What Physics Can't Tell Psychology (And What It Can)

We've been using the language of physics.

Manifolds. Curvature. Topology. Metrics. Geodesics. The whole apparatus of differential geometry, information geometry, category theory—mathematical tools forged in physics, imported into psychology.

This isn't accidental. The tools are powerful. They capture real structure. The geometry of belief states, the curvature of prediction landscapes, the topology of attractor basins—these aren't just metaphors. They're mathematical descriptions of how cognitive systems actually work.

But there's a danger.

Physics deals with simple systems under rigid constraints. Particles obey strict laws. Symmetries are exact. Conservation principles hold precisely. Predictions can achieve extraordinary accuracy because physical systems behave with extraordinary regularity.

Psychology deals with complex systems under soft constraints. People violate expectations. Symmetries are approximate. Regularities have exceptions. Predictions remain probabilistic because psychological systems behave with persistent variability.

Importing physics tools into psychology isn't automatically legitimate. The tools might not fit. The analogies might break. The precision might be false precision, imposing mathematical structure where reality is messier.

Understanding where physics illuminates psychology and where it misleads is essential for using these tools honestly. The geometry is real, but it's not the geometry of particles. The math applies, but not in the way physics applies math. Knowing the difference is knowing what we're actually saying when we talk about the mathematics of meaning.

What Physics Gets Right

Physics contributes several things that genuinely transfer.

The concept of state space. A physical system can be described by specifying its state—position, momentum, charge, spin, whatever variables are relevant. The space of all possible states is the state space. Physics taught us to think in these terms: systems as points in spaces, dynamics as trajectories through spaces.

This concept transfers to psychology beautifully. A cognitive system has states—configurations of belief, memory, attention, emotion. The space of possible states is a state space. Dynamics are trajectories. The concept doesn't depend on what the states are made of; it's a structural way of thinking that works for any system with states.

Geometric structure on state spaces. Physics didn't just introduce state spaces; it discovered that state spaces have geometry. Distances between states. Curvature of the space. Paths through the space. The structure of the space constrains what dynamics are possible.

This transfers too. Cognitive state spaces have geometry—not necessarily the same geometry as physical state spaces, but geometry nonetheless. The distances between belief states, the curvature of prediction landscapes, the topology of attractor regions—these are real structures, measurable in principle if not always in practice.

Variational principles. Physics discovered that many systems can be understood as optimizing something. A ball rolls downhill to minimize potential energy. Light travels to minimize travel time. Systems find paths through state space that extremize some quantity.

This transfers. Cognitive systems can be understood as minimizing prediction error (the free energy principle) or maximizing information (curiosity) or minimizing effort (cognitive efficiency). The variational framing is powerful because it lets you derive dynamics from principles rather than specifying them directly.

Invariants and symmetries. Physics discovered that what stays the same is as important as what changes. Conservation laws, symmetry principles, invariants under transformation—these are the skeleton that holds physical theories together.

This transfers. Coherence is an invariant—something that we're claiming stays the same (or should stay the same) across transformations, across scales, across manifestations. The functor network we discussed is a symmetry claim: that certain structures are preserved under mapping between levels.

These imports are legitimate. They're not physics applied directly to mind; they're structural tools that physics developed but that apply anywhere there are systems, states, and dynamics.

Where Physics Misleads

But other things don't transfer, and pretending they do creates confusion.

Exact symmetries become approximate symmetries. In physics, a symmetry is either exact or it's not a symmetry. Conservation of energy holds precisely, or it doesn't hold. The math requires exactness.

In psychology, nothing is exact. The patterns we call "structures" are statistical regularities, tendencies, approximations. A belief manifold isn't a precise mathematical object; it's a model of something messy. The symmetries and invariants are approximate, not exact.

This matters because physical reasoning often depends on exactness. If energy is almost conserved, you can't use conservation arguments reliably. If the symmetry mostly holds, the conclusions that follow from it might or might not apply.

Importing physical reasoning without acknowledging the shift from exact to approximate produces false precision. You derive conclusions as if the math were exact, but the reality is approximate, so the conclusions may not hold.

Rigid constraints become soft constraints. Physical systems obey laws that can't be violated. A particle doesn't choose to ignore momentum conservation. The constraints are rigid—built into the nature of reality.

Psychological systems face constraints that can be violated, at a cost. You can believe contradictory things; it's just uncomfortable. You can ignore evidence; it's just destabilizing. The constraints are soft—they push, but they don't absolutely restrict.

This changes what "constraint" means. In physics, a constraint eliminates possibilities. In psychology, a constraint makes possibilities more or less costly. The geometry isn't a hard boundary; it's an energy landscape. The manifold isn't a rigid structure; it's a distribution of probabilities.

Treating psychological constraints as rigid leads to false necessity. You conclude "this can't happen" when what's actually true is "this is unlikely" or "this has costs."

Universality becomes context-dependence. Physical laws are universal. The laws of thermodynamics apply to all thermodynamic systems. The laws of gravity apply to all masses. Universality is built into what makes something a "law."

Psychological regularities are context-dependent. What holds for one person might not hold for another. What holds in one culture might not hold in another. What holds at one developmental stage might not hold at another. The regularities are real but not universal.

This undermines generalization. Physical conclusions transfer automatically because physical laws are universal. Psychological conclusions transfer contingently—if the contexts are similar enough, if the population is similar enough, if the conditions are similar enough. The geometry might be different for different people.

Determinism becomes probabilism. Physical systems (leaving quantum mechanics aside) are often effectively deterministic. Given the state and the laws, you can compute the future state. The geometry determines the trajectory.

Psychological systems are stochastic. Even given complete knowledge of state and regularities, you can't compute the future with certainty. The geometry doesn't determine a trajectory; it defines a probability distribution over trajectories.

This changes what prediction means. In physics, prediction is about computing what will happen. In psychology, prediction is about estimating what's likely to happen. The geometry tells you where things probably go, not where they definitely go.

Simplicity becomes complexity. Physics often succeeds by finding the simple systems hidden inside apparently complex phenomena. Ideal gases. Point particles. Frictionless planes. The simplifications capture the essence; the complications are perturbations.

Psychology rarely simplifies so cleanly. The complications aren't perturbations; they're the point. Individual differences, contextual factors, developmental history—these aren't noise around a simple signal. They're constitutive of the phenomenon.

This limits what reductionism can achieve. In physics, you can often understand the whole by understanding the simple parts. In psychology, the parts don't capture the whole; the whole has irreducible complexity that survives any decomposition.

The Boundary Zone

Between the clearly legitimate transfers and the clearly illegitimate ones lies a murky zone where the status of the analogy is uncertain.

Metrics on psychological spaces. Physics has precise metrics—the Minkowski metric of spacetime, the Fisher metric on statistical manifolds. Can we put precise metrics on psychological spaces?

Maybe. The Fisher metric, which we discussed earlier, derives from information theory and applies wherever probability distributions are updated. If cognitive systems update beliefs through something like Bayesian inference, then the Fisher metric (or something like it) might apply.

But the application is uncertain. We don't directly observe the cognitive state space. We infer it from behavior, report, physiology. The metric we attribute to the space is a model, and models can be wrong. The geometry might be real without our model of it being accurate.

Conservation principles in psychology. Physics has conservation laws—energy, momentum, charge. Does psychology have conservation laws?

Probably not strict ones. But there might be soft conservation principles. Total attention might be approximately conserved—attention to one thing reduces attention to another. Metabolic resources for cognition might be bounded—effort in one domain costs capacity in another.

These soft principles are useful heuristics without being exact laws. Treating them as exact would mislead; ignoring them would miss real patterns. The boundary zone requires holding both possibilities.

Symmetry arguments in psychology. Physics uses symmetry to derive constraints. If a system is symmetric under rotation, certain things follow mathematically.

Psychology has approximate symmetries. The way you process information might be roughly symmetric under various transformations—e.g., scale-invariant, similar across certain contexts. But the symmetries are imperfect, and conclusions derived from them are tentative.

This is delicate. Symmetry arguments are powerful, but their power depends on the symmetry being real. An approximate symmetry might support approximately correct conclusions—or it might not. There's no general answer; you have to check.

Developmental Plasticity

One crucial difference between physical and psychological systems deserves special attention: developmental plasticity.

Physical systems don't develop, in the psychological sense. A particle is the same particle throughout its existence. A physical law doesn't change because of the particle's history.

Psychological systems develop. What you are now depends on what you were and what happened to you. The geometry of your belief space is shaped by experience. Your manifold's curvature was sculpted by development.

This introduces several non-physical features.

History-dependence. Two people in the same state now might have different dynamics because they have different histories. The state alone doesn't determine the trajectory; you need the whole developmental path.

Physics has something like this in hysteresis—systems whose current state depends on their history. But in physics, hysteresis is an exception. In psychology, it's the rule. Everything is shaped by what came before.

Critical periods. Some developments must happen during specific windows; if they don't, the possibility closes. This has no real physical analog. Physical systems don't have "windows" where certain changes are possible and then become impossible.

Plasticity variation. Some aspects of psychological systems are highly plastic—changeable throughout life. Others are less plastic—changeable in youth but fixed later. The degree of plasticity varies by what feature, at what age, under what conditions.

This means the geometry itself changes. The manifold that describes a person at 5 is not the manifold that describes them at 50. Development reshapes the space. The geometry is alive in a way that physical geometry isn't.

What We Can and Can't Claim

Given these asymmetries, what can we legitimately claim using physics-derived tools?

We can claim structural similarity. The structure of cognitive systems—states, transitions, trajectories—is analogous to the structure of physical systems. The same mathematical tools can describe both. The analogy is substantive, not superficial.

We can't claim physical precision. The structures are analogous, but the precision differs. Physical predictions can achieve extraordinary accuracy. Psychological predictions remain rough. The geometry is real but noisy.

We can claim useful modeling. Treating cognitive spaces as manifolds, treating coherence as geometric invariant, treating change as trajectory—these are useful models. They organize observations, suggest hypotheses, guide intervention. Useful doesn't require exact.

We can't claim the model is the reality. The manifold is a model of something. What it's a model of is messier than the model. The map is helpful; the territory remains complex.

We can claim that geometry matters. The geometry of a system—its curvature, its topology, its metric structure—constrains what the system can do. This holds for cognitive systems as for physical ones. Shape constrains dynamics.

We can't claim that geometry is all that matters. Physical systems are largely determined by geometry plus laws. Psychological systems have geometry but also have meaning, context, agency, development—features that don't reduce to geometry.

We can claim cross-scale structure. The functor network is real. Structure propagates across scales. Neural coherence relates to psychological coherence relates to relational coherence. The claim of structural correspondence is substantive and testable.

We can't claim the scales collapse. Psychology is not "just" physics. Meaning is not "just" geometry. The scales remain distinct even as they relate. Functors connect; they don't identify.

The Right Stance

The right stance toward physics-in-psychology is neither wholesale adoption nor wholesale rejection. It's careful, case-by-case evaluation.

Import the concepts. State space, geometry, dynamics, variational principles—these are useful ways of thinking that apply wherever there are systems with states.

Don't import the assumptions. Exactness, rigidity, universality, determinism—these are features of physical systems that psychological systems lack.

Use the math as scaffold. The mathematics of manifolds, curvature, topology—these organize thinking. They're scaffolds for reasoning, not final descriptions of reality.

Test the conclusions empirically. Mathematical arguments produce conclusions. Whether those conclusions hold for psychological systems is an empirical question, not a mathematical one. Derive it mathematically; verify it experimentally.

Maintain humility. The analogy between physical and psychological systems is productive but imperfect. We're using powerful tools in a domain where they don't perfectly apply. Expect some conclusions to be wrong. Be ready to revise.

This is what honest use of physics in psychology looks like. Not pretending minds are atoms. Not pretending the math is destiny. But also not abandoning the insight that structure matters, that geometry constrains, that the mathematics of meaning is possible even if it's not the same as the mathematics of motion.

Physics provides a lamp. We're carrying it into different terrain. The lamp illuminates; it doesn't transform the terrain into the lab where the lamp was made. What we see by its light is genuinely there. But we're not in the lab anymore.