Tuesday, October 24, 2017

Einstein, Sommerfeld and the Twin Paradox

A version of this article appeared Monday, October 28, 2013. The essay originally
appeared on Angelfire ca. 2005.
This paper was updated Dec. 10, 2009, Oct. 28, 2013. A minor addition citing
Henry Stapp was made Feb. 21, 2019.
Please use the control f function for easy access to points on this paper.
Please notify me of errors at Krypto46 attt Protonmail dottt commm

The paradox
Einstein's groundbreaking 1905 relativity paper, "On the electrodynamics of moving bodies," contained a fundamental inconsistency which was not addressed until 10 years later, with the publication of his paper on gravitation.

Many have written on this inconsistency, known as the "twin paradox" or the "clock paradox" and more than a few have not understood that the "paradox" does not refer to the strangeness of time dilation but to a logical inconsistency in what is now known as the special (for "special case") theory of relativity.

Among those missing the point: Max Born in his book on special relativity (1), George Gamow in an essay and Roger Penrose in Road to Reality (2), and, most recently, Leonard Susskind in The Black Hole War (3).

Among those who have correctly understood the paradox are topologist Jeff Weeks (zz2) and science writer Stan Gibilisco (4), who noted that the general theory of relativity resolves the problem. Isaac Asimov showed a good grasp of the paradox (3a), which he resolved by appeal to actual acceleration, while off-handedly exonerating Einstein of an oversight. (Not all of Asimov's arguments regarding the paradox are, however, persuasive.)

Dave Goldberg, a Drexel physics professor, correctly interprets the paradox in his book The Universe in the Rearview Mirror (zz1) and addresses a number of points raised in a previous version of this essay, coincidentally including correction of some (non-crucial) claims that I seem to have got wrong (and which I may or may not have excised).

As far back as the 1960s, the British physicist Herbert Dingle (5) called the inconsistency a "regrettable error" and was deluged with "disproofs" of his assertion from the physics community. Yet every "disproof" of the paradox that I have seen uses acceleration, an issue not addressed by Einstein until the general theory of relativity. It was Einstein who set himself up for the paradox by favoring the idea that only purely relative motions are meaningful, writing that various examples "suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest." [Electrodynamics translated by Perett and Jeffery and appearing in a Dover (1952) reprint.] In that paper, he also takes pains to note that the term "stationary system" is a verbal convenience only (7).

[It should be noted that Dingle's 1949 attempt at relativistic physics left Einstein bemused (6).]

But later in Elect., Einstein offered the scenario of two initially synchronized clocks at rest with respect to each other. One clock then travels around a closed loop, and its time is dilated with respect to the at-rest clock when they meet again. In Einstein's words: "If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be 1/2tv2/c2 slow."

Clearly, if there is no preferred frame of reference, a contradiction arises: when the clocks meet again, which clock has recorded fewer ticks?

Both in the closed loop scenario and in the polygon-path scenario, Einstein avoids the issue of acceleration. Hence, he does not explain that there is a property of "real" acceleration that is not symmetrical or purely relative and that that consequently a preferred frame of reference is implied, at least locally.

The paradox stems from the fact that one cannot say which velocity is higher without a "background" reference frame. In Newtonian terms, the same issue arises: if one body is accelerating away from the other, how do we know which body experiences the "real" force? No answer is possible without more information, implying a background frame.

In comments published in 1910, the physicist Arnold Sommerfeld, a proponent of relativity theory, "covers" for the new paradigm by noting that Einstein didn't really mean that time dilation was associated with purely relative motion, but rather with accelerated motion; and that hence relativity was in that case not contradictory. Sommerfeld wrote: "On this [a time integral and inequality] depends the retardation of the moving clock compared with the clock at rest. The assertion is based, as Einstein has pointed out, on the unprovable assumption that the clock in motion actually indicates its own proper time; i.e. that it always gives the time corresponding to the state of velocity, regarded as constant, at any instant. The moving clock must naturally have been moved with acceleration (with changes of speed or direction) in order to be compared with the stationary clock at world-point P. The retardation of the moving clock does not therefore actually indicate 'motion,' but 'accelerated motion.' Hence this does not contradict the principle of relativity." [Notes appended to Space and Time, a 1908 address by Herman Minkowski, Dover 1952, Note 4.]

However, Einstein's 1905 paper does not tackle the issue of acceleration and more to the point, does not explain why purely relative acceleration would be insufficient to meet the facts. The principle of relativity applies only to "uniform translatory motion" (Elect. 1905).

Neither does Sommerfeld's note address the issue of purely relative acceleration versus "true" acceleration, perhaps implicitly accepting Newton's view (below). And, a review of various papers by Einstein seems to indicate that he did not deal with this inconsistency head-on, though in a lecture-hall discussion ca. 1912, Einstein said that the [special] theory of relativity is silent on how a clock behaves if forced to change direction but argues that if a polygonal path is large enough, accelerative effects diminish and (linear) time dilation still holds.

On the other hand, of course, he was not oblivious to the issue of acceleration. In 1910, he wrote that the principle of relativity meant that the laws of physics are independent of the state of motion, but that the motion is non-accelerated. "We assume that the motion of acceleration has an objective meaning," he said. [The Principle of Relativity and its Consequences in Modern Physics, a 1910 paper reproduced in Collected Papers of Albert Einstein, Hebrew University, Princeton University Press.]

In that same paper Einstein emphasizes that the principle of relativity does not cover acceleration. "The laws governing natural phenomena are independent of the state of motion of the coordinate system to which the phenomena are observed, provided this system is not in accelerated motion." Clearly, however, he is somewhat ambiguous about small accelerations and radial acceleration, as we see from the lecture-hall remark and from a remark in Foundation of the General Theory of Relativity (1915) about a "familiar result" of special relativity whereby a clock on a rotating disk's rim ticks slower than a clock at the origin.

General relativity's partial solution
Finally, in his 1915 paper on general relativity, Einstein addressed the issue of acceleration, citing what he called "the principle of equivalence." That principle (actually, introduced prior to 1915) said that there was no real difference between kinematic acceleration and gravitational acceleration. Scientifically, they should be treated as if they are the same.

So then, Einstein notes in Foundation, if we have system K and another system K' accelerating with respect to K, clearly, from a "Galilean" perspective, we could say that K was accelerating with respect to K'. But, is this really so?

Einstein argues that if K is at rest relative to K', which is accelerated, the observer on K cannot claim that he is being accelerated -- even though, in purely relative terms, such a claim is valid. The reason for this rejection of Galilean relativity: We may equally well interpret K' to be kinematically unaccelerated though the "space-time territory in question is under the sway of a gravitational field, which generates the accelerated motion of the bodies" in the K' system. This claim is based on the principle of equivalence which might be considered a modification of his previously posited principle of relativity. By the relativity principle, Einstein meant that the laws of physics can be cast in invariant form so that they apply equivalently in any unformly moving frame of reference. (For example, |vb - va| is the invariant quantity that describes an equivalence class of linear velocities.)

By the phrase "equivalence," Einstein is relating impulsive acceleration (for example, a projectile's x vector) to its gravitational acceleration (its y vector). Of course, Newton's mechanics already said that the equation F = mg is a special case of F = ma but Einstein meant something more: that local spacetime curvature is specific for "real" accelerations -- whether impulsive or gravitational.

Einstein's "equivalence" insight was his recognition that one could express acceleration, whether gravitational or impulsive, as a curvature in the spacetime continuum (a concept introduced by Minkowski). This means, he said, that the Newtonian superposition of separate vectors was not valid and was to be replaced by a unitary curvature. (Though the calculus of spacetime requires specific tools, the concept isn't so hard to grasp. Think of a Mercator map: the projection of a sphere onto a plane. Analogously, general relativity projects a 4-dimensional spacetime onto a Euclidean three-dimensional space.)

However, is this "world-line" answer the end of the problem of the asymmetry of accelerated motion?

The Einstein of 1915 implies that if two objects have two different velocities, we must regard one as having an absolutely higher velocity than the other because one object has been "really" accelerated.

Yet one might conjecture that if two objects move with different velocities wherein neither has a prior acceleration, then the spacetime curvature would be identical for each object and the objects' clocks would not get out of step. But such a conjecture would violate the limiting case of special relativity (and hence general relativity); specifically, such a conjecture would be inconsistent with the constancy of the vacuum velocity of light in any reference frame.

So then, general relativity requires that velocity differences are, in a sense, absolute. Yet in his original static and eternal cosmic model of 1917, there was no reason to assume that two velocities of two objects necessarily implied the acceleration of one object. Einstein introduced the model, with the cosmological constant appended in order to contend with the fact that his 1915 formulation of GR apparently failed to account for the observed mass distribution of the cosmos. Despite the popularity of the Big Bang model, a number of cosmic models hold the option that some velocity differences needn't imply an acceleration, strictly relative or "real."

Einstein's appeal to spacetime curvature to address the frame of reference issue is similar to Newton's assertion that an accelerated body requires either an impulse imputed to it or the gravitational force. There is an inherent local physical asymmetry. Purely relative motion will not do.

Frank Close points out that in the quantum arena, unlike in SR, superconductivity shows that there is an absolute state of rest, That is, he writes, the superconductor is at rest relative to the electron but not the converse (9).

Einstein also brings up the problem of absolute relative motion in the sense of Newton's bucket. Einstein uses two fluid bodies in space, one spherical, S1 and another an ellipsoid of revolution, S2. From the perspective of "Galilean relativity," one can as easily say that either body is at rest with respect to the other. But, the radial acceleration of S2 results in a noticeable difference: an equatorial bulge. Hence, says Einstein, it follows that the difference in motion must have a cause outside the system of the two bodies.

Of course Newton in Principia Mathematica first raised this point, noting that the surface of water in a rapidly spinning bucket becomes concave. This, he said, demonstrated that force must be impressed on a body in order for there to be a change in acceleration. Newton also mentioned the issue of the fixed stars as possibly of use for a background reference frame, though he does not seem to have insisted on that point. He did however find that absolute space would serve as a background reference frame.

It is noteworthy that Einstein's limit c can be used as an alternative to the equatorial bulge argument. If we suppose that a particular star is sufficiently distant, then the x component of its radial velocity (which is uniform and linear) will exceed the velocity of light. Such a circumstance being forbidden, we are forced to conclude that the earth is spinning, rather than the star revolving around the earth. We see that, in this sense, the limit c can be used to imply a specific frame of reference. At this point, however, I cannot say that such a circumstance suffices to resolve the clock paradox of special relativity.

Interestingly, the problem of Newton's bucket is quite similar to the clock paradox of special relativity. In both scenarios, we note that if two motions are strictly relative, what accounts for a property associated with one motion and not the other? In both cases, we are urged to focus on the "real" acceleration.

Newton's need for a background frame to cope with "real" acceleration predates the 19th century refinement of the concept of energy as an ineffable, essentially abstract "substance" which passes from one event to the next. That concept was implicit in Newton's Principia but not explicit and hence Newton did not appeal to the "energy" of the object in motion to deal with the problem. That is, we can say that we can distinguish between two systems by examining their parts. A system accelerated to a non-relativistic speed nevertheless discloses its motion by the fact that the parts change speed at different times as a set of "energy transactions" occur. For example, when you step on the accelerator, the car seat moves forward before you do; you catch up to the car "because" the car set imparts "kinetic energy" to you.

But if you are too far away to distinguish individual parts or a change in the object's shape, such as from equatorial bulge, your only hope for determining "true" acceleration is by knowing which object received energy prior to the two showing a relative change in velocity. Has the clock paradox gone away?

The general theory undermined two claims of the 1905 special theory, according to Henry Stapp[x1], a quantum physicist. One was the Machian and "logical positivist" claim that only verifiable (testable) assertions should be accepted in physics. The other was the absence in nature of a preferred sequence of nows. "Furthermore, the universe we are living in has a global preferred reference frame, which defines instantaneous "nows" empirically. This frame has recently [1990s] been empirically specified to within several parts per million."

In other words, the original concept of "relative" has been somewhat modified. Velocities cannot be strictly relative to one another because, on the basis of a non-eternal Big Bang theory, each velocity must have received an infusion of energy ("true" acceleration) that brought it to its present level. Yet, the constancy of the velocity of light means that motions are still relative with respect to an observer.

Relativity of simultaneity
(Adapted from Wikipedia 02.21.2019.)

A popular picture for understanding the relativity of simultaneity is provided by a thought experiment similar to those suggested by Daniel Frost Comstock in 1910 and Einstein in 1917.

Consider one observer midway inside a speeding train car and another observer standing on a platform as the train moves past.

A flash of light goes off at the center of the car just as the two observers pass each other. For the observer on board the train, the front and back of the car are at fixed distances from the light source and as such, according to this observer, the light will reach the front and back of the car at the same time.

But, for the observer standing on the platform, the rear of the car is moving (catching up) toward the point at which the flash shone, and the front of the car is moving away from it. As the speed of light is finite and the same in all directions for all observers, the light headed for the back of the train will have less distance to cover than the light headed for the front. Thus, the flashes of light will strike the ends of the car at different times for that observer.

Curiously, it can happen that an observer cannot determine which spacetime event (as in two flashes of light) occurred first. Einstein and other relativists found no threat to strict causality in this fact, however. They assumed that it was always the case that an observer in the correct position could determine which event was first, thus preserving a necessary aspect of causality. Later Einstein could not accept the limitations on causality imposed by quantum mechanics. Yet, his own relativity theory at least posed a question as to the supposed universality of physical causality. In addition, his groundbreaking paper on the photo-electric effect extended the quantum hypothesis so as to make light corpuscles a plausible idea, an idea which in turn already undercut assumptions of reality and causality of classical physics.

In addition, Goedel's discovery of solutions to GR's equation that yielded closed time loops (a rocket could notionally fly into its own past) was dismissed by Einstein as not corresponding to physical reality. Yet Goedel argued that such a solution implied the non-reality of time -- a conclusion that obliterates standard mechanical causality.

Does GR resolve the clock paradox?
In any case, we now come to the question as to whether GR resolves the clock paradox.

GR resolves the paradox non-globally, in that Einstein now holds that some accelerations are not strictly relative, but functions of a set of curvatures. Hence one can posit the loop scenario given in Electrodynamics and say that only one body can have a higher absolute angular velocity with respect to the other because only one must have experienced an acceleration that distorts spacetime differently from the other.

To be consistent, GR must reflect this asymmetry. That is, suppose we have two spaceships separating along a straight line whereby the distance between them increases as a constant velocity. If ship A's TV monitor says B's clock is ticking slower than A's and ship B's TV monitor says A's clock is ticking slower than B's, there must be an objective difference, nevertheless.

The above scenario is incomplete because the "real" acceleration prior to the opening of the scene is not given. Yet, GR does not tell us why a "real" acceleration must have occurred if two bodies are moving at different velocities.

So yes, GR partly resolves the clock paradox and, by viewing the 1905 equations for uniform motion as a special case of the 1915 equations, retroactively removes the paradox from SR, although it appears that Einstein avoided pointing this out in 1915 or thereafter.

However, GR does not specify a global topology (cosmic model) of spacetime, though Einstein struggled with this issue. The various solutions to GR's field equations showed that no specific cosmic model followed from GR. The clock paradox shows up in the Weeks model of the cosmos, with local space being Euclidean (or rather Minkowskian). As far as this writer knows, such closed space geodesics cannot be ruled out on GR grounds alone.

Jeff Weeks, in his book The Shape of Space (zz3), points out that though physicists commonly think of three cosmic models as suitable for GR, in fact there are three classes of 3-manifolds that are both homogenous and isotropic (cosmic information is evenly mixed and looks about the same in any direction). Whether spacetime is mathematically elliptic, hyperbolic or euclidean, there are many possible global topologies for the cosmos, Weeks says.

One model, described by Weeks in the article linked above, permits a traveler to continue straight in a closed universe until she arrives at the point of origin. Again, to avoid contradiction, we are required to accept a priori that an acceleration that alters a world line has occurred.

Other models have the cosmic time axis following hyperbolic or elliptical geometry. Originally, one suspects, Einstein may have been skeptical of such an axis, in that Einstein's abolishment of simultaneity effectively abolished the Newtonian fiction of absolute time. But physicist Paul Davies, in his book About Time, argued that there is a Big Bang oriented cosmic time that can be approximated quite closely.

Kurt Goedel's rotating universe model left room for closed time loops, such that an astronaut who continued on a protracted space flight could fly into his past. This result prompted Godel to question the reality of time in general relativity. Having investigated various solutions of GR equations, Goedel argued that a median of proper times of moving objects, which James Jeans had thought to serve as a cosmic absolute time, was not guaranteed in all models and hence should be questioned in general.

Certainly we can agree that Goedel's result shows that relativity is incomplete in its analysis of time.

Mach's principles
Einstein was influenced by the philosophical writings of the German physicist Ernst Mach, whom he cites in Foundations.

According to Einstein (1915) Mach's "epistomological principle" says that observable facts must ultimately appear as causes and effects. Mach believed that the brain organizes sensory data into knowledge and that hence data of scientific value should stem from observable, measurable phenomena. This philosophical viewpoint was evident in 1905 when Einstein ruthlessly ejected the Maxwell-Lorentzian ether from physics.

Mach's "epistomological principle" led Mach to reject Newtonian absolute time and absolute space as unverifiable and made Einstein realize that the Newtonian edifice wasn't sacrosanct. However, in 1905 Einstein hadn't replaced the edifice with something called a "spacetime continuum." Curiously, later in his career Einstein impishly but honestly identified this entity as "the ether."

By rejecting absolute space and time, Mach also rejected the usual way of identifying acceleration in what is known as Mach's principle: Version A. Inertia of a ponderable object results from a relationship of that object with all other objects in the universe.

Version B. The earth's equatorial bulge is not a result of absolute rotation (radial acceleration) but is relative to the distant giant mass of the universe.

For a few years after publication of Foundations, Einstein favored Mach's principle, even using it as a basis of his "cosmological constant" paper, which was his first attempt to fit GR to a cosmic model, but was eventually convinced by the astronomer Wilem de Sitter (zz4) to abandon the principle. In 1932 Einstein adopted the Einstein-de Sitter model that posits a cosmos with a global curvature that asymptotically zeroes out over eternity. The model also can be construed to imply a Big Bang, with its ordered set of accelerations.

A bit of fine-tuning
We can fine-tune the paradox by considering the velocity of the center of mass of the twin system. That velocity is m1v/m1 + m2. So the CM velocity is larger when the moving mass is the lesser and the converse. Letting x be a real greater than 1 we have two masses xm and m. The algebra reveals there is a factor (x/x+1) > 1/(x+1). The CM velocity for earth moving at 0.6c with respect to a 77kg astronaut is very close to 0.6c. For the converse, however, that velocity is about 2.3 meters per femto-second.

If we like, we can use the equation

E = mc2(1-v2/c2)1/2

to obtain the energies of each twin system.

If the earth is in motion and the astronaut at rest, my calculator won't handle the quantity for the energy. If the astronaut is in motion with the earth at rest, then E = 5.38*1041J.

But the paradox is restored as soon as we set m1 equal to m2.

Notes on the principle of equivalence
Now an aside on the principle of equivalence. Can it be said that gravitational acceleration is equivalent to kinematic acceleration? Gravitational accelerations are all associated with the gravitational constant G and of the form g = Gm/r2. Yet it is easy to write expressions for accelerations that cannot be members of the gravitational set. If a is not constant, we fulfill the criterion. If in rx, x =/= 2, there will be an infinity of accelerations that aren't members of the gravitational set.

At any rate, Einstein's principle of equivalence made a logical connection between a ponderable object's inertial mass and its gravitational mass. Newton had not shown a reason that they should be exactly equal, an assumption validated by acute experiments. (A minor technicality: Einstein and others have wondered why these masses should be exactly equal, but, properly they meant why should they be exactly proportional? Equality is guaranteed by Newton's choice of a gravitational constant. But certainly, min = kmgr, with k equaling one because of Newton's choice.)

Also, GR's field equations rest on the premise (Foundation) that for an infinitesimal region of spacetime, the Minkowskian coordinates of special relativity hold. However, this 1915 assumption is open to challenge on the basis of the Heisenberg uncertainty principle (ca. 1925), which sets a finite limit on the precision of a measurement of a particle's space coordinate given its time coordinate.

Einstein's Kaluza-Klein excursion
In Subtle is the Lord Abraham Pais tells of a period in which Einstein took Klein's idea for a five-dimensional spacetime and reworked it. After a great deal of effort, Einstein offered a paper which took Klein's ideas presented as his own, on the basis that he had found a way to rationalize obtaining the five-dimensional effect while sticking to the conventional perceptual view of space and time denoted 3D+T (making one wonder what he thought of his own four-dimensional spacetime scheme).

A perplexed Pais notes that a colleague dismissed Einstein's work as unoriginal, and Einstein then quickly dropped it (7). But reformulation of the ideas of others is exactly what Einstein did in 1905 with the special theory. He presented the mathematical and physical ideas of Lorenz, Fitzgerald and Poincare, whom he very likely read, and refashioned them in a manner that he thought coherent, most famously by rejecting the notion of ether as unnecessary.

Yet it took decades for Einstein to publicly acknowledge the contribution of Poincare, and even then, he let the priority matter remain fuzzy. Poincare's work was published in French in 1904, but went unnoticed by the powerful German-speaking scientific community. As a French-speaking resident of Switzerland, it seems rather plausible that the young patent attorney read Poincare's paper.

But, as Pais pointed out, it was Einstein's interpretation that made him the genius of relativity. And yet, that interpretation was either flawed, or incomplete, as we know from the twin paradox.

Apologies for footnotes being out of order. Use the control f function. Any repetitions or other oddities below may be a result of a Blogger malfunction, which I have been unable to correct.
1. Einstein's Theory of Relativity by Max Born (Dover 1962).
2. Road to Reality by Roger Penrose (Random House 2006).
3. The Black Hole War by Leonard Susskind (Little Brown 2009).
3a. Understanding Physics (Dorset Press 1988 reprint of three 1966 volumes) by Isaac Asimov. See Vol. 2.
4. Understanding Einstein's Theories of Relativity by Stan Gibilisco (Dover reprint of the 1983 edition).
7. In his biography of Einstein, 'Subtle is the Lord':The science and the life of Albert Einstein (Oxford 1983), physicist Abraham Pais mentions the "clock paradox" in the 1905 Electrodynamics paper but then summarily has Einstein resolve the contradiction in a paper presented to the Prussian Academy of Physics after the correct GR paper of 1915, with Einstein arguing that acceleration ends the paradox, which Pais calls a "misnomer." I don't recall the Prussian Academy paper, but it should be said that Einstein strongly implied the solution to the contradiction in his 1915 GR paper. Later in his book, Pais asserts that sometime after the GR paper, Einstein dispatched a paper on what Pais now calls the "twins paradox" but Pais uncharacteristically gives no citation.
5. Though Dingle seems to have done some astronomical work, he was not -- as a previous draft of this page said -- an astronomer, according to Harry H. Ricker III. Dingle was a professor of physics and natural philosophy at Imperial College before becoming a professor of history and the philosophy of science at City College, London, Ricker said. "Most properly he should be called a physicist and natural philosopher since his objections to relativity arose from his views and interpretations regarding the philosophy of science."
6. Dingle's paper "Scientific and Philosophical Implications of the Special Theory of Relativity" appeared in Albert Einstein: Philosopher-Scientist edited by Paul Arthur Schilpp (Library of Living Philosophers 1949). Dingle used this forum to propound a novel extension of special relativity that contained serious logical flaws. Einstein, in a note of response, said Dingle's paper made no sense to him.
8. See for example Max Von Laue's paper in the Schilpp volume.
9. The Infinity Puzzle: Quantum field theory and the hunt for an orderly universe by Frank Close (Basic Books 2011).
zz1. The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality by Dave Goldberg (Dutton 2013).
zz2. Topologist Jeff Weeks on the twin paradox
http://www.math.uic.edu/undergraduate/mathclub/talks/Weeks_AMM2001.pdf
zz3. The Shape of Space by Jeff Weeks (Marcel Dekker 1985).
zz4. Michel Janssen's paper
https://netfiles.umn.edu/users/janss011/home%20page/rel-of-grav-field.pdf
x1. Mind, Matter and Quantum Mechanics, 3rd Edition by Henry P. Stapp (Springer 2009).
at No comments:

Wednesday, October 11, 2017

What hath 'Waverley' wrought?

The Waverley problem is outlined by Philosophy Professor Thomas C. Ryckman's excellent page

https://www2.lawrence.edu/fast/ryckmant/Russell's%20Theory%20of%20Descriptions.htm

on three of Bertrand Russell's puzzles for his Theory of Descriptions.

Russell observed that a sentence like,

(1) George IV wished to know whether Scott is the author of 'Waverley'

might well be true, even though

(2) George IV wished to know whether Scott is Scott

was no doubt false while, in fact,

(3) Scott was the (initially anonymous) author of Waverley

was true.

But if (1) is true, then it would appear that George IV was uncertain about the truth of (3) and if (2) were true, it would appear that George IV was uncertain about the truth of

(4) Scott is Scott

As Kurt Goedel observed [1],
The problem is: what do the so-called descriptive phrases (i.e., phrases as, e.g., "the author of Waverley " or "the king of England") denote or signify and what is the meaning of sentences in which they occur? The apparently obvious answer that, e.g., "the author of Waverley" signifies Walter Scott, leads to unexpected difficulties. For, if we admit the further apparently obvious axiom, that the signification of a composite expression, containing constituents which have themselves a signification, depends only on the signification of these constituents (not on the manner in which signification is expressed), then it follows that the sentence "Scott is the author of Waverley signifies the same thing as "Scott is Scott."
The ambiguities or apparent ambiguities of the English language in fact helped drive Russell's goal of formulating a more precise symbolical language that turned out to include the "exactly one" quantifier.

Russell pointed out [2] that his 1903 Theory of Descriptions is summarized by two definitions, written in logical symbolism, in Principia Mathematica (PM), which began publication in 1910. There the logician uses the existential quantifier and its "scope" (that part of the assertion to which the quantifier applies), along with the "exactly one" quantifier.
Let us put the problem thus:

George IV wished to know whether x = y.

But certainly the king did not wish to know whether y = y.

And yet x = y if and only if y = y.

So doesn't this suggest that His Highness actually did desire to know whether (x = y) <--> (y = y)?

I.e., George IV wished to know whether x = y, but not whether y = y -- even though (x = y) --> (y = y).

When Russell posed this question in 1903, he and Whitehead had yet to launch the revolution in logic wrought by Principia Mathematica. But by raising such questions in his early Theory of Descriptions and in PM, Russell sparked logicians to try to be more precise by what is intended by such words as identity, equality, equivalence and tautology. Various definitions and axioms have been used since.

Tarski [2a] gives Leibniz credit for this definition of equality or identity: x = y if and only if every property of x is held by y and every property of y is held by x.

Yet how does this differ from an equivalence, whereby x <--> y? In many cases, according to our position, equality and equivalence are indistinguishable.

For many writers, it is usual to apply the equivalence relation to constructs variously known as formulas, sentences, statements, propositions, predicates and sentential functions. These are the assertions in which a verb is implicit or explicit. For example, P may be the assertion 'x = y.' So we might write Ax Ey (x = y). So that assertion means "there is some y that is equal to any x."

Here, x and y are "bound" variables (i.e., they relate to the quantifiers).

Now though one ordinarily applies the equivalence relation to two assertions, P and Q, it is not customary to apply equivalence either to the variables of P(x,y) or of the terms of P(x,y), as in, say, P(x,t), where t is an instantiation of y (sometimes referred to as a "constant").

So then 'x = y' means that x and y are to be construed as interchangeable. That is, iff an instantiation of x is t, then an instantiation of y is t.

But, suppose x and y are variables representing assertions? In that case, we could properly write x <--> y.

Yet Russell's descriptions are verb-less "subjects" or "objects." How can descriptions then be construed as subject to equivalency? In the sense that if x is a description, then it is assumed that either 'x exists' or 'x doesn't exist.'

So the variables x and y of P(x,y) that are ordinarily said to be subject to the identity relation '=', are in our view first subject to the equivalence relation '<-->' -- where identity implies equivalence, but not the converse.

Note that all true assertions of an axiom-based theory are equivalent, in the context of the overall theory. For most, n/(n+1) <--> (n+1)/(n+2) is intuitively apparent. But '1 <--> pi' is not at all obvious. Another example: once we have established the set of natural numbers N -- perhaps by the von Neumann method -- we have that 1 <--> 2 and, further, 7 <--> 181.

So if we say description x <--> description y, we do not necessarily mean that they are identical.

Another example of non-identical equivalence: lim n-->inf. n/(n+1) = 1; lim n-->inf. (n+1)/n = 1. f(n) <--> f(n+1) but f(n) = f(n+1) only in the limit. Of course, we haven't bothered to offer a formal proof that f(n) <--> f(n+1), but if desired, the reader may do so, using some adequate number theory.

That is, we see that at any finite n, f(n) implies f(n+1) if and only if f(n+1) implies f(n) even though f(n) =/= f(n+1) except in the limit.

The Tarski/Leibniz identity definition fits nicely here: all properties held by f(n)'s limit point are held by f(n+1)'s limit point, and the converse. Yet, the non-identical curves approaching this point are equivalent in the sense that each intersects the other at infinity.

So, speaking loosely for a moment, King George had the "curve" x and the "curve" y and wished to know whether they intersected, thus establishing identity. He was uninterested in whether f(y) = y, as this fixed point provided no new information.

Well, how is the king to know whether "curves" x and y intersect? How is he to know whether x = y? Suppose he knows that Scott cannot be ruled out as the author of Waverley. In that case, Waverley implies a set of which Scott is a member and Scott is a member of a set that implies Waverley. So the asserted existence of the novel W strictly implies the asserted existence of the set S. Even so, the set S does not equal the novel W.

But the king in this case in fact wishes to know whether the author of Waverley and Scott pose an identity, which is here indistinguishable from an equivalence. That is, the king is curious as to whether it is so that the anonymous writer has all Scott's properties if and only if Scott has all the anonymous writer's properties.

This simple point brings us forthwith to the notion of deduction, as used by mathematicians and logicians. Rigorous deduction ordinarily requires a set of axioms (some systems include "axiom schema" which are tailored to questions of infinity). That is, we are to use a chain of modus ponens, or implication arrows, or nand connectives (they're all equivalent) to establish the truth value of a claim. This process is cited in the Deduction Theorem, which the reader can find online or in a number of logic textbooks.

The Deduction Theorem is required by our limited minds. If one had a large enough mind, one would not need to prove a claim holds by running through a chain of modus ponens. One could see the chain of reasoning and result all at once. An analogy can be drawn from the case of those people who can read a sentence in one glance as opposed to the majority who read word by word. And we also face the caveat of Goedel's proof that PM and related systems contain assertions that show that the Deduction Theorem can be put into a non-linear loop.

Well, we have discussed identity and equivalence. What about tautology? (Russell's protege, Wittgenstein, was among the first to use truth tables, as well as adopting the word tautology for logical redundancy, though it had long been used to imply rhetorical redundancy.z1)

A tautology describes the situation in which there are no false possibilities for a proposition. All truth table possibilities hold.

For example, the formula P --> (Q --> P) is always true. We simply rewrite this as

~P v (~Q v P), which is,

~P v ~Q v P = (P v ~P) v ~Q

Without resort to a truth table, we can see easily that, no matter what value P or Q has, the formula above always holds.

And so King George's mind was not so large that he could at a glance determine whether x = y is a tautology. Yet it certainly was large enough to determine that y = y is a tautology -- although we can in fact prove that y = y is tautological. That is, "=" implies "<-->" (but not the converse), and so we have y = y implies y <--> y. So y --> y & y --> y. Hence, (~y v y) & (~y v y), which holds for all truth table possibilities. (We have not reduced this last to ~y v y because that would introduce circularity into the proof.)

Of course, our proof relies on the rules of transformation of logical connectives. But a similar proof could have been supplied using either modus ponens arrows only or nand connectives only. This would seem like folderol to His Majesty, but it helps us see more clearly what is at issue in Russell's Waverley problem.

We have belabored the obvious, you say. Yes, but in doing so, we have made more precise our notions of identity, equality, equivalence and tautology. In addition, we have argued for the use of the equivalence relation for the variables of assertions, though then of course we wander into the issue of exactly what constitutes "first order predicate calculus" versus "second order predicate calculus." This last is left for further philosophical efforts.
Another take on this issue is the use of ordered pairs. Consider the set of people who may be writers, as in x e X. As each set is defined by its elements, each element differs from the next and so must have some specific property. If X were singleton, the king wouldn't be curious -- whatever the specific property. If not, then the king is aware that there is some element x that has the property "Scott" and another element y e Y with the property "sole author of Waverly."

So we have a set of ordered pairs, as in X X Y, with elements < x, k >, where k is the constant denoted as the "author of Waverly."

Thus the king's puzzlement is represented by the fact that the set X is non-singleton. He is asking which instantiation of x pairs with k, or which element of X X Y can be expressed as a true assertion, as in f(x,k).
Also see Morris Weitz [3] for a useful discussion of negations and the clarification of the Waverley puzzle and G.E. Moore [4] for a diligent, but largely outdated, overview.
1. "Reply to Criticisms" by K. Goedel in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.

2. "Russell's Mathematical Logic" by B. Russell in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.

2a. Introduction to Logic and to the Methodology of the Deductive Sciences by Alfred Tarski (Dover 1995 reprint of Oxford second edition 1946).

3. "Analysis and the Unity of Russell's Philosophy" by M. Weitz in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.

4. "Russell's 'Theory of Descriptions' " by G.E. Moore in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.

z1. A footnote in Elements of Symbolic Logic by Hans Reichenbach (Macmillan 1947) tells us:
Truth tables were used by L. Wittgenstein, Tractatus Logico-Philosophicus Harcour, Brace, New York, 1922, p. 93, and by E.L. Post, Amer. Journal of Math., XLIII, 1921, p. 163. Materially, the definition of propositional operations in terms of truth or falsehood was used earlier, for instance, in B. Russell and A.N. Whitehead, Principia Mathematica, Vol. I, 1910, p. 6-8. Furthermore, C.S. Pierce employed this definition...
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