Determinant

Definition and stakes

Portrait of Percy Alexander MacMahon Percy Alexander MacMahon,  1911 Encyclopædia Britannica, Volume 6… (1911)

“ We have to find the number of permutations such that exactly m of the letters are in places they originally occupied. We have the particular redundant product
(ax1 + x2 + ... + xn) ξ1 (x1 + ax2 + ... + xn) ξ2 ... (x1 + x2 + ... + axn) ξn
in which the sought number is the coefficient of am xξ11 xξ22 ... xξnn. The true generating function is derived from the determinant
and has the form
It is clear that a large class of problems in permutations can be solved in a similar manner, viz. by giving special values to the elements of the determinant of the matrix.
”
Source: Wikisource

Portrait of Victor Cousin Victor Cousin,  Lectures on the true, the beautiful and the good

“ The richness of determinations is a certain sign of the plenitude of being. Reflection distinguishes these determinations from each other, but it is not necessary that it should in these distinctions see the limits. In us, for example, does the diversity of our faculties and their richest development divide the me and alter the identity and the unity of the person? Does each one of us believe himself less than himself, because he possesses sensibility, reason, and will? ”
Source: Gutenberg

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