[ExI] Can a single observer make an infinite number of computation in a finite time? (Was Mathematical Realism)

[John Clark]

On Fri, Oct 9, 2026 at 11:25 AM Stuart LaForge via extropy-chat <
extropy-chat at lists.extropy.org> wrote:


>
>
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> *> One question he has is about whether the infinite digits of piexist for
> no other reason than some ancient Greek mathematicianpostulated the
> existence of pi. Here is what he asks: "If all of Pi'sinfinite digits did
> not exist, and have the exact values they happen tohave, then e^(2 * Pi *
> i) would not equal 1.*


*It is beyond dispute that starting with just a few simple axioms and the
laws of logic you will be able to reassemble some of the symbols of
mathematics until you get  e^(2 * Pi * i) -1 = 0, but do an infinite number
of digits of pi actually exist? Ultimately that is a cosmological question
not a mathematical one, and not just because only 62 digits of pi would be
necessary to compute the circumference of the observable universe with an
error of less than the Planck length. The big question is whether it's
possible, even in theory, for an individual observer to make an infinite,
and not just very large, number of computations in a finite amount of time.
If the universe is only capable of calculating a particular finite number
of digits of pi then talking about a digit larger than that number is
meaningless. *

*Any region with finite energy has a finite number of distinguishable
states due to the Bekenstein bound, so a computer confined to it can run
forever only by repeating states. Reversible computations can help but the
cosmological event horizon sets a temperature floor near 10^-30 K, so
Landauer erasure costs never entirely vanish, and that would seem to
indicate infinite computations are not possible; BUT if dark energy was
zero (or at least was ≥ −1/3) and the universe was not accelerating then
there would be no boundary and although you'd still have the CMB, that
would soon be red shifted away, so a temperature floor would not exist.*

*Barrow and Tipler, found that if dark energy reverses directions and
produces a big crunch then that recollapsing universe would undergo
infinitely many chaotic oscillations in finite proper time, with all
horizons erased. An individual observer could then in theory process an
unbounded amount  of information before the end. The trouble is it requires
General Relativity to remain accurate arbitrarily close to the point
singularity, which few expect, since Planck-scale physics should put a
finite cap on the state count. But nobody really knows if space is
continuous or what happens when you get super close to the Planck scale, so
maybe.  *

*In 1979 Freeman Dyson wrote a paper called Time Without End
<https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf> that said an
intelligent hibernating observer in step with the cooling background of the
universe could experience an infinite amount of subjective time from a
finite amount of energy in a finite amount of objective time if the
universe is not accelerating.  Today Dark energy is certainly present and
the universe is accelerating, so that would seem to be ruled out,
however if the current acceleration is just a transient phase of evolving
dark energy, then Dyson might be right after all and infinite computation
is possible. *

*It all depends on what dark energy does in the future. If it decays to
zero then Dyson-style infinite computation scenarios are possible. If dark
energy turns negative then you get a Big Crunch and a Tipler type situation
where an observer could perform an infinite number of calculations.  If
dark energy really is constant and the universe keeps on accelerating
forever then then any single observer would only be able to perform a
finite number of calculations. And if dark energy is getting stronger and
the acceleration is accelerating (in other words if the universe is a jerk)
then we get a Big Rip and infinite computation would be utterly impossible
and things are completely hopeless.*

*Today astronomers tell us that there is some indication dark energy is
getting weaker, but the evidence is not strong enough to claim a discovery,
you need 5 Sigma for that, and right now the Sigma is between 3 and 4.2,
depending on what experimental evidence you consider acceptable. But
whatever turns out to be true the question "do infinite digits of pi
exist?" is not a mathematical or a philosophical question, it is a question
that can only be answered by physics. *

*John K Clark*
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