The Physics of ‘As If’
Karl Friston1
1 Queen Square Institute of Neurology, University of College London
Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, where the task of writing the report was left to the team leader. Shortly thereafter, the physicist returned to the farm, saying to the farmer, “I have the solution, but it works only in the case of spherical cows in a vacuum.1
The physicists’ ‘spherical’ cow is an amusing place to start when considering the role of models in physics, particularly the physics of sentience afforded by the free energy principle (FEP). A careful reading of the foundational FEP literature will reveal an abundance of phrases like “as if” and “appears to”. So, what does “as if” imply for models of minds and brains? This is one of the central questions discussed by Kirchhoff in The Idealized Mind.
The notion of a ‘model’ in the FEP literature has a technical meaning. It refers to, and only to, a generative model. A generative model is just a probability distribution over the observations or sensations and their hidden or latent causes. Because the model can always be decomposed into a likelihood and a prior, the generative model is also a ‘map’ in the formal sense of a morphism from (hidden) causes to (observable) consequences. With a careful definition of what it is to be an observer—namely to possess a Markov boundary or blanket (Kirchhoff et al., 2018)—it is fairly straightforward to show that the dynamics on the interior of the blanket (e.g., brain) entails a generative model of the sensorium (the sensory sector of the Markov blanket). So, what does this mean?
It means that it looks “as if” the brain is making sense of sensory inputs under a generative model of how those inputs were caused, despite never having direct access to the cause-effect structures generating the sensorium. In what sense should we interpret “as if”? In this context, “as if” implies a mathematical equivalence that lends itself to multiple interpretations. In the FEP, this equivalence corresponds to showing that the self-organising, spatiotemporal neuronal dynamics can be read as an inference process (a.k.a., self evidencing) under a generative model or map (Hohwy, 2016). See Box 1: This is Kirchhoff’s conclusion in chapter 9. He doesn’t mean it in the joking sense of the spherical cow. He argues that generative models and the notion of the brain as engaged in probabilistic inference can only obtain by idealizing the brain. So, are there any spherical cow’s lurking in this reading or interpretation?
I think the answer is yes and no.2 To understand this we can revisit the spherical cow from the point of view of the renormalisation group. The renormalisation group allows one to formalise symmetries or invariances over scales. Crucially, one scale inherits lawfully from another by coarse-graining or compressing information to recover the same dynamics or physics at successive levels (Watson et al., 2022). This affords a description of a system at multiple scales, each perfectly apt to describe the physics at the scale in question. For example, for an astronomer, the movement of heavenly bodies (e.g., the moon) can be described exactly—with no approximation—by treating it as a spherical body, whose only attributes are position and velocity. However, if we wanted to land on the moon, one may need a finer scale of description, right down to the location of various craters and putative landing sites. The point here is that the statement that the moon moves “as if” it was a spherical body entails no approximation at the scale in question. The same applies to the FEP; namely, at a suitable scale, neuronal dynamics can be described exactly as an inference or sense-making process under a generative model (Friston, 2019). See Box 1.
The mathematical equivalence in Box 1 licences a teleological or functionalist interpretation of any brain at a suitable scale (i.e., at the scale of neuronal population or ensemble dynamics). From the perspective of the FEP, this reading is the only thing that licences the use of the word “model” or “map”. In turn, this means that all models or maps just are parts of my generative model—and possibly yours, if you are sufficiently like me. Interestingly, this includes models of you, models of me, scientific models, philosophy of science models and, of course, the FEP itself.
- Spherical cow – Wikipedia
- A technical but interesting qualification—to the above arguments—are the approxi-mations made in physics to simplify (descriptions of) things. Ubiquitous examples here are mean-field approximations, adiabatic approximations and the Laplace approxima-tion. Interestingly, the FEP can be applied under these approximations; for example, if we apply all three (mean field, adiabatic and Laplace) approximations to the dynamics in Box 1, we end up with predictive coding (Friston and Kiebel, 2009; Hohwy, 2012; Rao and Ballard, 1999) and the Bayesian brain (Clark, 2013, 2016; Knill and Pouget, 2004). Given that Bayesian inference is abductive—and there is no truth pointing—one could argue that these approximations or simplifications are part and parcel of self-evi-dencing. This is because the (log) evidence for our generative models can always be decomposed into accuracy minus complexity (Penny, 2012). Therefore, if we are self-evidencing, we may also minimise complexity, such that it “looks as if” we are making simplifying approximations.
References
Clark, A., 2013. Whatever next? Predictive brains, situated agents, and the future of cognitive science. The Behavioral and brain sciences 36, 181-204.
Clark, A., 2016. Surfing Uncertainty: Prediction, Action, and the Embodied Mind. Oxford University Press.
Friston, K., 2019. A free energy principle for a particular physics, eprint arXiv:1906.10184.
Friston, K., Da Costa, L., Sakthivadivel, D.A.R., Heins, C., Pavliotis, G.A., Ramstead, M., Parr, T., 2023. Path integrals, particular kinds, and strange things. Physics of Life Reviews 47, 35-62.
Friston, K., Kiebel, S., 2009. Predictive coding under the free-energy principle. Philosophical transactions of the Royal Society of London. Series B, Biological sciences 364, 1211-1221.
Hohwy, J., 2012. Attention and conscious perception in the hypothesis testing brain. Front Psychol 3, 96.
Hohwy, J., 2016. The Self-Evidencing Brain. Nous 50, 259-285.
Kirchhoff, M. 2025. The Idealized Mind: From Model-based Science to Cognitive Science. Cambridge, MA; The MIT Press.
Kirchhoff, M., Parr, T., Palacios, E., Friston, K., Kiverstein, J., 2018. The Markov blankets of life: autonomy, active inference and the free energy principle. J R Soc Interface 15.
Knill, D.C., Pouget, A., 2004. The Bayesian brain: the role of uncertainty in neural coding and computation. Trends Neurosci 27, 712-719.
Penny, W.D., 2012. Comparing Dynamic Causal Models using AIC, BIC and Free Energy. Neuroimage 59, 319-330.
Rao, R.P.N., Ballard, D.H., 1999. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects. Nature Neuroscience 2, 79-87.
Watson, J.D., Onorati, E., Cubitt, T.S., 2022. Uncomputably complex renormalisation group flows. Nature Communications 13, 7618.
Karl Friston’s response to Kirchhoff attempts to defend the Free Energy Principle against charges of idealization by invoking renormalization group theory. His argument is elegant, mathematically sophisticated, and fundamentally wrong. It demonstrates precisely the categorical confusion that plagues contemporary consciousness research, dressed in the respectable clothing of theoretical physics.
Friston’s Argument Reconstructed
Friston’s defense proceeds as follows:
1. The renormalization group allows us to describe systems at multiple scales, each perfectly suited to describe the physics at that scale.
2. An astronomer can describe the moon’s motion exactly, without approximation, by treating it as a spherical body with only position and velocity. This is not an idealization but the correct description at the astronomical scale.
3. Similarly, neural dynamics can be described exactly, without approximation, as an inference or sense-making process under a generative model at the appropriate scale (population or ensemble dynamics).
4. Therefore, saying the brain behaves “as if” it performs Bayesian inference involves no approximation at the relevant scale. The “as if” is mathematically exact.
5. Conclusion: The Free Energy Principle is not an idealization but the correct description at the appropriate scale of analysis.
This argument is seductive. It is also categorially incoherent.
The Fatal Disanalogy: Physical vs. Semantic Scale Invariance
Friston’s renormalization argument works for the moon because all scales of description remain within the same categorical domain: physics. Whether we describe the moon as a point mass, a sphere, or a cratered irregular body, we are always describing physical properties in physical terms, mass, position, velocity, shape, gravitational effects, surface topology.
The renormalization group preserves categorical coherence across scales. A coarse-grained physical description and a fine-grained physical description are both physical descriptions. They differ in resolution, not in logical category.
But when Friston applies this to the brain, something categorially illicit happens. At the fine-grained scale, we have neuronal activity: action potentials, neurotransmitter release, ion channel dynamics, electrical and chemical processes. These are physical-biological processes described in physical-biological terms.
At the “coarse-grained” scale, Friston suddenly introduces semantic and intentional vocabulary: the brain “performs inference,” “generates predictions,” “minimizes prediction error,” “has expectations,” “updates beliefs,” “represents hidden causes,” “constructs models.”
This is not a shift in scale within the same category. This is a jump between incommensurable logical categories. Physical processes do not become semantic processes through coarse-graining. Electrical activity does not become “expectation” by changing resolution. Chemical transmission does not become “belief updating” at population level.
The Category Error Concealed by Mathematics
Friston’s mathematical equivalence, that neural dynamics can be “read as” inference under a generative model, is genuine. But mathematical isomorphism does not establish categorical identity.
Consider: I can describe the dynamics of a pendulum using differential equations. I can also describe population growth using structurally identical differential equations. The mathematical form is the same. Does this mean the pendulum is “performing” population growth? Does the pendulum “reproduce”? Does it have “generations”?
Obviously not. The mathematical structure is shared, but the categories are different. Physical oscillation is not biological reproduction, no matter how isomorphic their equations.
Similarly, neural dynamics may be mathematically isomorphic to Bayesian inference, but this does not mean neurons are performing Bayesian inference, any more than pendulums are reproducing. The isomorphism is a property of our mathematical descriptions, not of the systems themselves.
Friston writes: “This reading is the only one that justifies the use of the terms ‘model’ or ‘map.'” But this is precisely backwards. The mathematical equivalence does not justify the semantic vocabulary. It merely shows that we can impose semantic vocabulary on systems that exhibit certain mathematical structures. But imposition is not justification.
What Renormalization Actually Requires
The renormalization group works for physics because physical properties are scale-covariant within the physical domain. Mass, energy, momentum, charge, these quantities transform in well-defined ways across scales, but they remain physical quantities at every scale.
For Friston’s argument to work, he would need to show that semantic properties are scale-covariant with physical properties. He would need to demonstrate that “expectation,” “inference,” “belief,” and “prediction” are legitimate coarse-grained descriptions of fine-grained physical processes in the same way that “spherical body” is a legitimate coarse-grained description of a fine-grained irregular surface.
But he cannot show this, because semantic properties are not physical properties at any scale. They are properties of a different logical type entirely.
The “Interpretation” Admission
Friston himself reveals the problem when he writes that neural dynamics “can be read as” inference under a generative model. Read by whom? Interpreted by whom?
When we say the moon moves “as if” it were a sphere, we do not mean it can be “interpreted” as spherical. We mean it IS spherical at that level of resolution. The craters are below the relevant scale. There is no interpretation required, only a choice of scale.
But when Friston says neural dynamics “can be read as” Bayesian inference, he admits that this requires interpretation. The neural activity does not become inference at population scale. It remains neural activity that we choose to interpret as inference.
This is the crucial difference: Physical properties survive coarse-graining because they are intrinsic to the system at every scale. Semantic properties do not survive coarse-graining, they are not intrinsic to the system at any scale. They are projections from an interpreting observer.
The Bayesian Apparatus: Category Error Compounded
Friston’s argument becomes even more problematic when we examine what Bayesian inference actually requires. Bayesian inference is not merely a mathematical structure. It presupposes:
1. A probability space defined over propositions
2. Prior beliefs expressible as probability distributions
3. Evidence that updates these beliefs
4. An epistemic agent with access to this evidence
5. The capacity to represent and manipulate probabilistic relations
None of these exist in the brain. Neurons do not have probability distributions “in” them. They do not represent propositions. They do not have epistemic access to evidence. They are not agents performing calculations.
The Bayesian framework is a modeling tool that requires an interpreting subject, us, the scientists, who impose a probability space, define what counts as evidence, and interpret neural activity as if it were updating beliefs. This interpretive apparatus is not in the brain. It is in our models of the brain.
Friston’s renormalization argument attempts to naturalize this interpretive apparatus by claiming it exists “at the right scale.” But changing scale does not create semantic properties where none existed. It only changes the resolution at which we observe physical properties.
The Performative Contradiction
There is a deeper problem: Friston’s entire argument is self-undermining. He claims that all models, including scientific models, philosophical models, and the FEP itself, are “merely parts of my generative model, and possibly yours if you are sufficiently similar to me.”
This is epistemological nihilism disguised as theoretical sophistication. If the FEP is merely part of Friston’s generative model, then it makes no claim about the world. It is not a scientific theory but an autobiographical report about how Friston models things.
But Friston does not present the FEP as autobiography. He presents it as an objective framework for understanding self-organizing systems. He claims neural dynamics actually do minimize variational free energy, not merely that he finds it useful to model them this way.
The moment he makes any objective claim, the moment he says “the brain actually does X”, he steps outside the relativist framework he has just constructed. He cannot have it both ways: either the FEP is an objective theory about how brains work (in which case his relativist dissolution fails), or it is merely Friston’s preferred modeling strategy (in which case it has no scientific authority).
What Friston Should Have Argued (But Cannot)
To defend the FEP against categorical confusion, Friston would need to show:
1. That semantic properties (expectation, prediction, inference, belief) are legitimate natural kinds that exist in physical systems independently of interpretation.
2. That these semantic properties have well-defined physical implementations that can be identified empirically.
3. That the mathematical isomorphism between neural dynamics and Bayesian inference reflects a genuine identity, not merely structural similarity.
4. That “reading” neural activity as inference is not an interpretive choice but a discovery of what is actually occurring.
He cannot show any of these things, because they are false. Semantic properties are not physical properties. Mathematical isomorphism is not categorical identity. And “reading as” is interpretation, not discovery.
The Real Physics of “As If”
Friston titles his response “The Physics of ‘As If.'” But there is no physics of “as if.” There is only physics. And there is interpretation of physics using semantic vocabulary that does not belong to physics.
When physicists say a gas behaves “as if” its molecules were point masses, they mean: at the relevant scale, the spatial extent of molecules is negligible compared to inter-molecular distances. The “as if” reflects a legitimate physical approximation within the physical domain.
When Friston says the brain behaves “as if” it performs Bayesian inference, he does not mean: at the relevant scale, the non-inferential aspects of neural activity are negligible. He means: we can describe neural activity using the vocabulary of inference if we choose to interpret it that way.
These are not the same kind of “as if.” The first is a physical approximation. The second is a categorical projection.
The Magnetism Example Revisited
To make the categorical error vivid, consider again the analogy I proposed to Kirchhoff: Imagine a theory of social bonds based on magnetism. Humans form relationships because they are drawn together by forces analogous to magnetic attraction. Social groups exhibit polarity like magnetic poles. Community formation minimizes “social-magnetic potential energy.”
Now imagine defending this theory using Friston’s renormalization argument: “At the appropriate social scale, human behavior can be described exactly, without approximation, as if it follows magnetic dynamics. The mathematical equivalence is exact at the level of social aggregates. This is not an idealization but the correct description at the social scale.”
This would be immediately recognized as absurd. But why? The mathematics might work perfectly. We could model social attraction with equations isomorphic to magnetic field equations. We could predict social clustering using magnetic potential energy minimization. The formalism could be empirically adequate.
The problem is not mathematical inadequacy. The problem is categorical incoherence. Magnetic forces are not social attractions, regardless of mathematical isomorphism. Physical attraction is not intentional bonding, regardless of predictive success. And no amount of renormalization group theory changes this fact.
Friston’s FEP commits exactly this error. It models intentional, semantic, goal-directed cognitive processes using the mathematics of thermodynamic potential minimization. The mathematics may be elegant. The predictions may be accurate. But the categories are wrong.
Why This Matters
This is not mere philosophical pedantry. Categorical confusion has real consequences for scientific progress.
When we model cognition as thermodynamic optimization, we systematically obscure what needs to be explained: How do physical processes give rise to semantic, intentional, phenomenal properties? By describing neural activity in semantic vocabulary from the outset (“the brain performs inference”), we presuppose what we should be explaining.
The FEP does not explain how neurons generate semantic content. It assumes they do and provides a mathematical framework for describing this assumed process. This is explanatory circularity disguised as scientific theory.
Moreover, the categorical confusion immunizes the theory against falsification. When empirical predictions fail, defenders can always retreat to the claim that the theory operates “at a different scale” or requires “the right interpretation.” The semantic vocabulary slides between literal mechanism and useful metaphor depending on dialectical pressure.
A categorially coherent theory would specify: Here is the physical mechanism. Here is how we measure it. Here is what counts as evidence for or against it. Here is the gap between physical mechanism and semantic properties, and here is why we cannot bridge it yet.
The FEP does none of this. Instead, it papers over the explanatory gap with mathematical isomorphism and declares victory through renormalization.
Conclusion: Mathematics Cannot Absolve Category Error
Friston’s renormalization defense fails because mathematical sophistication cannot overcome categorical incoherence. The brain is not “like” a Bayesian inference engine that happens to be implemented in neurons. It is not “as if” performing inference at the right scale. It is not “interpretable as” probabilistic computation through appropriate coarse-graining.
The brain is a physical-biological system whose dynamics can be described in physical-biological terms. We can model these dynamics using mathematical structures borrowed from information theory, thermodynamics, and probability theory. But modeling is not discovering. Isomorphism is not identity. And interpretation is not explanation.
When Friston claims that neural dynamics “can be read as” Bayesian inference at the population scale, he is admitting, not refuting, that this is an interpretive projection rather than a physical discovery. The renormalization group changes resolution, not categories.
The Free Energy Principle is not the physics of cognition. It is the mathematics we impose on cognition when we choose to interpret it through a particular theoretical lens. This lens may be useful for certain purposes, generating predictions, organizing data, suggesting experiments. But utility is not truth, and mathematical elegance is not explanatory adequacy.
Until the FEP can specify what physical properties at what physical scales constitute genuine Bayesian inference rather than merely permitting Bayesian interpretation, it remains a sophisticated form of the same categorical confusion that has plagued consciousness research for decades.
The spherical cow is amusing precisely because everyone recognizes it as an approximation. Friston’s claim that the brain is exactly, without approximation, a Bayesian inference engine at the right scale is not amusing. It is a category error elevated to theoretical principle, defended by mathematical sophistication, and immunized against criticism by systematic equivocation between literal mechanism and interpretive “reading.”
The physics of “as if” is just metaphysics dressed in equations. And metaphysics, however mathematically elegant, is not physics.
It makes me think of a recent paper criticizing the Bayesian brain hypothesis, PP, AIF and the FEP….the “as if” argument is also mentioned here…https://link.springer.com/article/10.1007/s00421-025-05855-6 Worth the read, but I would love to see a discussion where proponents of these approaches respond to the criticism raised (there is a fictitious discussion in the paper! but I would love to see more of this)…Anyone seen the paper?
Hi Evert
Many thanks for this comment and the link. Agree entirely that such a paper would be a welcomed addition to the literature.
Michael