Section 21 (first articulated 1.9.2021)

Fractals – Inverse Square

(See Event Particle)
(See Whitehead, “Cleopatra’s Needle”)

Pole (The External point-of-view)

Things can be viewed in two ways: first, from the external point of view, we see a set of objects, each moving in space, but each of those objects is taking on a real event from a series of possible events, i.e., if they are here, they are not there.¹ Even inanimate objects like a pole, which we pass by every day, we do not realize a difference, but every day something different is happening to that pole. In other words, the pole is a substrate going through a series of changes, or going through each possible event that can be associated with a pole.² (Cleopatra’s Needle)

Every moment the pole is going through a possible event, or rather, time itself is this series of possible events being realized one moment at a time.³ When we observe an object, we see it at one of those moments. As we commonly say, “this is not my best moment,” or “you caught me at a bad time,” these allude to the abstraction of catching something at one of its possible moments.⁴

We drive by the pole every day and do not realize any change, even though it is changing every day, until one day we recognize an evident change in the pole—we see rust, or it is broken in half. Here we are explicitly seeing one of its potential events as a real moment, but there was a buildup to this that we did not notice. That potentiality of one day breaking became a real event; it is “only a matter of time,” we say, before something like that happens.⁵

Each single object being externally perceived has the series of potential events integrated into itself at the present moment, which is part of a series of any one of its possible moments.⁶

When an object is in motion, you see it changing through these events, actualizing them at a rapid rate, but the resolution is so good that the same thing appears to be moving as itself, and change seems to be happening to it. But the thing in question undergoing the change has only been identified as one thing and not others. The thing you see as itself is a moment, part of an actualizing series.⁷

The same asteroid we see in space before it crumbles would not be identified as the same object after it enters the atmosphere and crumbles into smaller bits of rock. The asteroid that we identify as the same substrate that endures the change is part of many variables, meaning that it becomes many variables.⁸

We give something an identity based on its dominating mass. The thing with the bigger mass we identify as the object in question because, for one thing, it catches our attention. However, the mind, upon reflection, goes further and finds out the qualities of what is being perceived, and that is what identifies them as subjects of study.⁹

The Internal point-of-view

Second, from the internal point of view, is the perception of different things at different moments so as to have different experiences. When I turn my head one way, I see one thing and am blinded to another. When I sleep one day and wake up, I have a different experience than yesterday. When I look at a bunch of different details within the same reference frame, I am scanning each part of every object and being ignorant of the others.¹⁰

These changes in instances are the superficialities of what actually determines a possible event into the present moment. When the observer is witnessing their environment and therefore having thoughts about them—an animal, for instance, would feel or think a certain way and would automatically act on that, which therefore determines an event. When one cat sees another and becomes angry, it lashes out and attacks the other cat, and the once possible event of a cat fight is now a real situation.¹¹

Ordinary thinking views this as a random act, that is, there is no event before it happened. A cat fight is not an event that first exists; it is just a result of a couple of angry cats coming together to make it happen. However, we all know of a cat fight even if it is not currently happening in the present. Does a cat fight exist because at one point it has existed, or because there is a possibility of a fight to happen when two cats meet that makes us have the idea of a cat fight? The latter does not require a real event of a cat fight to be perceived in order for it to qualify as existing. The mere possibility of existing is enough for it to be part of existence.¹²

The question becomes: possibility, in terms of potentiality of a thing to happen at the present, is an objective criterion for a thing to exist. In reality, the potentiality of an event may only occupy one moment at a time—you cannot have two moments simultaneously occurring. However, it is exactly the latter that is presupposed by the definition of potentiality: that two or more events are simultaneously occurring. The only difference is, not any one of them occupies more reality than the others; they are all equally possible, so they all exist instantaneously.¹³

Space requires that two things happen within it at the same time. It is the limitation of space that discloses it, which limits the number of events that can occur at the same time.¹⁴

Making of an Event

An event is simply the result of a free body acting in a certain way. A man is a self-contained entity that simply got up and went to the washroom, and the washroom is its own unit built and readily available for use, and it is the coming together of these two objects that made the event.¹⁵

This is, however, not the way an event is conceived in the mind, because when we think of going to the washroom, we imagine the moment that characterizes the event before any of the pieces come together to form it. We imagine the man using the washroom before we imagine the washroom on its own and the man on his own as separate entities.¹⁶ The reactions—being the general notion of the event in the mind—are prior to the individual aspects or parts. The objects actually come together, and through their interaction, a production of motion occurs to form an event. The event is both the interaction of these individual aspects that come together and result in a scenario, and the construction of those relations into a general context—wherein the relations are more fundamental than the parts that compose them. The idea of the parts that compose it, that is, their positioning, is simultaneous with the parts themselves—the objects that take on those positions and their corresponding actions.

Moreover, there are no real entities that exist on their own without other entities. Every object conceived in the mind belongs to a context. But for sensation, you can derive a feeling from touching something independent from its context. The sense organ filters out the context and derives the feeling, even though being warm is part of a hot context that is part of iron, or whatever.¹⁷ The specialized functions of evolved sensations remove the necessary realization of context by presenting to the observer only a set of individuated objects that appear to interact with one another. The onlooker perceives these interactions without awareness that they are all part of the same fundamental event in time. Time, in the abstract sense, is the primary condition—meaning that things must be, or rather, must be present.

However, the question of how they are present, or which is present first, loses its meaning when confronted with the fact that all things are, at first, present simultaneously—at least in the mind. In thought, things are always first and foremost present all at once, instantaneously, and simultaneously. This indicates that things are also present externally, beyond the senses.

For the senses, all things are initially given at once; only afterward are they selected and dissected into separate parts. Yet once they are differentiated, sensation fails to recognize the original condition—that everything was first presented together as a unified whole before being divided. The understanding that operates through sensation assumes that things already come as distinct, serial parts and that the order among them is naturally given. In truth, this order is imposed by the very understanding that fails to realize this fact about itself.

Continuity

The external-point-of-view assumes that an object acts in a certain way—said to be its duration—that determines it to act this way. In other words, the object is simply actualizing what is set out by its duration. The internal point of view is a conception of different instances. When I look at one thing, there is a definite picture. When I turn my head, there is a motion of blurriness and uncertainty until I land on another certain picture. These determinations of what is conceived determine the path that the sequence will take to determine the particle that goes through each event. The particle is being determined by its duration, but in that duration, it is determining the direction in which the duration will determine it.¹⁸

When we see an instance, it is layered with an infinity of possible events that all mold and morph into each other, and their morphing into each other is what we perceive as the continuity of a thing in motion—that a car goes into the distance and becomes part of the horizon. But this continuity of a thing is just a sequence being played out at only a moment at a time it is being perceived. The only continuity we really perceive is of different instances concurring through each other. After watching the car go into the distance, you see a bird fly close to your head, and now the attention is there.¹⁹

Here we are trying to explain that the reference frame which observes a thing in motion, and the thing in motion, are the same continuum.²⁰

Footnotes

1. Cf. Aristotle, Physics IV.11, on potential motion and spatial occupation.
2. Alfred N. Whitehead, Process and Reality, Part II: “Actual Occasions” and persistence through change.
3. Henri Bergson, Time and Free Will, on time as the realization of potential moments.
4. Husserl, Lectures on Internal Time-Consciousness, §8.
5. Aristotle, Metaphysics Θ: potentiality (dunamis) and actuality (energeia).
6. Whitehead’s notion of “prehension” explains this self-contained temporal integration.
7. Bergson, Creative Evolution, Ch. II: critique of static perception.
8. Heraclitus, Fragment 12 (on flux and identity).
9. Kant, Critique of Pure Reason, A99–A110, on recognition through synthesis.
10. Merleau-Ponty, Phenomenology of Perception, on selective attention and perspective.
11. Spinoza, Ethics III, Proposition 2: on affect and bodily determination.
12. Modal realism: cf. David Lewis, On the Plurality of Worlds (1986).
13. Leibniz, Monadology, §31–33, on simultaneous potentialities.
14. Kant, Critique of Pure Reason, Transcendental Aesthetic.
15. Aristotle, Physics II: cause and effect as conjoined activity.
16. Hume, Treatise of Human Nature, Book I, on imagination and causation.
17. Locke, Essay Concerning Human Understanding, Book II: secondary qualities.
18. Bergson, Matter and Memory, Ch. I, on perception as temporal motion.
19. Whitehead, Science and the Modern World, on continuity as serial process.
20. Einstein & Minkowski: spacetime continuum — see Minkowski’s “Space and Time” (1908).

Ideas Pass Through Space as Events

Ideas pass through space, and this is the sequence of events that forms a temporal duration of time. We experience time as objects, in the sense that an object is processed by sense perception. However, this does not fully explain how objects exist in time in the first place. It is presupposed that objects are given to the senses. In order for an object to be received as an object by the senses, it must first exist as an object in time or have a temporal duration. As we say, “everything has its time,” meaning that everything decays or changes over time.

This idea is often extrapolated onto the object itself: a duration is ascribed to the object, and anything given a certain temporal length will eventually come to an end. Yet, we do not usually consider that the experience of a thing is itself a duration that can end. The duration of a thing ending and the experience of that ending are related: the former is an internal process, while the latter is external.

Unlike objects, ideas are not passively received; they are determined as the time during which objects will be received in space by the senses.

The object stands as a disclosed space within another disclosed space. This spatial extension is where physical substances can pass through the object as a medium between exterior and interior dimensions. Space, therefore, is the most ethereal substance, through which any identifiable substance—whether dense and heavy, or light and insubstantial—can travel.

If we consider thoughts as events, they are identifiable as objects in the sense of being occasions. Even if their physical components approach zero—having no mass, weight, or density—they are still as abstract as space itself. Space, being an abstract idea with the efficient function of disclosing and being disclosed by identifiable things, acts as the medium through which objects or events pass. In other words, any identity distinguished from another can pass through it in space.

Species of Time – Time Packets

A species of time is a packet of events following one another within a duration. This succession is not necessarily ordered toward a final end, although events often move toward some conclusion. What defines a duration is not the end of events but the necessity that one event follows another in rapid, nearly instantaneous succession. It is the sequence itself that constitutes duration.

A ‘species of time’ consists of a series of variations concerning the possibilities of what can occur within an event. They are called possibilities not because they lack content, but because each is only real at an instant—only one at a time. When one possibility is not realized in the present, others may exist from an ultimate, abstract viewpoint.

Every event in time is a packet of possibilities concerning the orientation of that event. These orientations serve as units of measure: each shift in conception is a unit that allows the event to move, either in duration and continuity or in locomotion from one position to another.

A species of time implies that an object identified as a single point in space is actually only one of its possible moments, which belongs to a larger set of possibilities. The object is identified as a single entity through continuity, but in reality, it is a species of moments: the object is both the common attribute shared by its moments and the common name used to denote its duration of activity.

Ordinarily, an object in time is seen as a distinct entity, separate from others, existing in a spatial void, and subject to gradual changes in its physical composition. We observe the passage of time in processes of “wear and decay.” From this observation, we abstract that time is an endless span in which objects flow continuously, like space, and that objects are acted upon by time, slowly changing and eventually corroding. This conceptualization arises from defining the universe primarily as a spatial domain. For instance, we commonly refer to the universe as “space,” saying “the rocket is going out into space” to denote departure from Earth.

Observational Limits and the Object

However, this spatial-temporal observation does not fully correspond with the way perceptual faculties derive objects of knowledge, nor how objects present themselves to these faculties. The faculties perceive and identify objects discontinuously: we can only see, hear, or smell a limited number of aspects at once, while the object presents certain aspects over others.[^1]

Even if our sensory capacities were perfect, we cannot perceive the entirety of an object at once. For example, when looking at the horizon with perfect vision, the curvature of the Earth eventually obstructs the view. Likewise, from the Moon, only one hemisphere of the Earth can be observed at a time. This limitation is inherent in geometry itself: a geometric figure is distinguishable from others because it exhibits particular sides, angles, and measurements. Geometry asks not why a figure is different from another, but what differences each figure exhibits. In inverse geometry, the focus shifts to how one figure transforms into another.

Footnotes

[^1]: Alfred North Whitehead, Process and Reality, provides a framework for understanding objects as occasions of experience and emphasizes the partial presentation of objects to perceptual faculties.

[^2]: Carl Jung, Psychological Types and Man and His Symbols, discusses the limits of perception and the incomplete apprehension of sensory input.

Here’s your “Squared Space” text with footnotes integrated and clearly marked throughout. I’ve added references and explanatory notes that link to mathematical, philosophical, and fractal concepts.

Squared Space

In mathematics, “to square” denotes the operation in which a number is multiplied by itself[^1]. In other words, a square is the result of multiplying a number by a quantity equal to itself. For instance, the square of 3 may be written as (3^2), which equals 9. Conceptually, this can also be expressed as (3 + 3 + 3 = 9), meaning 3 added to itself three times.

Squaring is equivalent to raising a number to the power of 2. The number 2 is used to symbolize the power of being squared because it is the first number that produces a result distinct from the number itself[^2]. For example:

[
2^1 = 2
]
[
2^2 = 4
]

By contrast, 1 raised to any power simply returns the original number, e.g., (1^9 = 1). Similarly, any number raised to the power of 1 yields itself.

Exponentiation is denoted by placing a smaller number above and to the right of a larger number[^3]. This notation has a geometric interpretation: from the perspective of a square, there is always space in its angle to accommodate another square. In this sense, a triangle can be seen as an abstraction derived from a square[^4].

The common visual illustration of squared numbers, with cubes placed side by side, is itself an abstraction. More accurately, squaring is closely related to the concept of fractions in mathematics. A fraction represents a part of a whole, and the operation of squaring reflects the fractal property of fractions[^5].

Fractals are patterns that are self-similar across different scales, created by repeating a simple process in a feedback loop[^6]. Fractals are a subset of Euclidean space for which the fractal dimension exceeds the topological dimension[^7]. This means that the space a figure occupies surpasses the form of the figure itself. The defining property of a fractal is that the object appears the same across multiple levels of observation. While fractals can extend infinitely, they maintain a finite form throughout this extension, contrary to the common belief that infinite objects lose their characteristic form.

The notion of a squared number, where the exponent sits atop the base, can be interpreted intuitively: it represents the multiplication of an event by the number of its possibilities in the mind[^8]. In this sense, the abstract element—the mind—can be considered the “square root” of a physical event. The present moment is, in a sense, square-rooted by its possible futures. The mind functions as a simulation of possible events. These possible events form a fractal-like structure, which is “squared” by the present: the present is defined in relation to its possible futures[^9].

In time, possible events form a fractal-like sequence that is the square root of the present. This indicates that the present moment contains a spatial extension that is not yet determined as a definite or singular event.

This fractal process occurs in the mind, or rather the mind is the superficial locus where the abstract properties of concrete events unfold[^10]. This supplements the earlier idea of a “time tube.” We have suggested that direction in space is quantitative rather than qualitative: when change occurs in space, there is no absolute change in the quality of an object. Rather, the relationships between entities, such as the position of a point on a plane, shift while the entities themselves remain uniform[^11].

By contrast, the direction of time is qualitative: it represents the transformation of one identity into another. Time is the movement from the abstract to the concrete—from the infinite possibilities of an event to its actualized, particular state[^12]. Temporal movement is not a change in location but a manifestation of differences across locations: in one location, one event occurs; in another, a different event occurs.

Every conception of a distinct entity carries with it an infinite series of possible moments, which together form its duration[^13].

Footnotes

[^1]: Euclid, Elements, Book II, defines geometric squaring as constructing a square on a given line segment.

[^2]: Raising a number to a power greater than 1 produces values distinct from the base; this is the foundational principle of exponentiation.

[^3]: Exponent notation, as in (x^n), is attributed to René Descartes (La Géométrie, 1637).

[^4]: Triangles and other polygons can be geometrically derived from the square; see Coxeter, Introduction to Geometry (1969).

[^5]: A fraction represents a part of a whole, linking squaring to fractal self-similarity.

[^6]: Mandelbrot, B. B., The Fractal Geometry of Nature (1982) introduces fractals and recursive self-similarity.

[^7]: Fractal dimension exceeding topological dimension implies the figure occupies more “space” than its traditional Euclidean dimension.

[^8]: Whitehead, A. N., Process and Reality (1929) describes events as occasions of experience and relates abstract potentiality to actualization.

[^9]: The mind as a simulator of possibilities is also discussed in Bergson, Time and Free Will (1889).

[^10]: The locus of abstract properties in the mind can be seen as the stage where potential events actualize.

[^11]: Direction in space as quantitative aligns with mathematical kinematics, where displacement is measured without altering the identity of the object.

[^12]: Direction in time as qualitative follows Bergson’s philosophy: temporal passage involves the change from potential to actual states.

[^13]: The infinite series of possible moments forming duration parallels the concept of “species of time” as in Whitehead’s process philosophy.

Inverse Square Law

Gravitational Force and Squaring

Gravitational force is a form of energy derived from motion, where the total measure of mass is inversely squared[^1]. The inverse square law describes the outcome of a process as the realization of a predetermined generality into a particular determination. Similarly, the time taken for a journey is inversely proportional to the speed of travel.

A square is a number multiplied by itself; in squaring, the self is multiplied by itself as an additional quantity[^2]. The measure of a singularity can only be indirectly deduced by measuring the mass of the object.

In physics, the inverse-square law states that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that quantity[^3].

The inverse-square law describes how the strength of certain physical effects—like gravity, light, sound, or electric force—changes as you move away from the source.

Key idea: The intensity of the effect decreases rapidly with distance because it spreads out over a larger area.

1. “Inversely proportional” means that as one thing increases, the other decreases.
   • If distance doubles, the effect becomes smaller.
   • If distance triples, the effect becomes even smaller.
2. “Square of the distance” means that the decrease happens according to the distance multiplied by itself.
   • Example: If the distance from a light source doubles (2×), the intensity of the light becomes 1/22=1/41/22=1/4 as strong.
   • If the distance triples (3×), the intensity becomes 1/32=1/91/32=1/9 as strong.
3. Why the square? Imagine the effect spreading out evenly in all directions.
   • In 3D space, the effect spreads over the surface of a sphere.
   • The surface area of a sphere is 4πr24πr2.
   • As the radius rr grows, the same total effect is spread over a bigger area, so the intensity decreases like 1/r21/r2.

Example:

• Gravity: The gravitational pull between two objects decreases as the square of the distance between them increases.
• Light: A bulb appears dimmer the farther you move from it because the same light energy spreads over a larger area.

Locus

The term locus generally refers to the way an activity forms a pattern that specifies the peculiarities of its operations without being limited to them[^4]. Aristotle noted in On the Heavens that the infinite is limited by the finite, as the infinite is always defined as the limit of the finite[^5].

Mathematically, a locus is defined as a curve or figure formed by all points satisfying a particular equation relating coordinates, or by a point, line, or surface moving according to defined conditions[^6]. One must ask: what are “defined conditions”? Are the conditions of a figure determined before the figure itself? Curves and figures consist of distinct points, but each point has the potential to belong to other figures as well. In this sense, the figure itself is a distinct point within the spectrum of all possible relations.

For example, in a circle, the center point is not central due to a fixed location but because the sphere assumes an equilateral magnitude, allowing any point to serve as the center. The radius demonstrates equilateral form, while the distinct points of the figure represent coordinates of all potential relations.

This is analogous to a flock of birds flying in formation: each bird’s potential positions are known in relation to the others. The coordinates of a figure are the distinct points of its form, capable of transformation while maintaining coherent motion.

Motion here aligns with Aristotle’s concept of becoming, not merely locomotion. Motion generates relations among parts, forming a distinct whole. Birds in a triangular formation, for example, move differently depending on their position, yet their relations create a stable geometric figure.

Subatomic Relations

At the subatomic level, the components of an atom are understood through the forces between them. A neutral atom contains a positively charged proton and a negatively charged electron. These components are bound to the nucleus via the Coulomb force, which determines the charges of the particles.

It is insufficient to assume that attraction and repulsion derive from pre-existing charges. Rather, the forces themselves constitute the distinction of charges. Equal charges produce repulsion due to self-identity, while opposite charges produce attraction[^7].

• If the charge is equal, the electrostatic force is repulsive.
• If the charge is opposite, the force is attractive.

The principle of identity explains repulsion: a self-identical entity excludes its other, producing inverse determination, while the opposite identity produces attraction.

Coulomb’s Law

Coulomb’s inverse-square law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them[^8]. The force acts along the straight line joining the charges.

Magnetic fields similarly operate as systems of repulsion and attraction, creating probabilistic interactions. Earth’s magnetic field, for instance, protects the planet by deflecting solar wind particles. Magnetism represents the fundamental capacity for “choice” or selection among possibilities[^9].

Matrix and Geometry

The concept of a matrix is relevant to the inverse-square law. In general terms, a matrix is “the material in which something develops; a surrounding medium or structure”[^10]. Mathematically, it is a rectangular array manipulated as a single entity.

In geometry, the 90-degree angles of a square define its dimension. A square is the relation between its angles, where the form of the angles determines the shape[^11]. Triangular angles, likewise, are triangular in nature; the angle itself embodies the shape of its relation.

Uncertainty and Form

The inverse-square law illustrates how each dimension contains other dimensions. A dimension is partly defined by how its quantity conforms to its quality. A matrix represents the structure of a dimension and the qualities describing its form.

The Flux Theorem states that externality is contained internally, as ideas contain matter. Our ideas define the dimensions we inhabit. The inverse-square relation, whereby self-identity interacts with opposite forces, models the dynamic between being and non-being, quantity and quality, intensity and generality[^12].

Evolution and Motion

Evolutionary divergence occurs when an individual appropriates the idea of the species and determines it in a particular manner, forming new lines of descent. Motion, in this sense, is defined by relational significance: the particular emphasizes one side of a relation. The locus is the condition revealing the structure of form through logical relations[^13].

Whitehead’s concept of the abstractive set applies here: a coordinate represents the potential extensive and temporal locus of form. Distinct points moving in perfect relation form a single figure; the figure exists as an abstractive set within each point of the relation[^14].

Footnotes

[^1]: Newton, I. Philosophiæ Naturalis Principia Mathematica (1687), Book I, Law of Universal Gravitation.

[^2]: Euclid, Elements, Book II, defines geometric squaring.

[^3]: Halliday, D., Resnick, R., Walker, J., Fundamentals of Physics, 10th Edition (2013).

[^4]: Aristotle, Metaphysics, Book IX.

[^5]: Aristotle, On the Heavens, 2.5.

[^6]: Coxeter, H. S. M., Introduction to Geometry (1969).

[^7]: Jackson, J. D., Classical Electrodynamics, 3rd Edition (1999).

[^8]: Coulomb, C. A., Théorie des Machines Simples, 1785.

[^9]: Bergson, H., Time and Free Will (1889).

[^10]: Oxford English Dictionary, Matrix definition.

[^11]: Euclid, Elements, Book I.

[^12]: Heisenberg, W., Physics and Philosophy (1958).

[^13]: Whitehead, A. N., Process and Reality (1929).

[^14]: Whitehead, A. N., Process and Reality (1929), on abstractive sets.

Consciousness as Degree

Breathing regulates consciousness.

Where does one body begin and another end? This question can be explained by considering how consciousness is the continuity of relations. On the one hand, the physical composition of one body distinguishes it from another. For example, the digestive system metabolizes vitamins by matching the composition of the nutrient with that of the organ. That organs require a certain composition akin to their own composition should not be a surprise. For example, vitamin C crystal takes on the appearance of a feathered bronze shield.¹

The molecular structures of these nutrients are taken by organs similar in molecular composition. The organism is a sublated form of the environment, just as a virus is a sublated form of the organism.² Viruses and bacteria can be seen as nature’s experimentation to advance cellular evolution. For example, vitamin D is appropriated by bones. Size, density, and other factors also play a role in distinguishing similar bodies. At the fundamental level, the distinction between bodies as separate individual entities becomes ambiguous.

Components Classified by Their Species

All individual components are scientifically classified by their species. The species informs the nature of its individual components by assigning common attributes as a common name. This fact is even more idiomatic at the atomic level. At the macroscopic level, perception evolves to pick out the individual from the species, and reason then derives the species in the individual. At the microscopic level, however, the discrepancy between individual components and the species is an elusive distinction.

An individual atom is an abstraction from a species of atoms. Hydrogen consists of billions of hydrogen atoms. The category of a hydrogen atom is meant to classify the quality of hydrogen generally. In our empirical observation, perception picks out the species of chimpanzees as individual monkeys. Yet in thought, there is no such thing as an individual chimp without the idea that encompasses all chimpanzees.³

The subatomic level seems to exhibit a nature synonymous with abstract thought rather than sensation. At the quantum level, the harmony between individual atoms is so perfect that there is no distinction between one and many; they appear as the very as-ethical composition of the element. The deeper we go into the fundamental composition of matter, the less there is a distinction between the individual component and the species. Whether one individual cell is said to operate separately from the whole of cells comprising the organ is only a question of maintaining the quality the species of cells promote. In our perception, we can gain an approximate distinction between an individual zebra and the species of zebras. But in less developed life forms, if we separate an individual ant from its colony, its lifespan is short.⁴

Consciousness as the Locus of a Spectrum

Consciousness is the locus of a spectrum. The term “spectrum” is the concept used to classify an activity in terms of its position on a scale between two extreme or opposite points. The proposition of a logical principle brings with it the inverse presupposition: nothing presupposes being. Their inverse relation forms a positive contradiction, which, having as its extremes the two inverse principles, constitutes the spectrum where all possible relations between being and nothing possess a position relative to the extreme points.

The middle point between being and nothing, which constitutes the range of the spectrum, is their synthesis: becoming. The principle of becoming is the transition of being into nothing and nothing into being, both of which are inverse relations with opposite positions relative to each other. This is why being comes out of nothing; it is the same as to say nothing comes out of being. Their becoming is the activity where they are the same moment.

Range

The concept of “range” is defined as “the distance within which something can be reached or perceived.” Becoming, as the range of nothing and being, is evaluative. In other words, becoming is the activity that sets an aim and determines the validity of the process towards actualizing that aim. For example, electromagnetic radiation is the arrangement of components as a progressive series according to wavelength. Color is exactly the bare quantitative notion of a qualitative distribution. It does not state why the distribution is, but that there is a distribution of differentiation.⁵

• A band of colors, as seen in a rainbow, produced by separation of the components of light by their different degrees of refraction according to wavelength.
• The entire range of wavelengths of electromagnetic radiation.
• An image or distribution of components of any kind arranged in a progressive series according to wavelength.
• An image or distribution of components of sound, particles, etc., arranged according to characteristics such as frequency, charge, and energy.
• The area or extent covered by or included in something.

Degree

The term degree refers to the amount, level, or extent to which something happens or is present.

• A unit of measurement of angles, one three-hundred-and-sixtieth of the circumference of a circle.
• There are degrees of physicality, degrees of being present, like contrast of color.
• A stage in a scale or series, in particular.
• A unit in any of various scales of temperature, intensity, or hardness.
• In mathematics, the class into which an equation falls according to the highest power of unknowns or variables present.

When sensory-deprived individuals lose distinctions, they may lose the boundary between their consciousness and that of another being (e.g., Alan Watts’ example of removing a mask and giving it to a fish underwater). The angles of degrees characterize the range of physical composition of consciousness.

Psychological and Evolutionary Implications

When we ask what has happened for eternity, the answer requires looking into the objective psychological condition of the world, given that substance is rational. The foundational psychoanalytical method for treating phobias is not to run away or ignore the affect. Instead, treatment requires directly confronting the object where the phobia is present (Peterson).

This principle extends beyond the human mind, as human complexes are based on internal self-conflicts that, when unresolved, develop into pathological neurosis. Human nature shares with nature generally the fact of self-contradiction. The process of evolution has produced organisms capable of navigating complex contradictions with proportional strength. The process of evolution, in this sense, produces the ideal specimen capable of dispensing with the complexity of the world.

Quantum Reality and Sensory Limitation

According to string theory, all things are fundamentally light waves. The motion of anything extends it into the light spectrum, which constitutes the physical composition of the object at the quantum level. Our senses, however, do not perceive the world in this way. If they did, all objects would appear as jumbled waves of light; the world would be one undifferentiated slab. Pre-Socratic philosophers argued that without void, the world would indeed be one single object. Sensory limitation is efficient: it allows perception to parse reality into discrete objects rather than continuous waves.

For example, a deaf person does not experience the absence of sound; instead, they experience uninterrupted sound, lacking the void necessary to distinguish one sound from another (Alan Watts). A magnetic field is essentially the probabilistic relation of attraction and repulsion, not being either, but the void whereby both contradict each other, forming a stalemate force.⁶

Position and Vector

An object’s position is its location relative to a reference point or origin of a coordinate system. Position is distinguished from a vector in that it does not necessarily point toward a direction, yet it is a vector quantity because it denotes the location of a direction. A vector is the direction of a location toward a definite point, but the location is also, at the same time, that toward which action is directed. Direction is not merely position insofar as having location, but position is the direction of some shifting point that itself is the location.⁷

Footnotes

[^1]: Crystallography and molecular geometry; see Hawkins, Structural Chemistry, 2018.

[^2]: Viruses and evolution: see Zimmer & Emlen, Evolution: Making Sense of Life, 2021.

[^3]: Philosophy of species and individuality; see *Aristotle, *Categories* and Physics.

[^4]: Ant colony studies: See Hölldobler & Wilson, The Ants, 1990.

[^5]: Spectrum, becoming, and logic: see Whitehead, Process and Reality, 1929.

[^6]: Electromagnetic spectrum, color distribution: see Griffiths, Introduction to Electrodynamics, 2017.

[^7]: Degree and measurement: see Stewart, Calculus, 2015.

[^8]: Alan Watts, The Way of Zen, 1957.

[^9]: Peterson, 12 Rules for Life, 2018.

[^10]: String theory and perception limits: see Greene, The Elegant Universe, 1999.

[^11]: Magnetic fields and probabilistic interpretation: see Feynman, The Feynman Lectures on Physics, Vol II, 1964.

[^12]: Whitehead, Process and Reality, 1929; see discussion on position and vector.

Time Tube

The idea of a time tube describes how mind ascends from the microscopic to the macroscopic. The distinction between micro and macro in the realm of time is not merely defined by size but rather by where the present conception of a moment lies on a temporal sequence. All potential events form a microscopic scale that reaches back into the mind from a concrete moment, and likewise, they extend outward from the concrete moment into a more abstract state of possibilities. These directions in time form a spherical loop.[^1]

At the end of each tube lies a particular event. How consciousness proceeds in time is through each layer—each layer being an event. What we observe as the present is simply the end of a time tube.

In an absolute framework of time, where one moment does not pass away while another takes its place, there is no distinction between the conception of one moment as opposed to another. Events can be outlined according to two theoretical models of spacetime:

1. The Continuous Model (Spiral Time)

In the first model, we can think of an object as it undergoes the experience of events—as a position in space that changes over time. Like an unchanging spiral shape in spacetime, the object’s movement can be seen as a line that curls through the temporal dimension.

We do not directly observe an orbit in space; rather, we infer it from an object occupying a pattern of positions. When we notice, every day, that the sun rises, we infer from that regularity that the Earth is rotating with respect to the sun. Likewise, observing the moon’s rise and set allows us to infer its orbit around the Earth.

We only perceive an abstraction of the orbit. The orbit, however, is at every point filled by every moment of the moon’s presence. The moon, as an object in a particle state, occupies every moment of its orbit simultaneously in the spacetime continuum.[^2]

The changes of an object through time are witnessed as a series of differences exhibited by the same object. For example, we notice one change at one moment and a different change at another, but we do not observe the actual sequence that maintains the continuity from one change to the next. The object maintains a certain stability as it passes through these changes.

This stability we identify as a particle state—the form that makes the object a self-contained free body in motion. However, the direction by which this object moves involves an identity of length, such as moving straight. So long as the particle state maintains this length of duration, it is preserved at every point along it.

We tend to think that “moving straight” is an inference derived from observing the object moving in a random direction—that it moves, and we infer straightness. However, to move straight, there must first exist a passage in nature—or the concept—that allows us to conceive the object moving in a straight line.[^3]

Whether it moves forward, backward, sideways, inward, or outward, these maintain the identity of that duration. The duration is the abstraction of capacity—it captures all the discernible differences that a particle state can take. This capacity then becomes the direction or the movement along which the particle state proceeds.

The quality for the length of duration to maintain an identity and not be broken into discrete pieces is the abstraction of a conception maintaining itself as it goes through, recognizes differences of experience, and records each conception of a change as a position of possibility for the event—or a possible outcome.[^4]

2. The Dimensional Model (Multidimensional Space)

Secondly, the differences making up the possibilities of events must occupy space for their occurrence. We observe an object at one point in space at a given time, and when it changes, it occupies a different point. It is always at one point, but never at every point.

In this model, every single change in the object’s position constitutes a whole new dimension—or is characterized by a distinct dimension in space. Ordinarily, we perceive the world as a single spatial dimension containing objects that are three-dimensional (if not higher-dimensional).

However, the world does not appear as a series of distinct dimensions layered simultaneously upon each other—each being a slightly different reiteration of the same thing. The world is not conceived statically, though things appear static and in motion; rather, the organism conceives its environment dynamically.

On some level, this means that as the organism moves, turns, and changes, the environment also moves, displaces, and transforms. We tend to think this occurs because of a collection of free bodies moving independently within the same space. Yet this so-called same space exhibits a series of different configurations.

If these differences were taken as distinct moments and overlaid upon each other, they would reveal a superposition—an overlapping of forms—that could be regarded as the very condition for perceiving new objects. In this sense, each temporal layer adds to the structure of perception, forming the tube of time through which consciousness advances.[^5]

Footnotes

[^1]: Compare with Minkowski’s concept of worldline in spacetime, where time is not linear but continuous and multidimensional.

[^2]: Einstein, A. (1916). Relativity: The Special and General Theory. The notion of simultaneity depends on the observer’s frame of reference.

[^3]: Aristotle, Physics, Book IV—discussion of continuity, motion, and the concept of place.

[^4]: Whitehead, A. N. (1929). Process and Reality. The “concrescence” of events corresponds to the continuity of duration described here.

[^5]: Heisenberg’s Uncertainty Principle suggests that the potential configurations of a particle exist simultaneously until measured; this is akin to overlapping dimensions of potential events within a “time tube.”

Reality as the Mechanism of Revelation

The wavelength is every single possibility of a particle state.[^1]

Reality is the mechanism of revealing truth—this is why it is called reality. What we observe as “reality” is not reality itself, but an abstraction of it. What truly is at any given moment is an instance of reality.

The meaning of instance in this sense does not merely denote a fast occurrence—as in the saying “it only takes an instant”—but rather a single or particular case, a moment that is still and static in conception. An instance is the fixed conception of an event. It is identical with the conception of a single set of things. An instance measures how long a particular configuration of things is held in thought before the conception changes to another set.

In other words, it is not the objects conceived that are immobile, but the conception itself which remains momentarily still, regardless of any external motion of what is conceived.

Reality, therefore, is not the sum of these instances, but rather the mechanism by which the actuality of these instances is revealed.[^2]

The Truth of an Event

The true meaning of an event is not merely what it signifies, but rather what potential arises from it.

Potential in this context is not necessarily good or ideal, though the word often carries such connotations. It is ideal only in the sense that it has not yet occurred—it is not yet abstracted or limited to a single conception. Potential is the possibility for an instance to be more than itself.

However, the function of potential is essential to the mechanism of reality, for it is through potential that the truth of an event is drawn out and revealed.

The Bookstore and the Liquor Store

To illustrate what is meant by the actuality of an event—the truth that reality mechanizes to bring out—consider the following example:

Imagine a bookstore that has existed in the same plaza for over twenty years. Inside, there is a large selection of books for purchase and a small café where people can sit and read while enjoying coffee. The café is typically busy, filled with readers and conversation.

One day, quite suddenly, the entire bookstore closes. In its place opens a large liquor store, equally as spacious and diverse in its selection—except, now, the product is alcohol rather than literature.

A psychoanalytic analysis of this social shift might suggest that the collective orientation of society has changed—that people have moved from the reflective activity of reading to the numbing activity of drinking. A more philosophical interpretation would be that the culture has turned from the nourishment of the intellect toward the sedation of consciousness.

In this case, the transformation of the bookstore into the liquor store expresses the actual truth of that cultural moment. The bookstore’s closure and replacement was not random; it was the manifestation of the underlying potential within society.

In other words, the bookstore turned into the liquor store because the liquor store was the true expression of what was already happening beneath the surface of the social psyche.[^3]

Footnotes

[^1]: Compare with quantum superposition—where a particle exists as a set of possible states until measured. The wavelength, as a distribution of possibilities, represents the “field” of potential realities. See: Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.

[^2]: This concept parallels Heidegger’s notion of aletheia—truth as “unconcealment.” See Heidegger, M. (1927). Being and Time. Reality is the ongoing process of revealing what is.

[^3]: This social transformation exemplifies Freud’s and Jung’s idea that unconscious tendencies externalize themselves in symbolic form. See Freud, S. (1917). Introductory Lectures on Psycho-Analysis. Also cf. Jung, C. G. (1952). Symbols of Transformation. The liquor store replaces the bookstore as the material symbol of a shift in collective desire.

Fractals

A single thing is whole things. The concept of the fractal is used to describe how all possible events are instantaneously present within any given thing.[^1]

The term fractal refers to “a curve or geometric figure, each part of which has the same statistical character as the whole.”[^2]

Having the “same statistical character” means that any part within a whole can be taken as the point from which the whole itself can be considered. Fractals are therefore structures in which similar patterns recur at progressively smaller scales.

The Function of the Fractal

The function of the fractal describes the relation between how a single component undertakes the extent of a full duration, and inversely how that duration—comprising the sum set of possible events—can be considered from the viewpoint of a single component.

In this sense, the fractal expresses a reciprocal identity between the part and the whole. Each component of reality is not simply a section of a larger process, but the entire process viewed from a particular scale of experience.

Fractality in Nature

The geometric form of an activity constituting the life of an object is fractal in nature.[^3] We see this in many natural structures—trees, seashells, snowflakes—and in the mathematical pattern of the Fibonacci sequence, which defines the self-similar proportions observed in living organisms.

Objects that do not immediately exhibit fractal-like structures also do so upon magnification. For instance, the surface of the skin appears homogeneous from a distance, but when magnified reveals a cellular fractal structure.

When we see fractal patterns in nature, we are witnessing an abstraction of a preexisting duration in time—a temporal process that has become spatially objectified into visible form.[^4]

When we perceive a tree, for example, we think of it as an individual tree, but this so-called individual tree is simultaneously the whole possibility of the experience of “tree.” The object is the experience of the tree. Yet the object is only a limited scope of that experience; its boundary marks the point where the conception of its duration ends.

This means that any single object, insofar as it is a fractal, is infinitesimally exponential—its form extends endlessly inward and outward in scale, even though perception grasps it as finite.

Fractal Motion and the Doppler Effect

The Doppler effect assumes a kind of directionality—a point-like particle heading toward or away from an observer as its wavefronts become condensed or stretched. We often conceive this motion linearly (from left to right), but when conceived in three dimensions, this becomes an outward–inward motion, motion extending infinitesimally onward.

The motion exhibited by a fractal is characteristic of change itself: what begins as a general, macro-scale transformation becomes inwardly intensified, reproducing itself at exponentially smaller scales.

Thus, when we witness a fractal object, we are already seeing the totality of all possible events that belong to that being—events which occurred or are occurring at different rates of time relative to our own point of observation.[^5]

Motion by way of change behaves fractally. A fractal is the form by which a series of events intertwine and complicate together to form the spatial extension of an object, so that the object may experience the duration of its own internal series of events.

In this sense, a fractal is the conception of a series of potential-like events disclosed within the same spacetime manifold. The leaves of a plant, for example, are the events of the plant itself.

Internal and External Observers

The physical substrate of a thing is simply the pure conception that undergoes the experience of its presupposed duration, the content of which is warped and molded by the form of its activity.

From an external observer, the duration of a fractal appears as the particular character that is identified with the object. However, there is also an internal observer, embedded within the structure of the object itself, undergoing the duration of that experience from within.

The motion of the observer is identical with the change the object undergoes through time. This includes both the extension of the object from which the conception is derived and the object being conceived. As the observer moves toward a potential event, the event rises like a sunrise over the horizon, meeting the observer midway.

In the grand scheme of the universe, there is no single center point, for every point is potentially the center. The factor of indeterminacy is itself a certainty: the potential avenues of future events are other beings.

What we call the structure of a single being is, in truth, the totality of all potential beings. The universe is not merely a place of objects but a place of beings. Objects are abstractions of beings—limited conceptions derived from the particular viewpoint of an external observer.[^6]

Footnotes

[^1]: The concept parallels Leibniz’s monads, each of which mirrors the whole universe from its own perspective. See Leibniz, G.W. (1714). Monadology.

[^2]: Mandelbrot, B. (1982). The Fractal Geometry of Nature. New York: W.H. Freeman.

[^3]: Compare with Whitehead’s process philosophy, where each “actual occasion” prehends the universe and expresses it according to its scale. See Whitehead, A.N. (1929). Process and Reality.

[^4]: This idea resonates with Bohm’s implicate order, in which space and time unfold from a deeper, enfolded totality. See Bohm, D. (1980). Wholeness and the Implicate Order.

[^5]: Fractal time models appear in chaos theory and relativistic physics, where self-similarity describes how temporal processes scale across magnitudes. Cf. Prigogine, I. (1997). The End of Certainty.

[^6]: The Presocratics, particularly Heraclitus, conceived of being as perpetual transformation (“You cannot step into the same river twice”), while Anaximander described the universe as a boundless (apeiron) in which all things exchange and re-balance.

Instantaneous Point Flash

When the extension of an activity is restricted so much that it becomes identical with a duration infinitesimally going onwards, Whitehead describes this as the occurrence of an “instantaneous point flash”—that moment when a process becomes infinitely limited and yet discloses the totality of its duration.[^1]

This flash point is not the whole of an activity captured simultaneously at an instant; rather, when it occurs, it means that the duration has reached its ideal limit, a moment of determination where the experience of each event unfolds in a definite order through time. At such a point, the process inverts inwards and becomes the origin from which the whole is disclosed.

All possible events are instantaneously present in this fractal manner. This idea may be understood in two complementary ways.

First, when all possibilities are present, we are contending with what is readily available—as though a number of objects were scattered across a plain, each representing a possibility. In this sense, possibility as instantaneity means whatever is presently available; whatever is here and now is already possible. But this conception can only include a finite set of objects.

Second, possibility extends beyond what is presently given, beyond what is directly within the reference frame of the observer. In this higher sense, possibility involves the ability to be present without yet being captured in the observer’s field. It is presence beyond perception—being that is, in principle, accessible but not yet actualized.

Perception and Logical Completion

In perception, every object appears to partially block off or be blocked off by another. Yet the mind sees where the eyes do not. Logic fills in the blind spots of vision.

For instance, perception does not directly see the water running beneath the bridge, but reason assures us it continues there. We need not look to confirm—it is enough to infer it. Thus, one does not have to fully see something to know it exists; one can partially perceive and rationally complete the rest. Most of the empirical knowledge of science operates in precisely this way: partial observation completed by logical deduction.[^2]

Instantaneity in the spatial domain therefore operates on the principle that possibility is confined within what is readily present—found both in the intention to have a certain set of objects, and in the limitations those objects impose on each other.

In the temporal domain, however, instantaneity involves both what is and what is not present. These two states exist simultaneously. Every single possible iteration of a thing exists within the same space, along with the iterations of all things in every possible way.

If this were experienced without limitation, there would be such an overlap of possibilities—such a dense clustering of action—that nothing could be picked out as distinct, nor could anything move at all. The moment there were any empty space, it would be immediately filled with infinite possibilities.

Limitation as Order

It is in the nature of space and time to impose a limitation on infinity, thereby giving rise to order.

Space cannot be infinite in the sense of an endless plain extending forever, because for that to occur, space must be disclosed by a conception maintaining it as infinite. Space, therefore, is disclosed within that conception—it is not independent of it. In this way, infinite space must be moving relative to a finite reference point. Otherwise, space itself would be an unmoving, finite object, and thus limited.

Hence, a finite conception serves as the reference that keeps space infinitely unfolding. The limitation of infinity is therefore a natural element of order.

The mind remedies the unlimited scale of all possibilities present in time by limiting them to a few things present in space. The observer conceives infinity precisely because there is a limit within itself by which that infinity is disclosed.

Infinity is conceived as the undifferentiated essence of everything—the primal ground from which distinct and finite conception emerges. Infinity is not the beginning of all things but rather their ideal energy, the potential for creation.

Thus, the beginning of anything must appear as the most finite and simple of all conceptions. Whether conceived as nothing (the absence of all things) or as the first minimal thing, the beginning is the finite condition through which infinity becomes conceivable.

The First Differentiation

This beginning is empty and void of qualities. It thus develops the first quality by realizing that the lack of all qualities is not a property of reality itself, but a limitation of its own awareness. Therefore, everything already exists, but consciousness blinds itself from it in order to regain the capacity to distinguish anything at all.

This process—of self-limiting to perceive—occurs in every instance of reality. Each thing that is “picked out” from the infinite is a moment of self-recognition. Infinity, as no-thing, identifies itself as the capacity to pick out something. From this limitation arises the ability to identify with any particular thing.

As soon as the first distinction is made, consciousness begins to select: first one thing, then another, and another. These selections form a series. Because the capacity to distinguish is infinite, the observer can pick any arbitrary part of infinity—even the infinite itself—and make that part the disclosed form from which infinity unfolds.

Thus, if the observer picks out a plant, that plant becomes the reference frame that discloses its own structure: the cells, the molecules, the atoms, and inversely, the cosmic environment those atoms compose. Each level is a reflection of the other; each part unfolds the whole.

If we abstract any part that makes up this whole fractal, we realize that each part is the same as the whole.

Mathematical Parallels

This can be expressed more formally.

In mathematics, a series is a set of quantities forming a progression determined by a common relation—whether by order (coming one after another) or by form (from the Latin serere, “to join or connect”).

The term infinite denotes what is not finished (in-finitus). It is greater than any assignable quantity or countable number. To be finite is to be finished, to have reached a limit. Infinity thus represents both the greatest extension and the negation of limitation.

Every definite series is disclosed by a greater quantity beyond it. The value of infinity is expressed as an exponent, a quantity representing the power to which another is raised (e.g., 2³ = 2 × 2 × 2). The exponent determines the multiplicative force of the number—it is, in essence, the expression of potential.

In another sense, an exponent is a realizer—someone or something that makes an abstract principle concrete (e.g., an exponent of truth). Likewise, a set in mathematics is a collection of distinct entities regarded as a unit, satisfying specific conditions. The Latin sectio (“to cut”) relates to this sense—each “section” is a cut from the infinite whole.

Footnotes

[^1]: Whitehead, A. N. (1929). Process and Reality. The concept of an “instantaneous point flash” refers to the limit of an actual occasion—where continuity is contracted into a single experiential unity.

[^2]: Cf. Kant, Critique of Pure Reason (1781), on the synthesis of intuition and understanding—where perception is completed by the categories of reason.

[^3]: Bergson, H. (1911). Creative Evolution. Bergson’s “duration” (la durée) parallels your use of infinitesimal continuity as the inner flow of experience.

[^4]: Bohm, D. (1980). Wholeness and the Implicate Order. Bohm’s notion of the implicate order captures this same instantaneity of all events enfolded together.

[^5]: Spinoza, B. (1677). Ethics. Proposition I, where all finite modes are expressions of a single infinite substance—each part reflecting the whole.

Mathematical Appendix: The Instantaneous Point Flash as a Limit Function

The metaphysical idea of the instantaneous point flash can be represented mathematically as the limit of a continuous process whose duration contracts infinitely toward zero while preserving the totality of its internal relations.

Let a process of experience be denoted as a function of time:

[
\phi(t): \mathbb{R} \rightarrow \mathbb{R}^n
]

where ( \phi(t) ) represents the state or configuration of an entity at time ( t ), within an ( n )-dimensional manifold of relations (physical, experiential, or informational).

We can model a duration as the interval between two temporal points ( t_1 ) and ( t_2 ):

[
\Delta t = t_2 – t_1
]

Now, the continuity of experience corresponds to the limit as Δt approaches 0, that is, as the differentiation between moments becomes infinitesimal:

[
\lim_{\Delta t \to 0} \frac{\Delta \phi}{\Delta t} = \frac{d\phi}{dt}
]

This derivative expresses change within continuity—the “motion of becoming.”

However, when the activity of change itself becomes infinitely restricted, so that even the differential relation ( d\phi/dt ) collapses into a single identity, the limit ceases to describe change and instead expresses a point of infinite potential:

[
\lim_{\Delta t \to 0} \phi(t + \Delta t) = \phi(t_0)
]

At this point ( t_0 ), the entire duration of the process is compressed into an instantaneous singularity of being. This is the mathematical image of Whitehead’s actual occasion or your instantaneous point flash—the moment when the continuous process of becoming reaches an absolute contraction of duration, revealing the totality of its internal relations in a single state.

We can visualize this limit in two interrelated ways:

1. From the macroscopic perspective, the process appears to evolve through time as a curve ( \phi(t) ).
2. From the microscopic perspective, each infinitesimal part of the curve (each point ( t_0 )) contains the entire structure of the curve itself—a fractal self-similarity:

[
\forall , \epsilon > 0, ; \exists , \delta > 0 \text{ such that if } |t – t_0| < \delta, \text{ then } |\phi(t) – \phi(t_0)| < \epsilon
]

This expresses the continuity condition—every small part of the process reproduces the pattern of the whole.

When ( \delta \to 0 ), we approach a state where the pattern of the whole is identical to the single point that represents it:

[
\phi_{\text{whole}} \equiv \phi(t_0)
]

This equivalence is the fractal identity of part and whole. The instantaneous point flash, therefore, is the limit condition of fractal recursion, where the infinite sequence of self-similarity terminates not by ceasing, but by converging absolutely upon itself.

Graphical Interpretation

If we imagine time as the x-axis and experience (or the configuration of states) as the y-axis, then:

• The curve ( \phi(t) ) represents the continuous becoming of a process.
• The derivative ( d\phi/dt ) represents its momentary rate of change—the motion of the present.
• The limit as ( \Delta t \to 0 ) represents the collapse of motion into being—the instantaneous point flash.

This flash is not static: it is the total compression of dynamism into a single state.

In this model, duration can be visualized as a spiral converging toward a center:
[
\lim_{r \to 0} f(r, \theta) = f(0)
]
Each rotation of the spiral represents a cycle of becoming; the center point represents the infinitesimal now—the limit where all rotations coexist simultaneously.

Relation to Fractal Geometry

In fractal dynamics, a self-similar function ( F(x) ) satisfies:
[
F(x) = a F(bx)
]
for scaling parameters ( a, b \in \mathbb{R} ).

As ( b \to \infty ), ( F(x) ) contracts into the same pattern at infinitesimal scales, expressing the identity of local and global structure.

The instantaneous point flash corresponds to the fixed point of this recursion:
[
F(x^) = a F(bx^) = F(x^) ] where ( x^ ) is the invariant point through which all scales converge.

This is the metaphysical center of the fractal: the point where every possible scale of reality—macroscopic and microscopic, finite and infinite—intersects.

Philosophical Summary

• The instantaneous point flash is the limit of continuity, the point where infinite becoming collapses into pure being.
• It represents the actualization of potential, where all events are instantaneously present in a single state.
• Mathematically, it is expressed by the limit operation and by the fixed point of self-similarity in fractal systems.
• Ontologically, it signifies the moment when duration and instant, infinity and finitude, become identical.

Further References

[^6]: Whitehead, A. N. (1929). Process and Reality.

[^7]: Mandelbrot, B. (1982). The Fractal Geometry of Nature.

[^8]: Deleuze, G. (1968). Difference and Repetition — especially on the differential element as pure becoming.

[^9]: Bohm, D. (1980). Wholeness and the Implicate Order — the implicate enfolding of all moments into the present.

[^10]: Cantor, G. (1895). Beiträge zur Begründung der transfiniten Mengenlehre — on the hierarchy of infinite sets, foundational to the concept of infinite contraction.