Mathematical Phenomenology

17 min readMar 21, 2021


Rafael Araujo

Phenomenology is the study of appearances. Let’s consider an example of the simplest kind: the situation of a person gazing attentively at some physical object, such as a coffee cup. There are two parts of the situation. One is the cup itself. Another is the seeing of the cup, the cup as it appears to the viewer. The former is a physical thing, and the latter a mental thing. (Perhaps the mental thing emerges from the physical thing that is a brain, but it is still of a different character and structure than the cup 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.

If the phenomenon arises from an object seen in the world, as in this example, then the phenomenon can be said to have spatial content: it includes an estimate of the object’s shape, a perceived hierarchical structure of parts, each with its own shape estimate and placement. The object is also placed in space relative to other objects, including the viewer’s own body. The space I refer to is phenomenal space — space as we experience it within our minds. Finally, if the object can be identified — say as a tree, cup, or shoe, then that identification is present within the phenomenon.

Is phenomenal space the same as “real” space as Euclid, Descartes, Newton, and Einstein have given us to understand it? Of course not! For one thing, the curvature discovered by Einstein does not come into the matter. But the differences go beyond that. All things in phenomenal space are finite in nature — they are approximations if that is the right word. No position has infinite precision, very far from it. In addition, perceived space is centered on the body of the perceiver, and contains only those things visible from that position and direction of gaze. Perceived space is only one kind of phenomenal space. There are also spatial memories, anticipations, and imaginings. But each of these is modeled on perceptual space, with a degradation in detail. Their structure is the same.

Phenomenal space is occupied by phenomenal objects, that is, by the phenomena that arise from perception of physical objects. Most of these are assigned categories (shoes, trees, cups, etc.). This partition of what is seen into individual phenomena with their categories is something imposed by the mind. A physicist’s space is occupied only by quantum fields.

Finally, phenomena contain patterns — regular repetitions, grids, textures, musical patterns, and so forth. As we will see shortly, those patterns cannot be identified with structure that occurs in the objects that give rise to the phenomena; the patterns live within the phenomena.

Despite this, phenomenal space retains some of the structure of outward space. It is still three dimensional — there is still up and down, side to side, and near and far. Rigid objects transform in the same way, via translations and rotations. Distance makes sense in both realms. Etcetera.

The important thing is to realize that phenomenal space is something real of its own kind, and distinct from Newtonian space. However, phenomenal space shares some mathematical structure with external space. The concepts that help us to understand the latter apply to the former.

Furthermore, We can investigate these properties without becoming entangled with qualia. Here is an explanation of the latter concept:

Phenomenal objects have color. Color is not a substance from physics, although there is a complex connection between color as experienced and the physics of light and its interaction with surfaces. Red, for example, is a name for a phenomenal property. It has its own specific nature, and is directly present to our minds when viewing a red object. It seems a primitive substance of our experience. It is ineffable in the sense that it could never be explained to a blind person. Communication about it among sighted people relies on identifying commonalities in experience. Colors are defined by pointing rather than describing. In the philosophy of mind red is called a “quale” (plural “qualia”). What is this mysterious substance, and how can it arise from neurons firing within the darkness of our skulls? This stands in for the general question: “how can experience arise from the physical world?”. This ancient problem has been known in recent times as the hard problem of consciousness.

My point is that we can investigate the structure of phenomena, such as spatial phenomena, without becoming entangled with qualia. We can start from this position: Yes, experience exists. Yes it possesses mathematical structure. Yes, we can investigate that structure. No, the problem of qualia is not thereby addressed. So, there is no claim that mathematical structure will ever give a complete account of phenomena — it seems evident from qualia that this is not possible. And perhaps there are other aspects than qualia which lie outside the grasp of mathematical thinking. Still, mathematical structure is present in phenomena, and can be investigated for its own sake. This kind of investigation is what I am calling mathematical phenomenology. The problem of phenomena is thereby partitioned into structure (mathematical) and substance (qualia), and the latter is left for another time.

Aside: there is also the project of investigating the phenomena involved in mathematical thought — that is, the phenomenology of mathematics. That is not the enterprise under consideration here. I am concerned, rather, with the use of mathematics in investigating phenomena of whatever kind.

As you’ll see later in this note, mathematical phenomenology, in combination with mathematical realism, can shed some light on the hard problem, even if qualia are not touched.

There is another aspect of mathematical structure of phenomena beyond 3D geometry: pattern. We are as adept at introducing this aspect into our phenomena as is the case for three dimensional geometry or categorization of objects. This pattern-imposition is present for perception of two-dimensional images, a simpler context within which to investigate the process than the natural 3D world. Consider the following image.

Image 1

This image 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 (eg ZF) set theory. That is, the phenomenon that the image induces in your mind has a mathematical structure, and that structure is built immediately upon attending to the image. If this were not the case, no such description would be possible.

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 1, I was able to detect its mathematical nature. To do this required some training in mathematics on my part, but the phenomenon in itself has this nature independent of the mathematical training of the individual in which it occurs. The phenomenon must have a 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.

Husserl calls the standpoint that we take in every day life when we are not philsophising “the natural attitude”. In this state of mind, it feels as if you see objects and images “as they are”; there is no thought of a phenomenon separate from what is seen. But, philsophising once more, we can see that the phenomenon and image have very different content, as the following example will show.

Images 2 and 3

Are the left and right images different? They are, but this requires study. The phenomena that arise immediately from the two images are identical. In fact there are 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”. Again this is close to a complete mathematical description. What is missing is the exact length and width of lines as a fraction of grid cell size, and the dimension of the grid by cell count. These details are present in the phenomenon as rough estimates, not exact quantities. I can say this because the phenomenon is present to me, so I can gauge its basic properties.

Another thing that I have left out: at any moment I am aware of the orientations of a few line segments at foveal position, but only if I attend to this, and only until I move my eyes (via a saccade ). The important thing to realize is that the phenomenon is a description of the image, not a copy. The nature of the the description, is, again, mathematical.

Despite the missing angles in the phenomenon, it does seem, in the natural attitude, that all of those lines are present to me, even if I do not know anything about them except the “randomly oriented” property. This shows that the feeling of presence, and actual presence in the phenomena, are two different things. Daniel Dennett calls the former “the representation of presence” and the latter, “the presence of representation”. To repeat: present in the phenomenon are 1) the description, and 2) a detailed capture of the part of the image which is projected on the fovea. The detail is overwritten on each shift of gaze. However, the description persists for longer, since it is held in short term visual memory. The representation of presence is the illusory feeling that images are present to us in full detail. See this paragraph for a concise formulation of Dennett’s views.

There is something else missing from the phenomenon: the patterns highlighted in red below:

Images 4 and 5

The images 4 and 5 are identical to images 2 and 3, except for the red highlighting . From this we see that patterns of a simple kind were present in the images 2 and 3, but not present in the initial phenomena. This makes the simple but important point that the patterns that we see reside within our minds. The presence of a pattern in the outer world, however simple, is no guarantee of its arrival in phenomena. It goes the other way too: patterns can appear in our phenomena that have no counterpart in the external world. This is apophenia. To paraphrase Luke 17:21, the kingdom of pattern is within you.

Neuroscience, Emergence, and Mathematical Realism

Now, I’ll switch gears entirely. Instead of taking the phenomenological view, in which phenomena are investigated from the inside as they appear to us, I will move to the exterior view — the view of the scientist who studies the brain as a physical system. But I will not forget the results of my phenomenological investigations. Even viewed from the outside, we need to make sense of why biological systems (we humans) report phenomena, and the forms that those reports take.

Consider again the various mathematical structures associated with images 1–5. One speculation is that these structures are encoded in the activity of my brain when I attend to the images. An equivalent formulation is that the structures are emergent entities of the brain’s activity.

What might be the nature of this emergence/encoding? It is premature to make any concrete claims. Only very simple kinds of encodings of mental content by neural activity are currently understood.

I don’t claim that the brain is like a computer in any way except that the ideas of encoding and emergence apply. But to explain what I mean by emergence, it is helpful to consider an analogy from computing. Consider a computer running CGI (computer generated imagery) software, and displaying a textured sphere. The activity of this system can be described at a number of levels:

Each of these levels is independent of those above and below it in that the laws that govern each level, and make its activity comprehensible to us, can be formulated without mentioning the other levels. The user of the CGI system need not know programming, the person who writes the source code needn’t know machine code, nor circuit design, nor the relevant physics, etc. These differing descriptions apply to the same object.

These levels of description exist objectively independent of any observer. One objective measure of emergence is Eric Hoel’s effective information (EI), which, roughly, characterizes the levels of a system by the degree to which they preserve information through time. If one system H is defined in terms of a lower level system L — as circuits(H) are defined in terms of physics(L) — the former may possess more effective information than the latter, even if the latter level incorporates much more detail. EI provides a measure of the degree to which a proposed level of description has its own coherence, independent of its basis in lower levels. The above example from computer graphics provides extreme examples. The middle levels are perfectly causal : the past determines the future via the laws of operation that pertain to the circuit, machine, and source code levels. The existence of these causal levels is an objective mathematical fact about the system, no observer needed. (By the way, high EI does not require determinism).

The brain might admit this sort of layered description. Indeed, it is already known to. Some neuroscientists study the brain at the level of biochemistry, others at the level of firing of individual neurons, others at the level of circuits involving many neurons. These levels are not as strictly separable as in the our computer graphics case, but a causal analysis at varying levels is still possible. Many such levels remain to be discovered, very likely.

To complete this investigation of phenomena from the outside, I require one more ingredient: mathematical realism.

Mathematical realism (also called mathematical platonism) is the view that mathematical structures exist independent of the mind. Other positions view mathematics and its objects as, in one sense or another, human creations or interpretations. Nominalists deny the existence of mathematical objects, and provide reformulations of mathematical statements in other terms. Intuitionists regard mathematical objects as mental constructions. Mathematical realism is not monolithic. For example, finitists take a realistic view of finite mathematical objects such as trillion digit numbers or the monster group, but not of completed infinities. Note that even the realism of finitists is wild in a way, in that it posits the existence of unbounded numbers of mathematical structures that are bigger and more complex than the physical universe.

Although I am a mathematical realist myself, I will not argue the case here, but discuss a consequence of realism : it provides a way of thinking about the substance of which phenomena consist.

A mathematical realist can visualize a sort of cloud of mathematics around physical things. This cloud consists of the mathematical objects that exhibit systematic correlations to the things in question. This supports the following way of talking about the mathematical structure of physical reality. One interprets “physical thing X exhibits mathematical structure Y”, as meaning “there exists the mathematical structure Y that exhibits such and such correlation to the physical object X”. This way of putting matters takes the realistic view towards the mathematical object: it is an existent in its own right which correlates to the physical. For example, for each elementary particle there is the corresponding Lie group. Actually, physicists almost always use the formulations of mathematical realism. This may seem like a distinction without a difference, but please read on.

Now consider a structure which exhibits levels of description, as in the computer graphics example above. The levels are mathematical structures clothing the physical system. From the mathematical realist’s point of view, these structures exist — and this is as true for the higher levels, as for the lower. Furthermore, they are, in the case of our CGI example, causal structures, whose behavior is determined through time by laws which operate at their own level.

The Hard Problem Reformulated

Now for the hypothesis: suppose that the brain admits multiple levels of description, and that the mathematical structures which we apprehend in phenomena actually exist in the mathematical realist’s sense, at some level. Might not these structures actually be the phenomena, at least in part? Such a hypothesis does not solve the hard problem of consciousness. It says nothing about qualia. But it is relevant to the problem, in that the question “why is it like something to be a biological system?” is replaced by “why is it like something to be the system of mathematical structures that emerge from brain/body activity?”. The latter question narrows the gap, in the sense that more similarity can be recognized between the entities on either side of the gap than is the case for the former question. In particular, the structures of the phenomena that we detect from the inside are already present in views from outside, with this reformulation of the hard problem. So the reformulation gets closer to an answer, perhaps (but only for us mathematical realists).

Phenomena Are Causal

If phenomena are partly constituted by mathematical structures, they are structures with causal force. Phenomena are accessible by a wide range of mental processes, such as attention, evaluation, memory, decision making, action selection and verbal report. Their properties can be queried by these processes. Phenomena resemble computing’s data structures in this respect — but I do not mean to imply any other similarity whatever.

Consciousness must exist at a single level, since its contents are unified, as described above. The global workspace model of consciousness, proposed by Bernard Baars in 1988, is relevant, because of its formlation of this unity. Stephen Dehaene and others have since proposed neuronal implementations of the global workspace architecture.

Do All Phenomena Have Mathematical Structure?

So far, the examples have been simple. But what about images such as these:

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 color 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 phenomena, again,are not copies of the images, but descriptions. In earlier examples I was able to articulate descriptions in English, and discern their mathematical nature. Here, I cannot do that. Clearly the systems of description 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 pattern of their recurence is formulated. This is a recursive process. At each stage of the creation of a description, details are replaced by summaries.

To repeat the question, do all phenomena have mathematical structure? There are two ways this might be so. First, it might be that mathematical structure is the only form that structure (as opposed to substance) takes. Are there known exceptions? Second, brains might rely on such structure for their role in existence — to preserve the life of their hosts. They might need to populate their higher level descriptions with mathematical structures (models) which allow them to minimize surprise, as Karl Friston would have it. This is an evolutionary account.

That said, it would be radically premature to conclude that all phenomena have a structure that can be captured mathematically. But some do!

Two Varieties of Causal Structures and Their Models

As remarked above, phenomena are causal structures in that we can act and report on them. Our examples have involved geometric and logical descriptions. The descriptions do not specify exact structures, but summarize them.

There is another kind of structure involved than descriptions. Phenomena of the spatial world (which arise from the auditory, kinesthetic, and tactile senses, in addition to vision) are embedded within a general theory of space and spatial action. This theory tells us that at all times we can shift our gaze and body to look at things more closely, and, more generally, navigate through a spatial context. Just as shapes and patterns are present to us via their descriptions, we are aware of the partial nature of our view of any object. We always know that there is more to the object than we can see (or touch) at any moment, and that by use of our body, if only by shifting our gaze, we can access a different part of the object.

The mode by which this theory operates is not yet understood, nor the form that it takes in the mind. However, it is clear that it is entangled with the mathematical theory of space, just as descriptions of individual objects are. I theorize that this aspect of mind exists within the same mathematical level as do descriptions of individual objects — the level of consciousness.

From the standpoint of mathematical realism, there are two aspects of the situation. The causal mathematical structures point to mathematical objects outside of the causal realm. For example, the description “a grid of short white line segments randomly oriented” is a description which applies to a set of mathematical objects: the infinite set of actual dispositions of line segments, with their exact orientations, to which the description applies. Similarly, the general theory of space refers (among other things) to what mathematicians think of when they think of space — an infinitely divisible continuum with three dimensions. In the jargon of mathematical logic these non-causal mathematical objects to which descriptions and theories refer are called their “models”. The causal structures are (presumably) finite, but their models are not.

Here is the point: in the natural attitude, a phenomenon feels present to us not as a description or theory, but as an object to which the description or theory refers . This a variety of Dennett’s “representation of presence”. Our awareness contains the apprehension that we are in the presence of objects in their endless detail, even if that detail never arrives in our minds.

That is, the felt presence of something amounts to our minds containing a description or theory of it. It feels as if the something is entirely there for us, though it never is. We feel that all those lines and their orientations are present to us in image 2, though these orientations are not to be found anywhere within our minds. Also we feel that space and in its unbounded divisibility is present to us, though what we possess in our minds is only a theory of space, not the structure itself in its infinite nature.

Let me close this note with a statement of the obvious: The structure of phenomena has been studied, under various rubrics, for more than a hundred years, by the phenomenologists themselves (such as Husserl), Gestalt psychologists, cognitive scientists , neuroscientists, and computer scientists. In many cases, the structures that have been illucidated by these studies is mathematical in nature.