Epistemic Horizons and the Foundations of Quantum Mechanics

In-principle restrictions on the amount of information that can be gathered about a system have been proposed as a foundational principle in several recent reconstructions of the formalism of quantum mechanics. However, it seems unclear precisely why one should be thus restricted. We investigate the notion of paradoxical self-reference as a possible origin of such epistemic horizons by means of a fixed-point theorem in Cartesian closed categories due to F. W. Lawvere that illuminates and unifies the different perspectives on self-reference.

💡 Research Summary

The paper investigates the notion of “epistemic horizons” – fundamental limits on the amount of information that can be obtained about a physical system – and proposes that these limits arise from paradoxical self‑reference. It begins by reviewing recent reconstructions of quantum theory that adopt two principles: (1) a finiteness assumption stating that there is a maximal amount of information that can be extracted from any system, and (2) an assumption of additional information asserting that, even when a system is maximally known, further information can always be acquired. At first glance these principles appear contradictory, but the author shows that they are compatible through Spekkens’ toy‑bit model. In this model a “toy bit” carries two classical bits, yet the knowledge‑balance principle permits knowledge of only one at a time, yielding six maximally known states and one completely mixed state. Complementary measurements (analogues of σ_z and σ_x) illustrate how acquiring new information necessarily disturbs previously known information, reproducing quantum‑like complementarity and demonstrating that new information can always be generated despite maximal prior knowledge.

The core of the argument is that paradoxical self‑reference creates epistemic horizons. The author invokes Lawvere’s fixed‑point theorem in Cartesian closed categories, which guarantees a fixed point for any self‑referential endofunction. Translating this into physics, if one could specify a complete state of a system that determines the outcomes of all future experiments, that specification would be a self‑referential function and thus must possess a fixed point. The existence of such a fixed point leads to a logical contradiction (a theorem that is provable if and only if it is not). Consequently, no complete specification can exist, and the second principle—always being able to acquire new information—is enforced.

The paper then links randomness to undecidability. Using algorithmic information theory, it discusses Chaitin’s Ω number, the halting probability of a universal Turing machine. Each bit of Ω encodes an undecidable proposition; Ω is both algorithmically random and uncomputable. This demonstrates that genuine quantum randomness cannot be reduced to ignorance or deterministic chaos; it is rooted in the existence of undecidable propositions. The author further presents an “algorithm plus infinite random bits” construction, showing that a non‑computable law can be decomposed into a deterministic part and a random part, mirroring the two‑tier dynamics of quantum theory (deterministic evolution of the wavefunction plus stochastic measurement outcomes).

Finally, the paper argues that the finiteness assumption implies a minimal phase‑space volume, which introduces a constant with the dimensions of action (ℏ). This minimal volume prevents arbitrarily precise localization in phase space, leading naturally to the Heisenberg uncertainty relations and to the deformation of the classical commutative algebra of observables into the non‑commutative Moyal algebra—precisely the algebra of operators on Hilbert space. Thus, epistemic horizons give rise to the core mathematical structure of quantum mechanics.

In summary, the author provides two complementary routes to quantum foundations: (1) Information finiteness → minimal phase‑space volume → ℏ and non‑commutativity; (2) Self‑reference → undecidability → perpetual generation of new information. By uniting epistemic‑horizon ideas with categorical fixed‑point theorems and algorithmic randomness, the paper offers a novel conceptual framework that explains complementarity, randomness, uncertainty, and non‑commutativity as inevitable consequences of fundamental limits on knowledge imposed by paradoxical self‑reference.