Dark Tales Of Lost Civilizations (44 page)

“Yes,” said the Director. “We know all about that. Technological artifacts of similar complexity and workmanship did not reappear until the fourteenth century, when mechanical astronomical clocks appeared in Europe. But we need to focus on the nano.”

“Exactly, sir. Instead of mechanical clocks or macroscopic computational components, we have bits and pieces of designed molecules. Instead of something to analog computer astronomical positions and Olympics, we have something digital and quantum to do, who knows what? These nanofragments are fifty times older than the Antikythera mechanism. There are many more parts. We don’t know if all the parts came from one nanodevice, or two, or a hundred. So what to do?

“Fortunately, the NSA and IID and Los Alamos supercomputers are, in principal, able to put together a jigsaw puzzle with a billion pieces, even if we can’t see the picture on the box they came in. But the combinations are staggering. Just to look at the different ways that ‘n’-identically sized spheres can be packed together in minimal rigidity, so that there are three contacts per particle and at least 3 x n-to-the 6th power total contacts, they could barely get up to ten spheres in the computer technology of 2010 A.D. Minimal rigidity is necessary, but not sufficient, for a structure to be rigid. Due to the large number of packings that must be evaluated, this analytical method is implemented computationally, and near n = 10 we reach the method’s computational limitations. There are exactly 750,352 minimally rigid 10 x 10 adjacency matrices of sphere packings in 3-dimensional space, for example. And the nanofragments are not even identical spheres, either,” said Dr. Weal.

“Bottom line?” said the Director.

“Bottom line, sir, we started with three categories of hypothetical explanation for the tablets.
Time travel
, or
secrets of the ancients
, or
contact with extraterrestrials
. We haven’t completely slammed the door shut on time travel, or on contact with extraterrestrials. But the one I would have said was least likely,
secrets of the ancients
, fits the evidence best out of everything my team can see.”

“Or maybe some combination?” said the Director. “If people were so advanced before the ice age, how can we exclude that they had paleocontact with extraterrestrials?”

“That’s above my pay grade.”

“Okay. Break for lunch and subcommittee data fusion. Report back at 1300 hours. Before we go home, I’ll have a briefing hacked together for the President.”

6. Heironymus Georg Zeuthen

“Now we’ll hear from the Math Semantics Committee. What do we know about what we
don’t know
?”

“Sir, that’s the right question,” said that committee’s chairman. “How to find math beyond what we already know, in the symbols of the tablets? Let me start with someone fairly obscure who died in 1920, but whose writings gave us some tips. Hieronymus Georg Zeuthen was born and began his education in Grimstrup, Denmark. The first research topic on which Zeuthen undertook was
enumerative geometry
.”

“The history of mathematics?” asked Dr. Daisy Weal.

“Yes,” said the committee chairman, “that would seem to have some clues. Hass said the following about Zeuthen’s style:

‘Zeuthen saw things intuitively. He constantly strove to attain an overall conception that would embrace the details of the subject under investigation and afford a way of seizing their significance.’”

“Such as?” the Director asked.

“Zeuthen was an expert on the history of medieval mathematics and produced important studies of Greek mathematics. He wrote numerous papers and books on the history of mathematics, some classics. Unlike many historians of science, Zeuthen explained the ancient texts like a colleague of the ancient mathematicians. In a major work in 1885, he looked in detail at the work of Apollonius on conic sections and showed that Apollonius used oblique coordinates. Zeuthen further argued Pythagoras himself discovered that 2 was irrational when computing the diagonal of a square. The passage from Plato’s
Theaetetus
where it states that Theodorus proved the irrationality of 3, 5, 7, 11, 13, and 17 was also carefully studied by Zeuthen. He suggested that the end of Theodorus’s proof somehow involved the continued fractions for 17 and 19, a conjecture which is very much in line with modern ideas about Greek mathematics. Zeuthen’s largest historical work was published in 1896 on work of Descartes, Viète, Barrow, Newton, and Leibniz, developing algebra, analytic geometry, and analysis.”

“Bottom line?”

“When doing the history of mathematics, he wanted to uncover ideas and motives of ancient masters. These ideas were usually formulated in an unfamiliar language, but the ideas themselves had not changed over time, so a modern mathematician can appreciate the work of a colleague two thousand years earlier. Zeuthen said that one cannot evaluate or understand mathematics of an earlier period on the basis of the mathematics of today. He thought it indispensable to be acquainted with the techniques and symbolism of former times to contrast those tools and what they could be used for with what they
had
actually been used for.”

“Is that how we recognized Clebsch-Gordan coefficients and tables of Calabi-Yau manifolds?”

“Basically. We looked for ideas around the border between what we know from the
Supercompactification Triality Theorem
of 2023, and what we know that we don’t know. Like making a map of the frontiers of ignorance, to get some sense of the shape of what we don’t know.”

“What did they know about physics, beyond Clebsch-Gordan and Calabi-Yau stuff?”

“They saw more deeply than us about how quantum mechanics and relativity connect. These are two of the three big scientific theories of the twentieth century, but the dirty little secret of physics is that they have yet to be reconciled. They seem inherently incompatible. That’s why string theory and quantum loop gravity were developed, yet they still fail to solve the deepest mysteries.

“So, what does all this mean for physics?” continued the math chairman. “The spin factors have an intriguing relation to special relativity, since its cone of positive elements can be revealed to be none other than that of a
future light cone
.”

The Time Travel Committee leaned forward in their chairs at the mention of ‘future light cone.’

“A future light cone is the path of a flash of light emanating from a single event and travelling in all directions of space-time. Furthermore, we can utilize a field of abstract algebra,
Jordan algebra
, which is infinite and has some interesting coincidences to the spin factors.”

“And what do we think in IID about coincidences?” asked the Director.

“Show me an alleged coincidence,” said Dr. Weal,” and I’ll show you someone with something to gain, and a reason to cover it up.”

“Okay. This is not a mere coincidence, but part of the frontiers of ignorance. This is the tip of a huge and still not fully fathomed iceberg,” said the math chairman.

“How does that mystery connect with the physics that these guys knew 3,750 years ago? Or 110,000 years ago?” asked the Director.

“Exceptional Jordan algebra remains mysterious,” said the math chairman. “For example, when it was first found that quarks come in three colors, Okubo and others hoped that 3 x 3 self-adjoint octonionic matrices might serve as observables for these exotic degrees of freedom.”

“Didn’t Zelmanov have something to say about that?” said the IID Director.

“Yes. In 1983, Zelmanov generalized the Jordan–von Neumann–Wigner classification to the infinite-dimensional case. There are two
27-dimensional exceptional Jordan algebras
. There’s the one described above, and its sister, which is defined the same way, but with the so-called split octonions taking the place of the octonions.”

“27-dimensional?”

“Yes, way beyond the 10 or 11 dimensions of string theory and M-theory. Further, Jordan algebra is built upon split octonions and then makes an appearance in string theory. Thus that leads us to the concept of a Hilbert space, which extends the methods of algebra and calculus from a standard two or three-dimensional plane into spaces with any infinite number of dimensions.

“My goal for next session is to tell you how the result of Jordan, von Neumann, and Wigner reappears as a classification of certain cones: cones that can be used to describe nonnegative observables, but also quantum
mixed states
. And here is where we’ll meet state-observable duality.”

“Bottom line?”

“If the deep ancients 110,000 years ago understood physics applications of the two 27-dimensional exceptional Jordan algebras, they might have been able to build a . . . ahem, a warp drive. A spaceship that can go completely outside on Einsteinian space-time and appear somewhere else, absurdly far away, in the universe. They may have reached a technological singularity before the ice age, and some of them might even have left Earth far behind.”

7. We Missed the Singularity by 110,000 Years

“We missed the singularity by 110,000 years?”

“Yes, Mister Director.”

“Then why is there not more evidence?”

The Chairman of the Math Semantics Committee nodded and said, “In higher latitudes, glaciers scraped away archeological traces of whatever cities there may have been. Along sea coasts, settlements were drowned and eroded when the ocean levels rose. Steel rusts and bricks wear. Most of the clay tablets were only preserved because they were in buildings that burned, and so the wet clay was fired, and became very hard and enduring. There must be more evidence, but in places and forms we have not yet found.”

“Are we abusing the terminology in saying that they reached the singularity?”

“Not in the sense of black holes, of course, but in the sense of Stanisław Marcin Ulam, a mathematician, whose work in Los Alamos was mainly concentrated in the theory of relativity as well as quantum theory. His broad scientific interest was not limited to mathematics and physics; it included also technology, computer science, and biology. He is now famous for his research in nuclear physics and also for other non-mathematical achievements; in particular, he developed original methods of propulsion of vessels moving above the Earth’s atmosphere.

“As to the singularity, we take
Vergil Ulam
in Greg Bear’s
Blood Music
, to be a double reference to the Roman poet Virgil and to Stanisław M. Ulam who, as we said, was a mathematician who participated in the Manhattan Project and originated the Teller-Ulam design of thermonuclear weapons. He also invented nuclear pulse propulsion and developed a number of mathematical tools in number theory, set theory, ergodic theory, and algebraic topology. Because Ulam was among the first to refer to the technological singularity—and possibly the originator of the metaphor itself—in May 1958, while referring to a conversation with John von Neumann:

‘One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.’

“Stanisław Ulam, while working at the Los Alamos National Laboratory in the 1940’s, studied the growth of crystals, using a simple lattice network as his model. At the same time, John von Neumann, Ulam’s colleague, was working on the problem of self-replicating systems. Von Neumann’s initial design was founded upon the notion of one robot building another robot. This design is known as the
kinematic model
. As he developed this design, von Neumann came to realize the great difficulty of building a self-replicating robot, and of the great cost in providing the robot with a
sea of parts
from which to build its replicant. Ulam suggested that von Neumann develop his design around a mathematical abstraction, such as the one Ulam used to study crystal growth. Thus was born the first system of cellular automata.”

“Okay, singularity it was. How advanced were they, 117,00 years ago?”

“They had nuclear physics, or at least advanced theory.
Quantum field theory
, or
quantum chromodynamics
, or whatever else gave them the lepton and baryon data. If that was from first principles, they need not have built gigantic accelerators such as the Tevatron or Large Hadron Collider or the Higgs-Hunter in the Australian nullarbor. String theory or M-theory or something beyond that which we have not rediscovered.”

“Are you sure we’re looking in the right places?”

“We’ve used the supercomputers to re-scan archived data on the surface of the moon, Mars and its moons, and asteroids and comet nuclei we’d mapped before. There are some anomalies, but no firm evidence that they expanded out into the solar system. But then again, since they had nanotechnology, we’d hardly see nanodevices from orbit.”

“Assessment of national security threat?”

“We recommend you tell the President there is no need for heightened alert of time machines popping up in the Rose Garden. And if there was, what good would our countermeasures be? Second, there is also no heightened alert for extraterrestrial invaders, though this in no way undermines IID’s long-standing charter. We’ll keep watching the skies, tapping every byte from the deep space network, the great orbital observatories, the neutrino detectors with their Cerenkov radiation photarrays, and isotope decay sensors in the liquid oxygen tanks below the Farside Lunar Observatory. We’re still working the protocols for archaeonanotechnology reconstruction. Obviously when and
if
we rebuild a working nanobot from before the ice age, we’d better be as close to one hundred percent sure as we can, that it won’t self-reproduce and escape and, you know,
gray goo
the world.”