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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?

  • water is a compound, you can divide it into hydrogen and oxygen. You can further divide hydrogen and oxygen into components...and further...and further...The Advaita Vedanta and Mahayana Buddhists also say that there is an indivisible unified 'substance'. – Swami Vishwananda Jun 11 '18 at 4:26
  • Yes, but what I meant by the example is that it's still water. Meaning if I divide water both portions of water are still h2o. It doesn't cease to be water because I pour it into two different cups. Also the question is less about my example, and more about what argument Leibniz has to motivate his theory on substance being indivisible outside of his hunch that bodies infinitely regress into smaller and smaller bodies. – Robert C Jun 11 '18 at 4:36
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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.

  • It seems to me that there is a distinction between a compound (aggregate) thing and a divisible thing. An aggregate being something that is made up of different things e.g. in this case that whatever is an aggregate is relying on the things that make it up to give it its reality. But a divisible thing doesn't necessarily seem to me to be an aggregate thing. Can there not be a thing that is self-caused and uniform that could be divisible but never divides itself? In that case it wouldn't derive its reality from any other thing, and even if it divided it would still be the same substance. – Robert C Jun 11 '18 at 7:41
  • An aggregate is indeed something that is made up of different things, but not necessarily of things of different nature. As you said it yourself, a thing that is divisible and uniform — i.e. composed of parts of the same form — is still an aggregate, only a homogenous one (i.e. generated from similar elements, as opposed to different ones, i.e. heterogenous). What you're confusing is aggregate with composite. Composite is indeed something composed of different elements. Regardless, a substance, in 17th C parlance, is what we today would call element, something undecomposable. – André Levy Jun 11 '18 at 9:51
  • It's simpler than this. Anything that is extended in time or space is divisible. – PeterJ Jun 11 '18 at 10:15
  • That's a corollary of what I said, @PeterJ. IOW, you can state it point blank, like you did, or prove it from first principles and the definitions I enunciated 😉 – André Levy Jun 11 '18 at 10:21
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    @RobertC I think Andre nails it. An aggregate would reduce to one substance or be a collection of substances. To me Leibnitz's argument means that Matter cannot be a substance and so is not what materialists imagine it to be. – PeterJ Jun 11 '18 at 14:21
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See Leibniz's Logical Conception of Substance :

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.

  • I don't understand how he goes from the complete individual concept (what is the first part of that quote) to somehow concluding (3). – Robert C Jun 11 '18 at 7:53

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