We can bring out the importance of this seemingly trivial point however if we turn again to Deleuze's reading of the calculus. I said originally that "the differential must differ in kind from the numbers that make up the primitive curve" - this was ambiguous and you were right to call me out on this. It's indeed far more precise to say that the derivative of f(x) yields another function f'(x): what I wanted to convey is that on Deleuze's reading, the difference between these two functions is not simply quantitative but rather qualitative. What does this mean? Negatively, that the differential cannot be a magnitude or a quantity: at the point at which dy/dx = 0/0, the value of the derivative is itself neither zero nor an infinitesimal. As Sean Bowden puts it, "dx represents only the cancellation of quantity in general"; instead, Deleuze's argument is that while it cannot be determined in the form of quantity, it can (only) instead be determined in "qualitative form". — StreetlightX
The existence of the difference, or relation, which cannot be expressed quantitatively can be demonstrated by the difference between spatial dimensions. The relation between spatial dimensions is incommensurable. The relation between the circle (2d) and the diameter (1d), is pi, which is an irrational ratio. If one takes two equal length line segments at a right angle to each other (representing two distinct dimensions), the diagonal (which crosses both dimensions), is again irrational.
Consider the difference between a straight line and a curved line. We could assume points on those lines to mark off segments. No matter how small the segment of line is that one marks off with the points, the segment of curved line will always be fundamentally different from the segment of straight line, and the relation between them is incommensurable. I believe there are two approaches to this problem. First, we could consider giving dimensionality to the point. Then a point on the curved line would be fundamentally different from a point on a straight line. But this would make "points" complicated, requiring different types of points for different applications, a 1d point, a 2d point, etc.. Furthermore, the non-dimensional point has been proven to be very useful, so there is very good reason to consider that it has some basis in reality. The second possibility then, is to maintain the non-dimensional point but to allow that the space between the points on the curved line is fundamentally different from the space between the points on the straight line. This requires that we reify space itself. We must allow that the space between points is something real, if we desire to maintain the use of non-dimensional points, and also allow for the reality of the non-quantifiable relation between different spatial dimensions. Space exists and it has real qualities which we do not know how to measure. We measure objects, but since objects are merely the way that space is represented to us, the unintelligible aspects of space render absolute accuracy impossible.
Now we approach the basis of the non-quantifiable relation. This is the relation between space (being now described as something real), and the non-dimensional point. In order to understand this relation we must give the non-dimensional point a position with respect to space. Without a position, it cannot be related to space. I believe its position is in time. The non-dimensional point is a point in time. Now we must reverse the relationship between space and time, which makes time the 4th dimension, such that time can have its proper relation to space, as the 0th dimension.
And what does it mean that the differential can be determined only in qualitative form? Simply that, as we've said, the derivative is never simply a value that correlates to a single, particular point on a primitive function, but instead defines the qualitative character of the function around a particular point. In Simon Duffy's words, "the differential relation characterises or qualifies not only the distinctive points which it determines, but also the nature of the regular points in the immediate neighbourhood of these points" (Duffy, "The Mathematics of Deleuze's Differential Logic and Metaphysics"). This is the import of the Aden quote above. — StreetlightX
This is very good, because if we consider the point in time, as the non-dimensional point, we can start to see the vague relationship between points in time, and the surrounding space. Recognize that we have reified space, such that it is something "real", in the sense that physical objects are real, but what we are actually looking for now is the real reality, the reality which is the "becoming" that lies beneath the object which has been identified as space, and is associated with the 0th dimension, time. Since the incommensurability has been identified as existing within the dimensions of space, the vagaries which exist around the non-dimensional points are proper to that object, space itself. So we must go deeper, into the non-dimensional points to find real quantity, or quantifiability.
Why is this reciprocal determinability of the differential important to Deleuze? For two reasons: first, not only does it provide a model for pure relationality, but second and even more importantly, this model itself has a distinctive trait that allows Deleuze to set himself against a position that his entire oeuvre pitches itself against: the idea that what exists prior to individuation is an indeterminate generality which is then progressively differentiated though limitation or negation (which itself calls for a correlative abandonment of any hylomorphic model of individuation). — StreetlightX
With respect to this then, my position is that the whole appearance of indeterminateness is due to the somewhat unintelligible nature of space. Because space has unintelligible aspects, we can conclude that space does not necessarily behave in the way that it should. The "way that it should", is the way that is determined by this underlying reality, the sphere which Deleuze is saying is completely determined. This is the realm of what Bohm calls "hidden variables". In his "Wholeness and The Implicate Order", he posits an underlying realm of activity, becoming, of which we see only a reflection of, in the spatial existence of objects, just like Plato's cave people only see a reflection of reality.