Reason: Spurious Quantification

The mistake of thinking that everything in the world can be quantified, which is contrary to all reason. It's a modern invention.

Author’s Preface

In this essay in the Reason series, I attempt to reflect on how language, mathematics, and measurement intersect with the world they describe—or try to describe. Although it seems clear that many aspects of the world can be quantified under some framing, I find myself increasingly uneasy with the assumption that anything important must be measurable, or that measurement captures the essence of the thing.

Even in the material world, quantification depends on selective abstraction. When we shift to the inner world—the world of thought, awareness, and subjective experience—I find it even harder to imagine that formal, numerical description could be meaningful in any straightforward way. This leads me to suspect that the generalization of quantification beyond its most effective domains may be unjustified, and that we have perhaps mistaken a narrow method for a universal principle.

Introduction

There is a growing tendency in modern intellectual and institutional life to assume that all real phenomena are, or should be, measurable. This assumption underlies much of contemporary policy, technology, science, and education. It is embedded in algorithms, statistical models, and management systems. And yet it is rarely questioned.

One might initially suppose this tendency emerged from the practical success of mathematical modeling in the physical sciences. That success is real, and its practical consequences are substantial. But it may also have led to an extrapolation—a creeping extension of quantitative logic into domains where it may not be applicable, or where its use distorts more than it clarifies.

This essay explores the limits of quantification, not as a technical failure but as a conceptual boundary. It aims to clarify why mathematical and linguistic representations, while useful, cannot capture the fullness of the world. In particular, it draws attention to the distinction between things and their abstracted properties, and to the widespread confusion between models and the objects they claim to describe.

Discussion

Representation vs. Reality

Language is representational. It names, describes, distinguishes, and interprets. But it does not contain the thing it names. A word points; it does not present. Mathematics is also representational. It describes relationships, transformations, and structures. But it too remains within the symbolic domain.

Both systems—natural language and mathematics—function as modes of symbolic reference. What they express is partial, filtered, and conditioned by the constraints of their form. Mathematics builds upon the foundation of natural language, refining it with greater internal rigor and precision, but it does not escape the representational condition. It is still language, albeit more narrowly structured and more selective in what it can express.

What these systems offer are approximations, not identities. They highlight aspects of the world, but never all of it. And they always do so under assumptions—framing decisions that determine what counts as relevant, tractable, or legible.

What Mathematics Can Describe

Mathematics excels at expressing regularity, proportion, structure, pattern, and so on. It describes systems under specified conditions. When applied to well-defined domains—idealized physics problems, constrained engineering environments—it allows for predictive and explanatory control.

But even in these settings, mathematics does not describe everything. It describes selected features under abstraction. The meaning of a variable or equation often depends on context that lies outside the formalism. And as systems become more complex, the assumptions that underpin mathematical modeling—continuity, linearity, equilibrium, independence—begin to fail.

Moreover, what mathematics can describe often depends on what is assumed to be measurable. When key features of a system are qualitative, relational, or historically contingent, their expression in mathematical terms becomes tenuous or speculative.

The Mystery of Awareness

There are domains in which quantification may not only be difficult but conceptually incoherent. Awareness, for instance, is not located in space or reducible to component parts. It has no clear units. It is not extended, accumulated, or directly observable from the outside. It is not data.

Understanding is similarly resistant. One may model certain behavioral outputs or simulate linguistic structure, but this does not constitute comprehension. Meaning and reference arise in ways that are situated, contextual, and tacit. It is difficult to see how these could be exhaustively described in mathematical form.

Reference itself—the act of pointing from a word or symbol to a thing—is not a property with magnitude. It is an act or function of consciousness, one that seems to presuppose the very understanding that formal systems try to reconstruct. It may be that these basic aspects of mind are simply outside the scope of quantification.

Do Physicists Make a Mistake?

Some physicists, particularly in theoretical contexts, appear to treat mathematics not just as a descriptive language but as the substance of reality itself. While such views may be intended metaphorically, they often seem to cross into ontological territory.

There are, of course, physicists who remain cautious about this. But in reading some accounts of cosmology, quantum theory, or the so-called "mathematical universe hypothesis," one does encounter what may be a category mistake: the treatment of a modeling language as the thing being modeled.

Even if some version of mathematical realism were correct, it would remain a leap to say that mathematics is reality in any exhaustive or literal sense. Such claims often bypass the selective, interpretive, and approximative nature of modeling. They may confuse the coherence of a system with the truth of its content.

Idealization and the Non-Ideal World

Mathematics depends on idealization. A perfect circle, an infinitely thin line, a massless particle—these are not features of the world but constructs of the system. They allow for tractable analysis but do not correspond directly to experience.

When mathematical descriptions succeed, they often do so because the idealizations are close enough to relevant aspects of the system. But they always involve simplification. The world, however, is not ideal. It is irregular, unstable, recursive, and noisy. It exhibits exceptions, contradictions, feedback, and discontinuity.

One might say that mathematical models describe ideal worlds perfectly. But the real world is not ideal. It departs from the assumptions required for closed-form description. This gap is not simply a matter of better instrumentation. It is built into the nature of abstraction.

Limits of Mathematical Applicability

There are parts of the world—especially complex, recursive, or historically determined systems—that resist mathematical formalization. This is not a limitation of the field, but a reflection of the nature of those systems. Many biological, ecological, or sociocultural systems involve interactions that are context-dependent and not well-described by fixed variables.

Even in the physical domain, large portions of real-world behavior defy comprehensive modeling. Turbulence, fracture mechanics¹, and many-body interactions continue to challenge predictive formalism. These are not arcane exceptions. They are widespread and materially significant.

And when one turns to domains such as ethics, interpretation, imagination, or intention, the difficulty becomes categorical. These domains appear to be structured around ambiguity, plurality, and irreducible specificity. It is not obvious that they could be meaningfully formalized without losing what is essential to them.

Mathematics Handles Only Certain Aspects

Mathematics can only describe aspects of reality that can be clearly framed, discretized, or ordered. It does not describe things in their full complexity, particularity, or embeddedness. It cannot easily capture open-endedness, contextual shift, or irregular contingency.

This does not make mathematics defective. It makes it selective. But when this selectivity is forgotten—when mathematical description is taken to be equivalent to full understanding—something important is lost. The aspects not measured may turn out to be those that matter most.

A Tree Is Not Its Model

A tree may seem like a simple object, but it serves as a useful example. One can measure its height, trunk diameter, canopy spread, leaf count, or biomass. One can model its growth over time. These are legitimate and useful abstractions.

But none of this constitutes the tree. The tree is a living organism, affected by weather, soil, season, and injury. Its shape, resilience, and presence reflect years of interaction with a specific environment. These features do not reduce easily to parameters.

Even a model as elegant as a fractal branching pattern, while suggestive, does not describe any particular tree. It is a type, not a thing. The tree remains a singular entity, not exhaustible by its measured properties.

Abstraction versus Object

Measurement always involves choice. One selects which properties to extract, how to define them, and what scale to use. These choices are shaped by the measurer’s purpose and the available tools. They are not neutral.

What results is not the object, but a constructed representation—a kind of summary or projection. It omits whatever was excluded in the framing process, whether texture, asymmetry, history, or interaction.

What is often forgotten is that these choices are made by people, within systems of interest and interpretation. The abstraction is only ever a slice, not the whole. It is a lens, not a mirror.

Quantification Omits Essence

One can quantify as many aspects as one wishes, but the result will still not be the thing itself. It will be a constellation of measures—selected, framed, and interpreted according to specific goals.

The essence of an object, its integrity as a presence in the world, is not the sum of its attributes. It is not recoverable by aggregation. That which gives something its particularity—its coherence, its embeddedness, its meaning—is not captured in the quantities we extract.

Language and Mathematical Expression

Mathematical expression does not eliminate the need for language. It builds on it. Every symbol must be defined, every operation interpreted, every result explained. We understand mathematics through natural language. That is how it is taught, shared, and discussed.

In that sense, mathematics remains a sub-language—a rigorously defined and useful one, but not self-sufficient. It cannot interpret itself. Its meaning depends on prior forms of understanding.

This suggests that mathematical clarity rests on a foundation of interpretive labor. That labor is linguistic, experiential, and shared. It is not formalizable within mathematics itself.

Are Some Scholars Cognitively Entrapped?

It is tempting, within specialized domains, to forget the limits of one's tools. When a formal system succeeds in describing part of the world, it becomes easy to imagine that all of the world should be accessible to the same approach. This is a kind of cognitive entrapment.

Certain communities—scientific, technical, philosophical—may, over time, become insulated from these questions. Their systems work within defined scopes, but the boundaries of applicability are left unexamined. In such settings, conceptual caution may give way to metaphysical confidence.

The result is often a narrowing of perspective, not an expansion of insight. The framework begins to dictate what counts as knowledge. What lies outside is treated as irrelevant, illegitimate, or unintelligible.

Summary

This essay has argued that the belief in universal quantification reflects a category mistake. It arises from conflating abstraction with essence, description with substance, and representation with reality.

Quantification can be useful, and in certain domains, indispensable. But most of the world—particularly in its subjective, historical, and relational aspects—cannot be meaningfully captured in numerical terms. Even in the physical domain, abstraction often omits what matters.

What we measure are not objects, but properties chosen under constraints. What remains unmeasured is not therefore unreal. It is often what gives the object its presence, identity, and significance.

Readings

Cartwright, N. (1983). How the laws of physics lie. Oxford University Press.
Annotation:
Nancy Cartwright’s influential critique challenges the view that physical laws offer universally true descriptions of the world. Instead, she argues that such laws are often idealizations that “lie” by omitting the messy, complex features of actual systems. This directly supports the essay’s claim that mathematics is not a mirror of reality, but a selective and often misleading abstraction. Cartwright emphasizes that the real world is irregular and context-sensitive, which aligns with the idea that quantification captures only partial aspects of phenomena.

Hacking, I. (1990). The Taming of Chance. Cambridge University Press.
Annotation:
Hacking’s historical-philosophical account of probability traces how statistical thinking reshaped our conception of knowledge and control. His treatment of how chance and statistical reasoning were culturally constructed supports the essay’s cautionary stance toward the reification of quantification. Hacking makes clear that probability and statistical inference are not pure reflections of reality but frameworks with historical, philosophical, and institutional underpinnings. This aligns with the essay’s implicit critique of spurious precision and unreflective formalism.

Polanyi, M. (1966). The tacit dimension. University of Chicago Press.
Annotation:
Polanyi argues that all knowledge contains a tacit component—that is, things we know but cannot fully articulate or formalize. This is crucial to the essay’s skepticism toward the idea that understanding or awareness can be quantified. Polanyi’s account reinforces the distinction between representation and the lived reality it attempts to describe. His idea that “we know more than we can tell” captures the irreducibility of subjective experience and contextual understanding, key themes in the argument against total quantifiability.

Daston, L., & Galison, P. (2007). Objectivity. Zone Books.
Annotation:
Daston and Galison provide a detailed history of the evolving ideal of objectivity in science, including how measurement practices and representational norms changed over time. They highlight the cultural and philosophical shifts that led to the privileging of mechanical objectivity and quantification. Their work supports the essay’s historical framing of quantification as a modern invention rather than a universal or timeless epistemic principle. It also gives context to why measurement and mathematization came to dominate certain academic and professional discourses.

Midgley, M. (2001). Science and poetry. Routledge.
Annotation:
Midgley’s book critiques scientism—the belief that science and its methods are the only valid way to understand the world. She argues for the legitimacy of non-quantitative, interpretive, and expressive modes of knowing. This is directly relevant to the essay’s claim that some realities (e.g., consciousness, meaning, particularity) may not be mathematizable without distortion. Midgley also warns against the metaphysical overreach of scientific language, echoing the concern that certain scholars become trapped in frameworks that exclude what cannot be measured.

Gigerenzer, G. (2002). Reckoning with risk: Learning to live with uncertainty. Penguin.
Annotation:
Gigerenzer challenges the overreliance on statistical risk models, particularly in medicine, economics, and public policy. He demonstrates how formal models often mislead, especially when applied to situations with high variability or unclear causality. This complements the essay’s argument about the dangers of extending mathematical reasoning into domains where it lacks conceptual fit. Gigerenzer’s work also connects to the theme of tacit understanding, as he emphasizes how human intuition often outperforms formal models in complex real-world scenarios.

Smith, L. A. (2007). Chaos: A very short introduction. Oxford University Press.
Annotation:
Smith’s primer on chaos theory illustrates how sensitivity to initial conditions and nonlinearity make certain systems inherently unpredictable—even if they are deterministic in principle. This has direct implications for the discussion of idealization and the limited scope of mathematical modeling. The presence of chaos in physical systems underscores the essay’s point that mathematical tools often fail not due to human error but because the world itself resists formal encapsulation. Smith’s clarity makes these ideas accessible and relevant to non-specialists.

James, W. (1907). Pragmatism: A new name for some old ways of thinking. Longmans, Green and Co.
Annotation:
Though older, James’s work remains highly relevant. His pragmatic theory of truth—as that which works in experience rather than what corresponds to an abstract ideal—directly contests the Platonic idealism that undergirds mathematical realism. James supports the idea that concepts, including mathematical ones, are tools, not essences. His pragmatic approach aligns with the essay’s core thesis: that descriptions are useful but partial, and should not be confused with the things they describe.

Suppe, F. (1977). The structure of scientific theories. University of Illinois Press.
Annotation:
This edited volume provides a philosophical treatment of how scientific theories are constructed, including essays on idealization, modeling, and the limits of empirical correspondence. Suppe and contributors discuss the difference between theory and world, model and object—exactly the kind of distinction the essay urges readers to keep in mind. It helps frame quantification as one epistemic tool among many, and underscores the importance of metatheoretical awareness.

1. Hitting a walnut with a hammer is a real-world example involving fracture mechanics—albeit in a simple, informal way.

Here's how:

1. Crack Initiation

The walnut shell already contains micro-cracks and natural weak points—typically along the seam or ridges. When force is applied by the hammer, the stress concentrates at these points.

2. Stress Concentration

The curved, brittle shell geometry focuses the impact stress. The hammer’s force isn’t spread evenly but is amplified at sharp edges, flaws, or thin sections of the shell—especially where the shell's curvature changes.

3. Crack Propagation

Once the stress intensity exceeds the shell’s fracture toughness, cracks grow rapidly—usually along existing flaws or seams—leading to sudden, brittle fracture. The shell splits, often explosively, due to the stored elastic energy.

4. Material Properties

The shell is a brittle material, meaning it doesn’t deform much before breaking. This is exactly the kind of material behavior that early fracture mechanics (like Griffith’s theory) was developed to understand.

The act of cracking a walnut with a hammer is governed by the same physical principles studied in fracture mechanics: how flaws lead to failure, how stress is concentrated, and how materials break under load. It's an everyday demonstration of a highly general physical principle.