At the Edge of Measurement Assumptions

When special relativity first started to make intuitive sense to me, I found one particular feature deeply intriguing.

I had already written about this while trying to understand why the speed of light is constant. What struck me was not merely the familiar statement that every inertial observer measures the same value (c).

It was something deeper.

Reality seems to behave as though there is a boundary between what mathematics allows us to describe and what physics allows us to regard as definite.

The idea I want to explore is roughly this:

When there is no possible physical relation by which some information could be defined or distinguished, reality does not merely hide that information from us. The corresponding quantity may have no physical definiteness at all.

That is very different from simply saying that we do not know something.

Suppose I place a ball inside a closed box. I may not know exactly where the ball is, but there is nothing incoherent about saying that it has some position inside the box. My ignorance does not make position meaningless.

But there seem to be other questions where the physical situation itself supplies no relation capable of defining the quantity we are asking for.

The first place I noticed this clearly was relativity.

Then I realised that quantum mechanics appears to contain something strangely similar.

And that led me to a more uncomfortable question.

We routinely describe physical systems using vectors, vector spaces, fields, operators, coordinates and manifolds. These structures are so successful that it is easy to stop treating them merely as mathematical language and begin imagining that reality itself must literally possess all of the same structure.

But does that follow?

I am no longer convinced that it does.

Which is slightly funny, because I named the linear algebra series I have been writing The Grammar of Reality.

Perhaps that name was more appropriate than I realised.

A grammar can be extraordinarily powerful.

But a grammar is not the thing it describes.

Relativity and Errors of Our Mental Model

A recent conversation made the issue particularly clear.

Imagine I am floating in space and throw a ball forward at speed (u) relative to myself.

If I am already moving at speed (v) relative to another observer, classical mechanics tells us that the other observer should measure the ball travelling at roughly

$$u+v.$$

So it is completely natural to ask why light should be different.

If a moving source emits light at (c) relative to itself, why should another observer not measure something like

$$c+v$$

or

$$c-v?$$

There is one important clarification first.

Saying that light travels at (c) relative to its source is not an alternative to special relativity.

Special relativity already says that.

The source is itself an inertial observer, and it also measures the light at (c).

The genuinely different proposal would be that the source frame has some privileged status, so that light leaves the source at (c), while observers moving relative to the source measure something different.

That is where the classical mental model becomes interesting.

For the thrown ball, there are two physical systems that continue to exist after the throw.

There is the thrower.

There is the ball.

Their separation changes with time.

That relation is physically accessible, so relative velocity between them is well defined.

The source also matters when light is emitted.

It determines where and when the electromagnetic disturbance is created.

But after emission, something changes.

The disturbance has left the source.

The source may later accelerate, stop, or even cease to exist.

Yet the already emitted disturbance continues.

So what physical structure requires the original source to remain the privileged frame of that disturbance?

There is nothing preventing us from writing a mathematical model in which it does.

The more interesting question is whether reality has supplied the physical relation required to justify that interpretation.

The source determines the creation event.

It does not automatically remain the reference frame for everything that follows.

Coordinates Make the Classical Picture Almost Too Easy

The classical intuition is easy to preserve because mathematics allows us to place a coordinate grid over the entire situation.

We write

$$x, y, z, t$$

and describe the disturbance as moving through that grid.

But the coordinate grid is a description.

Suppose we are moving toward or away from the source.

We can calculate a velocity relative to the source.

We can also calculate a velocity relative to some imagined propagation medium.

Mathematics is perfectly happy to let us do either.

But the ability to calculate a quantity does not establish that the quantity refers to something physically real.

Velocity relative to the source is grounded in an actual relation.

The source exists.

The observer exists.

Their changing relation can be measured.

Velocity relative to a medium is physically meaningful only if that medium supplies some distinguishable physical state of motion.

Sound gives us an obvious example.

Air has molecules, pressure, density and local motion.

It can flow.

So saying that sound travels at some velocity relative to the air refers to an actual physical relationship.

Now consider vacuum.

If we treat empty space itself as a propagation medium in the same physical sense as air, then we have quietly given the vacuum a physically meaningful state of rest.

And vacuum is not a local object sitting in one laboratory.

It surrounds stars.

It lies between galaxies.

It exists around moving observers throughout the universe.

So once we give the vacuum its own physically distinguished state of rest, we have effectively reintroduced a preferred frame.

We can certainly place a mathematical coordinate grid over vacuum.

But nothing in the physical vacuum identifies one particular inertial coordinate system as the one that is truly at rest while all the others move through it.

The grid came from us.

If we promote that grid into physical structure belonging to the vacuum, we have added something that the physical situation itself did not supply.

That is the mental-model error I am interested in.

Mathematics permits us to define more relations than reality necessarily gives us permission to interpret physically.

The equations may continue perfectly well.

But their symbols should not automatically be promoted into ontology.

Doppler Does Not Need the Source to Drag Light Along

The Doppler effect sometimes makes the source-relative picture feel more natural.

But it does not require light to retain the velocity of its source.

Suppose a source is moving away from us.

It emits one electromagnetic disturbance.

A short time later it emits another.

But the source has moved during that interval.

So the second emission occurs from a different location.

If the source is receding, successive emission events occur progressively farther away.

If it is approaching, they occur progressively closer.

That changes the spacing between successive wavefronts and therefore the observed frequency.

The important distinction is

$$\boxed{\text{source motion changes where and when successive disturbances are created}}$$

rather than

$$\boxed{\text{source motion changes the propagation speed of an already-emitted disturbance}}.$$

The source influences the emission pattern.

It does not need to remain attached to the emitted disturbance as its permanent kinematic reference.

A concise way to say it is:

The source determines where and when the disturbance is created. It does not remain the reference frame for its propagation.

Undefined Is Not the Same as Unknown

This is where relativity begins to suggest something broader.

Consider the question:

What is my absolute velocity through empty space?

Classically, that sounds like a perfectly meaningful quantity whose value we simply have not discovered.

But what would define it?

Velocity must be relative to something.

If vacuum supplies no physically distinguishable rest structure, then perhaps there is no hidden answer.

Perhaps the quantity is not unknown.

Perhaps it is undefined.

Unknown means

$$\text{the quantity has a value but I do not know it}.$$

Undefined means (as I would like to interpret in this blog)

$$\text{the physical relation required to assign such a value does not exist}.$$

That distinction is the heart of this article.

And it also reveals something important.

Where matter or radiation does supply a physically distinguishable relation, a frame can become meaningful.

The cosmic microwave background is a good example. We can define motion relative to the frame in which its large-scale radiation distribution is approximately isotropic.

But that is not an absolute frame built into spacetime.

It is a frame defined relative to an actual physical distribution.

That distinction fits the principle perfectly.

No relation, no physically selected frame.

A measurable relation appears, and a relational frame can appear with it.

The Same Pattern Appears in Quantum Mechanics

Quantum mechanics seems to confront us with something similar.

We write a quantum state as

$$|\psi\rangle.$$

We can expand it in a basis.

We can apply operators.

We can evolve it continuously.

The mathematics is extraordinarily precise.

But precision of representation does not automatically imply continuous physical definiteness of everything represented.

Within the interpretation being explored here, quantum mechanics seems to warn us against imagining that every mathematically expressible observable must secretly carry an ordinary classical value before measurement.

And here we reach the difficult part.

Measurement producing a definite outcome does not require us to say that nothing existed beforehand.

But it also does not require us to say that something existed beforehand as a completely well-defined classical state carrying all the properties that later become measurable.

I want to reject both easy pictures.

One extreme says, in effect:

$$\text{before measurement, there is nothing physically there}.$$

I do not think we need that.

The other says:

$$\text{before measurement, there is a completely definite hidden classical state}.$$

I do not think we are entitled to assume that either.

There is a third possibility, and it is much harder to think about:

$$\boxed{\text{physical existence without complete physical definiteness}}.$$

Something physical may exist and evolve without existing as a completely specified classical object.

That is uncomfortable because our intuition usually fuses existence and state-definiteness.

We tend to think that if something exists, it must already have a definite position, a definite trajectory, a definite collection of properties, and so on.

But perhaps that assumption itself is too strong.

Reality may permit existence without granting every mathematically expressible quantity a definite physical value.

Measurement would then do neither of the two simplistic things.

It would not create reality from nothing.

And it would not merely reveal an already complete hidden classical state.

It would establish a physically distinguishable relation in which some information becomes definite.

So the narrower claim is:

We should not assign a definite classical structure to an unmeasured process merely because our mathematical representation allows us to interpolate one.

When Classical Electromagnetic Structure Becomes Measurable

Consider the familiar drawing of an electromagnetic wave:

$$E(x,t),\qquad B(x,t).$$

We draw beautiful sinusoidal peaks and valleys filling space.

These fields are extremely successful mathematical descriptions.

But the success of the description does not by itself establish that every point of completely empty space possesses a tiny independently definite classical field value in exactly the ontological sense suggested by the picture.

Now take a macroscopic radio signal.

The energy of one photon is

$$E=h\nu.$$

At radio frequencies, each individual photon carries very little energy.

A macroscopic radio transmission can therefore involve enormous occupation numbers and stable collective phase and amplitude relations.

At that scale, field amplitude, phase, wavelength and polarization become experimentally resolvable to extraordinary accuracy.

The classical wave becomes a superb physical description.

This is important because the idea of undefinedness should not become an excuse to reject classical physics.

Quite the opposite.

Physical structure should be granted as much definiteness as the physical regime itself makes distinguishable.

Classical structure can emerge where the physical situation supports it.

The mistake would be assuming that because a classical representation works extremely well at one scale, every element of that representation must remain ontologically literal at every scale.

What about QFT?

At this point an obvious objection appears.

People often say that quantum field theory gives the vacuum a rich structure.

But we should look carefully at what that statement actually means.

Quantum Field Theory (QFT) certainly gives us a mathematically rich description of the vacuum state and predicts non-trivial observable consequences associated with it.

That does not mean QFT gives us a little mechanical picture of vacuum containing definite hidden objects, physical coordinate markers or a preferred state of rest.

This is where phrases such as “virtual particles popping in and out of existence” become dangerous if taken literally.

Virtual particles are not directly observed little objects inhabiting empty space. They are elements of perturbative descriptions.

Likewise, when we say that something can “emerge from the vacuum,” the physically definite part of the statement appears when an interaction produces an observable event.

A detector responds.

Another system interacts.

A correlation is measured.

A particle is detected.

These tell us that even what we call vacuum participates in non-trivial physics.

They do not automatically tell us that the vacuum itself contains a continuously definite hidden mechanical structure between those events.

So when someone says QFT gives the vacuum “structure,” I think the immediate question should be:

What exactly is physically distinguishable about that structure independently of an interaction?

The mathematical description can be rich.

The observable consequences can be rich.

But that is not the same thing as giving vacuum an internal ruler, clock or preferred frame.

Indeed, the ordinary QFT vacuum is Lorentz invariant rather than selecting one inertial state of rest.

So QFT does not obviously break the argument here.

If anything, it sharpens it.

The vacuum may support observable quantum effects without itself supplying a physically distinguished coordinate structure or state of rest.

Are Vector Spaces Reality or Representation?

This brings me back to linear algebra.

We represent physical states using vector spaces.

That gives us addition,

$$u+v,$$

scalar multiplication,

$$\alpha v,$$

operators,

$$Av,$$

eigenvectors, projections, spectral decompositions and much more.

It is an astonishingly effective grammar.

But must the universe of physical states itself literally be a vector space?

I am not sure.

There is nothing wrong with using vector spaces wherever the physics supports them.

The mistake would be assuming

$$\text{successful vector representation}$$

therefore

$$\text{reality itself is fundamentally a vector space}.$$

Those are different claims.

Perhaps linearity is local.

Perhaps it belongs to our representation of nearby changes, measurable alternatives or transformations.

Perhaps the deeper state structure is not itself linear.

Geometry gives us a useful analogy.

A curved manifold is not generally a vector space.

Yet at each point (p), it has a tangent space

$$T_pM,$$

which is a vector space.

Locally, linear algebra works beautifully.

We can differentiate.

We can define directions.

We can describe infinitesimal motion.

We can apply linear maps.

The global structure need not itself be linear for local linear mathematics to be exactly the right tool.

Perhaps something similar is true of physical state space.

There may be some deeper structure

$$\mathcal S$$

which is not itself a vector space, while sufficiently local descriptions admit linear spaces and operators.

That would not make linear algebra less important.

It would simply change what we mean by fundamental.

And perhaps The Grammar of Reality was the right name after all.

Grammar can be indispensable without being identical to what it describes.

General Relativity Does Not Begin With a Universal Grid

General Relativity (GR) becomes especially interesting from this perspective because it is often mentally pictured almost backwards.

The popular image is that GR gives us a giant spacetime grid and then tells us how mass bends it.

But GR does not require a physically privileged universal coordinate grid.

We can begin instead from things that are physically definable.

A clock here.

Another clock there.

A freely falling object.

A light signal.

One physical system relative to another.

These provide measurable relations.

Around one physical situation we can construct a local measurement framework.

Around another location we can construct another.

The coordinates are part of the representation of those measurements.

They are not little physical labels presumed to be embedded inside empty space.

That distinction matters.

We deal with measurements defined by measurable entities.

The mathematics then organises those measurements.

GR's Game Is Local Calibration

Once viewed this way, the conceptual game of GR becomes surprisingly clean.

At one location, physical systems allow us to define local quantities such as elapsed time, spatial separation, direction and free-fall motion.

At another location, another possible physical setup allows us to do the same.

Those local measurement systems do not have to remain calibrated against each other in one universal way from place to place.

A clock here and a clock there need not accumulate time identically.

A ruler defined locally does not have to extend into distant regions in the simple way flat-space intuition suggests.

A local inertial frame here does not have to extend unchanged everywhere else.

So rather than saying

there is one universal grid and gravity distorts it,

I find it more useful to say:

Local measurement systems can be defined wherever physical relations allow them, and GR tells us how those local measurements relate from one region to another.

Once those relations are known, we can predict how clocks compare, how freely falling objects move, how light connects physical events, how orbits behave and how gravitational lensing occurs.

Gravity is not an additional force painted onto a universal background.

It is encoded in how locally meaningful measurements relate across regions.

Curvature Is What Happens When Those Relations Accumulate

Suppose we have a physically defined local frame at one event.

We compare it with a neighbouring local frame.

Then another.

Then another.

Point by point, we study how these local measurement systems relate.

Now transport some locally defined direction or basis through a sequence of such regions and later compare it with the starting frame.

If the result depends on the path taken, that difference carries physical information.

That is what curvature captures.

The important point is not that we are trying to squeeze reality into a globally flat coordinate picture.

We are doing the opposite.

We allow each local measurement framework to be defined where the physics permits it.

Then we study how those frameworks relate.

Conceptually, the order is

$$\text{measurable entities} \rightarrow \text{local measurements} \rightarrow \text{local coordinate descriptions} \rightarrow \text{relations between them} \rightarrow \text{geometric predictions}.$$

That is very different from beginning with a giant invisible lattice filling the universe.

What About an Empty Region?

This becomes subtle when physicists talk about the geometry of an empty region.

Suppose there is no clock there.

No ruler.

No detector.

No material object.

Mathematically, GR can still assign coordinates and geometric quantities to that region.

That is useful.

But physically, we do not need to imagine that the vacuum itself contains invisible measuring devices or intrinsic coordinate labels.

The operational meaning is closer to this:

If an observable physical system were present here, what would it measure?

If a clock followed a possible worldline through the region, what proper time would it accumulate?

If a test body passed through, how would its motion relate to other physical systems?

If light crossed the region, how would physically defined emission and reception events relate?

The region does not have to contain an actual observer.

But the physical meaning of the local description comes from the possibility of observable relations being realised there.

So I would distinguish

$$\text{possible observable entity} \rightarrow \text{possible measurement} \rightarrow \text{mathematical description}$$

from the very different picture

$$\text{vacuum} \rightarrow \text{intrinsic hidden coordinate grid} \rightarrow \text{measurement}.$$

The first is enough.

We do not need the second.

Where Quantum Gravity Starts to Look Conceptually Strange

Quantum field theory and general relativity are not simply enemies.

QFT works extremely well on curved classical spacetime across a large range of physics.

The deeper difficulty appears when we try to make the geometry itself fully quantum.

And at that point I think we should stop before doing any calculation.

Because within the interpretation developed here, the phrase already raises a foundational question.

What did geometry mean in the GR picture we just built?

It encoded relations among possible local measurements.

A clock here.

A freely falling object there.

A light signal between events.

A locally meaningful frame defined through physical relations.

Now suppose the system that would ground that measurement framework is itself quantum.

What exactly are we assigning the local frame to?

A quantum system is not simply a classical object carrying a continuously definite collection of properties.

Its mathematical state can encode several possible outcomes and relations without those possibilities being equivalent to a single fully definite classical state.

So if we immediately give such a system a continuously definite local geometry, have we quietly reintroduced exactly the kind of definiteness that quantum mechanics warned us not to assume?

Within the logic of this article, we may already have crossed the boundary.

We may have gone from

$$\text{quantum physical situation}$$

to

$$\text{definite local measurement geometry}$$

without first establishing that the physical relations required to define that geometry are themselves definite.

Then an even stranger question appears.

If geometry is put into superposition, what exactly is in superposition?

Possible clock relations?

Possible causal relations?

Possible measurement outcomes?

Possible local calibrations?

Or a mathematical geometric object that we had already promoted into ontology before checking whether the physical relations required to define it existed?

I do not know the answer.

But I think the conceptual question comes first:

Are we trying to quantise a mathematical structure whose physical meaning originally came from definite local measurements after those very measurement-defining systems have become quantum-indefinite?

That does not prove that this is why quantum gravity is difficult.

It would be far too strong to claim that.

But it suggests a different starting question:

If geometry derives its physical meaning from possible measurements, perhaps we need to rethink what geometry means in a genuinely quantum regime before simply quantising the classical geometric variables.

Maybe the standard route is correct.

Maybe it is not.

But it should not be ontologically automatic.

A Possible Principle

The recurring pattern suggests a tentative principle.

I do not yet regard it as a finished axiom.

But the idea can be stated approximately as follows:

A mathematical quantity should be assigned physical definiteness only when the physical situation supplies, at least in principle, a distinguishable relation capable of grounding that quantity.

Or more simply:

Physics should not assign definiteness merely because its mathematical language allows us to write a variable.

There is a second part:

When physical distinguishability ends, mathematics may continue as representation, but its physical interpretation must become correspondingly constrained.

This does not restrict mathematics.

Mathematics should remain free.

We can construct coordinates.

Hilbert spaces.

Manifolds.

Vector fields.

Operators.

Bases.

Projections.

Abstract state spaces.

The restriction enters only when we turn around and say:

therefore reality itself possesses every one of these structures with equal definiteness.

That conclusion requires more than mathematical convenience.

It requires physical justification.

Twist With a Trick

There is, of course, another route.

Whenever physics becomes uncomfortable with classical intuition, we can often construct a story that restores some version of it.

Bohmian mechanics is an interesting example.

It introduces definite particle configurations and trajectories guided by the wavefunction. In that sense it restores a more classical-looking underlying ontology, although the resulting theory contains deeply nonclassical features of its own.

Something similar can be done with relativity.

We can introduce an ether or a preferred frame.

One possibility is a completely passive ether in the Lorentz–Poincaré spirit, arranged so that every observable prediction remains exactly the same as special relativity.

If no possible experiment distinguishes that preferred frame, it becomes impossible to experimentally choose between the two descriptions.

But then another question appears:

If nothing can distinguish the structure even in principle, what explanatory work have we gained by declaring that it physically exists?

Alternatively, we can make the preferred structure active.

Einstein-aether models, Hořava-style gravity and other Lorentz-violating theories allow preferred structure to influence observable physics.

Then the idea becomes testable.

Directional effects, preferred-frame behaviour, altered propagation or other deviations can in principle appear.

And those possibilities can be constrained by experiment.

There is something about this pattern that reminds me strongly of overfitting in machine learning.

Imagine that a model fails on a new test set.

Instead of reconsidering its assumptions, we add another mechanism to explain the failure.

Then another test arrives.

We add another mechanism.

Then another.

Eventually we may have a model flexible enough to accommodate almost anything, but we have stopped discovering the simpler principle that generalises.

Physics can fall into the same conceptual trap.

Quantum mechanics looks nonclassical, so we introduce hidden trajectories.

Relativity removes an absolute frame, so we introduce an undetectable preferred one.

And if we really want an escape route, we can almost always invent one.

If someone wants the chain of explanation ultimately to terminate at God, a story can be constructed around that.

If someone wants the explanation somehow to terminate at me, Isuru, the person writing this article, I am fairly sure an imaginative enough person could construct a case-by-case story around that too.

That sounds absurd.

But that is precisely the point.

Once unconstrained hidden structure is allowed, explanatory escape routes become almost unlimited.

The fact that a story can be invented is therefore not enough.

The harder question is whether the story is forced by distinguishable reality, or whether we introduced it simply because we were uncomfortable allowing something to remain undefined.

This is why I am less interested in finding another classical rescue mechanism and more interested in finding a fair axiom.

I want a modelling principle that applies to relativity, quantum mechanics, measurement and whatever comes next without requiring a new exception every time classical intuition fails.

Something like:

When reality provides no physically distinguishable relation capable of defining a quantity, do not manufacture additional physical structure merely to keep that quantity definite.

That principle may ultimately be wrong.

But at least it tries to generalise.

And thinking along these lines raises another strange question for me.

Are we, in some loose sense, fighting the kind of boundary exposed by Gödel's incompleteness theorem?

I do not mean that Gödel's theorem proves anything about quantum mechanics, relativity or physical measurement.

That would be far too quick.

But there is a resemblance that bothers me.

We construct a formal system powerful enough to describe something.

Then we discover that there may be boundaries on what can be established from within that formal structure.

In physics, perhaps we repeatedly make a related mistake: we expect our mathematical language to provide physical definiteness beyond what the physically available information can support.

Maybe those ideas are completely unrelated.

Maybe they are not.

I once raised a version of this question in a comment on the Fermilab YouTube channel and found the answer completely unsatisfying.

So I will leave Gödel alone here.

That deserves another article.

At the Edge

Special relativity may be telling us not to assign an absolute background where no physical background is provided.

Quantum mechanics may be telling us that physical existence does not necessarily imply complete classical definiteness.

Quantum field theory may be showing us that a mathematically rich vacuum can have observable consequences without supplying a physically distinguished rest structure.

General relativity may be showing us how extraordinarily far we can go by relating local measurement systems without requiring a physically privileged universal coordinate grid.

And quantum gravity may be warning us that those geometric concepts cannot simply be carried unchanged into a regime where the physical systems grounding measurement no longer possess classical definiteness.

Perhaps these are not unrelated accidents.

Perhaps they are manifestations of a more general modelling constraint.

Reality gives us information through physically distinguishable relations.

Mathematics allows us to organise, transform and extrapolate that information.

But mathematics can continue long after physical distinguishability has ended.

That is one of its greatest strengths.

It may also be one of the easiest ways to fool ourselves.

Perhaps the boundary we should care about is therefore not simply the edge of measurement.

It is the edge of measurement assumptions.

The point where we stop asking what reality has actually made definable and start assuming that whatever our mathematics can represent must also exist with equal physical definiteness.

Maybe some of our deepest conceptual problems begin exactly there.

And perhaps a useful principle for constructing new physics is surprisingly simple:

Let mathematics go as far as it wants.
But let physical definiteness go only as far as reality gives us something distinguishable to define.

And when reality provides no such distinction, perhaps the correct response is not to invent hidden structure merely to rescue our intuition.

Perhaps the correct response is to let the undefinedness itself constrain the mathematics.