Answer to Question #121184 in Geometry for Mabandi

Question

De Villiers (2003:5 - 10) classifies five different functions of proof within mathematical
statements. With the aid of examples, discuss the implications of the five functions of
proof to teaching and learning geometry.

Answer 1

a) Proof as of verification/conviction.

With few exceptions, tutors believe that any proof provides certainty for mathematicians and therefore only only authority in establishing the validity of a conjucture.

Research mathematicians seldom scrutinize the published proofs of results in detail. When investigating the validity of a new, unknown conjucture, mathematicians usually do not only look for proofs but also try to construct encounter-examples at the same time by means of quasi-empirical testing. Since such testing may expose hidden contradictions, errors or unstated assumptions. In attaining conviction, the failure to disprove conjectures empirically place just as important a role as the process of deductive justification.

b) Proof as a means of explanation.

Although it is possible to achieve quite a high level of confidence in the validity

of a conjecture by means of quasi-empirical verification (for example, accurate constructions and measurement, numerical substitution, and so on), this generally provides no satisfactory explanation why the conjecture may be true.It merely confirms that it is true, and even though considering more and more examples may increase one's confidence even more, it gives no psychological

satisfactory sense of illumination. No insight or understanding into how the

conjecture is the consequence of other familiar results. many mathematicians the clarification/explanation aspect of a proof is of greater importance than the aspect of verification. For instance, the well-known Paul Halmos stated some time ago that although the computer-assisted proof of the four colour theorem by Appel & Haken convinced him that it was true, he would still personally prefer a proof which also gives an "understanding".

c) Proof as a means of discovery.

There are numerous examples in the history of mathematics where new results were discovered or invented in a purely deductive manner; in fact, it is completely unlikely that some results (for example, the non-Euclidean geometries) could ever have been chanced upon merely by intuition and/or only using quasi-empirical methods. Even within the context of such formal deductive processes as axiomatization and defining, proof can frequently lead to new results. To the working mathematician proof is

therefore not merely a means of verifying an already-discovered result, but often also a means of exploring, analyzing, discovering and inventing new results.

d) Proof as a means of systematisation

Proof exposes the underlying logical relationships between statements in ways no amount of quasi-empirical testing nor pure intuition can. Proof is therefore an indispensable tool for systematizing various known results into a deductive system of axioms, definitions and theorems. it is in reality false to say at school when proving self evident statements such as that the opposite angles of two intersecting lines are equal, that we are "making sure". Mathematicians are actually far less concerned about the truth of such theorems, than with their systematization into a deductive system.

e) Proof as a means of communication.

One of the real values of proof is

that it creates a forum for critical debate. According to this view, proof is a unique way of communicating mathematical results between professional mathematicians, between teachers and students, and among students themselves. The emphasis thus falls on the social process of reporting and disseminating mathematical knowledge in society. Proof as a form of social

interaction therefore also involves subjectively negotiating not only the

meanings of concepts concerned, but implicitly also of the criteria for an

acceptable argument. In turn, such a social filtration of a proof in various

communications contributes to its refinement and the identification of errors, as well as sometimes to its rejection by the discovery of a counter-example.