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.
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.
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.]
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.
It's not about them, nobody wished them. They came as necessity.
> Everything can be done using approximations
Approximations are analytical animals.
> with bounded error
I don't know how to bound errors without analysis.
> in finite steps.
By usage I didn't mean occasionally computing. (This is not free of problems. BTW you'd be horrified to know even finite purely combinatorial mathematics leads to irrational polytopes, ending up with infinities.) But using as a cornerstone of theory that allows for reasoning of epistemic value i.e. one affording symbolic manipulations that behave in a guaranteed way. Taking differences instead of differentials doesn't matter (why people bother if it were that easy). Whereas Finitism just shifts infinities elsewhere ending up with ultrafilter.
> You have a very strange definition of “real”
I didn't realize I used one. Or that reality has a definition (old problem in philosophy).
Or that it matters. Irrespective of what reality is mathematics can be ① real, ② have different being still connected to reality, ③ or be unreal with some mechanism for reasoning to cross the boundary to reality. First and last are troubling. Issues with the middle one is why there is a philosophy of mathematics at all.
> “real numbers”
Real numbers of course are called so for being perversion that they are. Do try get rid of them. Two centuries people have failed and still try intensely. They need a cheer.
> Every mathematical use of completed infinities inherently involves “mysticism”
I don't see what you could possibly mean. Any specifics?
Other than a lazy trollish attempt at obverting my position. To remind it is that Fictionalism absolves one from studying internal mechanics of fictional things. (Other than stop such worries, what use of invoking it?) You say mathematics is the same. In fact worse: mysticism gets in at every other point, inherently. (Why do I even bother replying…) Is it a cult? I assure you whenever a garden variety idol has deep feelings about eternity and numbers, there is a pushback https://arxiv.org/abs/1211.0244
Mathematics is a study, deals in mysticism as much as accounting: when infinities arise, their need and utility are studied, employed and scrutinized as generously as quantitative easing. If you find baffling mysticism plaguing your understanding, there is no shortage of philosophical equivalents of Bitcoin.
Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence?
Maybe maths could help us understand biology or other non-maths fields better!
(Extra mystery: Why did we evolve phase-invariance?)
It’s not a coincidence, but it is only an approximation. Fourier analysis of sound assumes a continuous field of air pressure at each point in space, which evolves according to a continuous function; in reality, air consists of a large number of discrete particles, and “pressure” approximates the result of a large number of discrete, random collisions. We can prove mathematically that the more particles are in the system, the closer its behavior gets (with overwhelming probability) to the continuous ideal; and sound happens to operate at a sufficiently large scale that the discrepancy is far too small to make a difference, so we - and evolution - can just use the continuous ideal for our calculations. But that doesn’t require it to have any physical meaning.
On the other hand, at a lower level of abstraction, the most fundamental theories of physics known do tend to involve real numbers and continuous functions – from my limited understanding, that applies even to quantum mechanics in many cases, even though it’s known for discretizing quantities that were continuous in classical mechanics. However, it’s unknown – unknowable, even, at some point – whether these infinities are “real”, or themselves approximations of even more fundamental laws.
"Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence?
"
No, we model physical systems using math, so you should expect that a physical system involving dynamic systems to use Fourier transforms to effectively model that system. But the efficacy of math does not imply that mathematical objects exist.
"Maybe maths could help us understand biology or other non-maths fields better!"
Um, clearly. Math is extremely successful at describing many aspects of many sciences. But this does not imply mathematical objects exist and are describing the reality of what is occurring, despite providing accurate outputs.
Mathematics is tautological, by definition. In what sense anything within a tautological system can be an abstraction? It's a whole package. There is no à la carte Fourier transform without philosophical commitments.
Sure there is. I can use Fourier transforms because they are efficacious but this does not force ZFC upon my reality. Could they be apart of reality? Sure, and, if they are they produce efficacious tools. Could they not be apart of my reality and efficacious tools still be available? I don't see why not.
It depends on your reality of course. It might be the physical reality, whatever that is. Or your own-er reality which might not be logical for example. There are things invariant between realities of course, like the physical experiment and mathematics.
But if your reality upholds logic, you'd have to be very careful as not to take any consequence of your usage of the Fourier transform to be a part of your reality or its description. In which case why use it at all?
Using it is, like I said, an ontological commitment inviting infinities into your system. You may not ascribe them physical meaning, or may renormalize them out, but you can't deny them (do try! though attempts at mathematical ultrafinitism are plagued by problems, whereas finitism, or various less purifying forms of constructive mathematics lead to infinities in just slightly different places).
What you may deny is the reality of the description (that is of you perceiving your reality) whatsoever as Fictionalists do.
I just mean to say mathematical objects do not have to exist in order for them to be efficacious. We may just disagree on the ontology of mathematical objects. Or, I may just be misunderstanding your argument.
> 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.
Infinity has no physical counterpart or phenomenon. Though the Universe is vast and rapidly expanding, it is still finite. I don’t think supertasks are thinly veiled examples of wordplay, though. I think they’re useful thought experiments that may be the theoretical underpinning of some future form of computation.
> 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,
I tend to agree.
Thomson's lamp paradox assumes that time is continuous, i.e., that time intervals are infinitely divisible. Physicists do not know whether that's the case. They don't know whether the space is continuous either, so Zeno's paradox also stands on very shaky foundations.
this is a strawman. If I don't know something, that doesn't prove it doesn't exist.
The construction of the nat-nums you alude too relies on infinity as an axiom, so the construction itself doesn't prove jack.
For all practical intents and purposes "infinity", in a very narrow sense, is indistinguishable from arbitrary large numbers.
Your rhethoric question leads to contradiction: By the same line of reasoning, if I don't know what the weather will be next year, that must mean it wont exist, so time is finite, which contradicts your implications.
Crap, you're right. Wait, I mean, of course I meant volume when I said "size".
But I think you can seriously make the argument that not only is "largest number" an empirical question, but the answer depends on your units. I'm not sure that's more absurd than the stuff in TFA.
You aren't trolling, but it is trivial to come up with larger numbers that are, in a sense, just as practical. Memory suggests that if the universe has a finite size then there is a finite amount of information that can be stored in it. "The states of data that could be stored in the observable universe interpreted as a hard dive" is a much, much larger quantity than size of the universe in Plank Lengths.
However, what if we want to theorise about what, if anything, sits outside the observable universe? You seem to be suggesting that it will always be finite, which is optimistic to say the least. Just because we can't observe something doesn't mean that it isn't real, and that we won't get utility out of modeling it. Infinity might be out there.
Yes, and you can generalize this to BBN(v, N), which means N applications of BB to v. If we replace N with v, it becomes a single parameter BBN(v). Then we can apply this thing to itself, and so on. So, with any finite storage space, we can represent an arbitrarily large number in this fashion. Thus, finite storage does not set an upper bound on the number that can be represented.
I'm actually going to say that's not a real number until you actually build that busy beaver. For one thing, the Bekenstein Bound is doing to come into play.
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.)
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.
“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.
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.
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).
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?
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.
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.
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.
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.
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.
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.
These kinds of exercises are useful in the practice of learning how to think.
A commenter on the "Ask HN" question about good resources to learn about systems thinking mentioned the need to have a good understanding of stateful systems. Hilbert's Hotel is about state transition, it seems to me.
More prosaically, I thought of using Thompson's Lamp as an example to train some colleagues on how to use Matt Wynne's Example Mapping. The puzzles it presents might distract from the main point of the exercise though!
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.