Post by treeman on Sept 7, 2014 5:18:14 GMT -5
Once upon a time there was a wise man who stumbled upon a diamond in the rough;
But he couldn't believe his eyes, so he called it glass, and threw it away...
Who was this man? Farkas Bolyai was his name, and geometry was his game; the last in a long list of free thinkers who attempted the impossible - To prove that there are parallel lines in the universe.
I remember, when taking a final exam for a non-Euclidean Geometry class in graduate school, I was presented with the question:
Q: How can you know that there are parallel lines in the universe?
My answer?
A: Because I can walk from one end of this classroom to the other without hitting my head on the ceiling!
Farkas lost his mind. Alas, he was the man! He had discovered Non-Euclidean Geometry but didn't recognize it, he could not see it.
He had proven, in the late nineteenth century that the parallel postulate couldn't be proven, lest all the axioms of Euclidean Geometry become inconsistent.
(It wasn't until 1940 that Gödel would prove that "no" mathematical system can be complete.)
Truly, Farkas had discovered a diamond in the rough,
But couldn't believe his eyes;
So he called it glass, and threw it away...
Pity.
He had however, seen something that his forerunners had not; the assumptions in their arguments, and the circular reasoning of their and proofs; one and all.
Ptolemy (circa 130 AD) assumed that there was at least one line parallel to a line through a given point which is equivalent to Euclid's postulate - Circular reasoning en.wikipedia.org/wiki/Ptolemy
Proclus (410 - 485) assumed parallel lines are always equidistance which is an added assumption about parallel lines. en.wikipedia.org/wiki/Proclus
Wallis (1616 - 1703) proved the Parallel Postulate assuming a postulate about Similar Triangles which is equivalent to Euclid's postulate - Circular reasoning. en.wikipedia.org/wiki/John_Wallis
Saccheri (1667 -1733) worked with quadrilaterals, now called Saccheri Quadrilaterals, where the base angles are rights angles and the sides adjacent to the base are congruent.
Remember this one? Ha!
The question is:
What can be proven about the summit angles, <D and <C?
Without assuming the Parallel Postulate, it can be proven that the two summit angles are congruent.
Then, there are three distinct possibilities:
1) The summit angles are acute angles.
2) The summit angles are right angles.
3) The summit angles are obtuse angles.
What Saccheri finally wrote was:
"The hypothesis of the acute angle is absolutely false, because [it is] repugnant to the nature of the straight line!" (LOL!)
Clairaut (1713 -1765) proved the Parallel Postulate assuming a postulate about the Existence of Rectangles which is equivalent to Euclid's postulate - Circular reasoning.
Legendre (1752 -1833) worked with the Parallel Postulate assuming a postulate about the angle sum of a triangle being equal to 180 degrees, which is equivalent to Euclid's postulate - Circular reasoning.
Lambert (1728 -1777) worked with quadrilaterals, now called Lambert quadrilaterals, which have three right angles. The question is what can be said about the fourth angle? Etc…
It was because so many mathematicians had tried to prove Euclid's Parallel Postulate, that Klügel did his doctoral thesis in 1763 finding the flaws in 28 different proofs of this postulate.
The thesis led d'Alembert to call Euclid's Parallel Postulate "The scandal of geometry."
en.wikipedia.org/wiki/Jean_le_Rond_d'Alembert
When Farkas Bolyai learned that his son had turned his attention towards the fifth postulate, he lamented his own wasted life and feared for the sanity of his son!
The Hungarian Farkas Bolyai writes:
You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you; leave the science of parallels alone...
I thought I would sacrifice myself for the sake of truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind....
I turned back when I saw that no man can reach the bottom of the night. I turned back un-consoled, pitying myself and all mankind.
..... I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail.
The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness.
János Bolyai writes back:
It is now my definite plan to publish a work on parallels as soon as I can complete and arrange the material....
When you, my dear Father, see them, you will understand; at present I can only say nothing except this: that out of nothing I have created a strange new universe.
All that I have sent you previously is like a house of cards in comparison to a tower.
When János Bolyai learned from his father that his collegue Gauss had already done this work but would never publish, he became despondent and would never publish either.
It wasn't until 1829 when Lobachesky would be the first mathematician to publish an account of Non-Euclidean Geometry.
He was subsequently fired from his university post. Who was he to challenge Kant?
And then Riemann gave us elliptic geometry with the Riemann sum,
And then Poincaré of course, the models for hyperbolic geometry.
Farkas Bolyai had discovered what he was not looking for.
A whole new world where there could be an infinite number of parallel lines to any given line!
Imagine this...
Postulates in geometry are very similar to axioms, self-evident truths, and beliefs in logic, political philosophy and personal decision-making.
The five postulates of Euclidean Geometry define the basic rules governing the creation and extension of geometric figures with ruler and compass.
Together with the five axioms (or "common notions") and twenty-three definitions at the beginning of Euclid's Elements,
they form the basis for the extensive proofs given in this masterful compilation of ancient Greek geometric knowledge.
They are as follows:
1) A straight line may be drawn from any given point to any other point. 2) A straight line may be extended to any finite length.
3) A circle may be described with any given point as its center and any distance as its radius.
4) All right angles are congruent.
5) If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles,
then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles.
Postulate 5) - the so-called Parallel Postulate was the source of much annoyance, probably even to Euclid, for being so relatively prolix.
Mathematicians have a peculiar sense of aesthetics that values simplicity arising from simplicity, with the long complicated proofs,
equations and calculations needed for rigorous certainty done behind the scenes,
and to have such a long sentence amidst such other straightforward, intuitive statements seems awkward.
As a result, many mathematicians over the centuries have tried to prove the results of the Elements without using the Parallel Postulate, but to no avail.
However, in the past two centuries, assorted non-Euclidean geometries have been derived based on using the first four Euclidean postulates together with minor variations on the fifth.
Let me break it down right now.
Draw a circle.
There are an infinite number of points in the circle, right? So far - so good.
Now pick two points, a and b in the circle and draw a line L through them from one side of the circle to the other with the line stopping on the circumference of the circle.
Now we have what is referred to as a "CHORD".
Now, take a point C in the circle and draw a line through point C from one point on the circumference to another point on the circumference.
Notice that none of these new lines intersect line L.
All of these lines have an infinite number of points on them and they extend from infinity to infinity.
The circumference of the circle is at infinity and it has an infinite number of points on it.
We haven't violated any of the axioms of Eculidean Geometry, have we? No...
The line from point a, to point b, is consistent with axiom #1.
The line L is consistent with axiom #2.
The universe has an infinite number of points in it - so does the circle - axiom #3.
Our model does not violate axioms #4 or #5.
But look,
There are an infinite number of lines through point C (which is not on line L) in the same plane as line L, that are parallel to line L !!
All of these lines extend from infinity to infinity and none of them intersect line L,
And we have not violated any of the axioms of Eucledian Geometry!
BEHOLD
There are an infinite number of parallel lines to any given line in the universe,
And you can't prove otherwise...?...
Sometimes,
We are so close to the truth,
That we cannot see it...
...