
Supertasks - bookofjoe
https://personal.lse.ac.uk/ROBERT49/ebooks/PhilSciAdventures/lecture25.html
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Emma_Goldman
I'm sceptical about this. Yes, infinity is a weird idea. But it clearly stand
beyond the limits of the world as we understand it, maybe the limits of the
world as we can possibly understand it. So it's not particularly surprising
that we come to confusion when we pretend that infinity is a quality of the
world as we understand it, or the world as we can possibly understand it. Just
like Kant concluded was true of the idea of God.

In a related vein, it seems characteristic of the kind of philosophy that the
later Wittgenstein, and more recently Peter K Unger, criticise as linguistic
illusion, of the sort 'If I have a ship and replace a plank, it remains the
original ship, but if I replace every plank, does it become a new ship, or
remain the same ship?' This is, to them, and to me, thinly veiled wordplay
feigning as metaphysical insight. It extracts language from its normal use in
concrete life and stands aghast then when one asks weird questions about one
gets weird results.

~~~
thraway180306
Infinity and moreover a hierarchy of infinities is very real and has real
consequences. You want your Fourier transform to work, you have to quantify
infinities (this was in fact what prompted development of set theory: how to
make Fourier analysis precise).

Supertasks are real too. If you take Turing machine something like the
_lamplighter group_ pops up at the other end and you have no way to handwave
that away.

Unless of course you subscribe to Fictionalism as your philosophy of
mathematics, in which case stuff still has consequences just your
epistemological stance changes from studying them in favour to some kind of
mysticism that absolves you from the effort.

~~~
jacobolus
You don’t need any theory about infinity, hierarchical or otherwise, to get
Fourier transforms working. Everything can be done using approximations with
bounded error in finite steps. It’s just a pain in the butt to write proofs
that way.

(And indeed, in practice, we can’t perform any infinite processes when we are
computing Fourier transforms of real data, in either analog or digital
systems. All real-world systems are based on approximations and imperfect
models, full of measurement error, bugs in edge cases, etc.)

You have a very strange definition of “real”... or rather, it’s like the use
in “real numbers” (i.e. a pure thought experiment / set of abstract
manipulations of an abstract formal system with no physical embodiment), not
the use in “physical reality”.

Every mathematical use of completed infinities inherently involves
“mysticism”, and you seem to have embraced the absolution you mentioned.
[Which is fine... accepting Cantor’s paradise on faith and not worrying about
whether it is “real” or not has been very productive for mathematics, whether
or not most of the same results could have been found in a more cumbersome way
otherwise.]

~~~
cpsempek
thank you, you write what I was trying to say but was ultimately not clear
enough in expressing my thoughts. I don’t understand why people think that
because something is good at approximating values that it necessitates its
truth or existence.

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OscarCunningham
John Baez's "Struggles with the Continuum"
([https://arxiv.org/pdf/1609.01421.pdf](https://arxiv.org/pdf/1609.01421.pdf))
covers similar topics in more detail.

------
teh_klev
Full table of contents here:

[https://personal.lse.ac.uk/ROBERT49/ebooks/PhilSciAdventures...](https://personal.lse.ac.uk/ROBERT49/ebooks/PhilSciAdventures/toc.html)

Looks like I have some new bedtime reading.

~~~
gebeeson
Thanks for that link - I never would have come across that 'organically'. Now
I have some bedtime reading as well.

~~~
schoen
If you like this, you might also enjoy William Poundstone's book _Labyrinths
of Reason_. I found _Labyrinths of Reason_ a ton of fun, and it introduced me
to a lot of interesting ideas.

(Edit: At first I thought this book and Poundstone's book had a lot more
overlap in their contents than they seem to on further reflection, which makes
me _more_ confident in my recommendation.)

------
danielam
Worth noting is that Aristotle locates the source of Zeno's paradoxes in the
failure to distinguish between actuality and potentiality. Achilles does not
need to _actually_ cover an infinite number of lengths. A finite length is not
_actually_ an infinite number of lengths until we cut that length into an
infinite number of lengths, which, of course, is impossible. So until you
actually divide the length, the length is merely capable of being divided into
an infinite number of lengths, even though it is impossible to perform that
division to completion.

------
tzahola
“For an ideal, perfectly elastic ball, there are an infinite number of bounces
before the ball comes to a rest. Each bounce happens in less time than the
previous. “

False. A perfectly elastic ball would bounce back to the same height
indefinitely.

~~~
beiller
Ah I see you already wrote a similar comment as me. You are more correct. I
assumed the ball would bounce lower and lower, but never constantly on the
ground.

~~~
tzahola
If we put perfect elasticity aside (that is, the coefficient of restitution
[0] is < 1) then the ball would bounce back to exponentially decaying heights.
However, because of this, the duration of each bounce would also get shorter
and shorter in an exponential rate. Now, because the exponential series
converges to a finite value, they argue that the ball would bounce infinitely
many times in a finite amount of time, after which the ball would be standing
still on the ground (the bounce height having converged to 0).

[0]
[https://en.m.wikipedia.org/wiki/Coefficient_of_restitution](https://en.m.wikipedia.org/wiki/Coefficient_of_restitution)

~~~
beiller
Okay I guess I see the point is that the bouncing ball converges to a default
state, unlike a light switch, which converges to an unknown state. Is it the
force of "gravity" which causes this default state, so does the convergence
just cancel out all other influences on the system state perhaps?

~~~
tzahola
The ball's state can be described by its y coordinate, and it converges to
zero. The lightbulb's state can be described by Boolean variable (on/off), but
it doesn't converge, just oscillates between on and off with an exponentially
increasing frequency. If we have instead used a variable intensity light
switch, and turned it like 0%, 100%, 0%, 50%, 0%, 25%, 0%, 12.5% ..., with an
exponentially increasing frequency, it would have also converged to the 0%
state like the ball did.

So the variables that describe the system's state has to converge.

------
ilitirit
Vsauce did a video on this topic.

[https://www.youtube.com/watch?v=ffUnNaQTfZE](https://www.youtube.com/watch?v=ffUnNaQTfZE)

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wwarner
I'm surprised the author didn't consider a finite number of states when
mapping the concepts back to physics. Feels like infinity performs a useful
function in mathematical reasoning, but there is no reason to believe anything
we can really see or measure is boundless. None of this has any bearing on the
mathematics, which are interesting.

------
beiller
I think the statement is wrong about the bouncing ball light switch: "What
state is the lamp in after 1 minute is up? Because of how we've set things up,
the lamp will be on!". If the ball was perfectly elastic, it would bounce
forever, a smaller and smaller height, but the height never reaches 0
permanently.

------
conorcleary
You know the problem with the hotel room example? Until the last room's
occupant has packed up and moved into the newly created +1 room, there is no
open space for the newcomer.

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exikyut
I opened the comments half expecting comments _more_ inscrutable than the
discussion itself. Didn't expect the extant sentiment.

So, maybe, this question isn't so stupid:

...what's the point of this type of discussion? What real-world, non-
theoretically-based situations, would I use this type of understanding in?

I've tried to figure it out and I'm not coming up with anything relevant. I
know there _is_ relevancy to be had here, I can see that, I'm just not
figuring it out myself.

~~~
schoen
Supertasks have a connection to some interesting areas in computer science:

[https://en.wikipedia.org/wiki/Hypercomputation](https://en.wikipedia.org/wiki/Hypercomputation)

But you can also sort of repeat your question with regard to those areas of
computer science, because they don't appear to relate to computing devices
that we could physically build, even though they might clarify things like how
different groups of unsolvable problems relate to one another.

------
bakhy
It's a little funny to me to talk about infinity in the context of physical
systems. Where do you keep all those balls? :)

~~~
bookofjoe
In the box with Schrödinger's cat

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sureaboutthis
I love reading about these types of things.

I hate reading about these types of things.

~~~
andrewflnr
But if you vacillate between those positions at an exponentially increasing
frequency, what is your position at the end?

~~~
PuffinBlue
Enthusiastic apathy

