Natural vs. Representational Time

• Natural and representational time are distinct: natural time embodies the intrinsic, irreversible progression of physical processes, while representational time is the observer-defined coordinate system.
• In physics, the Hamiltonian dynamics illustrate timelessness, where clock time emerges from cyclic processes, and relativity differentiates between proper (natural) and coordinate (representational) time.
• Across disciplines, from biological rhythms to neural coding, understanding these two aspects of time provides practical insights into modeling complex systems and designing robust time-measuring mechanisms.

Natural and representational time are foundational but distinct concepts invoked across physics, philosophy, the biological sciences, and cognitive science. “Natural time” refers to temporal structures and processes intrinsic to physical systems, often associated with objective succession, dynamic evolution, or irreversibility. “Representational time” denotes temporal constructs, coordinates, or mechanisms introduced by observers for measurement, conceptualization, or modeling—typically exemplified by scientific timekeeping, formal mathematical time parameters, or subjective experience. This multidimensional distinction is deeply articulated across technical literature, ranging from timeless Hamiltonian formalisms and metrological clocks (0907.1707), dual-parameter models in biology (Bailly et al., 2010), and metaphysical and mathematical debates within relativity, quantum theory, and philosophy (Evans, 2010, Walton et al., 2020, Santo et al., 9 Apr 2024).

1. Timelessness, Hamiltonian Dynamics, and the Emergence of Clock Time

Modern analytical mechanics and field theory demonstrate that natural laws may be formulated independently of an explicit temporal parameter.

• Timeless Hamiltonian Framework: The action can be cast in the Maupertuis form, $A = \int p_i dq_i$, with stationarity condition $\delta A = 0$ and no explicit dependence on time. The evolution of a closed system is fully specified by relations among generalized coordinates and momenta $(q_i, p_i)$; the Hamiltonian $H(p,q)$’s invariance along trajectories underlines the core “timelessness” of fundamental dynamics (0907.1707).
• Parameter and Clock Time: Although the Hamiltonian structure defines a unique parameter $\varpi$ that labels evolution (via conserved energy), this parameter is not an observable until mapped to a measurable physical process. Metrological “clock time” $T$ emerges by associating the evolution parameter $\varpi$ with cycles of a physical subsystem exhibiting cyclicity (not necessarily periodicity) in phase space.
• Cyclicity and Stability: Real clocks, identified with cyclic subsystems, must satisfy a stability constraint (e.g., variance $\sigma = E^2 T(S_1)$) to ensure robust mapping between “parameter time” and experimental metric time. Thus, the observable passage of time in macroscopic experiments is always a coarse-grained, stable approximation built upon the underlying timeless, relational dynamics.

2. Dual and Multidimensional Temporal Structures in Physical and Biological Theory

Natural and representational times frequently bifurcate, or even multiply, in theoretical frameworks.

• Relativity Theory: Physical time in special and general relativity exhibits dual representations:
  • Coordinate time (frame-dependent, operational, representational) is set by clock synchronization in chosen inertial frames.
  • Proper time (frame-invariant, natural) is the arc length along timelike worldlines, given by $$T = \int_{s_1}^{s_2} \sqrt{\eta_{\mu\nu} dx^\mu dx^\nu}$$. Metaphysical tension arises between the static “block universe” (where past, present, and future are ontologically on par) and the locally dynamic progression manifest in proper time readings (Evans, 2010, Valente, 2013).
• Biological Systems: Linear, one-dimensional time is inadequate for biological phenomena such as circadian and metabolic rhythms. A two-dimensional manifold $M = \mathbb{R} \times S^1$ is constructed, with physical time $t$ and internal (cyclical) time coordinate $\theta$. The geometry supports representation and visualization of both irreversible progression (aging) and endogenous cycles (e.g., heartbeats) and explains interspecific scaling laws in allometry (Bailly et al., 2010).
• Cognitive and Relational Models: Some proposals separate sequential time (discrete, event-ordered, cognitive or physical updating parameter $n$ or $\sigma$) from relational time (measurable durations or intervals $t$ between events), aligning physical modeling with perceptual and epistemic structures (Östborn, 3 Nov 2024).

3. Stochastic, Computational, and Measurement-Theoretic Approaches to Time

Probabilistic and computational models provide frameworks that articulate the role of time at micro and macro scales, as well as its representation in scientific modeling.

• Natural Local Time in Stochastic Processes: Standard mathematical local time in stochastic process theory quantifies occupation in terms of spatial units, not clock time. “Natural local time” is defined to directly measure actual residence time per spatial neighborhood, integrating with respect to physical clock time (e.g., $\tau\ell_t^X(a)$, $T/L$). This measure is crucial for connecting microscopic dynamics (e.g., Brownian motion, skew diffusion at interfaces) to empirical residences and breakthroughs (Appuhamillage et al., 2012).
• Computational Frameworks and Irreversibility: Most mathematical models treat time as a parameter in an analytic, spatialized, reversible fashion, unable to account for emergence or novelty. A computational approach advocates modeling phenomena via algorithms that run in “natural time”—here, the sequential, irreversible step-by-step execution of computation embodies processual and creative aspects of natural time, thereby capturing biological evolution and creative emergence beyond the reach of time-symmetric formalism (White et al., 2019).

4. Philosophical, Metaphysical, and Logical Dimensions: Creative vs. Geometric Time

Advanced discussions in philosophy and foundations of mathematics/logic distinguish between multiple notions of time due to differing requirements of determinism, indeterminism, and epistemic access.

• Geometric Time: The predominant time in deterministic physics and classical mathematics; serves as a parameter labeling points/events in a block universe ($t$ in $M_4$), compatible with B-theory (tenseless, static ontology).
• Creative Time: Emerges when fundamentally indeterministic transitions occur—e.g., quantum events, chaotic systems under ontic indeterminacy—corresponding to the actualization of new information. The “present” is identified as the boundary where the future becomes fixed as the past. Constructive or intuitionistic mathematics, as well as multivalued tensed logic, can express this process (truth values “become” assigned as events are actualized) (Santo et al., 9 Apr 2024).
• Temporal Naturalism: Contrasts timeless (“block universe”) naturalism—where all events are equally real and dynamical passage is illusory—with temporal naturalism, which posits the real succession of present moments. Temporal naturalism argues for evolving laws, cosmological directionality, and an ontologically intrinsic “now,” underpinning not only physical processes but also qualitative experience (qualia) (Smolin, 2013).

5. Human Experience, Perceptual and Representational Time

Explanations of time in scientific and philosophical contexts must often account for both physical time and its perceptual or subjective correlates.

• Physical vs. Human Time: The evolving block universe (EBU) model posits that spacetime grows dynamically—the past is fixed, the future undetermined—providing a physical arrow of time. Simultaneously, the neural system (mind/brain) acts as an “imperfect clock” that coarsely samples the physical passage of time, leading to contextual and subject-dependent temporal experience. Quantum wave function collapse is identified as the process where information becomes fixed in the cosmos, irreversibly marking the passage of time (Ellis, 2022).
• Symbolic and Interpretive Frameworks: Time serves as both an artificial measure (symbolic, constructed) and a unifying bridge across scientific and humanistic disciplines. The “Universal Clock” metaphor is invoked to connect scientific measurement (coordinated, relativistic time) with the time of human existence (subjective temporality, cultural rhythms). Each field’s conception is partial, necessitating integrative analytical approaches (Kulikov, 2016).
• Dual Nature as Cognitive and Abstract Construct: Time is an abstraction that organizes the fundamentally ceaseless change of physical reality. While measurement and scientific discourse require formal time parameters (abstract bank, coordinate systems), the perceptual flow of “now” and the representation of memory and anticipation remain tied to cognitive and cultural frameworks (Radovan, 2014, Kulikov, 2014, Radovan, 2015).

6. Implications in Neural Systems and Artificial Intelligence

• Neural Representations and Temporal Codes: Neural population codes reflect both natural and representational aspects of time. Models trained for robustness (adversarial training, random smoothing) exhibit “representational straightening”: feature space trajectories of natural movies become more linear, simulating how biological vision organizes time for predictability and interpolation. This can be quantified using curvature metrics, and has direct correspondence to temporal coding observed in primate V1 (Toosi et al., 2023). Flexible, adaptive representations (as seen in neural encoding of fast and slow natural stimulus features) suggest multi-timescale compensation mechanisms evolved to support processing in a variable natural environment (Wang et al., 2023).

7. Synthesis: Integrative Perspectives and Future Directions

No single representation of time suffices to account for its multiplicity in natural processes, measurement, mathematical modeling, logic, and conscious experience. Distinctions between natural time (as irreducible succession, creative emergence, and causal efficacy) and representational time (as formal parameter, observable, or cognitive construct) are necessary for progress in physics, biology, philosophy, and neuroscience.

Domain	Natural Time	Representational Time
Hamiltonian Dynamics	Parameter from invariants, invariance under reparam.	Clock time from cyclic subsystems, counting cycles
Relativity	Proper time (worldline invariant)	Coordinate time (frame-dependent)
Stochastic Processes	Actual time spent (natural local time)	Quadratic variation/local occupation (mathematical local time)
Philosophy/Logic	Dynamic passage, creative time, A-theory	Geometric/static time, B-theory
Biological Systems	Cyclicity, scaling laws, global rhythms	Metric time, cognitive/subjective representations
Neural Coding/AI	Population code trajectories (natural sequences)	Embedding space, representational straightening

Recognition and rigorous modeling of both aspects clarify the logical foundations of physical theory, the design of biological and artificial timekeeping mechanisms, and the coherence of scientific explanations that span the micro to macro, deterministic to indeterministic, and objective to subjective domains.