Tony Phillips

Tony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

Ron Graham obituaries

"Ron Graham Dazzled Admirers With Math and Juggling Feats" was the headline for James Hagerty's July 17, 2020 obituary in the Wall Street Journal. Graham, who died July 6, was a pioneer in discrete mathematics. But there was much more, as Hagerty's headline implies. While at the University of Chicago, which he entered at age 15, he "helped support himself by performing in a circus act known as the Bouncing Bears," and he continued kinetic activity his whole life. "Blond, handsome and trim, he was always eager to amaze onlookers and excelled at gymnastics, trampoline tricks, table tennis, flying kites, riding unicycles and throwing boomerangs ... .' Motion stimulated his mind. 'I once had a flash of insight into a stubborn problem in the middle of a back somersault with a triple twist on my trampoline,' he said." Hagerty sketches Graham's mathematical expertise (he mentions Ramsey theory and graph theory) and his friendship and collaboration with Paul Erdös, "a wandering Hungarian mathematical genius;" he leaves us with one of Graham's sobering pronouncements: "The fact that we're humans is really pretty limiting. I mean, we didn't evolve to understand the structure of the space-time continuum or do things in 100,000 dimensions. We know how to stay out of the rain and keep [from] getting eaten by animals."

Kenneth Chang, writing for the New York Times (July 23), goes into more mathematical detail. "When they met, Dr. Graham and Dr. Erdös were among the few working in discrete mathematics, particularly in an area known as combinatorics — the mathematics of combinations.

  • "In an introductory probability class, a simple combinatorics problem might ask: If one pulls three balls at random out of a bag that contains six blue ones and four red ones, what are the chances that all three are red? (The answer is 1 out of 30.) Combinatorics proved to be important to the rise of digital technology in the 1970s. ...

"[Combinatorics] led to what became known as Graham's number, which was for a time the largest number used in a proof, according to the Guinness Book of World Records. The number came out of a problem [in a field] known as ... Ramsey theory, which states that in large systems there can never be complete disorder, that pockets of structure will appear within the apparent chaos.

  • "Dr. Graham was looking at cubes in which the lines between the corners were colored red or blue. In a three-dimensional cube, it is easy to color the lines so that no planar slice of the cube with four vertexes has edges all of one color. But mathematicians can also imagine cubes in four dimensions and greater, and so Dr. Graham wanted to know whether this property of being able to avoid slices of one color would persist in greater dimensions. 'The answer: no,' Dr. Graham explained in 2014 in an episode of Numberphile. 'If the dimension is large enough, you cannot avoid it. No matter how you color it, you cannot avoid it.' No one knows in precisely what dimension this unavoidability would kick in, but Dr. Graham calculated an upper bound for the answer — a number so huge that there is not enough space in the entire universe in which to write all of the digits." [In the video, Graham mentions that dimension 13 might be large enough, but that since the number of red-blue colorings of the edges of a 13-dimensional cube is $2^{33,550,336}$, we may never know.]

Dogs' ages in human years: a new rule

"Quantitative Translation of Dog-to-Human Aging by Conserved Remodeling of the DNA Methylome," by Trey Ikert (UCSD) and 10 collaborators, was published in Cell Systems on July 2, 2020, but the preprint posted on bioRxiv was picked up last November by Virginia Morell for Science, with the title: "Here's a better way to convert dog years to human years, scientists say." No more multiplying by 7; the new science-based algorithm is $$\mbox{age}_{\mbox{human}}= 16 \ln(\mbox{age}_{\mbox{dog}}) + 31. $$ (Natural logarithms are more than most people can do in their heads, but TheBark.com —"dog is my copilot"— has a handy table).

dog/human age

The new curve (blue), compared with the old multiply-by-seven rule (red) shows how dogs progress through juvenile and adolescent stages much faster than humans, but how the relative aging factor (by calculus, $16/\mbox{age}_{\mbox{dog}}$) decreases to 2 for an eight-year-old dog.

Morell explains: "The work is based on a relatively new concept in aging research," involving "chemical modifications to a person's DNA over a lifetime." In particular "one such modification, the addition of methyl groups to specific DNA sequences, tracks human biological age." It establishes what is called an epigenetic clock. "Other species also undergo DNA methylation as they age. Mice, chimpanzees, wolves, and dogs, for example, all seem to have epigenetic clocks. To find out how those clocks differ from the human version, [Ideker and colleagues] started with dogs.

"All dogs–no matter the breed–follow a similar developmental trajectory, reaching puberty around 10 months and dying before age 20. ... Ideker's team focused on a single breed: Labrador retrievers. They scanned DNA methylation patterns in the genomes of 104 dogs, ranging from 4 weeks to 16 years of age. Their analysis revealed that dogs (at least Labrador retrievers) and humans do have similar age-related methylation of certain genomic regions with high mutation rates; those similarities were most apparent when the scientists looked at young dogs and young humans or old dogs and old humans.

"Most importantly, they found that certain groups of genes involved in development are similarly methylated during aging in both species. That suggests at least some aspects of aging are a continuation of development rather than a distinct process—and that at least some of these changes are evolutionarily conserved in mammals.

"The research team also used the rate of the methylation changes in dogs to match it to the human epigenetic clock," hence the new formula (applies to dogs older than one, and most accurately to labradors).

Language learning: an emergent phenomenon?

A "white paper" in Le Monde (May 27, 2020) had the title: "Language, an explosive emergence." The authors, Wiebke Drenckhan and Jean Farago (Physics, Strasbourg) are reporting on "Random Language Model", published by Eric DeGiuli (ENS) (Phys. Rev. Lett 122, 128301) in March, 2019 (preprint here). Drenckhan and Farago begin by reminding us that the "miraculous gift" that allows very young children to pick up their native language, with an ease and speed that amazes adults attempting the same task, has long been a subject of scientific speculation. "The Principles and Parameters theory of the linguist Noam Chomsky proposes a 'pre-wiring' of our brain with universal grammatical principles." They object that this idea "implies an enormous waste and an ab initio complexity, whereas nature most often prefers simplicity and the emergence of complexity through interactions (leaf shapes, fur markings, ...)."

"[DeGiuli's theory] is built on the observation made by linguists that sentences have the structure of trees, with the peculiarity that the nodes on these trees (where a branch divides in two) correspond to invisible entities, namely, logical connectors between two functional elements."

tree bear cave

An English sentence with its "invisible" tree structure showing. Image adapted from Phys. Rev. Lett. 122, 128301.

Drenckhan and Farago: "The originality of DeGiuli's approach is to make this linguistic framework 'work' with rules from statistical physics, mimicking in a clever and considerably simplified way the learning process in babies. In this model, words and logical connectors pre-exist (actually, at least the first are the result of a parallel learning process). All sentences are possible, but not all have the same 'acceptability' — a kind of 'grade' which can be changed by the process in order to penalize aberrant grammatical constructions ... .

"The learning process itself is represented schematically by a random (but more and more limited as time passes) exploration of the values of the possible 'grades.' This exploration is not directed (as it would be during real learning) but it conserves an essential trait that can explain something that still puzzles today's linguists: when the randomness of the possible 'grades' becomes sufficiently weak, the corpus of 'acceptable' sentences shrinks drastically and starts to convey information.

"This threshold phenomenon, a phase transition, is well known in physics. ... Here it would be the underlying phenomenon explaining the very sudden appearance of language in children." [My translation - TP].

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

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