Bergson: Time, Duration & ML

• Bergson is a philosophical concept distinguishing quantified clock time from lived duration, emphasizing continuity through retention and protention.
• Its legacy influences debates in theoretical physics and mathematical biology, reconciling intuitive universal time with modern empirical findings.
• The open source 'Bergson' library applies state-of-the-art data attribution methods (MAGIC, SOURCE, TrackStar) to scalable machine learning systems.

Searching arXiv for the cited Bergson-related papers and recent context. Bergson denotes, in the cited literature, both philosopher Henri Bergson and a continuing conceptual vocabulary centered on duration, simultaneity, retention, and protention; in a distinct contemporary usage, it is also the title of an open source library for data attribution. Bergson’s philosophical position distinguishes quantified time from lived duration, later mathematical biology gives a minimal quantitative realization of an “extended present” through retention and protention, and contemporary machine learning uses “Bergson” for a scalable attribution framework implementing MAGIC, SOURCE, and TrackStar (Unnikrishnan, 2020, Longo et al., 2010, Quirke et al., 10 Jun 2026).

1. Duration, quantified time, and universal simultaneity

Bergson distinguishes “time” as a series of quantified instants from “durée,” which is continuous, qualitative, and directly experienced (Unnikrishnan, 2020). In this formulation, quantified time is the time of the physicist or mathematician, whereas durée is lived duration. Common intuition assumes a single universal time flowing equally for all beings and things, and an absolute cut through the stream of events marking simultaneity. For Bergson, this universal time need not be postulated metaphysically; it follows from the direct experience of durée.

The significance of this distinction is methodological as well as philosophical. Bergson’s account does not deny measurement, but it resists reducing temporality to an ordered set of instants. This suggests a persistent tension between operationalized clock time and the continuity of experienced temporality. In the later literature considered here, that tension becomes the point of contact between phenomenology, mathematical biology, and debates in theoretical physics.

2. Einstein’s 1922 Paris visit and Bergson’s critique of relativity

In April 1922, Einstein, invited by Paul Langevin, delivered several sessions at the Collège de France and a special session at the Sorbonne on 6 April 1922 organized by the Société Française de Philosophie; Bergson was asked, unplanned, to comment on the notions of time and simultaneity in Special Relativity (Unnikrishnan, 2020). According to the reconstruction in that paper, Bergson began by praising Einstein as providing “new physics and a new way of thinking,” then immediately emphasized that common sense assumes a unique time and asked whether this really conflicts with relativity.

The paper distinguishes Lorentz’s ether theory and Einstein’s 1905 theory. In Lorentz’s ether theory, one has an undetectable ether as a universal rest frame, moving rods contract, and moving clocks slow by real physical effects determined by their velocity through the ether. Poincaré’s “local time” is written as

$t' = t - \frac{v x}{c^2}.$

In Einstein’s theory, by contrast, the assumptions are the Principle of Relativity and the invariance of the one-way speed of light, yielding

$$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$

with

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$

Within this framework, Bergson’s critique was directed at what the paper calls a “multitude of times.” In the reconstruction offered there, he argued that reciprocal time dilation in Special Relativity makes it difficult to regard all frame-dependent times as equally real, and he hoped for a reinterpretation restoring a single universal real time. The same paper presents his criticism of Einstein’s embankment-and-train example by arguing that if one reverses the roles of train and embankment, one obtains the opposite result, and therefore simultaneity must agree in both frames, given an invariant light speed. It also reconstructs Bergson’s concerns about the twin paradox by emphasizing that in pure Special Relativity no inertial reason exists to privilege one clock over the other.

These arguments remain controversial in the history and philosophy of physics. What is clear from the cited account is that Bergson’s intervention was not a generic rejection of physics, but an attempt to determine whether the mathematical plurality of times in relativity could be reconciled with universal duration and simultaneity.

3. Retention, protention, and the “extended present”

Longo and Montévil introduce an abstract mathematical frame for biological time that explicitly invokes the phenomenological idea, associated with Husserl and Bergson, that the present is not an instantaneous point but extended (Longo et al., 2010). In their formulation, the interval $[t_0,t_1]$ is precisely the mathematical “extended present.”

Retention, understood as memory, is modeled as an exponential relaxation:

$$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$

Here $a_R$ is an amplitude constant and $\tau_R$ is the characteristic retention time. When $\tau_R \to 0$, the trace vanishes immediately. Virtual protention, understood as a purely prospective anticipation of a future event at $t_1 \ge t$, is modeled symmetrically as

$$V_P(t,t_1)=a_P \cdot \exp\!\left[\frac{t-t_1}{\tau_P}\right].$$

When $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$0, $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$1, corresponding to no capacity to anticipate.

A central postulate is that without any retention there can be no protention. The authors therefore define actual protention as the product of retention and virtual protention:

$$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$2

As a function of the present $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$3, $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$4 first rises because the future event $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$5 comes closer and eventually peaks at $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$6. One checks that $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$7, and when $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$8, $$x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right),$$9.

This formalism gives a minimal quantitative realization of Bergson’s claim that the present is always imbued with the immediate past and directed toward the immediate future. The resulting “extended present” is not merely descriptive: at each intermediate $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$0, the simultaneous values of $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$1 and $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$2 quantify how much of the past event remains in current cognition and how strongly the system is pre-tuned to the forthcoming event.

4. Global protention and biological inertia

The same framework defines a “global protention” by integrating actual protention over the entire extended present:

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$3

where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$4 (Longo et al., 2010). As a function of the delay $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$5, $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$6 has a single maximum at

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$7

The paper notes that in experiments on simple organisms, including amoebae, one indeed finds an optimal spacing between a conditioning stimulus at $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$8 and a test at $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$9 for maximal anticipatory effect.

Longo and Montévil then decompose protention into “physical inertia” and “biological inertia.” By algebraic rewriting,

$[t_0,t_1]$0

The first bracket,

$[t_0,t_1]$1

depends only on $[t_0,t_1]$2 and is formally the residual of a passive relaxation at $[t_0,t_1]$3. The second bracket,

$[t_0,t_1]$4

is the “biological inertia.” It depends only on constants and on the endpoints $[t_0,t_1]$5, not on $[t_0,t_1]$6, and measures the capacity of the system to “carry over” its protentional readiness from the moment of retention to the anticipated event.

The ratio

$[t_0,t_1]$7

controls the qualitative behavior. If $[t_0,t_1]$8, then $[t_0,t_1]$9 and $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$0 becomes almost $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$1-independent over $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$2; if $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$3, then $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$4 decays very fast with increasing delay and $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$5 is sharply peaked near $$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$6. The paper explicitly presents this as a way to make biologically explicit what Bergson described intuitively: the present is never empty but is always suffused with the just-passed and pulled toward the just-coming.

5. Bergson, universal time, and later physical reinterpretations

A further line of interpretation reassesses Bergson’s program in light of later developments in cosmological physics and precision timekeeping (Unnikrishnan, 2020). In that account, proper time is written as

$$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$7

and finite time dilation in Special Relativity is expressed as

$$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$8

The paper emphasizes Lorentz-transformation reciprocity: in Special Relativity each frame sees the other’s clocks as slow, and no asymmetric physical cause exists inside pure Special Relativity to favor one frame’s clocks.

The same paper then surveys later empirical and theoretical developments. It cites Rossi and Hall’s 1941 observations of cosmic-ray muons, the Hafele–Keating around-the-world atomic clock experiment, GPS range computation with the first-order Sagnac correction

$$R(t_0,t)=a_R \cdot \exp\!\left[\frac{t_0-t}{\tau_R}\right].$$9

and Sagnac’s 1913 interferometer with

$a_R$0

It also presents “Cosmic Relativity,” in which the comoving frame of the cosmic matter–radiation distribution functions as a privileged master frame and the cosmic microwave background rest frame is detectable through dipole anisotropy.

In that narrative, these developments are interpreted as restoring a universal time and absolute simultaneity in harmony with Bergson’s original intuition. The paper explicitly calls this the “surprising completion” of Bergson’s program on universal time, duration, and simultaneity. A plausible implication is that Bergson’s role in twentieth- and twenty-first-century thought is not limited to historical phenomenology; his concepts continue to function as test cases for how temporal experience, clock comparison, and cosmological structure are related.

6. “Bergson” in contemporary machine learning: an open source data attribution library

In contemporary interpretability research, “Bergson” is also the title of an open source library for data attribution that aims to enable faster progress in the field by providing techniques that scale to very LLMs and pre-training datasets (Quirke et al., 10 Jun 2026). The library natively supports on-disk gradient stores and multi-node distributed training, and introduces open-source implementations of three leading data attribution methods: MAGIC, SOURCE, and TrackStar.

Its architecture is organized as a composable pipeline with six primary stages: gradient collection; preprocessing; projection and compression; on-disk gradient store and indexing; attribution scoring; and validation and retraining. The design points include on-disk gradient stores that allow arbitrarily large datasets to be indexed once and queried many times, FAISS integration for sublinear-time approximate nearest-neighbor search for top-$a_R$1 dot-products, support for FSDP and DDP, a YAML-driven CLI and Python API, and native support for HuggingFace Transformers + Datasets, PEFT/LoRA adapters, and GRPO policy-gradient losses.

The implemented methods span exact unrolled differentiation, approximate unrolled differentiation, and compressed influence functions. For MAGIC, the training loss at step $a_R$2 is

$a_R$3

with gradient descent

$a_R$4

For TrackStar, classic influence functions are written as

$a_R$5

and the attribution score on compressed gradients is

$a_R$6

The scaling profile reported in the paper is explicitly large-model oriented. Bergson has been tested on models from Pythia-160M up to 12B in single-node FSDP and across two-node $a_R$7 clusters. On one A100, for Pythia-160M the end-to-end build+query for one MAGIC attribution scales nearly linearly in training tokens from $a_R$8 to $a_R$9 and remains within an order of magnitude of a plain training run of equal size. Gradient-collection wall-clock scales roughly as $\tau_R$0, enabling attribution on 12B-parameter models over 10M tokens in under an hour on $\tau_R$1s. The library also provides YAML-declarative pipelines, a deterministic trainer, a healthcheck tool, auto batch-sizing, evaluation-set templating, PEFT/LoRA integration, and GRPO-based attributions.

This contemporary usage is conceptually distinct from Henri Bergson’s philosophy, but it shows that “Bergson” now names both a major philosophical problem-space around temporality and a concrete software artifact in interpretability research. That juxtaposition is terminological rather than doctrinal; nonetheless, it places Bergson within a research landscape that spans phenomenology, mathematical biology, theoretical physics, and large-scale machine learning.