I find it difficult to get the big picture of GR, i.e. to connect the various components described by Reichenbach and Norton. These components are the principle of equivalence, the intrinsic curvature of spacetime that arises in the vicinity of masses, the spacetime and space-space sheets, the role of free fall etc. I'm also not sure I understand the notions Reichenbach discusses introducing the underlying concepts, like the conventionality of motion, the "transforming away of gravitation" and the distinction of local and astronomical inertial systems.
So the general context of GR is Einstein's aim to extend his theory of special relativity to gravitation, since it was incompatible with Newton's theory of gravitation. This new gravitational theory then postulates gravitation not as a force, but as an intrinsic curvature of spacetime around masses. It is no longer a force acting between bodies but a geometrical property of spacetime. This is often depicted using this image of masses in curved space like the surface of a blanket around a heavy object on it, which is inaccurate because the curvature is not merely a feature of space but also of spacetime, different sheets over time.
The basic idea of the principle of equivalence is that the effects of gravitation and inertial effects in an accelerated system (which in Newtonian physics seem to correspond coincidentally) are really the same thing, which arises from the fact that acceleration creates a gravitational field.
So is gravitation both inertial motion in curved spacetime and the effect of accelerated systems? I don't get this connection. Does acceleration create curvature of spacetime or is it equivalent to inertial motion in curved spacetime? Is it in this sense that free fall is a special kind of motion, in that it is both inertial motion in curved spacetime and accelerated motion creating such a field? And does GR abandon the idea of forces altogether, or does it merely postulate the equivalence of descriptions of forces in terms of curved spacetime?
The example of balls in free fall inside the earth in Norton's book shows how free fall is now described in terms of curved spacetime, where the trajectory of the balls traces geodics as the straightest lines in non-Euclidean geometry.
Lastly, I wonder what the role of the conventionality of motion for the theory is and what this concept of local inertial systems means?
Friday, 5 December 2014
Thursday, 27 November 2014
Reichenbach on Non-Euclidean Geometry and Space
Reichenbach on Non-Euclidean Geometry and Space
As far as I understand it, Reichenbach's main argument in his paper is that in order to get anywhere with geometry we need to impose a methodological rule which states/stipulates that there are no universal forces. Only when such a rule is established can we attempt to determine a geometrical framework that is not just an arbitrarily chosen system or one that is tailored to fit a preferred geometry by the compensation of universal forces. Of course, the stipulation of the rule itself makes geometric determination conventional from the outset.
Using his "hump" diagram (also explained in the slides), we can better understand the problem of universal force in a geometry. Imagine that people inhabit the curved surface G: Reichenbach supposes that they would be able to determine the curved shape by geometrical measurements; "noting the differences between their measurements and two-dimensional Euclidean geometry." In other words, people on G could determine their curved geometry from Euclidean by means of comparison. What is important is that they would measure the distance from A-B and B-C as identical/equal. However, if the measuring rods cast shadows onto the flat E plane below, A-B and B-C should be unequal. Now suppose that on E "a mysterious force varies the length of all measuring rods moved about in that plane, so that they are always equal in length to the corresponding shadows" projected from G. Consequently, the people on E would not be able to perceive the change, and they would obtain an equal measurement (a curved geometry) that is respective of the G measurement.
So, Reichenbach's question is: how can the force be detected if the nature of the geometry may not be used as an indicator?
If the force is heat, for instance, E would be distinguishable from G, as heat affects different objects differently. Thus heat is a "differential force," and is not so problematic in the determination of a geometry.
Thus suppose the force is something universal: affecting everything in the same way, and unable to be insulated against. In this case, E might be indistinguishable from G - so long as we find a physical description of objects to fit a certain geometrical framework, (such as the presence of a universal force in a curved "humped" geometry) all potential deficiencies in our account can be compensated for.
Hence for Reichenbach we must stipulate a rule which sets all universal forces to zero, otherwise we could compensate any geometrical system with a physical account, and vice versa. This makes geometric determination a matter of convention: we have freely stipulated away universal force, so we cannot be right or wrong. Furthermore, the definitions of physical objects that we use in a particular framework will need to be "coordinative": we say that "this concept" is coordinated to "this particular thing," and that concepts are "interconnected by testable relations" (so our words coordinate with real things in the world rather than other words/concepts)
At this point he introduces the idea of relativity: "the word "relativity" is intended to express the fact that the results of the measurements depend upon the choice of the coordinative definitions." So we can define a solid body as a physical body that can be pointed out, and a rigid body as one that is unaffected by differential forces, (universal forces unconsidered). Thus our physical definitions are stipulative/subject to choice. In short, our definitions are themselves conventions. Using the new definition of a rigid body, we can therefore stop worrying about the indeterminacy of a geometry in virtue of universal forces, as those forces do not affect the rigid bodies in our framework. I therefore take him to conclude that universal forces are stipulated out in virtue of the fact that rigid bodies will be unaffected by them in a geometry according to our coordinative definition.
He spends the 5th part of the chapter explaining what it means to be a rigid body in more detail, but I will skip over most of his arguments - (details from page 20 of his paper onwards). To summarize, he suggests that there are certain restrictions to be made upon the definition of a rigid body such as "the demand that the obtained metric retain certain older physical results, especially those of the 'physics of daily life,'" or that any definition of a rigid body will depend on a prior definition of a "closed system." He concludes, however, that there is no perfect way of restricting the definition: "this strictness is not possible." Thus "with a different definition physical laws would generally change"..."the laws are true relative to the definition of congruence by means of rigid bodies."
In the 6th part he describes ways in which universal forces might be detectable, but at the same time, indistinguishable from geometry. Below is a suggested hypothetical method for this: (sorry for potato quality of picture, I took it with my phone from the chapter itself due to lack of images on google...)
The line E-D would expand when heated and overtake the point A. However, similar models for the detectability of universal forces could be constructed in other geometries, so there is no single method that could demonstrate to us the nature of our own geometry beyond all doubt. Furthermore, universal forces would make the expansion of the line E-D undetectable due to the "destroying" of "coincidences," where "all objects are assumed to be deformed in such a way that the spatial relations of adjacent bodies remain unchanged." More info about this is on p.25. Basic point is that we wouldn't be able to tell whether or not a force was universal or not in any given geometry. Conclusion: although universal forces would be indistinguishable in any given geometry, they shouldn't be regarded with suspicion as they can be nonetheless (in theory) detectable. Having postulated their existence, we must stipulate them to zero before beginning any determination of a geometrical framework.
My thoughts:
1. I am inclined to agree with some of the comments raised by Norton that were mentioned in the lecture - in trying to correct for certain deficiencies in an account for geometry, we often find correlations and agreements about a particular one, e.g. curved, variable space in our case. So although things seem to be a matter of convention they are only so in a narrow sense - the selection of a geometry is not arbitrary but a result of well thought out and coherent reasoning
2. To add to the first point, the methods used for measurement which give us such coherent reasons for selecting/determining a geometry are varied and large in number, which to me strengthens the justification of a geometrical determination
3. I also agree with Norton that the introduction of a concept of universal forces is enough to render any system or quantity conventional, thus the introduction of such a concept here does nothing of particular significance, nor does it give us an interesting difference between fact and convention to the extent that we may question/revise our methods or determination of a geometry. I'm trying to think of a good example of a system of measurement that could be rendered conventional by the introduction of universal forces...any ideas?
As far as I understand it, Reichenbach's main argument in his paper is that in order to get anywhere with geometry we need to impose a methodological rule which states/stipulates that there are no universal forces. Only when such a rule is established can we attempt to determine a geometrical framework that is not just an arbitrarily chosen system or one that is tailored to fit a preferred geometry by the compensation of universal forces. Of course, the stipulation of the rule itself makes geometric determination conventional from the outset.
So, Reichenbach's question is: how can the force be detected if the nature of the geometry may not be used as an indicator?
If the force is heat, for instance, E would be distinguishable from G, as heat affects different objects differently. Thus heat is a "differential force," and is not so problematic in the determination of a geometry.
Thus suppose the force is something universal: affecting everything in the same way, and unable to be insulated against. In this case, E might be indistinguishable from G - so long as we find a physical description of objects to fit a certain geometrical framework, (such as the presence of a universal force in a curved "humped" geometry) all potential deficiencies in our account can be compensated for.
Hence for Reichenbach we must stipulate a rule which sets all universal forces to zero, otherwise we could compensate any geometrical system with a physical account, and vice versa. This makes geometric determination a matter of convention: we have freely stipulated away universal force, so we cannot be right or wrong. Furthermore, the definitions of physical objects that we use in a particular framework will need to be "coordinative": we say that "this concept" is coordinated to "this particular thing," and that concepts are "interconnected by testable relations" (so our words coordinate with real things in the world rather than other words/concepts)
At this point he introduces the idea of relativity: "the word "relativity" is intended to express the fact that the results of the measurements depend upon the choice of the coordinative definitions." So we can define a solid body as a physical body that can be pointed out, and a rigid body as one that is unaffected by differential forces, (universal forces unconsidered). Thus our physical definitions are stipulative/subject to choice. In short, our definitions are themselves conventions. Using the new definition of a rigid body, we can therefore stop worrying about the indeterminacy of a geometry in virtue of universal forces, as those forces do not affect the rigid bodies in our framework. I therefore take him to conclude that universal forces are stipulated out in virtue of the fact that rigid bodies will be unaffected by them in a geometry according to our coordinative definition.
He spends the 5th part of the chapter explaining what it means to be a rigid body in more detail, but I will skip over most of his arguments - (details from page 20 of his paper onwards). To summarize, he suggests that there are certain restrictions to be made upon the definition of a rigid body such as "the demand that the obtained metric retain certain older physical results, especially those of the 'physics of daily life,'" or that any definition of a rigid body will depend on a prior definition of a "closed system." He concludes, however, that there is no perfect way of restricting the definition: "this strictness is not possible." Thus "with a different definition physical laws would generally change"..."the laws are true relative to the definition of congruence by means of rigid bodies."
In the 6th part he describes ways in which universal forces might be detectable, but at the same time, indistinguishable from geometry. Below is a suggested hypothetical method for this: (sorry for potato quality of picture, I took it with my phone from the chapter itself due to lack of images on google...)
My thoughts:
1. I am inclined to agree with some of the comments raised by Norton that were mentioned in the lecture - in trying to correct for certain deficiencies in an account for geometry, we often find correlations and agreements about a particular one, e.g. curved, variable space in our case. So although things seem to be a matter of convention they are only so in a narrow sense - the selection of a geometry is not arbitrary but a result of well thought out and coherent reasoning
2. To add to the first point, the methods used for measurement which give us such coherent reasons for selecting/determining a geometry are varied and large in number, which to me strengthens the justification of a geometrical determination
3. I also agree with Norton that the introduction of a concept of universal forces is enough to render any system or quantity conventional, thus the introduction of such a concept here does nothing of particular significance, nor does it give us an interesting difference between fact and convention to the extent that we may question/revise our methods or determination of a geometry. I'm trying to think of a good example of a system of measurement that could be rendered conventional by the introduction of universal forces...any ideas?
Monday, 24 November 2014
On the consistency of Euclidean geometry
In class today I pointed out that non-Euclidean geometries are logically consistent if the Euclidean one is, and it sure seems to be. Jon asked whether Hilbert hadn't in fact proven the consistency of Euclidean geometry. Not having the facts loaded up in my head, I gave a hand-waving argument that he did not: Consistency proofs are always relative to some background theory that is assumed consistent, so at most Hilbert could have given a relative consistency proof like those of non-Euclidean geometry relative to Euclidean, perhaps taking set theory as the background. Here is a more concrete answer from the Stanford Encylopedia:
"For the axioms of geometry, [Hilbert proved consistency] by providing an interpretation of the system in the real plane, and thus, the consistency of geometry is reduced to the consistency of analysis. The foundation of analysis, of course, itself requires an axiomatisation and a consistency proof. Hilbert provided such an axiomatisation in (1900b), but it became clear very quickly that the consistency of analysis faced significant difficulties, in particular because the favoured way of providing a foundation for analysis in Dedekind's work relied on dubious assumptions akin to to those that lead to the paradoxes of set theory and Russell's Paradox in Frege's foundation of arithmetic." (http://plato.stanford.edu/entries/hilbert-program)
So rather than giving a set-theoretic model, as I suggested he might have, Hilbert gave a model in analysis, but the point is the same.
One might still take issue with my blanket statement, 'Consistency proofs are always relative.' In fact, it is sometimes possible to give a purely syntactic consistency proof, as Hilbert later tried to do -- a proof, that is, that some particular system of symbolic expressions and transformation rules can never lead to an expression of the form 'P and not P'. This is easily done, for example, if neither the axioms nor the rules of inference contain the expression 'not'! But such a proof is not very meaningful unless the formal system of logic is complete, i.e., unless all logical consequences of a theory can be derived from it syntactically. (This is usually cashed out in model-theoretic terms: P is a logical consequence of T if P is true in all models where T is true.)
But even given such a syntactic consistency proof in a complete logic, we might still say that the proof is relative, for we have then reduced the question of the consistency of our theory to that of an informal theory of formalisms, i.e., of symbols and transformations of symbol strings. As Hilbert thought, such a theory might be so simple and clear as to escape any serious worry of inconsistency. But this brings us full circle, for as I intended to suggest, Euclidean geometry is already beyond any serious worry of inconsistency. It would seem that the most one can do is reduce it to other theories that seem even less questionable.
"For the axioms of geometry, [Hilbert proved consistency] by providing an interpretation of the system in the real plane, and thus, the consistency of geometry is reduced to the consistency of analysis. The foundation of analysis, of course, itself requires an axiomatisation and a consistency proof. Hilbert provided such an axiomatisation in (1900b), but it became clear very quickly that the consistency of analysis faced significant difficulties, in particular because the favoured way of providing a foundation for analysis in Dedekind's work relied on dubious assumptions akin to to those that lead to the paradoxes of set theory and Russell's Paradox in Frege's foundation of arithmetic." (http://plato.stanford.edu/entries/hilbert-program)
So rather than giving a set-theoretic model, as I suggested he might have, Hilbert gave a model in analysis, but the point is the same.
One might still take issue with my blanket statement, 'Consistency proofs are always relative.' In fact, it is sometimes possible to give a purely syntactic consistency proof, as Hilbert later tried to do -- a proof, that is, that some particular system of symbolic expressions and transformation rules can never lead to an expression of the form 'P and not P'. This is easily done, for example, if neither the axioms nor the rules of inference contain the expression 'not'! But such a proof is not very meaningful unless the formal system of logic is complete, i.e., unless all logical consequences of a theory can be derived from it syntactically. (This is usually cashed out in model-theoretic terms: P is a logical consequence of T if P is true in all models where T is true.)
But even given such a syntactic consistency proof in a complete logic, we might still say that the proof is relative, for we have then reduced the question of the consistency of our theory to that of an informal theory of formalisms, i.e., of symbols and transformations of symbol strings. As Hilbert thought, such a theory might be so simple and clear as to escape any serious worry of inconsistency. But this brings us full circle, for as I intended to suggest, Euclidean geometry is already beyond any serious worry of inconsistency. It would seem that the most one can do is reduce it to other theories that seem even less questionable.
Friday, 21 November 2014
Putnam on time and physical geometry
Hello all.
Since it's Friday night and we don't yet have a blog post on Putnam, let me just ask you all to share your thoughts.
Putnam argues that the relativity of simultaneity in special relativity resolves one of the major problems of time. It shows, he argues, that the future is just as real as the past and present. Here's a short, rough version: There are no privileged observers. So if being real is determined by some spacetime relation, it must be a relation not to me but to "an observer". Now suppose that relation is: being in the present or past of the observer. There are observers in my present and past, so they are real. Events in their present and past are also real, because such events are in the present or past of "an observer". But some of those events are in my future! So events in my future are real. And by chains of observers, all events in my future are real.
In the lecture I briefly reviewed Sklar and Stein's alternative views. In short, Sklar suggests that 'real' might be relative and intransitive, while Stein separates 'real' from 'present' -- in fact, he proves that if 'real at' *is* transitive, and something in the past light cone of an event is real at that event, then all events in that past light cone are real at that event. So Stein's (noncommittal) suggestion is that all and only those events in the past light cone of an event are real at that event, and the simultaneity hyperplane doesn't enter into it.
So what do you think? Does Putnam's argument hold up? Where exactly does it fail?
Since it's Friday night and we don't yet have a blog post on Putnam, let me just ask you all to share your thoughts.
Putnam argues that the relativity of simultaneity in special relativity resolves one of the major problems of time. It shows, he argues, that the future is just as real as the past and present. Here's a short, rough version: There are no privileged observers. So if being real is determined by some spacetime relation, it must be a relation not to me but to "an observer". Now suppose that relation is: being in the present or past of the observer. There are observers in my present and past, so they are real. Events in their present and past are also real, because such events are in the present or past of "an observer". But some of those events are in my future! So events in my future are real. And by chains of observers, all events in my future are real.
In the lecture I briefly reviewed Sklar and Stein's alternative views. In short, Sklar suggests that 'real' might be relative and intransitive, while Stein separates 'real' from 'present' -- in fact, he proves that if 'real at' *is* transitive, and something in the past light cone of an event is real at that event, then all events in that past light cone are real at that event. So Stein's (noncommittal) suggestion is that all and only those events in the past light cone of an event are real at that event, and the simultaneity hyperplane doesn't enter into it.
So what do you think? Does Putnam's argument hold up? Where exactly does it fail?
Wednesday, 5 November 2014
Werndl’s new implication of chaos for unpredictability
In this paper, `What Are the New Implications of Chaos for
Unpredictability?’, Werdnl proposes a new definition for chaos, defends it, and
uses it to show that chaotic systems (now precisely defined) have a particular unpredictability
property---the new implication of chaos for unpredictability.
Werndl spends the majority or the paper defending her
proposed definition, that a system is chaotic just when it exhibits mixing (aka strong mixing) which is precisely defined in section 3.2, equation
(4). It’s easiest to see what mixing means once you’ve made the move of
equating a physical measure defined
on a phase-space to the probability
that an arbitrary system will be in the region of phase space (given an
appropriate normalisation of the measure). Werndl makes this move without
arguing for it except for saying that `it is quite natural under certain
assumptions’. Nevertheless, with this in hand we can gloss what it means for a
system to exhibit mixing: given two regions of the phase space A, and B; let T∞A be
the region of points resulting from allowing the points in A evolve, under the dynamics of the system for a very long time;
then the probability that an arbitrary system will be in both T∞A and B is equal to the product of the probabilities that a system is in A and B respectively. Another picture of this is that after large amounts
of time-evolution, any bundle of initial conditions becomes spread out evenly
over the whole phase space (although becoming highly filamentous in order to
preserve its initial measure).
Remembering, from elementary probability theory, that the
probability of two independent events occurring is the product of the
probabilities of each of them occurring in isolation, one is quickly led to
Werndl’s new implication for predictability. Suppose you knew that a system
started out in a particular region of the phase space, and that it had evolved
for a large amount of time. Suppose,
also, that you wanted to know what the likelihood is of it being in some other
region after the time-evolution. When the system is mixing, knowing the initial
conditions is no help whatsoever for finding where the system will end up. Since
the initial phase-space bundle has been spread all around the phase-space, the
probability of it ending up in some region is just the same as the probability
of an arbitrary system being in that region. This (along with her defence of
mixing as a definition of chaos) justifies her claim that: `a general new
implication of chaos to unpredictability is that for predicting any event … all
sufficiently past events are approximately probabilistically irrelevant.’
Finally, a note on Werndl’s project of defining chaos:
In this week’s seminar we argued about what worth there was
in defining chaos or given a vague
definition of chaos what worth there is in the concept at all. Werndl defends the
adequacy of her definition using the criteria due to Brin, Stuck, and Devaney
that `(i) [a proposed definition] captures the main pretheoretic intuitions about chaos, and (ii) it is extensionally correct’ [her emphasis]. I
think she successfully shows that her definition meets both of these criteria,
but this doesn’t provide support for redefining
chaos in the first place. I say redefining as, I submit, chaos is already
defined (albeit vaguely) by its use.
Werndl’s motivation for this redefinition seems to be to
answer the question: `What are the new implications of chaos for
unpredictability?’ which, in the abstract, she suggests ought already to have
been answered based on the commonplace views of chaos-theorists (of all
flavours). In my mind this isn’t sufficient, as it would be perfectly
permissible for the new implication to be exemplified by only some chaotic
systems (under a more permissive definition) or some chaotic systems and some
systems that aren’t (under a more restrictive definition).
The new implication could equally well (if not better) be
thought of as a `new implication of strongly mixing, measure preserving,
dynamical systems for predictability’ as an implication of chaos. It seems to
me that the real motivation for the redefinition is political, in that chaos is
a more sexy name with which to describe ones field---after all Jurassic Park didn’t
have a theorist of strongly mixing measure preserving dynamical systems, but
rather a chaos theorist.
Thursday, 30 October 2014
Defining Chaos
Hi guys,
Sorry for the late post. Batterman’s paper was quite dense for me since it was full of concepts I hadn’t encountered before and so took me a while to digest them.
In this paper Batterman criticises Stone and Ford separately. In this post I am going to mention some points which came to my mind during reading his critique of Stone (I may post another note for Batterman’s critique of Ford in the next couple of days) and comparing it with lecture slides.
1- Stone introduces three criteria for deterministic system: (a) there exists an algorithm which maps a state of system at any given time to a state at any other time (not probabilistic), (b) a given state is always followed by the same history of state transitions, (c) any state of system can be described with arbitrarily small nonzero error.
Batterman says condition (c) which is known as well-posedness is equivalent to the notion of continuous dependence which says: a solution depends continuously on the initial data if convergence of initial data entails convergence of solutions. It is not transparent to me that these two are equivalent. To make it clear we first need to find out what Stone means by the vague term “describe” in condition (c). We may want to say that by this term he means 'a description of the states of system which uses algorithm and some other states'. So we may reconstruct condition (c) as: 'any state of system can be described, with arbitrarily small nonzero error, using algorithm and some other state of system'.
Let’s compare this reconstruction with condition (d) which Stone introduces for defining the notion of predictability: "any state of the system can be generated from the algorithm with arbitrarily small nonzero error from any other state of the system”. These two sentences seem very similar except for their second quantifiers.
However, it seems, condition (d) and reconstruction of condition (c) are far from precise mathematical definition of continuous dependence and non-sensitive dependence which, in the paper, are respectively taken to be equivalent to (c) and (d). The mathematical definition of continuous dependence is in the slides, and the mathematical definition of non-sensitive dependence is the following:
∀δ, ∃ε, x0, ∀x1,t such that
dist(x0 , x1) < ε → dist(φ(x0, t), φ(x1, t)) < δ
One can see that condition (d) is far from this definition, at least in terms of its details: in condition (d) there is no mention of time variable. I don’t know why Batterman doesn’t criticise Stone for this.
2- Stone claims that determinism is a necessary condition for predictability. Batterman provides an example of an indeterministic system which is predictable to falsify Stone’s claim: "one can make decent predictions about quantum systems”. My question is: Can one make decent non-probabilistic predictions about quantum systems? If not, then I doubt this does count as a counterexample for Stone’s claim as I think what Stone means by prediction is non-probabilistic prediction.
3- Stone says "systems which do not admit closed form solutions have this property that the error present in the specification of the initial state can be amplified exponentially". The problem is Stone doesn’t provide any argument for this claim and I couldn’t find any criticism about this in Batterman's paper.
4- I pondered for a while on the difference between discontinuous dependence and sensitive dependence and finally reached the conclusion that having (ε, δ)-definition of the mathematical notion of limit in mind, one can say that the difference between discontinuous and sensitive dependence is analogous to the difference between (1) a situation in which a discontinuous function like f(x) doesn’t have any limit when x approaches a number like c and (2) another situation in which the limit of a function like g(x) goes to infinity as x approaches c.
5- In the slides it was said that exponential unstable systems may have continuous dependence but Batterman says they may have non-sensitive dependence. I was wondering which one of these claims is true and how one can prove either of them by using the mathematical definition of exponential instability: dist (φ(x0,t), φ(x1,t)) ≥ (dist (x1,x0))λt
P.S: I realised that in the paper Batterman does not assert that Stone's condition (d) is equivalent to non-sensitive dependence. So I was wrong in saying that these two are taken to be equivalent throughout the paper (However he explicitly claims condition (c) and continuous dependence are equivalent). He also doesn't mention anything about the relation between this condition and continuous dependence. He only claims requirement (d) can be satisfied by exponential unstable systems. So I was also wrong in claiming that Batterman says exponential unstable systems may have non-sensitive dependence.
P.S: I realised that in the paper Batterman does not assert that Stone's condition (d) is equivalent to non-sensitive dependence. So I was wrong in saying that these two are taken to be equivalent throughout the paper (However he explicitly claims condition (c) and continuous dependence are equivalent). He also doesn't mention anything about the relation between this condition and continuous dependence. He only claims requirement (d) can be satisfied by exponential unstable systems. So I was also wrong in claiming that Batterman says exponential unstable systems may have non-sensitive dependence.
Tuesday, 28 October 2014
Dainton, Teller, and the construction of neo-Newtonian spacetime
I told the class last week that I would
look again at Dainton’s discussion of the order in which neo-Newtonian
spacetime is constructed and the charge of emptiness against that
construct. I forgot to mention it in
class this week, so here are my thoughts.
I still think that it is not literally the
order in which the elements or components of neo-Newtonian spacetime are
constructed that is really at issue here.
Paul Teller argues that the inertial frames and trajectories of
neo-Newtonian spacetime are distinguished only by their lack of inertial
effects (effects like the tension on the cord between Newton’s globes, the
tendency of the water in Newton’s bucket to rise up the sides of the bucket,
and the feelings of pressure, weight, and perhaps queasiness that we feel in a
rapidly rising airliner). Since it is
(supposedly) only on the basis of lacking such effects that we pick out the
inertial trajectories, Teller claims that we are not really explaining the lack
of such effects (or their presence in other cases) by pointing out that those
in which the effects do not occur are inertial.
The suggestion seems to be that we are in effect saying nothing more
than ‘there are no inertial effects associated with these trajectories because
there are no inertial effects associated with them’.
Perhaps this can be understood as a matter
of order: We first observe that certain
trajectories or frames are not associated with inertial effects, and we then
label these the inertial frames/trajectories.
But what really matters in Teller’s argument is not the order but the basis, i.e., the reason for
distinguishing the trajectories in question.
In principle, we might have called them inertial first, and then
justified this later in terms of the lack of inertial effects.
In any case, I think Teller and Dainton
have both overlooked something important here.
It is not as if just any old trajectory counts as inertial just because
of an observed lack of inertial effects.
In fact, the inertial trajectories all have a special relation to one
another: They all exhibit uniform
relative velocity to each other. I think
this suggests that when we say that these trajectories are not associated with
inertial effects because they are inertial trajectories, we are not just
blurting out an empty tautology, but identifying something special about a
particular class of trajectories, something that is plausibly due to the
particular structure of the spacetime they inhabit and their relation to that
structure.
Announcement: I am about to post a corrected version of the differential equations "handout". The arguments in Equation (2) were reversed. Thanks to Somayeh for catching that.
Saturday, 25 October 2014
On the Notion of Cause 2
Hi again!
So rather than posting another long summation I’m
just going to point out a 5 interesting points about Russell’s meaning of cause
and effect.
- - The idea of a time-constant, which Russell
considers essential, is in fact dependent on multiple assumptions as to the
meaning of both cause and effect. Does Russell go far enough in his explanation
to be able to justify this?
- The sentence “What is essentially the same statement of the law of causation” isn’t allowed, why is Russell allowed to get away with that??? He builds the second half of his critique off this assumption (making some selective interpretations, something he then attacks Bergson for).
- Does the fact that an ‘event’ is only likely to occur once have any relevance to the possibility of a universal law of causality? For me it’s a separate point.
- “I deny is that science assumes the existence of invariable uniformities of sequence of this kind, or that it aims at discovering them”. This encapsulates Russell’s view on the matter, and his whole argument is based on this belief.
- “The principle "same cause, same effect," which philosophers imagine to be vital to science, is therefore utterly otiose.” I don’t think that Russell’s points really prove this. Surely he is only pointing out problems with this assumption but I don’t he think he goes as far as invalidating it.
- The sentence “What is essentially the same statement of the law of causation” isn’t allowed, why is Russell allowed to get away with that??? He builds the second half of his critique off this assumption (making some selective interpretations, something he then attacks Bergson for).
- Does the fact that an ‘event’ is only likely to occur once have any relevance to the possibility of a universal law of causality? For me it’s a separate point.
- “I deny is that science assumes the existence of invariable uniformities of sequence of this kind, or that it aims at discovering them”. This encapsulates Russell’s view on the matter, and his whole argument is based on this belief.
- “The principle "same cause, same effect," which philosophers imagine to be vital to science, is therefore utterly otiose.” I don’t think that Russell’s points really prove this. Surely he is only pointing out problems with this assumption but I don’t he think he goes as far as invalidating it.
Hope these statements are suitably provocative!!!
Thursday, 23 October 2014
On The Notion Of Cause
Hi everybody! Sorry for the late post!
On the notion of cause
In this paper Russell discusses the notion of cause as used
in science and philosophy. He states that his three aims as the following,
1. To discuss the flaws in the use of the term ‘cause’ in a scientific context
2. Suggest a more appropriate alternative to the ‘law of causality’
3. Explore some of the problems in theology and determinism that result from the improper use of ‘cause’
This first post will discuss the first of these points as
the others depend on the assumption of the first.
The cause of an event is often
considered to be a crucial part of scientific inquiry, with the likes of Ward
suggesting that causes are indeed the very ‘business of science’. In this piece
Russell contends that not only are ‘causes’ not the subject of advanced sciences
but that in reality there is no such thing. He first questions the
appropriateness of the word cause by using the then
available definitions of the word, taken from Baldwin’s dictionary. These are
as follows,
(1)
CAUSALITY
- The necessary connection of events in the time-series
(2)
CAUSE (notion of) - Whatever may be included in
the thought or perception of a process as taking place in consequence of
another process
(3)
CAUSE AND EFFECT - Cause and effect are
correlative terms denoting any two distinguishable things, phases, or aspects
of reality, which are so related to each other, that whenever the first ceases
to exist, the second comes into existence immediately after, and whenever the
second comes into existence, the first has ceased to exist immediately before
Russell discusses the appropriateness of each of these in
turn (and at length) so let us begin with the first.
Causality:
From the statement it is clear that the meaning of causality is
dependent on the meaning of necessary. Hence to fully understand the
implications of cause it is first required that we determine the meaning of
necessary. Necessary is defined as follows,
NECESSARY - That is necessary which not only is true, but
would be true under all circumstances. Some-thing more than brute compulsion
is, therefore, involved in the conception; there is a general law under which
the thing, takes place
Russell points out several problems with this statement. To
understand Russell’s critique it is required that we first understand his ideas
of a proposition and a propositional functions.
Proposition – A statement which is either true or false, no
other considerations can apply
Propositional Function – A statement containing a variable,
which is called the argument. Assigning a value to the variable creates a
proposition (the statement must then be either true or false).
[I’m curious as to
whether these were well established concepts or are they of Russell’s own
creation. This would seem to have a significant impact on the validity of his
argument, given that it relies on Baldwin making an informed inclusion of the
terms in his definition.]
With these definitions in mind let us re-examine the
definition of necessary. According to Baldwin for something to be necessary it
must be ‘true in all circumstances’. This implies that the subject must
therefore be a propositional function rather than a proposition. The example
given is that ‘if x is a man, then x is mortal’. As all men are mortal any
value for x which satisfies the first parameter must therefore satisfy the
second. Russell suggests an alternative definition for necessary on this basis,
NECESSARY – “is a predicate of a propositional function,
meaning that it is true for all possible values of its argument or arguments.”
However Baldwin’s definition states that for something to be
necessary it must also be true. This is impossible to rectify with Russell’s
ideas of propositions and propositional functions (which are mutually
exclusive) as the definition requires that a statement be both at the same time
(impossible because whilst a statement can become a proposition from a
propositional function it could not under Russell’s definition be both at the
same time).
[My personal thoughts
on this are that Russell is being ‘unnecessarily’ literal with Baldwin’s
definition. Whilst I accept the point that there is a conflict, it would appear
more obvious (and convenient!) to dismiss Baldwin’s double use of true as mere
emphasis. Dismissing the either, allows the definitions of causality and
necessary to fit.]
Russell resolves this conflict by requiring the argument of
the function to be identified within the statement. He therefore arrives at the
following definition which satisfies the implied meaning of causality,
NECESSARY - A proposition is necessary with respect to a
given constituent if it remains true when that constituent is altered in any
way compatible with the proposition remaining significant.
Armed with this definition Russell gives what he interprets
to be the universal law of causality (whilst making 100% sure he is not tied to
its authenticity!)
Cause
(Notion of):
Russell doesn't really spend much time discussion the 2nd
definition as it does not really concern the meaning of cause which is of
interest to him (so I won’t either).
Cause and
Effect:
Definition 3 appears to be the closet to what is
traditionally meant by causality in my opinion (or at least when it is used in
reference to science). The problem with the statement according to Russell is
the implied temporal contiguity. Treating the current and prior states as
distinct entities introduces the requirement for causal relations between the
two.
[The implications of this get a bit wordy but I’ll do my
best to simplify what I think Russell is trying to say.]
The definition seems to describe cause and effect as having
an associated finite time (not really instantaneous). Hence within the duration of the cause there can be said
to be earlier and later parts. If this is taken to be true then only the later
stages of the cause, those immediately preceding the effect, can be truly relevant
to the effect. The earlier parts of the cause are not contiguous to the effect
and so could therefore be altered without changing the effect. Really you’re
supposed to think that the effect is only dependent on a certain final part of
the cause.
This is clearly a bit ridiculous, and does not fit what
cause is supposed to be (at least in my opinion). Russell agrees and says it is
hardly acceptable that the effect should just spring from a cause at some
point. Cause and effect cannot therefore be contiguous in time. Russell then
introduces the time-interval to solve this.
I reckon this is enough for a first blog as it covers
Russell’s critique of Baldwin’s definitions. I shall put a second post up
tomorrow which will discuss the proposed solution to the 3rd
definition.
Friday, 17 October 2014
Maudlin and the spatiotemporal shifts
Hello all,
I've limited the scope of this post to Maudlin's
conception of Newtonian substantivalism and how it copes with the static and
kinetic spatiotemporal shift arguments. I had difficulty grasping the content
of sct. 4, and as such I think it would be more productive to focus on sct. 3,
which is relatively straight forward but yields some interesting questions and
responses. I'm not entirely sure of the best way to structure this, so I will
briefly summarise the text, highlighting any questions or concerns that
occurred to me.
Maudlin offers a view on the classical debate of
which holds that the key issue under consideration, the fact that Newtonian
absolute space has a "metaphysically distasteful" character of
ontologically distinct but empirically identical circumstances, is brought
about by a fundamental misunderstanding and subsequent misuse of the arguments.
He claims that the weight of Leibniz' spatiotemporal shift arguments is down to
a fallacious equivocation, and that upon investigation into their structures,
one finds that they are essentially different, and therefore cannot be
legitimately appealed to.
The arguments in brief: In a substantivalist
universe, where spatiotemporal properties of objects are defined absolutely in
relation with absolute space, and defined locally in relation with one another,
a static shift in time and/or space would essentially be unnoticeable, since
every observation made before and after the shift would be identical.
Similarly, a kinematic shift, in which the universe in its entirety would
undergo motion with repeat to absolute space, would also be unnoticeable for
the same reason. If the relations between bodies would not change, the
observations made within such a shifted universe would still hold as they had
before, and therefore the two universe pre-and-post-shift would be
indiscernible from one another. Following the Principle of the Identity of
Indescernibles, the two universes would be identical. "In short, both the
static and kinematic shifts (...) would result in ontologically distinct but observationally
indistinguishable states of affairs."
Maudlin goes on to expose the failure of the argument, which rests in an
“illusions engendered by imprecision concerning the notion of observationally
indistinguishable states of affairs.” The key here is that what is
observationally indistinguishable in the case of the kinematic shift is not
necessarily so in the case of the static shift. Essentially, within Newtonian
space-time, observation is possible if there is a change in spatiotemporal
property with respect to anything other than absolute space, so if two shifted
universe states differ only with respect to absolute space, the difference will
not be observable, and therefore the states will be observationally
indistinguishable. What this seems to come down to is that in the case of a
kinematic shift, we can say that we part of the actual world, but there are
certain aspects of it that are unknowable.
Similarly, in the case of the static shift, there will be no way to
observe certain spatiotemporal properties that would help us identify which
state of affairs we are part of. Maudlin then poses the following question: if
the entire universe were to shift statically in space or time, what could we
say about the actual state of the universe? In the kinematic case, he says,
physical questions about the actual nature of the universe can be asked but not
necessarily answered, whereas in the static case they cannot even be asked in
the first place without indexically picking out a spatiotemporal location, and
identifying the occupant of that location is a physical question that can be
answered.
Maudlin provides an example for this which amounts to the claim that it
knowledge of the spatiotemporal locations of things is presupposed in referring
to them. Were the universe shifted statically 15 billion years into the future,
he says, it is possible that someone just like him would be sitting in the same
place writing the same words, however the indexicals used in his writing would
ensure that the referents of his utterances would not be the same. In that
sense, it is an error on the part of Leibniz to assume that a static shift
would be undetectable; not because it could be observed, but because the
observer would no longer be around to observe it.
I agree with this to some degree in the case of a temporal shift, but I
am not convinced the same applies in the spatial shift. It is possible that the
entire universe from my point of view could shift spatially billions of
lightyears to the left, but if the relative distances and positions between the
occupants of the universe do not change, that is, were their positions to
change only with respect to “absolute” space, then the spatial location
contained in the meaning of my utterances of “here” or “there” would also have
to change, and there would be no observation possible that would make me aware
of that. In other words, it is true that “here” means two different things
pre-and-post-shift, but it seems dubious that my knowledge of that should track
the shift, as it were. In Maudlin’s words, “We can (…) formulate meaningful
counterfactuals about worlds where everything would be displaced from its
actual location, but we can also be assured that they are counterfactuals.”
Where does this assurance come from? It seems to me that taking spatiotemporal
properties as counterfactuals presupposes that the actual state can be
discerned through observation, which is precisely what remains to be shown.
Where Maudlin loses me is in his conception of what a static shift would
be like, so to close this post I would like to examine the ways in which this
is conceivable. The point of the static shift argument is to show that in
absolute space, a spatiotemporal change of position of the entirety of the
universe, that is, everything that exists within it, with respect to absolute
space, would be unnoticeable, and that therefore a view of space-time which
holds on to absolute space but admit the possibility of an infinity of possible
universes. But if we ignore PII (which, as Maudlin notes, is no longer taken
seriously), it seems that this is not a problematic consequence.
Wednesday, 15 October 2014
Questions and answers about neo-Newtonian spacetime.
Hi all,
Emma emailed me some questions about neo-Newtonian spacetime, and I thought it would be helpful to post my answers here. Please feel free to add remarks or follow-up questions in the comments.
--MWP
----------------
1. When you were going through the list of things that might be permitted under neo-Newtonian spacetime, I understood all of the answers except for the last one - if your spaceship is moving 100m/s relative to my spaceship why is that permitted if objects do not have spatial relations through time? Is it because we can manipulate the hyperplanes so that my spaceship could also be moving 100m/s relative to your spaceship? If so, I'm still not sure how 'm/s' is a permissible unit of measurement? To me, in neo-Newtonian spacetime 'metres-per-second' are forbidden or nonsensical.
Answer: It's not totally obvious, but we can reconstruct relative speed in neo-Newtonian spacetime. We have simultanaeity, distance at a time, and elapsed time. So two bodies (spaceships, say) have a distance at one time and another distance at another time. Lets say they have distance d at time t and distance d' at time t'. Then over the time interval from t to t' they have an average speed relative to each other, which is just the difference in distance divided by the difference in time: (d' - d) / (t' - t). So in neo-Newtonian spacetime we can reconstruct average relative speeds between bodies.
Notice that we get this "average" without adding up a bunch of values and then dividing by the number of values. We just take the ratio of the final differences. In fact, instantaneous speeds are derived from the average speeds by looking at averages over tiny time intervals. So we can also reconstruct instantaneous relative speeds in neo-Newtonian spacetime. We just don't have absolute speeds. Measuring the distance between the spaceships at different times won't tell us which one is moving or which one is moving faster.
2. Is space still absolute in neo-newtonian spacetime? If so, why can't objects move absolutely? I'm just picking up on your line: "Neo-Newtonian space time does not distinguish between absolute velocities." I think I have misunderstood the sentence. Does it mean that an object can move absolutely, but since we have no unit of measurement for velocity in the neonewtonian spacetime, there can be no such thing as a 'difference' in velocity between one object's worldline and another's'?
Answer: In neo-Newtonian spacetime, space is still absolute in the sense that, within each instant (each spatial hyperplane) there are different places. There is still an intrinsic difference between being here and being there. But objects cannot move absolutely because we have here no concept of being in the same place over time. Motion is change of place over time, and that makes no sense here. There is no rule (or even an unruly map) to identify a point in space now with a point in space at another time. So the statement "Neo-Newtonian spacetime does not distinguish between absolute velocities" means that, in neo-Newtonian spacetime, bodies don't even have absolute velocities, and there is not even a distinction between rest and uniform motion. Quite aside from the matter of stipulating a unit of measure for velocity, we can't even distinguish between some velocity and none at all. But, as explained above, there is such a thing as a difference in velocity, or a relative velocity. A body A has a definite relative velocity to any body B, but neither A nor B has a definite velocity on its own.
3. (continuation from 2) I think my problem is just trying to conceptualise the scenario: if absolute acceleration is permitted, but absolute velocity is not, what is the line moving through as it goes between hyperplanes, if not spatial distance? If I walk from here to the door in a neo-Newtonian spacetime, and am accelerating, my world line will curve upwards through various hyperplanes, yet at the same time, my worldline has not travelled any 'distance'. So here's my attempt to understand it: my worldline is moving through the hyperplanes as if through a series of pictures - so it's more like a dot-to-dot through fixed moments, than a representation of any smooth transition through time resulting in a spatial relation. Is that it?
Answer: No, a smooth transition is the right picture. Between any two hyperplanes there are (in the neo-Newtonian model) infinitely many more hyperplanes, forming a smooth continuum. As we pass from one hyperplane to another, we are always in some hyperplane. And I think you should think of a curved worldline as representing motion through spatial distance. It's just that you can't pick out any particular part of the curve as being nearly at rest or as being in very fast motion. There is no specific distance between temporally separate events, but there are relations between temporally separate events. The permissible alignments of spatial hyperplanes is constrained by the fact that all inertial worldlines must remain straight. So you might say there is sort of a loose relation of distance over time, even though it is not fully pinned down. In that sense, bodies do move through space, just not at any particular speed.
On the other hand, it might be better to think of bodies as moving through spacetime rather than through space. The curve traced by a body moves from sheet to sheet in a particular way, even though the distances between points on different sheets are undefined. It is not moving left-to-right or east-to-west, but forward in spacetime, in a manner that traces either a straight or a curved path.
I hope that helps, but I think that, with this question, we are dealing with cognitive discomfort more than questions of fact. You may have to just relinquish certain habits of thought.
Emma emailed me some questions about neo-Newtonian spacetime, and I thought it would be helpful to post my answers here. Please feel free to add remarks or follow-up questions in the comments.
--MWP
----------------
1. When you were going through the list of things that might be permitted under neo-Newtonian spacetime, I understood all of the answers except for the last one - if your spaceship is moving 100m/s relative to my spaceship why is that permitted if objects do not have spatial relations through time? Is it because we can manipulate the hyperplanes so that my spaceship could also be moving 100m/s relative to your spaceship? If so, I'm still not sure how 'm/s' is a permissible unit of measurement? To me, in neo-Newtonian spacetime 'metres-per-second' are forbidden or nonsensical.
Answer: It's not totally obvious, but we can reconstruct relative speed in neo-Newtonian spacetime. We have simultanaeity, distance at a time, and elapsed time. So two bodies (spaceships, say) have a distance at one time and another distance at another time. Lets say they have distance d at time t and distance d' at time t'. Then over the time interval from t to t' they have an average speed relative to each other, which is just the difference in distance divided by the difference in time: (d' - d) / (t' - t). So in neo-Newtonian spacetime we can reconstruct average relative speeds between bodies.
Notice that we get this "average" without adding up a bunch of values and then dividing by the number of values. We just take the ratio of the final differences. In fact, instantaneous speeds are derived from the average speeds by looking at averages over tiny time intervals. So we can also reconstruct instantaneous relative speeds in neo-Newtonian spacetime. We just don't have absolute speeds. Measuring the distance between the spaceships at different times won't tell us which one is moving or which one is moving faster.
2. Is space still absolute in neo-newtonian spacetime? If so, why can't objects move absolutely? I'm just picking up on your line: "Neo-Newtonian space time does not distinguish between absolute velocities." I think I have misunderstood the sentence. Does it mean that an object can move absolutely, but since we have no unit of measurement for velocity in the neonewtonian spacetime, there can be no such thing as a 'difference' in velocity between one object's worldline and another's'?
Answer: In neo-Newtonian spacetime, space is still absolute in the sense that, within each instant (each spatial hyperplane) there are different places. There is still an intrinsic difference between being here and being there. But objects cannot move absolutely because we have here no concept of being in the same place over time. Motion is change of place over time, and that makes no sense here. There is no rule (or even an unruly map) to identify a point in space now with a point in space at another time. So the statement "Neo-Newtonian spacetime does not distinguish between absolute velocities" means that, in neo-Newtonian spacetime, bodies don't even have absolute velocities, and there is not even a distinction between rest and uniform motion. Quite aside from the matter of stipulating a unit of measure for velocity, we can't even distinguish between some velocity and none at all. But, as explained above, there is such a thing as a difference in velocity, or a relative velocity. A body A has a definite relative velocity to any body B, but neither A nor B has a definite velocity on its own.
3. (continuation from 2) I think my problem is just trying to conceptualise the scenario: if absolute acceleration is permitted, but absolute velocity is not, what is the line moving through as it goes between hyperplanes, if not spatial distance? If I walk from here to the door in a neo-Newtonian spacetime, and am accelerating, my world line will curve upwards through various hyperplanes, yet at the same time, my worldline has not travelled any 'distance'. So here's my attempt to understand it: my worldline is moving through the hyperplanes as if through a series of pictures - so it's more like a dot-to-dot through fixed moments, than a representation of any smooth transition through time resulting in a spatial relation. Is that it?
Answer: No, a smooth transition is the right picture. Between any two hyperplanes there are (in the neo-Newtonian model) infinitely many more hyperplanes, forming a smooth continuum. As we pass from one hyperplane to another, we are always in some hyperplane. And I think you should think of a curved worldline as representing motion through spatial distance. It's just that you can't pick out any particular part of the curve as being nearly at rest or as being in very fast motion. There is no specific distance between temporally separate events, but there are relations between temporally separate events. The permissible alignments of spatial hyperplanes is constrained by the fact that all inertial worldlines must remain straight. So you might say there is sort of a loose relation of distance over time, even though it is not fully pinned down. In that sense, bodies do move through space, just not at any particular speed.
On the other hand, it might be better to think of bodies as moving through spacetime rather than through space. The curve traced by a body moves from sheet to sheet in a particular way, even though the distances between points on different sheets are undefined. It is not moving left-to-right or east-to-west, but forward in spacetime, in a manner that traces either a straight or a curved path.
I hope that helps, but I think that, with this question, we are dealing with cognitive discomfort more than questions of fact. You may have to just relinquish certain habits of thought.
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