If you drag a zero-dimensional point to move, the trajectory of the point you depict logically is a line. But in reality we can't move zero-dimensional points. Because a point is infinitesimally small without either width or length, we have no way to move it.
In addition, although you can draw a straight line on paper, the line drawn by any method has a width, so it is not a one-dimensional line. As in the case of the point just mentioned, even a thin line that is difficult to distinguish with the naked eye must have a width when viewed closely. Thus, a line drawn on paper is a long line with a width, that is, a two-dimensional plane.
Plato's teacher, Socrates, drank all the poisoned wine in prison, and the philosophical discussions with the students present at his death were compiled by Plato into the book Phaedo. This is primarily a work about how Socrates viewed death or the immortality of the soul, which also records the content of one-dimensional space. At the end of the day, in the real world, it is impossible to draw a real line without width (one-dimensional space) no matter how thin the line is drawn.
Plato believed that even if a "real line" could not be drawn in the real world, one could imagine it
But Plato thought it was important that even if there were no real lines in the real world, we could imagine lines without width. He believes that even if the eye cannot see the real line, it does exist.
About 20 years after Plato's death, Euclid was born, and he gave a stricter definition of the one-dimensional line. Euclid was a mathematician who was active in Alexandria, egypt, one of the most famous cities in the world at the time, facing the Mediterranean Sea. He systematically summarized the geometry of the time into 13 volumes and compiled the book "Geometric Origins".
The first volume of the work shows the 23 "definitions" and 5 "axioms" and 5 "postulates" that underlie geometry.
Axioms and axioms here refer to preconceived notions made to guide other propositions. Axioms are the most basic, absolute assumptions, while postulates are adaptive rather than absolute assumptions. Although the difference between the two is not very obvious.
As early as the first page of the Primitive Geometry, Plato's statement that "lines are long lines without width" is described as the second point of the definition. Not only that, but euclid further added the following to the description of the one-dimensional line:
(1) One end of the line is a point
(2) A straight line is the existence of a uniform arrangement of points
(3) One side of the face is a line
(4) Draw a line from any point to any other point to get a straight line
(5) A line of any length can be constructed from a line of limited length
The ancient Greek mathematician Euclid. His book The Primitives of Geometry had a great influence on the mathematical community that followed and is now
The Original Geometry, which was supposed to be included in the Library of Alexandria, was later lost or burned down by fire, leaving nothing but the manuscript. The current Geometric Primitives are written on the basis of codices, so the expression of these texts is slightly different in terms of editions, but the meaning of the expression is not changed. There is no preamble in the Primitive Geometry, which is customary in modern writings, and it begins coldly with the definition of "point without component". Moreover, the respective definitions and postulates are only listed, without additional explanations or as to why they are definitions and postulates. This seems to be rather incomplete.
In particular, when translating the book into Japanese, where (2) (definition 4 in the first volume) is to translate the original Greek text into English first, and then further translated into Japanese, there will be various expressions and explanations in the process, such as "a straight line is a horizontal line about the uniform distribution of the points on it" and "a straight line is a line about the uniform distribution of the points on it". The meaning of these translations themselves is also unclear
Euclid's Primitive Geometry
In simple terms, the points on a straight line exist uniformly and equally. But later mathematicians complained that the definition was not clear enough.
By the way, (3) is the definition 6, which means that one-dimensional lines can be represented according to two-dimensional faces.