Tag: geometry

Knot Theory and the Strangeness of Reality

The subfield of “knot theory” in math as a kind of geometry of “twistiness” gives us a deep “meta-intelligent” signal or lesson.

Meta-intelligent means “perspective-challenging” with or without full details of any subfield itself.

Consider this overview or comment on “knot theory” now:

“In mathematical knot theory, you throw everything out that’s related to mechanics,” Dunkel (MIT math professor) says. “You don’t care about whether you have a stiff versus soft fiber—it’s the same knot from a mathematician’s point of view. But we wanted to see if we could add something to the mathematical modeling of knots that accounts for their mechanical properties, to be able to say why one knot is stronger than another.”

But you immediately think: in the real world knots are not only twisted up in mathematically definable ways but are in fact actual shoelaces, neckties, ropes, etc, that have chemical and molecular properties before you describe their twist-and-tighten or slide-and-grip “shapes.”

Which is the real: the math or the “ropiness” of the ropes or the “laciness” of the laces?

The relationship between things and numbers is elusive.

Mathematicians have long been intrigued by knots, so much so that physical knots have inspired an entire subfield of topology known as knot theory—the study of theoretical knots whose ends, unlike actual knots, are joined to form a continuous pattern.

In knot theory, mathematicians seek to describe a knot in mathematical terms, along with all the ways that it can be twisted or deformed while still retaining its topology, or general geometry.

MIT mathematicians and engineers have developed a mathematical model that predicts how stable a knot is, based on several key properties, including the number of crossings involved and the direction in which the rope segments twist as the knot is pulled tight.

“These subtle differences between knots critically determine whether a knot is strong or not,” says Jörn Dunkel, associate professor of mathematics at MIT. “With this model, you should be able to look at two knots that are almost identical, and be able to say which is the better one.”

“Empirical knowledge refined over centuries has crystallized out what the best knots are,” adds Mathias Kolle, the Rockwell International Career Development Associate Professor at MIT. “And now the model shows why.”

As per usual in science, one is dazzled by the ingenuity of the quest and the formulations but puzzled by the larger implications since we can never decide whether math “made” us or we “made” (i.e., invented) math.

Education and “The Three-Body Problem”

The brilliant math-watcher, Ian Stewart, says of this classic physics problem, the Three-Body Problem:

Newton’s Law of Gravity runs into problems with three bodies (earth, moon, sun, say).

In particular, the gravitational interaction of a mere three bodies, assumed to obey Newton’s inverse square law of gravity, stumped the mathematical world for centuries.

It still does, if what you want is a nice formula for the orbits of those bodies. In fact, we now know that three-body dynamics is chaotic–so irregular that is has elements of randomness.

There is no tidy geometric characterization of three-body orbits, not even a formula in coordinate geometry.

Until the late nineteenth century, very little was known about the motion of three celestial bodies, even if one of them were so tiny that its mass could be ignored.

(Visions of Infinity: The Great Mathematical Problems, Ian Stewart, Basic Books, 2014, page 136)

Henri Poincaré, the great mathematician, wrestled with this with tremendous intricacy and ingenuity all his life:

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as “The Last Universalist,” since he excelled in all fields of the discipline as it existed during his lifetime.

Born: April 29, 1854, Nancy, France
Died: July 17, 1912, Paris, France.

We now think of applying in an evocative and not a rigorous mathematical way, the unexpected difficulties of the three-body problem to the n-body (i.e., more than three) problems of sociology or economics or history itself, and sense that social life is always multifactorial and not readily pin-downable, since “everything is causing everything else” and extracting mono-causal explanations must be elusive for all the planetary and Poincaré reasons and beyond.

This suggests to the student that novels are one attempt to say something about n-body human “orbits” based on “n-body” stances and “circumstances” with large amounts of randomness governing the untidy mess that dominates human affairs.

Words are deployed in novels and not numbers as in physics, but the “recalcitrance” of the world, social and physical, remains permanent.

Education and meta-intelligence would be more complete by seeing how the world, as someone put it, “won’t meet us halfway.” Remember Ian Stewart’s warning above:

“There is no tidy geometric characterization of three-body orbits…” and you sense that this must apply to human affairs even more deeply.

Education and Causality Changes

Scientific classification
Kingdom:Animalia
Phylum:Chordata
Class:Mammalia
Order:Primates
Suborder:Haplorhini
Infraorder:Simiiformes
Family:Hominidae
Subfamily:Homininae
Tribe:Hominini
Genus:Homo
Species:H. naledi
Binomial name:Homo naledi
Berger et al., 2015

The box above shows you, if you reflect for a moment, how involved and hierarchical taxonomy can be. The box refers to the tremendous fossil finds around 2013 outside Johannesburg, South Africa by Professor Lee Berger of Witwatersrand University and his team and associates in caves nearby.

The fossil finds are determined to be “homo” and not “australo” as you see in the table above (to the right of the word genus).

The PBS NOVA program “Dawn of Humanity” (2015), is about the story of these fossil finds and the interpretations of the finds which are deeply instructive for all knowledge-seekers, students, etc. because they leave behind any idea of a linear clearly branching “tree of life” in favor of the “bushiness” of evolution (no clear tree structure) and the whole process finally seen as a “braided stream.” This refers to a geological concept of the multiple pathways and reticulations of glacial ice and snow melt going down a mountain valley to a lake. The rivulets, channels, are crisscrossing in a “fluvial” flow pattern that is so complex one doesn’t know exactly which “exact” water went into the lake. If you say the lake is “homo sapiens” (humanity) and the swirling bushy tangled flow is the evolutionary raw material, some final causality is elusive.
When a “braided stream” (this kind of glacial water flow) metaphor gets fused with a “bushiness” one, then one sees that the random factors and endless crisscrossing obscures linear mono-causal explanations as we always imagined.

If you imagine a time when these concepts are applied to history, economy, and society you can begin to sense many “causality revolutions” in front of us where today’s textbooks will seem charmingly naive.

Every student, enrolled or not, should ponder the concepts of “braided stream” taxonomy (shown in the initial table above) and “bushiness” as opposed to tree structure. The student might also “walk around” these metaphors and ask what they imply for the “fractal geometry of nature” (a twig is like a little tree or branch on a bigger tree or branch and so on).