Two Observers, Two Truths: Why Physics Runs on Transformations

Johannes Falk on September 10, 2022 · 7 min read

Physical theories are written in classical two-valued logic, yet relativity and quantum mechanics routinely produce observer-dependent, mutually contradictory measurements. We argue that transformations are precisely the machinery that reconciles a two-valued science with a polycontextural world, in the sense of Gotthard Günther’s transclassical logic.

Imagine two physicists on two trains passing each other at half the speed of light. One physicist measures the time it takes for a laser pulse to travel from the carriage ceiling to the floor. The other observes this measurement from his own carriage and measures a different time. Both experiments are conducted correctly, and both measurements strictly follow the laws of physics. Yet, when the two physicists compare their results, they notice a discrepancy, and there is no basis for deciding who is right.

Such situations cannot be described using classical logic. A statement is either true or false; if my measurement contradicts yours, at least one of us must be mistaken. In the train example, however, special relativity teaches us that both measurements are correct. The same kind of tension appears in the quantum world, where light manifests as a wave in one experiment and as a stream of particles in another. Physics handles such situations as a matter of course.

In our paper Physics is Organized Around Transformations Connecting Contextures in a Polycontextural World (with Edwin Eichler, Katja Windt, and Marc-Thorsten Hütt, published in Foundations of Science), we look at this tension through the lens of a largely forgotten logical framework: Gotthard Günther’s theory of polycontexturality.

Binary logic presupposes a “view from nowhere”

Mathematics (and by extension, physics) adopts an assumption from Aristotle: truth is objective and independent of the observer. Something either is or is not—and which of these applies does not depend on who is asking the question. This two-valued (“binary”) logic, along with its associated objectivity, forms the foundation of Boolean algebra—and thus, fundamentally, of all the formal modes of thought and reasoning we use to formulate physical theories.

The problem, however, is that physics repeatedly yields results that depend on the observer. Heisenberg put it succinctly: “What we observe is not nature itself, but nature exposed to our method of questioning.” We are attempting to describe the world using a logic that rules out observer-dependent truth—yet we are dealing with a world that is permeated by precisely that. How can this possibly work?

Günther’s polycontextural logic

One answer comes from a thinker likely rarely read by physicists: the German logician and philosopher Gotthard Günther (1900–1984). Building on the theories of Hegel and Fichte (who recognized that the “thinking I” is usually part of the system it reflects upon and speaks about), Günther proposed that reality possesses a polycontextural structure. He argued that classical logic is not incorrect, but rather must be viewed as local.

According to his thesis, reality breaks down into contextures. These contextures are distinct domains within which ordinary binary logic applies without restriction. A statement may be true in one contexture while the same statement is false in another.

Problems arise only when statements from different contextures are compared as if they belonged to a single, global logic. From Günther’s perspective, reality is polycontextural: a network of many interconnected contextures, with no overarching perspective standing above them. The contextures form a heterarchy rather than a hierarchy—no observer standpoint stands above another.

The simplest interesting structure is what Günther calls the proemial relation: three interlocking contextures. In the first, a subject observes an object. In the second, this entire process of observation itself becomes the object of another observer. A third contexture mediates between the object as directly perceived and the object as seen through the perspective of another. This triangle constitutes the smallest logical setting in which self-reference and genuine subjectivity can be represented without contradiction.

Example 1: the two trains

Back to the trains. Inside their carriage, observer $O_1$ measures $t_1 = h/c$ for the falling laser pulse. From the other train, $O_2$ sees the pulse trace a diagonal of length $\sqrt{h^2 + s^2}$, and since the speed of light is the same for everyone, they get a different time. Each train is, in Günther’s vocabulary, a contexture: a domain with its own perfectly valid classical physics — and its own perfectly valid classical logic. Read through a single global logic, “$t_1$” and “$t_2$” contradict. Read polycontexturally, they are simply true in different places.

What stitches the two back together is the Lorentz transformation,

which is exactly an exchange relation between contextures: a rule that lets one observer map another observer’s subjective measurement into their own frame, without ever introducing an absolute time above both.

It should be noted that the Lorentz transformation is deterministic, and with due care, the entire situation can be forced back into an Aristotelian framework. Here, polycontexturality proves to be an illuminating redescription. In the realm of quantum mechanics, however, this approach appears less optional.

Example 2: wave or particle?

When light is passed through a double slit, it interferes: it is a wave with a measurable wavelength. When light strikes a metal, electrons are produced in the form of energy quanta: it is a stream of particles. Two experimental setups, two contradictory ontologies—and unlike trains, no clever shift in perspective resolves the contradiction. Einstein himself called them two contradictory pictures of reality that only work together.

In our framework, each experimental setup encompasses its own context. An observer who measures momentum (wavelength) and an observer who measures position do not disagree on a common fact; they generate facts in different logical domains. Heisenberg’s uncertainty principle

is then read as the quantitative statement that these contextures cannot be merged: sharpening one collapses the other. And once again there is a transformation doing the mediating work — the Fourier transform: a position wavefunction sharply peaked in space (small $\Delta x$) maps to a momentum distribution spread out everywhere (large $\Delta p$), and vice versa.

Quantum mechanics is also where the polycontextural reading stops being a metaphor. Measurement outcomes from incompatible experiments provably cannot be embedded into a single probability space (a single $\sigma$-algebra), and no-go results in the spirit of Frauchiger–Renner show that different observers can derive genuinely irreconcilable claims. A logic that structurally refuses a global truth assignment is a rather precise description of the situation.

What this is, and what it isn’t

This is a reinterpretation, not new physics. Nothing here changes a prediction of relativity or quantum mechanics, and the argument does not need you to abandon binary logic — quite the opposite, it explains why binary logic keeps working. What it offers is a different reading of the architecture of physical theory: Transformations let each observer project the polycontextural whole onto their own two-valued contexture, become aware of their own relativity, and correct for it.

Researchers engaging with Günther’s work are primarily based in sociology and systems theory. For them, the essay offers something unusual: examples from the hard quantitative sciences in which the formalism delivers tangible results. Conversely, it opens up a new perspective on familiar mechanisms for physicists—and suggests that fields dealing with distributed, observer-dependent classification (such as multi-agent systems or machine learning, with its demonstrably undecidable questions regarding learnability) might be the next candidates for a polycontextural approach. The same notion of interacting, subjective contextures can be turned into a concrete model of agents striving for consensus without a shared notion of “right” — which behaves like a statistical-physics system with a genuine phase transition and self-organised, sometimes stably mistaken, collective patterns. A natural next target is machine learning, where classifiers still largely live inside a single binary logic, and where undecidable statements are known to appear.

Frequently Asked Questions

Why can two observers measure different times and both be right?

Each observer occupies their own contexture: a domain with perfectly valid classical physics and classical logic. In the moving-trains example, $O_1$ measures $t_1 = h/c$ while $O_2$ measures a longer diagonal path. Read through one global logic these contradict; read polycontexturally they are simply true in different places.

Does quantum mechanics need polycontextural logic more than relativity does?

Yes. Special relativity is the gentle case, since the Lorentz transformation is deterministic and can be squeezed back into an Aristotelian frame. Quantum mechanics cannot: incompatible measurements provably resist a single probability space, and Frauchiger–Renner-style results show observers deriving genuinely irreconcilable claims.

How do transformations reconcile contradictory measurements?

A transformation is an exchange relation between contextures. The Lorentz transformation lets one observer map another’s measurement into their own frame without an absolute time above both; the Fourier transform links position and momentum descriptions. Each lets an observer project the polycontextural whole onto their own two-valued view and correct for their own relativity.