## 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.

“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.

## There Is No Fixed Entry Road into a Field

We keep emphasizing that the student must find a personal “driving question,” fortified by a sense of enchantment, to really “parachute” into a body of knowledge. Sitting in front of a nine-hundred page textbook, à la school and college, doesn’t do it.

Schools are mostly enervating at best and one has to productively rebel and “go underground” to “enter” a field as opposed to the effort one makes to survive quizzes and exams.

The sense or intuition that there is no set pathway or on-ramp or entry-road into fields or dimensions or aspects of knowledge but that it is always a self-created process of self-education is best expressed by a famous poem by the great Spanish poet, Antonio Machado (died in 1939):

XXIX

“Traveler, there is no path.
The path is made by walking.

Traveller, the path is your tracks
And nothing more.
Traveller, there is no path
The path is made by walking.
By walking you make a path
And turning, you look back
At a way you will never tread again
Only wakes in the sea.”

(Border of a Dream: Selected Poems of Antonio Marchado, Antonio Machado, Willis Barnstone (translator), Copper Canyon Press, 2003)

## Education and Holism from Day 1

A freshman enters a college whether in the U.S. or Japan or India or Brazil.

We keep trying to offer the view that holism must “rule” the discovery of a “major” or “field” in education.

This holism means placing on the “mental plate in front of the student” these dimensions of reality, considering that every student is first and always a person.

1. “This is your life.” Every person (the predecessor to being a student) is born, lives and dies. Your education is part of this larger truth and you the student, being shrewd in the largest sense, wants to “factor in” the larger frameworks of book-learning (i.e., life itself).
2. Shrewdness cannot be restricted to a ZIP code in a state that will be advantageous income-wise for orthodontists, divorce lawyers or pediatricians. Concern over “ZIP codes” when starting a career is fine and valid. However, it can’t be enough since the student is, as a person, more than a career “Olympic swimmer.”
3. Micro-shrewdness (i.e., career tactics and smarts) has to be supplemented by “macro-shrewdness” (“this is my life” thinking). You have to carry some “enchantment” from your education with you, or the life you have will be desiccated or insipid. Ultimately, you will “outsmart” yourself.

Max Weber (died in 1920), the great sociologist, says of that modern world that it involves “Entzauberung” (disenchantment, where only technical cost-benefit thinking is seen as valid). This Weber insight tells you that a student/person has to find something enchanting to carry him or her through life, its blows and its helter-skelter “shapelessness.”

This is why we “insist” on holism in education everywhere from Day 1 so that these various levels of shrewdness are contemplated together and overspecialized “rabbit holes” are seen more clearly in their limitedness. The process of overview-creating and overview-pondering in the life of every freshman must be combined with “career-cleverness” by itself.

## 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.

## Looking Backwards and Forwards at the Same Time

### Janus and Bi-Directional Smarts

The Roman god Janus looks backwards and forwards at the same time and learning to be somewhat Janus-like is very conducive in the metaintelligence (i.e., larger overview) quest.

There’s a useful French phrase, “reculer pour mieux sauter” which means like a high jumper, you have to take steps backwards to jump higher. In other words, learn to look bi-directionally at the world.

First look back, then forward.

Here’s a concrete example:

W. Arthur Lewis, the “father” of development economics, originally from the Caribbean, taught at Princeton. He won the Nobel in 1979 and wrote various classics such as Growth and Fluctuations, 1870-1913 (1978).

Lewis writes:

In this book we shall not be attempting to give formal or complete explanations of why fluctuations occurred. Like the captain of a ship navigating in stormy seas, we shall need to identify the waves, without needing an exhaustive theory of what causes waves.

When analyzing these fluctuations economists have identified four different cycles, distinguished by length of periodicity, each of which is named after the economist who first wrote about it:

(W. Arthur Lewis, Growth and Fluctuations, 1870-1913, 1978, page 19)

Lewis gives us a quick overview of how we got to the era covered by his book:

“The essence of the industrial and agricultural revolutions in the first three quarters of the nineteenth century was in new ways of doing old things—of making iron, textiles and clothes, of growing cereals, and of transporting goods and services. In the last quarter of the nineteenth century the revolution added a new twist—that of making new commodities: telephones, gramophones, typewriters, cameras, automobiles and so on, a seemingly endless process whose twentieth century additions include aeroplanes, radios, refrigerators, washing machines and pleasure boats.”

(Growth and Fluctuations, 1870-1913, page 29)

Professor Norman Stone in his masterpiece on WWI calls this late nineteenth century explosion of material change and inventions the greatest fast quantum leap in world history in transforming the world.

If one reads these lines with a “Janus mind” we wonder, looking forward from the Lewis book and its era:

1. How does his catchy metaphor of waves in the ocean relate to fluctuations and cycles? When Ben Bernanke (Fed Chair) describes recent decades as “The Great Moderation” does he mean to imply that Lewis-type waves disappeared or got much smaller?
2. Can computers and mobile phones really match cars and planes in profundity of impact? Or is it only the tremendous spread of mobile or smartphones in the Global South that can?

In fact, the recent economic history classic, Robert Gordon’s The Rise and Fall of American Growth argues against the assumption of endless technical change as a growth accelerator or endless frontier:

In the century after the Civil War, an economic revolution improved the American standard of living in ways previously unimaginable. Electric lighting, indoor plumbing, home appliances, motor vehicles, air travel, air conditioning, and television transformed households and workplaces. With medical advances, life expectancy between 1870 and 1970 grew from 45 to 72 years. Weaving together a vivid narrative, historical anecdotes, and economic analysis, The Rise and Fall of American Growth provides an in-depth account of this momentous era. But has that era of unprecedented growth come to an end?

Gordon challenges the view that economic growth can or will continue unabated, and he demonstrates that the life-altering scale of innovations between 1870 and 1970 can’t be repeated. He contends that the nation’s productivity growth, which has already slowed to a crawl, will be further held back by the vexing headwinds of rising inequality, stagnating education, an aging population, and the rising debt of college students and the federal government. Gordon warns that the younger generation may be the first in American history that fails to exceed their parents’ standard of living, and that rather than depend on the great advances of the past, we must find new solutions to overcome the challenges facing us.

A critical voice in the debates over economic stagnation, The Rise and Fall of American Growth is at once a tribute to a century of radical change and a harbinger of tougher times to come.

1. Why does one not read of the four cycles mentioned by Lewis (i.e., Kitchin) and the rest listed above in today’s business and financial press? Has there been some great discontinuity?

If you apply a “Janus mind” to the past (described by Lewis) and our sense of the future (described by techno-pessimists like Gordon) you get a more thoughtful sense of “the human prospect.”