Why Theories Break Down: Singularities, Idealisations, and the Call for Quantum Gravity

Physics has achieved remarkable success in modelling the universe, yet there are well-known thresholds where our theories cease to function. Black hole singularities, the Big Bang, and the infinite self-energy of electrons in quantum electrodynamics (QED) are all examples of this breakdown. What do these apparent “failures” tell us — about our models, about nature, and about the need for quantum gravity?

When Mathematics Meets Its Limits

At first glance, infinities or divergences might seem like a failure of mathematics. But in all of these cases, the mathematics itself is perfectly coherent: differential geometry handles curvature singularities, integrals in quantum field theory are well-defined, and limits can be treated rigorously. The problem does not lie in the consistency of the equations.

Rather, the issue arises when the assumptions embedded in our models — the idealisations we impose — push beyond what is physically meaningful. These idealisations allow us to simplify complex systems, but when extended to extreme scales, they generate results that are physically nonsensical: infinite densities, infinite curvature, or infinite energy.

---

Three Structural Factors Behind Breakdowns

From our analysis, three recurring structural assumptions are at the heart of these divergences:

1. The Use of Idealisations:

   • Models often treat matter or spacetime as perfectly smooth, homogeneous, or continuous.

   • Example: FLRW cosmology models the universe as a uniform fluid. Near t → 0, this idealisation leads to infinite energy density.

   • Relationally, the cut imposed by this assumption is “too sharp”: it attempts to resolve distinctions that the system’s potential cannot support.

2. Point or Dimensionless Particles:

   • Quantum field theories treat electrons, quarks, and other particles as dimensionless points.

   • Continuous fields interacting with these points produce infinities, such as the electron’s self-energy in QED.

   • Relationally, this is a cut that isolates an object from its relational context, over-localising it in a way the system cannot sustain.

3. Ignoring Minimal Physical Scales (Planck Length and Time):

   • Classical spacetime is treated as divisible without limit, ignoring the Planck scale.

   • Singularities in black holes or the Big Bang appear because equations are extrapolated to scales where spacetime itself likely loses operational meaning.

   • Relationally, the cut extends beyond the relational potential of the system.

---

Relational Ontology and the Nature of Singularities

Relational ontology reframes this problem elegantly:

• Reality is not a collection of pre-existing “things” with absolute properties, but a network of relations and potentialities.

• A singularity is not a point of infinite density; it is a region where our current construal (our chosen cut + idealisation) exceeds the system’s relational potential.

• Infinity, then, is a diagnostic marker: mathematics is signalling that the distinctions we are attempting to impose are too fine to be physically meaningful.

In this view:

• Black hole singularities, Big Bang singularities, and QED divergences are flags of cut misalignment, not literal infinities.

• The mathematics remains sound, the quantification remains systematic, but the interpretation breaks down because our assumptions no longer match the potential of reality.

---

Quantum Gravity as a Consequence of Idealisation

From this perspective, the persistent call for a theory of quantum gravity is not solely about “fixing gravity.” It is also a consequence of the idealisations that produce singularities:

• Infinite divisibility of spacetime in classical GR → singularities.

• Point particles in QED → divergences.

• Quantum gravity proposals (loop quantum gravity, string theory, causal sets) can be read as recalibrations of the cut, bringing the construal back in alignment with relational potential.

  • Discrete spacetime, extended particles, or minimal length scales prevent infinities from arising.

  • Infinities disappear not because the mathematics changes, but because the cut now respects the system’s potential.

---

Takeaways for Understanding Physical Theories

1. Infinities signal domain boundaries, not physical entities.

2. Singularities highlight the limits of our idealisations.

3. Quantum gravity emerges naturally as a way to realign the cut with reality, rather than as an arbitrary “fix” for broken mathematics.

By focusing on the structural assumptions — idealisations, point particles, and Planck-scale ignorance — we gain a clearer understanding of why our theories break down and how new approaches can restore coherence.

In short: infinities are messages, not mistakes — and the path forward lies in aligning our cuts with the relational potential of the systems we model.