In this note, I wish to convince you of three things, using a few examples. First, your perception of a patterned image, such as the one above, is not a copy of the image, but a hybrid of a general description, and transient bindings of that description to foveal copies. Second, the description is often mathematical in nature. Finally , I wish to show that the patterns that you see lie within your percepts. They might exist in the external world in some sense, but play no causal role in it except via these percepts.
All of this applies to natural as well as artificially generated images, in so far as the natural world includes patterns too.
When you gaze at an image, such as the one above, there are two things going on. First there is the image itself as a physical object (in this case an array of pixels on your screen). Another is the seeing of the image, the image as it appears to you. The former is a physical thing, and the latter a mental thing. (Perhaps the mental thing emerges from the physical thing that is your brain, but it is still of a different character and structure than the image itself.) The mental thing is called a phenomenon (plural phenomena), in the jargon of the school of philosophy known as phenomenology, founded in its modern form by Edmund Husserl early in the twentieth century. I will employ this jargon in what follows.
2. Phenomena arising from images are not copies or maps, but rely on general descriptions
Are the left and right images just above different? They are, but this requires study. The phenomena that arise immediately from the two images are identical-they look just the same. Considered as arrays of pixels, or grids of short lines, there are plenty of diffences: 69 differences in the angles of lines. This is not apparent because the common phenomenon to which the two images give rise does not contain the orientations of the individual line segments. Instead, the phenomenon is captured by the description “a square grid of randomly oriented white line segments on a field of black”. By randomly is meant: “with no pattern that I can detect”. This shows clearly that what is in the mind, the phenomenon, includes something in nature of a general description, and does not contain a copy, in any sense, of the image. By “general” I mean that it does not specify the image exactly, but only partially; many images fit the description. Also, I am not saying that the phenomenon is a description in the literal sense of a sequence of words, but that it yields such a description when our verbal powers are set to work, and hence includes within it something equivalent to the content of that description, a descriptive structure, we might call it.
A surprising aspect of this situation is that you feel so strongly that the angles of the line segments are present to you , even though they are missing from the phenomenon. The feeling of presence, and actual presence in phenomena, are two different things. Daniel Dennett calls the former “the representation of presence” and the latter, “the presence of representation”. See this paragraph for a concise formulation of Dennett’s views.
3. Phenomena arising from images are mathematical in nature (at least sometimes)
The image above exhibits a pattern of an abstract kind. With a glance, you know all about it. If I asked you to describe it, you would likely come up with something like: “It is a grid with short red horizontal lines at the centers of cells. White lines consisting of the left and top boundaries of grid cells zigzag up and to the right across the grid, but there is also a horizontal band half way down where all four boundaries appear as lines, and a vertical band half way across with the same property. ” All of these aspects are available in your experience of the image within a second or so of seeing it. This is very close to an exact mathematical description of the image, in that it could easily be rendered in mathematical terms by someone with knowledge of analytic geometry and in fully formal terms by someone who is also familiar with axiomatic methods (eg ZF set theory). That is, the phenomenon that the image induces in your mind includes content that gives rise to a precise description, which description can be formalized mathematically, and the phenomenon with its descriptive content is built immediately upon attending to the image. If this were not the case, no description like the one above would be possible for you. Its presence does not depend on your knowledge of mathematics; only its formalization would depend on this knowledge. This is a remarkable fact — that a phenomenon containing this mathematical precision should arise in a few instants (though articulating it takes longer). Note that the phenomenon arising from images 1 and 2 yields a precise mathematical description as well: “a square grid of randomly oriented white line segments on a field of black”.
Note that I am practicing phenomenology, not psychology, here. The primary technique of phenomenological investigation is to carefully attend to one’s own phenomena exactly as they present themselves. The technique also involves “bracketing”. Bracketing consists of placing aside any consideration of the external objects which give rise to phenomena, and even of whether there are such external objects. It is not those externals that are the topic of investigation, but the phenomena, just as they are. In considering the phenomenon associated with image 3, I was able to detect the mathematical nature of its descriptive content. To do this required some training in mathematics on my part, but the phenomenon in itself includes this nature independent of the mathematical training of the individual in which it occurs. The phenomenon must include this mathematical nature for everyone, because everyone can come up with the sort of description that I alluded to, but only those with some mathematical background will detect the description’s mathematical nature.
A visual phenomenon is not exhausted by its descriptive content. The description is also bound to the foveal image on each saccade (shift of gaze): each primitive feature in the foveal image (e.g. line or circle), is, as it were, labeled by its category and role within the description. The structure, when revealed in this form, appears in concrete rather than symbolic guise. That is, one feels that one is aware of the pattern itself, rather than a description of it. Let us call this “pattern-awareness” in distinction to “description-awareness”. The latter is present only when one is consciously constructing a verbal account of the pattern. In pattern-awareness the descriptive content, though not a topic of current awareness, must be present within the phenomenon as a mechanism for supporting the pattern-awareness; what we are aware of in this case is the binding — the incarnation of the pattern in the sensory input of the moment. Peripheral vision plays a role too: binding of the much lower resolution peripheral imagery takes place too.
A rebinding (and perhaps modification) of the description takes place with every saccade, since each saccade modifies the set of visible features. This is what constitutes visual presence: a combination of a descriptive structure together with the sequence of bindings to the detailed images that arise from saccades. A central aspect of the feeling of presence is the experience that wherever one redirects one’s gaze, and that redirection lies within the scope of the current description, then it can be rebound with only minor changes, changes that are small enough to be consistent with expected continuities. This is how the sense of a stable world arises from images that are always jumping around. There is evidence that memory of detailed imagery persists only for a fraction of a second (in iconic memory) over saccades; only the pattern description persists (in visual short-term memory). Thus visual perception is in part a matter of continual forgetting.
4. Patterns play no role in the world except via our awareness of them.
Images 4 and 5 are identical to images 1 and 2, except for the red highlighting . From this we see that patterns of a simple kind were present in the images 1 and 2, but not present in the phenomena arising from these images. This is a simple example of patterns existing in an image, but not in our awareness of it. In this sort of case, the patterns may exist in the external world in some sense, but play no role in that world — they have no causal efficacy (in the counterfactual sense). Such a pattern is like Berkeley’s unobserved tree, only more so. The pixels that make up the image jostle local photons, but there is no need to mention patterns in an account of this jostling. If a pattern makes it into someone’s awareness it can have plenty of effects (such as triggering a verbal description), but if it does not, there are none. One might say, paraphrasing Luke 17:21, that the kingdom of pattern is within you.
These observations apply, with a modification, to auditory patterns. A time window around the current moment plays a role similar to the foveal image. In vision, an image is captured and its description assembled via saccades, whereas the auditory window proceeds continuously with time across the patterned sound. But in both cases patterns which are present in the world may or may not be detected by the viewer or listener, and undetected patterns are similarly impotent in the world. Patterns are within you either way.
Numerous examples of images built from mathematical patterns can be found by search, on this subreddit, and on my own site devoted to the topic.
The central arguments of this note are complete. The last matter I wish to discuss is on the technical side, so I include it in an appendix.
Appendix: Do all phenomena arising from visual patterns have mathematical structure?
So far, the examples have been simple. But what about images such as this:
At first glance, this looks like 3 copies of the same image. In other words, the initial phenomena for the three images are the same. But inspection of local detail will show that most rectangular blocks differ in shade from image to image. At the pixel level, the images are entirely different. I doubt that you noticed that there are 9 vertical bands in the central image, but 8 on the side images. (Because 8 is above your subtizing threshold).
The common phenomenon arising from the three images includes a common description. In earlier examples I was able to articulate the descriptions in English and discern their mathematical nature. Here, I cannot. Clearly the descriptions for visual phenomena are immensely complex. One of many aspects: it seems that a dictionary of elements which recur (such as the vertical bands) is assembled, and then the nature of their recurence is formulated. This is a recursive process. At each stage of the creation of a description, details are replaced by summaries.
Be that as it may, it is my sense that the underlying structure of a description is of the same general kind as for the simpler examples. If we ever know what’s going on (e.g. by discerning the nature of the neural activity involved, and assuming that supervenience of the mental on the neural is still taken seriously), then I expect we will find mathematical structure. But this is only an intuition. So the answer to the question “Do all phenomena arising from visual patterns have mathematical structure?” must remain a maybe. Another question: what other sort of structure might be involved? When the term “structure” is used, is mathematics ever far behind?