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.

Meaningfulness versus Informativeness

The Decoding Reality book is a classic contemporary analysis of the foundations of physics and the implications for the human world. The scientists don’t see that physics and science are the infrastructure on which the human “quest for meaning” takes place. Ortega (Ortega y Gasset, died in 1955) tells us that a person is “a point of view directed at the universe.” This level of meaning cannot be reduced to bits or qubits or electrons since man is a “linguistic creature” who invents fictional stories to explain “things” that are not things.

The following dialog between Paul Davies (the outstanding science writer) and Vlatko Vedral (the distinguished physicist) gropes along on these issues: the difference between science as one kind of story and the human interpretation of life and self expressed in “tales” and parables, fictions and beliefs:

Davies: “When humans communicate, a certain quantity of information passes between them. But that information differs from the bits (or qubits) physicists normally consider, inasmuch as it possesses meaning. We may be able to quantify the information exchanged, but meaning is a qualitative property—a value—and therefore hard, maybe impossible, to capture mathematically. Nevertheless the concept of meaning obviously has, well… meaning. Will we ever have a credible physical theory of ‘meaningful information,’ or is ‘meaning’ simply outside the scope of physical science?”

Vedral: “This is a really difficult one. The success of Shannon’s formulation of ‘information’ lies precisely in the fact that he stripped it of all “meaning” and reduced it only to the notion of probability. Once we are able to estimate the probability for something to occur, we can immediately talk about its information content. But this sole dependence on probability could also be thought of as the main limitation of Shannon’s information theory (as you imply in your question). One could, for instance, argue that the DNA has the same information content inside as well as outside of a biological cell. However, it is really only when it has access to the cell’s machinery that it starts to serve its main biological purpose (i.e., it starts to make sense). Expressing this in your own words, the DNA has a meaning only within the context of a biological cell. The meaning of meaning is therefore obviously important. Though there has been some work on the theory of meaning, I have not really seen anything convincing yet. Intuitively we need some kind of a ‘relative information’ concept, information that is not only dependent on the probability, but also on its context, but I am afraid that we still do not have this.”

For a physicist, all the world is information. The universe and its workings are the ebb and flow of information. We are all transient patterns of information, passing on the recipe for our basic forms to future generations using a four-letter digital code called DNA.

See Decoding Reality.

In this engaging and mind-stretching account, Vlatko Vedral considers some of the deepest questions about the universe and considers the implications of interpreting it in terms of information. He explains the nature of information, the idea of entropy, and the roots of this thinking in thermodynamics. He describes the bizarre effects of quantum behavior—effects such as “entanglement,” which Einstein called “spooky action at a distance” and explores cutting edge work on the harnessing quantum effects in hyper-fast quantum computers, and how recent evidence suggests that the weirdness of the quantum world, once thought limited to the tiniest scales, may reach into the macro world.

Vedral finishes by considering the answer to the ultimate question: Where did all of the information in the universe come from? The answers he considers are exhilarating, drawing upon the work of distinguished physicist John Wheeler. The ideas challenge our concept of the nature of particles, of time, of determinism, and of reality itself.

Science is an “ontic” quest. Human life is an “ontological” quest. They are a “twisted pair” where each strand must be seen clearly and not confused. The content of your telephone conversation with your friend, say. is not reducible to the workings of a phone or the subtle electrical engineering and physics involved. A musical symphony is not just “an acoustical blast.”

The “meaning of meaning” is evocative and not logically expressible. There’s a “spooky action at a distance” between these levels of meaning versus information but they are different “realms” or “domains.”

Education and the “Knowability” Problem

There was a wonderful PBS Nature episode in 2006 called “The Queen of Trees” [full video, YouTube] which went into details about the survival strategy and rhythms and interactions with the environment of one tree in Africa and all the complexities this involves:

This Nature episode explores the evolution of a fig tree in Africa and its only pollinator, the fig wasp. This film takes us through a journey of intertwining relationships. It shows how the fig (queen) tree is life sustaining for an entire range of species, from plants, to insects, to other animals and even mammals. These other species are in turn life-sustaining to the fig tree itself. It could not survive without the interaction of all these different creatures and the various functions they perform. This is one of the single greatest documented (on video) examples of the wonders of our natural world; the intricacies involved for survival and ensuring the perpetual existence of species.

It shows us how fragile the balance is between survival and extinction.

One can begin to see that the tree/animal/bacteria/season/roots/climate interaction is highly complex and not quite fully understood to this day.

The fact that one tree yields new information every time we probe into it gives you a “meta” (i.e., meta-intelligent) clue that final theories of the cosmos and fully unified theories of physics will be elusive at best and unreachable at worst. If one can hardly pin down the workings of a single tree, does it sound plausible that “everything that is” from the electron to galaxy clusters to multiverses will be captured by an equation? The objective answer has to be: not particularly.

Think of the quest of the great unifiers like the great philosopherphysicist Hermann Weyl (died in 1955, like Einstein):

Since the 19th century, some physicists, notably Albert Einstein, have attempted to develop a single theoretical framework that can account for all the fundamental forces of nature–a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics. In particular, unification of gravitation and electromagnetism was actively pursued by several physicists and mathematicians in the years between the two World Wars. This work spurred the purely mathematical development of differential geometry.

Hermann Klaus Hugo Weyl (9 November, 1885 – 8 December, 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.

His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.

Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the “universalism” of Henri Poincaré or Hilbert, Weyl came as close as anyone.

Weyl is quoted as saying:

“I am bold enough to believe that the whole of physical phenomena may be derived from one single universal world-law of the greatest mathematical simplicity.”

(The Trouble with Physics, Lee Smolin, Houghton Mifflin Co., 2006, page 46)

This reminds one of Stephen Hawking’s credo that he repeated often and without wavering, that the rational human mind would soon understand “the mind of God.”

This WeylHawkingEinstein program of “knowing the mind of God” via a world-equation seems both extremely charming and beautiful, as a human quest, but potentially mono-maniacal à la Captain Ahab in Moby-Dick. The reason that only Ishmael survives the sinking of the ship, the Pequod, is that he has become non-monomaniacal and accepts the variegatedness of the world and thus achieves a more moderate view of human existence and its limits. “The Whiteness of the Whale” chapter in the novel gives you Melville’s sense (from 1851) of the unknowability of some final world-reality or world-theory or world-equation.

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 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 22: Education and Jean Piaget: Using “Moby Dick” as a Counterweight to Piaget

The Swiss psychologist Jean Piaget (who died in 1980) was perhaps the greatest theorist of cognitive development and education of the twentieth century.  His books are classics and his various explorations of childhood games, rules, knowledge, education, etc. are of outstanding quality.

A central work of Piaget’s for our purposes of educational deepening is his small masterpiece To Understand Is to Invent: The Future of Education.

We respectfully disagree with arguments put forth in this book which we see as overly narrow. Take these words on the uselessness of Eskimo knowledge: “We are like the old Eskimo who was asked by an ethnologist why his tribe so piously preserved certain rites, and answered that he could not understand what was the meaning of that, saying: ‘We preserve our old customs so that the universe will continue.’ ”

Piaget continues: “For primitive man, the universe is a great machine in unstable equilibrium where all is related to everything else (the social customs and physical laws are not differentiated one from another). If one removes even one of its pieces, even without knowing what purpose it has, the whole machine risks being thrown out of gear.” (Jean Piaget, To Understand Is to Invent: The Future of Education, Penguin Books, 1977, page 134)

Children too come under Piaget’s “disapproval” when he says a few paragraphs later:

“Every child has thought one day that the moon was following him, and, according to several primitive societies, the course of the heavenly bodies is ruled by the movement of men (in ancient China, for example, the Son of the Heavens insured the seasons by his moving about). The Chaldeans and the Babylonians made notable progress in freeing themselves from this initial egocentric vision and in understanding that the heavenly bodies have a trajectory which is independent of us…The Copernican revolution can be considered a most striking symbol of the victory of objective coordinations over the spontaneous egocentrism of the human being.” (Jean Piaget, To Understand Is to Invent: The Future of Education, Penguin Books, 1977, page 137-138).

Piaget, for all his acuity, sets up a rigid dichotomy between Western adults and primitive man and children. Primitive man is childish and children are primitive so they go together.

We find this extremely constraining and surprisingly, perhaps, point of Melville’s 1851 classic MobyDick as a counterexample to Piaget.

Ishmael, the narrator, is the only survivor of the shipwreck of the Pequod which is not only a ship but also a global university of sorts, a site of knowledge of all kinds: Ahab’s, Starbuck’s, Ishmael’s. Ishmael deeply respects the dignity and self-possession of the “primitive” sailor and harpooner Queequeg, whose coffin allows him not to drown. He (Ishmael) respects and finds moving the ‘primitive’ religious ceremonies of the native Queequeg for his god Yodo and Ishmael participates modestly and reverently.

Relentless dismissiveness of indigenous ways of seeing the world are dangerous and have led Western man to the current climate crisis and the complete paralysis in coping with it. Indigenous man’s basic belief that the “earth own us” and is Our Mother would be a healthy antidote to Western “techno-nihilism.” In Moby-Dick, Ishmael’s tolerance, openness, mildness, and cosmopolitan emotional life, saves him and this is a counterweight to Piaget-ism. He says in the beginning of the book that ships and voyages were “my Harvard and my Yale.”

Furthermore, childlike visions of the world (“the moon is following me”) are the basis, potentially of scientific advances later on since as Einstein and Feynman kept emphasizing, the trick in life and science is to “remain childlike all one’s life and keep asking all those children’s questions all through one’s life such as “why is the sky blue?” You will be told by the physics book Rayleigh Scattering, which explains the blue sky and then you ask, why is that? if we were wired differently would it still be blue? Is it blue or just seems so? What are colors like blue?  Why would cosmic evolution, if it pertains, evolve in this way (i.e., where Rayleigh scattering applies)?

In other words, to set us a rigid binary world where modern physicists are right and indigenous people and children are naive if not idiotic is not attractive to someone who wants a wide-angle and deep education and combine modern science, a great accomplishment, with Ishmael’s openness to other modes and types of being, another kind of great accomplishment, as Melville shows us.

This is especially true since the chapters in Moby-Dick, “The Whiteness of the Whale” and “The Doubloon” show us that finality in knowledge is not attainable and that modesty (i.e., Ishmael-ism) is what’s appropriate for man (e.g., open, inclusionary, tolerant views).

Real understanding (our goal) is to invent (following Piaget’s word) clusters of connected views, beyond specialization, and this would be the future of education. This can only be done by rescuing and including “childlike” and indigenous modes of understanding, a bit like Melville’s Ishmael.