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?
As a candidate for a universal force conventionalising something other than geometry: perhaps Lorentz-Fitzgerald contraction? The analogy would go something like: a universal force could render a space with one geometry observationally indistinguishable from one with no universal force but a different geometry, whilst Lorentz-Fitzgerald contraction renders a universe with a luminiferous ether observationally indistinguishable from one in which Lorentz-Fitzgerald contraction does not occur but there is no luminiferous ether.
ReplyDeleteRE Norton's response to Reichnbach:
I'm slightly skeptical of the claim that the conventionalism afforded by the possibility of universal forces is uninteresting. It is conceivable to me that some theoretician might, through postulating a universal force and a corresponding change in geometry make some kind of interesting progress. This might come through solving a problem previously intractable, discovering some new property of GR, or noticing an avenue for inquiry beyond GR. Perhaps the first two (and maybe even the third) of these things could have been done without resorting to a universal force, but if they are even aided by thinking in terms of a universal force then I think that makes the idea certainly an interesting one.
In fact I may be being a bit unfair to Norton here who really said that ``If we persist in allowing universal forces in our analysis, then we eliminate any interesting distinction between the conventional and the factual''. So I have merely suggested that Reichenbach's work could be fruitful, but not responded to Norton's ideas on conventionality. Here though I'm not entirely sure what Norton thinks: it seems that he thinks that universal forces are uninteresting because they lead to radical and complete underdetermination, but if this isn't an interesting sense of conventionality then what is? maybe Norton thinks that there is no interesting distinction between convention and fact, but then this isn't a critique of Reichenbach per se. I think I would have to see more about what Norton's opinion on conventionality in general is to evaluate his position.
I was thinking about Poincare’s claim that “geometrical claims are not experimental facts”. This means that ‘geometrical claims cannot be confirmed or falsified by experiment’. Let’s consider Poincare’s disk and the claim that “the length of its diameter is infinite". We want to check whether this claim is confirmed or falsified by experiment.
ReplyDeleteG: The length of the diameter of Poincare’s disk is infinite.
A1: The length of a line is the number of rods which can fit into that line.
————
Conclusion: An infinite number of "rods" can fit into the diameter of Poincare’s disk.
Observation 1: An infinite number of "temperature-sensitive rods" can fit into the diameter of Poincare’s disk.
Observation 2: Only finite number of "non-temp-sens rods" can fit into the diameter of Poincare’s disk.
Let’s compare our observational statements with our conclusion of the hypothesis G. In G we have the term “rod” while in our observational statements, we have the terms "temp-sens-rods” and “non-temp-sens rods”. Unless we determine the relation between these terms, we cannot judge whether our observations confirm or falsify G.
One can claim that by the term “rods” in the conclusion (and A1), one means ideal rods which are unaffected by any force. In this sense, the term “rods” is equivalent to “non-temp-send rods”. If we accept this interpretation of the term “rods” then observation 2 falsifies G.
Another interpretation of the term “rods” is “all the rods” i.e the union of “temp-sens rods” and “non-temp-send rods”. This interpretation also leads to the falsification of G. Because it suffices that there are a number of rods for which the conclusion doesn’t hold.
The third interpretation, which I reckon may be the one Poincare had in mind, is that what the term "rods" means, is a matter of choice. If we take it to mean "temp-sens rods” then observation 1 confirms G by HD method. Otherwise, observation 2 falsifies it. That’s why Poincare says the fate of G is a matter of convention.
Now let’s check if Einstein’s assertion that G+P (where P is some physics’ theories) can be tested by experiment.
P: The number of rods which fit into a certain line changes by temperature.
A2: The length of a line is fixed.
P+A2—> A1'
A1’: The length of a line is the number of non-temp sense rods that can fit into that line.
So the magic P does is to fix the meaning of the term “rods” in A1. Now having A1’ at hand we can reconstruct the first deduction as follows:
G: The length of the diameter of Poincare’s disk is infinite
A1’: The length of a line is the number of non-temp sense rods which can fit into that line.
——————
Conclusion: An infinite number of “non-temp-send rods" can fit into the diameter of Poincare’s disk.
One can see that this conclusion can be easily falsified by observation 2 and so G+P is a set of propositions that can be empirically tested.
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DeleteI am slightly confused about Norton's criticism. What Reichenbach claims is not that the definition involved in the specification of geometry of space is interesting or helpful. It is simply necessary and unavoidable to do that in order to be able to determine the geometry empirically. Any empirical test would not unambiguously detect a geometry for there could always be universal forces involved.
ReplyDeleteNow on the one hand Norton seems to suggest that it is not very interesting to distinguish these two factors, on the other hand he seems to think there are indeed empirical tests that lead the way to one space geometry without such definitions.
Wouldn't Reichenbach's answer to that simply be that this is not very different from what he suggests? We ignore the fact that we define universal forces to be equal to zero and we can work with and postulate a certain curved space geometry but we shouldn't claim to have found the true one completely empirically. But that is perhaps a mistake Norton would make.
Also, my reconstruction of Reichenbach's argument goes like this:
Delete(1) Geometry shows that knowledge of the actual geometry of our physical world an empirical matter of physics.
(2) Empirical experiments cannot reveal the true geometry without a convention with regard to the existence of universal forces.
(3) Therefore, our view of the actual geometry of our world is conventional, i.e. dependent to our definition with respect to the existence of universal forces.
The rest of the chapters is a more detailed analysis of what kind of definition this is, how it involves concepts regarding how our measuring devises behave (solid and rigid bodies) and how we should choose the definition such that we make things as easy as possible and compatible with the physics we have.