Geometry means something like earth measure and seems to have been given its fame and longevity by Euclid’s, the ancient Greek’s, The Elements which begins with 23 Definitions of things people measure, 5 Postulates, then 5 Axioms or Common Notions. These are followed by his famous Propositions whose validity is determined by acceptance of the Definitions, Postulates and Axioms. Modern Science with its increasing reliance on mathematics has been incorporating terms from Euclid’s Elements. Much of Newton’s and Galileo’s writings seem to presuppose the validity of Euclid’s Propositions. In more recent times Dedekind’s use of the geometer’s fundamental element, the line, to demonstrate his theory of number shows how long Euclidean geometric concepts have been needed. And even in more recent times mathematicians maintain Euclidean terms though with newer and more specific meanings. For instance postulates are no longer seen as speculative or hypothetical, they now represent mathematical rules. So for today’s mathematician postulates are the rules that one must follow to know how to justify and understand solutions to certain mathematical problems, not only as possibilities such as the possibility of drawing an imperfect line from a point to a point to prove it is really possible to do geometry even though a geometer’s objects were believed to represent mere abstract images of perfect shapes visible only from projections outside the cave of human perception. And Euclid’s first definition, that of a point, has come to mean something like a first element in its translation from the Greek as that which has no part, seems to confirm our popular conception of an atom.
Euclid’s First Definition
For some time now the generally accepted interpretation of Euclid’s first definition, the point, has disturbed me because as it’s been translated from the Greek, the word “outhen” meaning nothing/anything seems to have been overlooked with the result that the accepted definition of point is “that which has no part” not that which is not a part of anything/nothing (outhen); for point is the first abstraction needed to produce Euclid’s postulate # one: the possibility of drawing a straight line from point to point not a Pythagorean indivisible and fundamental atom-point out of which everything that comes into existence is the result of, but an arbitrary jot or mark.
Here is the Greek definition transliterated and interpreted:Zemeion estiv, ou meros outhen: point is, not part nothing; or grammatically: A point is not a part of anything.Further: a point is that which has no part; it is being defined as early Greek atomists might define an atom. According to Simplicius: Leucippus, Democritus, and Epicurus, say atoms are indivisible first principles that have no void in them. (Logically without void atoms are indivisible and so without parts.)
The denial of the reality of void or vacuum has been associated with Aristotle, the ancient Greek philosopher from Macedonia and teacher of Alexander the Great. His view that nature abhors a vacuum is assailed by modern science. Einstein’s memorable hypotheses seem to need empty space to stand as facts “the constancy of the velocity of light c” is true “in vacuo”.