Can Mathematics Describe Experience?

Can Mathematics Describe Experience?

by Gustaf Ullman

Physics and mathematics are extraordinarily powerful. They allow us to predict eclipses, explain why the sky is blue, and design quantum computers. In each case, a structured formalism—equations, symmetries, categories—captures what can be measured and tested. Yet there is a striking boundary: no matter how refined the mathematics becomes, it never seems to tell us what it is like to see red, to hear a melody, or to feel pain.

This boundary is not just a vague intuition. It arises from a structural feature of scientific theories, which I call operational closure. A physical theory is operationally closed if all admissible experiments and their combinations remain within the theory. For example, quantum mechanics is closed under sequential and parallel composition of processes, under conditioning on measurement outcomes, and under coarse-graining of statistics. Thanks to this closure, the theory is empirically complete: everything that can be tested within its domain is representable inside the formalism.

But completeness in this sense is not the same as descriptive adequacy for consciousness. The invariants of a theory—what remains the same across different experimental contexts—are structural. They tell us how a system will behave under tests. They do not and cannot tell us how the resulting state of consciousness feels from the inside. The difference is between characterization and description: science can characterize the conditions and correlates of experience, but description of phenomenal presence belongs to another register.

Why this matters

Many contemporary theories of consciousness, such as Integrated Information Theory (IIT) and Global Workspace Theory (GWT), implicitly promise more than this. They offer quantitative measures or structural criteria and often invite the thought that such quantities might be consciousness. My analysis shows why that leap cannot be justified. Even if a measure perfectly tracks all the conditions under which we report experiences, it still functions as a correlate, not a description of presence itself.

This is not to deny the value of such research. On the contrary, progress on correlations and mechanisms is real and essential. But we should be clear on what kind of progress it is. The explanatory gap is not a temporary lack of detail; it is a principled boundary marked by operational closure itself.

Relation to philosophy

These conclusions echo themes from phenomenology. Husserl emphasized that experience is given in a way that cannot be reduced to objective structures. My approach reframes that insight in mathematical language: operational closure shows why structural invariants cannot cross the threshold into phenomenal description. In this sense, formal mathematics and classical phenomenology converge on the same point from different directions.

Practical consequences

  • We can build ever more precise models of the neural and informational conditions of consciousness.
  • We can test these models experimentally with tools from neuroscience and quantum theory.
  • But we should not expect any model to yield the direct feel of experience as a mathematical output.

Recognizing this limit does not diminish science. Instead, it clarifies what science can and cannot do, and why the first-person dimension of experience retains its irreducible place.

Further reading

For those interested in the technical details, I have developed this argument in a research paper available on Zenodo: Operational Closure, Structure, and the Limits of Mathematical Description of Phenomenal Presence .

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