The deeper you dig, the more you run into things the equations simply do not contain — causes, the present moment, genuine chance. They aren't in the formulas, yet they're in your experience every day.
Physics does not need philosophy's permission to get the numbers right. But ask one more question — «what are these equations saying?» — and philosophy is already in the room. Are laws discovered or invented? Why is there no «cause» anywhere in the fundamental equations? Does «now» have any place in physics? And that quantum probability: is it our ignorance, or the world itself? None of these four is mysticism — each has experiment and mathematics biting into it, and each is still without consensus.
Are Laws Discovered or Invented?
Philosophy of science · What a law is
Intuition
You fit a pile of planetary positions to an equation. Is that equation something the universe already contained, or a product of your pen? Both extremes are uncomfortable. Call it an invention, and you have to explain why Mercury's perihelion obeys it so dutifully, including in corners nobody is watching. Call it a discovery, and you have to say in what form a law «exists», and where — presumably not floating in interstellar space waiting to be read.
Mechanism · three routes
① The Humean best system (David Lewis): all there is is the distribution of particular facts, and a «law» is the summary of all those facts that is both simplest and most informative — an optimal compression code. It pushes nothing around; it just describes extremely well.
② Law realism (Tim Maudlin, for one): laws are among the basic furniture of the world, and it is the law that makes the electron move that way. History is generated by laws, not summarised into them.
③ Structural realism (John Worrall, 1989): look at what actually survives in the history of science. Fresnel's optical formulas were built on the «ether» — the ether was thrown out entirely, yet his equations live on untouched inside Maxwell's theory. What gets replaced is always the stuff (ether, caloric); what survives is always the mathematical structure.
The counterintuitive part
This isn't word games; it decides what «the explanation is finished» means. If a law is only a summary, what makes the counterfactual «if I hadn't let go, the cup wouldn't have broken» true or false — there is no such fact in the world for you to summarise. Law realism has a much easier time here, at the price of admitting an invisible, ungraspable governing something. Every position pays tuition.
Cross-disciplinary reading · AI / mathematics
Compression is theory: minimum description length (MDL) and regularisation in machine learning are practically the algorithmic version of the best-system account — the model that best trades «fits well» against «is short to write» is declared the best regularity.
ML also warns you back: a model can compress its training distribution beautifully and fall apart on the next one — compressing well is not the same as catching the structure. That is exactly structural realism's complaint about pure description: however elegant the summary, ask whether it preserved the relations.
In one line: we are probably neither inventing things nor discovering things — we are discovering relations.
Think: If two completely different pieces of mathematics give exactly the same predictions, are they the same law?
This is the problem of theoretical equivalence — Lagrangian versus Hamiltonian mechanics, or different gauges of one theory, are ready-made examples. Operationally they are fully equivalent; a realist who wants to crown one of them «the true one» can only appeal to simplicity and similar criteria that no experiment can adjudicate. That is precisely why structural realism commits to structure and not to ontology.
Causation Is Not in the Equations
Causation · Russell 1913
Intuition
You let go, the cup falls — what could be more obviously causal? Yet open any fundamental equation and there is no quantity called «cause» in it. The equation only says «given the state now, the state at another time is that», and it works just as well run backwards. Hence Russell's 1913 jab: the law of causality is «a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm».
Mechanism
First fact: the fundamental laws are essentially time-symmetric. Replace the time variable with
t → −t
Read it as «swap time t for negative t» — that is, run the film backwards. The − in front is a minus sign (not a subtraction, not a hyphen). For nearly every fundamental interaction, the reversed process is still a legal solution of the equation — microphysics has no built-in «before and after». The one known exception is a tiny time-reversal violation in the weak interaction, far too small to account for the direction of causation you experience.
So where does the direction come from? Macroscopic «cause then effect» rides on the entropy gradient — the universe started in an extremely low-entropy state (an assumption usually called the past hypothesis), so records only point backwards and influence only pushes forwards. And where causation really lands is intervention: saying «X causes Y» means change X by hand and Y follows, not merely that the two move together.
Microphysics has no direction; the macroscopic one comes from entropy — and «what causes what» takes a hands-on intervention to settle.
The counterintuitive part
Causation isn't at the fundamental level — but that doesn't make it an illusion. Temperature isn't at the fundamental level either, and boiling water will still scald you. Causation is an effective structure that emerges robustly at the level where things are macroscopic, entropy-graded and manipulable — the same kind of real thing as temperature and pressure. Treating it as an illusion, and treating it as a basic law of the universe, are two ends of the same mistake.
Cross-disciplinary reading · AI / medicine / economics
AI: Judea Pearl's causal diagrams and do-calculus put «changing something by hand» into mathematics; a model that collapses under distribution shift has usually learned correlations rather than mechanism.
Medicine: a randomised controlled trial is an intervention in exactly the physical sense — randomisation severs every arrow running from constitution, income and health-seeking behaviour into «who takes the drug», so what's left may be called an effect.
Economics: you can't randomise policy, so you borrow an intervention — natural experiments look for precisely the variation that behaves «as if randomly nudged».
In one line: causation is not part of the fundamental laws; it grows on the entropy gradient.
Think: Why can we remember the past but not the future?
A record is two systems becoming correlated and staying that way. To form one, the medium must first sit in a low-entropy «ready to write» state — blank film, a clean disk, unwalked sand. Such low-entropy starting states exist in abundance only towards the past, so records can only accumulate in one direction: the direction of memory is the direction of entropy.
Time, and the Missing Now
Time · relativity · eternalism
Intuition
«Time flows» is one of the strongest intuitions a person has. But «flows» demands a rate — so how many seconds does time flow per second? The question gives the game away: flow is a borrowed metaphor that contradicts itself the moment you take it literally.
Mechanism
Relativity already abolished objective simultaneity (the «Special Relativity» and «Spacetime» pieces cover the mechanism): observers moving at different speeds slice four-dimensional spacetime into differently tilted «now» surfaces. Two events simultaneous for you have a definite order for someone flying past. With no unique «now», the cheapest picture is to take the universe as one whole four-dimensional block — the block universe (eternalism): past and future events exist just as this moment does, they simply aren't on your particular slice (Rietdijk and Putnam, 1960s).
The whole block sits there; each observer cuts their own «now», and how they cut it depends on their motion.
Where to be honest
The block universe does not mean «the future is already written, so effort is pointless». Your weighing, hesitating and deciding are themselves physical processes happening in that block, not decorations it routes around — the fatalist reading swaps concepts: «already exists» is not «settled for you by something else». And the other side deserves saying too: the block universe is the mainstream picture, not a verdict. Presentism (only this moment exists) still has serious defenders, and physicists such as Lee Smolin argue that time is the fundamental thing instead.
Neuroscience: your «now» is manufactured. Touch your nose and your toe and the signals reach your brain tens of milliseconds apart, yet you experience them as simultaneous — the brain realigns channels of differing latency into an integration window of roughly a hundred milliseconds. Subjective simultaneity was always a construction.
Engineering: distributed systems have no global clock, so nodes coordinate through a partial order of «what happened before what» (Lamport's happens-before). Engineers were forced by reality into the same conclusion relativity reached: there is no global «now», only causal order.
In one line: physics has time, but it has no «now».
Think: With no objective «now», what is that utterly certain feeling of «I am here, right now» pointing at?
It points at «me, on this slice». «Now» is an indexical, exactly like «here»: necessarily true when uttered, yet marking no special structure of the universe. Nobody asks where the universe's «here» is; «what time is the universe's now» is the same mistake.
Probability: Ignorance, or the World?
Probability · Born rule 1926
Intuition
The «randomness» of a die roll is fake. Know the release angle, the elasticity of the table and the air currents, and the outcome is fixed — probability here is just a ledger recording how much I don't know. The real question is whether quantum probability is the same kind of ledger.
Mechanism
Statistical-mechanical probability comes from ignorance about microstates, with deterministic dynamics underneath. Quantum theory is different: the Born rule says the probability of measuring a given outcome is the squared modulus of the wavefunction's amplitude.
P(outcome k) = |ψk|2
ψ is pronounced «psi» and is the wavefunction — it assigns each possible outcome a complex number called the amplitude. The vertical bars |…| mean «take the modulus» (the length of a complex number); they are not the digit 1 and not the letter l. The little 2 up top is a square. The whole line reads: square the length of the amplitude to get the probability of that outcome.
Can this too be read as ignorance — «it was settled all along, I just didn't know»? Bell turned that into an experimentally decidable question in 1964, and the experiments since (through the loophole-free versions of the 2010s, recognised by the 2022 Nobel Prize in Physics) return a verdict: local hidden variables are ruled out. «Local, pre-written hidden facts» will not do as an excuse.
One word, «probability» — and the two sides may not be referring to the same thing at all.
Why it matters
The interpretations answer completely differently: Copenhagen says it is fundamental randomness; many-worlds says every outcome happens and probability degenerates into self-locating uncertainty about «which branch am I in»; Bohmian mechanics keeps determinism at the price of flagrant nonlocality. So «what is probability» is a live question in physics, not a settled detail that popularisations skip. Anyone who tells you it's clear, from any camp, is overselling.
Cross-disciplinary reading · AI
Machine learning has turned this distinction into standard engineering practice: aleatoric uncertainty (noise inherent in the world) versus epistemic uncertainty (the model hasn't seen this). Sensor read-out noise in a self-driving car is the former — another ten thousand kilometres of data won't remove it; «never seen roadworks in snow» is the latter, and data will. Modelling them separately is what lets you answer «is labelling another batch actually worth it» — active learning only helps with the epistemic half. A question physics has argued over for a century becomes, here, a budget decision.
In one line: statistical mechanics' probability is our ignorance; quantum probability may be the world's.
Think: If quantum probability turned out to come from some kind of ignorance after all, what would the physical picture have to pay?
Nonlocality, or the experimenter's free choice. Bell tests do not rule out hidden variables as such — they rule out the package «local + measurement settings chosen independently». Bohmian mechanics saves determinism, and pays with hidden variables that must instantaneously «know» a distant setting. There is no free determinism.
Going deeper
Where exactly is the boundary of reductionism? Has «strong emergence» ever had an empirical case?
Weak emergence — high-level phenomena are in principle fixed by the low level, they just can't be computed and can't be stated without new vocabulary — has overwhelming evidence. Strong emergence — high-level regularities that in principle cannot be derived from the low level, or that reach down and govern it — has no accepted physical case to date. The «multiple realizability» often cited as evidence (one temperature realised by countless microstates) only shows that high-level concepts cannot be replaced by low-level vocabulary; it does not show the low-level dynamics being violated. So this is more an epistemic boundary: the low level isn't wrong, it just can't say what the high level needs to say.
Do physicists actually need to touch philosophy?
Feynman's line was that philosophy of science is as useful to scientists as ornithology is to birds. But history offers a hard counter-chain: Einstein read Mach's critique and only then dared ask what «simultaneous» actually refers to; his argument with Bohr produced the EPR paper; EPR forced Bell's inequality; Bell's inequality became an experiment; the experiment became today's quantum technology. A string of purely conceptual objections ended up as instruments in a laboratory. The measured claim: philosophy can't do your integrals, and shouldn't be a back door for mysticism; its use is flushing out questions hidden by sloppy words.
Why does the universe have comprehensible laws rather than sheer disorder?
Three candidate answers, none fully satisfying. ① Selection effect: a universe without stable regularities contains no observers to ask — not empty, but weak. ② Effective theory: whatever exists stably at large scales tends to display some low-dimensional effective regularity, and the universality discussed in «Phase Transitions and Criticality» gives that intuition some substance. ③ The definitional answer: we call the compressible part «laws» and push the incompressible part into «initial conditions», which makes the claim partly a tautology. This question may well be out of physics' range, but that doesn't make it a fake one.
Might the laws we've found be only «the ones we happened to be able to think of»?
The suspicion has substance. Human mathematical tools have long favoured linear, differentiable, low-order equations — exactly the small subset solvable with a pen; turbulence, protein folding and quantum many-body systems are called «hard» partly for failing that taste. Simulation and machine learning are widening the range of tractable shapes: some systems we can now predict to high accuracy without writing a single closed-form line. If large numbers of «regularities» come to exist only as incompressible models, «understanding» will need redefining — does predicting count, or must you be able to say it out loud?