Mathematical models can give you rigorous empirical predictions, but you cannot arrive at ontological statements about the nature of reality from mathematical postulates alone.

It is similar to an is/ought dilemma: an argument's conclusions cannot be stronger than its premises. Purely mathematical postulates will only get you to further mathematical conclusions. To say something about the nature of reality, you need an ontological postulate.

An ontological postulate is an assumption about the nature of reality and about how that reality relates to the mathematics. It goes beyond simply stating what the mathematical structure is or what predictions follow from it.

It is sort of like this: if you gave me a body of mathematics, I used it to make empirical predictions, and those predictions turned out to be correct, I would probably turn to you, astonished, and ask: Why are they correct? What does this mathematics mean? What does it represent?

Of course, you can be a pure pragmatist and not care about the answers to those questions at all. But almost no one is. Most people do care about ontological questions, such as "What is real?"

The important point, however, is that ontological postulates are not part of the mathematical framework itself. They are therefore not empirical in the same sense as the predictions generated by the mathematical model, and they cannot necessarily be experimentally proved or falsified.

That does not mean they are arbitrary. They still have to be consistent with the mathematics of the theory, and they still have to be coherent. Even if they cannot be empirically demonstrated, they must still be justified by some other means.

When I talk about "scientism," I am talking specifically about people who conflate ontological postulates with mathematical postulates, treating them as though they were one and the same and as though the former necessarily followed from the latter.

These people then provide no philosophical justification for their ontological postulates because they insist that those postulates have already been proved beyond any reasonable doubt by the mathematics and empirical evidence alone. In fact, if you ask them to justify their ontological assumptions, they will accuse you of being a science denier, a crackpot, or a crank simply for questioning the "scientific community."

Let me give you a simple example: fields.

Mathematically, a field is, roughly speaking, a quantity assigned to every point in some domain, such as 3D physical space. People often assume that these mathematical fields correspond to an ontological field: an invisible substance that permeates space and acts on particles, causing them to behave in particular ways. They therefore speak of fields as entities that exist in the world and have causal effects on particles.

However, it does not automatically follow from the empirical success of the mathematics of fields that we should interpret that mathematics as representing an ontological field, meaning an all-pervading invisible substance.

Suppose you turn on an electromagnet and observe iron filings arranging themselves along the familiar field lines. One interpretation is that the magnetic field acts on the iron filings and causes them to move into that configuration. But another interpretation is that the behavior of the iron filings follows from their own nature. The particles have certain dispositions to behave in certain ways under certain conditions, and the mathematics of fields describes those dispositions.

This is sometimes called a "dispositional" approach in the literature. The "nature" of an object is often described as its "disposition." To talk about the dispositional nature of particles is simply to talk about what those particles do given certain conditions, as a consequence of the laws governing their behavior, without proposing an additional invisible entity "from the outside" that acts upon them and causes them to move.

Notice what I have done here. I have questioned the ontological postulate that the mathematical field corresponds to an ontological field. But I have not questioned the mathematical field itself.

In other words, I am not questioning the efficacy of the mathematical model or its ability to predict what we observe. If I held this view, I would not be denying any empirical predictions made by the model, nor would I be denying any of the scientific evidence supporting those predictions. I would only be questioning what the mathematical model represents about reality.

This is just one example, and a relatively simple one at that. There are much more complicated examples when it comes to the ontological assumptions underlying theories of relativity or quantum mechanics.

Yet questioning an ontological postulate like this will quite often get you flamed, harassed, or attacked by large numbers of people because they equate ontological postulates with mathematical postulates. They then paint you as a "science denier," a "crank," a "flat-earther," or a "crackpot." They may accuse you of being "arrogant" or a "know-it-all" for "daring to question the scientific consensus." A newer insult I've encountered is the claim, made without evidence, that such ideas are the product of "AI psychosis."

If you point out that these personal attacks are unfair and that you'd like a real rebuttal to your argument, they then accuse you of "being upset that we did not automatically believe you without question and actually pushed back," even though their pushback never actually addresses the substance. This is because the person practicing scientism cannot address the substance because they don't know themselves why they believe those ontological postulates. They just take them as a given, without question, so they have no idea how to defend them other than to shut the skeptic down with personal attacks and appeals to popularity without ever actually addressing their argument.

For a real-life example of what I would consider scientism, consider Chad Orzel's TED-Ed talk on Schrödinger's cat. Schrödinger did not believe that the mathematical representation of a quantum superposition of states literally, ontologically, represents a system in multiple states at the same time. Orzel then presents Schrödinger's position as though rejecting this ontological interpretation of quantum superposition amounts to rejecting the science itself.

But Schrödinger obviously was not denying the empirical evidence or the mathematics. He was one of the people who developed the mathematics underlying quantum mechanics. His objection was to the interpretation of what that mathematics says about physical reality.

It is therefore disingenuous to frame disagreement over the ontology of quantum superposition as a rejection of science. Believing that a cat can literally be dead and alive at the same time in ontological reality is not a necessary condition for applying the mathematics of quantum mechanics or for building technologies based upon it.

When I refer to "scientism," then, I am referring to the failure to recognize that the mathematical postulates of scientific models are not, by themselves, sufficient to draw conclusions about the ontology of the world, the "truth" of the world.

To make ontological claims, you also need to put forward and defend ontological postulates. Those postulates are not themselves empirical observations, and they do not automatically follow from the mathematics.

The belief that ontological conclusions flow directly and necessarily from the mathematics is what causes people to become so dogmatic about them. A successful physical theory can effectively be elevated into a kind of religious faith: everyone repeats the associated ontological claims, those who question them are ostracized, and alternative possibilities are dismissed.

Yet when you ask for a philosophical defense of those ontological claims, there may be very little offered because the claims are simply assumed to have been empirically proven, despite the fact that the claims are non-empirical and cannot ever be empirically proven.

The irony is that none of this requires rejecting the science. You can accept the mathematics, accept the empirical evidence, accept the predictive success of the theory, and still ask a fundamentally different question: What, exactly, does this mathematical structure tell us about what reality is?

(Audio version of this post for those who can't be bothered to read it all.)

If you watch any pop-sci YouTube channel, you have probably heard the old story: physicists used to believe in this silly thing called the aether, but then the Michelson-Morley experiment disproved it in 1887, and Einstein replaced it with his theory of special relativity in 1905, which is "the most tested theory of all time."

But this is historical revisionism and does not accurately reflect the reality of what actually happened.

The 1887 experiment was only intended to test aether models where the aether is directly detectable. Its negative result could imply the aether doesn't exist, or it could imply it is only indirectly detectable.

Indeed, in 1904, the physicist Hendrik Lorentz put forward an aether model whereby the aether is not directly detectable, and thus was consistent with the Michelson-Morley result.

This model is mathematically equivalent to special relativity, and in fact Einstein's 1905 paper was explicitly in response to Lorentz's paper.

Since they are mathematically equivalent, the claim that "special relativity is the most tested theory of all time" makes no sense because no experiment can directly distinguish between Lorentz's 1904 model and Einstein's 1905. If you were to experimentally confirm time dilation, for example, well, both Lorentz's and Einstein's models predict that, so you didn't distinguish them.

Einstein's theory thus won out on philosophical grounds, not empirical grounds. Einstein had, at the time, the stronger philosophical argument: if the aether is not directly detectable, then maybe it doesn't exist, so maybe we should try constructing a model which makes the same empirical predictions as Lorentz's theory but without the aether.

And that is precisely what Einstein's 1905 paper achieves. This convinced basically every physicist to abandon Lorentz's ideas and adopt Einstein's ideas.

So, if Einstein's 1905 paper showed that the undetectable aether is not necessary to make the right predictions, what's the issue? Why do I even bring it up? Seems like all of this is completely reasonable.

I bring it up because of a caveat Einstein was well aware of himself: Lorentz's theory only renders the aether directly undetectable. It does offer a path under which its existence can be indirectly verified, and this bothered Einstein significantly, since he knew it would rule out his special relativity.

What is this indirect verification? The existence of Lorentz's aether can be indirectly verified if we can provide any experimental evidence, at all, that nature contains, anywhere, in any empirical observation, non-local effects.

Why?

Well, if two events occur so far apart and in such quick succession that not even light can traverse the gap, then special relativity predicts that different observers can disagree as to the temporal ordering of the events, measured on their clocks.

If no effect can travel faster than light, then these disagreements about the temporal ordering of events are always causally disconnected, and so it has no real implications.

If an effect can travel faster than light, then the causal sequence can become out-of-order in some perspectives, leading to causal paradoxes and logical inconsistencies in the theory.

This is only true for special relativity, however, because Einstein's theory treats the time-readings on clocks as equivalent to true, ontological time. Lorentz's theory treats the time-readings on clocks as an emergent feature and not fundamental, and so it is perfectly consistent for two observers to disagree about what event happened first without there being a paradox, because one of them can just be wrong and the other correct.

Why is all this relevant?

Newton's theory of gravity is unambiguously a non-local action-at-a-distance theory. Einstein knew that if he wanted his theory of special relativity to make sense, he needed to replace Newton's theory with a local theory of gravity, and so he did precisely that: developing the theory of general relativity.

However, Einstein also realized quantum mechanics is a non-local theory, and tried for the rest of his career to replace it with a local field theory, but never succeeded.

In fact, in 1964, the physicist John Bell proved it is impossible to replace it with a local theory without contradicting its empirical predictions. As Bell concludes at the end of his 1964 paper, either quantum mechanics is wrong, or the states of particles cannot be Lorentz invariant.

If they cannot be Lorentz invariant, they must be Galilean invariant, which then returns us to Lorentz's theory, as there is no absolute reference point in Einstein's theory, so Einstein's theory is incompatible with the requirement of Galilean invariance.

In 2022, the Nobel Prize was awarded for experimentally testing quantum mechanics and verifying that quantum mechanics is indeed not wrong, and, by extension, Lorentz's theory must necessarily be the correct theory, as the existence of the rest frame of the universe has been confirmed indirectly through proving the existence of non-local effects.

Did physicists respond to this by revisiting Lorentz's ideas? No! Because to do so would contradict the last century of ideology centered around the "revolution" that is special relativity.

So, instead, physicists took a different approach. Rather than agreeing with Bell's own conclusion that his 1964 theorem shows that nature is non-local and therefore rules out "locality," they instead claim that Bell's theorem rules out "local realism."

The implication here is that you could deny locality, but you could also deny realism.

"Reality doesn't exist, but thank God it is local!"

Despite a weird common misconception I've seen thrown about, "realism" has nothing to do at all with determinism or randomness. It is about object permanence, the idea that observable objects with observable properties continue to possess observable properties even when they are not being observed in the instantaneous moment of observation.

Bell's theorem does not assume determinism; it assumes merely that at least one of the particle's observable properties still contains definite values even when not being observed, regardless of whether or not those values evolve randomly or deterministically.

The denial of realism leads directly to the "measurement problem," because it creates a disconnect between the fact that you are claiming particles possess no observable (measurable) properties when not being observed (measured), but then this fails to explain why, when we try to observe (measure) them, they always empirically possess those properties.

This has led to a deluge of incoherent, incomprehensible papers on "interpretations" of quantum mechanics which all start with the premise that particles do not possess any observable properties at all when you are not looking at them and then work backwards from this conclusion to try and desperately make it coherent.

None of them succeed in this, and they readily admit they do not succeed in this. The popularization of statements like, "no one understands quantum mechanics" are all just passive admissions that these attempts to make sense of it in a non-realist framework are simply incomprehensible and nobody even knows what is being talked about anymore.

People use these kinds of phrases in an incredibly dishonest fashion. It gives the implication that quantum mechanics is objectively incomprehensible, and that is just a fact of life. But it's not true at all.

As Bell pointed out in 1982...

[T]he subjectivity of the orthodox version, the necessary reference to the 'observer,' could be eliminated...that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice

That is to say, the supposed "incomprehensibility" of quantum mechanics, the insistence that it is impossible to understand, the hyper-focus on the observer, measuring apparatus, stemming from its non-realist premises, are not fundamental features of the theory, but are explicit theoretical choices made by physicists.

These explicit theoretical choices all arise from a desperate attempt to preserve locality. The 2022 Nobel Prize shows empirically that, if the particles possess observable properties when not being observed (realism), the experimental results are impossible to explain unless those properties influence each other in a non-local fashion.

That directly renders special relativity non-viable. But, physicists use a loophole where they just argue the particles possess no properties when you're not looking at them.

You see, if you deny that anything exists when you're not looking at it, then the only possible way to prove non-locality is to prove non-locality precisely in what you are directly observing, which requires you to send a direct signal, i.e., superluminal communication.

Under a non-realist framework, you can't prove non-locality indirectly by showing merely that the results are impossible to explain without non-local influences if you did not directly observe those non-local influences. A non-realist framework only allows for the proof of non-locality if you can directly signal with it.

Of course, it has been proven for decades that quantum mechanics does not allow for signaling, so this loophole allows you to continue to claim quantum mechanics is local, but at the cost of denying that particles have real properties in the real physical world independently of you looking at them.

This denial of object permanence, the very basis of realism, then leads to all the absurd interpretations around the theory, interpretations which its own authors admit are entirely incomprehensible by continually repeating that "quantum mechanics is impossible to understand" over and over again.

Indeed, the complete incomprehensibility of these interpretations was proved in a paper by Frauchiger and Renner in the journal Nature titled "Quantum theory cannot consistently describe the use of itself." They proved in a simple four-qubit circuit that every major interpretation of quantum mechanics is logically inconsistent.

The only one which they analyzed that wasn't inconsistent was a realist interpretation, but when they applied this interpretation, they point out in the paper that the time-ordering of events matters, meaning you have to introduce an absolute time-ordering of events to resolve the inconsistency... meaning you need Lorentz's model.

All of this nonsense is just a desperate attempt to preserve the dominant philosophical paradigm for the last 100 years. As pointed out by Bell in his paper "How to Teach Special Relativity," Lorentz's and Einstein's views are mathematically equivalent, so changing from Einstein's to Lorentz's view does not require actually even changing any of the mathematics or the empirical predictions.

It is purely a philosophical reinterpretation of the same empirical results and the same mathematics, yet the reinterpretation allows you to avoid having to deny reality to make sense of violations of Bell inequalities, and to make sense of quantum mechanics in a trivially intuitive framework.

But, of course, if the distinction is only philosophical in nature and non-empirical, a lot of physicists just don't care, since many aren't particularly interested in philosophy and don't really care to know anything about the natural world. They are more pragmatic engineers and just want to build stuff, so they just want to know what the math says and can't care much about the ontological implications of said math.

If a person is a pure pragmatist, they wouldn't care about this debate anyway, one way or the other. So this rant is largely on the side of people who do care.

https://www.youtube.com/watch?v=1ud6LNtX014

https://www.stochasticnetwork.com//

It is often claimed that the bits in a quantum computer have no values until you look at them, then they "collapse" into a definite state. However, such a belief is not required by the mathematics but is a personal choice, as it is possible to represent any arbitrary quantum circuit as a Markov chain.

In classical probabilistic computing, you represent the bits using a probability vector we can call p⃗, which holds the current probability distribution for the bits, and we represent logic gates in terms of stochastic perbutations given by the Markov matrix Γ.

The evolution rule then just becomes:

• p⃗'=Γp⃗

The mathematical description does not track the definite values of the bits, but those are indeed believed to exist, because you can create a simulation whereby you start with well-defined values for the bits and stochastically perturb them to new definite values based on the probabilities given by Γ, and the statistics on such a system will reproduce p⃗ at every time step.

Quantum computers use a different evolution rule:

• ψ'=Uψ

Where U is a unitary matrix and ψ is the wavefunction. The main difference is that U and ψ are complex-valued, so it is not obvious what they physically represent or how this could relate to a Markov chain in any way.

Complex numbers aren't magic. They just represent a system with two degrees of freedom. That means we can get rid of the complex numbers by decomposing them into two separate real-valued vectors. Specifically, we can perform a polar decomposition on ψ.

• p⃗=|ψ|²
• φ⃗=arg(ψ)

Notice that, if we perform a polar decomposition on ψ, then one of the two degrees of freedom is quite literally a probability vector. Something interesting happens if we write an update rule for this p⃗ directly so we can evolve p⃗ directly without having to constantly convert back and forth between p⃗ and ψ.

We get this evolution rule:

• p⃗'=Γp⃗+q⃗ where Γ=|U|²

We get what looks like a classical Markov process but with a non-linear correction term called q⃗, which accounts for interference effects. The equation for computing q⃗ has a dependence upon φ⃗, and so φ⃗ acts like a memory effect which can cause the same logic gate to give different behavior if the value of φ⃗ is different at that time.

A prime example is the Hadamard gate. Apply it once then p⃗=[0.5; 0.5]. Apply it twice then the bit deterministically goes back to where it came from. That means it needs a way to "remember" where it came from, and that "memory" is stored in φ⃗. If you apply the Hadamard gate when the bit value begins with a value of 0, then φ⃗=[0; 0]. If you apply it when the bit value begins with 1, then φ⃗=[0; π].

Rather than using φ⃗ to compute q⃗, which is a correction term for Γp⃗, we can instead use φ⃗ to correct Γ, such that the same logic gate can give us a different Γ if the state of φ⃗ is different at a given time. If we do that, then we can represent each logic gate in terms of these corrected Γ, and thus fit the entire system to the evolution rule given by: p⃗'=Γp⃗.

There would be no need to correct Γp⃗ with q⃗ because we already built that correction into our corrected Γ. The method to correct Γ also turns out to be trivially simple, you just apply an algorithm called iterative proportional fitting which comes from optimal transport, which will adjust the initial Γ=|U|² as little as possible such that Γp⃗ maps you to the correct p⃗'.

That means we can fit the entire circuit to a Markov chain where the qubits have definite bit values at all times and each logic gate represents a stochastic pertubation upon the bits that merely permutes them randomly to a new definite configuration, and if you collect statistics on that stochastic process at every step, it will always match the Born rule distribution at that step.