to offer explanation’. Thus to become responsible is to be able to justify your actions to others.
Learning to be your own authority in mathematics, with being responsible through being able to justify, with learning how to convince yourself, a friend and a sceptic, probably reflects, or could reflect, perhaps even presage the same processes within a broader social context. Thus mathematics could speak to the experience of students at all ages.”
What constitutes a proof?
John Mason suggests that the learning of mathematical proof starts with convincing others: “Being convinced oneself is quite different from convincing someone else, particularly when that person is being sceptical. Learning to be sceptical oneself, to locate gaps and jumps in other people’s reasoning is important not just in mathematics, but as an involved citizen. As my remarks have suggested repeatedly, convincing, reasoning, proving, justifying are all emergent and developing processes, not rigid formats to be learned or followed."
Nitsa Movshovitz-Hadar broadly agrees with this view: “... mathematics knowledge is in principle not different than any other kind of knowledge, although, of course, the nature of the discipline is different. What, then, is knowledge? An answer, given by philosophers to this question, is that in order to be one’s knowledge a proposition must comply with three necessary (albeit not sufficient) conditions: (i) It must be true. (ii) one must believe it, and (iii) one must have justification for believing it."
She quotes Bertrand Russell: "'Minds do not create truth or falsehood. They create beliefs.' Given a statement p of a mathematical theorem, a learner should be able to relate to two basic questions: (a) 'Do you believe that p?' and, provided the learner’s answer to (a) is yes, (b) 'Why do you believe that p?'"
However, Hanna suggests: “It is imperative to make a clear distinction between a correct proof and a heuristic argument…"
She offers two conceptions of proof: "(1) Formal proof: proof as a theoretical concept 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, explora