We then introduce an error into the measurement of the initial velocity. Regardless of how small that error is, it will build over time until the error in predicted position is greater than the length of the cavity. — Banno

It's also worth noting that if our measurements of initial position and velocity are inherently imperfect, so will any subsequent measurements of position and velocity also be imperfect. Of course there will also be magnification of error, but how would we know what degree of the error is due to magnification of error and what is simply due to measurement error, if none of our measurements are perfect?

It's a wonder that them physicist get anything right.

Dang, it is a wonder and all!

Principle of infinite precision
1. Ontological – there exists an actual value of every physical quantity, with its infinite determined digits (in any arbitrary numerical base).
2. Epistemological – despite it might not be possible to know all the digits of a physical quantity (through measurements), it is possible to know an arbitrarily large number of digits. — Indeterminism, causality and information: Has physics ever been deterministic? by Flavio Del Santo

If we were to experimentally verify a theoretical value of ⅓ millijoule, 2½ millijoule, or π millijoule with infinite precision, then we'd be in the same predicament, yes? I mean, the particular number wouldn't make a difference?

So I don't see a relevant difference in kind between the marble int he tube in my example and a chaotic system. — Banno

The marble setup IS a chaotic system. I think you think I’m disagreeing with you more than I am.

I think you think I’m disagreeing with you more than I am. — Pfhorrest

Probably.

In fact “as soon as one realizes that the mathematical real numbers are not really real, i.e. have no physical significance, then one concludes that classical physics is not deterministic.
But what's the point?

Whether or not a real number is or is not a "real" number is beside the point, and not at all relevant. What is relevant is the "natural uncertainty in all observations", which some use of real numbers tends to veil with the pretense of what he calls "infinite precision".

This natural uncertainty is true of all all descriptions of initial conditions, so it applies to all inertial reference frames. Since the uncertainty develops exponentially with the passage of time, we rapidly become deficient in the capacity to distinguish between an improperly represented inertial reference frame, and an external cause in the occurrence which follows. — Metaphysician Undercover

Sqrt(2) is of course not a number, and is riddled with imprecision - can't do a thing with it. But wait! I can multiply it by the Sqrt(2). Probably equals somewhere in the vicinity of 2. Couldn't be 2, though, wa-ay too imprecise for that.

And then there's the Battleship U.S.S. Massachusetts, moored in Battleship Cove, Fall River, Mass. 680 feet, 36,000 tons, nine sixteen inch guns. She happens to be there when anyone looks. Which I do not understand. I know QM tells us that there's a chance that this instant it may be there, and the next in my back yard, 70 miles North and 35 miles inland, having quantum tunneled its way due to quantum uncertainty. But given, as you point out, that "uncertainty develops exponentially with the passage of time," and the battleship has been there for about fifty-five years, well, how do you account for its apparent permanence in place? Why isn't it appearing here and there and all over the place? For that matter, given that exponentially developing uncertainty, why isn't everything flashing into existence here at one moment and there at another?

I'm afraid you're in the nonsense business, MU. And that's an end of it!

At work yesterday I set up three jars and looked in the box and it was empty. On the shelf was all six. The universe has done something nice

Now the default position adopted in my high school physics class was that the error was introduced by a lack of precision in the measurement. The assumption was that there is indeed some real number that gives the exact velocity to infinite precision, and that the error represented the degree to which one could operationally approximate the actual velocity. The alternative explanation being offered by Del Santo is that the initial velocity does not correspond to some real number, but instead to some region of the real numbers. The boundaries of this region are also indefinite, but lies within the bounds of our arbitrarily accurate measurement. — Banno

There is another issue which needs to be considered, and that is the attempt to remove the margin of error through the manufacture of artificial initial conditions. This is what is done in experimentation, the apparatus is intentionally designed so as to supposedly give us the capacity to reproduce the same initial conditions over and over. This produces the idea that the error of measurement can be accounted for, or removed.

Consider your Galton box, the ball is channeled down the narrow throat, and positioned accordingly. This channeling is the creation of artificially limited initial conditions. If the ball is always dropped from the same height one might believe that the initial conditions have sufficiently been controlled. The point which that apparatus demonstrates is that no matter how well we control the initial conditions, it is always a simple matter to add an element of "chance" into such an apparatus which will render the outcome as unpredictable. This indicates that unpredictability is very likely an inherent feature of how we as human beings, produce and describe initial conditions. Therefore attempting to correct for the error is not the right approach, as it is an attempt to do the impossible, correct the uncorrectable. What is needed is a non-determinist approach which recognizes the reality of that unpredictability.

It's also worth noting that if our measurements of initial position and velocity are inherently imperfect, so will any subsequent measurements of position and velocity also be imperfect. — Janus

Sounds like determinism to me. Initial conditions lead to subsequent conditions. Banno is using determinism to show that determinism is false.

If determinism were false then we would get things right or wrong regardless of whether or not we had initial errors. Initial states of accuracy or inaccuracy would make no difference in subsequent states. We would never be able to establish a causal link between the initial state of being accurate or inaccurate with subsequent states to then say that the magnified errors were caused by our initial inaccuracy.

Reading notes: So it seems like there's a few things going on in the Anscombe essay. Here are some that stood out:

(1) Disentanglement of causality from necessity. Positive claim: there can be causes which do not follow of necessity.

(2) Disentanglement of determination from causality. Positive claim: Something can be determined without being caused. It strikes me that Anscombe is ultimately unconcerned with causality. It all but drops out of consideration in the second half of the paper. It was used as a 'way in' to talk about 'determination' and its obverse, 'indetermination'.

(2.1) 'Determination' cannot be thought outside of some given range of possibilities: "to give content to the idea of something’s being determined, we have to have a set of possibilities, which something narrows down to one – before the event". By distinction, causality is post-hoc: "But there is at any rate one important difference – a thing hasn’t been caused until it has happened".

(3) Conclusion: 'Indeterminism' must be admitted, at the very least, as a possibility. Interderminism meaning: given a set of outcomes, it cannot be specified, in advance, which will obtain.

(4) I have a huge question about the level of granularity - mereological and temporal - at which all these considerations are meant to apply. Are these conclusions meant to be the same for the Galton board, taken as a whole, and a single ball travelling along a Glaton board path? Why is each of these two cases individuated as such? What motivates this individuation? Why not consider some balls, and not others? Maybe two balls, rather than one; or why not the Galton board, and the path of one or two or three or all balls? How does taking into account these analytic 'cuts' - seemingly arbitrary, affect the analysis?

The question of 'givenness' ("given a range of possibilites...") has big implications on the status of in/determinism (epistemological? ontological? Something other?). Anscombe is ambigious about this, but intuits it when she discusses the temporality of determination (determined when?) and distinguishes - without coming back to it - between determination and what she at one point calls 'predetermination'. Want to say more about this later. Will just open the question for now, if anyone else can see the issue.

It's a wonder that them physicist get anything right. — Banno

Damn those environmental scientists and philosophy forum posters who keep telling us that humans are determining the future destruction of our planet by our ignorant actions.

(1) Disentanglement of causality from necessity. Positive claim: there can be causes which do not follow of necessity. — StreetlightX

There goes the Kalam cosmological argument.

...causality is post-hoc — StreetlightX

...but one ball or a thousand, the result is still a normal distribution. So despite causality being after the event we can predict the outcome. Is there a contradiction here?

Sounds like determinism to me. Initial conditions lead to subsequent conditions. Banno is using determinism to show that determinism is false. — Harry Hindu

Asserting determinism, or at least what is often referred to as "hard" or "rigid" determinism consists in claiming that from any set of conditions there can arise only one outcome at any subsequent time. An often cited "thought experiment" is that if it were possible to restart the evolution of the universe from the initial moment it came into being it would again unfold exactly, down to the minutest detail, as it has actually done this time.

This is an entirely groundless assumption. Under the aegis of Newtonian mechanics this may have seemed obviously true, but in the light of QM it seems not only vanishingly unlikely, but even just plain impossible.

See this. Harry's out of his depth.

I have no idea what Harry is on about there.

Simplest interpretation is that he doesn't understand a measurement's being accurate to within a certain error. Now that's Lesson 1 in physics. Same seems to be true of his use of "determinism", vacillating between cause and ascertain... the result was determined (caused) as against the result was determined (ascertained). Or something like that. He's just not on the page. Language issues.

I think you think I’m disagreeing with you more than I am. — Pfhorrest

Probably. — Banno

Just to clarify what it is that I was saying, in case that turns out to be useful to anyone in this thread:

Say we had a simulation of a Galton box, simulated using Newtonian physics, so no quantum stuff going on in the simulation (and the simulator sufficiently insulated from quantum noise, as most macroscopic stuff is, that it's completely negligible for our purposes).

That simulated Galton box would be strictly deterministic. You could play it forward, note where a given marble ended up, rewind it back to the beginning, then play it forward again and know in advance where that marble would end up.

But still, that system is nevertheless (probably -- I'm not super familiar with Galton boxes) still chaotic in that the tiniest deviation in the starting positions of the balls or pins or anything would result in huge changes in the outcome.

If we had to measure the positions of the balls a posteriori and program them into a different copy of the exact same simulator, we would necessarily have less than perfect measurement of them, and so have tiny differences in the starting conditions, and so see a vastly different result.

But so long as the simulator does actually have the exact positions it used the first time, it will give the same results every other time, over and over and over again.

It is conceivable in principle (though does not appear to be contingent fact) that our universe could be deterministic like that, and yet nevertheless chaotic in places (and therefore as a whole), meaning that if we have anything less than absolutely perfect knowledge of the present (which we can't have), then we could not predict the future, even if the universe were perfectly deterministic.

And then on top of all of that: predictor systems like us are inherently chaotic, so even if the universe was perfectly deterministic and otherwise completely non-chaotic, our attempts to predict it would make it chaotic and unpredictable.

So, the universe does not appear to be deterministic, as a matter of contingent fact.

But even if it were, it could still be chaotic, and so unpredictable in practice.

And even if it weren't otherwise chaotic, any attempt to use that non-chaotic determination would make it chaotic, and so unpredictable anyway.

...but one ball or a thousand, the result is still a normal distribution. — Banno

Certainly not one ball (you can't have a distribution of one ball).

So despite causality being after the event we can predict the outcome. Is there a contradiction here? — Banno

I don't think so - what motivates this question? And note that the question of prediction is almost entirely absent from the Anscombe paper. She mentions it twice, and both times they are ancillary to her main points. And there's the matter of being clear about what 'the outcome' refers to: surely you mean - 'the outcome' of a normal distribution of balls. But if determination is - as Anscombe argues - externally related to causality - it's not clear why one would think that a contradiction results from the predictability of outcome and the post-hoc nature of causality.

For any given initial location X of the ball in a Galton box, there will be some positive number delta(X) such that a measurement of initial conditions with error less than delta(X) can predict the outcome with certainty. That is provided the position X is not such that the lowest point of the ball is **exactly** above the highest point of the first pin (an 'unstable equilibrium position') and that no other unstable equilibria arise in the journey through to the final resting place of the ball. Since the probability of an item occupying a pre-specified exact position is zero, we can dismiss those cases.

However there is **no** positive number delta such that, for **any** initial location X of the ball, a measurement of initial conditions to accuracy delta allows certain prediction of the outcome.

For the mathematically inclined, that's because the function f that maps the initial position of the ball to its final location is continuous on the domain D that excludes only the set of measure zero comprising positions that lead to unstable equilibria. But f is not **uniformly continuous** on D.

Although I am not a hard determinist, I don't see anything in the notion of a Galton box in a context of classical physics, that defeats a belief in hard determinism. The unpredictability of the Galton box is just an instance of chaos theory, and chaos theory focuses on the consequences of practical limits in measurement accuracy, not on the theoretical impossibility of making a measurement with zero error. There will be some ridiculously small but nonzero error limit such that, if we could measure everything to within that limit, we could predict a Hurricane in Haiti from the flap of a butterfly's wing in Mongolia.
.
A thought experiment that generates similar questions about determinism is Norton's Dome, which is also based on an unstable equilibrium. While it is an interesting case to think about, it can't tell us anything about our world for the same reason as the Galton box, viz, the probability is zero of the ball being over the exact single point where the paradox arises.

Even if we were to conclude that predictability dissolves when objects are in an unstable equilibrium, I doubt it would discourage hard determinists. To go from 'hard determinism always holds' to 'hard determinism holds everywhere except in a special set of circumstances that has probability zero of ever arising' doesn't sound like much of a concession.

For any given initial location X of the ball in a Galton box, there will be some positive number delta(X) such that a measurement of initial conditions with error less than delta(X) can predict the outcome with certainty. — andrewk

That's a possible feature of how the balls are input, no? If you have a robotic arm capable of placing balls to arbitrary precision; like we can on paper by specifying an initial condition, then that's going to hold. If the causal processes that puts the balls into the Galton box by design does not constrain it in that manner; effectively evolving a volume of initial conditions forward through the box; then the output pattern is going to be close to binomial (on left vs right hole transitions) or approximately normal (on horizontal coordinate of box base) assuming the sample of initial conditions isn't really weird in some way.

The contrast is between:

(1) The Galton box is deterministic because there is a hypothetical arbitrary precision mathematical model of it that allows perfect prediction for every input trajectory that doesn't result in unstable equilibrium. Complete specificity of initial conditions. Does not actually occur in actual Galton boxes.
(2) The actual operation of the Galton box doesn't have that. Vagueness of initial conditions - a distribution of them.



The initial conditions of each ball in the Galton box are not specified to arbitrary precision, they're kinda just jammed in. So for the above Galton box and initial condition specification (kinda just jamming it in), and for any particular bead, it's true that we can't predict its trajectory. That sits uneasy with the hypothetical claim that we could if only it were specified to sufficient precision; that "if only" means we're no longer talking about the above box.

That sits uneasy with the hypothetical claim that we could if only it were specified to sufficient precision; that "if only" means we're no longer talking about the above box. — fdrake

:up:

Some additional thoughts:

For me, the most important take-away is the fact that for Anscombe, determination is 'possibility-relative': "We see that to give content to the idea of something’s being determined, we have to have a set of possibilities, which something narrows down to one – before the event".

But the question is: whence this set of possibilities, and not another set? With the Galton board, it seems 'natural' to pick out the relevant set of possibilities as distribution of balls among the pipes, but it's important to recognize just how arbitrary this is. It's certainly not a given of nature, for instance, that these possibilities must be thought together to the exclusion of all other possibilities (that the sky is blue at the time of the experiment, for instance). And if that is so, this means that 'determinability' (and with it, indeterminability) is itself not a 'natural' category - we cannot ask of nature, taken as a whole: 'is it determined or not?'.

Or, to introduce another distinction (whose applicability in these situations was brought to my attention by @fdrake): we cannot ask the question of determinability at a global level, only at a local one. And what 'picks out' or individuates a local situation ('set of possibilities') is, or must be, a question of motivation. Part of the problem with using a Galton board to think about this stuff is precisely because it is so arbitrary: the distribution of balls does not correspond to any particular effect which follows from that distribution - it does not couple to any system for which the distribution makes a difference.

The answer to the question of determinism could only be revealed with a thorough understanding of the nature of time, something human beings are very far from having.

Asserting determinism, or at least what is often referred to as "hard" or "rigid" determinism consists in claiming that from any set of conditions there can arise only one outcome at any subsequent time. An often cited "thought experiment" is that if it were possible to restart the evolution of the universe from the initial moment it came into being it would again unfold exactly, down to the minutest detail, as it has actually done this time.

This is an entirely groundless assumption. Under the aegis of Newtonian mechanics this may have seemed obviously true, but in the light of QM it seems not only vanishingly unlikely, but even just plain impossible. — Janus

So QM determines that determinism is impossible?

It certainly isn't a groundless assumption that events would be the same if the universe were restarted. It seems to me that the burden is on you for stating otherwise. Given the same conditions at every moment in time, the same effects will happen. It follows that given the same initial conditions, you get the same results. It doesn't follow that given the same initial conditions that you will get different results. In other words, logic itself would be useless in an indeterministic world. Your reasons determine your conclusion. You keep using determinism every time you make an argument where you conclusion follows your premise.

You and Banno seem to be making the mistake of believing that every instant of time is the same - as if the ball dropped at this moment is the same at some latter moment. The initial conditions at some moment are different than at some other moment, and ignorance is a factor because we could be oblivious to all the initial conditions that make a certain event happen. We might get it mostly right and it's just enough to make an accurate prediction, or we might get it wrong and then think that determinism is false. But it's not. It's just that you can't recreate a same moment in time at some other moment in time. But if you restarted the universe these events will happen at the same moment in the same way, and you will still be ignorant of all the initial conditions at any moment.

As for QM, what is it about QM that determines that determinism is impossible?

Simplest interpretation is that he doesn't understand a measurement's being accurate to within a certain error. — Banno

Sounds like more determinism. Seems like errors determine outcomes.

"We see that to give content to the idea of something’s being determined, we have to have a set of possibilities, which something narrows down to one – before the event". — StreetlightX

That role seems to be played by the initial conditions. For a given initial condition, there's a guaranteed outcome. For a range of initial conditions, there's a range of outcomes. Imprecise specification of an initial condition gives a range of outcomes consistent with (determined by? @Kenosha Kid) the range of inputs concordant with the imprecisions. The sleight of hand that makes determinism seem to be a system property seems to be the specification of an initial condition with sufficient precision; as if the specification of an initial condition was done externally to the dynamics of any actual Galton box.

Simplest interpretation is that he doesn't understand a measurement's being accurate to within a certain error. Now that's Lesson 1 in physics. Same seems to be true of his use of "determinism", vacillating between cause and ascertain... the result was determined (caused) as against the result was determined (ascertained). — Banno

I fail to see how a digital system used to measure an analog reality indicates that reality is indeterministic. It seems that what you are saying is what is indeterministic is our measurements, not reality. With that, I would agree. Measurements are like views, which could explain some of the results of the double-slit experiment. Taking measurements or views alters the effects. That doesn't mean that indeterminism is true, it means that our existence as observers and our measuring devices plays a causal role in the very events we are observing and measuring. Solving the mind-body problem I believe will provide the necessary link between classical mechanics and QM - between the macro and the nano, and unite them.

The sleight of hand that makes determinism seem to be a system property seems to be the specification of an initial condition with sufficient precision; as if the specification of an initial condition was done externally to the dynamics of any actual Galton box. — fdrake

Yes, exactly! I was thinking about this in relation to the little Galton board video you posted - funnily enough, the balls don't fall exactly into the 'normal' distribution - some lines are a little over, some are a little under. And it got me thinking - does this mean that this Galton board is a badly designed one? Well no - if the board were designed such that you really did get a perfect distribution, then then it's precisely the tweaking of inputs which guarantees consistency of output. In truth, the normal distribution is a totally ideal property: it's what the sum of infinite runs of the board would converge to, at the limit (ergodic property?).

So you're right: the 'search' for initial conditions ("if we just knew the initial conditions with enough precision...") can be nothing other than a fixing of initial conditions in order to make determinism a system property. I'm reminded here of Kant's 'intellectual intuition': that wherein knowledge and being coincide, available only to a God, who needs no mediation of the sensible (time and space). Or else Wittgenstein's meter rule: that which neither is nor is not a meter. The 'fixed' Galton board would be like that: neither a Galton board not not one.... it would be like, a gif of a Galton board, a moving image.

...but one ball or a thousand, the result is still a normal distribution.
— Banno

Certainly not one ball (you can't have a distribution of one ball). — StreetlightX

The probability of one ball falling in any particular bin is given by the normal curve.

The sleight of hand that makes determinism seem to be a system property seems to be the specification of an initial condition with sufficient precision; as if the specification of an initial condition was done externally to the dynamics of any actual Galton box. — fdrake

the 'search' for initial conditions... can be nothing other than a fixing of initial conditions in order to make determinism a system property. — StreetlightX

Yep.

@Pfhorrest, do I understand correctly that you disagree? I'm not following what you are saying about chaotic systems. For instance given any point on the complex plane we have an algorithm for finding out if it is a member of the Mandelbrot Set or not - so isn't membership determined for any given point? It is a member iff iterations of z_(n+1)=z_n^2+C converge.