Archive for November, 2013

Comparing Axiomatic Systems: Euclid & Lobachevsky

Uncategorized | Posted by Brian PCF
Nov 26 2013

Geometry seems a unique science. Those who practice what we would call “Natural Sciences” gather independent facts, try to find patterns, and then through cumbersome intermediate steps determine laws. On the other hand as geometry progresses in complexity everything that it creates is a new law, equally important from first to last. So long as it does not contradict anything that’s come before it is considered true no matter what its basis. Thus proving any system of geometry turns into simply avoiding contradicting oneself. Euclid did this elegantly by asking us to grant him the “Parallel Postulate” and then not using it for twenty eight propositions. Lobachevsky asks us to do this by proposing that the angle of parallelism is less than the sum of two right angles. Both of these systems seems true in that they do not contradict themselves, but they certainly contradict one another.

How does comparing these two systems inform us as to the necessity of questioning what we consider “laws” and what is the relative value of studying laws for their own sake versus questioning their “trueness” in comparison with other systems of laws? Said another way, what is the value of using our deductive reason to analyze deductive systems and our inductive reason to compare deductive systems?

Part I: Using deductive reason to analyze deductive systems.

Let us examine which deductions are shared by both Euclidean and Non-Euclidean systems and why. It seems implied that any of Euclid’s work still holds in Lobachevsky except for the fifth postulate (which gets us to the angle of parallelism being equal to two right angles) and anything that Lobachevsky specifically redefines. It must be different in some basic concepts otherwise there are no points in Lobachevsky. Where Euclid starts with a point, Lobachevsky starts immediately with the line and since he uses a different definition, this must be different than a Euclidean line:

Lobachevsky Theorem 1: A straight line fits upon itself in all positions. By this I mean that during the revolution of the surface containing it the straight line does not change its place if it goes through two unmoving points in the surface: (i.e., if we turn the surface containing it about two points of the line the line does not move.)

For Euclid’s definition of a line to work, we must include his definition of a point as well:

Euclid Definition 1: A point is that which has no part.
Euclid Definition 2: A line is breadthless length.

So it is now spelled out in the very beginning of Lobachevsky that rather than the Euclidean universe that is made up exclusively of points and lines and the relationships between the two, we start with a line and a surface with the potential for rotation as the fundamental building blocks of Lobachevsky’s geometry.

As Lobachevsky builds his system, we see several different concepts in his definitions, but none are that far afield of what is true within Euclid’s system until we get to Theorem 16 and find out what he calls the angle of parallelism can be either what Euclid states it is: the sum of two right angles, or possibly less than that. What he mathematically represents as:

If Euclid: ∏(p) = ∏/2.
If not: ∏(p) < ∏/2. So at this point, we take nothing for granted within Lobachevsky’s deductive system but the difference between him and Euclid is now becoming clear. In terms of pure deduction, he could just as easily propose that ∏(p) > ∏/2 or some random series of integers like ∏(p) = 4.2114591913951. At this point the key to hurdling the questions within pure deduction will be his ability for all resulting theorems to be consistent with his “if” statement.

As we continue to build to his true break from Euclidean geometry in Theorem 22, we find that sum of the angles of a triangle will be the same as his angle of parallelism in Theorem 16. That is, that the sum is less than ∏/2. Again he gives two options here, the Euclidean and what he calls the “imaginary geometry” which he says can apply to both rectilineal and spherical triangles.

However, the big deductive question in this is evidenced by Theorem 24 where he proposes that “The farther parallel lines are prolonged on the side of their parallelism, the more they approach one another.” Since we already have Theorem 17 “A straight line maintains the characteristic of parallelism at all its point” and Theorem 2 “Two straight lines cannot intersect in two points” we must throw out the most obvious question in Lobachevsky: do his straight lines curve? Then we must logically ask if all of his surfaces curve in some way? However, it seems in Theorem 22 that since he specifically differentiates between the rectilineal and the spherical triangles, his surfaces are consistent with the surfaces found in Euclid.

So we are only left with a few options in our deductive reasoning: first, that Lobachevsky’s prose can accurately describe a model that diagrams cannot. Second, we can wonder if this is similar to the parable of Meno’s ass: that if one line continues to move closer to the other line by halves, they will never meet.

Here we are at an impasse in our deductive capabilities. Both systems seem true in and of themselves, but both come to different conclusions about fundamental parts of geometry. It seems that to understand how true these systems are we must compare them in more detail to one another and use our inductive reasoning as much as possible.

Part II: Using inductive reason to compare deductive systems.

Geometry is pure deduction: everything follows from what comes before. To say this in another way is to restate our opening proposition: geometry seems very different than the natural sciences. Bacon called upon the natural sciences to use its inductive reason to find true laws, and while human nature has the same propensity towards error as in Bacon’s time, science has progressed and found rapidly appreciating application in understanding and making use of nature’s laws.

So how do we go about using our inductive reason to analyze these two systems of geometry? The first thing we can do is ask ourselves how to go about it: just because something that is designed to be purely deductive shows no obvious deductive flaws, how can we use to use inductive axioms to compare these deductive systems. Let us first look at an inductive axiom that seems to meet Bacon’s tests and has held up to several inductive and deductive challenges since its first introduction: Darwin’s survival of the fittest by means of natural selection.

How does Lobachevsky hold up against Euclid (and vice versa) in the application of this axiom? Though it is outside the scope of this paper, it would seem that we are uniquely fortunate in trying this model of examination because rather than large historical gaps being present through missing fossils, we see every deductive step in Euclid and Lobachevsky’s work numbered out for us. We can look at those individual evolutions of the species and then look at how it affected the larger genera of science. We can rather simply ask the question (the answer may not be so simple), do these systems act within science as a dominant species or one that looks like it will become extinct? As Darwin describes the difference: “new and improved varieties will supplant and exterminate the older less improved and intermediate varieties; and thus species are rendered to a large extent defined and distinct objects. Dominant species belonging to larger groups tend to give birth to new and dominant forms.”

While Euclidean geometry helped give birth to Newton’s physics, it was the Non-Euclideans (including Lobachevsky) that gave birth to Reiman which helped produce Einstein’s Theory of General Relativity which showed the flaws in Newton’s formulation of gravity and the movement of celestial bodies.

What is interesting and brings us full circle in a sense is that both are still accepted as “true” within certain parameters. When describing a body moving through space, so long as that body is moving in a unified gravitational field and nowhere close to the speed of light, Newton’s system works flawlessly. If we need to predict the movement of a body through space with inconsistent gravitational field and/or moving close to the speed of light however, we must use Einstein’s model to make an accurate prediction.
So it seems that even in the natural sciences, there are models that are sometimes true and sometimes less true.

Conclusion:

What one can potentially learn from comparing the two are first: that even that which stands for centuries and is taught as doctrine must be tested, changed, and potentially improved upon. If we do not attempt this we not only limit ourselves to profound discoveries that may aid centuries of scientific development, we also contradict what Bacon warns us against as the greatest pitfall to human understanding: “Those who have taken upon them to lay down the law of nature as a thing already searched out and understood, whether they have spoken in simple assurance or professional affectation, have therein done philosophy and the sciences great injury.”

Second is proof of the concept that there can be more than one answer. Euclidean geometry still “works” in a Euclidean model. Non-Euclidean Geometry works in a Non-Euclidean model. Both are tools that have applications in different fields and have value in and of themselves.
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Reference:
Full text of Euclid’s Elements here.
Full text of Lobachevsky’s Theory of Parallels here.

The Structure of Euclid’s Elements, Book I

Uncategorized | Posted by Brian PCF
Nov 11 2013

The structure of Euclid’s Elements can be explained simply by “lack of chaos”. Everything that comes about does so orderly and methodically. There is no new thing. There is no room for the addition of a new element, like in the periodic table. There is no new word that can be invented like Shakespeare did for language. Everything is just either a point, a line, or something brought about by the relationship of the two. There is no discovery of something new in that sense in Euclid, only something new in terms of the relationships between points and lines.

This leads to the second great structure or theme of Euclid’s Elements: his invitation to find flaws or explain it better. The arguments are so elegant, they are so “obvious” once they are explained, an amateurish mind might think “I can prove that wrong” or “I can do that better”. Euclid leaves that avenue open by explaining every point succinctly and referencing every conclusion exactly.

Section I: The Definitions

The definitions provide a type of alphabet for Euclid’s language. All of the propositions that he constructs will share these basic building blocks. What’s most striking about the first two definitions is that they are defined by what they are not:

1. A point is that which has no part.
2. A line is breadthless length.

These first two definitions demonstrate that Euclid’s universe is very much abstract: meaning it can’t exist in a real physical world that we can see or construct, only in our imagination. And yet the construction of the remainder of the definitions, and hence all of the propositions as well, rely on these two things: lines and points.

So two question to return to are: a) is Euclid creating an entire system that depends on us believing that things exist that we cannot prove exist and/or cannot physically comprehend, and b) do we accept a system that must be tied together using things that can’t exist in the physical universe as a starting point? While these questions sound similar, we will see they lead us to different conclusions.

But what are the definitions themselves? They are a type of relational shorthand. They are all made up of points and lines but in order to simplify his terms Euclid creates (or at least clearly defines) concrete descriptions based on relations between points and lines. These definitions serve a similar role to early propositions: once we have an agreed upon conception of these ideas, we can simply refer to them and we will understand their meaning specifically and in the context of their use.

Section II: The Postulates

The postulates, or one translation: “grant me this”, help move the definitions from abstract to real or at least easier to understand. It is interesting that Euclid uses different verbs for each of the first three postulates:

1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center or distance.

Why not just use “draw” or “produce” or “describe” for each postulate? Why use different terms? The most likely answer is because Euclid thinks to make these different things more understandable, we must do specific things to make them so. Producing or drawing a circle will not do, it must be “described.” Drawing or describing a straight line continuously in a straight line cannot be done, it must be “produced”. Likewise with a straight line from any point to any point, it must be “drawn”.

While this logic seems circuitous, one finds the difference necessary once within the propositions. Without them, their use and definitions become less clear. The most obvious of example is the circle. To “draw” or “produce” a perfect circle is something commonly regarded by artist’s as impossible. But we can create something, say “this is a circle” and then with our definitions mesh this with the context and meaning within the proposition.

Section III: The Common Notions

The Common Notions might be easier understood if Euclid called them “Common Sense Notions”. These are things which if not indisputable, would be very hard to dispute. One can imagine that Euclid created the whole of the Elements, but some competing school of thought decried his work for not laying out, for instance, that a “whole is greater than the part”. So while these things are obvious, the theme of construction tightly woven with what’s come before must be satisfied and hence these common sense sayings or notions must be included.

Section IV: The Propositions

What is a “proposition”? Using only what we read contextually from Euclid, a proposition seems to be something constructed through what is already there. But while they rely on something already established, they also create something new. Once the proposition is satisfactorily constructed (whether it is “quod erat demonstrandum” or “quod erat faciendum”) it is now part of our building blocks and can be used to construct something more.

As to the specific propositions, the overarching theme is simple to complex. More building blocks are available as we move along and Euclid makes use of them. Circles are the easiest to understand at first, so that is what Euclid uses first. But we must get to triangles for all the equalities we can learn from them. While circles are understandable, they are also much less useful than triangles since the only things we can relate them to are lines, points and triangles. Euclid never uses them in Book I for rectilineal angles, parallel lines, parallelograms, or squares. Yet triangles can be used to demonstrate relationships for all these shapes.

The trickiest part of Euclid’s first book seems to be the idea of parallel lines. While he gives them a meaning in Definition 23, he has to ask his reader to grant him some further concepts to support their existence in Postulate 5:

That if a straight line falling on two straight lines make the interior angles on that same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angle less than two right angles.

One could call this a clunky description. It is certainly inelegant when one compares it to the definition of a point or a line.

Euclid is also hesitant to use Postulate 5, waiting until Proposition 29 to reference it. At this point, he already has two propositions (27 and 28) that make use of parallel lines without using Postulate 5. So is it because he feels uncomfortable with the idea of parallel lines that he wants to get as many propositions constructed as possible before having to use them?

Here we find ourselves in a similar situation as when we finished the first two definitions. We are presented something which we must accept to continue on, but which we cannot really understand except in a hazy abstraction. So questions remain and relate to those from a point and a line: a) is Euclid creating an entire system that depends on us believing that things exist that we cannot prove exist and/or cannot physically comprehend, and b) do we accept a system that must be tied together using things that can’t exist in the physical universe as the starting point?

The answer to the former must be yes. He is asking us to either believe or at least agree with him on the definition of a point, a line, and that parallel lines will not meet. As to the latter, the most likely reply is “what is defined in abstraction must be agreed upon in abstraction.” Meaning that we must omit Euclid from the physical universe if he does cannot relate it to the physical universe.

Section V: Conclusion: Where does his structure leave us?

The only limit placed upon us implicitly at the end of Book I is “use what has come before.” Euclid has demonstrated that one can create a great deal out of a point, a line, and the relationship between the two. We can construct whatever we want from here, so long as it is not contradictory to what has come before.

But even if it is contradictory, we can examine our constructions and Euclid’s constructions exactly and determine which is more true. No conclusion is left unexplained and every step is laid out. So one can work within the bounds that Euclid has laid out and accept his definitions, postulates, common notions, and propositions and build on them. Or one can attempt to prove these wrong.