Further reflections on mathematics
> If the thing in question be created, the definition must (as we have said) comprehend the proximate cause.
Where does a circle come from, where does it rest? I ask these aloud, am told at first by a pleasant girl that they come from the sundial, then this proving mistaken for its imperfect form, from the wheel. The notion of cyclicality is different from that of circularity. The word chronological comes from reference to the mill, the similarity of root sound to grain being no coincidence: time the devourer with scythe and gnashing teeth.1 This great sacrifice of order to ourselves. The harvest as buffalo jump.
The tree, I think; that is where the circle comes from, that likely also the wheel from the first forms of movement in this way. Consider other means of moving vast figures: to roll it upon logs, or to walk it with ropes. The latter is the algebraic method, and this in the fullness of its metaphor: to move through ropes and points. These ropes anchor and execute transformations of the original figure.
The geometric method then, what is that? Here the object rolls, there is a continuous aspect to this movement. This is a thing determined by contact between two, by the movement of an object over a surface. The algebraic likewise implies a surface, but this is framed discontinuously in terms of points. Consider the continuous function: it is a method which seems to produce a curve, but what is that really?2
There are a series of points with a sufficient regularity that we blur discontinuity and consider them as motion.3 Supposing we had never encountered a curve, then what would be the defining trait which held together this regularity? It would, far as I can tell, be a non-entity. The defining feature of a shape is its negation, here the shape tells us what is not like this; that this excludes the structure we take to be a curve.4
It is defined ultimately in this way because it is necessarily defined incompletely. The incompleteness of an entity is its life, that only the dead are complete. This is the forward-facing aspect of the idea, by virtue of which it can touch the world.
The edges are not defined by rule or system but rather by taste and discernment. While the structure of mathematics may be mechanical—hence rational—ultimately it is rooted in an intuitive base with the sole credential of being repeatedly and improbably true. This the foundation of all mathematics.
Here the question is how this moves, how does development occur with regularity in mathematics when it rests ultimately upon intuition? There is the sense that we are describing some sort of truthfulness, something that is beyond us. This is true, and it is for this reason that it matters most where we get the circle.
The circle is seen twofold as a matter of morphological computation and physical constraint. For the latter, there is something in the basic law of things, something below or above our notion of physical law, which tends towards the circle—as also the curve—and such constraints manifest regularly in space.
This can be seen in the tree, and this form corresponds with its peculiar characteristic of rolling.5 All of this can be explained in its own terms but not beyond that. All here takes by necessity an axiomatic basis in the structure of these shapes, these symbols, and this is necessarily taken for granted.
There are certain shapes that we find in the world. I do not believe that the nature of our ideas is divinely created, but I cannot find any better cause for the nature of the world. The argument here is that the nature of the world is such as to allow not only the development of life but further of its peculiarity, precisely what it is we find.
I recall once writing on theology, on the teleological and cosmological problems: that the universe is such, in terms of its fundamental constants, as to allow life; and that the universe exist at all. My argument was that these collapse into a single term: that anything is is indistinguishable from the fact that it is this way and not some other.
Some say that we can explain the whole in terms of processes, and the extent of this truth will illustrate out point: the process cannot justify itself. The process does not cause itself, rather is posited as cause. The process follows from how things are and draws lines of string between aspects in time and space. This is its power.
This string construction, however, as any other, is held together entirely by that to which it is attached; in this case, nothing. The fact it holds together is our basic requirement. We need only suspend our disbelief and he will handle the rest.
Per Bongiovanni (2014): “These parallel lines of development are reconcilable in the basic meaning of the PIE root *g̑erH- ‘crush, grind, wear down,’202 which by extension denotes also the material substance that is crushed or ground, viz. grain, and also, in a corollary sense, the physiological condition obtained as a consequence of the act of wearing down, viz. old age, wherefore it seems only natural to conceive of time here as the causative agent. This begins to explain the symbolism of the rotating sky as a cosmic mill in Scandinavian and other mythologies.”
There is a fruitful tension at the heart of topology: that the continuous space is defined with reference to a discontinuous set. This sleight of hand is as productive here as elsewhere.
This seen in Zeno, or likewise in Bergson: “Faced with our impotence to reconstruct the movement with these points, we insert other points, believing that we can in this way get nearer to the essential mobility in the movement. Then, as this mobility still escapes us, we substitute for a fixed and finite number of points an “indefinitely increasing” number—thus vainly trying to counterfeit, by the movement of a thought that goes on indefinitely adding points to points, the real and undivided motion of the moving body.”
This in Spinoza’s letter to Jellis: “it is plain that the whole of matter considered indefinitely can have no figure, and that figure can only exist in finite and determinate bodies. For he who says, that he perceives a figure, merely indicates thereby, that he conceives a determinate thing, and how it is determinate. This determination, therefore, does not appertain to the thing according to its being, but, on the contrary, is its non-being. As then figure is nothing else than determination, and determination is negation, figure, as has been said, can be nothing but negation.”
Consider the theory that ‘sphere’ may be defined in terms of movement: “Connections with σπαίρω (spaírō, “to gasp”) or Proto-Indo-European *sperH- (“to kick, rebound, move convulsively”, the original sense would be "something that rebounds") have been suggested, but the aspiration of σπ- to σφ- is unexplained and the semantic development is dubious.”