Education and Word and Number Hidden Vagueness

These mini-essays help students of any age to re-understand education in a deeper and more connected way.

They look for “circum-spective” intelligence. (Not in the sense of prudential or cautious but in the sense of “around-looking.”)

One of the things to begin to see is that explaining things in schools is misleading “ab initio” (i.e., from the beginning).

Let’s do an example:

In basic algebra, you’re asked: what happens to (x2 – 1)/(x – 1) as x “goes to” (i.e., becomes) 1.

If you look at the numerator (thing on top), x2 is also 1 (since 1 times 1 is 1) and (1 – 1) is zero. The denominator is also (1 – 1) and zero.

Thus you get 0 divided by 0.

You’re then told that’s a no-no and that’s because zeros and infinities lead to all kinds of arithmetic “bad behavior” or singularities.

You’re then supposed to see that x2 – 1 can be re-written as (x – 1)(x + 1) and since “like cancels like,” you cancel the x – 1 is the numerator and denominator and “get rid” of it.

This leaves simply x + 1. So, as x goes to 1, x + 1 goes to 2 and you have a “legitimate” answer and have bypassed the impasse of 0 acting badly (i.e., zero divided by zero).

If you re-understand all this more slowly you’ll see that there are endless potential confusions:

For example: you cannot say that (x2 – 1)/(x – 1) = x + 1 since looking at the two sides of the equal sign shows different expressions which are not equal.

They’re also not really equivalent.

You could say that coming up with x + 1 is a workaround or a “reduced form” or a “downstream rewrite” of (x2 – 1)/(x – 1).

This reminds us of the endless confusions in high school science: if you combine hydrogen gas (H2) with oxygen gas (O2) you don’t get water (H2O). Water is the result of a chemical reaction giving you a compound.

A mixture is not a compound. Chemistry is based on this distinction.

Math and science for that matter, are based on taking a formula or expression (like the one we saw above) and “de-cluttering” it or “shaking loose” a variant form which is not identical and not the same but functionally equivalent in a restricted way.

A lot of students who fail to follow high school or college science sense these and other “language and number” problems of hidden vagueness.
School courses punish students who “muse” to themselves about hidden vagueness. This behavior is pre-defined as “bad woolgathering” but we turn this upside down and claim it is potentially “good woolgathering” and might lead to enchantment which then underlies progress in getting past one’s fear of something like math or science or anything else.

One is surrounded by this layer of reality on all sides, what Wittgenstein calls “philosophy problems which are really language games.”

Think of daily life: you say to someone: “you can count one me.” You mean trust, rely on, depend on, where count on is a “set phrase.” (The origin of the phrase and how it became a set phrase is probably unknowable and lost in the mists of time.)

“You can count on me” does not mean you can stand on me and then count something…one, two, three.

In other words in all kinds of language (English, say, or math as a language) one is constantly “skating over” such logic-and-nuance-and-meaning issues.

The genius Kurt Gödel (Einstein’s walk around buddy at Princeton) saw this in a deep way and said that it’s deeply surprising that languages work at all (spoken, written or mathematical) since the bilateral sharing of these ambiguities would seem deadly to any clarity at all and communication itself would seem a rather unlikely outcome.

You could also say that drama giants of the twentieth century like Pinter, Ionesco and Beckett, intuit these difficulties which then underlie their plays.

All of this together gives you a more “composite” “circum-spective” view of what is really happening in knowledge acquisition.

Education and Pre-Understanding

To embark on an education in any field, physics, say, is enervating because the student (in high school) say, enters a strange ocean with “zillions” of names and laws, units of measurement (amps, ohms, coulombs, faradays, etc.) which are very intricate and confusing.

A student does start swimming in this ocean via school “coercion” (i.e., how one will be punished for “failing.”)

There’s a much deeper and useful and practical way to create a pathway into fields: looking for a pre-understanding of what the field is like by taking one particular question or “head-scratcher” and start to delve into it, welcoming any initial sense of not-being-sure, as part of the fun of it, the enchantment.

Consider this article from 2001 in Scientific American:

“Can somebody finally settle this question: Does water flowing down a drain spin in different directions depending on which hemisphere you’re in? And if so, why?” [Archived PDF]

If you start to worry about the water swirling down your kitchen sink or bathtub, you are inevitably faced with the puzzling discussions of something called Coriolis forces, named after the French scientist of this name:

“In physics, the Coriolis force is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise (or counterclockwise) rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.”

The Coriolis force is called a pseudo-force or fictitious force which is already quite puzzling. It seems to push an ant walking across a 78 RPM record in motion on the turntable in unexpected ways and affects the swirling motions of storm phenomena (hurricanes, cyclones, etc.).

The student would immediately sense that at the heart of physics—using this Coriolis force as an indicator—there’s an unbelievable intricacy—but also the sense that these explanations (i.e., forces versus pseudo-forces) that are not entirely convincing and might well be overturned or re-done by someone with a deeper grasp of the problem, in the future. There’s an “ad hoc-ness” (i.e., the explanations and units and theories and proofs seem somehow “circular” or “tautological” in a way that eludes us, as we wait for a clearer theory).

A person walking across a moving merry-go-round or carousel and the complexity of the pushes and pulls experiences “shoves” that are unfamiliar and the water going down the drain in the bathtub awaits a better theory. There is a subfield called “turbulent flow” and that would need to be brought into it. Weather phenomena like tsunamis, cyclones, etc. are turbulences that are complex and our theories are both unbelievably intricate but perhaps subject to revision.

All of this might be an enchanting “gateway” into physics and would give the student a pre-understanding of physics’s “style of thinking and explaining.” In other words, to “parachute” into a field you need the parachute of some particular puzzling example which you use as a “private gateway” into the way people in that field think and act.

Just to go through the years of high school and college in an endless and mindless “slog” with the “failure gun” of coercion pointed at you, is a tremendously soul-destroying way to educate oneself. You have to “go underground” and find your own pre-understanding and its twin brother or clone, enchantment.

Education and the Problem of Dishonest History

One reason a kind of educational repair or re-education is so necessary is that the simplest truths of world history are never presented clearly and openly.

Here’s an aspect of “global inequality” that is completely overlooked or considered taboo:

One dimension or axis of world history is the world-historical “land question”—which groups “grabbed” gigantic pieces of the land surface of the earth and which didn’t.

Thus: Canada, China, Russia, Brazil, America and India represent territorial “mega-grabs” which typically means “world heft” is in the hands of these big countries which are “monstrous” compared in size to the Andorras, Portugals, Liechensteins and Jamaicas of this world. This question of “who grabbed what” is not allowed in high school or college and is usually “swatted away” by phrases like “manifest destiny.”

The problem is of course that any Putin can and will say that absorbing part or all of Ukraine is Russia’s “manifest destiny.”

A Putin can also invent his own regional “Monroe Doctrine” (i.e., stay out of my sphere of influence as randomly defined by me) and thus we have local (in this case, Russian) reinventions of America’s “Manifest Destiny” and the “Monroe Doctrine.”

This inchoate “relativism” at the heart of human affairs guarantees instability and mayhem and “historical inequality” (i.e., who gets to be “anarcho-lawless” and who doesn’t).

There can’t be a real education without putting on the table, in front of him or her, on their “educational plate” all of these truths, from the personal to the impersonal to the world historical.