Tag: arithmetic

Movies As Education: Books and Selves

La Notte (English: The Night) is a 1961 Italian drama directed by Michelangelo Antonioni. The film stars Marcello Mastroianni, Jeanne Moreau, and Monica Vitti (with Umberto Eco, the novelist, appearing in a cameo).

Filmed on location in Milan, the film depicts a day in the life of an unfaithful married couple and their deteriorating relationship.

In 1961, La Notte received the Golden Bear (at the Berlin International Film Festival, the first for an Italian film) and the David di Donatello Award for Best Director.

La Notte is the central film of a trilogy, beginning with L’Avventura (1960) and ending with L’Eclisse (1962).

The movie follows Giovanni Pontano (Marcello Mastroianni), a distinguished writer, and his beautiful wife Lidia (Jeanne Moreau) as they visit their dying friend Tommaso Garani (Bernhard Wicki) who is hospitalized in Milan. Giovanni’s new book, La stagione (The Season), has just been published, and Tommaso praises his friend’s work.

La Notte reflects the director’s intuition that “you are what you read,” and books create a kind of thread through the story.

The dying, hospitalized patient has recently published an article on the famous philosophical writer Theodor Adorno. At the party the couple drifts into, the works of the Austrian–Jewish writer, Hermann Broch, are mentioned. Essentially, in a depressing glitzy world of lost and semi-lost souls, reading and books constitute a kind of emotional life raft or direction-finding compass, at least potentially. Antonioni frequently uses this motif.

We find this kind of reading and books-centered view of people interpreting their (bewildering) worlds in the works of the French thinker Charles Péguy (who died in battle during World War I in 1914):

“The Jew,” he declares in a passage that has become famous, “is a man who has always read, the Protestant has read for three hundred years, the Catholic for only two generations.”

(quoted in Consciousness and Society, H. Stuart Hughes, Vintage Books, paperback, 1958, page 355)

Charles Péguy is also central to Louis Malle’s classic French film Au revoir les enfants (English: “Goodbye, Children”).

If we “zoom out” and look for a meta-intelligent lesson, we can say that reading, writing, and arithmetic, the three basics mentioned in the phrase we all know, are very deeply entwined with who we are. Stories explain us to ourselves, and stories involve books and reading in our “Gutenberg world.”

The replacement of these by various (post-Gutenberg) screens and games may or may not be thought of as a variant since they constitute a kind of “pseudo-participation” and not participation based on perusal.

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 the Question of Intuition

An intuition pump is a thought experiment structured to allow the thinker to use his or her intuition to develop an answer to a problem. The phrase was popularized in the 1991 book Consciousness Explained by Tufts philosophy and neuroscience professor, Daniel Dennett.

We argue in this education-completing book, that our intuitions are puzzling in a way that “intuition pump” talk does not cope with at all.

Let’s go immediately to the example of simple versus compound interest in basic finance.

You borrow $100.00 for a year at an annual interest of 100%, without compounding and hence simple. A year passes and you owe the lender the initial $100 plus one hundred percent of this amount (i.e., another hundred). In a year, you owe $200.00, and every year thereafter, if the lender is willing to extend the loan, you owe another hundred to “rent” the initial hundred.

This is written as A+iA, where A is the initial amount (i.e., $100.00) and i is the interest. This can be re-written as A(1+i)n where n is the number of years. Thus, if n=1, you owe: A(1+i), which is 100×2 (i.e., the $200 we just saw). There’s nothing tricky in this.

You then are introduced to compound interest (i.e., where the interest accumulates interest). You can see where compounding by 6 months (semi-annually, or half a year) or 12 months involves dividing the n (the exponent over 1+i) by 12 months, two half-years or 365 days. You could routinely go to days and hours and minutes and seconds and nanoseconds and you could calculate interest payments compounding for each case.

But here is where your intuition falters and fails: suppose you compound continuously?

You get to the number e as growth factor where e=2.71823

Simple algebra does show that at 100% interest, $100 of a loan becomes $100 multiplied by e1 (hundred percent=1) or just e (i.e., you owe $100e).

This gives you $271.82.

So what has happened?

At one hundred percent simple interest you owe $200.00 to the lender. Continuous compounding means you owe $271.82. Instead of owing $100 in interest, you owe $171.82. Your interest bill has gone up by $71.82 or about 72 percent.

Does that seem intuitive? Probably not.

How could one ever apply an “intuition pump” to this arithmetic? We get to the 72% increase in interest by using e which has nothing very intuitive about it. Thus it’s not clear that “intuition pumps” will work here.

You use compound interest arithmetic to get a number which you would never have been able to estimate based on standard intuition since like the 22/7 or 3.14 for π (pi), there’s nothing to “recommend” 2.71823 in and of itself. This means that the link between computational arithmetic understanding and your “gut” or “sixth sense” is feeble at best.

By exploring this way of thinking you could deepen your “meta-intelligence” (i.e., perspective-enhancement). The British economist Pigou (Keynes’s teacher) says that people have a “defective telescopic facility” (i.e., have a poor or even erroneous sense of time-distance).

How one might strengthen one’s sense of time-distance or “far horizons” is not clear.

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.

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).