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

[Jason Resch]

On Sat, Oct 10, 2026, 9:56 AM John Clark via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

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

It only becomes a cosmological question if you subscribe to a sort of
local/subjective constructivism. This appears to be your unstated
assumption. But note that other views exist in the philosophy of
mathematics.


* 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. *
>

I would say it is independent of us, but not meaningless. We speak of other
possible universes in a greater multiverse that we will never access, but
that idea is not meaningless. For example, without that idea how else does
one explain fine tuning without invoking intelligent design? Also, of you
were to (re)appear in such another universe, it might have consequences for
what you can expect to experience, and hence would also be meaningful to
you whether or not than multiverse exists.

The same sorts of conclusions follow from a large mathematical reality,
which if it exists independently of minds and matter, could host minds and
provide the appearances of matter.

So I depart from you here in concluding that it 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.*
>

The amount of memory available is also an important consideration when
talking about infinite computations, as eventually you will need to repeat
states after exponential time. But perhaps there is further good in that,
if running the computation once was good.


* 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. *
>


This reminds me of the question: all astronomical questions being equal, is
it more likely we exist in a universe that hosts infinite observers, vs.
any finite number?

I think the same type of question could be asked about time (at least
assuming we might currently be simulations in an implausibly remote future).

Jason
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