Already Aristotle (Physics III, 8, 206) objected to Zeno that, among magnitudes, there exists infinity by addition (I can always find an even number greater than the preceding one) but not by division, insofar as the infinity of the subintervals into which a unit of length is divisible is always contained in a limited totality (never greater than one) which may constitute the object of an empirical intuition.
In other words, if, cosmologically speaking, there is never perhaps a Firstness that is not the result of a previous Thirdness, cognitively speaking there is a limit to our perceptive abilities, which experience as undivided something that, cosmologically speaking, is in posse capable of being further divided. What is in posse belongs to cosmology. What is in actu belongs to the agent subject.
What happens when we put ourselves in the place of a perceiving subject? Zellini (2003: 26–27) reminds us that:
Adolf Grünbaum [(1969)] recently demonstrated that the measured structure of physical time justifies applying the arithmetical theory of limits to the solution of the paradox. Human awareness of time has a base limit of perceptibility, that is, a minimal threshold beyond which temporal intervals vanish into inconceivable smallness. If we consciously tried to contemplate ‘all’ the intervals of the series (a), it would be realized concretely as a countable infinity of mental acts, and the duration of each of these would be larger than the minimal threshold that time allows. But this insuperable ‘minimum’ is an Archimedean quantity: when added to itself infinite times, it yields an infinite result. Consequently, the mental contemplation of the entire series would result in an impossibly unlimited period of time. This would happen, for example, if one ‘counted’ the intervals of (a) one by one, assigning to each of them an ordinal number. This would take more time than the necessary minimum just to conceive or pronounce them. (But it is absurd, Aristotle objected [Physics 8, 8, 263a–263b], to maintain that whatever moves, moves while counting.) In reality, by raising doubts about the possibility of traversing the interval (0–1), Zeno exploits the unacceptable delay that is implied by reducing the series (a) to the corresponding mental acts of the counting process, but he fails to make clear that this process does not reproduce exactly the measurement of the physical time involved in the actual traversal.
Thus, Grünbaum finds Zeno’s argument illegitimate because it uses what is basically an inevitable confusion between two incompatible forms of thought. He explains that we do not experience the intervals into which we subdivide the traversal in any measure that corresponds to their actual nature. Rather, we derive our impression of their duration from the time needed for our acts of mental contemplation, which for each fraction of the distance must perforce exceed our minimal threshold or limit.
In other words, our perception is not mathematical but ingenuous, just as our perception of the supposed movement of the sun is ingenuous and not astronomical. Zellini (1980: 44) reminds us that the existence of a threshold of observability is a postulate both of physics and of the psychology of perception.
Zellini also appeals to Hume: our imagination must be capable of reaching a minimum beyond which we cannot conceive of further subdivisions. We can speak of the thousandth or ten-thousandth part of a grain of sand, but (apart from the fact that we cannot see it—which from the point of view of perception is no small matter) we can’t even imagine it except with the same dimensions as the grain of sand itself: “The idea of a grain of sand is not distinguishable, nor separable into twenty, much less into a thousand, ten thousand, or an infinite number of different ideas.”
“Put,” said Hume, “a spot of ink upon paper, fix your eyes upon that spot, and retire to such a distance, that at last you lose sight of it; ’tis plain, that the moment before it vanish’d the image or impression was perfectly indivisible” (Treatise of Human Nature, I, 2, 27) At a certain point, the spot will become invisible, because it is too far away, but when it is on the point of disappearing, it will still be visible as a punctual and indivisible minimum. As is the case for the ideas of the imagination, an ultimate conceivable term is given for our sense impressions, whereby we go directly from nothing to a minimal perceivable reality not resolvable into smaller parts.
Hume might have added that—while it may be true that under the microscope the same ink blot would reveal a universe of bacteria that made it look like a painting by Kandinsky—from the point of view of our perceptual abilities, it is a black spot, nothing more or less.
If it can be granted that for Peirce the Ground is what I referred to as primary iconism, let us bear in mind that the Ground is an element, a marker, a quality that is (for whatever reason) being isolated and considered in itself. By whom is it isolated? Potentially isolable, it becomes isolated when a subject isolates it, from a certain point of view, and at that point it becomes the terminus a quo of an inferential process, in an upward and not a downward direction—toward the series of relationships, in other words, that bind that spot to me and to my perceptual interests, not toward the series of the infinite possible decompositions of the spot itself.
This, it seems to me, is exactly what happens when Peirce tells us that we feel the blackness of the ink as Firstness. It is possible that—to be able to recognize that what strikes our senses is a quality of blackness—the brain deep down performs an immense number of successive operations. I also agree with Paolucci (2005) that, for the empirical concept of dog as well, the Kantian intellect may make use, not of images, but of a flowchart. But, aside from the fact that the brain too, as a computational machine, must come to a stop at a certain point in order to be able to transmit “blackness,” at the level of conscious perception we are not aware of that additional fractalization. There is a threshold on this side of which we perceive or sense “black” as Firstness, primary iconism (or whatever you choose to call it), and that is the starting point for our all subsequent inferences.
Commenting on Hume, William James (1987: 1061) declared: “Either your experience is of no content, of no change, or it is of a perceptible amount of content or change. Your acquaintance with reality grows literally by buds or drops of perception. Intellectually and on reflection you can divide these into components, but as immediately given they come totally or not at all.”
Zellini also cites Wittgenstein (Notebooks, 18, 6, 15):
If the complexity of an object is definitive of the sense of the proposition, then it must be portrayed in the proposition to the extent that it does determine the sense. And to the extent that its composition is not definitive of this sense, to this extent the objects of this proposition are simple. THEY cannot be further divided.…
What I mean is: if, e.g. I say that this watch is not in the drawer, there is absolutely no need for it to FOLLOW LOGICALLY that a wheel that is in the watch is not in the drawer, for perhaps I had not the least knowledge that the wheel was in the watch, and hence could not have meant by “this watch” the complex in which the wheel occurs. And it is certain—moreover—that I do not see all the parts of my theoretical visual field. Who knows whether I see infinitely many points?
Let us suppose that we were to see a circular patch: is the circular form its property? Certainly not. It seems to be a “structural” property. And if I notice that a spot is round, am I not noticing an infinitely complicated structural property?…