1. 15 external internal relation

External and internal relations

section 12 (first update 12.21.2020)

Atomism and organicism designate alternative views about the ultimate nature of reality. They are competing ontologies concerned with the same fundamental question: What is the nature of Being? The key distinction between them lies in their different conceptions of the relations between the ultimate components of reality—namely, what does it mean to be a component?¹

Before we examine these perspectives—each of which every person implicitly subscribes to, whether consciously or not—we must first define what is meant by a “component.”

A Component

A component is a self-identical unity differentiated from an external opposition whose only relation is the lack of relation.² The initial relation between two variables is duality: each is identical with itself but not with the other. This self-subsistence—the capacity to maintain one’s own identity—must exist before differentiation into separate entities is possible.³

Yet being different from each other also implies that they must share at least one commonality: each is the same within itself, and the same in the respect that it is different from the other. This circularity establishes, from either side, the two-ended relation each thing must have with itself.

In the first instance, before a component has any relationship with something “other,” it must first have a self-relation. However, this raises a problem: if a thing must be self-identical to be different from something else, and the reason for its self-identity is that it shares with the other the property of being different and sharing nothing in common, then what makes the self-identity truly unique? It cannot be merely difference, because we must then ask: What is difference itself? The answer cannot simply be “a lack of sameness.” The fundamental difference must already be something shared within each individual component that allows it to bear a relation to the other.⁴

The relation between the one and the other thus consists of two forms: external relations and internal relations.

The Difference is Not a Trait

Difference is not reducible to any specific trait or feature an object possesses. The feature that distinguishes one thing from another—say, one species from another—is often precisely what groups the members of that species together. For example, all rhinoceroses have a horn at the center of their heads, but this trait does not differentiate them from other species that also possess horns.⁵ Distinction based solely on traits fails to explain the deeper principle of difference. The true principle is dimensional: objects and their features are distinguished on the basis of two fundamental dimensions each possesses within itself.

External Relation —Internal Relation — Difference and Sameness

The external relation is the shared dimension among all objects. Contrary to the intuitive meaning of the term, an “external” relation is a common feature two distinct objects share—it is the dimension in which two or more things interact while each maintains its own self-subsistence. In this sense, the external relation is the sameness among things: the fact that they are all external to one another.⁶

The internal relation is the dimension within each component that makes it truly different from all other components. It is the self-relation any object must possess to be identical with itself. The internal relation is what makes a thing the same as itself yet different from everything else.⁷

However, external relations are the true difference by our common sense of the term “difference” because, as the name suggests, the difference between a self-identical unity lies outside that identity, thereby presenting an “other” or an opposition to it. While internal relations also exhibit an element of difference—namely, the variability within the same component itself, whether in terms of its physical constitution (e.g., atoms) or its abstractions (e.g., thoughts)—this difference remains disclosed within the same identity. Internal difference is thus infinite in scope, yet it remains internal to the unity, making internal relations the principle of sameness concerning the nature of a component.

When a component is said to exhibit sameness, this means that it possesses a self-identical principle that grants it a kind of infinity in relation to components considered external—i.e., its relations with other components in an environment. However, these characterizations are not as fixed as we might assume. The issue of the organism and its environment introduces a contradiction: at certain levels of reality, we cannot clearly distinguish where an environment ends and an organism begins, and vice versa.

Footnotes

This recalls Leibniz’s principle of the identity of indiscernibles—no two things can share all the same properties and still be distinct. ↩

Cf. Aristotle, Metaphysics, esp. Book Δ, on the various senses of “being.” ↩

This framing parallels Hegel’s analysis of identity and difference in Science of Logic. ↩

Spinoza’s concept of causa sui—self-causation—relates closely to the notion of self-subsistence. ↩

Hegel refers to this as “identity in difference” (Science of Logic, Quality). ↩

Aristotle uses similar reasoning in Categories, distinguishing essential and accidental predicates. ↩

See Whitehead, Process and Reality, on “external relations” in contrast to “internal relations.” ↩

Atomism

Atomism is the worldview that conceives reality as fundamentally composed of independent, self-subsisting units. Each unit — whether called an “atom,” “monad,” or “particle” — possesses its essential nature in itself, apart from the relations it has with other units.¹ In this view:

• Essence precedes relation — A component’s identity is defined prior to, and independently of, any interaction it has with other components.
• Relations are external — Interactions occur only after these self-contained components are already formed; they are like collisions between billiard balls whose internal nature remains unchanged by the encounter.
• Change is mechanical — Motion, collision, or rearrangement of components explains the appearance of change, but the fundamental building blocks themselves do not depend on the whole for their nature.²

Historically, Atomism appears in Democritus, Epicurus, and in modernity with thinkers such as Descartes (res extensa) and early Newtonian mechanics, where bodies interact through forces but retain their own intrinsic essence regardless of those forces.³

Organicism

Organicism is the worldview that conceives reality as a network of internally related components, such that the identity of each part is constituted by its relation to the whole.⁴ An organism’s parts are not detachable “atoms” in the atomist sense; they are functional moments within a living unity.

• Relation constitutes essence — A component’s identity cannot be understood apart from the environment, structure, or system it inhabits. Remove the relation, and the “component” as such ceases to exist as that kind of thing.
• Relations are internal — Each part is internally determined by its connection to other parts; the “whole” is not merely an aggregation of pre-existing parts but the very ground from which the parts emerge.
• Change is developmental — Transformation is understood as a process in which the whole and its parts co-evolve, like organs in a living body or concepts in a dialectic.⁵

Organicism finds roots in Aristotle’s conception of form and teleology (the organism as a whole with parts ordered toward an end), in Hegel’s dialectics (where moments gain identity only through their place in a rational whole), and in systems theory and ecology, which treat organisms and environments as co-defining.⁶

Key Contrast

Feature	Atomism	Organicism
Essence vs. Relation	Essence is prior to relation	Relation constitutes essence
Nature of Relations	External (do not alter essence)	Internal (define essence)
View of Change	Mechanical rearrangement	Developmental transformation
Metaphor	Billiard balls	Living organism
Classical Roots	Democritus, Newton	Aristotle, Hegel

Footnote:

1. Democritus, Fragments.
2. Descartes, Principles of Philosophy, Part II.
3. Newton, Principia Mathematica.
4. Hegel, Science of Logic, “Doctrine of Essence.”
5. Aristotle, Parts of Animals, Book I.
6. von Bertalanffy, Ludwig. General System Theory.

Atomism (External Relations) vs organicism (Internal Relations)

There are two fundamental worldviews about the nature of reality that conceive the essential relations between components in two different ways.¹

External Relations – *“Atomism embraces all those ontologies which assume these relations are external, meaning by this that the essential qualities of an ultimate component (an ‘atom’) exist independently of its relations, and that an ultimate component possesses qualities without being itself a quality.”*²

External relations explain how forms interact with each other after they have been generated as particular objects. They define the nature of matter insofar as they examine the relative relations between separate forms. The science of external relations, for example, is most famously exemplified in Einstein’s theories of general and special relativity.³ The interactions between forms in external relations often involve contradictions, simply because each form bears a particular structure. The tendency of matter toward self-merging arises from the way differing structures contradict each other and subsequently reach a state of mutual accommodation.⁴

Internal Relations – *“Organicism embraces all those ontologies which assume relations are internal, meaning by this that an ultimate component’s essential qualities are the outcome of its relations, and that the component is itself a quality—an adjective—of its situation.”*⁵

This means that one can only know what a thing is by defining it as being part of its relationships. Internal relations define the foundational conduct of logic: the proposition of one thing cannot be made without the presupposition of another. For example, if not-P is proposed, then P is invariably presupposed. The proposition of not-P necessarily implies P as something distinct, and vice versa. Logic, in this sense, is an indivisible substance because its relations form structures wherein, if one component is removed, the entire system collapses.⁶

Footnote:

1. For a general discussion of metaphysical worldviews, see …
2. Winslow, Ted. [Exact work or lecture title here].
3. Einstein, Albert. Relativity: The Special and General Theory.
4. Hegel, science of logic
5. Winslow, Ted. [lectures].
6. On the dependency structure of logical propositions, see Aristotle’s Metaphysics and Organon.

Self-Determination and Real Potentiality

The approaches of Organicism give a central role to self-determination as the key factor for understanding internal relations. The idea of the “self” undergoing a process of “determination” refers to what any self-identical component can do—namely, determine itself—and, conversely, to what it means: namely, to be determined.¹

Self-determination is conceived in such a way that a component can only condition what is possible, rather than fully determine what occurs.² Any distinct component finds itself already situated within a pre-determined context, but within that environment it can alter and transform itself into something other than what it originally was. Thus, self-determination defines what it means to be a component: it is the ongoing process of conditioning possibility within an already given structure.³

A component, in this sense, generates what Alfred North Whitehead calls “real potentiality”, the field within which self-determination operates.⁴ Real potentiality, unlike either a pure actuality or a mere potentiality, is not fixed but changes in congruency with the modifications of the internal relations that constitute it.

• A reality on its own is the present moment as it occurs into being, the way it is observed in its immediacy.
• A potentiality, by contrast, is only an “ideal moment” always outside the current moment we call the present. It is the horizon towards which the present is always moving, yet which it never fully arrives at.

The concept of real potentiality unites these inverse poles—reality and potentiality—into an intentionally altered, self-determined activity of individuals. This implies, first, that general activity itself consists of the relations among separate and distinct individuals, and, second, that individuals themselves, as distinct aspects of the same relation, participate in determining what will later happen to them within that activity.⁵

This introduces a profound conundrum at the heart of metaphysics: the tension between free will and determinism. If individuals are determined by relations, how can they simultaneously determine themselves? The answer suggested by organicist metaphysics is that freedom does not negate determination but rather arises within it: self-determination is the mediation between necessity and possibility.⁶

Footnotes

1. Hegel, Science of Logic, trans. A. V. Miller (London: George Allen & Unwin, 1969), “Doctrine of Essence,” on self-determination.
2. Aristotle, Metaphysics, Book Θ, on potentiality and actuality.
3. G. W. F. Hegel, Phenomenology of Spirit, trans. J. N. Findlay (Oxford: Oxford University Press, 1977), §17.
4. Alfred North Whitehead, Process and Reality, ed. David Griffin and Donald Sherburne (New York: Free Press, 1978), Part II, Ch. 2.
5. Charles Hartshorne, The Logic of Perfection (LaSalle, IL: Open Court, 1962), esp. chs. 1–2.
6. Robert C. Neville, Creativity and God: A Challenge to Process Theology (New York: Seabury Press, 1980), on freedom as determination-through-relations.

Organicism and Human Self-Determination

Karl Marx, Theses on Feuerbach, Thesis 3:

“The materialist doctrine that men are products of circumstances and upbringing, and that, therefore, changed men are products of other circumstances and changed upbringing, forgets that it is men who change circumstances and that the educator must himself be educated.”

Great question. Marx’s line — “the educator must himself be educated” — is deceptively simple, but it carries a deep critique of determinism and “mechanical” materialism. Let’s unpack it step by step:

1. The background

Marx is responding to Ludwig Feuerbach and other “materialists” of his time who argued that human beings are essentially the product of their circumstances and upbringing. In this view:

• If people live in poverty, ignorance, or oppression → their ideas and behavior are determined by those conditions.
• To “improve” people, you simply need to change their environment.

This is a deterministic materialism: people are passive effects of external causes.

2. Marx’s critique

Marx says this overlooks something crucial:

• Circumstances do shape people, yes.
• But people themselves also act back upon circumstances.

If you say: “people must be changed by educators,” then — who educates the educators? If educators are also just products of circumstances, they too need educating. This leads to an infinite regress unless you admit that people are also active agents capable of changing conditions themselves.

Thus: “the educator must himself be educated.”

3. Meaning of the phrase

It means:

1. Human agency cannot be erased. People aren’t just passive results of forces; they also change those forces.
2. Revolutionary praxis (the conscious activity of changing society) is the missing piece in Feuerbach’s determinism. Humans shape themselves through changing their world.
3. The relationship between individual and environment is dialectical, not one-directional. Circumstances make men, but men also make circumstances.

4. Wider implications

• This quote is foundational for Marx’s theory of praxis: the unity of thought and action.
• It also grounds his critique of determinism — history isn’t mechanically predetermined but shaped by human struggle.
• In your project on internal vs external relations, this fits perfectly: Marx insists that the “internal relation” of man and environment is reciprocal, not fixed as a one-way causality.

Organicism is fully compatible with the existence of self-determination in individuals. Individual self-determination is, in fact, the very mark of self-determination itself: the individual manifests freedom not in isolation from relations, but through them.

Karl Marx explicitly rejects, on grounds of incoherence, determinist accounts of the relation between character and circumstances.¹ As he puts it, such “materialistic doctrines” overlook the fact *“that circumstances are changed by men and that the educator must himself be educated.”*² In other words, such doctrines implicitly and inconsistently assume that some individuals (namely, the “educators” who are supposed to shape the conditions of others) stand outside the causal relations which, according to their own theory, determine the characteristic ideas and behaviors of everyone else.

From an organicist standpoint, the development of human beings is best understood as the interdependent unfolding of freedom and reason.³ Freedom and reason are not external additions to human life but are the essential characteristics of the human individual. Yet these characteristics are not innate in a fixed sense: they are the outcome of internal relations, of the dialectical process through which individuals both shape and are shaped by their conditions.

At the heart of this process lies the potential for rationality—the defining feature of human beings.⁴ Rationality, in this sense, is not merely a biological endowment but the telos of human self-determination: the active realization of freedom through participation in reason.

Footnotes

1. Jon Elster, Making Sense of Marx (Cambridge: Cambridge University Press, 1985), esp. ch. 3 on determinism and freedom.
2. Karl Marx, Theses on Feuerbach, Thesis 3, in The German Ideology, ed. C. J. Arthur (London: Lawrence & Wishart, 1970).
3. G. W. F. Hegel, Philosophy of Right, trans. T. M. Knox (Oxford: Oxford University Press, 1967), §§4–6, on freedom as the essence of spirit.
4. Aristotle, Nicomachean Ethics, Book I, on the rational capacity as the essence of human nature.

Kant defines “Man” as the potentially rational animal.¹ Full realization of this potential is only possible within fully free relations of mutual recognition between what Marx calls “universally developed individuals”, within “an association in which the free development of each is the condition of the free development of all.”²

The “universally developed individual” is a concept of the ideal potentiality of Man as a human being capable of actualizing their essential purpose. There are two ways a human being can actualize their purpose:
i) They may achieve their own individual goal or vocation, such as pursuing a profession recognized in society.
ii) They may participate in the wider realization of human freedom as such — not merely in their individual accomplishments, but in contributing to and sharing in relations of universal freedom.

On the other hand, full freedom requires full rationality. The latter is the basis of what Whitehead calls “the essence of freedom” — “the practicability of purpose.”³

The organicist accounts of human development frequently treat it as the interdependent development of human freedom and reason. This distinguishes many organicist approaches to economics from orthodoxy. Economists whose orientation is organicist often reject the assumption that human behavior is everywhere and always rational. John R. Commons, for instance, dissents from what he associates with Rousseau and orthodox economics — the view of an original state of liberty and rationality of human beings. Instead, he claims that “historically it is more accurate to say, as Malthus said, that man is originally a being of passion and stupidity for whom liberty and reason are a matter of slow evolution of moral character and the discipline enforced by government.”⁴

Ideal relations only become part of real potentiality at the end of a long process of historical development. The intermediate stages of this process are characterized by less than full freedom and rationality (and hence by irrationality — an irrationality taken into account in Hegel’s, and following him, in Marx’s, notion of the “passions”).⁵ Though incompatible with full development, each stage is compatible with some development. Hegel’s account of the master/slave relation illustrates this in a way relevant to interpreting Marx’s conception of relations of production as “basic.”⁶

This development widens real potentiality to include the practicability of freer relations at the next stage. There is thus set up a tendency for progressive movement, as Whitehead puts it, “from force to persuasion.”⁷ This process is best conceived as a process of education, or Bildung. Goethe’s Faust gives poetic expression to an essential aspect of this process — the development of knowledge of the ideal.⁸ The ideal relations at which the process is aimed create a context in which individual character is fully self-determined, the product of educators radically different from those presupposed by the materialistic doctrine criticized by Marx.⁹

The necessity for development to take place through stages arises from the internal relations between them. Institutional arrangements of a particular kind — say, capitalist or socialist — are only part of the real potentiality of a particular context where historically produced relations make possible the development of the kind of individuality those arrangements require. Socialism in Marx’s sense, for example, has almost certainly never yet been part of the real potentiality of any community. Attempts to impose it have, therefore, invariably failed: the arrangements which resulted, though called “socialist,” have never been socialist in Marx’s sense. Moreover, such attempts have often (perhaps always) produced arrangements inferior to others which were also part of the community’s real potentiality. Similar remarks could be made about proposals to introduce “free market” arrangements — of a kind that not only have not yet been, but in addition, could never be part of any community’s real potentiality — into the collapsed “socialist” economies of Eastern Europe.

Ontological premises have important implications for method. For example, where relations are internal, identity changes with changes in relations. This has direct relevance to the use of mathematical methods. For instance, one thing plus another thing only makes two things where at least partial identity (that required to keep what Whitehead calls “the relevant principle of individuation” undisturbed) is maintained through the process of addition.¹⁰ Whitehead gives as an example of processes of addition which do not meet this requirement the addition of a lighted match to a stick of dynamite.

Similarly, the “variable” in algebra designates an entity which must remain self-identical to the extent required for the purposes of the argument. Consequently, the use of algebra is only “rigorous” (in the sense of “logically justified”) where this condition is satisfied. This is true of any form of logical reasoning which makes use of the variable. Taking our example once more from Whitehead: “for the purposes of inheriting real estate the identity of the man of thirty with the former baby of ten months is dominant whereas for the purposes of navigating a yacht it is the differences that are essential.”¹¹ Some processes characterized by organic interdependence will meet the requirements for the applicability of mathematical forms of reasoning such as arithmetic and algebra; others will not. The point is that, where relations are internal, maintenance of the requisite degree of self-identity cannot be taken for granted.

Footnotes

Alfred North Whitehead, Process and Reality (1929). ↩

Immanuel Kant, Anthropology from a Pragmatic Point of View (1798), where he defines the human being as the rational animal, though always only potentially so. ↩

Karl Marx and Friedrich Engels, The Communist Manifesto (1848), ch. 2. ↩

Alfred North Whitehead, Process and Reality (1929), on the essence of freedom. ↩

John R. Commons, Institutional Economics (1934), and Thomas Malthus, Essay on the Principle of Population(1798). ↩

G.W.F. Hegel, The Phenomenology of Spirit (1807), on the “passions” as the material through which Reason realizes itself. ↩

Ibid., Master/Slave dialectic. ↩

Alfred North Whitehead, Adventures of Ideas (1933). ↩

Johann Wolfgang von Goethe, Faust, Part I (1808). ↩

Karl Marx, Theses on Feuerbach (1845), esp. Thesis 3: “The educator himself must be educated.” ↩

Alfred North Whitehead, The Concept of Nature (1920). ↩

The common folk expression “I can cut the tension with a knife” is actually not just an empty idiom, but rather expresses something physically tangible to the mind, though not felt by the senses. This expression is an unconscious recognition of internal relations.¹ That there might exist a level of reality which physically constitutes the observable world, but which cannot itself be directly observed. In this sense, we may designate this field of reality—unobservable yet rationally conceivable—as “consciousness.” However, the precise manner in which this relates to individual consciousness, as opposed to a universal consciousness, still requires further explanation.

The law of consciousness may be stated as follows:

Any set of variable external relations corresponds to an invariable internal relation. For example, in any social interaction, there is an external relation — e.g., two people talking with sound to each other, apparently saying something. Yet beneath this there is an internal relation, wherein a certain thought, accompanied by a feeling, is silently experienced by each party engaged in the discussion, without the other person knowing. Each participant experiences something inwardly that the other does not perceive, even though outwardly both share the common knowledge that they are discussing the same topic. In other words, both experience the same external relation but inhabit different internal relations, which constitute the deeper, lived dimension of consciousness.²

Inversely, this difference in internal relations is also the common reality they share: namely, that both individuals have thoughts and are thinking. Even though they think differently, they still participate in Reason, which observes itself through the other. These internal relations extend even to more fundamental components, such as atomic structures and their forms. By contrast, external relations may be likened to a particular, finite, and specific chair that represents the Form of a chair. The Form of a chair, in turn, is the internal relation of all particular chairs—their essential “idea,” and therefore their essence.

Footnotes

G.W.F. Hegel, Phenomenology of Spirit (1807), describes consciousness as the dialectic between subjective inwardness and objective outwardness, wherein thought and feeling mediate external relations. Similarly, Alfred North Whitehead, Process and Reality (1929), emphasizes that internal relations are constitutive of experience and cannot be reduced to external appearances. ↩

This distinction between the sensible and the intelligible dimensions of experience has roots in Plato’s Timaeusand Republic, where perception is contrasted with the inner reality apprehended by reason. The idiom’s endurance can be seen as a folk recognition of the same tension between outward relations and inward states. ↩

Internal relations assert that the essential nature of a component is entirely dependent on its relations; without these relations, the component cannot exist. In contrast, external relations view the interaction between objects as independent of the specific variables involved: once internal relations stabilize as components, these components inversely relate to one another, each imposing its own self-identity onto the self-identity of the other. What we perceive as the interaction between two variables is, in fact, predicated on a more fundamental self-relation—the underlying cause that generates a specific kind of object with a distinct identity, which can later interact with identities different from itself.

External relations define matter and mechanical nature because perception recognizes appearances of attraction and repulsion, action and reaction, or cause and effect. However, this is primarily dependent on the manner in which these compounds were generated in the first place through internal relational processes. These processes constitute the pure activity—the pre-relational ground—not even the relation mediating two things, but the pure relation prior to their generation. This potentiality for existence is a real precondition for actual existence. For example, in arithmetic, the pure relation is not the result obtained when two or more specific numbers are added together; rather, it is the concept of addition itself—the capacity and potential for any set of numbers to be combined—regardless of the specific result they yield.

Footnotes:

1. Alfred North Whitehead’s Process Philosophy:
   Whitehead’s philosophy emphasizes the interdependence of entities through internal relations, where the identity of each entity is constituted by its relationships with others. This contrasts with external relations, which treat entities as independently existing and interacting. Whitehead associates internal relations with extension and externality with prehension, suggesting that the achieved result (the superject) is the atomic realization of an occasion in its ultimate externality to the rest of the world. 
2. Kant’s Distinction Between Internal and External Relations:
   Immanuel Kant distinguishes between internal and external relations in his metaphysical framework. Internal relations are those that are essential to the identity of the entities involved, while external relations are contingent and do not affect the intrinsic nature of the entities. This distinction plays a crucial role in Kant’s understanding of how entities relate to one another within the bounds of human experience. 
3. Conceptual Foundation of Arithmetic Operations:
   In arithmetic, the concept of addition is not merely the act of combining specific numbers but represents a general capacity or potential for combining any set of numbers. This abstract concept underlies the specific instances of addition and serves as a fundamental principle in arithmetic operations. 

The total amount of relations in any system always exceeds the total of its individual components. In addition, it is the capacity to be added—not the actual result—that emerges from addition. But why would the relation always be such that the whole is greater than the parts? Consider two objects sharing a single relation—for example, two points lying on the same line. If we measure quantity, the line (relation) may appear “less” than the two points. A single line may be assigned a value of 1, while the two points collectively exceed 1. However, this reasoning conflates quantity with quality. If we recognize that the point and the line are different qualities—the line representing extension, the point representing discrete measure—their quantitative measures are fundamentally incomparable. A line is an extension of an indefinite number of points, which is always “more” than any definite set, whereas a point measures discreteness and certainty. A known value is always less than an unknown or indeterminate value.¹

Indeterminate measures are “greater” in potential than definite measures because they embody all possible outcomes relative to a definite starting point. For example, if we begin with 2 as a definite measure, an indeterminate measure may be greater than 2, less than 2, or infinitely less than 2. Yet any combination of possibilities—including negative values—ultimately yields a sum that surpasses the initial definite measure. Potentiality, therefore, always exceeds definiteness in terms of what can emerge. There must first exist an abundance of potential values before any actual combination, such as 1 + 1, occurs.²

The pure and essential relation in nature is thought. Thought is entirely engaged with itself, eternal, and unbounded, yet it produces components that interact in specific ways. When one ball hits another, causing it to move, this exemplifies an external relation. Internal relation, in contrast, encompasses the entire event as a possible moment in a network of parallel variations.³

A purely materialist view interprets cause and effect linearly: the effect arises from the cause, and the cause necessarily produces the effect. Yet this explanation only accounts for temporal succession, not the deeper reason why the effect follows the cause. In internal relations, thought is infinite activity: to think is simultaneously to act, producing particular relations manifested as compounds. A compound, composed of two or more interconnected elements, is itself a relation. Internal relations operate through infinite contradiction, which externalizes as observable compounds.⁴

When thought encounters a contradiction, the contradiction abstracts into a relation, forming a compound. This compound represents a particular idea structuring the material substratum into finite nature. Finite nature, in turn, exhibits external relations, while internal relations subsist as infinite. An idea is a relation—a connection that hosts variable variations of the same substance. For instance, a plane divisible into discrete points remains a unified entity containing all distinct points. The arrangement of these points—the composition of the whole—is the essence of internal relation.⁵

Each relation consists of two indivisible, inverse parts. Every object is, fundamentally, an idea. Even the idea of an abstract color like white is contradicted by black. Internal relations imply that no component exists independently of its inverse: if I think of white, the corresponding concept of black is simultaneously evoked. Every idea carries an inverse correlation; this is akin to quantum entanglement, where one component’s state immediately relates to another complementary state.⁶

Footnotes:

Bohm, David. Wholeness and the Implicate Order. Routledge & Kegan Paul, 1980. Connection to quantum entanglement as a model for inverse relations of ideas. ↩

Whitehead, Alfred North. Process and Reality. Corrected ed., Free Press, 1978, pp. 18–23. See discussion of internal relations and the primacy of potentiality over actuality. ↩

Peirce, Charles Sanders. Collected Papers of Charles Sanders Peirce, Vol. 2, Harvard University Press, 1931, pp. 234–236. Discusses indeterminacy and generality in logical relations. ↩

Hegel, G.W.F. Science of Logic, trans. A.V. Miller, Humanity Books, 1998, §§54–56. On internal and external relations as they manifest in reality. ↩

Aristotle. Metaphysics, Book VII, trans. W.D. Ross, Oxford University Press, 1924. On form, substance, and the generation of compounds. ↩

Whitehead, Process and Reality, pp. 40–45. On the notion of “nexus” as internal relations producing actual entities. ↩

A foundational fact of neurology is that the organization of neural structures in the brain enables the rational faculties to reason about things that the mind has experienced. Each neural network is configured to consider what is possibly true in the environment (reality). In other words, neurological brain systems are rooted in reality; they are not merely abstract. They may represent previous paths of experience accumulated by ancestors and epigenetically passed down to their offspring. In other words, what parents have experienced may potentially be transmitted to the child as new neurological pathways. Almost like a unconscious natural memory system.

One cannot think of “not something” without simultaneously invoking that “something”; any negation of a thought merely generates another thought. For example, when I think of “not a cat,” I have necessarily thought of a cat and perhaps also a dog. The brain’s neurological system is structured so that ideas trigger other ideas, but only those that are possible and based on real or potential objects of experience.¹

These ideas are not mere abstractions; at the most fundamental level, they correspond to the primary conditions of matter. They manifest in the form of neurons, electrically charged cells with definite orientations. The understanding that matter can exhibit different magnitudes of the same quality suggests that all physical things share an underlying abstract principle, which is apprehended through sensation.² Ideas, in this sense, represent the forms that essential energy assumes as it passes through pathways of potentialities. Each pathway exists in parallel, yet simultaneously relates to others.

These structures take on an organic form, analogous to the organization of a cell, whose evolutionary processes have led to the development of the brain. The brain, as an organic compound within the universe, fulfills the ontological definition of a component: it is both a self-identical, discrete unity and a complex system of internally related parts. Its internal relations extend inward toward an infinitesimally small point in spacetime, coordinating the integration of diverse extensions of activity.³

Footnotes:

Kandel, Eric R., Schwartz, James H., and Jessell, Thomas M. Principles of Neural Science, 5th edition, McGraw-Hill, 2013. Describes the structural and functional integration of neurons in the brain. ↩

Edelman, Gerald M. Neural Darwinism: The Theory of Neuronal Group Selection. Basic Books, 1987. Discusses how neural networks generate possible ideas through selective activation. ↩

Whitehead, Alfred North. Process and Reality, corrected edition, Free Press, 1978, pp. 18–25. On the metaphysical interpretation of potentiality and ideas as forms of actual entities. ↩

When examining the difference between internal and external relations, we must be careful not to confuse the terms. The term internal should not be mistaken for interior. The term interior denotes an external relation situated inside another external relation. For example, my stomach is an interior, not an internal, because it exists inside me. The internal relation of an organ like the stomach—or any object—is actually its own individuality, a quality distinct from the rest of the body; it is not an arm or a leg. The term internal refers instead to the essential activity sustaining the object as an external entity. Quantum science can be characterized as a science of internal relations because it concerns the generation of natural phenomena and their original causal processes. Philosophically, it is the cause without being caused.^1

Law of Moderation

Aristotle proposed the law of non-contradiction and the law of the excluded middle to elaborate the principle of mediation.^2 In this principle, something can both be and not be, which is a resolution, not merely a contradiction—a resolution of being and non-being. Any concrete object is a moderation between extremes. Moderation does not imply symmetry but rather the unity of extremes that produces the identity of a thing. The law of non-contradiction, for example, is only an abstraction, a moment of the fundamental truth that a thing both is and is not the same thing. For instance, at the present moment, I am here but not there; yet, reflecting over the day as a whole, I was both here and there, though not at the exact same time, but within the same temporal framework.^3

Feeling of Time

Time has an interesting feature: when we are late, time seems to move faster; when we have all the time in the world, it seems to slow down. This is not due to the actual speed of time—which remains constant—but due to our perception of its duration. Time never stops; it is constant activity. Measures of time—seconds, minutes, hours—are abstractions imposed on an ongoing process of events, relations, and happenings. Every material object is a unity of being and non-being, undergoing generation and degeneration, such that its organization or form is continuously modified. In the visible universe, moderation is an essential principle of any object. Yet moderation presupposes an imbalance between objects in their external relations. What is moderate for one object may be extreme for another. This imbalance creates the observable order in the universe. For example, why does the gravity of the sun not collapse all surrounding planets?^4

Internal and External Relations

External relations characterize the atom as the explicit many taking the implicit form of the One. The external relation presupposes the many as independent from each other; the One manifests implicitly through this multiplicity. Internal relations provide the logic and reason explaining why reality appears as it does. Observing a landscape, we see many distinct objects, each self-identical, yet each containing internal differences. A landscape unites these distinct objects, with the only commonality being that it is larger than the individual objects it contains. The One takes on the Many, but the Many presupposes the One. This logic explains how consciousness perceives the world.^5

The internal dialectic presupposes its outer projection. The outer projection is determined by internal relations. In quantum physics, internal relations are sublated into the external relations of the atom. Newtonian laws of motion formalize external relations:

1. An object remains at rest or in uniform motion unless acted upon.
2. Force equals mass times acceleration (F = ma).
3. For every action, there is an equal and opposite reaction (F = -F).

These laws describe external interactions without accounting for the deeper process of internal relations that generate the objects themselves. Einstein expanded this to include macroscopic interactions, such as light and gravity, showing that external relations are invariably underpinned by internal relations. In the quantum state, external relations manifest internal relations; internal relations, in turn, manifest as external relations.^6

Newton and Einstein analyze motion analytically, observing it as it appears in the external world. However, these accounts do not accommodate motion as free determination. Whitehead defines free determination as the capacity of a component or process to realize potentialities beyond mechanical necessity.^7 Kant provides a similar analysis through laws of repulsion and attraction, while Hegel’s logic emphasizes the dialectical movement within matter. Quantum mechanics, in contrast to classical mechanics, characterizes motion as free determination, reflecting the interplay of internal relations that generate the observable world.^8

Footnotes:

1. Whitehead, Process and Reality, 1929, pp. 52–53.
2. Aristotle, Metaphysics, Book IV, 1005b–1006a.
3. Ibid.; cf. Hegel, Science of Logic, Part I, §93.
4. Whitehead, The Concept of Nature, 1920, Ch. 2; cf. Einstein, Relativity: The Special and General Theory, 1920.
5. Ibid.; cf. Kant, Critique of Pure Reason, A51/B75.
6. Einstein, Relativity, 1920, Ch. 3; Heisenberg, Physics and Philosophy, 1958, Ch. 5.
7. Whitehead, Process and Reality, 1929, pp. 46–48.
8. Kant, Metaphysical Foundations of Natural Science, 1786; Hegel, Science of Logic, Part I, §§92–94.

Logic characterizes what Peirce calls the “growth of concrete reasonableness”, which manifests as the “ultimate aesthetic ideal” in nature.^1 For Peirce, aesthetics concerns the kind of conduct likely to promote the growth of rationality. In this sense, logic is the guiding principle for rational development.

According to Aristotle, logic concerns the relations necessary for form and matter.^2 The action of logic is the relation itself, which does not settle in one place but produces components that form in specific ways. For example, in a geometrical context, logos signifies the ratio between shapes. The ratio is potentially measurable in general, while the ratio between two particular shapes has specific, concrete measurements.

Logic can be defined in terms of internal relations, described as follows:

Peirce compares logic with mathematics in this way:

Where a mathematical calculus aims to reduce the number of intermediate steps necessary to reach a conclusion, a logical calculus expands the number of steps in order to better demonstrate the validity of the argument.^3

Mathematics seeks the resolution with the fewest steps, whereas logic aims to trace every possible step. According to Peirce, logic underpins mathematics because all possible paths in a calculus must first be shown before the most efficient solution can be determined.

Internal relations constitute the necessary logical activity that generates form. Once internal relations generate forms, these forms begin to relate externally. External relations describe how forms interact after their generation. Internal and external relations are two sides of the same coin: the former explains the generation of form, while the latter explains the changes the form undergoes after generation. However, external relations are ultimately dependent on internal relations.

For example, consider throwing one ball at another: the resulting motion is an external relation. The internal relation is the logical necessity that one ball striking another requires movement. The conclusion that one ball hitting another causes motion must first exist as a logically valid principle before it can manifest physically. Without this abstract rational principle, there is no reason why the physical interaction would occur. The physical event demonstrates the abstract principle: one element is the warp, the other the woof.

In terms of internal relations, a component cannot exist in one way without another component existing in a corresponding way. By contrast, external relations allow one component to exist independently of another. For instance, internally, the idea of white automatically implies the idea of black; imagining white without black is impossible. Every idea in internal relations has an invariable correlation. In external relations, however, a white wall does not presuppose a black dog; they can exist independently. This, however, assumes the prior existence of both the white wall and the black dog.

Mathematically, internal relations are expressed via binary operators such as addition, subtraction, multiplication, and division. In mathematics, the operator indicates how numbers should relate. In logic, however, the operator is more fundamental than the numbers themselves. For instance, the number 2 is abstracted from the relation of 1 added to another 1. This relation requires that 1 multiply with 1 as another 1, and their addition signifies the emergence of 2. In logic, the constitution of any number depends on the nature of the binary operator; numbers alone do not inherently require addition or subtraction.

Logical binary operators form an interconnected system: presupposing multiplication implies the presupposition of addition. Internal relations determine how logic operates in nature, giving rise to external relations that are subsequently formalized by the human understanding.^4

Footnotes:

1. Peirce, Charles S., Collected Papers of Charles Sanders Peirce, Vol. 1, Harvard University Press, 1931, pp. 473, 85.
2. Aristotle, Metaphysics, Book IV, 1005b–1006a; Organon, Prior Analytics, 24a.
3. Peirce, Charles S., Collected Papers, Vol. 1, 473.
4. Whitehead, Alfred N., Process and Reality, 1929, pp. 46–48; cf. Kant, Critique of Pure Reason, A51/B75.

According to Peirce, “Metaphysics recognizes an inner and outer world, a world of time and a world of space” (CP, Philosophy and the Conduct of Life, 36).^1 The logical rule that captures this relationship is that “the outer is the expression of the inner” (Hegel, Phenomenology of Spirit, 262).^2 Space and time are fundamental determinations of logic in nature.

Time is the quality of self-identical determination, meaning that it is the determination by which an activity is self-contained (Metaphysics VI.1018a; Lawson-Tancred).^3 The magnitude of time is discrete because self-identity assumes a state that makes it a bounded unit. In this way, time represents the potential for activity to take on particular properties (Hegel, Philosophy of Nature, 201).^4

Space is the quality of self-external determination, meaning that it is the potential for activity to be outside of identity (Hegel, Philosophy of Nature, 197).^5 Space is always external because it is the determination that remains different (Metaphysics VI.1018a; Lawson-Tancred). The magnitude of space is continuous because it extends beyond other determinations.

The negation of each principle against the other defines their inherent qualities. Identity is the negation of externality, and externality is the negation of identity; this defines both their function (Hegel, Philosophy of Nature, 203).^6 Without their internal relation, both concepts dissipate as empty abstractions. Self-identity exists only insofar as it presupposes its difference as externality; similarly, externality exists only insofar as it is outside something identical. Identity and externality are real only through each other, just as P presupposes not-P (Metaphysics IV.3.1005b; Lawson-Tancred).^7 From each perspective, however, both appear entirely distinct and not the same, even though they presuppose each other for their respective existence.

For example, the identity of a period has the plane as its externality. The period and the plane are necessary because of their relative relation to each other. Whether the period exchanges to become the plane, or the plane exchanges to become the period, indicates only that their relative relation is based on the contradiction between one as identity and the other as externality.

A black period in a white plane, or a white period in a black plane, expresses the difference between the determinations of externality and identity; in either variation, both principles appear as distinct forms. The principle of identity is only self-identical because its difference exists externally. Identity presupposes externality as its difference, so that it can remain distinct from any difference. Hence, difference is inherent for identity, and identity is inherent for difference. Their contradiction, e.g., P or not-P (Metaphysics V.6; Ross),^8 makes them separate determinations opposed to one another. Yet their opposition constitutes their similarity, because the only quality they share is their mutual opposition (Aristotle, On Interpretation 9–10; Edghill).^9

The excluded middle, or the principle of opposition, asserts that if identity is true, its negation (difference) must also be true (Metaphysics V.5.7; Lawson-Tancred).^10 Moreover, the excluded middle posits that the opposition itself generates a distinct form, one different from the two opposing principles (Hegel, Philosophy of Nature, 202). It is the logic of the inverse of the inverse, or the negation of the negation, in which identity and difference derive each other’s nature and become interdependent.

Within this framework, each principle reaffirms its own truth from the perspective of the other. From the standpoint of difference, identity also possesses difference, because identity is difference relative to difference. Similarly, from the standpoint of identity, difference is self-consistent, maintaining distinction from identity. In this way, difference exists because of identity, and identity exists because of difference.

Space and time are only meaningful in relation to each other. In special relativity, space is never discussed independently of time. The concept of spacetime arises from their relation. Together, spacetime produces the dimension in which identical activity is encapsulated by externality, while externality represents the continuity of identical activity into further determinations external to one another (U. Oregon, Relativity).^11

Footnotes:

1. Peirce, Charles S., Collected Papers of Charles Sanders Peirce, Vol. 1, “Philosophy and the Conduct of Life,” 36.
2. Hegel, G. W. F., Phenomenology of Spirit, trans. A. V. Miller, Oxford University Press, 1977, §262.
3. Hegel, G. W. F., Metaphysics, VI.1018a, Lawson-Tancred translation.
4. Hegel, Philosophy of Nature, 201.
5. Hegel, Philosophy of Nature, 197.
6. Hegel, Philosophy of Nature, 203.
7. Hegel, Metaphysics, IV.3.1005b, Lawson-Tancred translation.
8. Hegel, Metaphysics, V.6, Ross translation.
9. Aristotle, On Interpretation, 9–10, Edghill translation.
10. Hegel, Metaphysics, V.5.7, Lawson-Tancred translation.
11. University of Oregon, Introduction to Relativity Theory, spacetime section.

Internal and External Relations, Identity, and the Logic of Nature

The distinction between internal and external relations is central to understanding both the generation of form in nature and the way objects interact. Internal relations are the essential logical structures that sustain an object as a self-identical entity. They define the potential for relations, the essence of identity, and the necessary correlation between components. External relations, in contrast, describe the observable interactions between already generated forms, such as the motion of one ball causing another to move. Yet even external relations presuppose internal relations: without the logical possibility of interaction, the physical impact itself would have no necessary reason for occurring.^1

Internal relations are not reducible to spatial or temporal positions, nor should they be confused with the “interior” of a body. For example, a stomach is an interior organ, but its internal relations are its essential qualities that distinguish it from other organs while maintaining the unity of the organism. Quantum science exemplifies a study of internal relations because it concerns the generative processes of phenomena and the causal potentials underlying physical events. Philosophically, this can be seen as “cause without being caused,” where the activity of reality is self-determining at its foundation.^2

Charles Peirce describes logic as the “growth of concrete reasonableness” which manifests as the “ultimate aesthetic ideal” in nature (CP, Philosophy and the Conduct of Life, 36).^3 For Peirce, logic is not merely a formal system; it governs the rational development of ideas and the conduct that supports the growth of reason. Aristotle similarly views logic as the study of relations necessary for form and matter: “logos” in a geometrical context is the ratio between shapes, generalizable as potential relations, while particular measurements instantiate these ratios.^4

Internal relations give rise to external relations. For instance, when one ball hits another, the external relation of motion is an actualization of the internal logical necessity: for the interaction to occur, it must first be possible as a rational principle. Without this internal relation, there is no reason why the impact should produce motion.^5 Similarly, in thought, the concept of white presupposes the concept of black; every idea carries its inverse as an essential correlate. External relations allow components to exist differently from each other, but internal relations dictate that their essential characteristics are mutually determined.^6

In mathematics, the binary operator exemplifies internal relations. Addition, subtraction, multiplication, and division are not merely operations on numbers; they define the relations that constitute numbers themselves. The number 2 is an abstraction of the relation between 1 + 1; the operation itself precedes the objects it relates. Logic, therefore, is the generative principle that produces forms, while external relations are the instantiated interactions between those forms.^7

Space and time are further manifestations of internal and external relations. According to Peirce, “Metaphysics recognizes an inner and outer world, a world of time and a world of space” (CP, Philosophy and the Conduct of Life, 36), and Hegel asserts that “the outer is the expression of the inner” (Phenomenology of Spirit, 262).^8 Time is the quality of self-identical determination: it is discrete, bounded, and represents the potential for activity to take on particular properties (Metaphysics VI.1018a; Hegel, Philosophy of Nature, 201).^9 Space is the quality of self-external determination: continuous, unbounded, and representing the potential for activity to extend beyond itself (Hegel, Philosophy of Nature, 197).^10

Identity and difference are defined through their internal relation. Identity presupposes externality, while externality presupposes identity; each exists only in relation to the other. The principle of opposition, or the excluded middle, ensures that identity (P) and difference (not-P) are mutually constitutive (Metaphysics V.5.7; Hegel, Philosophy of Nature, 202). From the perspective of difference, identity is itself differentiated, and from the perspective of identity, difference is self-consistent. This mutual presupposition underlies the formation of space, time, and the external manifestations of objects in the universe.^11

Newtonian physics captures external relations in their mechanical form. The three laws of motion describe observable interactions between objects: inertia, acceleration proportional to force, and action-reaction (f = –f). However, these laws abstract from the internal relational structures that make motion possible. Einstein’s relativity unifies space and time, revealing that external relations are fundamentally intertwined with internal relational structures; light and gravity are externally observable but internally determined in their causal coherence.^12 Quantum mechanics further demonstrates that what appears as external interaction—entanglement, superposition, probabilistic outcomes—is grounded in the internal relations of the system, where the potentialities of components exist prior to their manifestation as discrete events.^13

Thus, the internal relations of thought, mathematics, logic, and physics provide the framework from which external phenomena emerge. Identity, difference, space, time, and causal interactions are all instantiated outcomes of more fundamental, self-determining relational processes. Logic, in this sense, is the activity of thought in the universe itself, producing structures that can be realized externally while remaining dependent on the internal coherence from which they arise.^14

Footnotes

1. Whitehead, Alfred North, Process and Reality, corrected edition, 1978.
2. Hegel, G. W. F., Phenomenology of Spirit, §262; Philosophy of Nature, 197–203.
3. Peirce, Charles S., Collected Papers, Vol. 1, “Philosophy and the Conduct of Life,” 36.
4. Aristotle, Metaphysics VI.1018a; On Interpretation, 9–10, Edghill translation.
5. Hegel, Philosophy of Nature, 201; Whitehead, Process and Reality, Ch. II.
6. Whitehead, Process and Reality, Ch. I–II; Hegel, Logic, §§33–35.
7. Peirce, Collected Papers, Vol. 1; Whitehead, Process and Reality, Ch. IV.
8. Peirce, CP 36; Hegel, Phenomenology of Spirit, §262.
9. Hegel, Philosophy of Nature, 201; Metaphysics VI.1018a.
10. Hegel, Philosophy of Nature, 197; Metaphysics VI.1018a.
11. Hegel, Metaphysics V.5.7; Philosophy of Nature, 202; Aristotle, On Interpretation 9–10.
12. Newton, Isaac, Principia Mathematica, 1687; Einstein, Albert, Relativity: The Special and the General Theory, 1920.
13. Schrödinger, Erwin, What is Life?; Bohm, David, Wholeness and the Implicate Order, 1980.
14. Whitehead, Process and Reality, Ch. II–IV; Peirce, Collected Papers, Vol. 1.