanna 2016). She referred to this diagram in support to the claim “
If one were to take the position that an explanation is simply a deductive argument, then all proofs would automatically be explanations”. This is the consequence of a quick reading of the paper where I used this diagram (
Balacheff, 2010, p.130), but with a text making explicit the meaning of the three sets and the corresponding perspective.
In the said 2010 paper, entitled “
Bridging proving and knowing in mathematics”, I postulated the following: “
the explaining power of a text (or non-textual ‘discourse’) is directly related to the quality and density of its roots in the learner’s (or even mathematician’s) knowing.” I then added explicitly that such a text or discourse is “
an “explanation” of the validity of a statement from the subject’s own perspective.” The following intended to position the three expressions: explanation, proof and mathematical proof.
“What is produced first is an “explanation” of the validity of a statement from the subject’s own perspective. This text can achieve the status of proof if it gets enough support from a community that accepts and values it as such. Finally, it can be claimed as mathematical proof if it meets the current standards of mathematical practice. So, the keystone of a problématique of proof in mathematics (and possibly any field) is the nature of the relation between the subject’s knowing and what is involved in the ‘proof’.” (my today emphasis)
Setting this framework was cautious enough not to restrict mathematical proof to logic in a narrow sense but to “the current standards of mathematical practice”. I must recognize that using such a diagram was a bit risky, and misleading for a quick reading.
I first developed this approach at the end of the 80s. Taking the perspective of the learner’s knowing, I chosen the word “explanation” instead of “argumentation” to account for the genuine effort of the learner to respond to the “why” a statement or a result is valid based on his or her “existing knowledge” – as Gila Hanna refers to it; the ground of the claim for validity being the functional organization and semantic value of the statements as opposed to what Duval called their epistemic value. Indeed, the ultimate aim of an explanation is to modify the epistemic value of the statement or result which initially is best qualified as a conjecture. Transforming one's own personal explanation of the validity of a statement into a proof (or a mathematical proof) is a complex process not always successful nor possible. When reading a proof, the reverse process necessary to get from it an explanation of the claimed validity is in itself an issue. It means constructing the links between the content and structure of the proposed proof and the reader’s own existing knowledge. It is in this manner that I understand the issue of the explaining power of a proof.
Hanna G. (2016)
reflections on proof as explanation. In: 13th International Congress on Mathematical Education. Hamburg, 24-31 July 2016 [https://www.researchgate.net/publication/316975364]
L'argumentation mathématique, un concept nécessaire pour penser l’apprentissage de la démonstration 
