This month’s topics:
Quipus and quantum computing
Frank Wilczek’s column in the the Wall Street Journal (April 14, 2022) had the title “A Quantum Leap, With Strings Attached; The Inca system of quipu—tying a series of knots to record information—is providing a surprising model to modern physics and quantum computing.”
This textile document is a typical quipu in that its data is numerical and recorded in a decimal system. (For another example and more details, see Nicole Rode’s YouTube video from the British Museum). While they were used by earlier pre-Columbian Andean cultures, most of the surviving specimens date from the period of Inca domination, c. 1400–1532 CE. (We can only guess what the numbers recorded on quipus were actually counting —these civilizations left no written records).
Wilczek compares quipus with the information storage and transmission systems we encounter today: “written human language, the binary code of computers and the DNA and RNA sequences of genetics,” and remarks that the Andean system involves “something unique: topology, the science of stable shapes and structures.” In fact the difference between one knot and another, which is contrastive in quipus, to borrow a term from linguistics, is one of the most basic examples of a purely topological concept.
The connection between quipus and “modern physics and computing theory” comes precisely through topology. The equivalents of Andean cords are the world-lines of particles. Wilczek asks us to suppose our particles are only moving in two dimensions, and that we add a third dimension to represent time. Then as time progresses the successive positions of a particle trace out a curve: this is its world-line. And if several particles are observed at once, their world-lines can tangle (“these are not our ancestor’s strings”) and form what mathematicians call braids.
“There are certain particles, called anyons, whose quantum behavior keeps track of the braid that their world-lines form [and therefore could be used to store information]. The anyon world-lines form a quantum quipu.” Wilczek’s terminology has to be taken with a grain of salt. Knots and braids are very different mathematical objects, although fundamentally related (see the drawing above and Alexander’s Theorem on Wikipedia); quipus use one and not the other. Nevertheless it is striking that the topology of curves in space turns up both in an antique recording system and in the latest quantum science.
The science really is very new. Anyons were only experimentally detected two years ago (Wilczek himself had conjectured their existence, and named them, some 40 years back). He tells us that “the simple quantum quipus that were produced in those pioneering experiments can’t store much information” but that only last month “Microsoft researchers announced that they have engineered much more capable anyons.” This is presumably the research described in the Microsoft Research Blog on March 14.
Geometry, a human language?
Siobhan Roberts contributed Is Geometry a Language That Only Humans Know? to the March 22, 2022 New York Times. The subtitle is more specific: “Neuroscientists are exploring whether shapes like squares and rectangles — and our ability to recognize them — are part of what makes our species special.” The neuroscientists in question are Stanislas Dehaene (Université Paris-Saclay and Collège de France) and his collaborators.
The first part of Roberts’s article concerns the research that Dehaene and his team published last year in PNAS: “Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity.” In a typical experiment they report, subjects were presented with a display of polygons. Five of the six were similar, differing only in size and orientation; the sixth was like the others except that its shape had been changed by moving one vertex. Subjects were asked to pick out the oddball.
The “normal” polygons were chosen from a family of eleven quadrilaterals that can be ranked, starting with a square, by how unsymmetrical they are.
The first experiment, involving 605 French adults, showed that the number of errors they made “varied massively” with the lack of symmetry/orthogonality/parallelism of the “normal” exemplar.
The team repeated the experiment with French kindergarteners and with Himba adults (“a pastoral people of northern Namibia whose language contains no words for geometric shapes, who receive little or no formal education, and who, unlike French subjects, do not live in a carpentered world.”) The results correlated strongly with those of French adults. “Both findings converge with previous work to suggest that the geometric regularity effect reflects a universal intuition of geometry that is present in all humans and is largely independent of formal knowledge, language, schooling, and environment.”
The experimenters had access to a colony of baboons (Papio papio) in the south of France; they managed to train the baboons to where they had a “clear understanding of the task”—they could recognize the oddball apple in a group of watermelon slices, and even a regular hexagon in a group of non-convex polygons, but “although error rates differed across the 11 shapes, with a consistent ordering across baboons, […] they correlated weakly and nonsignificantly with the geometric regularity effect found in human populations.”
After speaking with Moira Dillon (a psychologist at New York University) Roberts puts this research in a historical context: “Plato believed that humans were uniquely attuned to geometry; the linguist Noam Chomsky has argued that language is a biologically rooted human capacity. Dr. Dehaene aims to do for geometry what Dr. Chomsky did for language.” But Frans de Waal (a primatologist at Emory University) cautioned her: “Whether this difference in perception amounts to human ‘singularity’ would have to await research on our closest primate relatives, the apes.”
Roberts reviews connections between this research and work in artificial intelligence, and then moves on to Dehaene et al.‘s latest project, essentially figuring out what in the human mind makes geometric regularity so significant. Here’s a clue, quoting from Dehaene: “We postulate that when you look at a geometric shape, you immediately have a mental program for it. You understand it, inasmuch as you have a program to reproduce it.” The team explored an algorithm, DreamCoder (the authors overlap with Dehaene’s collaborators) that “finds, or learns, the shortest possible program for [drawing] any given shape or pattern.” Then they tested human subjects on the same shapes. “The researchers found that the more complex a shape and the longer the program, the more difficulty a subject had remembering it or discriminating it from others.”
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