Review | A calculus for self-reference

Introduction

Self-reference is one of those motifs that never disappears: the axioms hiding inside the explanation, the brain writing its own theory, the observer folded into what is observed, the cell “computing its own computer,” and the ouroboros—snake and eater, tail and mouth—closing a circle that somehow generates novelty instead of mere repetition. Francisco Varela opens his paper by insisting that these are not curiosities but facts “always before our eyes,” especially in living systems where self-production is not an exception but the very style of organization. Self-reference, in other words, is not a decorative metaphor; it is an organizing principle that shows up in biology, cognition, and the formal sciences alike.

Yet self-reference has historically been treated as an anomaly because of what it does to language. Antinomies are expected when language turns on itself. The Epimenides paradox (Epimenides the Cretan says, “that all the Cretans are liars”) compresses into Cantor-style diagonal constructions, which then become the engine behind formal limitations in logic and mathematics. What makes these situations hard is always the same, the distinction between actor and acted-upon, operator and operand, collapses. My first guess is that this collapse in a way that resonates strongly with Pattee’s epistemic cut: there is an irreducible duality between the act of expression and the content addressed; self-reference blends these two “immiscible” components of cognition and thereby appears “peculiar” to our knowledge.

Because of this discomfort, the modern default has been to avoid inconsistency at all costs, accept incompleteness rather than contradictions. We build our infrastructures in systems that forbid the paradox, even though discourse itself is self-referential and even though many real systems (especially living ones) are organized by circular causation. Spencer-Brown’s Laws of Form is pivotal precisely because it digs “deeper than truth” into the act of distinction, and because it identifies self-reference with re-entry (an expression re-entering its own indicative space). But Spencer-Brown also admitted that he had only indicated a direction; re-entry breaks the comfortable connection not only to arithmetic but also to algebra; once higher-degree equations are allowed, classical transformations begin to generate contradictions. How far can we go in accepting contradiction instead of avoiding it? In today’s review, I will examine the calculus for self-reference that Francisco Varela developed in the 1970s to overcome precisely this gap: to construct a calculus that contains self-reference rather than quarantining it.

Formalizing self-reference

Varela’s motivation begins with a precise technical diagnosis. Spencer-Brown’s calculus of indications works cleanly so long as one stays within its primary arithmetic and primary algebra. Trouble appears when we allow higher-degree equations, basically when an expression re-enters its own space. Spencer-Brown claimed that the primary algebraic initials hold “for all equations whatever their degree,” but Varela exhibits a counterexample: the moment you permit re-entry, a transformation that is valid in the primary algebra can become invalid, producing a contradiction when substituted into a second-degree equation. The takeaway is not that re-entry is “bad,” but that the existing algebra is not rooted deeply enough to accommodate it. So the problem is structural, the calculus needs an additional value if it is to treat re-entry as a first-class citizen rather than an exotic exception.

The key step is to take Spencer-Brown’s ‘imaginary’ value seriously. Varela introduces a third state in the form—distinct from the marked and unmarked states—arising “autonomously by self-indication.” He denotes this autonomous state with a special mark (the self-cross) and proposes to treat it as a bona fide value in an extended arithmetic. For readability, one can represent the three values as unmarked (0), marked (1), and autonomous (a). The point is not merely to add a symbol, but to add a domain; a state governed by laws that cannot be reduced to the dual laws of marked/unmarked. This single move defines Varela’s principal intention: build an extended calculus of indications fully compatible with higher-degree equations and thus capable of handling self-referential forms at a deep enough level.

Varela then becomes a mathematician in the strict sense. He lays down context, definitions, and notations, and he introduces four initials (axioms) meant to determine the Extended Calculus of Indications—dominance, order, constancy, and number. In the paper’s own notation, the spirit of these initials is that (i) the marked value dominates in mixed contexts, (ii) nesting/ordering rules simplify expressions, (iii) the autonomous value is invariant under the “negation-like” operation induced by the calculus, and (iv) repeated autonomous marks condense (capturing self-indication as idempotent). While the technical details live in Varela’s proofs, the conceptual result is crisp: every finite expression simplifies to a simple expression (a marker), and this simplification is unique. Those two theorems do the heavy lifting. Together, they secure consistency; the three values are “not confused,” and identity/value/consequence rules follow as evident corollaries.

With the arithmetic stabilized, Varela constructs the Extended Algebra. He takes three results proven earlier as new initials—occultation, transposition, and autonomy—and derives a family of propositions yielding a richer algebraic calculus than Spencer-Brown’s primary one. Importantly, Varela notes that some identities valid in the primary algebra become invalid in the extended setting; that is not a defect but a signature of the new domain’s structure. He also observes that a subset of the derived consequences is compatible with three-valued Boolean arithmetic, foreshadowing the later logical interpretation: once you introduce a third value, you necessarily loosen classical bivalence. The crucial test, however, is whether the algebra matches the arithmetic in expressive power. Varela proves completeness; every valid arithmetic form is demonstrable in the algebra. Philosophically, this is the paper’s most provocative formal stance. Instead of responding to Gödelian constraints by retreating into a purely incomplete-but-consistent world, Varela builds a system that remains consistent while being algebraically complete for the extended arithmetic, precisely by admitting the autonomous state.

Only then does he explicitly return to re-entry. Allow any expression to re-enter its own indicative space at odd or even depth and you have introduced indeterminacy: the value of a re-entering functional form cannot be obtained merely by fixing variable values. Varela classifies this indeterminacy by introducing the notion of degree (how deep re-entry occurs), together with two rules of lexicographical consistency that let one rewrite re-entries unambiguously such that the payoff is structural. Any higher-degree expression can be reduced to degree at most three (with auxiliary variables recording what is re-entering), and—most strikingly—there are only six basic indeterminate forms generated by re-entry. Thus self-reference does not explode into unclassifiable chaos; it collapses into a small, manageable family of canonical behaviors.

From self-reference to organizational closure

The third part of Varela’s paper begins with a philosophical thesis that is easy to miss if one reads only the theorems. The world of indication contains not just two obvious domains (marked and void) but a third, less obvious domain of autonomous self-reference that cannot be reduced to the laws of the dual domains. If we refuse to include this third domain explicitly, we end up confronting it only as paradox and pathology. But if we include it, the paradox becomes a phenomenon with its own lawful behavior. This is why Varela claims that what looks contradictory from the standpoint of the primary arithmetic becomes constitutive once autonomy is admitted as a value.

In this light, the self-cross is not a trick; it is the paradigmatic self-referential form. In the logical reading of the calculus, where the mark behaves like negation, a self-cross becomes equivalent to its own negation, an antinomic form. Instead of treating that as a reason to banish self-reference, Varela treats it as a reason to name a new state: autonomy is precisely that which is “unmodified by indication” (or by negation, in the logical interpretation). The calculus then yields an unexpected service. It can decide when an expression that appears self-referring actually is; not all re-entering expressions take an autonomous value (some reduce to mark or blank) so the calculus distinguishes true self-reference from merely complicated syntax.

The paper’s most suggestive move, in my opinion, is to connect autonomy to time. Spencer-Brown interpreted re-entry as an imaginary value seen “in time” as an alternation of marked and unmarked states. Varela’s extension no longer needs that interpretation for consistency (autonomy is a value of the arithmetic itself) but Varela insists that we cannot eliminate time so easily. The double nature of self-reference—operator and operand interlocked—cannot be conceived outside a temporal process in which two states alternate. “Our cognition cannot hold both ends of a closing circle simultaneously; it must travel through the circle ceaselessly” (p. 20). Hence a peculiar equivalence. Self-reference cannot be conceived outside time, and time “comes in whenever self-reference is allowed” (p. 20). Re-entry also generates an “excursion to infinity” (infinite recursion), linking infinity and time in the behavior of closed self-referential systems: productions of productions, descriptions of descriptions, and so on. From this perspective, a self-cross can be viewed as oscillation; other re-entering expressions become modulations of a basic frequency, a topic Varela explicitly flags as open for future investigation.

Once time is in play, Varela’s calculus begins to look less like a curiosity of formal logic and more like a template for biological organization. In living systems, the producer-produced duality is ubiquitous: a cell produces the constraints that produce the cell; an organism maintains the boundary that maintains the organism; symbols and dynamics co-define each other in what Pattee later called semantic closure and what Rosen formalized as closure to efficient causation. Varela’s language of autonomy, an emergent third state arising from the mutual negation/closure of two domains, maps naturally onto these closure notions. It also resonates with Rosen’s modeling relation, where the describer and the described are not separable without remainder. Self-reference forces a relational, time-unfolded picture in which model and modeled co-determine the space of valid descriptions. In this sense, Varela’s claim that self-reference brings time with it reads less like metaphor and more like a deep constraint on any theory that aims to treat living organization as intrinsically self-producing.

Conclusion

Varela’s achievement is twofold. Mathematically, he takes Spencer-Brown’s hint about re-entry and turns it into a fully constructed theory: a consistent extended arithmetic with a genuine third value (autonomy), an extended algebra proven complete with respect to that arithmetic, and a classification theorem showing that re-entry yields only a finite family of indeterminacies. Philosophically, he refuses the traditional reflex to treat self-reference as merely anomalous. The antinomy is not a defect to be eliminated but a domain to be recognized—autonomous not pathological. That reversal is precisely what makes the calculus relevant beyond logic; it offers a disciplined way to confront circularity wherever it appears.

The interpretive arc, however, is the paper’s lasting provocation: autonomy, re-entry, and time are mutually implicated. If we read autonomy as oscillation, self-reference becomes intrinsically temporal; if we read time as what is required to “see” autonomy, then time emerges as the price of closing circles in cognition and organization, explaining why life is capable of conceiving time! This is not merely a philosophical flourish; it suggests a concrete research program in which self-referential dynamics might be characterized by frequencies (or timescales) and higher-degree re-entries by modulations, an idea Varela highlights but leaves open. It is also the point where the calculus touches living systems most directly, because organizational closure is always conceived in time—maintenance, repair, reproduction, and interpretation are processes, not static relations. Thus, we need a process ontology (instead of a substance ontology) to describe life!

Open gaps remain, and they are productive. Varela shows that admitting a third value forces abandonment of the law of excluded middle; he notes that consistent logics without tertium non datur can still reconstruct most classical mathematics, but the broader epistemological consequences deserve more development. Likewise, the bridge from formal autonomy to embodied autonomy (cells, nervous systems, observers) is sketched more than constructed. The calculus supplies a deep grammar of self-reference, but applying it to concrete systems requires an additional layer connecting formal re-entry to physical constraints and timescales. Still, that is exactly why this paper remains a landmark and I wanted to share it with all of you. Rather than banishing self-reference to protect our formalisms, Varela enlarges the formal domain until self-reference has a lawful place—and in doing so, he gives theoretical biology and epistemology a rare gift: a calculus where circularity is no longer a scandal, but a generative principle, just as life does.