Chapter 11Note on Wolfram's principle of computational equivalence

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Below is a bit of philosophy of mathematics. I am well aware that readership will be quite limited.

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Stephen Wolfram has discussed his "principle of computational equivalence" extensively in his book A New Kind of Science and elsewhere. Herewith is this writer's understanding of the reasoning behind the PCE:

1. At least one of Wolfram's cellular automata is known to be Turing complete. That is, given the proper input string, such a system can emulate an arbitrary Turing machine. Hence, such a system emulates a universal Turing machine and is called "universal."

2. One very simple algorithm is Wolfram's CA Rule 110, which Matthew Cook has proved to be Turing complete. Wolfram also asserts that another simple cellular automaton algorithm has been shown to be universal or Turing complete.

3. In general, there is no means of checking to see whether an arbitrary algorithm is Turing complete. This follows from Turing's proof that there is no general way to see whether a Turing machine will halt.

4. Hence, it can be argued that very simple algorithms are quite likely to be Turing complete, but because there is no way to determine this in general, the position taken isn't really a conjecture. Only checking one particular case after another would give any indication of the probability that a simple algorithm is universal.

5. Wolfram's principle of computational equivalence appears to reduce to the intuition that the probability is reasonable -- thinking in terms of geochrons -- that simple algorithms yield high information outputs.

Herewith the writer's comments concerning this principle:

1. Universality of a system does not imply that high information outputs are common (recalling that a bona fide Turing computation's tape halts at a finite number of steps). The normal distribution would seem to cover the situation here. One universal system is some algorithm (perhaps a Turing machine) which produces the function f(n) = n+1. We may regard this as universal in the sense that it prints out every Turing machine description number, which could then, notionally, be executed as a subroutine. Nevertheless, as n approaches infinity, the probability of happening on a description number goes to 0. It may be possible to get better efficiency, but even if one does so, many description numbers are for machines that get stuck or do low information outputs.

2. The notion that two systems in nature might both be universal, or "computationally equivalent," must be balanced against the point that no natural system can be in fact universal, being limited by energy resources and the entropy of the systems. So it is conceptually possible to have two identical systems, one of which has computation power A, based on energy resource x, and the other of which has computation power B, based on energy resource y. Just think of two clone mainframes, one of which must make do with half the electrical power of the other. The point here is that "computational equivalence" may turn out not to be terribly meaningful in nature. The probability of a high information output may be mildly improved if high computation power is fairly common in nature, but it is not easy to see that such outputs would be rather common.

A mathematician friend commented:

I'd only add that we have very reasonable ideas about "most numbers," but these intuitions depend crucially on ordering of an infinite set. For example, if I say, "Most integers are not divisible by 100", you would probably agree that is a reasonable statement. But in fact it's meaningless. For every number you show me that's not divisible by 100, I'll show you 10 numbers that are divisible by 100. I can write an algorithm for a random number generator that yields a lot more numbers that are divisible by 100 than otherwise. "But," you protest, "not every integer output is equally likely under your random number generator." And I'd have to agree, but I'd add that the same is true for any random number generator. They are all infinitely biased in favor of "small" numbers (where "small" may have a different meaning for each random number generator).

Given an ordering of the integers, it is possible to make sense of statements about the probability of a random integer being thus-and-so. And given an ordering of the cellular automata, it's possible to make sense of the statement that "a large fraction of cellular automata are Turing complete."

My reply:

There are 256 cellular automata in NKS. The most obvious way to order each of these is by input bit string, which expresses an integer. That is, the rule operates on a bit-string stacked in a pyramid of m rows. It is my thought that one would have to churn an awfully long time before hitting on a "universal." Matthew Cook's proof of the universalism of CA110 is a proof of principle, and gives no specific case.

As far as I know, there exist few strong clues that could be used to improve the probability that a specific CA is universal. Wolfram argues that those automata that show a pseudorandom string against a background "ether" can be expected to show universality (if one only knew the correct input string). However, let us remember that it is routine for functions to approach chaos via initial values yielding periodic outputs.

So one might need to prove that a set of CA members can only yield periodic outputs before proceeding to assess probabilities of universalism.

Perhaps there is a relatively efficient means of forming CA input values that imply high probability of universalism, but I am unaware of it.

Another thought: Suppose we have the set of successive integers in the interval [1,10]. Then the probability that a randomly chosen set member is even is 1/2. However, if we want to talk about an infinite set of integers, in line with my friend's point, the probability of a randomly selected number being even is meaningless (or, actually, 0, unless we invoke the axiom of choice). Suppose we order the set of natural numbers thus: {1,3,5,7,9,2,11,13,15,17,4...}. So we see that the probability of a specific property depends not only on the ordering, but on an agreement that an observation can only take place for a finite subset.

As my friend points out, perhaps the probability of hitting on a description number doesn't go to 0 with infinity; it depends on the ordering. But, we have not encountered a clever ordering and Wolfram has not presented one.

Chapter 10Drunk and disorderly: The fall and rise of entropy

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Some musings posted Nov. 20, 2010

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One might describe the increase of the entropy ⁰ of a gas to mean that the net vector -- sum of vectors of all particles -- at between time t ₀ and t _n tends toward 0 and that once this equilibrium is reached at t_n , the net vector stays near 0 at any subsequent time. One would expect a nearly 0 net vector if the individual particle vectors are random. This randomness is exactly what one would find in an asymmetrical n-body scenario, where the bodies are close together and about the same size. The difference is that gravity isn't the determinant, but rather collisional kinetic energy. It has been demonstrated that n-body problems can yield orbits that become extraordinarily tangled. The randomness is then of the Chaitin-Kolmogorov variety: determining future position of a particular particle becomes computationally very difficult. And usually, over some time interval, the calculation errors increase to the point that all predictability for a specific particle is lost.

But there is also quantum randomness at work. The direction that an excited photon exits an atom is probabilistic only, meaning that the recoil is random. This recoil vector must be added to the other electric charge recoil vector associated with particle collision -- though its effect is very slight and usually ignored. Further, if one were to observe one or more of the particles, the observation would affect the knowledge of the momentum or position of the observed particles.

Now supposing we keep the gas at a single temperature in a closed container attached via a closed valve to another evacuated container, when we open the valve, the gas expands to fill both containers. This expansion is a consequence of the effectively random behavior of the particles, which on average "find less resistance" in the direction of the vacuum.

In general, gases tend to expand by inverse square, or that is spherically (or really, as a ball), which implies randomization of the molecules.

The drunkard's walk
Consider a computerized random walk (aka "drunkard's walk") in a plane. As n increases, the area covered by the walk tends toward that of a circle. In the infinite limit, there is probability 1 that a perfect circle has been covered (though probability 1 in such cases does not exclude exceptions). So the real question is: what about the n-body problem yields pi-randomness? It is really a statistical question. When enough collisions occur in a sufficiently small volume (or area), the particle vectors tend to cancel each other out.

Let's go down to the pool hall and break a few racks of balls. It is possible to shoot the cue ball in such a way that the rack of balls scatters symmetrically. But in most situations, the cue ball strikes the triangular array at a point that yields an asymmetrical scattering. This is the sensitive dependence on initial conditions associated with mathematical chaos. We also see Chaitin-Kolmogorov complexity enter the picture, because the asymmetry means that for most balls predicting where one will be after a few ricochets is computationally very difficult.

Now suppose we have perfectly inelastic, perfectly spherical pool balls that encounter idealized banks. We also neglect friction. After a few minutes, the asymmetrically scattered balls are "all over the place" in effectively random motion. Now such discrete systems eventually return to their original state: the balls coalesce back into a triangle and then repeat the whole cycle over again, which implies that in fact such a closed system, left to its own devices, requires entropy to decrease , a seeming contradiction of the second law of thermodynamics. But the time scales required mean we needn't hold our breaths waiting. Also, in nature, there are darned few closed systems (and as soon as we see one, it's no longer closed at the quantum level), allowing us to conclude that in the ideal of 0 friction, the pool ball system may become aperiodic, implying the second law in this case holds.

(According to Stephen Wolfram in A New Kind of Science, a billiard ball launched at any irrational angle to the banks of an idealized, frictionless square pool table must visit every bank point [I suppose he excludes the corners]. Since each point must be visited after a discrete time interval, it would take eternity to reach the point of reversibility.) And now, let us exorcize Maxwell's demon, which, though meant to elucidate, to this day bedevils discussions of entropy with outlandish "solutions" to the alleged "problem." Maxwell gave us a thought experiment whereby he posited a little being controlling the valve between canisters. If (in this version of his thought experiment) the gremlin opened the valve to let speedy particles past in one direction only, the little imp could divide the gas into a hot cloud in one canister and a cold cloud in the other. Obviously the energy the gremlin adds is equivalent to adding energy via a heating/cooling system, but Maxwell's point was about the very, very minute possibility that such a bizarre division could occur randomly (or, some would say, pseudo-randomly).

This possibility exists. In fact, as said, in certain idealized closed systems, entropy decrease MUST happen. Such a spontaneous division into hot and cold clouds would also probably happen quite often at the nano-nano-second level. That is, when time intervals are short enough, quantum physics tells us the usual rules go out the window. However, observation of such actions won't occur for such quantum intervals (so there is no change in information or entropy), and as for the "random" chance of observing an extremely high-ordering of gas molecules, even if someone witnessed such an occurrence, not only does the event not conform to a repeatable experiment, no one is likely to believe the report, even if true.

Truly universal?
Can we apply the principle of entropy to the closed system of the universe? A couple of points: We're not absolutely sure the cosmos is a closed system (perhaps, for example, "steady state" creation supplements "big bang" creation). If there is a "big crunch," then, some have speculated, we might expect complete devolution to original states (people would reverse grow from death to birth, for example). If space curvature implies otherwise, the system remains forever open or asymptotically forever open.

However, quantum fuzziness probably rules out such an idealization. Are quantum systems precisely reversible? Yes and no. When one observes a particle collision in an accelerator, one can calculate the reverse paths. However, in line with the Heisenberg uncertainty principle one can never be sure of observing a collision with precisely identical initial conditions. And if we can only rarely, very rarely, replicate the exact initial conditions of the collision, then the same holds for its inverse.

Then there is the question of whether perhaps a many worlds (aka parallel universes) or many histories interpretation of quantum weirdness holds. In the event of a collapse back toward a big crunch, would the cosmos tend toward the exact quantum fluctuations that are thought to have introduced irregularities in the early universe that grew into star and galactic clustering? Or would a different set of fluctuations serve as the attractor, on grounds both sets were and are superposed and one fluctuation is as probable as the other? And, do these fluctuations require a conscious observer, as in John von Neumann's interpretation?

Of course, we face such difficulties when trying to apply physical or mathematical concepts to the entire cosmos. It seems plausible that any system of relations we devise to examine properties of space and time may act like a lens that increases focus in one area while losing precision in another. I.e., a cosmic uncertainty principle.

Conservation of information?
A cosmic uncertainty principle would make information fuzzy. As the Heisenberg uncertainty principle shows, information about a particle's momentum is gained at the expense of information about its position. But, you may respond, the total information is conserved. But wait! Is there a law about the conservation of information? In fact, information cannot be conserved -- in fact can't exist -- without memory, which in the end requires the mind of an observer. In fact, the "law" of increase of entropy says that memories fade and available information decreases. In terms of pure Shannon information, entropy expresses the probability of what we know or don't know.

Thus entropy is introduced by noise entering the signal. In realistic systems, supposing enough time elapses, noise eventually overwhelms the intended signal. For example, what would you say is the likelihood that this essay will be accessible two centuries from now? (I've already lost a group of articles I had posted on the now defunct Yahoo Geocities site.) Or consider Shakespeare's plays. We cannot say with certainty exactly how the original scripts read.

In fact, can we agree with some physicists that a specified volume of space contains a specific quantity of information? I wonder. A Shannon transducer is said to contain a specific quantity of information, but no one can be sure of that prior to someone reading the message and measuring the signal-to-noise ratio.

And quantum uncertainty qualifies as a form of noise, not only insofar as random jiggles in the signal, but also insofar as what signal was sent. If two signals are "transmitted" in quantum superposition, observation randomly determines which signal is read. So one may set up a quantum measurement experiment and say that for a specific volume, the prior information describes the experiment. But quantum uncertainty still says that the experiment cannot be described in a scientifically sensible way. So if we try to extrapolate information about a greater volume from the experiment volume, we begin to lose accuracy until the uncertainty reaches maximum. We see that quantum uncertainty can progressively change the signal-to-noise ratio, meaning entropy increases until the equilibrium level of no knowledge.

This of course would suggest that, from a human vantage point, there can be no exact information quantity for the cosmos.

So this brings us to the argument about whether black holes decrease the entropy of the universe by making it more orderly (i.e., simpler). My take is that a human observer in principle can never see anything enter a black hole. If one were to detect, at a safe distance, an object approaching a black hole, one would observe that its time pulses (its Doeppler shift) would get slower and slower. In fact, the time pulses slow down asymptotic to eternity.

So the information represented by the in-falling object is, from this perspective, never lost. But suppose we agree to an abstraction that eliminates the human observer -- as opposed to a vastly more gifted intelligence. In that case, perhaps the cosmos has an exact quantity of information at t _a. It then makes sense to talk about whether a black hole affects that quantity.

Consider a particle that falls into a black hole. It is said that all the information available about a black hole is comprised of the quantities for its mass and its surface area. Everything this super-intelligence knew about the particle, or ever could know, seemingly, is gone. Information is lost and the cosmos is a simpler, more orderly place, higher in information and in violation of the second law... maybe.

But suppose the particle is a twin of an entangled pair. One particle stays loose while the other is swallowed by the black hole. If we measure, say, the spin of one such particle we would ordinarily automatically know the spin of the other. But who's to tell what the spin is of a particle headed for the gravitational singularity at the black hole's core? So the information about the particle vanishes and entropy increases. This same event however means the orderliness of the universe increases and the entropy decreases. So, which is it? Or is it both. Have no fear, this issue is addressed in the next section.

Oh, and of course we mustn't forget Hawking radiation, whereby a rotating black hole slowly leaks radiation as particles every now and then "tunnel" through the gravitational energy barrier and escape into the remainder cosmos. The mass decreases over eons and eons until -- having previously swallowed everything available -- it eventually evaporates, Hawking conjectures.

A question: suppose an entangled particle escapes the black hole? Is the cosmic information balance sheet rectified? Perhaps, supposing it never reached the singularity. But, what of particles down near the singularity? They perhaps morph as the fields transform into something that existed close to the cosmic big bang. So it seems implausible that the spin information is retained. But, who knows?

Where's that ace?
There is a strong connection between thermodynamic entropy and Shannon information entropy. Consider the randomization of the pool break on the frictionless table after a few minutes. This is the equivalent of shuffling a deck of cards.

Suppose we have an especially sharp-eyed observer who watches where the ace of spades is placed in the deck as shuffling starts. We then have a few relatively simple shuffles. After the first shuffle, he knows to within three cards how far down in the deck the ace is. On the next shuffle he knows where it is with less accuracy. Let's say to a precision of (1/3)(1/3) = 1/9. After some more shuffles his potential error has reachs 1/52, meaning he has no knowledge of the ace's whereabouts.

The increase in entropy occurs from one shuffle to the next. But at the last shuffle, equilibrium has been reached. Further shuffling can never increase his knowledge of where the ace is, meaning the entropy won't decrease.

The runs test gives a measure of randomness ¹ based on the normal distribution of numbers of runs, with the mean at n/2, "Too many" runs are found in one tail and "too few" in another. That is, a high z score implies that the sequence is non-random or "highly ordered." What however is meant by order? (This is where we tackle the conundrum of a decrease in one sort of cosmic information versus an increase in another sort.) Entropy is often defined as the tendency toward decrease of order, and the related idea of information is sometimes thought of as the surprisal value of a digit string. Sometimes a pattern such as HHHH... is considered to have low information because we can easily calculate the nth value (assuming we are using some algorithm to obtain the string). So the Chaitin-Kolmogorov complexity is low, or that is, the information is low. On the other hand a string that by some measure is effectively random is considered here to be highly informative because the observer has almost no chance of knowing the string in detail in advance.

However, we can also take the opposite tack. Using runs testing, most digit strings (multi-value strings can often be transformed, for test purposes, to bi-value strings) are found under the bulge in the runs test bell curve and represent probable randomness. So it is unsurprising to encounter such a string. It is far more surprising to come across a string with far "too few" or far "too many" runs. These highly ordered strings would then be considered to have high information value.

So, once the deck has been sufficiently shuffled the entropy has reached its maximum (equilibrium). What is the probability of drawing four royal flushes? If we aren't considering entropy, we might say it is the same as that for any other 20-card deal. But, a runs test would give a z score of infinity (probability 1 that the deal is non-random) because drawing all high cards is equivalent to tossing a fair coin and getting 20 heads and no tails. If we don't like the infinitude we can posit 21 cards containing 20 high cards and 1 low card. The z score still implies non-randomness with a high degree of confidence.

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0.Taken from a Wikipedia article:
The dimension of thermodynamic entropy is energy divided by temperature, and its SI unit is joules per kelvin.

In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication."

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1. We should caution that the runs test, which works for n1 > 7 and n2 > 7, fails for the pattern HH TT HH TT... This failure seems to be an artifact of the runs test assumption that a usual number of runs is about n/2. I suggest that we simply say that the probability of that pattern is less than or equal to H T H T H T..., a pattern whose z score rises rapidly with n. Other patterns such as HHH TTT HHH... also climb away from the randomness area slowly with n. With these cautions, however, the runs test gives striking results.

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2.Taken from Wikipedia:
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication."

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically-distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

A fair coin has an entropy of one bit. However, if the coin is not fair, then the uncertainty is lower (if asked to bet on the next outcome, we would bet preferentially on the most frequent result), and thus the Shannon entropy is lower. Mathematically, a coin flip is an example of a Bernoulli trial, and its entropy is given by the binary entropy function. A long string of repeating characters has an entropy rate of 0, since every character is predictable. The entropy rate of English text is between 1.0 and 1.5 bits per letter,[1] or as low as 0.6 to 1.3 bits per letter, according to estimates by Shannon based on human experiments.

Chapter 9My aboutface on the Shroud

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This article appeared a number of years (ca. 2005) before I reposted it in 2013 and again in 2020.

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Sometimes I am wrong

I haven't read The Da Vinci Code but...
. . . I have scanned a book by the painter David Hockney, whose internet-driven survey of Renaissance and post-Renaissance art makes a strong case for a trade secret: use of a camera obscura technique for creating precision realism in paintings.

Hockney's book, Secret Knowledge: rediscovering the lost legacy of the old masters, 2001, uses numerous paintings to show that European art guilds possessed this technical ability, which was a closely guarded and prized secret. Eventually the technique, along with the related magic lantern projector, evolved into photography. It's possible the technique also included the use of lenses and mirrors, a topic familiar to Leonardo da Vinci.

Apparently the first European mention of a camera obscura is in Codex Atlanticus.

I didn't know about this when first mulling over the Shroud of Turin controversy and so was quite perplexed as to how such an image could have been formed in the 14th century, when the shroud's existence was first reported. I was mistrustful of the carbon dating, realizing that the Kremlin had a strong motive for deploying its agents to discredit the purported relic.

See my old page

Science, superstition and the Shroud of Turin
http://www.angelfire.com/az3/nuzone/shroud.html
Also,
https://needles515.blogspot.com/2020/09/science-superstition-and-shroud-of-turin.html

But Hockney's book helps to bolster a theory by fellow Brits Lynn Picknell and Clive Prince that the shroud was faked by none other than Leonardo, a scientist, "magician" and intriguer. Their book The Turin Shroud was a major source of inspiration for The Da Vinci Code, it has been reported.

The two are not professional scientists but, in the time-honored tradition of English amateurs, did an interesting sleuthing job.

As they point out, the frontal head image is way out of proportion with the image of the scourged and crucified body. They suggest the face is quite reminiscent of a self-portrait by Leonardo. Yet, two Catholic scientists at the Jet Propulsion Lab who used a computer method in the 1980s to analyze the image had supposedly demonstrated that it was "three-dimensional." But a much more recent analysis, commissioned by Picknell and Prince, found that the "three-dimensionalism" did not hold up. From what I can tell, the Jet Propulsion pair proved that the image was not made by conventional brushwork but that further analysis indicates some type of projection.

Picknell and Prince suggest that Leonardo used projected images of a face and of a body -- perhaps a cadaver that had been inflicted with various crucifixion wounds -- to create a death mask type of impression. But the image collation was imperfect, leaving the head size wrong and the body that of, by Mideast standards, a giant. This is interesting, in that Hockney discovered that the camera obscura art often failed at proportion and depth of field between spliced images, just as when a collage piece is pasted onto a background.

Still the shroud's official history begins in 1358, about a hundred years prior to the presumed Da Vinci hoax. It seems plausible that either some shroud-like relic had passed to a powerful family and that its condition was poor, either because of its age or because it wasn't that convincing upon close inspection. The family then secretly enlisted Leonardo, the theory goes, in order to obtain a really top-notch relic. Remember, relics were big business in those days, being used to generate revenues and political leverage.

For if Leonardo was the forger, we must account for the fact that the highly distinctive "Vignon marks" on the shroud face have been found in Byzantine art dating to the 7th century. I can't help but wonder whether Leonardo only had the Mandylion (the face) to work with, and added the body as a bonus (I've tried scanning the internet for reports of exact descriptions of the shroud prior to da Vinci's time but haven't succeeded).

The Mandylion refers to an image not made by hands. This "image of Edessa" must have been very impressive, considering the esteem in which it was held by Byzantium. Byzantium also was rife with relics and with secret arts -- which included what we'd call technology along with mumbo-jumbo. The Byzantine tradition of iconography may have stemmed from display of the Mandylion.

Ian Wilson, a credentialed historian who seems to favor shroud authenticity, made a good case for the Mandylion having been passed to the Knights Templar -- perhaps when the crusaders sacked Constantinople in 1204. The shroud then showed up in the hands of a descendant of one of the Templars after the order was ruthlessly suppressed. His idea was that the shroud and the Mandylion were the same, but that in the earlier centuries it had been kept folded in four, like a map, with the head on top and had always been displayed that way.

The other possibility is that a convincing relic of only the head was held by the Templars. A discovery at Templecombe, England, in 1951 showed that regional Templar centers kept paintings of a bearded Jesus face, which may well have been copies of a relic that Templar enemies tried to find but couldn't. The Templars had been accused of worshiping a bearded idol.

Well, what made the Mandylion so convincing? A possibility: when the Templars obtained the relic they also obtained a secret book of magical arts that told how to form such an image. This of course implies that Leonardo discovered the technique when examining this manuscript, which may have contained diagrams. Or, it implies that the image was not counterfeited by Leonardo but was a much, much older counterfeit.

Obviously all this is pure speculation. But one cannot deny that the shroud images have a photographic quality but are out of kilter with each other and that the secret of camera obscura projection in Western art seems to stem from Leonardo's studios.

The other point is that the 1988 carbon analysis dated the shroud to the century before Leonardo. If one discounts possible political control of the result, then one is left to wonder how such a relic could have been so skillfully wrought in that era. Leonardo was one of those once-in-a-thousand-year geniuses who had the requisite combination of skills, talents, knowledge and impiety to pull off such a stunt.

Of course, the radiocarbon dating might easily have been off by a hundred years (but, if fairly done, is not likely to have been off by 1300 years).

All in all, I can't be sure exactly what happened, but I am strongly inclined to agree that the shroud was counterfeited by Leonardo based on a previous relic. The previous relic must have been at least "pretty good" or why all the fuss in previous centuries? But, it is hard not to suspect Leonardo's masterful hand in the Shroud of Turin.

Of course, the thing about the shroud is that there is always more to it. More mystery. I know perfectly well that, no matter how good the scientific and historical analysis, trying to nail down a proof one way or the other is a wil o' the wisp.

Table of content of Mind Journeys

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This is the revised edition of November 2022, containing a new 18th chapter.
This e-book contains about 100,000 words.

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• Not suited for schoolwork
• Chapter 1: Toward a signal model of perception
• Chapter 2. Ghost slips Ryle's grasp (Part A)
• Chapter 2. Ghost slips Ryle's grasp (Part B)
• Chapter 2. Ghost slips Ryle's grasp (Part C)
• Chapter 2. Ghost slips Ryle's grasp (Part D)
• Chapter 2. Ghost slips Ryle's grasp (Part E)
• Chapter 3. The subtle error in Roe v Wade
• Chapter 4. The knowledge delusion. A rebuttal of Dawkins
• Chapter 5. Do dice play God?
• Chapter 6. Einstein, Sommerfeld and the twin paradox
• Chapter 7. The cosmos cannot be fully modeled as a Turing machine
• Chapter 8: On Hilbert's sixth problem
• Chapter 9. My aboutface on the Shroud
• Chapter 10. Drunk and disorderly. The fall and rise of entropy
• Chapter 11. Note on Wolfram's principle of computational equivalence
• Chapter 12. Proto-integers and (very) naive sets
• Chapter 13. In search of a blind watchmaker
• Chapter 14. Plato and Cantor vs. Wittgenstein and Brouwer
• Chapter 15. Time thought experiments
• Chapter 16. Brahman as Unknown God
• Chapter 17. Four colors suffice (proposed short proof)
• Chapter 18. Chasing Schrödinger's cat
• Appendix A to Chapter 1. Synchronicity anecdotes
• Appendix B to Chapter 1. 'Experiments' in synchronicity

Chapter 8On Hilbert's sixth problem

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This essay was recovered from a now defunct internet account.

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There is no consensus on whether Hilbert's sixth problem: Can physics be axiomatized? has been answered.

From Wikipedia, we have this statement attributed to Hilbert:

6. Mathematical treatment of the axioms of physics. The investigations of the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

Hilbert proposed his problems near the dawn of the Planck revolution, while the debate was raging about statistical methods and entropy, and the atomic hypothesis. It would be another five years before Einstein conclusively proved the existence of atoms.

It would be another year before Russell discovered the set of all sets paradox, which is similar to Cantor's power set paradox. Though Cantor uncovered this paradox, or perhaps theorem, in the late 1890s, I am uncertain how cognizant of it Hilbert was.

Interestingly, by the 1920s, Zermelo, Fraenkel and Skolem had axiomatized set theory, specifically forbidding that a set could be an element of itself and hence getting rid of the annoying self-referencing issues that so challenged Russell and Whitehead. But, in the early 1930s, along came Goedel and proved that ZF set theory was either inconsistent or incomplete. His proof actually used Russell's Principia Mathematica as a basis, but generalizes to apply to all but very limited mathematical systems of deduction. Since mathematical systems can be defined in terms of ZF, it follows that ZF must contain some theorems that cannot be tracked back to axioms. So the attempt to axiomatize ZF didn't completely succeed.

In turn, it would seem that Goedel, who began his career as a physicist, had knocked the wind out of Problem 6. Of course, many physicists have not accepted this point, arguing that Goedel's incompleteness theorem applies to only one essentially trivial matter.

In a previous essay, I have discussed the impossibility of modeling the universe as a Turing machine. If that argument is correct, then it would seem that Hilbert's sixth problem has been answered. But I propose here to skip the Turing machine concept and use another idea.

Conceptually, if a number is computable, a Turing machine can compute it. Then again Church's lamda calculus, a recursive method, also allegedly could compute any computable. So are the general Turing machine and the lamda calculus equivalent? Church's thesis conjectures that they are, implying that it is unknown whether either misses some computables (rationals or rational approximations to irrationals).

But Boolean algebra is the real-world venue used by computer scientists. If an output can't be computed with a Boolean system, no one will bother with it. So it seems appropriate to define an algorithm as anything that can be modeled by an mxn truth table and its corresponding Boolean statement.

The truth table has a Boolean statement where each element is above the relevant column. So a sequence of truth tables can be redrawn as a single truth table under a statement combined from the sub-statements. If a sequence of truth tables branches into parallel sequences, the parallel sequences can be placed consecutively and recombined with an appropriate connective.

One may ask about more than one simultaneous output value. We regard this as a single output set with n output elements.

So then, if something is computable, we expect that there is some finite mxn truth table and corresponding Boolean statement. Now we already know that Goedel has proved that, for any sufficiently rich system, there is a Boolean statement that is true, but NOT provably so. That is, the statement is constructible using lawful combinations of Boolean symbols, but the statement cannot be derived from axioms without extension of the axioms, which in turn implies another statement that cannot be derived from the extended axioms, ad infinitum.

Hence, not every truth table, and not every algorithm, can be reduced to axioms. That is, there must always be an algorithm or truth table that shows that a "scientific" system of deduction is always either inconsistent or incomplete.

Now suppose we ignore that point and assume that human minds are able to model the universe as an algorithm, perhaps as some mathematico-logical theory; i.e., a group of "cause-effect" logic gates, or specifically, as some mxn truth table. Obviously, we have to account for quantum uncertainty. Yet, suppose we can do that and also suppose that the truth table need only work with rational numbers, perhaps on grounds that continuous phenomena are a convenient fiction and that the universe operates in quantum spurts.

Yet there is another proof of incompleteness. The algorithm, or its associated truth table, is an output value of the universe -- though some might argue that the algorithm is a Platonic idea that one discovers rather than constructs. Still, once scientists arrive at this table, we must agree that the laws of mechanics supposedly were at work so that the thoughts and actions of these scientists were part of a massively complex system of logic gate equivalents.

So then the n-character, grammatically correct Boolean statement for the universe must have itself as an output value. Now, we can regard this statement as a unique number by simply assigning integer values to each element of the set of Boolean symbols. The integers then follow a specific order, yielding a corresponding integer. (The number of symbols n may be regarded as corresponding to some finite time interval.)

Now then, supposing the cosmos is a machine governed by the cosmic program, the cosmic number should be computable by this machine (again the scientists involved acting as relays, logic gates and so forth). However, the machine needs to be instructed to compute this number. So the machine must compute the basis of the "choice." So it must have a program to compute the program that selects which Boolean statement to use, which in turn implies another such program, ad infinitum.

In fact, there are two related issues here: the Boolean algebra used to represent the cosmic physical system requires a set of axioms, such as Hutchinson's postulates, in order to be of service. But how does the program decide which axioms it needs for itself? Similarly, the specific Boolean statement requires its own set of axioms. Again, how does the program decide on the proper axioms?

So then, the cosmos cannot be fully modeled according to normal scientific logic -- though one can use such logic to find intersections of sets of "events." Then one is left to wonder whether a different system of representation might also be valid, though the systems might not be fully equivalent.

At any rate, the verdict is clear: what is normally regarded as the discipline of physics cannot be axiomatized without resort to infinite regression.

So, we now face the possibility that two scientific systems of representation may each be correct and yet not equivalent.

To illustrate this idea, consider the base 10 and base 2 number systems. There are some non-integer rationals in base 10 that cannot be expressed in base 2, although approximation can be made as close as we like. These two systems of representation of rationals are , strictly speaking, not equivalent.

(Cantor's algorithm to compute all the rationals uses the base 10 system. However, he did not show that all base n rationals appear in the base 10 system.)

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Two persons commented.

Unknown said,
Hilbert's Sixth has been solved.

Anonymous said,
A) Your assertion that ZF must contain some theorems which cannot be proved is rather sloppy. A theorem is a proposition which can be proved. You meant to say that ZF contains some propositions which cannot be proved, and whose negations cannot be proved either. This is incompleteness.
B) Goedel's Incompleteness Theorem has nothing to do with whether Physics can be axiomatised. It might have something to do with whether Theoretical Physics can ever be complete. But it might not. After all, Theoretical Physics does not have to include all of ZF. Set theory might go way beyond physical reality, so that Theoretical Physics would not actually contain all of mathematics, and thus Goedel's theorem would not apply. In fact, Goedel also proved a completeness theorem: First order Logic is complete. Now, why would a physicist want second-order quantifiers? So, who knows.

C) Forget Boolean algorithms and Turing Machines. No algorithm can be exactly modelled by a physical machine: there is always some noise. E.g., no square wave, and hence no string of bits, can be completely reliably produced in the real world, and if it could be produced, the information in it, the «signal», still could not be reliably extracted in an error-free fashion by any finite physical apparatus.

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Chapter 7The cosmos cannot be fully modeled as a Turing machine

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¶ This essay, which had been deleted by Angelfire, was recovered via the Wayback Machine.
¶ Please note: The word 'or' is usually used in the following discussion in the inclusive sense.

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Note A, added March 3, 2021
The unspoken assumption behind Principia Mathematica, Lamda calculus, Goedelian recursion theory, Boolean circuit logic and Turing machinery is either

¶ Naive linear time and naive linear space, or
¶ Newton's absolute time and absolute space.

Adopting an Einsteinian view of spacetime or a Hawkensian notion of time becoming asymptotic to a line intersecting an unreachable singularity certainly knocks out any Turing type model.

(We note that Goedel came to view time as a nonexistent illusion in Parmenides's sense. In fact, Goedel was a modern version of Zeno of Elea.)

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Note B, added March 4, 2021

On the cosmic scale, the time issue calls into question the concept of thermodynamic entropy^1a At bottom, standard entropy assumes Newtonian mechanics, which requires one or the other of the assumptions about time given above. If we modify the physics behind entropy with quantum mechanics, we enter the quagmire of quantum weirdness, which certainly undermines both naive and Newtonian concepts of time.

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Many subscribe to the view that the cosmos is essentially a big machine which can be analyzed and understood in terms of other machines.

A well-known machine is the general Turing machine, which is a logic system that can be modified to obtain any discrete-input computation. Richard Feynman, the brilliant physicist, is said to have been fascinated by the question of whether the cosmos is a computer -- originally saying no but later insisting the opposite. As a quantum physicist, Feynmen would have realized that the question was difficult. If the cosmos is a computer, it certainly must be a quantum computer. But what does that certainty mean? Feynman, one assumes, would also have concluded that the cosmos cannot be modeled as a classical computer, or Turing machine.¹

Let's entertain the idea that the cosmos can be represented as a Turing machine or Turing computation. This notion is equivalent to the idea that neo-classical science (including relativity theory) can explain the cosmos. That is, we could conceive of every "neo-classical action" in the cosmos to date -- using absolute cosmic time, if such exists -- as being represented by a huge logic circuit, which in turn can be reduced to some instance (computation) of a Turing algorithm. God wouldn't be playing dice. A logic circuit always follows if-then rules, which we interpret as causation. But, as we know, at the quantum level, if-then rules only work (with respect to the observer) within constraints, so we might very well argue that QM rules out the cosmos being a "classical" computer.

On the other hand, some would respond by arguing that quantum fuzziness is so miniscule on a macroscopic (human) scale, that the cosmos can be quite well represented as a classical machine. That is, the fuzziness cancels out on average. They might also note that quantum fluctuations in electrons do not have any significant effect on the accuracy of computers -- though this may not be true as computer parts head toward the nanometer scale. (My personal position is that there are numerous examples of the scaling up or amplification of quantum effects. "Schrodinger's cat" is the archetypal example.)

Of course, another issue is that the cosmos should itself have a wave function that is a superposition of all possible states -- until observed by someone (who?). (I will not proceed any further on the measurement problem of quantum physics, despite its many fascinating aspects.)

Before going any further on the subject at hand, we note that a Turing machine is finite (although the set of such machines is denumerably infinite). So if one takes the position that the cosmos -- or specifically, the cosmic initial conditions (or "singularity") -- are effectively infinite, then no Turing algorithm can model the cosmos. So let us consider a mechanical computer-robot, A, whose program is a general Turing machine. A is given a program that instructs the robotic part of A to select a specific Turing machine, and to select the finite set of initial values (perhaps the "constants of nature"), that models the cosmos.

What algorithm is used to instruct A to choose a specific cosmos-outcome algorithm and computation? This is a typical chicken-or-the-egg self-referencing question and as such is related to Turing's halting problem, Godel's incompleteness theorem and Russell's paradox.

If there is an algorithm B to select an algorithm A, what algorithm selected B? -- leading us to an infinite regression. Well, suppose that A has the specific cosmic algorithm, with a set of discrete initial input numbers, a priori? That algorithm, call it Tc, and its instance (the finite set of initial input numbers and the computation, which we regard as still running), imply the general Turing algorithm Tg. We know this from the fact that, by assumption, a formalistic description of Alan Turing and his mathematical logic result were implied by Tc. On the other hand, we know that every computable result is programable by modifying Tg. All computable results can be cast in the form of "if-then" logic circuits, as is evident from Turing's result.

So we have

Tc <--> Tg

Though this result isn't clearly paradoxical, it is a bit disquieting in that we have no way of explaining why Turing's result didn't "cause" the universe. That is, why didn't it happen that Tg implied Turing who (which) in turn implied the Big Bang? That is, wouldn't it be just as probable that the universe kicked off as Alan Turing's result, with the Big Bang to follow? (This is not a philisophical question so much as a question of logic.)

Be that as it may, the point is that we have not succeeded in fully modeling the universe as a Turing machine.

The issue in a nutshell: how did the cosmos instruct itself to unfold? Since the universe contains everything, it must contain the instructions for its unfoldment. Hence, we have the Tc instructing its program to be formed.

Another way to say this: If the universe can be modeled as a Turing computation, can it also be modeled as a program? If it can be modeled as a program, can it then be modeled as a robot forming a program and then carrying it out?

In fact, by Godel's incompleteness theorem, we know that the issue of Tc "choosing" itself to run implies that the Tc is a model (mathematically formal theory) that is inconsistent or incomplete. This assertion follows from the fact that the Tc requires a set of axioms in order to exist (and hence "run"). That is, there must be a set of instructions that orders the layout of the logic circuit. However, by Godel's result, the Turing machine is unable to determine a truth value for some statements relating to the axioms without extending the theory ("rewiring the logic circuit") to include a new axiom.

This holds even if Tc = Tg (though such an equality implies a continuity between the program and the computation which perforce bars an accurate model using any Turing machines).

So then, any model of the cosmos as a Boolean logic circuit is inconsistent or incomplete. In other words, a Turing machine cannot fully describe the cosmos.

If by "Theory of Everything" is meant a formal logico-mathematical system built from a finite set of axioms [though, in fact, Zermelo-Frankel set theory includes an infinite subset of axioms], then that TOE is either incomplete or inconsistent. Previously, one might have argued that no one has formally established that a TOE is necessarily rich enough for Godel's incompleteness theorem to be known to apply. Or, as is common, the self-referencing issue is brushed aside as a minor technicality.

Of course, the Church thesis essentially tells us that any logico-mathematical system can be represented as a Turing machine or set of machines and that any logico-mathematical value that can be expressed from such a system can be expressed as a Turing machine output. (Again, Godel puts limits on what a Turing machine can do.)

So, if we accept the Church thesis -- as most logicians do -- then our result says that there is always "something" about the cosmos that Boolean logic -- and hence the standard "scientific method" -- cannot explain.

Even if we try representing "parallel" universes as a denumerable family of computations of one or more Turing algorithms, with the computational instance varying by input values, we face the issue of what would be used to model the master programer.

Similarly, one might imagine a larger "container" universe in which a full model of "our" universe is embedded. Then it might seem that "our" universe could be modeled in principle, even if not modeled by a machine or computation modeled in "our" universe. Of course, then we apply our argument to the container universe, reminding us of the necessity of an infinity of extensions of every sufficiently rich theory in order to incorporate the next stage of axioms and also reminding us that in order to avoid the paradox inherent in the set of all universes, we would have to resort to a Zermelo-Frankel-type axiomatic ban on such a set. Now we arrive at another point: If the universe is modeled as a quantum computation, would not such a framework possibly resolve our difficulty?

If we use a quantum computer and computation to model the universe, we will not be able to use a formal logical system to answer all questions about it, including what we loosely call the "frame" question -- unless we come up with new methods and standards of mathematical proof that go beyond traditional Boolean analysis.

Let us examine the hope expressed in Stephen Wolfram's New Kind of Science that the cosmos can be summarized in some basic rule of the type found in his cellular automata graphs.

We have no reason to dispute Wolfram's claim that his cellular automata rules can be tweaked to mimic any Turing machine. (And it is of considerable interest that he finds specific CA/TM that can be used for a universal machine.)

So if the cosmos can be modeled as a Turing machine then it can be modeled as a cellular automaton. However, a CA always has a first row, where the algorithm starts. So the algorithm's design -- the Turing machine -- must be axiomatic. In that case, the TM has not modeled the design of the TM nor the specific initial conditions, which are both parts of a universe (with that word used in the sense of totality of material existence).

We could of course think of a CA in which the first row is attached to the last row and a cylinder formed. There would be no specific start row. Still, we would need a CA whereby the rule applied with aribitrary row n as a start yields the same total output as the rule applied at arbitrary row m. This might resolve the time problem, but it is yet to be demonstrated that such a CA -- with an extraordinarily complex output -- exists. (Forgive the qualitative term extraordinarily complex. I hope to address this matter elsewhere soon.)

However, even with time out of the way, we still have the problem of the specific rule to be used. What mechanism selects that? Obviously it cannot be something from within the universe. (Shades of Russell's paradox.)

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Footnotes
1a. For more on this, see The Janus Point -- A New Theory of Time by Julian Barbour (Basic Books 2020).

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1. Informally, one can think of a general Turing machine as a set of logic gates that can compose any Boolean network. That is, we have a set of gates such as "not", "and," "or," "exclusive or," "copy," and so forth. If-then is set up as "not-P or Q," where P and Q themselves are networks constructed from such gates. A specific Turing machine will then yield the same computation as a specific logic circuit composed of the sequence of gates.

By this, we can number any computable output by its gates. Assuming we have less than 10 gates (which is more than necessary), we can assign a base-10 digit to each gate. In that case, the code number of the circuit is simply the digit string representing the sequence of gates.

Note that circuit A and circuit B may yield the same computation. Still, there is a countable infinity of such programs, though, if we use any real for an input value, we would have an uncountable infinity of outputs. But this cannot be, because an algorithm for producing a real number in a finite number of steps can only produce a rational approximation of an irrational. Hence, there is only a countable number of outputs.

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¶ Dave Selke, an electrical engineer with a computer background, has made a number of interesting comments concerning this page, spurring me to redo the argument in another form. The new essay is entitled On Hilbert's sixth problem.

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¶ Thanks to Josh Mitteldorf, a mathematician and physicist, for his incisive and helpful comments. Based upon a previous draft, Dr. Mitteldorf said he believes I have shown that, if the universe is finite, it cannot be modeled by a subset of itself but he expressed wariness over the merit of this point.

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Discussion of Wolfram cellular automata was added to Draft 3. The notes at the beginning of the chapter were added in Draft 4.

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Chapter 6Einstein, Sommerfeld and the twin paradox

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This essay is in no way intended to impugn the important contributions of Einstein or other physicists. Everyone makes errors.

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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 relativity¹, George Gamow in an essay and Roger Penrose in Road to Reality², and, most recently, Leonard Susskind in The Black Hole War.³

Among those who have correctly understood the paradox are topologist Jeff Weeks (see link above) and science writer Stan Gibilisco⁴, who noted that the general theory of relativity resolves the problem.

As far back as the 1960s, the British physicist Herbert Dingle⁵ called the inconsistency a "regrettable error" and was deluged with "disproofs" of his assertion from the physics community. (It should be noted that Dingle's 1949 attempt at relativistic physics left Einstein bemused.⁶) 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

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 oberserver 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, |v_b - v_a| 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.

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 interesting to note here 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?
Now does GR resolve 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, 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 (see Janssen above) 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 r^x, 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, m_in = km_gr, 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 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 Abraham Pais notes that a colleague dismissed Einstein's work as unoriginal, and Einstein then quickly dropped it⁷. 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.

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Footnotes

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Apologies for footnotes being out of order. Haven't time to fix.

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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).
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 (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 1949 in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp. Dingle used this forum to propound a novel extension of special relativity which 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 Albert Einstein: Philosopher-Scientist edited by Paul Arthur Schilpp (1949).

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This paper was updated on Dec. 10, 2009