Yin-ed but not quite Yang-ed or is it vice versa?

The Left Hand of the Electron


Some of my articles stir up more reader comment than others, and one of the most effective in this respect was one I once wrote in which I listed those who, in my opinion, were scientists of the first magnitude and concluded by working up a personal list of the ten greatest scientists of all time.
Naturally, I received letters arguing for the omission of one or more of my ten best in favor of one or more others, and I still get them, even now, seven and a half years after the article was written.
Usually, I reply by explaining that estimates as to the ten greatest scientists (always excepting the case of Isaac Newton, concerning whom there can be no reasonable disagreement) are largely a subjective matter and cannot really be argued out.
Recently, I received a letter from a reader who argued that Archimedes, one of my ten, ought to be replaced by Euclid, who was not one of my ten. I replied in my usual placating manner, but then went on to say that Euclid was 'merely a systematizer' while Archimedes had made very important advances in physics and mathematics.
But later my conscience grew active. I still adhered to my own opinion of Archimedes taking pride of place over Euclid, but the phase 'merely a systematizer' bothered me. There is nothing necessarily 'mere' about being a systematizer.[Sometimes there is. In all my non-fiction writings I am 'merely' a systematizer. - Just in case you think I'm never modest.]

For three centuries before Euclid (who flourished about 300 B.C.) Greek geometers had labored at proving one geometric theorem or another and a great many had been worked out.
What Euclid did was to make a system out of it all. He began with certain definitions and assumptions and then used them to prove a few theorems. Using those definitions and assumptions plus the few theorems he had already proved, he proved a few additional theorems and so on, and so on.
He was the first, as far as we know, to build an elaborate mathematical system based on the explicit attitude that it was useless to try to prove everything; that it was essential to make a beginning with some things that could not be proved but that could be accepted without proof because they satisfied intuition. Such intuitive assumptions, without proof, are called 'axioms'.
This was in itself a great intellectual advance, but Euclid did something more. He picked good axioms. To see what this means, consider that you would want your list of axioms to be complete, that is, they should suffice to prove all the theorems that are useful in the particular field of knowledge being studied. On the other hand they shouldn't be redundant You don't want to be able to prove all those theorems even after you have omitted one or more of your axioms from the list; or to be able to prove one or more of your axioms by the use of the remaining axioms. Finally, your axioms must be consistent. That is, you do not want to use some axioms to prove that something is so and then use other axioms to prove the same thing to be not so.
For two thousand years, Euclid's axiomatic system stood the test. No one ever found it necessary to add another axiom, and no one was ever able to eliminate one or to change it substantially - which is a pretty good testimony to Euclid's judgment. By the end of the nineteenth century, however, when notions of mathematical rigor had hardened, it was realized that there were many tacit assumptions in the Euclidean system; that is, assumptions that Euclid made without specifically saying that he had made them, and that all his readers also made, apparently without specifically saying so to themselves.
For instance, among his early theorems are several that demonstrate two triangles to the congruent (equal in both shape and size) by a course of proof that asks people to imagine that one triangle is moved in space so that it is superimposed on the other. -That, however, presupposes that a geometrical figure doesn't change in shape and size when it moves. Of course it doesn't, you say. Well, you assume it doesn't and I assume it doesn't and Euclid assumed it doesn't - but Euclid never said he assumed it.
Again, Euclid assumed that a straight line could extend infinitely in both directions but never said he was making that assumption. Furthermore, he never considered such important basic properties as the order of points in a line, and some of his basic definitions were inadequate-
But never mind. In the last century, Euclidean geometry has been placed on a basis of the utmost rigor and while that meant the system of axioms and definitions was altered, Euclid's geometry remained the same. It just meant that Euclid's axioms and definitions, plus his unexpressed assumptions, were adequate to the job.

Let's consider Euclid's axioms now. There were ten of them and he divided them into two groups of five. One group of five was called 'common notions' because they were common to all sciences:
(1) Things which are equal to the same thing are also equal to one another.
(2) If equals are added to equals, the sums are equal.
(3) If equals are subtracted from equals, the remainders are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.

These 'common notions' seem so common, indeed so obvious, so immediately acceptable by intuition, so incapable of contradiction, that they seem to represent absolute truth. They seem something a person could seize upon as soon as he had evolved the light of reason. Without ever sensing the universe in any way, but living only in the luminous darkness of his own mind, he would see that things equal to the same thing are equal to one another and all the rest.
He might then, using Euclid's axioms, work out all the theorems of geometry and, therefore, the basic properties of the universe from first principles, without having observed anything.
The Greeks were so fascinated with this notion that all mathematical knowledge comes from within that they lost one important urge that might have led to the development of experimental science. There were experimenters among the Greeks, notably Ctesibius and Hero, but their work was looked upon by the Greek scholars as a kind of artisanship rather than as science.
In one of Plato's dialogues, Socrates asks a slave certain questions about a geometric diagram and has him answer and prove a theorem in doing so. This was Socrates' method of showing that even an utterly uneducated man could draw truth from out of himself. Nevertheless, it took an extremely sophisticated man, Socrates, to ask the questions, and the slave was by no means uneducated, for merely by having been alive and perceptive for years,he had learned to make many assumptions by observation and example, without either himself or (apparently) Socrates being completely aware of it.
Still as late as 1800, influential philosophers such as Immanuel Kant held that Euclid's axioms represented absolute truth.

But do they? Would anyone question the statement that 'the whole is greater than the part'? Since 10 can be broken up into 6 + 4, are we not completely right in assuming that 10 is greater than either 6 or 4? If an astronaut can get into a space capsule, would we not be right in assuming that the volume of the capsule is greater than the volume of the astronaut?
[Can a TARDIS fit inside a TARDIS? -Will Self]
How could we doubt the general truth of the axiom? Well, any list of consecutive numbers can be divided into odd numbers and even numbers, so that we might conclude that in any such list of consecutive numbers, the total of all numbers present must be greater than the total of even numbers. And yet if we consider an infinite list of consecutive numbers, it turns out that the total number of all the numbers is equal to the total number of all the even numbers. In what is called 'transfinite mathematics' the particular axiom about the whole being greater than the part simply does not apply.
Again, suppose that two automobiles travel between points A and B by identical routes. The two routes coincide. Are they equal? Not necessarily. The first automobile traveled from A to B, while the second traveled from B to A. In other words, two lines might coincide and yet be unequal since the direction of one might be different from the direction of the other.
Is this just fancy talk? Can a line be said to have direction? Yes, indeed. A line with direction is a 'vector' and in 'vector mathematics' the rules aren't quite the same as in ordinary mathematics and things can coincide without being equal.
In short, then, axioms are not examples of absolute truth and it is very likely that there is no such thing as absolute truth at all. The axioms of Euclid are axioms not because they appear as absolute truth out of some inner enlightenment but only because they seem to be true in the context of the real world.
And that is why the geometric theorems derived from Euclid's axioms seem to correspond with what we call reality. They started with what we call reality.
It is possible to start with any set of axioms, provided they are not self-contradictory, and work up a system of theorems consistent with those axioms and with each other, even though they are not consistent with what we think of as the real world. This does not make the 'arbitrary mathematics' less 'true' than the one starting from Euclid's axioms, only less useful, perhaps. Indeed, an 'arbitrary mathematics' may be more useful than ordinary 'common-sense' mathematics in special regions such as those of transfinites or of vectors.
Even so, we must not confuse 'useful' and 'true'. Even if an axiomatic system is so bizarre as to be useful in no conceivable practical sense, we can nevertheless say nothing about its 'truth'. If it is self-consistent that is all we have a right to demand of any system of thought. 'Truth' and 'reality' are theological words, not scientific ones.

But back to Euclid's axioms. So far I have only listed the five 'common notions'. There were also five more axioms on the list that were specifically applicable to
geometry and these were later called 'postulates'. The first of these postulates was:
(1) It is possible to draw a straight line from any point to any other point.
This seems eminently acceptable, but are you sure? Can you prove that you can draw a line from the Earth to the Sun? if you could somehow stand on the Sun safely and hold the Earth motionless in its orbit, and somehow stretch a string from the Earth to the Sun and pull it absolutely taut, that string would represent a straight line from Earth to Sun. You're sure that this is a reasonable 'thought experiment' and I'm sure it is, too, but we only assume that matters can be so. We can't ever demonstrate them, or prove them mathematically. And, incidentally, what is a straight line? I have just made the assumption that if a string is pulled absolutely taut, it has a shape we would recognize as what we call a straight line. But what is that shape? We simply can't do better than say, 'A straight line is something very, very thin and very, very straight', or, to paraphrase Gertrude Stein, 'A straight line is a straight line is a straight line-'
Euclid defines a straight line as 'a line which lies evenly with the points on itself,' but I would hate to have to try to describe what he means by that statement to a student beginning the study of geometry.
Another definition says that: A straight line is the shortest distance between two points.
But if a string is pulled absolutely taut, it cannot go from the point at one end to the point at the other in any shorter distance, so that to say that a straight line is the shortest distance between two points is the same as saying that it has the shape of an absolutely taut string, and we can still say 'And what shape is that?'
In modern geometry, straight lines are not defined at all. What is said, in essence, is this: Let us call something a line which has the following properties in connection with other undefined terms like 'point', 'plane', 'between', 'continuous', and so on. Then the properties are listed.

Be that as it may, here are the remaining postulates of Euclid:
(2) A finite straight line can be extended continuously in a straight line.
(3) A circle can be described with any point as center and any distance as radius.
(4) All right angles are equal.
(5) If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

I trust you notice something at once. Of all the ten axioms of Euclid, only one - the fifth postulate is a long jaw-breaker of a sentence; and only one - the fifth postulate - doesn't make instant sense.
Take any intelligent person who has studied arithmetic and who has heard of straight lines and circles and give him the ten axioms one by one and let him think a moment and he will say, 'Of course!' to each of the first nine. Then recite the fifth postulate and he will surely say, 'What!'
And it will take a long time before he understands what's going on. In fact, I wouldn't undertake to explain it myself without a diagram like the one below.

Consider two of the solid lines in the diagram: the one that runs from point C to point D through point M (call it line CD after the end points) and the one that runs through points G, L, and H (line GH). A third line, which runs through points A, L, M, and B (line AB), crosses both GH and CD, making angles with both.
If line CD is supposed to be perfectly horizontal, and line AB is supposed to be perfectly vertical, then the four angles made in the crossing of the two lines (angles CMB, BMD, DML, and LMC) are right angles and are all equal postulate 4). In particular, angles DML and LMC, which I have numbered in the diagram as 3 and 4, are equal, and are both right angles.
(I haven't bothered to define 'perfectly horizontal' or 'perfectly vertical' or 'crosses' or to explain why the crossing of a perfectly horizontal line with a perfectly vertical line produces four right angles, but I am making no pretense of being completely rigorous. This sort of thing can be made rigorous but only at the expense of a lot more talk than I am prepared to give.)
Now consider line GH. It is not perfectly horizontal. That means the angles it produces at its intersection (I haven't defined 'intersection') with line AB are not right angles and are not all equal. It can be shown that angles ALH and GLB are equal and that angles HLB and GLA are equal but that either of the first pair is not equal to either of the second pair. In particular, angle GLB (labeled 2) is not equal to angle HLB (labeled 1).
Suppose we draw line EF, passing through L, and that line EF is (like line CD) perfectly horizontal. In that case it makes four equal right angles at its intersection with line AB. In particular, angles FLB and ELB are right angles. But angle HLB is contained within angle FLB (what does 'is contained within' mean?) with room to spare. Since angle HLB is only part of FLB and the latter is a right angle then angle HLB (angle 1) is less than a right angle, by the fifth 'common notion'.
In the same way, by comparing angle ELB, known to be a right angle, with angle GLB (angle 2), we can show that angle 2 is greater than a right angle.
The 'interior angles' of the diagram are those on the side of line GH that faces line CD, and those on the side of line CD that faces line GH. In other words, they are angles 1, 2, 3, and 4.
The fifth postulate talks about 'the interior angles on the same side,' that is, 1 and 4 on one side and 2 and 3 on the other. Since we know that 3 and 4 are right angles; that 1 is less than a right angle, and that 2 is more than a right angle, we can say that the interior angles on one side, 1 and 4, have a sum less than two right angles, while the interior angles on the other have a sum greater than two right angles. The fifth postulate now states that if the lines GH and CD are extended, they will intersect on the side where the interior angles with a sum less than two right angles are located. And, indeed, if you look at the diagram you will see that if lines GH and CD are extended on both sides (dotted lines), they will intersect at point N on the side of interior angles I and 4. On the other side, they just move farther and farther apart and clearly will never intersect.
On the other hand, if you draw line JK through L, you would reverse the situation. Angle 2 would be less than a right angle and angle 1 would be greater than a right angle (where angle 2 is now angle JLB and angle 1 is now angle KLB). In that case interior angles 2 and 3 would have a sum less than two right angles and interior angles 1 and 4 would have a sum greater than two right angles. If lines JK and CD were extended (dotted lines), they would intersect at point O on the side of interior angles 2 and 3. On the other side they would merely diverge further and further.
Now that I've explained the fifth postulate at great length (and even then only at the cost of being very un-rigorous) you might be willing to say, 'Oh yes, of course. Certainly! It's obvious! '
Maybe, but if something is obvious, it shouldn't require hundreds of words of explanation. I didn't have to belabor any of the other nine axioms, did I?
Then again, having explained the fifth postulate, have I proved it? No, I have only interpreted the meaning of the words and then pointed to the diagram and said, 'And indeed, if you look at the diagram, you will see-'
But that's only one diagram. And it deals with a perfectly vertical line crossing two lines of which one is perfectly horizontal. And what if none of the lines are either vertical or horizontal and none of the interior angles are right angles? The fifth postulate applies to any line crossing any two lines and I certainly haven't proved that.
I can draw a million diagrams of different types and show that in each specific case the postulate holds, but that is not enough. I must show that it holds in every conceivable case, and this can't be done by diagrams. A diagram can only make the proof clear; the proof itself must be derived by permissible logic from more basic premises already proved, or assumed. This I have not done.

Now let's consider the fifth postulate from the standpoint of moving lines. Suppose line GH is swiveled about L as a pivot in such a way that it comes closer and closer to coinciding with line EF.(Does a straight line remain a straight line while it swivels in this fashion? We can only assume it does.) As line GH swivels toward line EF, the point of intersection with line CD point N) moves farther and farther to the right.
If you started with line JK and swiveled it so that it would eventually coincide with line EF, the intersection point O would move off farther and farther to the left. If you consider the diagram and make a few markings on it (if you have to) you will see this for yourself.
But consider line EF itself.When GH has finally swiveled so as to coincide with line EF, we might say that intersection point N has moved off an infinite distance to the right (whatever we mean by 'infinite distance') and when line JK coincides with line EF, the intersection point O has moved off an infinite distance to the left. Therefore, we can say that line EF and line CD intersect at two points, one an infinite distance to the right and one an infinite distance to the left.
Or let us look at it another way. Line EF, being perfectly horizontal, intersects line AB to make four equal right angles. In that case, angles 1, 2, 3, and 4 are all right angles and all equal. Angles 1 and 4 have a sum equal to two right angles, and so do angles 2 and 3.
But the fifth postulate says the intersection comes on the side where the two interior angles have a sum less than two right angles. In the case of lines EF and CD crossed by line AB, neither set of interior angles has a sum less than two right angles and there can be an intersection on neither side.
We have now, by two sets of arguments, demonstrated first that lines EF and CD intersect at two points, each located an infinite distance away, and second that lines EF and CD do not intersect at all. Have we found a contradiction and thus shown that there is something wrong with Euclid's set of axioms?
To avoid a contradiction, we can say that having an intersection at an infinite distance is equivalent to saying there is no intersection. They are different ways of saying the same thing. To agree that 'saying a' is equal to 'saying b' in this case is consistent with all the rest of geometry, so we can get away with it.
Let us now say that two lines, such as EF and CD, which do not intersect with each other when extended any finite distance, however great, are 'parallel'.
Clearly, there is only one line passing through L that can be parallel to line CD, and that is line EF. Any line through L that does not coincide with line EF is (however slightly) either of the type of line GH or of line JK, with an interior angle on one side or the other that is less than a right angle. This argument is sleight of hand, and not rigorous, but it allows us to see the point and say: Given a straight line, and a point outside that line, it is possible to draw one and only one straight line through that point parallel to the given line.
This statement is entirely equivalent to Euclid's fifth postulate. If Euclid's fifth postulate is removed and this statement put in its place, the entire structure of Euclidean geometry remains standing without as much as a quiver.
The version of the postulate that refers to parallel lines sounds clearer and easier to understand than the way Euclid puts it, because even the beginning student has some notion of what parallel lines look like, whereas he may not have the foggiest idea of what interior angles are. That is why it is in this 'parallel' form that you usually see the postulate in elementary geometry books.
Actually, though, it isn't really simpler and clearer in this form, for as soon as you try to explain what you mean by 'parallel' you're going to run into the matter of interior angles. Or, if you try to avoid that, you'll run into the problem of talking about lines of infinite length, of intersections at an infinite distance being equivalent to no intersection, and that's even worse.

But look, just because I didn't prove the fifth postulate doesn't mean it can't be proven. Perhaps by some line of argument, exceedingly lengthy, subtle and ingenious, it is possible to prove the fifth postulate by use of the other four postulates and the five common notions (or by use of some additional axiom not included in the list which, however, is much simpler and more 'obvious' than the fifth postulate is).
Alas, no. For two thousand years mathematicians have now and then tried to prove the fifth postulate from the other axioms simply because that cursed fifth postulate was so long and so unobvious that it didn't seem possible that it could be an axiom. Well, they always failed and it seems certain they must fail. The fifth postulate is just not contained in the other axioms or in any list of axioms useful in geometry and simpler than itself.
It can be argued, in fact, that the fifth postulate is Euclid's greatest achievement. By some remarkable leap of insight, he realized that, given the nine brief and clearly 'obvious' axioms, he could not prove the fifth postulate and that he could not do without it either, and that, therefore, long and complicated though the fifth postulate was, he had to include it among his assumptions.
So for two thousand years the fifth postulate stood there long, ungainly, puzzling. It was like a flaw in perfection, a standing reproach to a line of argument otherwise infinitely stately. It bothered the very devil out of mathematicians.
And then, in 1733, an Italian priest, Girolamo Saecheri, got the most brilliant notion concerning the fifth postulate that anyone had had since the time of Euclid, but wasn't brilliant enough himself to handle it- Let's go into that in the next chapter.

The Left Hand of the Electron





Chaos Quantum Logic Cosmos Conscious Belief Elect. Art Chem. Maths

The Left Hand of the Electron File Info: Created 22/11/2013 Updated 13/1/2014 Page Address: http://leebor2.100webspace.net/Zymic/l-hand10.html