Misconceptions About Geometry "Postulates"

The avoidance of assumptions is one of the clearest indicators of intelligence, accuracy, and consistency.  Despite being at the foundation of all sound philosophy, the rejection of assumptions is championed by a select few at most, and assumptions (whatever arbitrary nature they take in a given irrationalist's worldview) are often defended and even celebrated.  Indeed, many children are explicitly taught that assumptions are sound and necessary, particularly when it comes to geometry.  This counterproductive endeavor is on full display when students are introduced to postulates.

The typical description of various postulates, whether those postulates are correct or not, is that they are foundational ideas assumed to be true without proof and that they are also used to prove other things.  This is not only an inaccurate representation of the foundational premises of sound geometry, but it is also a completely inaccurate representation of epistemology itself.  Logical axioms--self-evident and self-verifying necessary truths that must be affirmed to be denied--are not proven by other premises, but they are not mere assumptions.  Instead, they prove themselves.  For this reason, they cannot be false or uncertain no matter what else is.  Many geometry postulates do not have this property, and thus they cannot legitimately be treated as geometric axioms (not that axioms have to be assumed at all to begin with!), contrary to the claims of almost all geometry educators.  Consider the postulate that only one line can run between two points.

Can their only be one straight line through two respective points in space?  Of course this is the case, but this is not something that has to be assumed, much less something that is unverified or unverifiable.  This fact is also not a self-evident and self-verifying axiom (like the law of identity): in order to know that only one line can connect two given points, one must already know what points and lines are and then reason out why such a thing must be true.  It is not as if there is no way to demonstrate that multiple lines cannot run through two points!  Even a self-evident thing does not have to be assumed, but this "postulate" is not self-evident.  It can be reduced down to other logical and geometric facts, meaning it is not the utter foundation of shapes.

When it comes to the postulate that any three points fall on the same plane, the verifiability is similar.  It is true that any three points must always fall on the same plane (and are thus coplanar) if a plane is infinite, having no spatial boundaries whatsoever.  If a plane was finite, then it could not true that any three points must be contained within it.  No matter how large a finite plane is, there is always the possiblity that two points are within it and that the third is outside of it (or that all three are outside of it), even if only by a miniscule distance.  Either way, the truth of the matter is not a self-evident, self-verifying thing; it stands upon premises that are more foundational.  One must understand the basic concept of infinite space and the finite nature of points to realize that a plane must contain three points.

These and other geometric facts do not have to be assumed.  Moreover, no one who believes that assumptions are necessary understands the basic aspects of sound epistemology.  The very nature of an assumption is that it is uncertain; if an idea is assumed to be true (even if it ultimately is true), nothing that builds on that idea is known to reflect reality.  The fact that so many people truly think that an assumption could ever ground actual knowledge does not and cannot legitimize even a single assumption.  Rather, it affirms the fundamental unintelligence of each person who would claim that such an impossibility is true.  There is not a single instance where an assumption is anything more than a blind, irrational leap into a darkness that could be at least somewhat illuminated by the necessary laws of logic.