A² + B² = C²: Can You Prove It?

Ask any student, "What is the Pythagorean Theorem?" and they will say without blinking, A² + B² = C²! Everybody knows that! However, in the LaRouche Youth Movement (LYM) Saturday night meeting in Washington, D.C., I asked the question, "Can you prove it!"

In a quote from Presidential pre-candidate Lyndon H. LaRouche, Jr. released on Dec. 16, 2002 and in an article, "Believing Is Not Necessarily Knowing," (Jan. 17, 2003 Vol. 30 No. 2 edition of the Executive Intelligence Review), he states: "The root of the increasingly catastrophic failure of U.S. education policy of practice, over the recent thirty-odd years, may be summed up in five points of a general indictment:

1. Sense perception is not necessarily knowing.
   
   
2. Learning is not necessarily knowing.
   
   
3. General accepted opinion, academic or other, is not a standard for the definition of truth.
   
   
4. Today's teachers have not necessarily intended to educate or test their students in a manner suited to human beings.
   
   
5. The radically reductionist fad known as 'Information Theory,' as associated with the influence of Bertrand Russell devotees Norbert Wiener and John v. Neumann, was always a hoax. Thus, the educational methods adapted to the 'Information Theory' hoax, such as the related so-called 'New Math,' are also a hoax."

Now, many students will agree that they are not learning anything, and to add on to that, that they do not know anything. However, as organizers in LYM, we come across a few students who are loyalists to the "Information Theory/New Math" hoax. One might ask, "How do these youth, from various formal educational backgrounds, deal with these Matrix-watching loyalists?" The answer is, "We kick their butts with the Platonic Method!" As Mr. LaRouche has said, it is the case that much of modern-day education, especially in the area of mathematics is untruthful because students might derive a "right answer," but have no idea how it is generated, or for that matter what it really means in the physical universe. So, the question remains, what do you really know?

The meeting started with a strategic briefing by a LYM member, followed by a presentation on Solon of Athens by one old and two new LYM members, and a pedagogical on The Pythagorean Theorem, which I presented.

I started out by asking, "Who knows who Pythagoras is? And what do you know about him?"; as one can imagine, people gave the typical responses. I interjected with, "Can you tell me what this formula really means, without numbers?" Whispers came from the audience, "You have to plug in a number, for at least two variables." After hearing the whispers I asked a member from the audience to pick two numbers, and she did, 9 and 6 (see Fig. 1).

Then I asked people to "plug in the numbers for the variables in the formula," and they did. After which I said, "Okay, what does this answer tell you?" One reply from the audience was, "nothing!" There was a silencing pause in the room, a mixture of confusion and enlightenment transformed into big Kool-Aid smiles, which suffused almost everyone's face, when most said, "Yeah, you are right, this answer tells you nothing!" "Let's get ready to rumble!"

The rules of the ring: If you have a square A and a square B, can you prove that A + B will give you a square of side C (see Fig. 2) In other words, given three squares, where the length of the side of square C is the hypotenuse of a right triangle, and where the sides of squares A and B are opposite and adjacent, respectively, to the hypotenuse, can you prove geometrically that A² + B² = C², using only a straight edge and using three arbitrary measurements for the lengths of the sides of squares A, B, and C, respectively?

After ringing the bell, I asked contenders to step into "the ring," the white board. Contender 1 was a female member of LYM, who said that if you cut square B into four right triangles and place them in square C, along the perimeter, the leftover square in the center, X, will equal square A (see Fig. 3). Contender 2, another member of LYM, set up the audience conceptually by reviewing the method of doubling of the square. Further, he stated that if you divide squares A and B into right triangles, they will fit into square C (see Fig. 4).

Contender 2 credited his method to the geometric progression found in the doubling of the square. Contender 2 sat down with a grin and both fists in the air as if he were the champion. Knowing that I still had time on the clock, I gave his solution a once-over and I said, "Are you assuming that squares A and B are equal? What if these squares are not equal, then what?" Contender 2 slowly lowered his fist from the air; determined to be down, but not out, he said, "Well, yes, if A and B are equal then it works, but for the other cases, I don't know."

Contender 3—now this one was a hustler, making the audience believe she did not know the answer, while letting Contender 4 jump up to be smacked down. Contender 4 tried to take off of Contender 1's answer when Contender 5 shouted from the sidelines, "What if square A is changed to a really small square, then what, what happens to square X?" Cascades of "yeahs!" slapped Contender 4 ever which way, "whap, whap!" He was down for the count, TKO! Contender 5 jumped in the ring to "finish him off"—or so he thought—when a "seasoned" contender jumped up and said, "What does that have to do with anything?"

"Seasoned" Contender tagged another "seasoned" contender into the ring to take on Contender 5.

Meanwhile, while everyone in the audience was working on a proof for the Pythagorean Theorem, Contender 3 was plotting her entrance back into the ring, for a second round. In the midst of the action, Contender 3 upper-cuts Contender 5 and the two "seasoned" Contenders out of the ring (see Fig. 7).

Calling for time out, I pointed out to people the flaws in mere learning, as opposed to knowing, or actual knowledge. Quoting Contender 1, "For those familiar with the fighting technique of Mike Tyson, the eating of ears is not necessary, but a voracious appetite for ideas is!"