Reason: Probability Reconsidered as I Move Toward a Theory of the World — Part I

Once again, I opine beyond my pay grade. I expect I am wrong, at least in details, but maybe the overall thrust is correct. Other’s are often wrong, why should I be any different? I present my case.

Author’s Preface

In this series of essays in the Reason Series, I return to one of the oldest yet most persistent questions: how do we understand the world? The approach here is not technical in the narrow sense, though it engages with mathematics, probability, and logic. Rather, it is concerned with the conditions of description—the way language, models, and conceptual frameworks allow us to grasp, represent, and argue about reality.

My focus throughout is on the tension between the inner world of awareness and the outer world of events, and on how mathematics and probability function as specialized dialects of language to bridge the two. I am less concerned with whether the descriptions are “true” in a Platonic sense than with how they work: how they are invented, how they are stabilized, and how they succeed or fail in practice.

This is Part I of a four-part sequence:

· I – Postulates Toward a Theory of the World – setting out the starting axioms that ground reasoning about causality, variability, and the relation between inner and outer worlds.

· II – Invention and Mathematical Rules – examining how mathematical systems are human constructions, invented, tested, and maintained through rules and conventions.

· III – Meta-assumptions and Curve Fitting – exploring the assumptions that underlie proofs and models, and showing how probability distributions arise from the pragmatic act of fitting curves to observed regularities.

· IV – Situations and Nomological Machines – considering how causal capacities are stabilized within bounded contexts, and how these constructed “machines” provide the conditions in which probabilistic and mathematical reasoning can be meaningfully applied.

What unifies these essays is not a single doctrine but a stance: that knowledge is always situated, that mathematics is a language among languages, and that probability is not a property of the world but a way we describe patterns under carefully delimited conditions.

Introduction

Part I – Postulates Toward a Theory of the World

This part begins with axioms, provisional postulates taken as givens. They set the stage for reasoning about the world as objectively real, causally structured, and accessible only through the inner medium of awareness and language. These foundations ground the discussion of determinism, variability, and probability.

Part II – Invention and Mathematical Rules

Here the focus turns to mathematics itself, not as revelation but as human invention. Rules are proposed, tested, and retained when they work. Mathematics emerges as a dialect of language—compact, abstract, and powerful, but always invented and verified within human practice rather than discovered in a Platonic realm.

Part III – Meta-assumptions and Curve Fitting

This section addresses the hidden assumptions behind proofs, models, and probability distributions. Curve fitting illustrates how mathematical descriptions depend on judgments of stability, context, and independence. What often appears as necessity is instead the outcome of meta-assumptions about how the world is framed and about where idealizations may hold.

Part IV – Situations and Nomological Machines

Finally, attention turns to the role of situations and boundaries. Probabilistic reasoning only works within bounded machines—contexts where capacities are stabilized and variability is constrained. Nomological machines, as Cartwright describes them, are both physical arrangements and descriptive constructs. They provide the ground on which mathematics, language, and probability can operate meaningfully.

Part I - Postulates Toward a Theory of the World

To construct a theory of how the world works—and how we describe it, with whatever degree of precision we can manage—it is necessary to begin with foundational postulates. These function as axioms: not proven, but taken as givens, as grounds for discussion.

The stance is that the world exists objectively and is causally structured. Arguments against determinism collapse into self-defeat, because reasoning itself requires stable connections between evidence and conclusion. Yet causality is not simple; it is tangled, situational, and dependent on boundaries we impose. In practice, innumerable factors operate, but only some predominate within a given bounded context. Understanding therefore requires drawing boundaries, naming objects, and setting conditions so that causes can be investigated.

From this perspective, here are the current axioms for reasoning about experience and reality. They are not thought to be complete, only provisional.

Axioms

1. There is an inner world and an outer world.
Experience divides into two domains: the subjective field of awareness and the objective realm of events. Both must be recognized to make sense of human knowledge.

2. The outer world exists; denying it is self-defeating.
Any attempt to argue that the external world is unreal already presupposes engagement with it through speech, reasoning, and survival.

3. Survival depends on manipulating the outer world, sometimes successfully, often not.
Human life is constrained by the need to act effectively in the environment, though success is never guaranteed and often partial.

4. Causality is real and can be used to act effectively, unlike Hume’s skeptical reading of “constant conjunction.”
While regularities may be described as conjunctions, effective action demonstrates that real causal connections allow prediction and intervention.

5. Causality is complex and entangled; only parts are ever grasped, sometimes enough for survival.
No one apprehends the whole web of causes. What is grasped is fragmentary, but often sufficient for practical purposes.

6. The inner world is the ground of awareness, of all being; nothing about the outer world is known except through it.
Perception, memory, and thought all arise within the field of awareness, so access to the world is always mediated by the inner domain.

7. The inner world depends causally on the body and thus on the outer world.
Consciousness is not free-floating but bound to the physiology of the organism, itself embedded in and dependent on its environment.

8. Language is not identical with thought, but in humans it is tightly bound to thought and communication.
Language structures expression and dialogue, but thought can precede or exceed words; the two remain interwoven in practice.

9. The “thing in itself” is unknowable; knowledge concerns abstractions of it.
Reality as it exists independently of perception cannot be grasped directly. What is known are conceptual and perceptual representations.

10. Words are ambiguous, fuzzy, and metaphorical, dim reflections of reality, yet sufficient to act upon it.
Language never maps the world with precision. Its value lies in enabling action, communication, and partial understanding despite its limits.

Inner World and Outer World

Do not confuse the inner world of language and explanation with the outer world of what actually happens. What happens in the outer world is deterministic. We then attempt to explain it by abstracting through words and through mathematics, which is a specialized form of words. I did not say the world was not deterministic—that was someone else’s invention.

Variability

The Brute Fact of Variability

The brute fact of the world is variability in all outcomes. Sometimes we can explain it, but most of the time we cannot—until experiments and controls allow us to uncover limited explanations. In the long run, variability remains a persistent obstacle.

When Probabilities Enter

Probabilities are needed when variability exceeds what our causal models can account for. Complexity and complications force us to turn to probability as a way of predicting in the aggregate. But this only works in certain situations. In many cases, the instability of the system prevents smooth curves or neat equations. The data may be too unstable, too diffuse, too spiky for probability to provide a tractable description.

Determinism and Abstraction

Every event and outcome involves variability, even in domains habitually labeled deterministic, such as objects falling under gravity. Newton’s laws are abstractions that rely on idealized conditions—what Nancy Cartwright would call a nomological machine. In reality, countless perturbations occur. A feather does not fall at the same rate as a lead weight. The context, boundaries, and conditions must always be taken into account, even in cases that appear strictly deterministic.

The Problem of Framing

Framing is essential: how boundaries are drawn, how situations are defined, and how stability is assumed over time. Probabilities rely on stability. Without it, no equation can reliably predict outcomes. Prediction depends as much on how the situation is framed as on the causal processes themselves.

Long-Range Patterns

In some cases, causal structures allow long-range patterns to be detected. These can support aggregate prediction, even when individual events remain unpredictable. The mathematics applies to the continuum of outcomes, not to specific occurrences.

Variability and Causal Factors in Ohm’s Law

Variability in a “Simple” Law

Even something as clear-cut as Ohm’s Law, which appears not to require probabilities, involves variability. Causal factors are not uniformly effective. Some dominate, others are negligible, and many are irrelevant or uncorrelated. In practice, reasoning requires identifying which factors matter most.

Specialized Conditions

Ohm’s Law is not a universal truth but a relation that applies under well-defined conditions. Manufacturing tolerances in resistance do not alter the law itself, but other influences—such as temperature, local electromagnetic fields, or the type of material—can affect outcomes.

Material Effects and Exceptions

Certain components, such as diodes, change their resistivity depending on the voltage applied. These cases illustrate how the law functions within a restricted domain. Outside those boundaries, variability emerges, sometimes in ways that the law does not capture.

Limits of Precision

Even within controlled conditions, variability persists. Measuring instruments introduce error, and external fields can alter resistivity or induce additional voltages. These factors ensure that Ohm’s Law never works with perfect precision, but only as an approximation within specified frames.

Probabilities, Gravity, and Nomological Machines

Limited Role of Probability

There is little value in applying probabilistic reasoning to Ohm’s Law, which functions within narrow, well-defined conditions. Gravity, however, provides a different case. Here probabilities may be invoked, though it is often more straightforward to examine the causal factors directly.

Causal Influences in Falling Bodies

Air resistance illustrates how causal factors complicate outcomes. A feather falls differently from a lead ball, and a human body falling from an airplane presents further variation. The rate of descent depends on body angle, posture, or whether a wingsuit is worn.

Framing Through Nomological Machines

Such examples show that variability is inseparable from context. To reason about gravity—or any other domain—one must specify the conditions, constraints, and boundaries that make a stable description possible. This is what Nancy Cartwright calls defining a nomological machine.

Causality

Causality and Determinism

Understanding causality underlies prediction. When variability is small, predictions become reliable, and we call this determinism. Yet even events considered deterministic contain some degree of variability—though often so minor it can be ignored.

Defining Causality in Practical Terms

Causality is not a deductive abstraction but a practical recognition of patterns in everyday life. Touching fire results in a burn; throwing a rock sends it through the air; steering a car keeps it on the road. These observations embody causality as lived experience. Hume’s attempt to treat causality as a purely deductive concept misses the point. Human beings navigate the world by recognizing causal regularities, not by constructing logical proofs. The mystery lies not in its existence, but in how deeply it is woven into experience. Claims that quantum physics shows “no causality” are incoherent, since the very identification of regularities presupposes causal order.

Regularities in Physical Laws

Causality in the world produces regularities. Sometimes the variability is minimal, sometimes great. Newton’s laws and Ohm’s law exemplify such regularities. These are idealizations that deliberately exclude many complicating factors. Within their domains they work well enough to be verified empirically, but they do not capture every detail of reality. Newton’s laws in particular are strong idealizations, while Ohm’s law omits fewer but still ignores some factors. In both cases, breakdowns can be found, but within the contexts where they apply, they remain highly effective.

Determinism

Variability and Framing

Every event in the world has variability, even those we habitually label as deterministic, such as falling bodies. Newton’s laws are abstractions that rely on idealized conditions—what Nancy Cartwright calls nomological machines. In reality, perturbations are everywhere: a feather falls differently from a lead weight; air resistance changes with context. Determinism requires framing—defining boundaries, conditions, and time spans. Without stability across time, predictive equations fail.

Determinism, Randomness, and Probabilities

Philosophers sometimes speak of propensities, an ambiguous idea that seems to occupy a space between causes and randomness. Determinism holds that everything has a cause. Randomness can be taken in two ways: either as caused but unknown due to complexity and human limits, or as truly uncaused. Whether there is a third possibility remains doubtful, though perhaps this reflects a failure of language or imagination.

Induction and Inference

The case for determinism is ultimately inductive. Everyday survival shows us repeated causal patterns, even when variability obscures them. By imposing controls, causes often become visible where none were seen before. We never grasp the “thing in itself,” only aspects of it through abstraction. Still, the world consistently exhibits causal repeatability. Physicists complicate this picture with quantum indeterminacy, but the claim of uncaused events sits uneasily with observed regularities. Einstein’s refusal to believe that “God plays dice” reflects this tension. What quantum experiments show may not be acausality, but rather our failure to identify all relevant factors.

The Dappled Universe Problem

Some philosophers and physicists propose a dappled universe: partly caused, partly random. Yet uncaused events that also show regularity make little sense. If some events are caused and others not, where is the boundary? Is randomness confined to the microscopic world, while the rest remains caused? No clear dividing line exists, which makes the proposal incoherent under ordinary reasoning.

Language and the Outer World

It is crucial not to confuse the inner world of language and explanation with the outer world of events. What actually happens is deterministic. Words and mathematics, as specialized abstractions, are tools we use to explain that reality.

Propensities

Propensities and Neoplatonic Interpretations

Some traditions treat probability as though it were a Neoplatonic feature of the world, as if it inhered in things themselves. To capture this, philosophers coined the term propensity. It is unclear whether this simply restates causality in different words or posits a mysterious third property of reality. A more defensible view is that causality is complex and entangled, producing patterns of variability. These patterns may appear probabilistic in the long run, though not necessarily in the short run.

Questioning Platonic Alternatives

The alternative to this causal interpretation seems to be a Platonic one: probabilities existing as independent entities. If that is the “third possibility,” it is unconvincing.

Variability Versus Probability

It is important not to confuse variability with probability. Quantum physics may require probabilistic models, but that does not imply a quasi-Platonic property of “propensities” or inherent randomness. The debate continues, but treating randomness as an uncaused property of the universe adds little clarity.

Probabilities as Inhering in the Universe

Some thinkers go further, arguing that probabilities themselves inhere in the universe. This is presented as more than just recognition of causal regularities. Yet the claim remains incoherent: no one can give a workable account of what such inherent probabilities would mean.

Randomness and the Statistical Enterprise

If randomness is assumed to inhere in the universe, then statistics supposedly depends on “true randomness.” Statisticians sometimes claim that without it, statistical reasoning collapses—though rarely defining what “true randomness” means. This creates confusion about what probability represents and how to use it. The statistical enterprise can then look like a house of cards: useful when it works, but vulnerable when theory outruns evidence.

Contemporary Belief in Platonism

Modern statisticians sometimes echo these Platonic tendencies. For example, Stark has argued that only true randomness justifies probabilistic calculation, without clearly explaining what “true randomness” entails. His examples of what is not random were unconvincing. This was a case of quantifauxcation: making sweeping claims without solid grounding.

Overlooked Possibility

Is there another possibility for the nature of probability, besides causality and Platonic propensities? Perhaps, but if so it is elusive. The appeal to propensities has not provided a convincing third way.

Popper and Probabilities

Karl Popper introduced the idea of propensities, but the proposal seems remarkably weak. A coin flip, for instance, does not depend on some metaphysical chance property of the coin; it depends on the entire system—hand, air resistance, surface, and so forth. Boundaries and contexts must be drawn before probabilities can be meaningfully applied. To ignore this is to work with an empty abstraction. That Popper overlooked such basics is surprising.

The Philosophical Notion of Propensities

Philosophers may continue to use the term, but its meaning remains vague. At best, propensities can be understood as shorthand for causal factors that, within defined frames, lead to stable long-run outcomes. But they are always embedded in contexts, boundaries, and language—never free-floating metaphysical entities.

Polanyi Versus the Platonic Tradition

By contrast, Michael Polanyi offered a more grounded approach. As a chemist and polymath, he exposed the sterility of Platonic abstractions and the self-referential habits of many mathematicians and philosophers of mathematics. His successors extended his insights into mathematics, showing how knowledge must remain situated in practice rather than in imagined metaphysical properties.

Series continues at:

I – Postulates Toward a Theory of the World – setting out the starting axioms that ground reasoning about causality, variability, and the relation between inner and outer worlds.

II – Invention and Mathematical Rules – examining how mathematical systems are human constructions, invented, tested, and maintained through rules and conventions.

III – Meta-assumptions and Curve Fitting – exploring the assumptions that underlie proofs and models, and showing how probability distributions arise from the pragmatic act of fitting curves to observed regularities.

IV – Situations and Nomological Machines – considering how causal capacities are stabilized within bounded contexts, and how these constructed “machines” provide the conditions in which probabilistic and mathematical reasoning can be meaningfully applied.