t in formal logic, which may be thought of as the ideal which actual mathematical practice only approximates, and (2) Acceptable proof: proof as a normative concept that defines what is acceptable to qualified mathematicians.
Practising mathematicians have long agreed that proofs may have different degrees of formal validity and still gain the same degree of acceptance… The acceptance of a theorem by practising mathematicians is a social process which is more a function of understanding and significance than of rigorous proof… mathematicians undeniably insist on using their judgment when deciding whether or not to accept a new proof… mathematicians seem to adhere to Bateson's view that 'the advantages in scientific thought come from a combination of loose and strict thinking, and this combination is the most precious tool of science'… A proof is valued for bringing out essential mathematical relationships rather than for merely demonstrating the correctness of a result.
The distinction I have made between proofs that prove and proofs that explain is similar to that which Bolzano makes between "making certain" (Gewissmachung) and "building a foundation" (Begruendung).”
Mariotti suggests why this shift from inductive/abductive reasoning based on experience, to deductive reasoning based on logic, is difficult: “The difficulty emerges from controlling the complex relationship between mathematical validation, rooted in the frame of a theoretical system, and common sense validation, rooted in empirical verification."
Besides direct acquisition of information that is mostly related to factual evidence and attained through experience, human culture has developed a complex way of obtaining information and knowledge, which is not direct, but is rather mediated by means such as language, logic and reasoning. As a consequence of this mediation, the structural unity between cognition and adaptive reactions has been broken.
The interest in the case of abduction lies in the fact that, according to experimental evidence, exploration supporting a conjecture is very often accompanied by arguments showing this structure, so that passing from conjecturing to proving would require transformation from an abductive into a deductive structure. This passage presents difficulties that seem to require specific didactical treatment to overcome."
Balacheff explores the difficulties in moving from argumentation to mathematical proof in more detail: "Teachers organize situations in order to “convince” or “persuade” learners (in the vocabulary of Harel & Sowder).
Argumentation seems the best means to this end. It works both as a tool for teaching and as a tool for doing mathematics for a long while. But then learners suddenly face an unexpected revelation: In mathematics you don’t argue, you prove… It must be remembered that at stake is not truth but the validity of a statement within a well-defined theoretical context.
The arguments in such a discussion involve three types of critical considerations: the search for certainty, the search for understanding and the requirements for a successful communication. The complex nature of proof lies in the fact that any effort to improve a proof on one of these dimensions may change its value on the other two. There is no clear standard to decide on the correct balance.
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; it can be claimed as mathematical proof if it meets the current standards of mathematical practice.
Students have to learn mathematics as social knowledge, they are not free to choose the meanings they construct. These meanings must not only be efficient in solving problems, but they must also be coherent with those socially recognized
Being the product of a choice, actions are considered in terms of validity and the adequateness of effect. Action is related to its aim, and a contradiction becomes apparent when the aim 