Language, Mathematics, and Their Limits

How Our Tools of Description Constrain What We Can Perceive

The Prison of Categories

Language shapes reality as much as it describes it. The very structure of our conceptual tools—the words we use, the grammatical forms we inherit, the mathematical frameworks we employ—creates a kind of cognitive prison that can make certain ways of seeing nearly impossible while making others seem inevitable. This is not merely an abstract philosophical concern; it lies at the heart of our difficulty in understanding consciousness, both our own and that of other species.

Consider the fundamental bias embedded in Indo-European languages toward what linguists call “subject-predicate” structure. While the strong version of the Sapir-Whorf hypothesis remains controversial, the basic fact that many Indigenous languages are verb-based or otherwise fundamentally different from Indo-European structures is well-established, and it seems clear enough that language has considerable effect on the way we think. Nearly every sentence we construct implies a world of separate subjects acting upon distinct objects: “The brain produces consciousness,” “The dolphin makes sounds,” “The scientist observes behavior.” This grammatical architecture predisposes us to see reality as composed of isolated entities engaged in external relationships, making it extraordinarily difficult to conceptualize the kind of participatory, relational reality that process philosophers like Whitehead describe or that the fluid, acoustic world of cetaceans might exemplify.

The philosopher Benjamin Lee Whorf famously argued that the Hopi language, with its radically different temporal structure, enables ways of thinking about time and change that are virtually impossible in English.¹ While the strong version of the Sapir-Whorf hypothesis remains controversial, the basic insight is undeniable: the conceptual tools we inherit both enable and constrain what we can think. When it comes to consciousness—particularly forms of consciousness radically different from our own—these constraints may be especially limiting.

The Reification Trap

One of the most pervasive errors in our thinking about consciousness stems from what Alfred North Whitehead called “the fallacy of misplaced concreteness”—the mistake of treating abstract concepts as if they were concrete realities.² Our language systematically turns processes into things, relationships into objects, and dynamic patterns into static entities. We speak of “having” consciousness as if it were a possession, of “the” mind as if it were an object or location, of “a” thought as if it were a discrete object that could be isolated and examined.

This reification trap becomes particularly evident when we attempt to understand cetacean intelligence. We ask questions like “How intelligent are dolphins?” as if intelligence were a substance that could be measured and compared across species. We wonder “What do whales think about?” as if thoughts were discrete objects floating through their minds. We investigate “dolphin communication” as if communication were a process separate from the beings who engage in it, rather than a fundamental aspect of their being-in-the-world.

The reality may be far more dynamic and relational. The acoustic environment in which cetaceans live is one of constant flux, where sound waves overlap, interfere, and create complex three-dimensional patterns of information. In such a world, the boundaries between self and environment, between communication and perception, between individual and collective awareness may be fundamentally different from the discrete, object-oriented reality that our language assumes.

Recent research in cetacean acoustics suggests their communication systems have extraordinary complexity. Dolphins not only use signature whistles that function as individual names, but they can mimic and modify these whistles in ways that may convey complex relational information. Whales engage in coordinated singing that can span hundreds of miles and involve multiple individuals in what appears to be collaborative composition. Some researchers have documented what seems to be real-time collaborative hunting coordination through acoustic signals that suggest a level of collective decision-making that challenges our individual-centered assumptions about consciousness.³

Mathematics as Incomplete Map

Mathematics represents humanity’s most successful attempt to create a universal language for describing reality. Its power is undeniable—mathematical equations can predict eclipses, guide spacecraft to distant planets, and reveal the fundamental structure of atoms. Yet mathematics, for all its precision and universality, captures only certain aspects of reality: quantity, form, pattern, and relationship. It systematically excludes quality, meaning, value, and subjective experience.

This exclusion is not a bug in the mathematical system; it is a feature. Mathematics achieves its power precisely by abstracting from the full richness of experience to focus on those aspects that can be precisely quantified and manipulated. But when we mistake this powerful tool for a complete description of reality, we fall into what Whitehead called “the fallacy of simple location”—the assumption that the mathematical description exhausts what there is to be described.⁴

The implications for understanding consciousness are profound. A complete mathematical description of a brain—every neuron, every synapse, every electrochemical event—would still miss what the philosopher David Chalmers calls “the hard problem”: why and how any of this objective, mathematical description is accompanied by subjective, qualitative experience.⁵ The map, however detailed and accurate, remains fundamentally different from the territory it describes.

This limitation becomes even more acute when we consider cetacean consciousness. The three-dimensional acoustic world in which dolphins and whales live may involve forms of spatial and temporal processing that our mathematically-oriented, vision-based conceptual frameworks are poorly equipped to capture. While we can measure the frequency, amplitude, and directionality of their vocalizations, we have no mathematical tools for describing what it might be like to navigate through acoustic imaging, to “see” through biosonar, or to participate in the kind of acoustic collective awareness that whale songs might represent.

The Need for New Modes of Description

If consciousness is indeed fundamental to reality, and if forms of consciousness exist that are radically different from our own, then we may need to develop new modes of description that can honor both precision and interiority, both objectivity and subjectivity. This does not mean abandoning mathematics and empirical science, but rather supplementing them with additional tools that can address the qualitative dimensions of reality.

Such tools might include phenomenological analysis that maps the structures of experience with the same care that anatomy maps the structures of the body. They might involve contemplative methodologies that train attention to discern subtle features of consciousness that normally escape notice. They might require collaborative research protocols that honor the subjectivity of research participants—including, potentially, non-human participants—rather than treating them merely as objects to be observed.

Most radically, understanding cetacean consciousness might require developing new forms of communication that transcend the current limitations of human language. If dolphins and whales inhabit a world of three-dimensional acoustic images, collective awareness, and fluid boundaries between self and environment, then meaningful communication with them might require technologies and methodologies that we can barely imagine. Some researchers are already exploring computer-mediated interfaces that might allow dolphins to communicate through visual symbols or musical patterns, but such efforts remain in their infancy.⁶

Cetacean Communication: Beyond Human Frameworks

The study of cetacean communication has revealed systems of complexity that challenge our basic assumptions about the nature of language and meaning. Unlike human language, which relies heavily on discrete symbolic units (words) arranged in linear sequences, cetacean communication appears to be fundamentally analog, continuous, and multidimensional.

Dolphin signature whistles, for example, are not simply “names” in the human sense. They appear to be complex acoustic patterns that can be modified in real-time to convey information about emotional state, social context, and relational dynamics. When dolphins mimic each other’s signature whistles, they are not merely repeating sounds but engaging in a form of acoustic empathy that may involve temporarily taking on aspects of another individual’s identity or perspective.

Humpback whale songs present an even more profound puzzle. These elaborate compositions can last for hours, involve complex musical structures with themes and variations, and are shared and modified across entire populations. The songs change over time in ways that suggest cultural transmission and collective creativity. Some researchers have proposed that whale song may function not just as communication but as a form of acoustic architecture—a way of structuring the three-dimensional space of the ocean through sound.⁷

Perhaps most intriguingly, there is growing evidence that some cetacean vocalizations may convey information that goes far beyond anything possible in human language. The echolocation clicks of dolphins not only allow them to navigate but may also convey detailed information about their internal states to other dolphins. The possibility that cetaceans can literally share their perceptual experiences through sound—that one dolphin might be able to “show” another what it sees through echolocation—suggests forms of communication that transcend the symbol-based nature of human language entirely.

The Translation Problem

The challenge of understanding cetacean communication illuminates a deeper philosophical problem: the possibility that there may be forms of consciousness so different from our own that they are, in principle, untranslatable into human conceptual frameworks. This is not simply a matter of complexity or unfamiliarity, but a more fundamental issue of incommensurable ways of being conscious.

The philosopher Thomas Nagel raised this possibility in his famous essay “What Is It Like to Be a Bat?” Nagel argued that while we can study the objective features of bat echolocation in great detail, we cannot possibly know what the subjective experience of echolocating is like because it involves sensory modalities and spatial experiences that are completely foreign to human consciousness.⁸ The same challenge applies, perhaps even more dramatically, to cetacean consciousness.

Consider the implications of living in a truly three-dimensional acoustic environment. For terrestrial mammals like ourselves, space is fundamentally structured by gravity, surfaces, and the limitations of our visual and tactile senses. We inhabit a world of “up” and “down,” of solid boundaries and discrete objects. But for a dolphin, space might be fundamentally fluid, relational, and penetrable through sound. The very concepts of “self” and “other,” “inside” and “outside,” “here” and “there” that structure human experience might be irrelevant or misleading when applied to beings who inhabit such a different reality.

This suggests that understanding cetacean consciousness might require not just new scientific methodologies but new forms of imagination—what we might call thinking toward forms of consciousness that have no human analogues. Such thinking would need to be both rigorously grounded in empirical observation and radically open to possibilities that transcend our current conceptual categories.

Expanding Our Descriptive Capacity

The limitations of language and mathematics in addressing consciousness do not mean that these tools are useless, but rather that they are incomplete. A truly comprehensive understanding of reality—one that can encompass both the objective features that science studies so successfully and the subjective features that constitute our most intimate experience—may require what we might call “conceptual evolution.”

Just as biological evolution has produced increasingly sophisticated forms of consciousness, conceptual evolution might produce increasingly sophisticated tools for understanding consciousness. Such tools might include:

Hybrid methodologies that combine first-person contemplative investigation with third-person empirical observation, allowing researchers to study consciousness from both the inside and the outside simultaneously.

Phenomenological cartography that maps the structures and dynamics of different forms of conscious experience with the same precision that geographical cartography maps the features of physical terrain.

Interspecies communication technologies that might allow meaningful dialogue with forms of consciousness radically different from our own, potentially revealing aspects of reality that are invisible to purely human investigation.

Process languages that can describe dynamic, relational realities without falling into the subject-object dualism that pervades Indo-European languages.

Qualitative mathematics that might extend mathematical precision to include experiential dimensions currently excluded from formal description.

Such developments would not replace current scientific methodologies but would expand them to encompass the full spectrum of reality. They would allow us to maintain the rigor and precision that have made science so successful while opening up domains of investigation that are currently inaccessible to formal study.

The Cosmic Implications

The limitations of our current descriptive tools have implications that extend far beyond the study of cetacean consciousness. If reality includes qualitative dimensions that are fundamental rather than emergent, then our mathematical descriptions of the cosmos may be missing crucial features of what actually exists. The universe that emerges from purely quantitative science—a collection of particles and forces governed by mathematical laws—may be as incomplete as a description of music that includes only the physics of sound waves while ignoring melody, harmony, and aesthetic meaning.

This suggests that the universe itself might be far stranger, richer, and more meaningful than our current conceptual tools allow us to perceive. The convergent evolution of large-brained consciousness in multiple lineages may be evidence not of unlikely accidents but of the universe’s fundamental tendency toward increasing complexity, awareness, and meaning—tendencies that our current scientific language can describe only indirectly, through their objective manifestations.

If this is correct, then developing new tools for understanding consciousness is not just an academic exercise but an essential step in humanity’s ongoing effort to understand its place in the cosmos. The cetacean presence in our oceans may represent not just a fascinating biological puzzle but a crucial clue to aspects of reality that we are only beginning to glimpse.

As we will explore in the next section, even the foundations of physical science—quantum mechanics—seem to resist simple mathematical interpretation and may require concepts that bridge the apparent gap between mind and matter. The mechanistic worldview that has dominated Western thought for centuries may be giving way to something far more participatory, relational, and alive.