Paradoxes Prove Causal Finitism? The Pros and Cons (w/@MajestyofReason) (Ep. #208)

Philosopher Joe Schmid joins me to talk about casual finitism. We chat about his recent paper on arguments for casual finitism and some of Joe's concerns about casual finitism.

Timestamps:
0:00 Opening
0:58 Introduction
3:14 CF — Causal Finitism
5:30 The paper.
19:04 LRP — Littlewood Ross paradox
27:55 TLP — Thompson’s Lamp Paradox
34:36 JLU — Joes Laser Urn (I took the liberty of naming it, maybe it has another name in the article.)
38:10 Elevator pitch of the arguments.
42:48 Six Reasons against CF
1:14:00 Wrapping up!

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[HyperFocusMarshmallow]

0:00 Opening
0:58 Introduction
3:14 CF — Causal Finitism
5:30 The paper.
19:04 LRP — Littlewood Ross paradox
27:55 TLP — Thompson’s Lamp Paradox
34:36 JLU — Joes Laser Urn (I took the liberty of naming it, maybe it has another name in the article.)
38:10 Elevator pitch of the arguments.
42:48 Six Reasons against CF
1:14:00 Wrapping up!

[grumpylibrarian]

Joe, congrats on having your paper accepted and reaching pre-publication status in a Springer journal. I don't know much about Erkenntnis, but Springer has a great reputation, so I would presume this is an accomplishment to be proud of.

I am going to address some disagreements I had with your paper. I did not watch this video, and instead went straight to the paper when I was made aware it existed. So if I cover something you address in the video, then there you go.

The underlying problem you have is that you're trying to measure a fixed result *after* an infinite number of steps. This is generated artificially in the Ross-Littlewood and Thomson's Lamp thought experiments, by using a convergent set of an infinite number of units of time with a fixed clock stop point. This "feels" the same as traversing an infinite series of steps that cover an infinite amount of time, but it's qualitatively different, because these experiments have a concept of "before" (initial state) and "after" (clock strikes noon). If the infinite steps actually take infinite time, there is no "after, " or if you're extrapolating this backward to the past to make your final case for Causal Finitism, there is no "before." The inability to get a determinate answer at a point outside of the range does not imply that it would be indeterminate at any point inside the range. For every point in time across the range (finite range in the experiment but infinite range in the case you're trying to disprove), the urn had a determined and even finite number of balls within it, and the lamp is definitively either on and off.

You may very well be thinking of Zeno's "Achilles and the tortoise" thought experiment, where Achilles gives the tortoise a head start in a foot race. It seems to Zeno that Achilles could never catch up with the tortoise; once he'd reached the tortoise's starting point, the tortoise had moved a small but positive distance ahead of that point. Once Achilles reaches the next point, the tortoise has moved another small distance ahead, and so on. This is a case of an infinite number of steps as well, but as long as Achilles actually moves faster on average than the tortoise and the finish line is sufficiently distant, then Achilles WILL catch and surpass the tortoise. We can even calculate the precise point in space and time where this will occur if both are moving at fixed speeds. What is the qualitative difference between this footrace and Ross-Littlewood? The fact that this infinite series of steps is convergent. At each step, the distance and time to traverse the step had shrunk a proportional amount. Some basic Calculus II math will sum that series pretty easily.

I give you a lot of credit for recognizing that 1 + -1 + 1 + -1... isn't convergent. Failure to recognize that is the source for the videos "1 + 2 + 3 +4 ... = -1/12" videos that pop up from time to time on Youtube, and it's a trap very similar to the 1=2 "proof." The Riemann series theorem states that "if an infinite series of real numbers is conditionally convergent (does not converge absolutely), then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges." This is a direct description of the Ross-Littlewood problem that's pretty easy to ascribe directly. For Thomson's Lamp, it requires recognizing that 1 + -1 + 1 + -1... = (1 + 1 + ...) - (1 + 1 +...), or infinity minus infinity, which is undefined. As addition is commutative, the terms can be rearranged to arrive at any sum. As these sums are indeterminant, your attempt to measure "after" an infinite causal chain is also indeterminate.

If these don't convince you, then let me present it from another viewpoint. If all causal chains must be finite per your CF definition, then time would have a boundary in the future as well as the past. While there might indeed be physical reasons to bound time at either or both ends (mathematicians tend to love infinities and physicists tend to hate them), I fail to see any logical reason why time would need to cease. Just as when extrapolated to the past, at any point in the future, no matter how far into the future, the universe will be in a defined state, perhaps even calculable per Laplace's demon.

[HyperFocusMarshmallow]

Detailed timestamps:
0:00 Start
0:17 Opening banter. Is red Joe’s favorite color? 😱
———
0:58 Main topic: CF — Causal Finitism, specifically some topics from Joe’s article: ASBSAFCF — “A step by step argument for Causal Finitism”. (Link in description)
1:30 PN — Preliminary notes.
1:30 PN1: the article (ASBSAFCF) is exploratory.
2:16 PN2: All things considered, Joe is about agnostic on the truth of CF.
———
3:14 CF — Causal Finitism. What is it?
———
5:30 The paper. Very rough outlook.
Principles stipulated in the article:
6:36 SP — Step Principle
13:36 IP — Ineffectiveness principle
15:06 RIP — Removal Ineffectiveness principle
———
Paradoxes:
19:04 LRP — Littlewood Ross paradox (setup) (Infinite Urn, actions: combinations of [add ball, remove ball given an index, reindex ball given an index], actions occur at some continuous (real) time, countable infinite sequence of actions 1/2-exponentially bunching up at T0=‘noon’.
20:53 LRP. Process 1
22:53 LRP. Process 2
24:40 LRP.P1 conclusion
25:38 LPR.P2 conclusion
26:25 LPR compare P1 & P2
26:57 LPR conclusion: violation of SP (Paradox!)

27:55 TLP — Thompson’s Lamp Paradox
(Lamp with states on, off. Countably infinite toggle on/off exponentially bunching up at T0.) [Details glossed over for time] (See article for more.)

29:55 Variants by Joe (Only one variant presented, if I’m not mistaken.)
32:23 Preamble to JLU
34:36 JLU — Joes Laser Urn (I took the liberty of naming it, maybe it has another name in the article.)
Infinitely many lasers cleverly spaced and triggered such that their beams arrive to trigger something like a Littlewood-like Urn. (At least that’s my interpretation based on Joe’s verbal description, it’s not fleshed out in detail in this presentation, so check the article for more, I definitely will later.)
———
38:10 Elevator pitch of the arguments followed by a bit of reflection.
———
Leaving the article, to focus on reasons against CF.
42:48 Six Reasons for thinking CF is not the best solution to these paradoxes.
43:00 1. Reservations about extant arguments. They simply aren’t super-convincing.
46:00 2. Big problems with the arguments and further unwanted implications by attempting to solve them with causal finitism.
47:57 3. Causal finitism seem to be metaphysically profligate (wildly extravagant)
51:49 4. The arguments rely on applying either intuitions or modal principles far removed from ordinary experience.
56:35 5. “Companions in guilt”-type argument. (Preamble)
57:17 ——— Battery out quick break. Channel plug.
58:52 5. “Companions in guilt”. (The argument) Comparing: “Nothing causal depend on infinitely many things” to “Nothing depends on infinitely many things”. (Pretty nice verbal presentation!)
1:10:00 6. Compare weirdness of infinite past to weirdness of finite past.

1:13:07 Summary of arguments against CF and reflections.
———
1:14:00 Wrapping up!

[ygfddgghhbvdx]

Any state of system must involve time. So in principle no states can identically revisited. STATE=(state, time).

[HyperFocusMarshmallow]

I have a question for Joe. I’ll preface it by saying that I’m more of a physicist than mathematician and I’m more of a mathematician than I am a philosopher so bear with me.

But I want to put on my mathematical hat for a second.

Take the little-wood Urn example.
You describe two cases.
One, which you say results in an empty Urn and the other which results in an Urn with some strange super finite elements with an infinite string of zeros.
My question is: how did you tell?

More specifically which notion of mathematical limit do you use and why?

I know your argument isn’t meant to be mathematical in that sense, but rather philosophical with a hint towards philosophy of physics.

But let’s keep the mathematical hat on for a bit.
In mathematics, limits are defined as their very own special thing. I’d contend that something like a notion of limit is needed for these discussions to make sense. And it seems to me that any coherent notion of a super task must invoke some notion of limit otherwise whatever comes after is just not defined. Maybe that can be disputed. But I’ll run with it for now.

There may in fact be some notion of limit that would give the same result for your two examples in the Urn task. It would depend on the details of the limit. The more general notion of limit uses the notion of topology on both the domain and target of a map. A sequence is just a special kind of map. But it might actually be that we care more about limits on the real continuum rather than for our little sequence. There are lots of free parameters that you weren’t specific about.

A given sequence or causal-chain might be divergent on a specific notion of limit while being convergent on another (divergent doesn’t mean that the values drift apart, just that there doesn’t exist a limit).

Ok, now with some mix of hats.
It seems plausible to me that convergent infinite causal-chains might be alright while divergent ones are at least part of what we shouldn’t allow. Mathematically if a state of a physical theory at some point in time is supposed to be determined by a divergent sequence that seems pathological. The limit gives nothing useful. Then beyond that point the state would simply not be defined and we would probably have to throw out the theory beyond that point, since it fails to make mathematical sense and it fails to be predictive.
It’s an important theoretical problem to show that certain physical theories keep evolving into valid states given certain initial conditions. For certain of our best physical theories there are strong theorems of that sort that can be proven. For others there might not be.

Anyway, the stuff about the notion of limit was the most important point.

It seems like you want to call the two sequences in the Urn example essentially equivalent(in some sense) by appealing to the step principle. There might be a corresponding way to set up limits to force them to have the same limit.

I’m curious what you think of these concerns. I’d be happy to discuss it further if you find the line of reasoning interesting.

[jaskitstepkit]

I find Joe's explorations interesting as they weigh both sides but I still think that CF has a better edge. The reason is that finitist does not cause those paradoxes and is seen in physics meaning, it's more intuitive and simple. Physicists often say if your equations show infinite you have done something wrong. Surely not all infinite structures seem impossible some of them do seem unproblematic but when we try to apply many of them to our normal world we get nonsense. Any material with infinite mass or size would literally destroy our world. Continual structures like allegedly spacetime are only potentially infinite not actually so they cause no problem. Modern phychis in quantum mechanics however do show the spacetime to be discreet ( String theory and quantum loop gravity ). Finite past do have some problem in the nature of its original cause but it is far more intuitive to say something ultimately grounds the cause and effect chain of our world that an infinite amount of explanations that ultimately have not explainations as a whole which seems wrong.

[AlexADalton]

Haven't watched yet, but just dropping in to say I love Joe Schmid. Looking forward to this. Was really liking his co-hosted stuff with Elephant Philosophy. Maybe one day...

[yaterodst]

I had a thought on Joes two scenarios. I think both scenarios result in the same paradox. There are infinitely many balls in each urn but you cannot find any natural number in either urn. Each natural numbers step gets a 0 added to the end. So you cant find any natural number in that urn either. So I think the end states are not actually different.