Tag: algebra

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.

Essay 27: Extracting Signals from Noise

We wish to help students “parachute into” or sneak up on mathematics before any “rocket science” (i.e., focus on high school and don’t get lost in the weeds).

Think of square roots. The square root of 4 is 2, of 9 is three of sixteen is 4.

But the moment someone asks:  what about the square root of 17, you will find that crystalline simplicity and obviousness are long since gone.

If you “keep your nerve” and calmly get into the “complexity jumps” you may well find it enchanting that numerical understanding has to be coaxed forth and doesn’t offer itself up readily.  But why would the physical universe, if it is really mathematical in its very “fabric,” be so ready to “jump away from you” in its complexity?

Go back to the square root of 17. Suppose you disallow logarithms and log tables. Suppose you say that “approximation theory” is too approximate. There’s no HP Scientific Calculator. One insight you might need is that 17 is “not far” from 16 so that the square root must be 4 plus a little. Call this “little” x.

Then (4+x) (4+x) = 17. You solve for x with the quadratic formula you had in high school. If you don’t remember the formula or cannot derive it, you’d have to look it up which might be disallowed in this “game.”

If you stand back, “meta-intelligently” (i.e., asking, “what does this tell me?”), you wonder whether the universe and its math fabric are an endless “onion” of such layers and complexity and not “boil-down-able.”

Another such example is Grandi’s series.

Grandi’s series and its trickiness:

In 1703, the mathematician Luigi Guido Grandi studied the addition: 1 – 1 + 1 – 1 + … ( 1-1, infinitely many, always +1 and –1).  You find if you group the numbers in certain valid and legitimate way, you could different results.  How can that be?

In mathematics, the infinite series, 1 − 1 + 1 − 1 + ⋯, can also be written:

Grandi’s series

It is sometimes called Grandi’s series, after Italian mathematician, philosopher, and priest Luigi Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. (On the other hand, its Cesàro sum is ½.)

One obvious method to attack the series (

One obvious method to attack the series (1 − 1 + 1 − 1 + ⋯) is to treat it like a telescoping series and perform the subtractions in place:

(1 − 1) + (1 − 1) + (1 − 1) + … = 0 + 0 + 0 + … = 0

On the other hand, a similar bracketing procedure leads to the apparently contradictory result:

1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + … = 1 + 0 + 0 + 0 + … = 1

Thus, by applying parentheses to Grandi’s series in different ways, one can obtain either 0 or 1 as a “value.” (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)