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.

Essay 27: Extracting Signals from Noise

We wish to help students “parachute into” or sneak up on mathematics before any “rocket science” (i.e., focus on high school and don’t get lost in the weeds).

Think of square roots. The square root of 4 is 2, of 9 is three of sixteen is 4.

But the moment someone asks:  what about the square root of 17, you will find that crystalline simplicity and obviousness are long since gone.

If you “keep your nerve” and calmly get into the “complexity jumps” you may well find it enchanting that numerical understanding has to be coaxed forth and doesn’t offer itself up readily.  But why would the physical universe, if it is really mathematical in its very “fabric,” be so ready to “jump away from you” in its complexity?

Go back to the square root of 17. Suppose you disallow logarithms and log tables. Suppose you say that “approximation theory” is too approximate. There’s no HP Scientific Calculator. One insight you might need is that 17 is “not far” from 16 so that the square root must be 4 plus a little. Call this “little” x.

Then (4+x) (4+x) = 17. You solve for x with the quadratic formula you had in high school. If you don’t remember the formula or cannot derive it, you’d have to look it up which might be disallowed in this “game.”

If you stand back, “meta-intelligently” (i.e., asking, “what does this tell me?”), you wonder whether the universe and its math fabric are an endless “onion” of such layers and complexity and not “boil-down-able.”

Another such example is Grandi’s series.

Grandi’s series and its trickiness:

In 1703, the mathematician Luigi Guido Grandi studied the addition: 1 – 1 + 1 – 1 + … ( 1-1, infinitely many, always +1 and –1).  You find if you group the numbers in certain valid and legitimate way, you could different results.  How can that be?

In mathematics, the infinite series, 1 − 1 + 1 − 1 + ⋯, can also be written:

Grandi’s series

It is sometimes called Grandi’s series, after Italian mathematician, philosopher, and priest Luigi Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. (On the other hand, its Cesàro sum is ½.)

One obvious method to attack the series (

One obvious method to attack the series (1 − 1 + 1 − 1 + ⋯) is to treat it like a telescoping series and perform the subtractions in place:

(1 − 1) + (1 − 1) + (1 − 1) + … = 0 + 0 + 0 + … = 0

On the other hand, a similar bracketing procedure leads to the apparently contradictory result:

1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + … = 1 + 0 + 0 + 0 + … = 1

Thus, by applying parentheses to Grandi’s series in different ways, one can obtain either 0 or 1 as a “value.” (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)