数学教育における「証明」についての基礎的研究 : 「定理」と「証明」についての考察(1)

抄録

　　In this paper, I analyze a process of knowing a theorem through some kinds of concrete operations, like action proofs. For this sake, I intend to use the theory of excluding the third term, invented by Imamura, Hitoshi, a Japanese philosopher.

[fig. 1]

[fig. 2]

　　Consequently, this process consists of four phases.
　　We can display each phase by a kind of diagrams. In this diagram, O(a), O(b), etc mean operations on concrete objects a,b,etc. And O'(a), O'(b), etc mean how to manipulate concrete objects. Then, the fig.1 mean that O(b) represents O(a).
　　The first phase can be pictured like fig.2.
　　This diagram means, if O(b) represents O(a) then the manipulating way of the object b is represented by O(b) being able to represent O(a). It seem natural that we transform the fig.2 to fig.3. These two figures will display the first phase. And the second phase, pictured by the fig.4,

[fig. 3]

[fig. 4]

where only two manipulations O(a) and O(b) are pictured. In this diagram, there are no distinctions between O(a) and O(b). And all objects are lacking distinction.
　　We can draw fig.5 for the third phase.
　　In this phase, one object "a" is picked up in order to represent the other manipulates O(b), O(c), O(d), etc. And the same time, each manipulate can represent O'(a).
　　Finally, the fourth phase, we can picture like fig.6. In the diagram, O(*) mean the form of O(a), O(b) etc that is the manipulating way itself. Usually, we give expression to this O(*) through a kind of algebraic symbols etc.
　　And then, we can obtain a kind of formal proofs.

[fig. 5]

[fig. 6]

　　In these phases, there exist at least two problems. The first, whether it is natural to transform to fig.3 from fig.2 or not. The second, what is the condition to be able to transform to fig.6 from fig.5.
　　We have to do further studies on these problems.