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.