There is a vast literature on mathematical proof; this post is an attempt to synthesise some of it into extracts that I have found interesting. It is the first of two posts, the second being on the pedagogical implications of this theory.
Why learn about proof?
Gila Hanna cites many purposes of proofs, including verification and explanation: “But in the classroom, the fundamental question that proof must address is surely ‘why?’. In the educational domain, then, it is only natural to view proof first and foremost as explanation, and in consequence to value most highly those proofs which best help to explain... Mathematicians clearly expect more of a proof than justification; they would also like it to make them wiser.”
Maria Mariotti broadly agrees: “Proof clearly has the purpose of validation - confirming the truth of an assertion by checking the logical correctness of mathematical arguments - however, at the same time, proof has to contribute more widely to knowledge construction. If this is not the case, proof is likely to remain meaningless and purposeless in the eyes of students.”
Guershon Harel & Larry Sowder suggest the process of proof is a search for certainty:
This is roughly in line with Humberto Maturana's view of proof: "If the criterion of acceptability applies, the (re)-formulation is accepted and becomes an explanation... the emotion or mood of the observer shifts from doubt to contentment, and he or she stops asking over and over again the same question."
For Nicolas Balacheff, proof requires a shift in epistemological position: “To construct a proof requires an essential shift in the learner’s epistemological position: passing from a practical position (ruled by a kind of logic of practice) to a theoretical position (ruled by the intrinsic specificity of a theory)." [More about Balacheff's views on the process of proof below.]
Dave Hewitt also talks of a shift, in awareness and attention: “Where attention is placed and of what one is aware are important within any proof. Proof is not only about properties, but is also about awareness of properties... Attending to proof can force a student to examine what they do know and what they do not know about something... I believe it is the attention to properties which helps develop someone's learning in mathematics, because that is what mathematics is about.”
Mike Ollerton (private correspondence) agrees that proof is fundamental to learning mathematics: "We certainly need to distinguish between repetitive demonstration and proof though I am not sure I understand what 'formal' proof is or what it means. For me I see proof as deeper conceptual understanding; the ultimate process and something that goes beyond generality. Having said that I want to see proof as part and parcel of what it means to "think and behave mathematically" at any age."
As does Alan Bell: “It might be said that this is the essence of mathematics - the art of dealing with the general by working with the particular.”
John Mason suggests that learning about proof has wider (social and personal) implications: “Children have been exposed to a variety of reason-giving situations from a variety of types of authorities… For young children, authority is external, expressed and warranted through adults who know more, or implicit within the structure of 'how things are'.
However, belief alone is inadequate as a warrant in mathematics. Mathematics offers an entirely different form of warrant for belief, and source of authority. One of the features of mathematics is that it lends itself, indeed it depends upon, a shift from external authority to the authority of one’s own reasoning. However this is not an easy shift to make…
Algebra is much more than the principal language of (mathematical) reasoning, because it is also the study of the structure of that reasoning itself, and of the structures within which that reasoning is fruitful. As such I consider it to be both within the reach of, and vital for, every active citizen...
To be responsible means literally to be able to spond from the Latin, meaning ‘