I'am reading John Taylor's Classical Mechanics chapter 1 page 20 where he proves the principle of conservation of momentum which states "If the net external force $F^{ext}$ on an $N$-particle system is zero,the system's total momentum $P$ is constant.
I'am not sure if i'am having trouble understanding the usage of summation notation or the entire derivation (which includes the concept) itself.Anyways here it is.A five particle system labelled by $\alpha$ or $\beta = 1,2,...,5$. The particle $\alpha$ is subject to four internal forces,shown by small arrows and denoted $F_{\alpha \beta}$ and the net external force shown by the large arrow and denoted by $F^{ext}_{\alpha}$.
Thus the net force on particle $\alpha$ is $$F_{\alpha} = \sum_{\beta \neq \alpha}F_{\alpha \beta}+F^{ext}_{\alpha}$$
According to Newton's second law,this is the same as the rate of change of $P_{\alpha}$:$$\dot{P}_{\alpha} = \sum_{\beta\neq\alpha}F_{\alpha\beta}+F^{ext}_{\alpha}$$
Then he considers the total momentum of the $N$-particle system,$$P=\sum_{\alpha}P_{\alpha}$$ Differentiating wrt time we get $$\dot{P}=\sum_{\alpha}\dot{P}_{\alpha}$$ or,substituting for $\dot{P}_{\alpha}$,$$\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+\sum_{\alpha}F^{ext}_{\alpha}$$ .The above expression is what is bothering me .I need someone to give me an intuition behind the double summation sign. How come it's $\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+\sum_{\alpha}F^{ext}_{\alpha}$ and not just $\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+F^{ext}_{\alpha}$. He eventually concludes that $$\dot{P}=\sum_{\alpha}F^{ext}_{\alpha}\equiv F^{ext}$$.