Elsewhere, I derive real numbers in order to transform that inverse of exponentiation known as "logarithm" from a partial inverse into a total inverse. However, due to geo-bias, this isn't how students are quasi-axiomatically taught. Rather they are taught that root extraction is exponentiation's only inverse. Logarithm seems "like something the cat drug in". (I learned logarithm-as-inverse from a 1900 book written by a Princeton U. math professor -- a book given me by my father-in-law, a Princeton grad.)Traditionally, the problem began thus. Given a right triangle with unit sides, its diagonal (by the Pythagorean formula) is the square root of 2 Trouble! The triangle's diagonal is incommensurable with the triangle's unit side. That is, you can't combine several diagonals to match exactly some number of combined sides!
This incommensurability was interpreted as saying, "Geometry and arithmetic are incompatible" -- like Punch and Judy! But do you dig what just happened? An arithmetic problem was "kidnapped by" geometry, "revealing" arithmetic to be geometrically deficient! GEE! GEO-BIAS!
Geo-bias pervaded Plato's Academy, with its inscription, "Let no one ignorant of geometry enter here!" And this geo-bias STALLED an industrial revolution! Why? Because Plato's Academy also proclaimed, "Motion is simply geometry set to time". Hence, you cannot arithmetize motion -- cannot speak of speed, velocity, force, momentum, energy, and other kinematic-dynamic measures of theoretical mechanics!
Yet "The Alexandrian Age" (330 B.C.-640 A.D.) boasted an effective technology, with steam cars and robots! In Alexandria and Rome, engineers and technicians were usually Greek slaves or freedmen. Aristocratic disdain for labor inhibited significant experimentation. And why develop "nonsensical" mechanical theory? The mechanical revolution stalled!
But the 12th and 13th centuries witnessed "The Medieval Revolution". Needing to raise food, materials for clothing and such, yet save time for prayer and meditation, monks (who didn't disdain labor) received Papal permission to read ancient ("pagan") manuals describing labor-saving devices. Soon thousands of windmills and water mills covered Europe and Britain. (So many peasants ground grain at Cistercian Order mills that prostitutes picked up their clients from the lines.)
What we call "The Industrial Revolution", beginning around 1776, was "The Thermodynamic Revolution", using steam engines to power mechanical devices formerly powered by air, water, human or animal labor.
However, mechanical theory began to develop in the Post-Renaissance period, partially weakening geo-bias. But this all meant that slavery endured longer than necessary! Slavcery was abolished only when shown not to be cost-effective. (Our "Civil War" was between the Southern "Mechanical Industrial Revolution" -- little more advanced than that of 13th century monks! -- and New England's "Thermodynamic Industrial Revolution".)
And geo-bias engenders gender bias, putting girls and woman at a disadvantage. Even thus misjudged (I didn't did it!), I'll show that geometric perception is often irrelevant. First a critical case affecting standardized testing which shows downplaying of linguistic ability.
Alfred Binet (1857-1911) originated his test to determine which learning-disabled students might benefit from remediation. In his first test, girls scored higher than boys. The linguistic proportion was diminished so that boys scored higher than girls. This bias continues in standardized testing!
Overemphasized geometric perception? We've known this in mathematical physics since 1788!
Please recall that geo-bias thwarted Medieval theoretical mechanics by claiming that motion cannot be arithmetized. Arabic scholars, using optical and pharmaceutical paradigms, encouraged arithmetization. Geo-bias weakened during the Renaissance, as in applying the law of falling bodies, but returned with Sir Isaac Newton (1643-1727). Using limits (described elsewhere at this Website), Newton developed "fluxions" (his version of the "calculus"). However, in his masterpiece, Principia Mathematica, Newton lost his nerve and downplayed calculus in formulating a law of gravitation to explain the solar system, stating major results in terms of synthetic geometry. Even when translated from Latin into English, only specialists understand many of Newton's derivations in Principia.
Ludicrously, texts still cite "Newton's" rotational differential equations of motion. These appeared around 1740 (after Newton's death), the work of the great Swiss mathematician, Leonhard Euler (1707-1783), the most prolific of all mathematicians. Euler developed our present analytic geometry (bypassing synthetic geometry) and initiated non-Newtonian mechanics, free of geo-bias.
Extending Euler, the great French mathematician, Joseph Lagrange (1736-1783), published in 1788 Mechanique Analytique, revamping mechanics in non-Newtonian form, replacing force by energy. In his preface, Lagrange boasted that his book lacked a single diagram! ("Geo-bias, get lost!").
Lagrange developed the algebraic "Lagrangian", yielding equations of motion for planets or elementary particles. Hamilton (founder of modern generatics) derived from this "The Hamiltonian". Frontier physicists work in non-Newtonian Lagrangians or Hamiltonians.
Championed by Hilbert, Emmy Noether (1882-1935), greatest woman mathematician, showed how to derive conservation laws from the Lagrangian and predict particles as carriers of this conservation. "Noether currents" are constantly researched.
But exemption from geo-bias penetrates neither classroom nor SAT. Problems statable in generative-algebraic form are arbitrarily proclaimed in axiomatic-geometric form, disadvantaging girls.
Not the subject but the "majority" demands it. THE GAME IS FIXED!
Consider, please, these hypotheses.
- "Girls generally have better hearing than boys, hence excel in the language arts."
- "Boys excel in geometric perception."
A Yale U. MRI experiment on brain functioning showed that women process language with both sides of the brain, while men predominantly process language in the left brain. The experimenters also found superior female linguistic ability and superior male processing of geometric percepts. They also claimed that males are superior in mathematical reasoning (distinguished from calculation).
I'll bravely predict in TESTABLE form.
Personal experience explains my bravado (providing a "second opinion") A "Mad Russian" professor, Ervard Kogbetliantz, taught two of my three "calculus" semesters at Columbia University. Kogie thought we were all engineers. ("You will build bridges. They will fall down and kill people! Please don't say I was your teacher!") Kogie was fanatical about 3-D perception, having invented 3-D chess (now popular with computer nerds). An old weekly Life Magazine displayed Kogie glaring demonically through tiers of his 3-D chess board. He frequently challenged our "3-D ability", picking on me.
- It will be found that the allegedly "superior mathematical reasoning" was cast in axiomatic-deductive (nongenerative) form.
- Women trained in generative mathematical reasoning will display ability equal to or better than axiomatic-deductive performances of men.
Once Kogie drew a complicated curve on the blackboard and asked one student after another to identify it. Coming to me, "No use asking you!" Stung, I retorted, "I'll bet I could if you gave me the equation!"
"Oh, you're so smart! Well, here's the equation." Writing it on the blackboard, he went on. I busily set about factoring it. Then I brushed past Kogie to the blackboard and drew the intersection of two planar curves, one in the X-Y plane, the other in the Y-Z plane.
Kogie's eyes popped. "How you do that?" I showed him my factoring. His face changed.
"Class, I apologize to Mr. Hays. I have been peeking on him because he has no geometry-mind. Like me. But now I see he has algebra-mind. Which is better! I can only think in three dimensions. But algebra-minds can think in any number of dimensions!" And, from then on, Kogie stopped "peeking" on me.
Hypothesis (testable?): Most men have "geometry-minds"; most women have "algebra-minds" .
Third opinion. Dyscalculia is a dysfunction (in calculating), resembling dyslexia (in reading). Peter Rosenberger, Director of the Learning Disorders Unit at Massachusetts General Hospital in Boston, found that dyscalculic but nondyslexic children, although "poor" in arithmetic, may do well in algebra. Rosenberger offers the explanation: "Algebra is clearly language".
The linguistic ability of girls might give them an advantage, not only in algebra, but also in the calculus, vector analysis, and other subjects taught generatively -- number bone connected to the word-bone! -- in the choice offered to them in MATHEMATICS ARITHMETIZED.
However, in a geo-biased learning environment, which downplays linguistic ability and overvalues geometric perception, capable girls may suffer.
And some of the proponents of geo-bias and meritocracy have launched a new attack: on those who have been thought to be "learning disabled".
I'm not only calling upon WOMEN to defend the RIGHTS OF GIRLS but THE RIGHTS OF TROUBLED CHILDREN OF BOTH GENDERS!