[A thing such as a book is finite: it extends only so far, and some day it will decompose to dust. So it has determinate limits spatially and temporally.
But the book turns into dirt, which becomes a tree, and on and on. So we might broaden our perspective of duration and consider what underlies all things no matter when in time. The book-dirt-tree-etc would just be one part it, in an endless chain of changes. Hence, we can imagine that the underlying substance has no beginning or end. However, when we think of the lifespan of the book, from its manufacture to its decomposition into dust, we can think that it lasted a determinate period of time. It had a duration. But if what underlies everything has no beginning or end, then it has no duration. In that sense it is eternal and infinite. But it is not an infinite duration, because durations have beginnings and ends.
Also, we might think that the book is a set of pages. And the pages are made of fibers, and the fibers of molecules and on and on infinitely. And we might think that the book is part of a library, which is part of a city, which is part of a state, which is part of a continent, which is part of a planet, and on and on until we get to everything. We might also then think that every division was arbitrary, for it could have been otherwise, the divisions never held for long, and etc. (see the Zeno entry for an extended explanation of this example.) So what underlies everything can be thought of as indivisible, even though we may use our imaginations to draw lines in the world around us to create distinguishable entities. Really, everything is all one substance, according to this perspective.
We might also distinguish the book in our hand from the book "in our mind." In a sense, there seem to be two primary dimensions, Thought and Extension. The book extends in space, the idea of it does not. There is the extending book, and also the idea book, but there is only one book between the two. So we might say that there is one substance that expressed itself qualitatively in two ways, extensively and conceptually. One of substance's natures is its being extended, the other is its being thought or ideated. One such nature is one attribute or quality of substance. We may think of substance as extending, or we may think of it in terms of its ideas. So each attribute is a way that substance is conceived.
If substance is truly infinite, then we might think that substance expresses itself in an infinity of attributes, even though we as finite beings only have access to two. So substance would express itself as Thought, Extension, Quality 3, Quality 4, and so on to infinity.
We considered how the book was manufactured, then decayed, then became a plant, and so on. And we thought that we were mistaken to even distinguish the book from everything else, at least on a fundamental level.
When we shape dough for the desired bread, we are modifying the dough's form or shape. The dough stays the same, but we might modify it to make it a pizza crust, a pie crust, a bread loaf, and so on. In the same way, we might consider that the book was a modification of substance that became reshaped again as soil.
Using our imaginations, we drew a boundary to distinguish the book from the air around it, even though we also considered that fundamentally both paper and air are substance. And they are the same substance that has been modified in different ways. When we draw these boundaries, we may imagine the eternal indivisible substance as having finite parts. For, the book extends only so far, and lasts only so long, even though what underlies the book as well as everything else is not limited in any way.
Recall that substance, as infinite, expressed itself in an infinity of ways, which is to say that substance has an infinity of attributes. Again, we only have access to two: thought and extension. But that also means that when we designate a modification of substance as a book, we know that this mode corresponds to the idea for that book. But as well, if there are an infinity of other attributes, then this modification is producing the equivalent of the book in all the other ways which are not explicitly available to us. Nonetheless, they are there, implicitly. One finite mode implies an infinity of other modes, so it implies the infinity of substance. In a sense, substance's infinite nature is enveloped in every mode, such as the book. So the all the infinity of other attributes are involved in each finite mode. Thus when we see some finite thing like a book, we can know that explicitly we see something finite, but implicitly we see the infinity of the one substance. It is not hidden. It is fully there. Every dimension of substance is fully expressed, only some parts are implicit and others are explicit. By the deductive power of our rational faculties, we can know this infinity that substance expresses even without any other means to access to it.]
Finite and infinite are tied together by the relation of implication and involvement. Spinoza explains this relation in his 12th Letter on the Infinite. Here Spinoza claims that people have misunderstood the infinite because they failed to analyze the different sorts of infinity and the different ways of considering it. (20a) [For more on these particulars, see Spinoza's 12th Letter and Gueroult's commentary,.]
Spinoza illustrates his notion of the infinite using a geometrical example. [Below I replicate Duffy's rendition.]
To understand why Spinoza uses this figure, we should examine first its appearance in his Principles of Cartesian Philosophy. [For a more extensive analysis, see the entry on the 12th Letter.] Imagine water moving around this circular course:
The water moves from A to B, and from B to A, in a circle. If the space were even all the way around, we would expect the water to move the same speed the whole time. But here we see that more water has get through the narrow space at B. So it will have to go faster to go through it. Hence the water moves faster at B than it does at A.
The outline of a circle is perfectly continuous. These circles are off-set, creating differences in distance between the one and the other. And because
1) both are circles
2) circles have continuously changing outlines,
3) the circles are off-set,
we can then conclude that there is a continuity of difference between the one and the other. So if we were to pick any point of the water's course, we know it will be moving at a different speed than the water at any other point of the water's course.
Consider Spinoza's purely geometrical example.
On the left, the semi-circles are evenly set with each other. Hence the distance between one outline and the other is the same all the way around. On the left we have off-set circles. Hence the distance between the two is everywhere different. Again, there is a continuity of an infinity of differences.
Hence we return to the diagram from the 12th letter, in Duffy's rendition.]
Spinoza says that
1) the inequalities of the spaces between the two is infinite, and
2) the variations of the speed of matter travelling through that space is infinite.
(Duffy 20bc)
Spinoza will show us a new way to understand the infinite.
We might normally think something is infinite when it is so extremely great that no matter how large we think it, the infinite is always greater. In this conception of the infinite, which Spinoza opposes, we cannot assign a magnitude to any infinity, so we cannot say that any infinity is larger or smaller than another one.
Here Spinoza notes that the inequalities and varations are infinite not because we cannot determine where the space ends. For, we know where it terminates: lines AB and CD (the maxima and minima.) Also, we cannot just say that the circles are infinitely large and that hence the space between them is infinitely large. For, even if we take a very small finite slice of the infinitly large circles, there will still be within that finite slice an infinity of inqualities and variations.
[We are dealing with extensive space. We noted before that substance has an infinity of attributes. We add that each attribute itself is infinite. So there is nothing that limits extension itself. Thus if we take some part of extension, a slice of the non-concentric circles for example, we can perform Zeno's operation and continually subdivide it. But we will never come to any smallest extensive pieces. For, if it extends, it can be divided. In a sense that makes every finite piece of extension infinite, which seems absurd, but this is Spinoza's point. We have to change our understanding of infinity. So instead of infinity being too large to be described in finite terms, Spinoza proposes another kind of infinity.
We might first consider a number like pi. We know that it is an irrational number that cannot be expressed in numerical digits, because there would be no end to those digits. Does that mean that pi is infinite? No, because pi is smaller than 4 and larger than 3. Other determinate values surround it. But because there is no end to the digits, there is no way to express pi in its entirity using numerical digits. In a sense, number does not apply perfectly to pi, even though it is a fundamental proportional-value in the world around us. So pi exists, but it cannot be numerically expressed.
Spinoza wants us to see infinity in a similar way. There are boundaries to the slices of the non-concentric circles. But the amount of variations between them has no limits. So the internal variation of a mode cannot be expressed in number. This is Spinoza's infinite. It is not infinite divisibility like Zeno's infinite. Rather, it is indivisibility. What underlies the divisible objects that extend is something that is indivisible. But it was a creation of our imagination to divide-up substance's modifications to see the book as an individual part. Likewise every division of extension is a creation of our imagination. Hence it can go on for ever. For, what underlies extension keeps "giving" more extension, as it were. Take some away, substance gives more, because you cannot really take anything away from substance. It is indivisible. Spinoza's infinite is the infinite indivisibility of substance. That is so say, it is the infinite intensity of substance, as we will see in another entry.]
Substance is infinite by nature. Substance causes its own modifications. And these caused modes then are infinite in the way we just described. The reason modes are infinite, then, is because their cause is infinite. We may use our imaginations to designate finite regions of infinite substance, like the book, and divide it infinitely. But the book is finite. Only its underlying substance is infinite.
Then, there is a related infinity. Modes were infinite because their cause is infinite. But, what makes up a mode is an infinity of continuous variation. And we may say that in a larger slice of the circles, there is a larger infinity of variation. Not because there are more divisions. For we are not talking about divisions. Divisions can be expressed in number, and this sort of infinity cannot be expressed in number. But a larger extending object expresses more of substance's infinity, because it contains more of its indivisibility. For, it exhibits more internal variation. Hence the infinities that finite modes express can be compared in terms of more-or-less. And they are infinities even though they are bound by maxima and minima. (21b)
Duffy, Simon. The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze. Aldershot: Ashgate Publishing, 2006.
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