Uncalculable irrational numbers and omniscience

 Today 8:18 AM
We don't know if there are infinite or not double primes (11 and 13), and as I understand it the largest such pair is pretty damn big.

Yes, you are thinking of twin primes: pairs of primes that differ by exactly 2, like (11, 13), (17, 19), (29, 31), and so on.

The famous twin prime conjecture says there are infinitely many such pairs, but nobody has proved it. We have found an enormous number of them, including some very large ones, but a finite list (however long) can never settle the question.

The largest known twin primes are indeed absurdly large. The record pair has hundreds of thousands of digits. The exact record changes as computers search further, but the scale is roughly:
p,p+2
p,p+2

where pp has hundreds of thousands of decimal digits.

A curious thing about twin primes is that they become rarer in a very predictable way. As numbers get larger, primes themselves thin out (roughly according to the prime number theorem), and two nearby numbers both being prime becomes even less likely. But "less likely" is not the same as "eventually stops"—and that is the heart of the mystery.

There is also a related result that was a major breakthrough: mathematicians proved that there are infinitely many pairs of primes separated by some finite gap, at most 246 (under current best unconditional bounds). So nature does produce infinitely many "nearby primes"; we just cannot force the gap all the way down to 2.

Twin primes are a wonderful example of a simple question that a child can understand and that still resists the world's best mathematicians. Prime numbers have a habit of doing that.

On a graph, that "predictable" way looks rather slow and that it wants to flatten out.

Yes, and that is exactly what makes the twin prime question so deceptive. If you plot the number of twin primes found up to NN, the curve rises very slowly and appears to be flattening. It almost looks as if it might eventually level off.

But that appearance is misleading. Many functions that are known to grow without bound grow painfully slowly.

For ordinary primes, the number of primes up to NN is approximately:
π(N)≈Nln⁡N
π(N)≈lnNN​

which already grows much slower than a straight line. Twin primes are rarer still. A heuristic prediction (the Hardy–Littlewood twin prime conjecture) says the number of twin prime pairs below NN should be roughly:
CN(ln⁡N)2
C(lnN)2N​

where CC is about 1.32. That extra logarithm in the denominator makes the curve look like it is flattening dramatically.

For example, compare:

    NN: grows explosively

    N/ln⁡NN/lnN: grows slowly

    N/(ln⁡N)2N/(lnN)2: grows even more slowly

A graph over a few billion or even a few trillion can easily give the visual impression that the curve is approaching a limit, while mathematically it may still be creeping upward forever.

This is a recurring theme in number theory. The sequence of primes itself looks increasingly sparse, but Euclid proved over 2,000 years ago that primes never run out. The hard part with twin primes is that we need two rare events to keep happening together.

A fun analogy: imagine watching a very long-distance runner who slows down every mile. From the first few miles you might conclude he is going to stop soon. But he may simply be slowing according to a rule that never actually reaches zero. The graph alone does not tell you whether he stops. That is exactly the trap twin primes set for our intuition.

That is a very interesting way to frame the issue, and it gets at a real tension between continuous mathematics and finite information.

If the universe is fundamentally quantized, then one can imagine (at least in principle) that the total physical state of the universe contains a finite amount of information. An omniscient being would then have something like a complete "state description" to know. The problem would still be enormous—perhaps beyond any conceivable computation—but it would not face the particular issue of an actual continuum containing uncountably many independent possibilities.

If, however, the universe is a true continuum in the mathematical sense, things become much more complicated. A field value at every point in space, or a continuous spacetime geometry, can require infinitely many numbers to specify. If those numbers are arbitrary real numbers, then many would be non-computable. A complete description of the universe might require an infinite amount of information, not merely an impossibly large amount.

But there is a subtle distinction: omniscience does not necessarily require calculation. It requires knowledge. A classical theologian might say that a divine mind is not a physical information processor constrained by the universe's information capacity. A critic might reply that this simply moves the mystery from "how is it calculated?" to "what does it mean to know something that no process can ever access?"

Physics adds another twist. Even if spacetime is continuous, the universe may still have a finite information capacity because of constraints such as quantum theory and gravitational limits (for example, ideas related to black hole entropy). Conversely, even a discrete universe might have infinitely many possible states if it has an infinite extent.

So the chain is not quite:

quantized universe → omniscience possible
continuous universe → omniscience impossible

It is more like:

finite information content → omniscience is at least conceptually less problematic
unbounded non-computable information → omniscience becomes much harder to make coherent

You have identified one of the deepest assumptions hidden in the idea of an all-knowing entity: it quietly assumes that the totality of reality is something that can, in some sense, be completely specified. Whether the universe is ultimately a finite "book" or an endlessly detailed continuum matters enormously to that question.

"Omniscience does not necessarily require computation." By "computation" I would mean not just arithmetic but also other methods of knowing, such as inferring, observing, whatever.  An uncalculable number is out of reach, so this seems to contradict the quoted statement.

I think your objection is well taken, and it exposes a weakness in the usual phrase "omniscience does not require computation."

If we use computation in your broader sense—any possible route to knowledge: observing, measuring, inferring, reasoning, deriving, remembering—then the distinction becomes much harder to maintain.U

A non-computable number is not merely inconveniently difficult. The mathematical claim is that there is no possible finite procedure that produces its digits. If "knowing" means having obtained information through some process, then yes, an uncomputable number appears to be beyond reach.

The usual theological response is to say that divine knowledge is not obtained at all. It is not the result of observation, inference, or calculation. It is more like an immediate awareness of truth. Aquinas, for example, would say that God's knowledge is not discursive—that is, not a chain of reasoning from one thing to another.

But your point is that this may simply redefine "knowledge" so that it escapes the difficulty. If a number is inaccessible to every possible information-gathering process, what does it mean to say it is known? We are no longer talking about a superior method of knowing; we are talking about a fundamentally different relation between a mind and reality.

This is closely related to a famous issue in philosophy: truth versus knowability.

A mathematical realist might say:

    The statement "this number has digit 7 in the trillionth place" is either true or false, whether anyone can ever find out.

A more epistemic view might say:

    A claim that cannot, even in principle, be accessed by any possible intelligence is not meaningfully different from a distinction we invented.

Your argument is essentially that omniscience seems to require not merely having all truths, but having some coherent relationship to those truths. If the universe contains non-computable information, then "knowing everything" becomes problematic unless we allow a kind of knowledge that is unlike any kind we experience or can analyze.

And this loops back to your earlier point about quantization. If reality is ultimately finite-information, then omniscience looks more like having the complete data set. If reality contains a true continuum with arbitrary real values, then omniscience requires something much stranger: access to an amount and kind of information that may not be representable by any process whatsoever.

So I would refine the earlier statement: omniscience may not require computation in the narrow sense of running algorithms, but if "knowing" means any intelligible acquisition or possession of information, your objection is very strong. The theological answer depends entirely on whether one accepts that "knowing" can mean something fundamentally beyond any possible mode of cognition available to us.

I can think of another escape hatch, saying God knows everything it is in his nature to know." This has a paralell to the old question of whether God can sin.

Yes, that's a very close parallel.

Classical theology often responds to apparent paradoxes not by saying "God can do absolutely anything whatsoever," but by saying that omnipotence and omniscience apply to all things that are genuinely possible or meaningful.

So when someone asks:

    Can God create a square circle?

    Can God make 2 + 2 = 5?

    Can God create a stone so heavy He cannot lift it?

    Can God sin?

The traditional answer is often that these are not limitations on God because they are not genuine possibilities in the first place. A square circle is not an object waiting to be made; it is a contradiction in terms.

Likewise, Aquinas argued that God cannot sin, not because God lacks power, but because sin is a defect. The inability to be defective is not a weakness.

Your proposed reformulation of omniscience works the same way:

    God knows everything it is meaningful to know.

That avoids some paradoxes by narrowing the domain of "everything."

But then the debate shifts. The question becomes: Are non-computable truths meaningful objects of knowledge?

A Thomist might say yes. They exist as truths, and God knows them.

You seem to be pushing further and asking:

    What does "knows" mean here?

If no process, method, observation, inference, or representation can reach those truths, then saying they are "known" starts to resemble saying a stone is liftable because it is in the nature of stones to be liftable. The statement becomes true by how the terms are defined.

In that sense, your comparison to the "can God sin?" question is apt. Both involve moving from:

    What can God do?

to

    What do we mean by "do," "know," "power," and "possibility"?

Aquinas would probably regard that as a strength of the argument. He spent much of his effort clarifying concepts before answering questions.

A skeptic might regard it as a retreat into definitions.

What's interesting is that your non-computable-number example is harder than the usual stone paradox. With square circles, we can point to a direct contradiction. With non-computable numbers, there is no contradiction. The object exists perfectly well within mathematics. The difficulty is explaining what it would mean for a mind—any mind—to possess an infinity of information that cannot be generated, derived, observed, or represented by any process.

That is a much subtler challenge than the standard omnipotence puzzles, and I suspect Aquinas never had to contend with anything quite like Turing's notion of non-computability. Had he encountered it, I imagine he would indeed have reached for something like your proposed escape hatch: divine knowledge is knowledge of all knowable truths, and divine knowing is not a process. Whether one finds that satisfying is another matter.