A plenum is a space filled through and through, with no true gaps or voids in between things, though there may be different stuff (or different amounts of stuff) at different points in the space. This is in contrast to the concept of a space where objects have definite hard boundaries, where there is stuff inside those boundaries and there is truly absolutely nothing outside their boundaries, between the objects.

I'm trying hard to think of a real-world example or analogy, and the best I can think of right now is the difference between a digital signal and an analog signal. A digital signal is always either 100% on or 100% off, 1 or 0. But an analog signal is never truly "off": it's just different degrees of amplitude. The later is like a plenum, the former is like atoms in a void.

Leibniz doesn't seem to be arguing here that space is a plenum, just assuming that we already know it is, because he doesn't seem to give any reasons for it in the quoted bit. He's just saying that since it is (every point in space is filled with some thing or other, none of it is truly empty), if every thing was exactly the same as every other thing, there would be no discernible change, because if you swap two identical things you can't tell the difference. If space were not a plenum, that would not be the case, because you would still have the arrangement of identical things and the gaps between them changing around.

It's like saying that since every point in an image is filled with some pixel or another, if all pixels were the same color we wouldn't see any change as they moved around. But if there was such a thing as a non-pixel, like if screens were transparent except where pixels got filled in, then even if all pixels were the same color we could still see patterns changing in where there were or were not pixels.

Thank you.

According to contemporary physics isn't it popular now to conjecture that space really is made up of "pixels", since it has the notion of a Planck Length, a value less than which is meaningless?

That is an unresolved question in contemporary physics. General relativity treats space as continuous, while quantum mechanics treats everything as discrete (which is what the "quantum" part means). The two have yet to be reconciled with each other, but yeah it is generally expected that there will be found some way of quantizing space and time (and thereby gravity), we just don't know for sure what it is yet.

But also, both theories in effect treat space as a plenum. In GR, space itself is a significant feature of the universe, so since there is space everywhere definitionally, and it's all got some curvature and energy and gravitational effects and so forth, then no space is really empty in GR, even if there was "nothing" in it. But meanwhile in QM, every particle is viewed as an excitation in a quantum field of infinite extent, where a field is by definition a mathematical object which has a value everywhere, so "field" is almost a synonym for "plenum"; there aren't, for example, hard boundaries on an electron, but rather some degree of "electron-ness" everywhere, of which electrons as we know them are sharp little local concentrations; and likewise with all other particles.

Very enlightening and fascinating. Thanks.

where a field is by definition a mathematical object which has a value everywhere, so "field" is almost a synonym for "plenum" — Pfhorrest

Nice exposition. A few additional comments: There are various kinds of fields. A mathematical field is more an algebraic structure with operations, like a group or ring. Complex or Euclidean vector fields have associated with each point therein a direction and a magnitude. A force field is an example. A time-dependent force field has the property that each point has a variable force vector that varies with time. Shifting magnetic fields are an example. Most of space, I imagine, has some sorts of fields running throughout, without meaning a material substance. On the other hand, plenum might refer to the aether (if you think it might exist).

Monads now exist as proper mathematical entities in non-standard analysis.

(I once searched my math pedigree and found that Friedrich Leibniz (father of GWL) was my earliest ancestor)