Meta-probability: Language, Knowledge, and the Limits of Statistics

A sketch of possibilities for more nerdish and partially informed discussions. I have had these discussions with my assisant many times. What do you think of the title; too commercial?

Author’s Preface

A long, long time ago, in a place far, far away, I studied statistics. That wasn't my major, I just studied it. I took a graduate course. I actually did fairly well in terms of the class average. I was the top scorer. I beat out my self-described polymath office mate, Mr. Korlenchuk, who was a math major. Ha ha. In any case, I learned squat. Some applied stuff. Some matrix algebra. But I never really looked at the foundations, the conceptual underpinnings of statistics and probability.

So now, five decades later, I'm looking at it again via conversations with my assistant, idiot savant that my assistant is. So I'll get some things wrong, but that's life. We all get things wrong, routinely. That's the nature of the world. However, I've had time to reflect on statistics and probability and its application to the soft sciences, so-called. In particular, psychology, being my former field of study.

I've been spurred on by reports of the replication crisis in psychology and have come to think on that very, very deeply and explore that topic (thar’s some as would call it thinking.) And I've had pages and pages of LLM AI output, which I've tried to synthesize into articles, essays: I've done a few. And now I'm trying to do it better, more thoroughly. So hence, I'm looking at the topics that I need to cover in some coherent fashion. Hence, this outline here, prepared with the assistance of my assistant. We've made some very reasonable progress on what I'm trying to accomplish.

For partial context see

Introduction

Probability is often presented as a settled mathematical discipline, but in practice it is a linguistic model for dealing with uncertainty, built on assumptions that do not always fit the world in which it is applied. This work explores probability not as timeless truth but as a linguistic construct—useful, limited, and sometimes misleading—especially when extended from controlled, closed systems into open, unstable domains such as the soft sciences. The aim is to clarify what probability describes, when it works, and why it fails, with particular attention to the conceptual and practical difficulties exposed by the replication crisis in psychology.

Discussion

1. Understanding of the World of Probabilities

This could serve as the overarching title or framing device for the series. The idea is to take stock of probability not simply as a branch of mathematics but as a way of talking about knowledge, uncertainty, and the limits of prediction. Traditional treatments often assume probability is a settled discipline with unproblematic applications, but this project suggests otherwise: that probabilistic reasoning needs to be re-examined in light of its practical uses, failures, and underlying assumptions. The aim is not to dismiss probability but to situate it properly—less as timeless truth and more as a contingent tool.

2. Knowledge and Truth

A distinction can be drawn between knowledge, which involves what is believed and justified at a given time, and truth, which refers to what actually corresponds to reality regardless of belief. Probability has been invoked in both arenas: to express confidence in what is known and to suggest closeness to truth. But the relationship is not straightforward. One may have high probability judgments based on evidence and still be wrong about the truth. Conversely, truths may exist that are beyond current knowledge or probabilistic description.

3. The three worlds: the inner world, the outer world, and the Platonic world

Human thought operates within three overlapping domains. The inner world is the realm of subjective experience—emotions, sensations, reflections. The outer world is the empirical environment, the things encountered and tested through observation. The Platonic world is the imagined domain of ideal forms, abstract mathematics, and logical constructs. Probability straddles these worlds: conceived abstractly, applied empirically, and interpreted subjectively. Confusion often arises when categories from one world are mistaken for those of another—for example, when mathematical idealizations are assumed to perfectly mirror real-world processes.

4. There is an objective world

Despite the roles of language, perception, and abstraction, something exists independently of interpretation. Stones fall whether or not one believes in gravity; the sun rises regardless of cultural stories. This is a rejection of pure relativism or solipsism. Models and descriptions may be flawed, but they are always attempting to connect with an underlying reality. Acknowledging this objective world sets boundaries for how probability and knowledge claims can be evaluated.

5. Thought, awareness, phenomenology

Before probability, before mathematics, there is awareness. Human beings live in the flow of experience—seeing, hearing, feeling—long before concepts are applied. Phenomenology, the reflective study of direct experience, reminds us that thinking begins here. Any model, including probability, is layered on top of this ground of consciousness. To talk about knowledge without acknowledging this base is to miss the fact that all understanding is rooted in lived experience.

6. Language, what language does

Language is the medium through which thought is shared, structured, and extended. It allows individuals to describe the world, to reason together, and to pass knowledge across generations. But language does more than mirror reality; it shapes perception, highlights some features, and hides others. Words are never neutral labels but come with associations and implied structures. Probability as a language is no different: it does not simply record uncertainty but constructs ways of speaking about it.

7. Language describes the world

At its most obvious, language serves to describe. “The cat is on the mat” or “The dice landed on six” are attempts to map words to events. Yet this descriptive function is imperfect: words generalize, oversimplify, or mislead. Probability descriptions—like “a fair coin has a fifty percent chance of heads”—likewise aim to capture aspects of the world but often depend on idealizations. The usefulness of language lies not in flawless correspondence but in providing workable approximations of reality.

8. The creative origin of languages

Language did not spring fully formed; it emerged through human history, gradually evolving as a tool for survival, coordination, and storytelling. Early languages were spoken long before writing, and they carried the structures of thought that shaped human societies. Just as spoken language emerged from the practical needs of life, probability language arose from practical problems—gambling, risk, decision-making. To understand probability is to see it as part of this broader trajectory of linguistic invention.

9. Math as a language, a specialized language

Mathematics can be viewed not as metaphysical truth but as a highly specialized language, a compact and precise set of symbols and rules for expressing relationships. Like natural language, it enables description, but it operates at a level stripped of ambiguity. Still, it is a human construction: equations do not exist in nature in the way rivers or trees do. The symbols are useful for capturing patterns, but they are nonetheless artifacts of human invention.

10. Trio of possible views of models

Models can be approached in at least three ways. First, as theoretical constructs, they are judged internally for consistency and derivation from assumptions. Second, as applications, they are tools to solve practical problems. Third, as validated instruments, they are tested against empirical outcomes. Confusion arises when these roles are conflated—when a mathematically consistent model is assumed to be empirically accurate, or when an applied success is mistaken for a proof of universal validity.

11. Models as language, ideal but limited

Models are best understood as a kind of language. They create simplified versions of reality, highlighting certain features while ignoring others. This makes them powerful for certain purposes, but inherently limited. A weather model, for example, may capture air pressure and temperature patterns but cannot fully encompass the complexity of local microclimates. Recognizing models as limited linguistic constructs prevents the mistake of treating them as literal representations of reality.

12. Meta-assumptions of models

Every model carries hidden background assumptions—about stability, independence, measurability, and so forth. These assumptions are often invisible to the user but decisive for the outcome. In probability, a meta-assumption is that the world can be described as a set of repeatable events or outcomes. Without this, the system collapses. The applicability of models to the real world therefore cannot be taken for granted; it must be critically assessed.

13. Argument as language, proof is not Platonic

A logical proof is often thought of as a window into eternal truths, but in practice it is a structured use of language meant to persuade. Proofs convince because they follow agreed rules, not because they reveal timeless entities floating in a Platonic realm. Probability proofs, such as derivations of the central limit theorem, should be seen in this light: persuasive demonstrations that depend on shared conventions, not on metaphysical inevitability.

14. There is an objective world (reprise)

The reminder bears repeating: whatever the limitations of language or models, an external reality exists. If a coin is tossed, it lands somewhere, even if predictions about the outcome remain uncertain. This is the anchor against which linguistic and mathematical constructions must be tested. Without affirming an objective world, the entire enterprise of probability loses coherence.

17. Probability as a linguistic model

Probability is not a property of the world itself but a way of talking about uncertainty. Saying a coin has a fifty percent chance of heads is not to describe a physical force in the coin but to express expectations about outcomes. The language of probability thus functions as a model for reasoning, not as a mirror of reality. Recognizing this prevents the category mistake of treating probabilities as objective features like mass or length.

18. Applicability of probability as a system

Probability works well in some domains and poorly in others. It is powerful in controlled settings like dice games, where outcomes are enumerable and repeatable. But in open or chaotic systems—such as ecosystems, economies, or human behavior—it often falters. In such cases, other frameworks, like chaos theory or qualitative description, may better capture what is going on. Applicability depends not on the elegance of the mathematics but on the match between assumptions and the real world.

19. Open versus closed worlds

Closed systems are those where all possible outcomes can be listed in advance—rolling dice, drawing cards, flipping coins. Open systems are unbounded and evolving, like weather, language change, or human history. Many real-world contexts lie on a continuum between stability and instability. Probabilistic reasoning assumes closure, but most of the world is open. Recognizing this distinction highlights both the strength and weakness of probability as a framework.

20. Computations, combinations, permutations

The mechanics of probability begin with counting. Combinations and permutations are ways of enumerating possible arrangements of outcomes. If six horses race, how many finishing orders are possible? If a coin is flipped three times, how many sequences arise? Such computations define probability in terms of ratios: one set of outcomes over the total possible. But this neatness holds only when the system is stable and well-defined, which real-world systems often are not.

21. Control and variability

In seeking to understand the world, humans try to impose control—fixing conditions, eliminating noise, holding variables constant. In the ideal of determinism, complete control would mean probability collapses to certainty. But the world resists: variation persists, unpredictability intrudes. Probability becomes the figure against the ground of variability, a way of articulating the tension between what can be controlled and what cannot.

22. Randomness and determinism

Debates about probability often hinge on whether events are caused or uncaused. Some define randomness as the absence of cause, while others see it as the presence of hidden but inaccessible causes. The paradox is that the world appears structured yet unpredictable. To call something “uncaused but structured” may be self-contradictory, but it captures the human sense that regularity exists alongside surprise. Probability sits uneasily between these perspectives.

23. Ratios of events, outcomes, the whole system

Probability is formally defined as a ratio: the number of favorable outcomes over the number of possible outcomes. Yet this definition depends on how the system itself is framed. What counts as an “outcome”? What if bias is built into the system? A die weighted toward sixes still produces outcomes, but not with equal likelihood. Thus, probability ratios are never neutral; they reflect the underlying structure of the system and the choices made in modeling it.

24. Stable versus unstable domains

Some domains lend themselves to repeatable observation and measurement. Physics experiments with controlled variables fall into this category. Other domains—psychology, economics, climate—are unstable, with shifting conditions and feedback loops. Applying probability in unstable contexts risks producing misleading numbers. Distinguishing between stable and unstable domains is therefore crucial for judging when probability is appropriate.

25. Computation and distributions

Once outcomes are counted, they are often organized into distributions: the normal curve, the binomial, the Poisson, and so on. These provide templates for reasoning about variability. Yet distributions are themselves idealizations, requiring assumptions that may not hold in practice. For example, assuming a normal distribution in data where outcomes are skewed produces misleading results. The reliance on distributions shows both the power and the fragility of probabilistic reasoning.

26. The central limit theorems, multiple

The central limit theorem is often invoked as a justification for probability in the real world. But in fact there are multiple versions, each with technical conditions—independence, identical distribution, finite variance—that rarely hold outside controlled systems. The popular idea that “everything tends toward normality” is an oversimplification. Recognizing the limits of the theorem prevents false confidence in its universal applicability.

27. Measuring the unmeasurable

Psychological states, attitudes, or emotions are often treated as if they can be measured with numbers. Scales and questionnaires assign values, but these are proxies, not true measures. Unlike length or mass, there is no physical standard against which “anger” or “happiness” can be compared. The attempt to quantify the unmeasurable produces spurious precision, raising doubts about whether probability has any rightful place in such domains.

28. Operational definitions

To make unmeasurable things measurable, sciences like psychology rely on operational definitions: defining intelligence, for example, as “what IQ tests measure.” This strategy enables research but at the cost of circularity and arbitrariness. The numbers generated reflect the definitions more than the phenomena. Probability and statistics applied to such constructs therefore rest on shaky foundations.

29. Ordered classifications or ordinal scales

Many variables in the social sciences are ranked but not truly measured: satisfaction levels from “very dissatisfied” to “very satisfied,” for example. These ordinal scales impose order but not equal spacing. Yet in practice, they are treated as if they were numerical, with averages and variances computed. Some defend this pragmatically; others criticize it as hand-waving. Either way, it exposes the fragile bridge between language categories and quantitative claims.

30. Varieties of statistics

Statistics is not a single monolith but a family of approaches: descriptive, inferential, nonparametric, multivariate, and more. Each has its own assumptions and uses. But all rely on probability to some degree, and all face the same challenges of applicability. The variety itself is evidence that no single framework suffices for the diversity of real-world problems.

31. Frequentist statistics in their various varieties

Frequentist methods interpret probability as long-run frequency. This works in theory for stable systems but becomes problematic in unstable or non-repeatable domains. Null hypothesis significance testing (NHST), the most common tool, is particularly fraught: it produces p-values that are widely misunderstood and often misused. Many critics have argued that NHST is not only flawed but irrational as a basis for knowledge claims.

32. Bayesian statistics. There are problems with that

Bayesian methods reframe probability as degrees of belief updated by evidence. While elegant in formalism, they rest on controversial assumptions: that all possible hypotheses can be enumerated, that prior beliefs can be quantified, and that numbers can capture subjective states. Whether people truly “believe” in numerical priors or merely use them as figures of speech is questionable. The conflation of belief with number is a deep conceptual problem.

33. Signal detection theory. There are problems with that

Signal detection theory arose in contexts like radar and psychophysics, aiming to model decisions under uncertainty. It distinguishes “hits,” “misses,” “false alarms,” and “correct rejections,” using thresholds and noise assumptions. While useful in engineering, its application in psychology raises questions: are perceptual decisions really like signals in noise, or is this a misleading analogy? Its seeming rigor may mask fragile assumptions.

34. There are other likelihood-based theories

Beyond frequentist and Bayesian approaches, there are alternative likelihood-based methods, such as likelihoodism or fiducial inference. These have their own technical justifications but remain less widely used. Their relative obscurity shows both the ongoing dissatisfaction with mainstream approaches and the lack of consensus about how probability should be interpreted. The field remains unsettled.

35. Practice trumps theory

Despite theoretical objections, practitioners often proceed pragmatically. If a method seems to produce workable results, it is used. Medical trials, engineering reliability estimates, and business forecasts all rely on statistics not because the theory is flawless but because the practice appears useful. This pragmatic orientation recognizes the gap between theory and real-world application.

36. But how do you validate the practice?

The difficulty lies in knowing whether pragmatic success is genuine. Replication is never exact; dice rolls differ slightly each time, and psychological studies are never repeated under identical conditions. At best, approximations are made. Probability language itself arises from this recognition: that knowledge is bounded, uncertainty is pervasive, and outcomes must be described in terms of likelihood rather than certainty. But whether those descriptions are trustworthy remains an open question.

Summary

Probability serves as a linguistic model that enables discussion of uncertainty, but its usefulness is bounded by the assumptions on which it rests. It can clarify patterns in controlled, closed systems, yet often falters when applied to open, unstable domains where replication and stability cannot be secured. The central lesson is that probability should be seen not as timeless truth but as a contingent tool, powerful in some contexts and misleading in others, especially within the soft sciences where its limits are most exposed.

Here is an extended annotated reading list that pulls together the major strands of your project: probability and statistics, replication and psychology, philosophy of science, language and meaning, and measurement theory. It is intentionally wide-ranging so it can serve as a backbone bibliography for the essay series.

A Little Light Reading

Probability, Statistics, and Models

Cartwright, N. (1999). The dappled world: A study of the boundaries of science. Cambridge University Press.

— Argues that scientific laws and probabilistic models work only in patchy, local domains rather than universally, offering a critical lens on the scope of statistics.

De Finetti, B. (1974). Theory of probability (Vol. 1). Wiley.

— Presents Bayesian probability as subjective judgment, foundational for understanding debates over whether probability reflects the world or merely belief.

Feller, W. (1968). An introduction to probability theory and its applications (Vol. 1, 3rd ed.). Wiley.

— A rigorous mathematical text, influential for formal probability, useful for contrasting technical precision with applied limitations.

Gigerenzer, G. (2002). Reckoning with risk: Learning to live with uncertainty. Penguin.

— Explains how ordinary reasoning often misinterprets statistical results, showing the gap between formalism and public understanding.

Hacking, I. (1975). The emergence of probability: A philosophical study of early ideas about probability, induction and statistical inference. Cambridge University Press.

— A historical-philosophical treatment of how probability emerged as both a mathematical idea and a tool for reasoning about uncertainty.

Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge University Press.

— Expands Bayesian methods as extensions of logic, useful for appreciating both the appeal and the fragility of probabilistic reasoning.

Kolmogorov, A. N. (1956). Foundations of the theory of probability (2nd ed.). Chelsea Publishing.

— Establishes the modern axiomatic system of probability, a cornerstone of formalism that highlights the gulf between abstract rules and real-world application.

Savage, L. J. (1954). The foundations of statistics. Wiley.

— Introduces subjective expected utility theory and Bayesian decision-making, pivotal for linking probability with rational choice.

Taleb, N. N. (2007). The black swan: The impact of the highly improbable. Random House.

— Popular but incisive critique of overconfidence in probabilistic models, focusing on rare and disruptive events that models often ignore.

Ziliak, S. T., & McCloskey, D. N. (2008). The cult of statistical significance: How the standard error costs us jobs, justice, and lives. University of Michigan Press.

— Critiques the dominance of significance testing and the uncritical reliance on p-values in social science and economics.

Replication, Psychology, and Soft Science

Ioannidis, J. P. A. (2005). Why most published research findings are false. PLoS Medicine, 2(8), e124. https://doi.org/10.1371/journal.pmed.0020124

— Seminal article that crystallized concerns about reproducibility in biomedicine, with implications across the sciences.

Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus and Giroux.

— Explains how human reasoning departs from formal probability through heuristics and biases, crucial for understanding psychology’s limits.

Meehl, P. E. (1967). Theory-testing in psychology and physics: A methodological paradox. Philosophy of Science, 34(2), 103–115.

— Explores the problem of weak theories and low replication in psychology, foreshadowing the crisis of confidence decades later.

Open Science Collaboration. (2015). Estimating the reproducibility of psychological science. Science, 349(6251), aac4716. https://doi.org/10.1126/science.aac4716

— Large-scale project attempting to replicate published psychological studies, widely cited as evidence of the replication crisis.

Polanyi, M. (1966). The tacit dimension. University of Chicago Press.

— Stresses the role of tacit knowledge in science, relevant for understanding why codified statistical methods cannot fully capture human knowing.

Language, Thought, and Meaning

Boas, F. (1911). Handbook of American Indian languages. Smithsonian Institution.

— Early anthropological work illustrating how language influences thought, rejecting simplistic linguistic determinism.

Cromby, J. (2012). Beyond belief. Theory & Psychology, 22(6), 766–785. https://doi.org/10.1177/0959354312461546

— Critiques psychology’s reliance on linguistic artifacts as evidence of mental structures, resonating with the argument that psychology studies language more than mind.

Hayakawa, S. I. (1949). Language in thought and action. Harcourt, Brace and World.

— Accessible and enduring account of how language shapes reasoning, directly relevant for framing probability as a linguistic model.

Lakoff, G., & Johnson, M. (1980). Metaphors we live by. University of Chicago Press.

— Explores metaphor as foundational to human thought, illuminating how probabilistic language is structured by metaphorical framing.

Whorf, B. L. (1956). Language, thought, and reality: Selected writings. MIT Press.

— Posthumously collected essays on the relationship between language and perception, important for debates about linguistic relativity.

Measurement Theory and Its Limits

Michell, J. (1999). Measurement in psychology: A critical history of a methodological concept. Cambridge University Press.

— Argues that psychology has misused the concept of measurement, applying numbers where no true quantitative structure exists.

Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103(2684), 677–680. https://doi.org/10.1126/science.103.2684.677

— Defines the widely adopted four scales of measurement (nominal, ordinal, interval, ratio), influential but also criticized for conflating classification with measurement.

Trendler, G. (2009). Measurement theory, psychology, and the revolution that cannot happen. Theory & Psychology, 19(5), 579–599. https://doi.org/10.1177/0959354309341926

— Argues that psychology cannot meet the standards of genuine measurement as achieved in the physical sciences.

Velleman, P. F., & Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician, 47(1), 65–72. https://doi.org/10.1080/00031305.1993.10475938

— Critiques Stevens’ typology, arguing that it oversimplifies the complexity of data types and their statistical treatment.

Philosophy of Science and Knowledge

Kuhn, T. S. (1962). The structure of scientific revolutions. University of Chicago Press.

— Classic account of paradigm shifts in science, illustrating how scientific practice is historically and socially contingent.

Peirce, C. S. (1878). How to make our ideas clear. Popular Science Monthly, 12, 286–302.

— Introduces the pragmatic maxim, linking meaning to practical effects, relevant to understanding probabilistic claims as tools rather than eternal truths.

Popper, K. R. (1959). The logic of scientific discovery. Hutchinson.

— Lays out falsifiability as the demarcation of science, influential in debates about the validity of probabilistic inference.

Putnam, H. (1975). Mathematics, matter and method. Cambridge University Press.

— Essays on philosophy of science and mathematics, including reflections on the interplay between formal systems and empirical reality.