HPS 0410 | Einstein for Everyone |

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What does it all mean?

The Conventionality of Simultaneity

John
D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

This chapter continues the discussion of the last chapters on morals that we might try to draw from special relativity. It develops one candidate moral that combines a general idea about theories and also a notion specific to time. The general idea is that some elements in a theory are chosen conventionally. That means that they do not reflect something real in the world, but merely our choices in how we describe the world. The notion specific to time is that judgments of simultaneity even in one inertial reference frame are just such a convention.

**We learn from special relativity that judgments of
simultaneity are not just dependent on the inertial frame. Within a
single inertial frame, the judgment of which spatially separated events
are simultaneous is a matter of convention; that is, we are free to
stipulate it as we like.**

The relativity of simultaneity is one of the central results of the special theory of relativity. We have analyzed it here at length. There is a second thesis concerning simultaneity deriving from special relativity, which we have barely discussed so far. It is the thesis of the conventionality of simultaneity. The two are different and should not be confused.

The relativity of simultaneity asserts that judgments of the simultaneity of distant events must change as we move between inertial frames of reference. However it presumes that within a single inertial frame there is one correct judgment to be made. | The conventionality of simultaneity pertains to judgments of simultaneity of distant events in just one inertial frame. In asserts that there is no single, correct judgment of simultaneity. Rather, in each inertial frame, we have broad freedom in assigning simultaneity to pairs of events. In the same frame of reference, one person may assign relations of simultaneity one way; another person may do it differently. Within some limits, neither is factually wrong, according to the conventionality thesis, for there is no unique fact of simultaneity in the world. |

The table summarizes the difference.

Relativity of Simultaneity | Conventionality of Simultaneity | |||

Inertial frame 1 | Unique relation of simultaneity 1 | Inertial frame 1 | Many candidate relations of simultaneity 1a, 1b, 1c, ... | |

Inertial frame 2 | Unique relation of simultaneity 2 | Inertial frame 2 | Many candidate relations of simultaneity 2a, 2b, 2c, ... | |

... | ... | ... | ... |

The term convention in the thesis is intended to convey the idea that the selection is freely chosen. That many people make the same choice is no reflection of a factual state of affairs.

Think of the conventions concerning the side of the road upon which we drive. In the US and most of Europe, we drive on the right hand side of the road. In Great Britain and many Commonwealth countries, we drive on the left hand side of the road. Neither choice is factually correct; there is no cosmically pre-ordained, correct side of the road upon which to drive. We have just agreed in some countries to drive on one side of the road; and in others we agree to drive on the opposite side.

The thesis of the conventionality of simultaneity asserts that the selection of just which events are simultaneous, even in just one inertial frame of reference, is a convention in the same sense.

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Image:
http://fr.wikipedia.org/wiki/Fichier:Beijing_traffic_jam.JPG

The argument for the conventionality of simultaneity draws directly on Einstein's analysis of simultaneity in his famous 1905 "On the Electrodynamics of Moving Bodies." In the first section of the paper, Einstein introduces the relativity of simultaneity. To do it, he considers two clocks, an A clock and a B clock at different places in space and at rest in the same inertial system.

He then asks how can we synchronize the two clocks, so that the A-time of clock A agrees with the B-time of clock B. That is, how can we know that we have set the clocks in proper synchrony so that clock A ticks 0, 1, 2, ... and clock B ticks 0, 1, 2, ... at exactly the same moment.

Einstein plans to synchronize the clocks by means of light signals exchanged between the clocks. At this crucial moment in the development, Einstein asserts that there is no factually correct way to set the clocks. We set them by introducing a definition:

"So
far we have defined only an 'A-time' and a 'B-time,' but not a common
'time' for A and B. The latter can now be determined by establishing *by
**definition* that the 'time' required for light to travel
from A to B is equal to the 'time' it requires to travel from B to A."

To underscore that he is using a definition, Einstein has emphasized the words "by definition" and he calls the section "The Definition of Simultaneity."

The definition works in the familiar
way. A light signal is sent from the A clock to the B clock. When it
arrives at the B clock, it is immediately reflected back to the A
clock. The definition requires that the two trips,
forward and return, take the same time. That means that the
reflection at the B clock is simultaneous with an event temporally
half way between the emission and return of the light signal from
the A clock. Hence the clocks are set to the same time for these two
events. This synchronization is shown in the figure. A light signal leaves the A-clock at A-time 0 and returns at A-time 4. Hence the reflection occurs at A-time 2. The clock at B is set to B-time 2 for this reflection event. If Einstein intends that this procedure implements a definition of simultaneity, then it must be the case that other definitions are possible. That is, it is chosen conventionally. It is not a discovery, for then only one simultaneity relation can be found. For example, we define an ordinary or statute mile as 5,280 feet. That is a definition, since others are possible. There is an alternative nautical mile, which turns out to be around 6,076.1 feet. In contrast, the height of Mount Everest is 29,029 feet. It is discovered. We cannot choose a different value. |

The claim of a convention is clear. However Einstein has offered rather little argumentation in its favor. Rather he mostly declares that the rule used to synchronize the clocks is a definition. He does not show that no other definitions are possible. | Einstein considers only one other possibility, that the A-clock sends out a signal at A-time 0 and that the B-clock and all other others in space set themselves to 0 when that signal arrives. The flaw in this arrangement, he notes, is that it treats the A-clock as somehow special, making its time control everyone else's. |

Einstein himself seems to have given little more attention to the notion that the simultaneity of distant events must be introduced as a definition. While he gave many later accounts of special relativity, they rarely mention the need for a definition. The important exception is his 1916 popular book on relativity theory. (Relativity: The Special and the General Theory.) In Part I, Section 8, he elaborates. One might expect that no other definition is possible since all Einstein's definition amounts to is this: light takes the same time to travel from A to B as from B to A. That is, the speed of light in the direction AB is the same as in the direction BA.

Einstein responds that this equality of times or equality of speeds cannot be known factually, for we need to have clocks already synchronized at different places to know what is the speed of light when light travels from one place to another. He concludes:

"It would thus appear as though we were moving here in a logical circle."

From the short treatment given by Einstein, it is far from clear that there really is no definite fact as to which events are simultaneous within one inertial frame. It is also far from clear that we must always be trapped in the sort of logical circle Einstein sketched. Can we really use other definitions?

Two elaborations supported the conventionality thesis, Reichenbach's causal theory of time and his ε.

An early appeal of the conventionality thesis was that it
appeared to conform especially well with Reichenbach's causal
theory of time, discussed above. It asserts
that temporal relations simply are nothing more that causal relations: For
event E_{2} to be later than event E_{1} just means that a
process at E_{1} can causally affect a process at E_{2}.
Reichenbach understood relativity theory to assert that no real
propagation goes faster than light; light is what he called a "first
signal."

Returning to Einstein's clocks A and B , it follows that none of the events at clock A between the emission of the light signal and its return can causally affect what happens at clock B at the event of the reflection of the light signal. For any such causal influence, in either direction, would need to propagate faster than light.

To complete the analysis, we merely need to understand simultaneity to consist in failure of possible causal interactions. Then all the events at A between the emission and return of the light signal could be deemed simultaneous with the reflection of the signal at the B. We are free to choose which we declare simultaneous; any choice is as good as any other, just as the conventionality thesis asserts.

In brief, the causal structure of the Minkowski spacetime, that is, the catalog of which events can causally affect which others, leaves the relation of simultaneity incompletely determined. That incompleteness is the conventionality of simultaneity. It results from the finiteness of the speed of causal propagation, which is limited to the speed of light.

The possibility of using other synchrony conventions was explored by Hans Reichenbach, Adolf Gruenbaum and others at some length. They looked at what happens if we try to use another definition. Instead of selecting a time half way between the emission and return of Einstein's 1905 procedure, what if we choose one that is, say, ε ("epsilon") fraction along, where ε is just some number we pick between 0 and 1. The figure shows the difference between the standard synchrony of ε=1/2 chosen by Einstein (on the left) as his definition and another choice of ε=1/4 (on the right). | Adolf Gruenbaum http://www.pitt.edu/~pittcntr/Images/Donuts/2007-08/grunbaum_11-30-07/grunbaum19_hi-res.jpg |

ε = 1/2 |
ε = 1/4 |

The figures above show how the standard and non-standard judgments of simultaneity propagate through the spacetime. Standard ε=1/2 synchrony gives us the familiar hypersurfaces of simultaneity. Non-standard ε=1/4 synchrony gives us the tilted hypersurfaces shown.

These figures above show how different conventions lead to differences in judgments of simultaneity. The figures look something like those that represent the relativity of simultaneity since the effect involves a tilting of hypersurfaces of simultaneity. The similarity is misleading. As has been noted above, the conventionality of simultaneity is not the relativity of simultaneity. The conventionality thesis pertains to just a single frame of reference. The relativity of simultaneity arises when we shift between inertial frames of reference. The figure at left illustrates the relativity of simultaneity not the conventionality of simultaneity. In it, we have shifted to a new inertial frame of reference. The worldlines representing the point A and B in space are now inclined to the vertical. (The represent a speed of 1/4c.) The Einstein synchrony convention is used. That can be seen through the fact that the time taken for the outward and return trips for light are equal, that is, two units of time are taken for each on the frame's clocks. |

If the standard choice of ε=1/2
really is freely chosen, then the alternative choice of ε=1/4 should
not get us into any trouble. That is, we should generate no
contradictions within the theory, although we may arrive at some
unexpected results. Strange things do happen when we use a non-standard synchrony convention, such as ε=1/4. One is shown in the figures. With ε=1/2, light takes 2 units of time to go forward from A to B and 2 units to return from B to A. With the non-standard ε=1/4, things are quite different. Light takes one unit of time to go from A to B. (Departs A at A-time 0; arrives at B at B-time 1; 1-0=1) But the light takes three times as long to return from B to A. (Departs B at B-time 1; arrives at A at A-time 4; 4-1=3.) Therefore, the speed of light in the BA direction is three times greater than in the AB direction. What happened to the light postulate and the constancy of the speed of light? According to the conventionality of simultaneity, the one-way speed of light, such as from A to B, varies according to our conventional choice of ε. The two-way speed of light, such as the average speed for the round trip from A to B to A, is not affected by the choice of ε. The constancy asserted by the light postulate applies to this two-way speed. It retains the familiar average value of c. What has happened here is that judgments of simultaneity play a role in determining the speed of light. We measure its one-way speed over some distance with two, synchronized clocks, one at each end. So if we change our conventions for synchronizing clocks, we end up changing our judgments of the speed of light. |

This strangeness will be repeated in many places. We have seen the judgments of simultaneity play a role in determining many physical magnitudes. All these magnitudes will manifest analogous strange effects if we use a non-standard synchrony.

All speeds will be altered, not just that of light. The length of a moving rod will vary according to the direction in which it moves. The figures below show show two rods moving uniformly in opposite directions. The rods and their motions are the same in both figures. All that has changed is that we are judging their lengths at some instant using different synchrony conventions in thte two figures.

This figure below shows the case of
standard synchrony, ε=1/2. We see that, under standard synchrony,
the two rods are moving in opposite direction at the same speed. The
two rods coincide at the instant of time 2 and have the same
length. ε=1/2 |
The figure below shows the same two
rods, moving in exactly the same way. However we are now surveying
their motions and lengths with a non-standard synchrony of ε=1/4.
The two rods coincide at the instant of time 2. However it is clear
from the figure that they do not have the same length. The rod
moving to the right is now found to be longer
than the rod moving to the left. ε=1/4 |

The slowing of clocks will also vary according to their direction. Under standard synchrony, we can reduce the amount of slowing of a moving clock to as small an amount as we like just by moving the clock very slowly. That means that a very slowly moving clock will remain synchronized with the other clocks in the frame, as long as those clock are synchronized by the standard rule. What if we had synchronized the clocks non-standardly, that is, with an ε other than 1/2? Then the slowly moving clock would no longer remain synchronized with the other clocks it passes in the frame.

Turn this around and describe it from the perspective of the clocks spread through the frame: under non-standard synchrony, there is an additional direction related shift in the reading of a clock that slowness of transport cannot eradicate.

We need not go into further details of the many changes that non-standard synchrony requires. For our purposes, all that matters is that there is a simple procedure for determining what these changes are. One first describes the process using the space and time coordinates of standard synchrony. One then replaces these standard coordinates with those coordinates associated with the non-standard synchrony. What results is a new description that conforms with the non-standard synchrony.

The new descriptions will be strange. Light will be propagating at different speeds in different directions. But that strangeness is all there is. Mere strangeness does not make them incorrect. There will be no contradictions, if the original descriptions were contradiction free. Equally importantly, everything measurable will come out the same in the two descriptions, so that no observation of a matter of fact can decide which is the correct one.

Finally, one might be inclined to insist that we must employ standard synchrony, for no account of light can be accepted if it has light propagating at different speeds in different directions. Space, we believe, is isotropic, that is, the same in all directions. The conventionalist response is that the anisotropy of the one way speed of light is merely an oddity of description, not a factual error, for the differences in the one-way speeds reflects nothing physical. The differences merely reflect our decision to synchronize clocks in different ways.

If one does not subscribe to the causal theory time, then recovering the conventionality thesis from it is unpersuasive. Then, the strongest support of the conventionality of simultaneity might be the ε procedure sketched above. It assures us that we can always find a consistent redescription of processes that conforms with any chosen non-standard synchrony.

However that procedure reveals a great weakness. Choosing different synchrony conventions has been reduced to assigning time coordinates to events in arbitrary ways. There is something quite trivial about that. We label events with four numbers: 3 space coordinates and one time coordinate. They are just labels and, as long as we keep track of the system used, we can relabel the events any way we please.

If all the thesis is about is the freedom to label events as we please, then the thesis is much less than it initially seemed. It is not a thesis specifically about time. We can also relabel the spatial coordinates as we please, as long as we keep track of the system used. We can relabel any physical magnitude in any odd way we please. The descriptions that result will seem strange, but as long as we keep track of the system used, the strangeness is merely an oddity of description, not the revealing of an important, overlooked convention.

The relativity of simultaneity is regarded as settled physics and is accepted by anyone competent in relativity theory. The thesis of the conventionality of simultaneity, however, remains controversial. Most prominent among objections to the thesis is a result proved in 1977 by David Malament. His result creates difficulties specifically for the connection between the causal theory of time and the conventionality thesis. | David Malament |

Malament showed that, once we have specified a particular inertial frame of reference, there is only one (non-trivial) relation of simultaneity definable from the causal structure of a Minkowski spacetime. That reverses the expectation of the causal theorists. The causal structure seemed to leave a lot of freedom for the assignment of the relation of simultaneity. Malament showed that there was no freedom. The one definable relation proves to be standard ε=1/2 synchrony. | For a simple account of the proof, see my "Philosophy of Space and Time" Ch. 5, Part III, in M. H. Salmon et al, Introduction to the Philosophy of Science. |

The talk of "definable" may at first seem abstruse. The notion is pretty simple, however, and another example gives the basic idea.

We believe that ordinary gases are nothing more than a lot of molecules in rapid motion. As a result, all the factual properties of gases ought to be traceable back to molecular properties. Take the temperature of the gas. If two gases have different temperatures, then there must be something different about the molecules. A familiar result for ideal gases is that the temperature of the gas just is fixed by the average of the squared speeds of its molecules.

That is, gases are nothing but molecules in motion, so all their properties can be defined in terms of molecular properties. (For experts in philosophy of physics: may I be permitted to set aside the present, fierce debates over reduction and emergence in the interests of simple pedagogy? Thank you!)

The same reasoning applies to time. If temporal relations are nothing more than causal relations, then we ought to be able to define a temporal relation like simultaneity in terms of causal relations. When we carry out the exercise, Malament shows, we find that there is a unique simultaneity relation definable in causal terms. Just as the molecular constitution of a gas fixes its temperature, it seems that the causal structure of a Minkowski spacetime fixes the unique simultaneity relation appropriate to each inertial frame of reference.

Malament's result turns the original intuition of the conventionality thesis on its head. The thesis came originally from the fact that causal connections in a Minkowski spacetime are more limited than in classical, Newtonian spacetimes: nothing propagates causally faster than light. That opened a freedom in assigning relations of simultaneity that is the conventionality thesis. Malament's result now suggests that the causal structure does not admit that freedom after all. Only one simultaneity relation, standard synchrony, is definable non-trivially by it.

The weakness of Malament's result resides in the delicate selection of just which structures are allowed when we define a simultaneity relation. The analysis only returns a unique result if we keep pretty much precisely to Malament's list. Small deviations from it are sufficient to destroy the uniqueness, whether we add or subtract from the list.

For example, Peter Spirtes showed that uniqueness is destroyed merely by allowing the definition to distinguish past from future.

Adolf Gruenbaum has objected that Malament's analysis assumes that a simultaneity relation will divide the spacetime into non-overlapping sets of mutually simultaneous events. (These turn out to be the the familiar hypersurfaces of simultaneity.) Drop this assumption, he points out, and the uniqueness fails. What results, however, is an odd relation of simultaneity that is not transitive. That is, if event A is simultaneous with B and event B with C, then it does not follow that event A is simultaneous with event C.

For a helpful survey, see Allen Janis, "Conventionality
of Simultaneity", The Stanford Encyclopedia of
Philosophy. |
For a recent addition to the literature, see
Robert Rynasiewicz, "Simultaneity,
Convention, and Gauge Freedom." |
The conventionality of simultaneity has been the focus of protracted debate in the philosophy of science literature. It is difficult to come to a clear conclusion on it. The debate has faded in recent years, but not because one side has won. Rather the combatants have just become weary of the fight. On each side are philosophers of physics with a deep and rich command of relativity theory. So the real question is why the disagreement can persist. The two sides must differ over something so foundational that it is not entering explicitly into the discussion. |

Two remarks may help.

*First* is an oddity of the notion itself. The
conventionalists urge that the conventionality of simultaneity is an
important physical insight. It is, purportedly, something more important
than the many minor conventions routinely employed in science, such as our
decision that a mile is 5,280 feet. Yet the conventionality of
simultaneity seems to have no consequences outside
the debate over the conventionality itself.

Compare that to another result that may be characterized as a non-trivial convention. According to special relativity, when we select an inertial reference, we do so for descriptive convenience. There is no factually preferred inertial reference frame. That contrasts with a Lorentz-style ether theory. In it, there is a special frame of reference, the ether rest frame. In making the choice of inertial frame conventional, we convey the fact that there is no ether rest frame. We thereby force our physics to become relativistic. That is, we must reformulate the laws of physics so that they come out the same in all inertial frames of reference.

If the choice of a simultaneity relation is conventional, there is no corresponding consequence for our physics. That means that no further issue has emerged that can decide the debate. Here it is unlike the issue of an ether reference frame. Ether theories fell from favor because no one could find a way to discern just which was the preferred ether frame of rest. There is no corresponding failure associated with conventionality of simultaneity. Indeed we have the reverse situation. Standard synchrony is readily distinguished from all others by the Einstein synchrony rule.

*Second*, I suspect a connection with a different
issue in philosophy of science. It concerns competing views of the
structures posited by spacetime theories.

At one extreme is an antirealist attitude. According to it, all that matters is what we observe. This observational content is exhausted by catalogs of facts like "this light signal arrives here when this clock reads 2." The geometry of the spacetime plays a subsidiary role only in helping us to catalog which observations are possible and which are not. In the most extreme construal, the geometry is just a useful fiction. | At the other extreme, a realist construes spacetime theories as giving a literal account of the world. So when a spacetime theory posits a Minkowskian geometry for spacetime, we are to understand that factually the spacetime has that geometry. The geometry is as real as tables and chairs. |

The difference between conventionalists and non-conventionalists is definitely NOT merely the difference between these two views. However there is a partial alignment.

The conventionalists
would incline to more antirealist views. This is clearest with
Reichenbach's causal theory of time. All there really is to time,
according to that theory, are the causal relations among events,
which are in turn mapped out by light signals. Anything more is
factually superfluous. That applies to relations of simultaneity
and, presumably, much else in the Minkowski spacetime geometry. Similarly, the different ε based descriptions of processes in spacetime are judged equally good, even though some are more complicated, because they conform to the same observations. That conformity is a major source of support for the conventionalist thesis. |
The non-conventionalists
would incline towards realist views. If one takes Minkowski geometry
as factually the geometry of spacetime, as real as tables and
chairs, then one automatically takes its natural simultaneity
relation as factual. That natural relation is just standard
synchrony. In geometrical terms, it comes out as the spacetime analog of the ordinary geometrical notion of perpendicularity, here usually called "orthogonality." If we take the worldline of an inertially moving body, there is only one set of hypersurfaces of simultaneity orthogonal to it under a Minkowskian spacetime geometry: those of standard synchrony. |

In brief, the more of a realist you are concerning spacetime geometry, the less you can incline towards the conventionality of simultaneity.

In balance, my view is more sympathetic to the non-conventionalist view. I incline towards the realist view of Minkowski geometry. So I find our collected observations in spacetime strong evidence for something factual behind the observations. The situation is not so different from our detection of the ionosphere in the earth's upper atmosphere. We know it is there, even though we cannot see it, since it reflects radio waves back to earth. We can map its shape and location by those reflections. Correspondingly, we use observations in space and time to map out the geometric structure of the spacetime. What we map out is a geometry that includes an orthogonality relation, which I take to be the unique notion of simultaneity provided by the spacetime geometry.

http://commons.wikimedia.org/wiki/File:Circuit_ondes_réfléchies.svg

To conclude this discussion, let us get some perspective on the importance of this debate. In the second half of the 20th century, it was the most prominent debate in philosophy of space and time. Yet now the fire and fury has all but dissipated and the debate rarely reappears.

In my view, there are two reasons for why this debate seems much less important now that it did in the past.

First, the conventionality thesis is one that can be mounted only within the special theory of relativity. That was Einstein's theory of 1905. In spacetime terms, the analysis depends on the wide-reaching uniformities of the Minkowski spacetime of special relativity. They are homogeneous, isotropic and remain unchanged as we boost between different inertial motions.

By 1915, as we will learn in later chapters, special relativity was eclipsed by a much richer and more adventurous theory of space and time: the general theory of relativity. That theory provides for spacetimes with much richer structure. They lack the full range of uniformities of a Minkowski spacetime. As a result, it does not seem possible to mount a convincing argument for the conventionality of simultaneity in them.

For example, general relativity provides us the spacetimes of the expanding universes of modern cosmology: the Friedman-Lemaître-Robertson-Walker spacetimes. In them we lose even the relativity of simultaneity. They divide up spacetime into a unique set of spatial hypersurfaces that comprise the "now."

These spacetimes have an absolute relation of simultaneity and a distinguished motion, the expanding motion of the cosmic fluid formed by the galaxies, viewed on the largest scale.

In sum, whatever fundamental moral we may have drawn about the conventionality of simultaneity, it is limited specifically to the special theory of relativity. It has no general applicability. There is no such convention in later theories of space and time and, in particular, in the spacetime structure that we think better represents our universe.

The second reason is an assumption essential to the thesis of the conventionality of simultaneity: it is possible in a non-arbitrary way to divide the content of the theory into the conventional and the factual parts. Sometimes it is possible to identify unequivocally some conventional element in a theory. The obvious examples are the stipulations of units. Just how heavy is one kilogram? For a long time, that was given by stipulating that a particular, treasured mass in Paris is the kilogram by definition.

The difficulty is that there are many ways of presenting a body of facts about the world. What is a convention in one presentation can become a fact in another.

For example, the present definition of a meter is the distance light travels in vacuo in 1/299,792,458 th second. Then it turns out to be a rather useful fact that the distance from the equator to the North Pole is exactly 10,000 kilometers. This useful fact, however, was originally no fact at all, but a stipulation. The meter was originally defined as 1/10,000,000 that distance along the meridian through Paris.

In geometry, we might define a straight line as the line of shortest distance between two points. Then it turns out as a factual matter, that a straight line maintains a constant direction in space. We can invert this and define a straight line as one that always maintains a constant direction in space. Then we will find as a factual matter that the straight line is the shortest distance between two points.

The the thesis of the conventionality of simultaneity depends on taking certain sorts of assertions in special relativity as factual and then seeking a sharp boundary that contains them. The assertions delimited as factual concern the departure and arrival times of idealized light signals propagating in vacuo and the coincident readings of clocks that are idealized to run perfectly. These assertions have a factual feel to them. However under closer scrutiny, the large number of idealization presumed means that they are themselves theoretical statements that may have various mixtures of fact and convention buried within them.

As the theories we consider become more complicated, it becomes harder to see that there is a uniquely correct way of dividing the factual from the conventional. There are many ways of axiomatizing complicated theories such as quantum mechanics. The axioms of one system will contain its own definitions, which are conventional stipulations. Yet another axiom system might use different definitions and end up inverting what is factual and conventional. The locating of a sharp division between the factual and the conventional itself looks to be conventional.

- The various philosophical morals people have tried to draw from relativity theory.
- How to identify a clear thesis and the argument that supports it.
- How to criticize the statement of a thesis and the argument that supports it.
- Your own view of which philosophical morals can be drawn from relativity theory.

February 11, 2013. February 9, 2017. October
10, 2018. January 30, 2022. December 21, 2023.