Randomness, Determinism, and What Finite Observers Can Justifiably Infer.

The Evidence Boundary - Can an observer determine whether the events in its world are genuinely indeterministic, rather than the outputs of an unknown deterministic mechanism?

There is a summary version of this paper:

What Randomness Is, and What Finite Evidence Can Say: The Evidence Boundary

Can any observer, using finite observations and experiments, determine whether the events of its world are genuinely undetermined, rather than the output of some hidden deterministic mechanism?

T333TEduardo Bergel

Abstract

Can an observer determine whether the events in its world are genuinely indeterministic, rather than the outputs of an unknown deterministic mechanism? The strongest defensible answer is neither a simple yes nor a simple no. No finite record can deductively entail ontic indeterminism when the deterministic alternatives are left unrestricted: any finite transcript can be generated by a deterministic program, and a sufficiently unconstrained deterministic theory can be tailored to reproduce any finite experiment. Yet this logical fact does not make evidence powerless. Particular deterministic theories can be falsified; broad deterministic families can be excluded under explicit causal assumptions; quantum experiments can certify unpredictability relative to specified adversaries; and simplicity, likelihood, intervention structure, and explanatory integration can rationally favor one theory over another. The central distinction is therefore not merely between real and apparent randomness. It is between logical proof, model-relative certification, empirical evidence, and metaphysical fact. Exact empirical equivalence blocks discrimination within a stated experimental domain, but it does not turn rival theories into the same theory, nor does it show that there is no underlying fact. The proper conclusion is an epistemic boundary, not an ontological erasure.

Keywords: randomness; determinism; algorithmic information; Bell inequalities; pseudorandomness; underdetermination; causal inference; scientific evidence

1. The question has several different meanings

The word random is used for properties that can come apart.

A sequence may be statistically random-looking: its frequencies, runs, correlations, and other observable features resemble those expected from a chosen probability model. A sequence may be algorithmically random: its prefixes resist compression relative to a specified universal description language. An event may be unpredictable for an observer: the observer lacks enough information or computational power to forecast it. Finally, an event may be ontically indeterministic: the complete physical state and laws do not fix a unique outcome.

None of these properties entails all the others.

A deterministic process can be statistically impeccable and practically unpredictable. A cryptographically secure pseudorandom generator is designed to achieve exactly that relative to a class of feasible tests. Conversely, an indeterministic process can produce a conspicuous pattern. A genuinely chancy coin could land heads a thousand times. That outcome would be extraordinarily improbable under a fair-coin model, but its improbability would not retroactively make the underlying process deterministic.

Algorithmic randomness introduces another separation. An infinite sequence can pass every effective statistical test associated with a computable probability measure and still be considered as a fixed mathematical object. Martin-Löf’s framework makes “randomness of an individual sequence” mathematically precise by defining randomness relative to effective tests and a measure [1]. But this is a descriptive property of the sequence. It does not, by itself, identify the causal history by which the sequence entered the physical world.

The central question of this essay is therefore narrow:

From finite observations and interventions, what can an observer justifiably infer about whether the underlying physical process was ontically indeterministic?

The word justifiably matters. Scientific knowledge rarely consists of assumptions-free deduction. It normally consists of inference within a framework of causal, statistical, and methodological commitments. Any serious answer must distinguish the impossibility of absolute proof from the possibility of strong evidence.

2. What a finite record can never prove by itself

Let an observer possess a finite transcript

$h=(x_1,x_2,\ldots,x_n),$

where the entries may include measurement settings, outcomes, timestamps, and every other recorded variable. There is always a deterministic program that prints exactly this transcript and halts. There is also a deterministic infinite process whose first $n$ outputs equal the transcript and whose later outputs follow any chosen continuation.

This yields a minimal but rigorous proposition.

Proposition 1: finite-record compatibility

For every finite observational transcript $h$, there exists at least one deterministic mechanism compatible with $h$.

The proof is trivial: encode $h$ in a program and instruct the program to print it. The philosophical consequence is not trivial:

No finite transcript, considered without restrictions on admissible mechanisms, logically entails that its source was indeterministic.

This is the strongest general impossibility result available from the finitude of evidence alone. It is also much weaker than several conclusions often drawn from it.

First, compatibility is not explanatory adequacy. A program that contains the entire transcript may be almost as long as the transcript. It does not reveal a compact law, expose a mechanism, or successfully predict data that were not used in its construction. It is a deterministic completion of the evidence, but perhaps a terrible scientific theory.

Second, one matching transcript does not establish empirical equivalence. A theory must describe what would occur under alternative experiments, not merely reproduce the path that happened to be observed. A prerecorded answer sheet can match one examination. It does not thereby possess the same capacities as a system that answers every possible examination question according to a stable rule.

Third, the proposition concerns an unrestricted class of deterministic mechanisms. Science rarely compares an unrestricted class with a single rival. It compares specified models or constrained families: local hidden-variable theories, finite-state generators, Markov processes, collapse theories, pilot-wave theories, and so on. Once a deterministic family is constrained, data may exclude it.

The correct conclusion is thus precise:

Finite evidence cannot eliminate every logically possible deterministic explanation. It can eliminate many scientifically substantive deterministic explanations.

That distinction will govern everything that follows.

3. Compatibility is not equivalence

Suppose two theories, $T_1$ and $T_2$, are compared over a set of accessible experiments $\mathcal E$. For each experiment $e\in\mathcal E$, let

$P_{T_i}(o\mid e)$

be the probability that theory $T_i$ assigns to outcome $o$.

The theories are operationally equivalent on $\mathcal E$ when

$P_{T_1}(o\mid e)=P_{T_2}(o\mid e)$

for every accessible experiment and every possible outcome.

If this equality holds, no decision rule that receives only data from $\mathcal E$ can discriminate between the theories better than the prior information already allowed. In Bayesian language, the likelihood ratio is one for every possible dataset. If a hypothetical model label $H$ indicates which theory is true and the data distribution is identical conditional on either value of $H$, then the mutual information between $H$ and the data is zero.

This is a genuine information-theoretic barrier. But notice where its force comes from: operational equivalence is a premise, not a conclusion obtained merely because one finite record is shared.

Causal analysis sharpens the point. Two models can generate the same passive observational distribution and respond differently to interventions. Modern causal modeling therefore distinguishes observational equivalence from interventional equivalence: matching what is seen under ordinary conditions is weaker than matching the consequences of deliberate changes to the system [2]. A serious deterministic rival to an indeterministic theory must reproduce not only the actual transcript but the distributions generated under all relevant interventions.

This distinction exposes a common mistake in arguments based on a predetermined infinite sequence. A fixed sequence $s$ is a point in a space of possible sequences. A stochastic process is a probability measure $P$ over that space. If the fixed sequence is represented probabilistically, it corresponds to a point mass $\delta_s$, which assigns probability one to $s$ and zero to every other sequence. Unless the stochastic process is itself degenerate, $\delta_s\neq P$.

A sequence may be typical or Martin-Löf random with respect to $P$, but the sequence is not the measure. A realized history and a law assigning probabilities to possible histories are different mathematical objects.

A deterministic theory can nevertheless reproduce a stochastic theory’s observable probabilities by introducing hidden variables. Let $\lambda$ denote a hidden initial condition, let $\mu$ be a distribution or typicality measure over possible $\lambda$, and let outcomes be generated deterministically by

$o=F(\lambda,e).$

The observable probabilities are then the pushforward of $\mu$ through $F$:

$P(o\mid e)=\mu{\lambda:F(\lambda,e)=o}.$

With a suitable $F$ and $\mu$, this induced distribution may coincide with that of a stochastic theory. Bohmian mechanics provides a physically developed example of deterministic dynamics combined with a quantum-equilibrium distribution that recovers the statistical predictions of nonrelativistic quantum mechanics [3].

But the measure $\mu$ now carries essential explanatory weight. Is it objective probability, epistemic uncertainty, typicality, a lawlike equilibrium condition, or a distribution over an ensemble of possible worlds? Deterministic dynamics alone does not answer that question. The apparent removal of chance may simply relocate probability into the initial-condition measure and its interpretation.

Operational equivalence therefore licenses a restrained conclusion:

If deterministic and indeterministic theories make exactly the same predictions for every accessible intervention, the accessible evidence cannot decide between them.

It does not license the stronger claims that the theories are mathematically identical, that their causal structures are the same, or that no fact distinguishes them.

4. Why evidence survives the failure of proof

The impossibility of assumption-free proof does not abolish rational comparison. Scientific inference depends on more than bare logical compatibility.

For two specified theories, Bayes’ rule gives

$\frac{P(T_1\mid h)}{P(T_2\mid h)} =
\frac{P(h\mid T_1)}{P(h\mid T_2)}
\frac{P(T_1)}{P(T_2)}.$

Data change the relative credibility of the theories whenever their likelihoods differ. If their likelihoods are exactly equal, the data do not update their relative odds, although other evidence or prior commitments may still differ. For composite theory classes, the result also depends on how probability mass is distributed across the models inside each class. An unrestricted class that contains one specially tailored model for every possible dataset cannot be evaluated sensibly by taking only its maximum likelihood: such a class wins by construction through overfitting.

Description-length approaches make the same point in another language. A theory should be charged for the complexity of the model as well as for the residual information needed to encode the data. Rissanen’s minimum-description-length program formalized model selection in terms of the combined length of a model and the data encoded with that model [4]. A deterministic program that merely stores an incompressible transcript receives no free scientific victory from assigning that transcript probability one; the transcript has simply been moved into the model.

This does not mean that simplicity alone proves indeterminism. A compact stochastic law and a hardcoded deterministic history may have similar total descriptive costs once the realized outcomes are encoded. Nor is there a unique, assumption-free prior over all possible theories. The point is narrower:

“A deterministic explanation exists” and “the evidence supports that deterministic explanation” are different claims.

Science routinely rejects possibilities that remain logically compatible with the data because they are ad hoc, unstable under intervention, needlessly complex, disconnected from other successful theories, or dependent on conspiratorial parameter choices. These judgments are fallible and framework-dependent, but they are not arbitrary.

The epistemic landscape can therefore be divided into three levels:

1. A finite record cannot deductively exclude every deterministic completion.
2. A finite record can strongly disfavor or falsify specified deterministic models.
3. No model comparison is free of background standards concerning admissibility, causal structure, simplicity, and prior plausibility.

The first level is a logical limit. The second is ordinary science. The third explains why ordinary science remains possible despite the first.

5. Quantum experiments: powerful constraint, conditional certification

Quantum theory provides the most important physical test case because it does more than produce irregular data. It imposes structural restrictions on any deeper deterministic account.

Bell’s theorem shows that no theory satisfying the relevant locality and measurement-independence conditions can reproduce all quantum correlations [5]. Later experiments have observed Bell-inequality violations while closing the major locality and detection loopholes, sharply excluding broad classes of local hidden-variable explanations [6]. These results are not merely failed searches for patterns. They rule out a mathematically defined causal architecture.

Bell-based randomness protocols go further. An observed Bell violation can certify that outcomes possess conditional min-entropy relative to an adversary, without requiring a detailed model of the internal devices. This is the sense in which the protocol is “device independent.” Pironio and collaborators demonstrated the connection between Bell violation and certified random-number generation [7].

Yet device independence is not assumption independence. The certification depends on a framework that includes conditions such as the validity of the relevant quantum or no-signalling constraints, isolation from unauthorized information leakage, correct causal separation, and sufficient independence between measurement choices and the devices’ hidden state. Randomness amplification weakens the required quality of the initial choice source, but it begins from some nonzero independence or partial freedom rather than from a completely predetermined source [8].

This matters because Bell’s theorem does not eliminate determinism in the abstract. It eliminates deterministic theories that retain the relevant package of locality and independence assumptions. Deterministic alternatives remain logically available by changing the package.

Bohmian mechanics retains determinism by accepting nonlocal dynamics [3]. Measurement-dependent models weaken the independence between hidden variables and measurement settings; explicit deterministic models can reproduce singlet correlations under such relaxations [9]. Retrocausal approaches alter the assumed direction of dependence. Everettian quantum mechanics retains unitary evolution while interpreting probability through branch-relative uncertainty rather than unique stochastic collapse.

These alternatives are not empirically or conceptually costless. They differ in causal structure, ontology, explanatory strategy, compatibility with relativistic principles, and the role assigned to probability. Bell experiments therefore produce real evidence: they make large regions of deterministic theory space untenable and force surviving theories to accept specific structural commitments.

The defensible conclusion is neither “quantum mechanics proves metaphysical chance without assumptions” nor “quantum evidence tells us nothing about determinism.” It is this:

Quantum experiments certify randomness and exclude deterministic explanations relative to explicit causal and physical assumptions. They do not deductively eliminate every conceivable deterministic ontology.

That is not a weakness peculiar to quantum mechanics. It is how empirical inference works whenever evidence bears on theories through a background model of instruments, interventions, and causal independence.

6. Algorithmic information: a boundary on certification, not on evidence

Algorithmic information theory gives exact tools for discussing the complexity of individual strings.

For a finite string $x$, its Kolmogorov complexity $K(x)$ is the length of the shortest program, in a fixed universal description language, that outputs $x$. A string is algorithmically incompressible when $K(x)$ is close to its length. Most strings of a given length are incompressible by a counting argument: there are too few short programs to describe all long strings.

But $K(x)$ is not computable in general. A universal procedure that always found the shortest program would solve instances of the halting problem. Chaitin also showed that any sufficiently strong, sound, computably axiomatized formal system has a system-dependent ceiling beyond which it cannot prove arbitrary statements asserting high program-size complexity [10].

These results matter, but their philosophical scope must be stated carefully.

They imply that there is no universal algorithm for certifying the exact incompressibility of every finite record. They also imply limits on what a fixed formal system can prove about specific strings. They do not imply that no string can ever be shown to resist a particular compressor, no deterministic regularity can ever be discovered, or every finite record has a short deterministic explanation.

Every finite string has a deterministic print program. For an incompressible $n$-bit string, however, the shortest such program is roughly $n$ bits long. Logical compatibility with determinism remains, but compression has failed to reveal a concise deterministic law.

For infinite sequences, Martin-Löf randomness is defined relative to a probability measure and the class of effective tests [1]. Two cautions follow.

First, no finite prefix establishes that the infinite continuation is Martin-Löf random. Every finite binary string extends both to computable infinite sequences and to Martin-Löf-random infinite sequences. A finite observation can therefore never settle the algorithmic character of the entire future.

Second, an algorithmically random sequence is not identical to the probability measure relative to which it is random. The measure defines which effective exceptional sets count as negligible; the sequence is one point that avoids those sets. Confusing sample path with probability law reproduces, in mathematical form, the confusion between realized history and generative mechanism.

Algorithmic information also corrects two persistent myths about infinity.

Irrationality is not algorithmic richness

A real number may have an infinite, nonrepeating expansion and still possess a very short description. The prefixes of $\pi$ and $\sqrt 3$ can be generated by fixed algorithms given the desired precision. Their first $n$ digits therefore have complexity no greater than the description of the algorithm plus a description of $n$, up to a constant. Infinite expansion does not imply an infinite reservoir of independent information.

Almost all real numbers, relative to ordinary measure, are noncomputable and algorithmically random. But the use of real-valued coordinates in a physical theory does not show that physically instantiated states are generic in that measure-theoretic sense. A continuum state space permits information-rich points; it does not force nature to occupy one, nor does it show that arbitrary digits are physically accessible.

Finite description is not finite state

A deterministic system becomes eventually periodic when its complete state space is finite. By the pigeonhole principle, some total state must repeat, after which deterministic evolution repeats the same trajectory.

A finitely described program need not have a finite state space. With unbounded memory, a short program can generate a nonperiodic computable sequence forever: the digits of $\pi$, the Thue–Morse sequence, or an enumeration of the primes. It is therefore false that a finite seed or finite program must eventually cycle unless the entire evolving state is confined to finitely many possibilities.

Chaos does not create algorithmic information from nothing

A deterministic chaotic system can magnify tiny differences in initial conditions and make long-term prediction practically impossible. But if the dynamics, initial state, and time parameter have short descriptions, then the state reached at that time also has a description consisting of those inputs plus the simulation rule. Computable deterministic evolution can redistribute and expose information contained in initial conditions; it does not manufacture unlimited algorithmic information ex nihilo.

Chaos, unpredictability, and ontic indeterminism are therefore distinct. A deterministic world may be operationally opaque without being algorithmically supplied by an incompressible initial real number. Conversely, an indeterministic law may be simple even though its realized history is incompressible.

7. Pseudorandomness is relative to an observer class

Pseudorandomness is not a defective imitation of a single absolute property. It is a relation among a generator, a distribution, and a class of tests.

A sequence is pseudorandom for an observer when the observer’s admissible procedures cannot distinguish it from the target random distribution with significant advantage. Change the observer’s information or computational resources and the judgment may change.

Complexity theory makes this relativity precise. Nisan constructed generators that stretch a relatively short random seed into a longer sequence that appears random to algorithms operating within a specified space bound [11]. The theorem does not say that the output is indistinguishable to every possible observer. It says that a carefully defined class of resource-bounded computations cannot tell the difference beyond the stated bound.

This has two implications.

First, practical unpredictability can be extremely strong even when the source is deterministic. For cryptography, simulation, randomized algorithms, and secure systems, the relevant question is often not whether a sequence is ontically uncaused, but whether an adversary with specified side information and resources can predict or distinguish it.

Second, resource bounds do not by themselves prove a universal limitation on every embedded observer. An observer implemented within a system can still discover many facts about that system’s mechanisms. Software can inspect source code. A computer can reveal the seed and recurrence of its own pseudorandom generator. Scientists can infer dynamical laws from repeated interventions. Gödelian and computational limits constrain complete self-knowledge; they do not imply that no subsystem can understand any process that implements it.

Embeddedness becomes decisive only when it is joined to a stronger condition: all accessible evidence is distributed identically under the rival theories. In that case the obstacle is not that the observer is physically smaller than the world. The obstacle is that the available information channel carries no discriminating signal.

Thus two very different limits must not be conflated:

• Resource-relative indistinguishability: the signal exists, but the observer lacks the resources to extract it.
• Operational equivalence: no discriminating signal exists within the stipulated experimental domain.

Greater intelligence or computation may overcome the first. It cannot overcome the second without expanding the domain of access.

8. What an observer can know

The debate becomes clearer when “knowledge of randomness” is divided into levels.

This hierarchy prevents two symmetrical errors.

The first error is triumphalism: treating a successful randomness test or Bell-based certificate as direct access to metaphysical openness without acknowledging the assumptions that connect the data to that conclusion.

The second error is nihilism: treating the absence of assumption-free proof as evidence that experiments, theory comparison, and causal constraints are epistemically worthless.

Both errors impose the standards of formal deduction on a domain governed by ampliative inference. Scientific evidence is not rendered empty because a sufficiently flexible skeptical alternative can absorb it. If that standard were adopted consistently, almost every claim about unobserved mechanisms, remote objects, past events, and causal structure would disappear.

A more defensible position is fallibilist realism. There may be a fact about whether an event was fixed by the prior physical state. Evidence may bear on that fact indirectly through the success and failure of theories. Yet no finite dataset forces the conclusion independently of all assumptions about admissible laws, causal independence, simplicity, and the reliability of experimental practice.

9. Does empirical equivalence erase the fact?

Suppose two complete theories are operationally equivalent for every experiment physically available to every observer in a world. One says that a unique future was fixed; the other says that several futures were genuinely possible. Is there still a fact of the matter?

The evidence alone does not decide this philosophical question.

A verificationist may argue that a difference with no possible empirical consequence is not a genuine physical difference. On this view, the rival descriptions are alternative representations of one operational structure, and asking which is “really” true adds nothing.

A realist may answer that causal and modal facts need not be directly observable. Whether an event could have occurred otherwise may be a real feature of the world even if all actual records are shared. Scientific theories frequently posit structures whose warrant is indirect, deriving from explanatory unity, counterfactual coherence, or integration with broader theory rather than from isolated observation.

Operational equivalence alone cannot settle the dispute because the dispute concerns the criterion of facthood itself. But one negative conclusion is available:

Empirical equivalence does not logically imply ontological identity.

Two theories may agree on every observable distribution while disagreeing about ontology, laws, counterfactuals, or explanation. Whether those disagreements represent substantive physical alternatives or surplus structure requires an additional philosophical argument.

The epistemic result should therefore remain separate from the metaphysical one:

• If theories are operationally equivalent, accessible experiments cannot choose between them.
• It does not follow solely from that equivalence that there is no truth about which ontology obtains.

Underdetermination is a relation between evidence and theory. It is not automatically a relation of identity between theories.

10. A disciplined research program

The problem of ontic randomness is not exhausted by saying that unrestricted deterministic alternatives are always logically possible. That observation sets the boundary of absolute certification; the scientific work begins inside it.

10.1 Map operational equivalence under interventions

The relevant unit of comparison is not a single transcript but a family of distributions over possible interventions. Research should characterize when deterministic and stochastic causal models are observationally equivalent, when interventions separate them, and when equivalence persists across every physically implementable intervention. The result would be a hierarchy of identifiability rather than a binary verdict.

10.2 Quantify the costs of surviving deterministic theories

Bell violations show that deterministic completions cannot retain every classical causal assumption. The next question is comparative: how much measurement dependence, nonlocality, retrocausal structure, fine-tuning, or additional ontology is required? These costs should be represented mathematically where possible rather than dismissed rhetorically. Hall’s measurement-dependent construction is an example of turning a vague “conspiracy cost” into a quantified relaxation [9].

10.3 Separate physical finiteness from computational finiteness

A finite entropy bound, a discrete state space, a finite description, and a finite total number of states are different claims. Their consequences for recurrence, simulation, and randomness differ sharply. Physical theories should specify which form of finiteness is asserted before conclusions about periodicity or information capacity are drawn.

10.4 Clarify the semantics of probability

Deterministic theories often recover probabilistic predictions through ignorance, typicality, equilibrium measures, branch weights, or self-locating uncertainty. Stochastic theories may treat probabilities as propensities, lawlike chances, or primitive transition weights. A comparison of determinism and indeterminism is incomplete until it states what probabilities mean in each theory and why the relevant measure governs rational expectation.

10.5 Treat certification as a layered achievement

Randomness certification should report its assumptions as part of the result: the adversary model, causal separation, independence conditions, trusted components, and physical principles invoked. This does not weaken certification. It makes clear what has actually been established and which skeptical alternatives remain outside the theorem’s scope.

11. Conclusion

No finite observation can deliver assumption-free proof that nature is ontically indeterministic. For any finite record, a deterministic mechanism can be constructed that reproduces it. If a deterministic and an indeterministic theory are exactly equivalent across every accessible intervention, then no evidence drawn from that domain can discriminate between them.

Those claims define a genuine boundary. They do not justify a collapse into agnosticism about everything.

A transcript-printing program is not automatically a good explanation. A fixed history is not the same mathematical object as a probability measure. A deterministic hidden-variable theory requires more than deterministic equations if it is to recover empirical probabilities: it also requires a measure and an account of why that measure is relevant. Algorithmic incompressibility is difficult to certify universally, but that does not make every deterministic explanation short. A continuum permits information-rich states, but irrational coordinates do not themselves contain independent random bits. A finite program need not cycle unless its complete state space is finite. Resource-bounded pseudorandomness shows that deterministic unpredictability can be rigorous, but it does not imply that embedded observers are incapable of discovering mechanisms.

Quantum experiments provide the clearest illustration of the balanced conclusion. They exclude broad classes of local deterministic theories and enable powerful model-relative randomness certification. They do not eliminate every deterministic ontology without assumptions. What they offer is not metaphysical certainty but sharply structured evidence.

The central principle is therefore:

Unrestricted alternatives block deduction; explicit restrictions make evidence possible.

The world may contain genuine chance. There may be a determinate fact about whether a particular event was fixed by what came before. Finite observers can investigate that question, constrain its answers, and rationally prefer some theories over others. What they cannot do is force a final metaphysical verdict from finite data while refusing every premise that gives the data evidential meaning.

That is not the defeat of inquiry. It is the condition under which inquiry becomes intellectually honest.

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