Leibniz seems to have this notion that unity is a necessary property of substance, and that substance cannot be divisible. Why is this the case? Couldn't it be the case that substance was like water in the sense that when you divide it it's still the same substance? It seems like he uses this idea to motivate parts of his theory for monads, but what is the argument he makes for substance itself necessarily being unified/indivisible?
At the time of Leibniz, substance didn't mean what we typically think of it today; it meant that which underlies everything else (sub•stance):
In contemporary, everyday language, the word “substance” tends to be a generic term used to refer to various kinds of material stuff (“we need to clean this sticky substance off the floor”) or as an adjective referring to something’s mass, size, or importance (“that is a substantial bookcase”). In 17th century philosophical discussion, however, this term’s meaning is only tangentially related to our everyday use of the term. For 17th century philosophers the term is reserved for the ultimate constituents of reality on which everything else depends (Tad Robinson, Internet Encyclopedia of Philosophy).
Thus, of course substance had to be indivisible, for otherwise it would compounded of other underlying (sub•standing) things.
In §8 of the Discourse on Metaphysics, Leibniz gives one of his most important accounts of the nature of individual substance. [...] in every true predication, the concept of the predicate is contained in the concept of the subject. “Since this is so,” Leibniz claims, “we can say that the nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.”
In other words, x is a substance if and only if x has a complete individual concept, that is, a concept that contains within it all predicates of x past, present, and future.
The consequences that Leibniz draws from the logical conception of substance and the doctrine of marks and traces are remarkable. In the following section (§9) of the Discourse on Metaphysics, we are told they include the following:
(1) No two substances can resemble each other completely and be distinct.
(2) A substance can only begin in creation and end in annihilation.
(3) A substance is not divisible. [...]
Unfortunately, Leibniz's reasons for drawing these consequences are not in all cases obvious. [...] If we consider the complete individual concept as that which allows us to pick out and individuate any individual substance from an infinity of substances, then we realize that, if the individual concepts of two substances, a and b, do not allow us (or God) to distinguish the one from the other, then their individual concepts are not complete. That is, there must always be a reason, found within the complete individual concept of substances and issuing from the free decree of God, that a is discernible from b. And this fact points to another important fact about the interpretation suggested above: it is not only the case that each substance has a complete individual concept–the essence of the substance as it exists in the divine mind–but for every essence or complete individual concept there is one and only one substance in a world.
Consider again Leibniz's view :
The nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed (§8 of the Discourse on Metaphysics).
If we imagine to "divide" an individual substance in two, what we get are two individual substances sharing all the predicates (i.e. properties).
But, according to Leibniz's principle of The Identity of Indiscernibles :
if, for every property F, object x has F if and only if object y has F, then x is identical to y,
we are forced to conclude that the two "divided" substances are identical, i.e. they are one and the same.