It happens sometimes. People just explode. Natural causes.
January 2, 2025 9:02 PM Subscribe
"Even quite simple Newtonian systems can harbor uncaused events and ones for which the theory cannot even supply probabilities. Because of such systems, ordinary Newtonian mechanics cannot license a principle or law of causality. Here is an example of such a system fully in accord with Newtonian mechanics. It is a mass that remains at rest in a physical environment that is completely unchanging for an arbitrary amount of time--a day, a month, an eon. Then, without any external intervention or any change in the physical environment, the mass spontaneously moves off in an arbitrary direction, with the theory supplying no probabilities for the time or direction of the motion." [John D. Norton,
Department of History and Philosophy of Science, University of Pittsburgh; via Up and Atom (YouTube, Nebula)]
The linked piece is based on section 3 of Norton's Causation as Folk Science [Philosopher's Imprint, November 2003, Volume 3, No. 4, pp. 1-22] so it's by no means new work. Food for thought, though. I always enjoy seeing serious thinkers take issue with the idea of treating causality as some kind of Holy Spirit, a presence deemed necessary even in cases where the contortions required to justify it over-complicate perfectly useful physical models for no good reason.
Up and Atom also provides links to the following documents:
The Dome: An Unexpectedly Simple Failure of Determinism [PDF] in which Norton provides answers to many of the objections that others have raised against The Dome as an inherently acausal model:
As well as links to all of Norton's peer-reviewed publications, his site has a Goodies section (because "some things are just too much fun") that's well worth browsing. I very much enjoyed What is Time? Or, Just What do Philosophers of Science Do?
The linked piece is based on section 3 of Norton's Causation as Folk Science [Philosopher's Imprint, November 2003, Volume 3, No. 4, pp. 1-22] so it's by no means new work. Food for thought, though. I always enjoy seeing serious thinkers take issue with the idea of treating causality as some kind of Holy Spirit, a presence deemed necessary even in cases where the contortions required to justify it over-complicate perfectly useful physical models for no good reason.
Up and Atom also provides links to the following documents:
The Dome: An Unexpectedly Simple Failure of Determinism [PDF] in which Norton provides answers to many of the objections that others have raised against The Dome as an inherently acausal model:
Prepared for the Symposium "The Vagaries of Determinism and Indeterminism," PSA 2006: Philosophy of Science Association Biennial Conference, Vancouver, November 2006.and also a copy of Causation as Folk Science [PDF] from Norton's own University of Pittsburgh site in case the link via Philosopher's Imprint doesn't work for you.
Because of the specific shape of the dome at its apex, Newton’s equations of motion tell us that a mass at rest at the apex can spontaneously be set into motion. It has been suggested that this indeterminism should be discounted since it draws on an incomplete rendering of Newtonian physics; or it is “unphysical”; or it employs illicit idealizations. I analyze and reject each of these reasons.
As well as links to all of Norton's peer-reviewed publications, his site has a Goodies section (because "some things are just too much fun") that's well worth browsing. I very much enjoyed What is Time? Or, Just What do Philosophers of Science Do?
This question, we are told, is "...a deep question, and it has no simple answer." That is wrong. It is a trite question that gains the mere appearance of depth cheaply by hidden presumptions.
Never met the man, but he strikes me as playful in ways that so, so many published philosophers are so, so tediously not.
posted by flabdablet at 9:57 PM on January 2 [3 favorites]
posted by flabdablet at 9:57 PM on January 2 [3 favorites]
This has led me down a rabbit hole. Today I learned about jerk, snap, crackle, and pop.
Some of his critics think jerk is relevant to this topic.
posted by night_train at 3:05 AM on January 3 [1 favorite]
Some of his critics think jerk is relevant to this topic.
posted by night_train at 3:05 AM on January 3 [1 favorite]
So if we rewrite Newton’s laws so that they no longer require things to stay the same, we reveal that correctly understood, they allow things to change?
What’s the point of sitting your ball on a dome? If it moves, it moves: you don’t need to dramatise and extend the motion. Wasn’t invalidating the whole of physics by introducing arbitrary motion exciting enough?
I probably just need to accept that all this is not actually counterintuitive. :)
posted by Phanx at 4:00 AM on January 3
What’s the point of sitting your ball on a dome? If it moves, it moves: you don’t need to dramatise and extend the motion. Wasn’t invalidating the whole of physics by introducing arbitrary motion exciting enough?
I probably just need to accept that all this is not actually counterintuitive. :)
posted by Phanx at 4:00 AM on January 3
I'm always here for "Repo Man."
posted by chavenet at 4:44 AM on January 3 [3 favorites]
posted by chavenet at 4:44 AM on January 3 [3 favorites]
Some of his critics think jerk is relevant to this topic.
Since jerk appears nowhere in any causal interpretation of Newtonian mechanics, I think they're simply wrong. In any case, jerk is unproblematic on the dome. Snap isn't - it has a step discontinuity at t=T - but again, snap plays no part in any account of causality derivable from Newtonian mechanics.
As Norton points out on page 6 of "From The Dome: An Unexpectedly Simple Failure of Determinism" linked above, in Newtonian mechanics the cause of an acceleration can only be a force, and the force applied to our infinitesimal slippery spherical cow at all times up to and including the activation time T is exactly zero.
At all times t > T there is a perfectly well-defined non-zero applied force, and a corresponding acceleration; but there is no applied force at time T, and yet the equations of Newtonian mechanics admit of a class of solutions to the cow's equations of motion where time T, with T chosen completely arbitrarily, is the last at which she remains at rest.
Norton:
The entire point of this thought exercise is that no such rewriting is required. Newtonian mechanics as it already exists include equations of motion that demonstrably allow for solutions that describe spontaneous, uncaused motion, given suitable initial conditions; the dome is one such demonstration. These solutions are indeed surprising - most people probably will find their existence counterintuitive - but they're physically implausible only to people who insist a priori that every event must have a cause or causes, regardless of how ill-specified any such cause might need to be.
If you're one such person, give Causation as Folk Science a careful critical read and see what you think afterwards.
posted by flabdablet at 5:13 AM on January 3 [3 favorites]
Since jerk appears nowhere in any causal interpretation of Newtonian mechanics, I think they're simply wrong. In any case, jerk is unproblematic on the dome. Snap isn't - it has a step discontinuity at t=T - but again, snap plays no part in any account of causality derivable from Newtonian mechanics.
As Norton points out on page 6 of "From The Dome: An Unexpectedly Simple Failure of Determinism" linked above, in Newtonian mechanics the cause of an acceleration can only be a force, and the force applied to our infinitesimal slippery spherical cow at all times up to and including the activation time T is exactly zero.
At all times t > T there is a perfectly well-defined non-zero applied force, and a corresponding acceleration; but there is no applied force at time T, and yet the equations of Newtonian mechanics admit of a class of solutions to the cow's equations of motion where time T, with T chosen completely arbitrarily, is the last at which she remains at rest.
Norton:
If we care to graft causal language onto the spontaneous motion, we can express quite concisely what makes it puzzling. We expect a change of motion to have an initiating cause, a first cause. Since the motion starts at t=T, we expect that first cause to be active at that moment t=T. In Newtonian physics, the only admissible candidate is a net force. Yet at t=T, there is no net force acting on the body. So there is no first cause for the motion. In trying to accommodate how the motion can fail to have a first cause, it is helpful in reflecting on the moment of spontaneous motion, t=T, not to think of it as the first moment of the motion, but to think of it as the last moment of rest. Thus the time interval in which the mass moves, t>T, has no first moment at which a first cause could first act.So if we rewrite Newton’s laws so that they no longer require things to stay the same, we reveal that correctly understood, they allow things to change?
Or perhaps there is a simpler solution. The form (N1) of Newton’s first law is not as he originally stated it. His versions, in their time-honored, archaic translations are (Newton, 1729, Vol. 1, p. 13)Law 1. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.Does the motion (4) really conform with Newton’s version of the first law? It seems to. Newton’s version of the law applies to the motion (4) only in the time interval t ≤ T, for that is the time interval during which no net force acts on the mass. During that time interval, the mass remains at rest, as Newton’s law required.
Law 2. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
The complication is that the phrasing “compelled to change” suggests that changes of motion must be brought about by forces acting at the same time as the change, if not even earlier. Yet we have just seen that no force acts on the mass at the time t=T at which the motion changes, while something seems to be changing at t=T. Although the acceleration a(0)=0, the fourth time derivative of r has a discontinuity at t=T. It isd4r(t)/dt4 = 1/6 for t > T (6)where the quantity proves to be not well-defined at t=T.
= 0 for t < T
= ? for t = T
It is by no means clear to me that this amounts to a violation of Newton’s form of the first law. Newton’s wording suggests, but does not clearly assert, that forces must be first causes. More importantly, the position, velocity and acceleration of the mass are all vanishing at t=T and that seems sufficient to meet Newton’s requirement of “state of rest” or “uniform motion.” If it is not, then we will be creating difficulties with other canonical examples. Take the motion of a simple harmonic oscillator, a mass on a spring. In suitable units, its displacement x may be given as a function of time t byx(t) = sin t, dx(t)/dt = cos t, d2x(t)/dt2 = -sin t, d3x(t)/dt3 = -cos t (7)The mass will pass through a position of vanishing net force at t=0, when x(t) = 0 and the acceleration d2x(t)/dt2 = 0. We normally think of the mass at just that one moment as moving inertially—there is no net impressed force, so the velocity is constant, in the sense that the acceleration vanishes. Yet at that same moment, the third derivative does not vanish: d3x(t)/dt3 = -1.
The entire point of this thought exercise is that no such rewriting is required. Newtonian mechanics as it already exists include equations of motion that demonstrably allow for solutions that describe spontaneous, uncaused motion, given suitable initial conditions; the dome is one such demonstration. These solutions are indeed surprising - most people probably will find their existence counterintuitive - but they're physically implausible only to people who insist a priori that every event must have a cause or causes, regardless of how ill-specified any such cause might need to be.
If you're one such person, give Causation as Folk Science a careful critical read and see what you think afterwards.
posted by flabdablet at 5:13 AM on January 3 [3 favorites]
What’s the point of sitting your ball on a dome?
To constrain its possible states of motion in exactly such a way as to make the Newtonian equations describing them allow for solutions that describe spontaneous, uncaused, unpredictable motion that breaks determinism.
Note well: breaking determinism merely denies the primacy of causality as the ultimate organizing principle, asserting that "there's a reason for everything" is in fact false. It does not break physics, despite the pull quote in Up and Atom's YouTube thumbnail. If it did, Schrödinger's Cat would not be a thing.
posted by flabdablet at 5:27 AM on January 3 [1 favorite]
To constrain its possible states of motion in exactly such a way as to make the Newtonian equations describing them allow for solutions that describe spontaneous, uncaused, unpredictable motion that breaks determinism.
Note well: breaking determinism merely denies the primacy of causality as the ultimate organizing principle, asserting that "there's a reason for everything" is in fact false. It does not break physics, despite the pull quote in Up and Atom's YouTube thumbnail. If it did, Schrödinger's Cat would not be a thing.
posted by flabdablet at 5:27 AM on January 3 [1 favorite]
the time interval in which the mass moves, t>T, has no first moment at which a first cause could first act
As an apparently casual aside, Page 19 of Causation as Folk Science includes this little gem:
posted by flabdablet at 5:39 AM on January 3
As an apparently casual aside, Page 19 of Causation as Folk Science includes this little gem:
Analogous problems arise in the case of big bang cosmology. The universe exists for all cosmic times t>0, and its state at each time might be represented as the cause of the state at a later time. However there is no state at t=0 (loosely, the moment of the big bang) and the demand that there be a first cause for the process must conjure up causes that lie outside the physics.Speaking as one who has never been convinced of the physicality of t=0, I find this intensely pleasing.
posted by flabdablet at 5:39 AM on January 3
Physics is an absolutely incomplete, possibly incorrect in some ways, description of the universe.
So, in as much as this thought experiment tests our understanding of our understanding of the universe (physics) it is interesting.
That's why it's called theoretical physics.
Does it matter in day to day life? No. Might it someday? Perhaps! (Ask an applied physicist.)
Thinking about shit like this is how we got from banging rocks together to pocket supercomputers, neutron bombs and self-replicating RNA vaccines.
posted by seanmpuckett at 5:41 AM on January 3 [1 favorite]
So, in as much as this thought experiment tests our understanding of our understanding of the universe (physics) it is interesting.
That's why it's called theoretical physics.
Does it matter in day to day life? No. Might it someday? Perhaps! (Ask an applied physicist.)
Thinking about shit like this is how we got from banging rocks together to pocket supercomputers, neutron bombs and self-replicating RNA vaccines.
posted by seanmpuckett at 5:41 AM on January 3 [1 favorite]
I can't see all the causes, and what I can see may not be all the causes; I don't have complete knowledge.
My model is worse, "all models are wrong by some are helpful" guides me to accept that there's a pragmatic limit to the level of detail in the model. And Newtonian mechanics is a consistent model -- that has no equations built-in for the story you're telling. That's over-reach, you use the model to describe and predict pre-existing observations, say about this harmonic oscillator.
...that "harmonic oscillator" model has an especially terrible explanation, lacking any accommodation that the circles of position, force, speed and momentum all sit tidily with the sine and cosine of the model. The first law doesn't apply: there's an unbalanced driving force.
posted by k3ninho at 5:43 AM on January 3
My model is worse, "all models are wrong by some are helpful" guides me to accept that there's a pragmatic limit to the level of detail in the model. And Newtonian mechanics is a consistent model -- that has no equations built-in for the story you're telling. That's over-reach, you use the model to describe and predict pre-existing observations, say about this harmonic oscillator.
...that "harmonic oscillator" model has an especially terrible explanation, lacking any accommodation that the circles of position, force, speed and momentum all sit tidily with the sine and cosine of the model. The first law doesn't apply: there's an unbalanced driving force.
posted by k3ninho at 5:43 AM on January 3
I used to have a light grasp of Newtonian mechanics but it's faded with time. This post is encouraging me to revisit and try to understand exactly what he is saying.
posted by night_train at 6:07 AM on January 3
posted by night_train at 6:07 AM on January 3
Does it matter in day to day life?
I'd argue quite strongly that it does, having been quite convinced for some decades now that the belief that everything happens for a reason causes far more suffering than it avoids.
The first law doesn't apply: there's an unbalanced driving force
Point is that exactly at those instants where the oscillating mass is at minimum displacement and maximum speed, instants which happen twice per cycle, there isn't an unbalanced driving force. If there were, the oscillation would not be harmonic.
Plausibly, the cause of the acceleration experienced at all times other than those instants is the interaction between the oscillating mass's velocity at those instants and the physical construction of the system that constrains its motion. And sure, the motion of a harmonic oscillator is well-behaved and nicely predictable.
The point of mentioning the harmonic oscillator at all is just to highlight the fact that the motion's jerk being nonzero at the instants of minimum displacement and maximum speed plays no part in any causal account of the system's behaviour.
The mass on the dome is also a system where a mass's motion is constrained by the physical construction of its surrounds. Unlike the harmonic oscillator, the resulting motion is not necessarily predictable despite having a well-defined Newtonian equation of motion. But like the harmonic oscillator, the values (or lack thereof) of jerk or snap or crackle or pop at any given instant do not play any part in any causal account of that motion that derives its legitimacy from Newton.
It may well be interesting that snap becomes undefined at activation time T, but no more so than that T is also the last time at which the mass remains at rest. The points not to forget here are that (a) T is completely arbitrary - we can set it to whatever we like and the solution to the mass's equation of motion still works and (b) the direction in which the mass moves off the dome's apex is completely unpredictable until after it has begun to move, which it's doing at all times t > T.
What this setup actually is: a nice, clear, physically plausible example of spontaneous symmetry breaking that doesn't require venturing terribly far into a dark mathematical forest before it can even be thought about.
posted by flabdablet at 6:19 AM on January 3 [2 favorites]
I'd argue quite strongly that it does, having been quite convinced for some decades now that the belief that everything happens for a reason causes far more suffering than it avoids.
The first law doesn't apply: there's an unbalanced driving force
Point is that exactly at those instants where the oscillating mass is at minimum displacement and maximum speed, instants which happen twice per cycle, there isn't an unbalanced driving force. If there were, the oscillation would not be harmonic.
Plausibly, the cause of the acceleration experienced at all times other than those instants is the interaction between the oscillating mass's velocity at those instants and the physical construction of the system that constrains its motion. And sure, the motion of a harmonic oscillator is well-behaved and nicely predictable.
The point of mentioning the harmonic oscillator at all is just to highlight the fact that the motion's jerk being nonzero at the instants of minimum displacement and maximum speed plays no part in any causal account of the system's behaviour.
The mass on the dome is also a system where a mass's motion is constrained by the physical construction of its surrounds. Unlike the harmonic oscillator, the resulting motion is not necessarily predictable despite having a well-defined Newtonian equation of motion. But like the harmonic oscillator, the values (or lack thereof) of jerk or snap or crackle or pop at any given instant do not play any part in any causal account of that motion that derives its legitimacy from Newton.
It may well be interesting that snap becomes undefined at activation time T, but no more so than that T is also the last time at which the mass remains at rest. The points not to forget here are that (a) T is completely arbitrary - we can set it to whatever we like and the solution to the mass's equation of motion still works and (b) the direction in which the mass moves off the dome's apex is completely unpredictable until after it has begun to move, which it's doing at all times t > T.
What this setup actually is: a nice, clear, physically plausible example of spontaneous symmetry breaking that doesn't require venturing terribly far into a dark mathematical forest before it can even be thought about.
posted by flabdablet at 6:19 AM on January 3 [2 favorites]
they're physically implausible only to people who insist a priori that every event must have a cause or causes
that and people who expect Newtonian mechanics to be deterministic, ie that you should be able to predict the motion of the ball, right?
another weird thing to me is that there's no obvious way to even give a probability distribution for T. It's any time from 0 to infinity, so it can't be a uniform distribution- which means it could be anything. At least quantum mechanics spits out probabilities if you square the amplitude of the wave function.
posted by BungaDunga at 6:22 AM on January 3 [1 favorite]
that and people who expect Newtonian mechanics to be deterministic, ie that you should be able to predict the motion of the ball, right?
another weird thing to me is that there's no obvious way to even give a probability distribution for T. It's any time from 0 to infinity, so it can't be a uniform distribution- which means it could be anything. At least quantum mechanics spits out probabilities if you square the amplitude of the wave function.
posted by BungaDunga at 6:22 AM on January 3 [1 favorite]
that and people who expect Newtonian mechanics to be deterministic
Seems likely to me both that those are the same people and that Newton was one of them.
posted by flabdablet at 6:26 AM on January 3
Seems likely to me both that those are the same people and that Newton was one of them.
posted by flabdablet at 6:26 AM on January 3
> I'd argue quite strongly that it does, having been quite convinced for some decades now that the belief that everything happens for a reason causes far more suffering than it avoids.
I'm really sympathetic to your argument, but applying this Newtonian thought experiment as some kind of salve to the caustic woo at the heart of Calvinism is ... optimistic. But okay!
Anyway, I have a porcelain sink that does this. I put the soap dish, dry and motionless, on the dry and horizontal backsplash, and some arbitrary time later it is in the basin. There's probably delightful non-quantum explanations for it but I just prefer to think of it as Shit Happens.
posted by seanmpuckett at 6:55 AM on January 3 [1 favorite]
I'm really sympathetic to your argument, but applying this Newtonian thought experiment as some kind of salve to the caustic woo at the heart of Calvinism is ... optimistic. But okay!
Anyway, I have a porcelain sink that does this. I put the soap dish, dry and motionless, on the dry and horizontal backsplash, and some arbitrary time later it is in the basin. There's probably delightful non-quantum explanations for it but I just prefer to think of it as Shit Happens.
posted by seanmpuckett at 6:55 AM on January 3 [1 favorite]
Or, seanmpuckett, cat happened....
(Someone let schrodinger's cat outof the box.)
posted by mightshould at 8:04 AM on January 3 [2 favorites]
(Someone let schrodinger's cat outof the box.)
posted by mightshould at 8:04 AM on January 3 [2 favorites]
the belief that everything happens for a reason causes far more suffering than it avoids.
I think you trip up a bit on how many people apply a semantic equivalence of "cause" for "reason."
When people say "everything happens for a reason" (where their use of the word reason is tied to agency and motive on the part of something capable thereof) and then defends their position with Newtonian mechanics, they've already mushed two separate things together as though they were interchangeable (and they're not!), and you not only can but *should* dismiss their point as aesthetic rather than empirical.
Put me into the determinist camp. We may not *know* the cause of an observable event, but all observable events have a cause. Very, very few have a reason.
posted by tclark at 8:38 AM on January 3 [1 favorite]
I think you trip up a bit on how many people apply a semantic equivalence of "cause" for "reason."
When people say "everything happens for a reason" (where their use of the word reason is tied to agency and motive on the part of something capable thereof) and then defends their position with Newtonian mechanics, they've already mushed two separate things together as though they were interchangeable (and they're not!), and you not only can but *should* dismiss their point as aesthetic rather than empirical.
Put me into the determinist camp. We may not *know* the cause of an observable event, but all observable events have a cause. Very, very few have a reason.
posted by tclark at 8:38 AM on January 3 [1 favorite]
all observable events have a cause
What causes radioactive decay events? Is there any good reason for asserting that they are not simply spontaneous?
posted by flabdablet at 9:44 AM on January 3 [3 favorites]
What causes radioactive decay events? Is there any good reason for asserting that they are not simply spontaneous?
posted by flabdablet at 9:44 AM on January 3 [3 favorites]
I thought this might be about a n=5 body problem or something similar, where you get what's often called "deterministic chaos", with the strong sensitivity on initial conditions and the much maligned butterfly effect, etc. And there's really neat stuff there. You can have planets going about their business, looking for all the world like they are in stable orbits around a star, and then at some point, one will just pop off on a trip to infinity. The notion of 'determinism' there is itself highly strained, and the only meaningful notion of the "cause" for this excursion is the entire state of the system as it evolved over time. So sure, you can call it causality if you'd like, but then you have to just accept that everything is caused by... everything.
posted by SaltySalticid at 9:47 AM on January 3 [1 favorite]
posted by SaltySalticid at 9:47 AM on January 3 [1 favorite]
What causes radioactive decay events? Is there any good reason for asserting that they are not simply spontaneous?
We know radioactive decay is not purely spontaneous because at sufficiently large statistical quantities, the half-life is observed, rather than decay being a linear function. In the case of radioisotopes, the nucleus is in a metastable state, and has a probability of decay within a specific window of time. I contend that the probability is the cause. I contend that you and Norton semantically merge cause and reason, and while radioactive decay (or the quantum decay of any metatstable system) is statistically bound with probabilities does not mean that the passage of time itself is not a perturbation on quantum systems, and that perturbation has statistical effects which we -- I emphasize this because it's important -- reliably can measure in systems with sufficient number of particles.
posted by tclark at 9:59 AM on January 3 [2 favorites]
We know radioactive decay is not purely spontaneous because at sufficiently large statistical quantities, the half-life is observed, rather than decay being a linear function. In the case of radioisotopes, the nucleus is in a metastable state, and has a probability of decay within a specific window of time. I contend that the probability is the cause. I contend that you and Norton semantically merge cause and reason, and while radioactive decay (or the quantum decay of any metatstable system) is statistically bound with probabilities does not mean that the passage of time itself is not a perturbation on quantum systems, and that perturbation has statistical effects which we -- I emphasize this because it's important -- reliably can measure in systems with sufficient number of particles.
posted by tclark at 9:59 AM on January 3 [2 favorites]
I further more contend that the Dome scenario is Norton saying that his ball and dome are bovine spheres except when he decides that rather than Newtonian mechanics he applies a quantum statistical probability property that conveniently applies macroscopically, and therefore with a grand gesture simply sweeps away everything we thought about causality.
I'm not convinced by Norton in the slightest, and remain on team determinism.
posted by tclark at 10:02 AM on January 3 [1 favorite]
I'm not convinced by Norton in the slightest, and remain on team determinism.
posted by tclark at 10:02 AM on January 3 [1 favorite]
when he decides that rather than Newtonian mechanics he applies a quantum statistical probability property that conveniently applies macroscopically
Where has he done that? All his workings-out are strictly Newtonian as far as I can see.
posted by flabdablet at 10:12 AM on January 3 [4 favorites]
Where has he done that? All his workings-out are strictly Newtonian as far as I can see.
posted by flabdablet at 10:12 AM on January 3 [4 favorites]
I contend that the probability is the cause.
That strikes me as reaching.
posted by flabdablet at 10:14 AM on January 3 [3 favorites]
That strikes me as reaching.
posted by flabdablet at 10:14 AM on January 3 [3 favorites]
I'm not going to spend any more of my time on this thread, and should have known from the beginning what I was engaging with when you wrote "I always enjoy seeing serious thinkers take issue with the idea of treating causality as some kind of Holy Spirit."
Just pretend I wasn't here because this clearly is a waste of my time and yours.
posted by tclark at 10:16 AM on January 3
Just pretend I wasn't here because this clearly is a waste of my time and yours.
posted by tclark at 10:16 AM on January 3
Well, don't think we're going to get answers on this but it hasn't been a waste for me. Phase transitions are one. But "Strange Attractor" is another. An "end state" that's a dynamical orbit that is reached *no matter* where you start.
Causality can bite it.
posted by aleph at 10:25 AM on January 3 [1 favorite]
Causality can bite it.
posted by aleph at 10:25 AM on January 3 [1 favorite]
edit: Should have added:
https://en.wikipedia.org/wiki/Strange_attractor
posted by aleph at 10:37 AM on January 3
https://en.wikipedia.org/wiki/Strange_attractor
posted by aleph at 10:37 AM on January 3
Norton makes a pretty good case for the usefulness of thinking of all kinds of attractor, not merely the strange ones, as final causes for the behaviour of systems that exhibit them.
posted by flabdablet at 10:45 AM on January 3 [2 favorites]
posted by flabdablet at 10:45 AM on January 3 [2 favorites]
We know radioactive decay is not purely spontaneous because at sufficiently large statistical quantities, the half-life is observed, rather than decay being a linear function.
My stats are rusty as fuck, but I seem to recall learning that seeing a half-life emerge in the behaviour of an ensemble is a consequence of the probability of any individual decay event being constant for any given time interval, and therefore independent of how long the individual decayable item has remained in its undecayed state. This, to me, looks far more indicative of a model where individual decays are spontaneous and in-principle unpredictable than one where something more systematic is going on under the hood.
So I can see how it would be reasonable to say that a probability distribution of that kind is the cause of the half-life behaviour seen in a sufficiently large ensemble, but I don't understand how it's meaningful to assert that any individual decay event is caused by its probability. For example, I don't see how the mere fact that there's a one in six probability of rolling a three can reasonably be said to have caused any particular dice roll that came up three to do so.
posted by flabdablet at 11:13 AM on January 3 [6 favorites]
My stats are rusty as fuck, but I seem to recall learning that seeing a half-life emerge in the behaviour of an ensemble is a consequence of the probability of any individual decay event being constant for any given time interval, and therefore independent of how long the individual decayable item has remained in its undecayed state. This, to me, looks far more indicative of a model where individual decays are spontaneous and in-principle unpredictable than one where something more systematic is going on under the hood.
So I can see how it would be reasonable to say that a probability distribution of that kind is the cause of the half-life behaviour seen in a sufficiently large ensemble, but I don't understand how it's meaningful to assert that any individual decay event is caused by its probability. For example, I don't see how the mere fact that there's a one in six probability of rolling a three can reasonably be said to have caused any particular dice roll that came up three to do so.
posted by flabdablet at 11:13 AM on January 3 [6 favorites]
I take great joy in seeing people dive into this. I know Dr. Norton (John); have known him for years. He is wicked smart, a leader in a very niche field, and know that he loves nothing more than to find the fun in hard science and theory. His sense of humor is unique, to be sure. Best known as an Einstein scholar, but he dabbles all over. This is the man whose third voicemail option was once 'press 3 to hear a duck quack" ... and that is what you got. Literally. I let him know he is 'on the Blue.'
posted by buffalo at 11:32 AM on January 3 [5 favorites]
posted by buffalo at 11:32 AM on January 3 [5 favorites]
My stats are rusty as fuck, but I seem to recall learning that seeing a half-life emerge in the behaviour of an ensemble is a consequence of the probability of any individual decay event being constant for any given time interval, and therefore independent of how long the individual decayable item has remained in its undecayed state. This, to me, looks far more indicative of a model where individual decays are spontaneous and in-principle unpredictable than one where something more systematic is going on under the hood.
I'd seen it described as a Poisson process though this seems to be an approximation, and it's actually best modeled as a binomial distribution:
I'd seen it described as a Poisson process though this seems to be an approximation, and it's actually best modeled as a binomial distribution:
The fundamental assumption that leads to the use of Poisson distribution is that the number of decays in non-overlapping time intervals is statistically independent. The statistical independence implies that the knowledge about the number of decays in one interval does not influence the knowledge about the number of decays in any other interval. This assumption is violated for nuclear decay.posted by BungaDunga at 11:45 AM on January 3 [3 favorites]
"...as final causes for the behaviour of systems"
Yeah. It does seem similar in a weird way. Dissipative systems is my best guess on how life grows out of not-life.
posted by aleph at 11:49 AM on January 3 [1 favorite]
Yeah. It does seem similar in a weird way. Dissipative systems is my best guess on how life grows out of not-life.
posted by aleph at 11:49 AM on January 3 [1 favorite]
Plenty of simple differential equations have initial value conditions which admit multiple solutions-- I'm not sure it really says anything deeply meaningful that a superficially plausible physical interpretation can then be contrived for such a scenario.
Norton explicitly states this is how the dome was conceived: "I originally concocted the dome example by starting with a text-book example of a system that violates a Lipshitz condition and has multiple solutions; and then worked backwards to a plausible physical instantiation."
In general I think there's a bit of a slight-of-hand quality to his arguments: he clearly wants us to think that this thought experiment has some relevance to the real physical world, but when the dome is challenged on grounds like "the world is governed by more than just F=ma" or "this force field is not physically realizable" he retreats back into claiming he is only talking about abstract differential equations and his critics are being unsophisticated when they interpret the dome too literally.
posted by Pyry at 12:54 PM on January 3 [3 favorites]
Norton explicitly states this is how the dome was conceived: "I originally concocted the dome example by starting with a text-book example of a system that violates a Lipshitz condition and has multiple solutions; and then worked backwards to a plausible physical instantiation."
In general I think there's a bit of a slight-of-hand quality to his arguments: he clearly wants us to think that this thought experiment has some relevance to the real physical world, but when the dome is challenged on grounds like "the world is governed by more than just F=ma" or "this force field is not physically realizable" he retreats back into claiming he is only talking about abstract differential equations and his critics are being unsophisticated when they interpret the dome too literally.
posted by Pyry at 12:54 PM on January 3 [3 favorites]
I think it serves to highlight the oddity of treating the accuracy/consistency of Newtonian mechanics as evidence for the universe being deterministic, when there's an example of the model in its own right being nondeterministic in this way. Saying it's inapplicable because the circumstance is "unphysical" is in a sense getting things the wrong way around - it's like claiming that 3+3=6 is only reliable and trustworthy when you're adding three apples to three apples, and you can't expect it to work on integers.
posted by NMcCoy at 1:39 PM on January 3 [5 favorites]
posted by NMcCoy at 1:39 PM on January 3 [5 favorites]
This was very interesting and I'm not sure what to think of it.
Is it possible that the Newton's laws we know and love are not complete (yes, I know about Relativity and QM. I'm talking about "not complete" even in their own realm) and that a "proper" formulation of the laws would exclude this?
posted by It's Never Lurgi at 1:40 PM on January 3
Is it possible that the Newton's laws we know and love are not complete (yes, I know about Relativity and QM. I'm talking about "not complete" even in their own realm) and that a "proper" formulation of the laws would exclude this?
posted by It's Never Lurgi at 1:40 PM on January 3
Given the prevalence of Gödel incompleteness-type situations amongst various systems, I have a hunch that it's entirely possible that the only solution to 'fixing' this is to add a law that says the equivalent of "cheeky nondeterministic constructions are hereby banished from the domain of Newtonian physics, not to be considered any further" which I suspect may not satisfy an intuitive demand for "completeness".
posted by NMcCoy at 1:49 PM on January 3 [1 favorite]
posted by NMcCoy at 1:49 PM on January 3 [1 favorite]
Hah! Wait until this guy finds out that Maxwell's equations admit a solution for light going backward in time that physicists just kinda ignore because light just doesn't seem to do that.
So what this guy has done is construct a very slightly rounder cone and hidden the discontinuity.
Now with a cone, your intuition might serve you better. If someone talks about setting an infinitely tiny ball on the top of an infinitely pointy cone, well you could do a similar calculation to this and come to the conclusion that the ball would start rolling instantly.
But then you ask the question "which direction?" and the model falls apart.
Because you've said you're setting an infinitely tiny ball on an infinitely tiny point, but also need to know what side of it it's on to tell which way it's going to roll, so maybe you could assume that due to symmetry, it's just going to say there, but your mathematical model says it has to move somewhere, so there is a contradiction. And you can get into the formalities of discontinuities and limits, but at least you're thinking in the right direction what does it mean to be infinitely small or infinitely pointy and not interrogating Newton's Laws for answers.
This "dome" is actually a similar situation, just with one more level of smoothness applied so that it's easier to get lost in the math. Here, you don't quite have a pointy bit, but as you approach the tip at r = 0, the change in the force with the change in position becomes infinitely large. Mathematically speaking, dF(r)/dr ~ d(r^1/2)/dr ~ 1/r^(1/2). The upshot of this is that no matter how infinitesmally small you miss 0 by, or how tiny a nudge the ball gets from literally anything in the universe, the ball will roll away in a finite amount of time.
Nevermind the pull of the moon or Jupiter or any of the other bodies whose gravitational affects we deem too insignificant to include in the model of this balls trajectory: there is no scale that does not matter here. A single neutrino on the other side of the universe is enough to disrupt this whole process and derail our calculation!
Really, this isn't much different the infinitely pointy cone now is it? Except here the math tells you that the force would be 0 if you somehow did manage the equivalent of dropping an infinitely small ball onto the tip of an infinitely pointy cone. But if you existed in such a universe, then you could do the experiment of dropping an infinitely tiny ball on and infinitely tiny cone to see what happens and find out what calculus and Newtons laws couldn't tell you about that universe. But that universe is not this one.
Because ultimately, physics is not some thought exercise, it is an experimental science. All the math is in the service of predicting how the universe behaves via models, which are necessarily approximate and inexact. And near as we can tell from the best models we make, our universe is a messy one, an inexact one, a non-deterministic one. We've known that for hundreds of years: quantum physics, chaos theory - now there's an interesting one. Chaotic systems have divergent behavior over finite regions! You don't need to try to drop an infinitely tiny ball on an infinitely tiny point, you can actually physically realize them in this universe even with nominally deterministic processes!
Stuff like this is just getting lost in the weeds well before you get to the really interesting parts.
posted by Zalzidrax at 2:56 PM on January 3 [6 favorites]
So what this guy has done is construct a very slightly rounder cone and hidden the discontinuity.
Now with a cone, your intuition might serve you better. If someone talks about setting an infinitely tiny ball on the top of an infinitely pointy cone, well you could do a similar calculation to this and come to the conclusion that the ball would start rolling instantly.
But then you ask the question "which direction?" and the model falls apart.
Because you've said you're setting an infinitely tiny ball on an infinitely tiny point, but also need to know what side of it it's on to tell which way it's going to roll, so maybe you could assume that due to symmetry, it's just going to say there, but your mathematical model says it has to move somewhere, so there is a contradiction. And you can get into the formalities of discontinuities and limits, but at least you're thinking in the right direction what does it mean to be infinitely small or infinitely pointy and not interrogating Newton's Laws for answers.
This "dome" is actually a similar situation, just with one more level of smoothness applied so that it's easier to get lost in the math. Here, you don't quite have a pointy bit, but as you approach the tip at r = 0, the change in the force with the change in position becomes infinitely large. Mathematically speaking, dF(r)/dr ~ d(r^1/2)/dr ~ 1/r^(1/2). The upshot of this is that no matter how infinitesmally small you miss 0 by, or how tiny a nudge the ball gets from literally anything in the universe, the ball will roll away in a finite amount of time.
Nevermind the pull of the moon or Jupiter or any of the other bodies whose gravitational affects we deem too insignificant to include in the model of this balls trajectory: there is no scale that does not matter here. A single neutrino on the other side of the universe is enough to disrupt this whole process and derail our calculation!
Really, this isn't much different the infinitely pointy cone now is it? Except here the math tells you that the force would be 0 if you somehow did manage the equivalent of dropping an infinitely small ball onto the tip of an infinitely pointy cone. But if you existed in such a universe, then you could do the experiment of dropping an infinitely tiny ball on and infinitely tiny cone to see what happens and find out what calculus and Newtons laws couldn't tell you about that universe. But that universe is not this one.
Because ultimately, physics is not some thought exercise, it is an experimental science. All the math is in the service of predicting how the universe behaves via models, which are necessarily approximate and inexact. And near as we can tell from the best models we make, our universe is a messy one, an inexact one, a non-deterministic one. We've known that for hundreds of years: quantum physics, chaos theory - now there's an interesting one. Chaotic systems have divergent behavior over finite regions! You don't need to try to drop an infinitely tiny ball on an infinitely tiny point, you can actually physically realize them in this universe even with nominally deterministic processes!
Stuff like this is just getting lost in the weeds well before you get to the really interesting parts.
posted by Zalzidrax at 2:56 PM on January 3 [6 favorites]
All right, none of this is meant to be snark and I'm sincerely curious to engage with this argument. First of all, count me in the camp who argue that the snap discontinuity is a valid criticism. There is an external force in this model, and it's the scientific philosopher who specifically defined a time T, and artificially grafted together two different physical models relating to this time T without stating why these two models should be connected. Just because Newton didn't concern himself with continuous differentiability doesn't mean we shouldn't do so as well with physical theory derived from Newton. And part of my gripe is that after grafting these two models together, the argument is just about the definition of a real number, that there's no smallest time next to T where a force could happen. I have vague thoughts about that there should be a continuous C^infinity function from one part of the definition to the other.
posted by indexy at 5:33 PM on January 3
posted by indexy at 5:33 PM on January 3
... he clearly wants us to think that this thought experiment has some relevance to the real physical world, but when the dome is challenged on grounds like "the world is governed by more than just F=ma" or "this force field is not physically realizable" he retreats back into claiming he is only talking about abstract differential equations and his critics are being unsophisticated when they interpret the dome too literally.
Couple of somewhat disconnected thoughts on this.* One thing is that it's a lot cleaner to talk about whether this or that theory is deterministic than it is to talk about whether the world is deterministic. We get around to the world by way of the theories, so the claim that the world is deterministic ends up being that our best theory is deterministic. And then we get a bunch of questions: What's really required in order for a theory to be deterministic? How natural or artificial are the requirements? Are there interesting examples of deterministic and non-deterministic theories? Are the theories that have historically been thought to be deterministic actually deterministic? How can we tell for sure that a theory is deterministic? And so on.
Way back in 1987 (so, 16 years prior to his folk causation paper for those keeping track), Norton, in collaboration with John Earman, argued that "within a very broad class of spacetime theories" -- including general relativity -- substantivalism about space-time entails non-determinism. They get there by way of a corollary to the gauge theorem. In their paper, Norton and Earman make the following point having to do with metaphysical commitments:
So, now come back to the dome. It's not a physical example. It's telling us something about a theory. (A theory, just to remind us all, that Norton doesn't think is literally true. So what's going on here cannot be as simple as "This theory has [or lacks] property F, so the world has [or lacks] F.") Of course, we can amend the theory to rule out the kind of non-determinism that shows up in the dome example. But what's our motivation for doing so? Is it because we're already committed to determinism as a matter of metaphysics or because we have properly physical reasons for imposing the extra constraints? And what do we learn about the target, causation, if we amend or don't amend the theory that Norton is considering?
* Caveats and disclaimers: I work on causation. But I'm not a philosopher of physics; I mostly work on causal cognition and on methodology in statistics and the social sciences. I don't care much about determinism; unlike Norton, I think determinism is a bit of a red herring in the causation conversation. I also disagree with Norton about whether causation is fundamental to science, but I take a line that he doesn't consider: causation is fundamental to science because it's a presupposition of experimental practice, in particular the practice of isolating experimental set-ups and having proper controls.
posted by Jonathan Livengood at 9:15 PM on January 3 [2 favorites]
Couple of somewhat disconnected thoughts on this.* One thing is that it's a lot cleaner to talk about whether this or that theory is deterministic than it is to talk about whether the world is deterministic. We get around to the world by way of the theories, so the claim that the world is deterministic ends up being that our best theory is deterministic. And then we get a bunch of questions: What's really required in order for a theory to be deterministic? How natural or artificial are the requirements? Are there interesting examples of deterministic and non-deterministic theories? Are the theories that have historically been thought to be deterministic actually deterministic? How can we tell for sure that a theory is deterministic? And so on.
Way back in 1987 (so, 16 years prior to his folk causation paper for those keeping track), Norton, in collaboration with John Earman, argued that "within a very broad class of spacetime theories" -- including general relativity -- substantivalism about space-time entails non-determinism. They get there by way of a corollary to the gauge theorem. In their paper, Norton and Earman make the following point having to do with metaphysical commitments:
If a metaphysics, which forces all our theories to be deterministic, is unacceptable, then equally a metaphysics, which automatically decides in favour of indeterminism, is also unacceptable.In the folk causation paper, there's a similar metaphysical target that is problematic in a broadly similar way. Remember that at the beginning of the folk causation paper, Norton sets out a dilemma that he argues for in the "negative" part of the paper. (Lots of people entirely forget that there's a "positive" part of the paper at the end!) Either a commitment to a principle of causality has some empirical content such that it rules out some physical theories (and hasn't itself been ruled out by experiment) or it doesn't. Norton is skeptical that any interesting principle of causality can be formulated -- based largely on an induction from historical attempts and failures. But if the principle of causality doesn't rule out any physical theories, then it's an "empty honorific" and not to be regarded as "fundamental." It's on the second horn that I think we can apply the line from the earlier paper: we ought to accept or reject a principle of causality for reasons of physics, not on the basis of commitments that can be dropped without empirical consequences.
Determinism may fail, but if it fails it should fail for a reason of physics, not because of a commitment to substantival properties which can be eradicated without affecting the empirical consequences of the theory.
So, now come back to the dome. It's not a physical example. It's telling us something about a theory. (A theory, just to remind us all, that Norton doesn't think is literally true. So what's going on here cannot be as simple as "This theory has [or lacks] property F, so the world has [or lacks] F.") Of course, we can amend the theory to rule out the kind of non-determinism that shows up in the dome example. But what's our motivation for doing so? Is it because we're already committed to determinism as a matter of metaphysics or because we have properly physical reasons for imposing the extra constraints? And what do we learn about the target, causation, if we amend or don't amend the theory that Norton is considering?
* Caveats and disclaimers: I work on causation. But I'm not a philosopher of physics; I mostly work on causal cognition and on methodology in statistics and the social sciences. I don't care much about determinism; unlike Norton, I think determinism is a bit of a red herring in the causation conversation. I also disagree with Norton about whether causation is fundamental to science, but I take a line that he doesn't consider: causation is fundamental to science because it's a presupposition of experimental practice, in particular the practice of isolating experimental set-ups and having proper controls.
posted by Jonathan Livengood at 9:15 PM on January 3 [2 favorites]
Wait until this guy finds out that Maxwell's equations admit a solution for light going backward in time that physicists just kinda ignore because light just doesn't seem to do that.
Classical electrodynamics and classical (i.e. Newtonian) mechanics are both time-reversible theories, a fact of which Norton is demonstrably aware. Causation as Folk Science, page 11:
Doing so only when it suits us is no way to run a railroad, which is Norton's point.
There are loads of standard scenarios in physics textbooks that describe other idealizations of physical processes in which jerk, snap et al become ill-defined, cases that nobody seems to have a problem with. Among the most obvious is the case of a mass colliding elastically with another, in which the forces on both of them change instantaneously at the point of collision.
In any case, Norton's argument is not about the real-world constructibility of theoretical models. As Jonathan Livengood points out, it's about the extent to which theories such as classical mechanics get appealed to in order to reinforce a notion of determinism, aka universal causality, that they do not in fact support.
his critics are being unsophisticated when they interpret the dome too literally
It's not a question of sophistication, merely one of staying on topic.
near as we can tell from the best models we make, our universe is a messy one, an inexact one, a non-deterministic one. We've known that for hundreds of years: quantum physics, chaos theory - now there's an interesting one.
In fact neither quantum physics nor chaos theory have been things for hundreds (plural) of years. The very first published paper on anything recognizable as quantum physics was from Max Planck in 1900. Chaos theory first popped its head above the parapet a little earlier than that but not by much.
Even so, either of those should already have been enough to sink the boot well into the backside of Laplace's Demon, and yet it persists. In this very thread, even.
posted by flabdablet at 9:55 PM on January 3 [1 favorite]
Classical electrodynamics and classical (i.e. Newtonian) mechanics are both time-reversible theories, a fact of which Norton is demonstrably aware. Causation as Folk Science, page 11:
There is a simple way to see that the spontaneous motion of the mass is actually not that strange. Instead of imagining the mass starting at rest at the apex of the dome, we will imagine it starting at the rim and that we give it some initial velocity directed exactly at the apex. If we give it too much initial velocity, it will pass right over the apex to the other side of the dome. So let us give it a smaller initial velocity. We produce the trajectory T1 of Figure 1b. The mass rises towards the apex, but before it arrives it loses its motion, momentarily halts and then falls back to the rim. So we give it a little more initial velocity to produce trajectory T2. The mass rises closer to the apex but does not reach it before momentarily halting and falling back. We continue this process until we give the mass just the right initial velocity so that it rises up and momentarily halts exactly at the apex. In this last case, we have ended up with the mass momentarily at rest at the one force free point on the dome, the one point where, if it is at rest, the mass can (but need not) remain at rest. So let us imagine that it does remain at rest once it arrives. We now have a trajectory in which the mass rises up to the apex, halts there and remains there at rest for any arbitrary time period we care to nominate.Just because Newton didn't concern himself with continuous differentiability doesn't mean we shouldn't do so as well with physical theory derived from Newton.
An important feature of Newtonian mechanics is that it is time reversible, or at least that the dynamics of gravitational systems invoked here are time reversible. This means that we can take any motion allowed by Newton's theory and generate another just by imagining that motion run in reverse in time. So let us do that with the motion we have just generated. That reversed motion corresponds to a mass that remains at rest at the apex of the dome for some arbitrary time period and then spontaneously moves off to- wards the rim. And that is just a qualitative description of one of the solutions of (3).
This time-reversal trick is powerful, but we must be cautious not to overrate it. It is best used just to make the acausal behavior plausible, while the proper mathematical analysis of (1), (3), and (4) proves it. The reason is that there is a loophole. The spontaneous motion can happen only on domes of the right shape, such as those of Figure 1a. It cannot happen on others such as a hemispherical dome. The time-reversal argument fails for these other cases, for a reason that is easy to overlook. As we proceed through the trajectories T1, T2, … on a hemispherical dome, the time taken for the mass to rise to its momentary halt increases without bound. The final trajectory we seek, the one that momentarily halts at the apex, turns out to require infinite time. This means that the mass never actually arrives. Its time reverse displays a mass that has been in motion at all past times, without any spontaneous launches. The corresponding time for the dome of Figure 1a, however, is finite, so the analysis does succeed for this case.
Doing so only when it suits us is no way to run a railroad, which is Norton's point.
There are loads of standard scenarios in physics textbooks that describe other idealizations of physical processes in which jerk, snap et al become ill-defined, cases that nobody seems to have a problem with. Among the most obvious is the case of a mass colliding elastically with another, in which the forces on both of them change instantaneously at the point of collision.
In any case, Norton's argument is not about the real-world constructibility of theoretical models. As Jonathan Livengood points out, it's about the extent to which theories such as classical mechanics get appealed to in order to reinforce a notion of determinism, aka universal causality, that they do not in fact support.
his critics are being unsophisticated when they interpret the dome too literally
It's not a question of sophistication, merely one of staying on topic.
near as we can tell from the best models we make, our universe is a messy one, an inexact one, a non-deterministic one. We've known that for hundreds of years: quantum physics, chaos theory - now there's an interesting one.
In fact neither quantum physics nor chaos theory have been things for hundreds (plural) of years. The very first published paper on anything recognizable as quantum physics was from Max Planck in 1900. Chaos theory first popped its head above the parapet a little earlier than that but not by much.
Even so, either of those should already have been enough to sink the boot well into the backside of Laplace's Demon, and yet it persists. In this very thread, even.
posted by flabdablet at 9:55 PM on January 3 [1 favorite]
So what this guy has done is construct a very slightly rounder cone and hidden the discontinuity.
I think you're on to something here. Now it is a bit of a different kind of discontinuity in that the force on the point mass is continuous at the top of the dome whereas it wouldn't be at the tip of of a cone. But there still is a kind of pointiness to the dome and the point mass is very pointy indeed. I'm almost positive using a ball rather than a point mass as in the video destroys the non-determinacy. (To be fair point masses don't show up well on video). I suspect you would have non-determinacy if you balanced a second Norton's dome upside down of the first so long as there was no friction, the pointiness would be well-disguised then, but the equations of motion far nastier.
posted by mscibing at 10:15 PM on January 3
I think you're on to something here. Now it is a bit of a different kind of discontinuity in that the force on the point mass is continuous at the top of the dome whereas it wouldn't be at the tip of of a cone. But there still is a kind of pointiness to the dome and the point mass is very pointy indeed. I'm almost positive using a ball rather than a point mass as in the video destroys the non-determinacy. (To be fair point masses don't show up well on video). I suspect you would have non-determinacy if you balanced a second Norton's dome upside down of the first so long as there was no friction, the pointiness would be well-disguised then, but the equations of motion far nastier.
posted by mscibing at 10:15 PM on January 3
causation is fundamental to science because it's a presupposition of experimental practice, in particular the practice of isolating experimental set-ups and having proper controls
I'd argue that seeing experimental practice as fundamental to science is one of the things that's holding a lot of excellent science back.
Controlled experiments are widely held to be the gold standard, and results from the branches of science that actually engage with the parts of our world that are too messy to admit of such experiments are widely held to be vastly less worthy of respect. I have lost count of the number of times I've heard fields like sociology or biology or climate science dismissed as "not real science" on no better basis than that they're more about careful observation and cross-checked modelling than controlled experiment.
It has long seemed to me that controlled experiment is feasible only in tightly restricted domains of inquiry, and that holding systematic inquiry into those domains and those domains only to be definitional of "real" science is no more than an exercise in hubris. It's like handing out medals to the drunks looking for their car keys only under the street lights because that's where they'd be easiest to see.
My own metaphysics holds that determinism is bunk except in the restricted and tautological sense that only what will happen will happen. The universe is exceedingly messy, repeatability is the luxurious exception rather than the general rule, such repeatability as we do find is more likely to be a consequence of statistics than anything else, and this somewhat unfortunate aspect of the way things are deserves to get taken way more seriously than it generally is.
Einstein observed that the most incomprehensible thing about the world is that it is comprehensible.
When I was about eight years old it struck me with the force of revelation that I didn't need to go to school tomorrow because I already knew everything, and just to make sure this was really true, I spent the next hour thinking of things I knew, and making sure for each one that yes, that's a thing I know. See? I know everything!
I hear echoes of that experience in Einstein's remark. Turns out, quite a lot of what I knew at eight years old was facts about the multiplication table for integers 1..12. It's just so weird how easy that makes multiplying smallish numbers together.
I'm almost positive using a ball rather than a point mass as in the video destroys the non-determinacy.
It really doesn't. All you've done by using a ball is put the centre of mass of the sliding object a fixed distance away from the surface of the dome, so all you need to do to construct a model that's completely equivalent to the point-mass version is make another dome whose surface is displaced inward and normal to the surface of the original by the radius of your ball. The equation describing the dome profile gets messier, but all of that mess disappears when you back-construct the surface that's actually constraining the motion of your test mass's centre of gravity.
If you wanted to stipulate that the ball rolls rather than slides then you'd need to model its moment of inertia as well. Again, that would make things messier but I would be astonished to learn that it would make such a huge mess as to render it impossible to construct a dome profile with a force vs displacement relationship equivalent to Norton's original.
posted by flabdablet at 10:41 PM on January 3 [1 favorite]
I'd argue that seeing experimental practice as fundamental to science is one of the things that's holding a lot of excellent science back.
Controlled experiments are widely held to be the gold standard, and results from the branches of science that actually engage with the parts of our world that are too messy to admit of such experiments are widely held to be vastly less worthy of respect. I have lost count of the number of times I've heard fields like sociology or biology or climate science dismissed as "not real science" on no better basis than that they're more about careful observation and cross-checked modelling than controlled experiment.
It has long seemed to me that controlled experiment is feasible only in tightly restricted domains of inquiry, and that holding systematic inquiry into those domains and those domains only to be definitional of "real" science is no more than an exercise in hubris. It's like handing out medals to the drunks looking for their car keys only under the street lights because that's where they'd be easiest to see.
My own metaphysics holds that determinism is bunk except in the restricted and tautological sense that only what will happen will happen. The universe is exceedingly messy, repeatability is the luxurious exception rather than the general rule, such repeatability as we do find is more likely to be a consequence of statistics than anything else, and this somewhat unfortunate aspect of the way things are deserves to get taken way more seriously than it generally is.
Einstein observed that the most incomprehensible thing about the world is that it is comprehensible.
When I was about eight years old it struck me with the force of revelation that I didn't need to go to school tomorrow because I already knew everything, and just to make sure this was really true, I spent the next hour thinking of things I knew, and making sure for each one that yes, that's a thing I know. See? I know everything!
I hear echoes of that experience in Einstein's remark. Turns out, quite a lot of what I knew at eight years old was facts about the multiplication table for integers 1..12. It's just so weird how easy that makes multiplying smallish numbers together.
I'm almost positive using a ball rather than a point mass as in the video destroys the non-determinacy.
It really doesn't. All you've done by using a ball is put the centre of mass of the sliding object a fixed distance away from the surface of the dome, so all you need to do to construct a model that's completely equivalent to the point-mass version is make another dome whose surface is displaced inward and normal to the surface of the original by the radius of your ball. The equation describing the dome profile gets messier, but all of that mess disappears when you back-construct the surface that's actually constraining the motion of your test mass's centre of gravity.
If you wanted to stipulate that the ball rolls rather than slides then you'd need to model its moment of inertia as well. Again, that would make things messier but I would be astonished to learn that it would make such a huge mess as to render it impossible to construct a dome profile with a force vs displacement relationship equivalent to Norton's original.
posted by flabdablet at 10:41 PM on January 3 [1 favorite]
The snap discontinuity goes away, while I think the basic setup remains, if we start our point mass perched on a C∞ bump function.
But all of this is statement about one theory or another, where there are more theories than there are worlds. When you set a negatively stable initial condition, that's not theory's problem.
posted by away for regrooving at 12:14 AM on January 4
But all of this is statement about one theory or another, where there are more theories than there are worlds. When you set a negatively stable initial condition, that's not theory's problem.
posted by away for regrooving at 12:14 AM on January 4
There are doubtless more theories than worlds and likely always will be, because when it comes right down to it there's only one world and the things we wish to know about it are too complicated for any single theory to describe in ways that can actually be applied in practice. But all of that is a sideline to the main thrust of the linked paper, which is about determinism and the routine abuse of theory to prop up a belief in it.
Today's most successful physical theories do not imply determinism; quantum mechanics, in particular, explicitly denies it. And yet, the belief that at some level determinism must be true - that Laplace's Demon is in principle completely sound - persists, and one particular theory - classical Newtonian mechanics - is cited more often than any other as justifying that belief.
What Norton is about here is providing an existence proof for models that lie squarely within classical Newtonian mechanics and yet do not imply determinism, making the use of that body of theory in that way illegitimate. Arguments about the physicality of the specific and unusually simple model he's constructed in order to make this point are therefore somewhere between tangential and obfuscatory.
The point is whether or not determinism has any scientific basis, i.e. some basis ultimately derived from scrupulous observation of the world as we find it, as opposed to being a belief that people just take on faith because they find it intuitively appealing.
The reason I led this post with links to Norton, rather than links to Up and Atom, is that I think Up and Atom's clickbait claim that Norton's work "breaks physics" is pretty much entirely at odds with what it actually does seek to do.
posted by flabdablet at 2:39 AM on January 4 [1 favorite]
Today's most successful physical theories do not imply determinism; quantum mechanics, in particular, explicitly denies it. And yet, the belief that at some level determinism must be true - that Laplace's Demon is in principle completely sound - persists, and one particular theory - classical Newtonian mechanics - is cited more often than any other as justifying that belief.
What Norton is about here is providing an existence proof for models that lie squarely within classical Newtonian mechanics and yet do not imply determinism, making the use of that body of theory in that way illegitimate. Arguments about the physicality of the specific and unusually simple model he's constructed in order to make this point are therefore somewhere between tangential and obfuscatory.
The point is whether or not determinism has any scientific basis, i.e. some basis ultimately derived from scrupulous observation of the world as we find it, as opposed to being a belief that people just take on faith because they find it intuitively appealing.
The reason I led this post with links to Norton, rather than links to Up and Atom, is that I think Up and Atom's clickbait claim that Norton's work "breaks physics" is pretty much entirely at odds with what it actually does seek to do.
posted by flabdablet at 2:39 AM on January 4 [1 favorite]
If it breaks some of the unjustifiable assumptions that so many people seem to have made about physics, so much the better.
posted by flabdablet at 2:46 AM on January 4
posted by flabdablet at 2:46 AM on January 4
It really doesn't. All you've done by using a ball is put the centre of mass of the sliding object a fixed distance away from the surface of the dome, so all you need to do to construct a model that's completely equivalent to the point-mass version is make another dome whose surface is displaced inward and normal to the surface of the original by the radius of your ball.
I think what's going on with the curvature near the peak will mean the displaced surface will have some small cusps and self-intersection.
I do think the dome can work with an extended object, just that a sphere is the wrong shape: you probably need a curvature discontinuity on the object where it is initially touching the dome, and allow the object to slide. It's not the moment of inertia that I'm concerned about: there's funny stuff going on with the second derivative of h with respect to r that I don't think is going to be possible with rolling or with a sphere.
I do find the dome interesting for how mild the idealization and non-physicality needed to make it work is: Norton makes a strong case that it is nothing out of the norm for working in Newtonian mechanics.
posted by mscibing at 4:23 PM on January 4
I think what's going on with the curvature near the peak will mean the displaced surface will have some small cusps and self-intersection.
I do think the dome can work with an extended object, just that a sphere is the wrong shape: you probably need a curvature discontinuity on the object where it is initially touching the dome, and allow the object to slide. It's not the moment of inertia that I'm concerned about: there's funny stuff going on with the second derivative of h with respect to r that I don't think is going to be possible with rolling or with a sphere.
I do find the dome interesting for how mild the idealization and non-physicality needed to make it work is: Norton makes a strong case that it is nothing out of the norm for working in Newtonian mechanics.
posted by mscibing at 4:23 PM on January 4
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posted by Jonathan Livengood at 9:52 PM on January 2 [4 favorites]