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.

Two Kinds of Extra Understanding: Pre and Post

We argue here in this proposal for an educational remedy that two dimensions of understanding must be added to “retro-fit” education.

In the first addition, call it pre-understanding, a student is given an overview not only of the field but of his or her life as well as the “techno-commercial” environment that characterizes the globe.

Pre-understanding includes such “overall cautions” offered to you by Calderón de la Barca’s 17th century classic Spanish play, Life is a Dream (SpanishLa vida es sueño). A student would perhaps ask: “what would it be like if I faced this “dreamlike quality” of life, as shown by the Spanish play, and suddenly realized that a life of “perfect myopia” is not what I want.

Hannah Arendt warns similarly of a life “like a leaf in the whirlwind of time.”

Again, I, the student ask: do I want such a Hannah Arendt-type leaf-in-the-whirlwind-like life, buried further under Calderón de la Barca’s “dream state”?

But that’s not all: while I’m learning about these “life dangers,” all around me from my block to the whole world, humanity does its “techno-commerce” via container ships and robots, hundreds of millions of vehicles and smartphones, multilateral exchange rates, and tariff policies. Real understanding has one eye on the personal and the other on the impersonal and not one or the other.

All of these personal and impersonal layers of the full truth must be faced and followed, “en face,” as they say in French (i.e., “without blinking”).

Call all this pre-understanding which includes of course a sense of how my “field” or major or concentration fits into the “architecture of knowledge” and not in isolation without connections or a “ramification structure.”

Post-understanding comes from the other end: my lifelong effort, after just about all that I learned about the six wives of King Henry VIII and the “mean value theorem”/Rolle’s theorem in freshman math, have been completely forgotten and have utterly evaporated in my mind, to re-understand my life and times and book-learning.

Pre-and post-understanding together allows the Wittgenstein phenomenon of “light falls gradually over the whole.”

Without these deeper dimensions of educational remedy, the student as a person would mostly stumble from “pillar to post” with “perfect myopia.” Education mostly adds to all the “fragmentariness” of the modern world and is in that sense, incomplete or even disorienting.

Education in this deep sense is supposed to be the antidote to this overall sense of modern “shapelessness,” to use Kierkegaard’s term.

Education and the Pursuit of Improved Overviews

Professor Sherman Stein was a prominent mathematician and popularizer, and his book, Mathematics: The Man-Made Universe, is a modern classic. The subtitle “The Man-made Universe” already tells you that you’re looking at a clear exposition of “humans made math” in contrast to the “mathematics fundamentalism,” à la Professor Max Tegmark of MIT, whose tone seems to say mathematics allowed for reality and us.

This is of course a perfect “argument without end.” This is the kind of argument that should help a student to rethink their assumptions and not obsess about some once-and-for-all final understanding which can become an “idée fixe” (i.e., fixed idea in French, indicating being overly rigid or stuck).

In the preface to Professor Stein’s mathematics survey classic, he writes:

“We all find ourselves in a world we never made. Though we become used to the kitchen sink, we do not understand the atoms that compose it. The kitchen sink, like all the objects surrounding us, is a convenient abstraction.

Mathematics, on the other hand is completely the work of man.

Each theorem, each proof, is the product of the human mind. In mathematics all the cards can be put on the table.

In this sense, mathematics is concrete whereas the world is abstract.”

(Sherman Stein, Mathematics The Man-Made Universe, Dover Publications, “Preface” Third Edition, page XIII, 1999)

Meta-intelligence tells you if views of what is real, what is concrete, what is abstract, what is man-made, what is mathematical, are so radically different depending on the interpreter or analyst, it makes prudent sense to keep various views in one’s mind and modify them or juggle them as you go along. Our ability as a species to nail down for eternity what the nature of mathematics, humans and kitchen sinks are and how they all interrelate, is elusive and tangled up in language, as Wittgenstein keeps saying.

Education: Linguistic and Arithmetic Elusiveness

We wish to sensitize the student to the obvious-but-hard-to-see truth that both language and arithmetic have slippery natures built into them and seeing this clearly is a part of deeper education, our mission here.

Take four simple statements and see that they’re entwined and “confusing.”

  1. You can count (i.e., numeracy).
  2. You can count (depend) on me.
  3. You don’t count (i.e., importance).
  4. Count (include) me in.

When a person says, “you can count on me” do they mean that you will be standing on me and then go, “one apple, two apples, three apples” (i.e., count in the everyday sense). No, obviously not. “On” in this context is not physical or locational, but figurative. Ask yourself: how is it that you know the difference and nuances of all these meanings given that the word count and the preposition “on” seem straightforward but are really “polyvalent.”

Wittgenstein tells us that philosophy and its conundrums are ultimately based on “language games.”

When Gadamer (Heidegger’s student) tells us that “man is a linguistic creature” he means, among other things, that man “swims” in this ambiguity ocean every moment and puns and jokes aside, handles these ambiguities automatically, somehow. How does a child acquiring language get the sense of all this? It’s difficult to understand and explain. Language is both our nature and somehow beyond our grasp.

The same slipperiness, in a different way, holds for arithmetic and numbers. You can immediately see that the square root of 16 is 4 (plus or minus) but if you are asked, “what is the square root of seventeen?” you’d be “at sea” without a calculator. If you’re now asked, what is the square root of -17 (negative seventeen), you would probably be lost.

These would seem to be very basic “operations” and yet are baffling in their way and parallel the “sudden difficulties” in language use and orientation and clarity.

Deep and “meta-intelligent” education, which we promote here, begins by seeing, among other things, that both our ability to function while “swimming” among words and numbers is puzzling if you look at them “freshly.”

It’s also not so easy to define exactly what reading and writing are in the first place or why exactly the smile in Leonardo da Vinci’s Mona Lisa painting is enigmatic.

When one glimpses the truth that we are surrounded by obvious things that are never really obvious, one pauses and thinks. This is where (self) “re-education” begins, especially if “enchantment” (genuine magical fascination) accompanies the thinking.

Interesting Intuition from Marx

There’s an intriguing and puzzling quote from Marx which is very informative in a completely unexpected way, when Marx says:

Life is not determined by consciousness, but consciousness by life.”
(The German Ideology, International Publishers, 1970, page 47)

This seems to fit Marx’s obsession with practical circumstances as the “driver” and ideas and subjective states of mind as secondary or even derivative. For Marx, culture and consciousness are “epiphenomena” like the foam on a wave.

In a different way, modern philosophers have their own versions of this:

  1. For Wittgenstein, “forms of life” come first before all else.
  2. For Husserl, “the life-world” comes before theory.
  3. For Heidegger, “being-in-the-world” comes before theory.

Marx’s reduction of everything to material circumstances as primary causes of everything would seem to these other philosophers as a kind of extremist monomania on Marx’s part, as when he says:

We do not set out from what men say, imagine, conceive, nor from men as narrated, thought of, imagined, conceived, in order to arrive at men in the flesh. We set out from real, active men, and on the basis of their real life-process we demonstrate the development of the ideological reflexes and echoes of this life-process. Their material life-process dominates.

The phantoms formed in the human brain are also, necessarily, sublimates of their material life-process which is empirically verifiable and bound to material premises. Morality, religion, metaphysics, all the rest of ideology and their corresponding forms of consciousness, thus no longer retain the semblance of independence.

Karl Marx with Friedrich Engels, The German Ideology, page 47

This idea from Marx is both suggestive and obsessive and maniacal at the same time, what the French call an “idée fixe” or fixation.

It is more accurate to say perhaps that life and consciousness are a “double helix.”

Education and Wittgenstein “Language Games”

It is instructive for a student to get a grip on the whole question of “language games” à la Wittgenstein, who says that these “games” (i.e., ambiguities) are central to thinking in general and thinking about philosophy in particular.

Let’s make up our own example and step back from the meaning of the preposition “in.”

The comb is in my back pocket has nothing to do with the “in” of “he’s in a good mood” or “he’s in a hurry” or “he’s in a jam or pickle” or “he’s in trouble.” Furthermore, in modern deterministic neuroscience language, a good mood is a footnote to brain and blood chemicals so that means that a good mood is in you via chemicals and not you in it.

Does the word “jam” here mean difficulty or somehow the condiment called jam? You don’t know and can never without more information (i.e., meaningful context).

Imagine we take a time machine and are standing in front of the home of Charles Dickens in London in his time say in the 1840s. They say he’s working on a new novel called Oliver Twist.

Someone says: a novel by Dickens is a kind of “fictional universe.” Shall we say that because Dickens is in his home (at home) in London (though in London is itself confusing since London as a city is not like a pocket to a comb or wallet) his fictional universe is “in” the universe which might be a multiverse according to current cosmological speculations? That’s not what we mean. The fictional universe of Dickens is a shared cultural abstraction involving his stories, characters, people absorbing his tales, his mind and our mind, books and discussions. A fictional universe is as “weird” as the other universe. The preposition “in” does not begin to capture what’s going on which is socio-cultural and not “physicalistic.”

We begin to intuit that everyday language which we use and handle as the most obvious thing in the world in constant use, is completely confusing once you look at it more clearly.

Einstein’s friend at Princeton, Kurt Gödel, looked into language as a logical phenomenon and concluded that it’s entirely puzzling that two people could actually speak and understand one another given the ambiguities and open-endedness of language.

A language-game (German: Sprachspiel) is a philosophical concept developed by Ludwig Wittgenstein, referring to simple examples of language use and the actions into which the language is woven. Wittgenstein argued that a word or even a sentence has meaning only as a result of the “rule” of the “game” being played. Depending on the context, for example, the utterance “Water!” could be an order, the answer to a question, or some other form of communication.

In his work, Philosophical Investigations (1953), Ludwig Wittgenstein regularly referred to the concept of language-games. Wittgenstein rejected the idea that language is somehow separate and corresponding to reality, and he argued that concepts do not need clarity for meaning. Wittgenstein used the term “language-game” to designate forms of language simpler than the entirety of a language itself, “consisting of language and the actions into which it is woven” and connected by family resemblance (German: Familienähnlichkeit).

The concept was intended “to bring into prominence the fact that the speaking of language is part of an activity, or a form of life,” which gives language its meaning.

Wittgenstein develops this discussion of games into the key notion of a “language-game.”

Gödel saw that language has deep built-in ambiguities which were as puzzling as math and logic ones:

Gödel’s (died in 1978) incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modeling basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.

Take any simple sentence: say, “men now count.”

Without a human context of meaning, how would you ever decide if this means count in the sense of numeracy (one apple, two apples, etc.) or something entirely from another domain (i.e. males got the vote in a certain country and now “count” in that sense).

When you say, “count me in” or count me out,” how does that make any sense without idiomatic language exposure?

If you look at all the meanings of “count” in the dictionary and how many set phrases or idioms involve the word “count,” you will immediately get the sense that without a human “life-world” (to use a Husserl phrase), you could never be sure of any message or sentence at all involving such a fecund word.

One task of real education is to put these difficulties on the student’s plate and not avoid them.

Linguistics as such is not what’s at issue but rather a “meta-intelligent” sense of language, written or spoken as highly mysterious with or without the research into vocal cords, language genes (FOXP2, say) or auditory science and the study of palates or glottal stops and fricatives, grammars and syntax.

Seeing this promotes deep education (i.e., where understanding touches holism in an enchanting way).

Essay 51: “The Whole:” a Quick Second Look

We started this book with a quote from Wittgenstein “Light dawns gradually over the whole” and argued that the meaning of the “whole” is and will be elusive forever.

That is as it should be:

Think of the final pages of John Dewey’s classic book, The Quest for Certainty.  You’ll sense how Dewey oscillates between the “pin-down-ability” of the “whole” and its eternal slipperiness:

“Diversification of discoveries and the opening up of new points of view and new methods are inherent in the progress of knowledge.  This fact defeats the idea of any complete synthesis of knowledge upon an intellectual basis.  The sheer increase of specialized knowledge will never work the miracle of producing an intellectual whole.  The astronomer, biologist, chemist, may attain systematic wholes, at least for a time, within his whole field.

“Man has never had such a varied body of knowledge in his possession before, and probably never before has he been so uncertain and so perplexed as to what his knowledge means, what it points to in action and in consequences.”

(Dewey, The Quest for Certainty, Capricorn Books, 1960, pages 312/313)

Wholeness, Dewey senses, like the white whale in Moby-Dick, “won’t sit for a portrait.”   That is why the student should take an eternally “non-rigid” answer to these questions which are “arguments without end” and that’s fine.

Essay 37: The Language Phenomenon in Education

Wittgenstein (1889–1951) identifies language as the principal “confusion-machine” within philosophy:

“Philosophy is a battle against the bewitchment of our intelligence by means of language.”

The philosopher’s treatment of a question is like the treatment of an illness.

“What is your aim in philosophy?—To show the fly the way out of the fly-bottle.”

Education if deep and meaningful would put language itself in front of a student to understand the “bewitchment” and to perhaps “escape from the fly-bottle.” The fly-bottle is roughly “the captive mind syndrome” described by Czesław Miłosz, the Polish poet-thinker.

There are various aspects of this language-watching:

Hans-Georg Gadamer (Heidegger’s successor, who died in 2002) writes:

“It is not that scientific methods are mistaken, but ‘this does not mean that people would be able to solve the problems that face us, peaceful coexistence of peoples and the preservation of the balance of nature, with science as such. It is obvious that not mathematics but the linguistic nature of people is the basis of civilization.’”

(German Philosophy, Oxford University Press, 2000, pages 122/123)

This is readily seeable. Imagine Einstein and Kurt Gödel walking near the Princeton campus. They speak to each other in German, their native tongue which they both “inhabit.” Gödel communicates the limits to logic and Einstein the limits to modern physics such as quantum mechanics. They bring in Bohr and Heisenberg and the “Copenhagen Interpretation” as a counter-view. They refer to equations and experiments and conjectures and puzzles, current papers and conferences.

They take “communicative action” by use of speech using German as a means.

There are two levels here that are always confused: the ontological (i.e., all the why-questions people ask using language) and the ontic level, all the how-questions people pose using mathematics and laboratory results (e.g., Higgs boson).

Gödel once made the observation that if you look at language as a kind of logical system, it’s absolutely puzzling that people can communicate at all since language is so utterly ambiguous and “polyvalent.”

Take the sentence: “Men now count.” Out of context, does it mean count as in the sense of numeracy, one, two, three apples in front of me or do you mean perhaps that men in a certain country were given the right to vote and now “count” politically. Without the context and the ability to contextualize, no sentence by itself makes certain sense at all.

This is partly why Wittgenstein sees philosophy problems as “language games.”

Heidegger coming from “being-in-the-world” as foundational, and calls language “the house of being.”

You inhabit a native language the way you “inhabit” a family home or a home town. You flow through.

When a child of ten plays marbles (as analyzed by Piaget) and his native language (say French) comes pouring out of him in a spontaneous gusher, how can we really explain it since the child doesn’t look up syntactical rules and grammatical definitions when he speaks. The words flow.

Heidegger retorts that language speaks you in other words, you’re channeling the language in a way a songwriter explains how a song comes to him. In the end, it’s something spontaneous and not propositional like grammar is.

A moment’s reflection shows you how “slippery” language is: 

A man driving to New York says to you, “the car died on me halfway there.”  He does not mean the car was “on” him physically. To die on doesn’t really mean perish forever, it means, on average, stopped to function in a way that usually can be fixed in the garage.  It means this reparable conking out of the car gave him a big headache and aggravation as he waited for the Triple A people to get there and do the paperwork. You visualize all these layers and twists.

Again, without a human context, the sentence “the car died on me” makes little sense. Without a human context, “the sky is blue” makes incomplete sense too. Does a camel or cricket see a blue sky?

A full education would explore these dimensions of language and this has nothing to do with bringing back Latin or Greek or studying a foreign language to meet a Ph.D. requirement.  Formal linguistics à la Chomsky, Fodor, Katz, etc. is not what’s being discussed, as interesting as all that might be.

It also is not about language genes such as FAP-2 or how vocal cords work since these questions are ontic (i.e., how does it work?) and not ontological (i.e., what does something mean or imply?). Thinking about language in an engineering sense with the human mouth as a “buccal cavity” is quite legitimate and a voice coach might do well to do that.  We are talking about something else:  the centrality of language in human self-understanding, functioning and the making of meaning.

Essay 6: Enchantment as an “Engine of Education”

We started this book mentioning Wittgenstein’s assertion, “Light dawns gradually over the whole.”

There are two “players”—light (illumination) and the whole.

The learner, especially the deeper variety of learner, then has two quests: the flashlight or searchlight that gives off the light and the “problem” of defining “the whole.”

We argue in this book only something called “enchantment” (seeing the magic in some question or phenomenon or thing) can be the engine that gives you the impetus to go on in this double search.

For example:

  1. Think of the opening line in the great novel from 1959, The Last of the Just, which won the Goncourt Prize, the highest literary award in France.

The opening line, which serves as a kind of “overture” for the entire book, is: 

“Our eyes register the light of dead stars.”

The author uses this as a figure of speech which captures the lasting influence of people who came before you who somehow are “stars” in the sense of principal actors in your mental life. When you begin the novel, you don’t know if the writer is going to use this concept not as a statement about stellar objects in the sky, as understood by astronomy or cosmology or optics, but in the personal influence sense, as he does. 

This is a beautiful “overture” because it links the physical to the personal in a “dual metaphor.” There’s a secondary poetical device since stars could mean shiny objects in the sky or people as in “movie stars.”

Great writing has this “enchanting” quality and it addresses a deep human hunger for so-called “words to live by.”

  1. Go back to our elementary math example where 1=.9 recurring.

A student gets intrigued by this and senses “how can that be? how can you add these decimal nines infinitely?

In fact, this is a deep and “enchanting” question. If you look into something called infinitesimals (smallest math “objects”) you will find that this issue is still an “argument without end” to use Pieter Geyl’s phrase.

Furthermore: If something is or seems to be “an argument without end,” what does that imply about our ability to “nail” things down in our minds?  That’s an enchanting question in itself which resonates with the Descartes “epistemology” and certitude quest we have seen previously.

Then there’s the other elusive “player” in the Wittgenstein sentence: “the whole.”

Does one mean the whole of a novel or math problem? The whole of the world of metaphors and numerical thinking (i.e., math)? Does one mean everything that exists? It’s not a set or static “thing.”

The point is not to decide any of this in a “once-and-for-all” way. The point is only to allow the enchantment engine to carry the student into these realms and domains without insisting on an eternal “final answer.”

This is why this kind of meta-intelligent self-education or re-education parts company with quests such as Stephen Hawking’s, to “know the mind of God” as mentioned in the last lines of his 1988 book, A Brief History of Time.

Enchantment gives you some pre-understanding which pulls you higher and you can relax the insistence on finality or absolute certainty which characterizes the whole trajectory from Descartes through Husserl, who died in 1938 (think of his book, Cartesian Meditations) through contemporary “scientism” such as exemplified by Hawking with his undoubted analytical genius.