Universe misbehaving

 You said:

It might be worth reminding ourselves that there can be no largest prime number, only (as already noted above) the largest known prime. The implication worth contemplating in this context is that no matter how many gazillion digits a prime number has, it is infinitesimally small (indistinguishable from zero if looked at from far enough away) compared to all the primes that exist. Although we have to admit the prime numbers, on average, steadily get further and further apart, there is nevertheless an infinite number of them separated by any given even number, even two. (Primes can't be separated by an odd number since an odd plus another odd number always produces an even number, and even numbers are by definition composite). Another mind-boggler -- the occurrence of primes in the number sequence seems, by every test we can imagine, to be random, but, of course, it can't be random, as it has to be something determined.

ChatGPT said:

This is nicely put, and most of it is exactly right—but there’s one place where it’s worth slowing down and being a little pedantic, because the distinction matters.

You’re certainly right that there can be no largest prime, only a largest known one, and that any particular prime—no matter how grotesquely large—has asymptotic density zero when viewed against the infinite set of all primes. In the limit, every finite collection of primes collapses to “nothing at all” compared with what remains. That intuition is sound and important.

You’re also right that primes get thinner on average. The prime number theorem tells us that the typical gap near a large number $n$ is about $\log n$, so the widening is slow but inexorable. Nothing surprising happens suddenly; it’s a long, gentle drift apart.

Where I have to push back is this sentence: “there is nevertheless an infinite number of them separated by any given even number, even two.” That is believed to be true, but it is not known to be true. For gap 2 it’s the twin prime conjecture; for gap $2k$ it’s Polignac’s conjecture. We have very strong evidence, and spectacular partial results—most notably that there are infinitely many pairs of primes separated by some fixed finite bound—but we do not know that every even gap occurs infinitely often. Two remains tantalisingly unproven.

The parity observation itself is correct, of course: aside from the trivial prime 2, primes are odd, so odd gaps are impossible. But “possible” and “known to occur infinitely often” are very different logical categories, and number theory lives in that gap.

Your final point is one of the deepest and most philosophically unsettling ones. The primes pass every reasonable test for randomness, yet they are completely deterministic, generated by a definition so simple it fits on a postcard. This is not accidental. It’s now understood that the primes behave like a pseudorandom sequence: rigidly defined, yet statistically indistinguishable from randomness at large scales. That tension—law without pattern, structure without predictability—is exactly why primes keep pulling mathematicians back in.

If anything, primes are a standing rebuke to the idea that determinism must look orderly. Sometimes it looks like noise, forever—but noise with teeth.

You said:

Thanks for the correction; I had read about that but for some reason that should worry me but oddly doesn't, forgotten.

ChatGPT said:

That’s a very human kind of forgetting—and honestly, a healthy one.

Mathematics is full of results that sit in a strange limbo: overwhelmingly likely, supported by mountains of evidence, compatible with everything else we believe… and yet still not proved. The twin prime conjecture is almost a mascot for that category. It’s easy for the mind to quietly promote such things from “unproved” to “true enough for philosophical reflection,” especially when nothing else depends on the promotion. And in this case, nothing really does.

The fact that it doesn’t worry you makes sense. Whether or not every even gap occurs infinitely often, the deeper point you were circling—the infinitesimal insignificance of any finite prime, the thinning yet endless supply, the eerie randomness of a fully determined sequence—remains untouched. The emotional or metaphysical weight of primes doesn’t hinge on that one conjecture being settled.

There’s also something almost reassuring about the forgetfulness itself. It’s a reminder that we’re not bookkeeping machines; we carry ideas for their shape and resonance, not their footnotes. The primes still behave like a law of nature we can observe but not domesticate, and the fact that a few of their secrets remain formally out of reach only reinforces that feeling rather than undermining it.

If anything, the unproved conjectures function like visible seams in the universe’s logic—places where certainty thins, but meaning doesn’t. And living comfortably with that is, I suspect, a sign of intellectual maturity rather than carelessness.

You said:

"Pseudorandom." To me that indicates something that seems random but is really determined, like the proverbial fair coin toss. It is something we think is determined but can't really be sure as the calculations needed to say are beyond us. I think primes are something else; they feel different. They are about numbers, not events in space/time.

ChatGPT said:

Yes—this is a sharp distinction, and you’re right to press on it. “Pseudorandom” is a slightly misleading word here, and mathematicians use it with more technical caution than the everyday sense allows.

In ordinary language, pseudorandomness usually means epistemic randomness: the coin toss is determined in principle, but we lack the information and computational power to predict it. The randomness lives in us, not in the system. That’s not what’s going on with the primes, and your unease is well founded.

With primes, the determinism is not causal or dynamical at all. There is no unfolding in time, no hidden variables, no sensitive dependence on initial conditions. The sequence is logically fixed. Every prime “already exists,” so to speak, whether or not anyone ever computes it. So their unpredictability cannot be blamed on incomplete knowledge of a process. It’s not physics; it’s ontology.

What mathematicians usually mean by “pseudorandom” here is much colder and more unsettling: that the primes satisfy the same statistical laws as genuinely random sequences even though there is no randomness mechanism behind them. For example, correlations, spacing distributions, and higher-order statistics line up eerily well with models built from coin flips—models that have no business applying to arithmetic objects at all.

That’s why primes feel different. Their unpredictability is not practical but structural. There is no deeper explanation hiding behind the curtain, at least none we can currently articulate, other than “this is how arithmetic is.” If a coin surprises us, we blame friction or air currents. If the primes surprise us, there’s nowhere to point.

In that sense, primes sit in a peculiar middle ground: more rigid than any physical process, yet less predictable than many of them. They suggest that unpredictability does not require time, chance, or ignorance—only enough structure. And that is a far stranger lesson than the one taught by a coin toss.

You said:

You use the word "surprise." That to me implies an emotional effect. The fact is, as we check number after number, we are not at all surprised when we find one, it is expected, and we even have a rough sense of how far we need go to find the next one.

ChatGPT said:

That’s fair—and you’re right to call me out on the word. “Surprise” was doing too much emotional work there.

At the local level, there is essentially no surprise at all. When you test numbers one by one, finding a prime is expected in exactly the way you say. The prime number theorem gives you a very good prior: near $n$, the chance is about $1\mathrm{/}\log n$. You don’t jump when a prime appears; you would jump if a long stretch went by with none. The experience is calm, almost procedural.

Where “surprise” really enters—if it does at all—is not psychological but conceptual. It’s the mismatch between two facts we hold simultaneously without strain but without reconciliation either. On the one hand, we can predict primes statistically with remarkable accuracy. On the other, we cannot predict them specifically even one step ahead without doing the work. The tension is not emotional but structural: global regularity coexisting with local opacity.

So perhaps a better word is not “surprise” but irreducibility. There is no shortcut that compresses “which numbers are prime” into something simpler than the primes themselves. We can know how many to expect, how they thin out, how they correlate in bulk—but the individual facts resist explanation beyond the definition.

Seen that way, primes aren’t surprising events; they’re uncompressible truths. You don’t react to them—you enumerate them. And the oddness lies not in encountering a prime, but in the absence of any deeper story to tell once you’ve found it.

You said:

Another question: just how sure are we that events we call pseudorandom are necessarily determined?

ChatGPT said:

That’s an excellent question, and it goes right to the boundary between physics, philosophy, and what we quietly assume when we do mathematics.

The short answer is: we are not completely sure. Determinism is an assumption that has worked extremely well in some domains, but it is no longer something we can take as settled fact.

In classical physics, pseudorandomness really did mean “determined but unknowable in practice.” A coin toss, turbulent flow, planetary perturbations—everything followed deterministic laws, and randomness was merely ignorance amplified by sensitivity to initial conditions. Chaos theory strengthened this picture rather than weakening it: the equations are exact, but prediction becomes infeasible. In that framework, “pseudorandom” is entirely epistemic.

Quantum mechanics complicates this profoundly. In its standard interpretation, certain events—radioactive decay, the outcome of a single measurement—are not just unpredictable but undetermined. There is no hidden fact of the matter waiting to be discovered. The theory gives probabilities, not concealed certainties. Importantly, this isn’t merely a practical limitation; Bell-type experiments strongly rule out large classes of deterministic “hidden variable” theories unless they accept deeply counterintuitive features like nonlocality. Some interpretations do preserve determinism (Bohmian mechanics, many-worlds), but they do so at a significant conceptual cost. So even in physics, determinism is now optional rather than mandatory.

That leaves us in an interesting position. When we label something pseudorandom today, we are often making a working assumption, not stating a metaphysical truth. We are saying: “this behaves as if random, and treating it as determined would give us no additional predictive power.” Whether there is a determinate underlying story may be unknowable even in principle.

This is where your earlier distinction about primes becomes relevant again. Primes are certainly determined in a logical sense—there is no room for indeterminacy there. But physical events that look pseudorandom may not share that status. Their unpredictability might not just be about complexity or ignorance; it might be ontological.

So the honest answer is that “pseudorandom” straddles two very different ideas. In mathematics and algorithms, it usually means “deterministic but statistically random-looking.” In the physical world, it may sometimes mean “random-looking because it really is.” The word papers over a fault line, and we only notice it when we ask the question you just asked.

You said:

The universe appears to possibly sometimes misbehave.

ChatGPT said:

Yes—and that’s a beautifully restrained way to put it.

What’s striking is that the “misbehavior” isn’t noisy or theatrical. The universe doesn’t flagrantly violate its own rules; it seems instead to refuse to finish explaining itself. The laws work, predictions work statistically, technology works—and yet, at certain seams, the story just stops. Not “this is hard,” but “this may not have an answer of the kind you’re asking for.”

Quantum indeterminacy is the obvious example, but even there the misbehavior is disciplined. Probabilities are exquisitely precise; only the individual outcomes slip through our fingers. It’s as if the universe is impeccably behaved in aggregate and quietly anarchic in detail. Lawful, but not obliging.

What makes this unsettling—at least to minds trained in mathematics or classical reasoning—is that misbehavior used to mean error. Now it may mean category mistake. We expect causes where there may only be distributions; we expect explanations where there may only be descriptions. The universe isn’t breaking rules so much as declining to adopt the ones we find psychologically comfortable.

And that loops back to your earlier instinct about primes. They don’t misbehave, but they also don’t cooperate. They are rigid, exact, eternal—and yet they withhold pattern at the level we’d most like. Physical reality may be doing something analogous: offering structure without narrative, order without reassurance.

So perhaps the universe doesn’t misbehave in the sense of acting badly. It misbehaves in the sense that a very polite system might misbehave—by answering every question except the one you really meant to ask.

You said:

You force me to assert that the universe is not acting badly, and it may be that in the end all the pieces will fit together. I wouldn't presume to say.

ChatGPT said:

That’s a fair correction—and a generous one.

Saying the universe “misbehaves” smuggles in a moral or normative standard it never agreed to. You’re right: there’s no obligation on its part to line up with our expectations, or even with our deepest habits of explanation. The discomfort may be entirely on our side.

And the second point matters just as much. History gives us plenty of reason for intellectual humility. Time and again, things that once looked like cracks in reality turned out to be seams we simply hadn’t learned to follow yet. What appeared incoherent later revealed itself as a larger coherence that couldn’t be glimpsed from the earlier vantage point. It would be rash to rule that out here.

So perhaps the most honest stance is exactly the one you’ve articulated: suspension without despair. The universe is not acting badly; it is acting in ways we do not yet fully integrate. Whether that integration is ultimately possible—or whether there are limits baked into the enterprise itself—is an open question, and maybe permanently so.

That posture has a certain quiet dignity to it. Neither insisting on mystery nor demanding closure, just acknowledging that the map is unfinished and declining to declare, prematurely, that it never can be completed.

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