A young woman complained to the famous American psychologist and philosopher William James: "Algebra is just a low form of cunning!"
Paraphrasing The King of Siam (expostulating in the movie The King and I, and in other versions): "Algebra is a puzzlement!"
Why the name? "Algebra" is from Arabic, allegedly meaning "bone-setting".
Whatsit? Well, I've never seen "algebra" adequately defined. I'll do so, later.
Another confusion: MATHEMATICS INCLUDES MANY DIFFERENT ALGEBRAS ("algebra of statements", "algebra of vectors", "algebra of combinations", "algebra of group operations", etc.) What most people label "algebra" should be called "Numerical Algebra" or "Arithmetic Algebra", or say, "NUMALGEBRA".
Some PUZZLEMENT derives from the way ARITHMETIC is TAUGHT. The QUASI-AXIOMATIC ("The LORD GOD commanded!") presentation of ARITHMETIC and THE NUMBER SYSTEMS looks as if "the Stork delivered them". Also, this formulation seems to encourage CHEATING.
Thus, for NATURAL or COUNTING NUMBERS, the subtraction 2 — 3 is meaningless ("can't take away 3 oranges from a heap of 2"). But then you're told you can perform this subtraction as
2 - 3 = -1, by putting a funny sign in front of a number -- which would be CHEATING!
Again, for NATURAL NUMBERS or INTEGERS, you can't perform the division 5 ÷ 2
(" You can't divide 5 people into two equals groups"). But then you're told you can perform this
division as 5/2 by putting a funny slash mark between the numbers -- which would be CHEATING.
And similar puzzles arise.
Actually, THE GENERATIVE METHOD described, herein (under "Arithmetic Redux"), shows how
Arithmetic is constructed and how those problems above are BYPASSED without CHEATING. But the GENERATIVE METHOD -- which began with Pythagoras (580-496 BC), 2500 years ago -- IS NOT TAUGHT!
And part of the PUZZLEMENT derives from "some of the funny rules of Arithmetic".
For example, THE RULE OF SIGNS (first encountered in MULTIPLYING INTEGERS):
"Why is negative times negative a positive number? Crazy!"
Another PUZZLEMENT is the DIVISION of a FRACTION (RATIONAL NUMBER) by a FRACTION (RATIONAL
NUMBER): (a/b) ÷ (c/d) = (a/b) · (d/c).
"Why do you invert the divisor fraction, then multiply? What sense does that make?"
If you learn ARITHMETIC by quasi-axiomatic ("stork-borne" or "The Lord God commanded") methods,
you wonder why "they made up such crazy rules". (Inability to explain these and other critical
quandaries also explains why a "School of Social Constructivism" has grown up in recent years, arguing that ANY MATHEMATICAL SYSTEM IS JUST A SOCIAL CONSTRUCTION -- on par with Table Etiquette. So, students are jammed between Moses and Miss Manners. Well, mainstream mathematicians and teachers who complain against "Social Constructivism" -- or the slandering of "algebra" -- have only feckless homework to blame!)
However, when you learn GENERATIVE ARITHMETIC, you see why -- and will see why in "Arithmetic
Redux".
(As a second opinion, the great British logician-mathematician,
The INTEGER RULE begins in the NATURAL NUMBER SYSTEM: you want A DEFINED DIFFERENCE MINUS A
DEFINED DIFFERENCE TO EQUAL A DEFINED DIFFERENCE. This is nothing but the "sacred" CLOSURE RULE OF MATHEMATICS. Without CLOSURE ("All in The Family"), MATHEMATICS (as we know it) WOULD NOT EXIST, BUT WOULD DEGENERATE INTO SOMETHING RESEMBLING TABLE ETIQUETTE!
Similarly, that funny division rule is required in INTEGERS (and imported to RATIONALS): A DEFINED QUOTIENT DIVIDED BY A DEFINED QUOTIENT EQUALS A DEFINED QUOTIENT.
AGAIN, a grave problem in the TEACHING OF ARITHMETIC arises when THE REAL NUMBER SYSTEM is explained "exogamously" (not-All-in-the-Family) as arising GEOMETRICALLY ("the diagonal of a square"), instead of explaining it "endogamously" or ARITHMETICALLY as MAKING TOTAL ONE OF THE INVERSES (LOGARITHM) OF EXPONENTIATION -- with logarithm seeming "like something the cat drug in". ARITHMETIC-EXPLAINING-ARITHMETIC banishes some of the PUZZLEMENT.
So, no funny stuff, no cheating, no unnecessary PUZZLEMENT.
Returning to our earlier theme, when Arithmetic follows Alice into The Looking-Glass World, it morphs as "Numalgebra", to remorph as Arithmetic when Alice remorphs. So
Dig? arithmetic ⇒ numalgebra ⇒ arithmetic. (Back in the same day!)
Whathuhek is "x"? (Yes, perhaps x reminds you of The Mad Hatter. But keep cool with Alice, and you'll funnel the tunnel.
What good is ALGEBRA (in any form)? The following files that answer that question.
But, for now, why does ALGEBRA seem such a puzzlement?
Referring back to Alice, I claim that THE ALICE-BOOKS say: "PROPER LITTLE GIRLS [such as Alice Lidell, Carroll's inspiration] SHOULD LEARN THAT THE REAL WORLD IS NOT ALWAYS PROPER!" (The books make this point by "overkill".) Similarly, ALGEBRA IS TO TEACH "PROPER" STUDENTS THAT THE REAL WORLD IS NOT ALWAYS FILLED WITH "PROPER" PROBLEMS!
Actually, you'll discover that THOSE PROBLEMS ARE VERY PROPER IN THE REAL WORLD -- outside THE LOOKING-GLASS.
BACK.
Bertrand Russell
(1872-1972) said axiomatics "has all the advantages" over generatics "that theft has over honest labor" -- assuming what should be generated from a basis of elements.)
Don't howlgebra
At "Algebra".
Get withmetic
Arithmetic.