We use the validity of Addition and Multiplication for a hidden variables theory. First, we provide an example that the two operations Addition and Multiplication do not commute with each other as revealed by the analyses that are performed in a finite set of numbers. Our discussion leads to an initial conclusion that Sum rule and Product rule do not commute with each other in a hidden variables theory. If we accept this conclusion, we do not get the Bell- Kochen -Specker paradox. In more detail, quantum mechanics may accept the hidden variables theory. Next, we discuss the validity of operators under an assumption that Sum rule and Product rule commute with each other. In this case, we indeed get the Bell- Kochen -Specker paradox. We got the non-classicality of macroscopic experimental data observed in the Stern-Gerlach experiment and the double-slit experiment. If we detect |↑> and then we detect |↓>, the experiments cannot accept the hidden variables theory. We considered whether we can assign the predetermined “hidden” result to numbers 1 and -1 as in results of measurements with the number of measurements finite (e.g., twice) in the experiments. It turned out that we cannot assign the predetermined hidden result to such results of measurements. The next conclusion indicates interestingly that the Stern-Gerlach experiment cannot accept classical mechanics. The double-slit experiment had led to the same situation, and they were indeed quantum mechanical phenomena.