## Chapter 37

Example of Hypothesis Selection in Discovering

Example 37.1 Suppose you are working with matrices a,b,c etc. If an NCProcess command
discovers that ab = bc and, at that point, you realize that it is reasonable to assume that no
eigenvalue of a is an eigenvalue of c, then b must be zero from an analytic argument.
Therefore, one can add the polynomial equation b = 0 to the collection of polynomial equations and
continue. As research progressed, one would add more and more analytic observations at the points
where it became clear what those observations were.

The syllogism corresponding to the above type of argument would go as follow.

Fact 37.2 If 〈 whatever hypothesis 〉, then ab = bc.

Fact 37.3 If a, b and c are matrices, no eigenvalue of a is an eigenvalue of c and ab = bc, then
b = 0.

Fact 37.4 If 〈 whatever hypothesis 〉 and b = 0, then 〈 desired conclusion 〉.

Conclusion 37.5 If 〈 whatever hypothesis 〉, then 〈 desired conclusion 〉.

Fact 37.2 would be seen by using the algorithms in this paper. Fact 37.3 is an observation from
analysis. Fact 37.4 would be seen by using the algorithms in this paper. Conclusion 37.5 is a tautological
conclusion of the Facts 37.2, 37.3 and 37.4.