Many modern scientists and mathematicians make the grievous error of presuming that entities in the real world have spatial qualities fundamentally different from those that the illustrious geometer Euclid discovered them to possess. Theories of “non-Euclidean geometry” have been devised, based on a rejection of Euclid’s fifth postulate—the parallel postulate—which states that no two mutually parallel lines will ever intersect. Applications of “non-Euclidean geometry” have resulted in absurd theories conceiving of “space” or “the universe” as “curved” and not at all consistent with our basic sensory perceptions.

In Praxeology and Certainty of Knowledge, I defended Euclidean Geometry’s universal and irrefutable truth against non-Euclideans’ rejections of it.

The axioms of Euclidean geometry correspond to the physical world, whereas the axioms of geometric systems contrary to Euclid's do not. (That is, they are not true axioms, since they can be elementarily refuted in the course of ubiquitous daily observation.) The human body can be measured by using three and only three spatial parameters – known as dimensions: any system of measurement claiming more or less than three dimensions will fail to adequately describe man's physical form. All parts of the human body have boundaries, describing which necessitates the Euclidean constructs of points, lines, and planes. Furthermore, all human movement and interaction with other entities occurs three-dimensionally. Every possible path of motion can be described by adding three mutually perpendicular vectors of the proper magnitudes. Moreover, all spatial measuring instruments can only be built with Euclidean postulates at the foundation of their design.

No measurement can ever refute the validity of Euclidean geometry, since measuring tools themselves – as well as the bodies and movements of those who measure – are predicated upon the axioms of Euclid's system. If the spatial qualities of humans and all the objects they observe and interact with can be described and measured only through Euclid's system, there is no point in asserting that any non-Euclidean geometry can also be true: it cannot be true if it describes nothing that exists!

A reader of my essay e-mailed me, challenging me on this position. This article is an adapted version of the interesting discourse that followed—in which I elaborate my defense of Euclidean geometry, explain how every valid application of “non-Euclidean geometry” is in fact consistent with Euclid’s system, and prove Euclid’s parallel postulate by following the definition of a parallel line. 

Real-World Applications of Non-Euclidean Geometry?

In defense of non-Euclidean geometry, the questioner cited its apparent practical usefulness—despite its axioms’ lack of correspondence with reality:

Isn't it true that Riemannian geometry, which does not use the parallel axiom, has eventually found many applications in the "physical world", like explaining black holes and heavy stars? Isn't it also used in computer imaging? Aren't there often axioms that have no correspondence with everyday experience but prove useful in developing theorems that later explain something otherwise not explainable in the world?

The mere ability of a theory to explain some part of observable reality does not qualify it to be a valid theory. One could state with legitimacy that the theory is imperfect but partially valid—if that theory clashes with certain ubiquitous or self-evident knowledge but explains some known occurrences. Such a situation only implies that some of the theory’s assumptions correspond to reality, whereas others do not. However, then the burden of proof is on the advocate of the theory to show that the theory's correct predictions do not derive from those of its assumptions that are false. It is, I claim, possible to isolate the elements of a theory that work and have valid applications from those elements that are contradictory and inevitably inhibit the theory's functionality.

For Non-Euclidean forms of geometry, one must always specify context. It is true that a shape drawn, say, on the surface of an ellipsoid, will not have the same characteristics as an autonomous shape. Riemannian geometry might indeed account for those different characteristics. Yet this is not because Euclidean geometry is wrong but because the shapes are, in fact, different. Both shapes are still entities that fundamentally obey the laws of Euclidean space. The "triangle" on the surface of an ellipsoid is drawn on the surface of a three-dimensional entity whose shape can be explained by Euclidean constructs. When non-Euclidean geometry tries to extrapolate its observations beyond shapes on actual three-dimensional surfaces, however, it comes into conflict with the true axioms of Euclidean geometry; those applications are, therefore, wrong. The non-Euclidean "axioms" are the result of such false application. If they contradict Euclid's axioms, then they falsely extrapolate particular observations to make ubiquitous generalizations (say, about the "shape" or "curvature" of space—absurd concepts both).

In short, the valid uses of non-Euclidean geometry stem not from its fundamentally flawed axioms but from its application to particular contexts where the non-Euclidean propositions do not clash with Euclidean ones. It would be interesting if mathematicians and scientists accepted my challenge to find out exactly which parts of non-Euclidean geometry are thus consistent with ubiquitous observation and which are not. Such a conceptual re-systematization would constitute an enormous forward leap for the theoretical integrity and applicative power of mathematics.

Irrefutable Demonstration of Euclid’s Parallel Postulate

After reading my reply, the questioner raised another challenge:

Many thanks for your answer. It does raise another question, though – about "true axioms." How do you know that two Euclidean parallels will not cross when extended to infinity?

The truth of the parallel postulate is quite evident, once one recognizes how parallel lines are defined: two parallel lines are lines with the exact same slope, with every point on line 1 (L₁) located at a set distance from every corresponding point on line 2 (L₂).

Let us assume that L₁ and L₂ have slope m. At the y-axis, L₁ has coordinates (0,y₁), and L₂ has coordinates (0,y₂). The vertical distance between L₁ and L₂ along the y-axis is (y₂-y₁).

If the x-coordinates of both L₁ and L₂ are increased by any value t (positive or negative), then their y-coordinates will be increased by mt, since their slopes are m. Thus, at x=t, L₁ has coordinates (t,y₁+mt), and L₂ has coordinates (t, y₂+mt). The vertical distance between L₁ and L₂ at x=t is (y₂+mt)-(y₁+mt) = (y₂- y₁).

Thus, at any x=t, the vertical distance between two distinct parallel lines will always be the same. Since we defined y₂ as unequal to y₁, this means that this distance will never be zero, which means that the parallel lines can never intersect, no matter how far they extend.

Euclid was right, and this is quite elementary to show—provided one keeps in mind what a parallel line is.

The Concept of Distance as Primary to the Concept of Parallelism

In response to my irrefutable demonstration of the parallel postulate, the questioner raised the following contention:

You are using the results of Euclidian geometry ("distance", "corresponding point"), based on the parallel axiom, to prove that the parallel axiom is "true". I meant "results" or "other postulates". For centuries, mathematicians have tried to derive the fifth postulate from the first four, but without success. There is no way to prove the fifth postulate before developing Euclidean geometry based on the fifth postulate. Indeed, this is what an axiom is: something that can't be proven.

Quite the contrary: the concept of "distance"—especially of "distance of separation"—is primary to not only the concept of "parallel lines" but the concept of "line" itself. The very idea of spatial dimensions can only come about from the ubiquitous observation that all entities and all parts of all entities are separated from each other by distance. I explain this thoroughly in A Rational Cosmology: Chapter III. (In that chapter, I also describe the manner in which Euclidean geometry perfectly describes the spatial qualities of entities.)

The parallel postulate is axiomatic because it is implied in the definition of parallel lines, so that one cannot contradict the postulate without violating the definition. Parallel lines can only be defined as lines with the same slope. Without that definition, the very concept of "parallelism" is just an empty word. Of course, "slope" must be defined before "parallelism" can be—and the definition of slope follows from the definition of a line, which follows from the ubiquitous observation of distance separation between entities.

I did not prove that the parallel postulate was true, but rather irrefutably demonstrated its truth by showing that it cannot be contradicted without defying the very definition of parallel lines. Thus, like action is implied in the nature of man—so is the parallel postulate implied in the nature of parallel lines. The attempt to contradict this truth implies the absurdity of stating that "parallel lines are not parallel."

Here, I have briefly defended the ability of Euclidean geometry to perfectly describe the spatial attributes of all entities. I have shown that those attributes of non-Euclidean geometry that have applicative relevance to reality are consistent with Euclid’s axioms. Furthermore, I have proved Euclid’s parallel postulate—the most heavily contested of his axioms—and shown how this postulate must follow from the very definition of a parallel line, so that the very attempt to contradict it would only violate the nature of parallelism as derived from the nature of lines and, ultimately, distance of separation. 

Euclid was correct: we live in a knowable, measurable universe accurately observable through the human senses.

G. Stolyarov II is a science fiction novelist, independent filosofical essayist, poet, amateur mathematician, composer, contributor to Enter Stage Right, The Autonomist, Le Quebecois Libre, and the Ludwig von Mises Institute, Senior Writer for The Liberal Institute, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress. His newest science fiction novel is Eden against the Colossus. His latest non-fiction treatise is A Rational Cosmology. Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.