I put on the Peirce lens – the real, radical, synechistic one.
It sees the continuum as ontologically primary, points only as a posterior abstraction, and reality as a relational, mediating whole (Thirdness), permeated by vagueness (Firstness) and resistance (Secondness).
Then I slowly, carefully, and without haste walked through modern exact science: mathematics, logic, physics, quantum field theory, gravitation, complexity, information theory, category theory, topos theory—everything I know.
I took notes wherever something felt surprising—where science itself suddenly appeared coherent, almost obvious, or strangely incomplete.
Afterwards, I revisited all notes, examined them from every angle, checked for inconsistencies, hidden point-assumptions, implicit discretizations, and genuinely relational structures, and assembled them into a coherent picture.
The truly living, forward-moving mathematics of the last 50 years (topos theory, synthetic differential geometry, homotopy type theory, locale theory, non-Archimedean analysis) has effectively moved away from Cantorian point-set ontology. It operates with local structures, relations, gluing conditions, and nilpotent infinitesimals—exactly what Peirce would have demanded.
Surprise: Mathematics has not abandoned the continuum. It has merely reformulated it in a language Peirce would have intuitively understood (“from which points are obtained by abstraction”). The point is no longer a primitive object, but a derived—and often problematic—construct.
Almost all major open problems in physics lie exactly where the model is forced to treat something as a point or as a sharp discretization:
- Singularities in general relativity
- UV divergences and renormalization in QFT
- The measurement problem and collapse in quantum mechanics
- The emergence of time and the arrow of time
- The origin of physical constants (α, ħ, c, G)
- Why exactly three generations of fermions?
Under the Peirce lens, all these problems suddenly appear as symptoms of the same mistake: we are trying to describe something that is inherently not made of points using tools that presuppose points. Renormalization is not a technical fix—it is a desperate attempt to undo the abstraction after the continuum has already been cut apart.
Modern physics is already moving in Peirce’s direction—but incoherently and without naming it:
- Relational Quantum Mechanics, QBism, and categorical quantum physics explicitly emphasize relations over things
- Sheaf-theoretic and topos-based approaches to QFT exist
- Entropic gravity, emergent spacetime, holographic principles, and AdS/CFT attempt to derive spacetime from something more relational and continuous
- The renormalization group itself is a deeply Peircean concept: laws (habits) emerge and stabilize across scales through mediation
Yet all of these approaches remain fragmentary. They lack a single coherent, relational, point-free ontological foundation of the kind Peirce would have demanded.
If one keeps the lens on consistently, a clear, sobering, yet hopeful picture emerges:
- The true Peircean continuum is not directly computable, because every computation is an abstraction and thus a fragmentation
- But this insight is not the end of science—it is its deepest corrective
- Modern exact science is already beginning to move away from point-based ontology in small, incoherent steps—just without naming it properly or drawing the full philosophical consequences
- The most fruitful advances of the coming decades will occur exactly where scientists begin to treat relational, mediating, continuum-like structures as primitive, rather than as derived approximations
The “magic numbers” of the periodic table, the fine-structure constant, the three generations, the renormalization group, the emergence of time—all of these suddenly look, under this lens, like traces of a deeper relational dynamics, not arbitrary parameters.
We cannot “compute” the true continuum.
But we can learn to build our models in ways that abstract less violently.
That is not a small difference.
It is the transition from a physics of things to a physics of mediation.
Written with the Peirce lens on—after a long, honest walk through modern science.