Dear Bertrand D J-F Thébault, I would start my answer from offtopic. Neither points, nor lines or numbers are inhabitants of out world (Universe, Reality). There is no way to detect, measure or establish experimentally what concept of line is correct: as composed of points but discrete, or as being continuous but not being able to be constructed from points. Line, point and number are mathematical abstracts, and the proper definition depends on chosen mathematical construction.

You mentioned Euclid's Elements. However, modern axiomatic of Euclidean geometry is given by David Hilbert in 19th century. In this axiomatic, there is no "definition" of points and lines, both are undefined objects. This means, that in Hilbert's axiomatic of Euclidean geometry, line cannot be "constructed" from points, it is distinct type of objects. The defined relations of points and lines are:

- Point lies in a line, or

- Point does not lie in a line

Next mathematical construction is linear algebra. This construction is based on the notion of Real Numbers (by Cantor, Dedekind, Weierstrass etc). The set of real numbers R is continuous in a very specific mathematical understanding of this term. Also, a set of real numbers satisfies all axioms of 1-dimensional Euclidean geometry. The geometricity of real numbers is emphasized by phrase "real axes" giving it geometrical interpretation. This means that RxR can be a proper model of 2-dimensional Euclidean geometry and RxRxR can be a proper model of 3-dimensional Euclidean geometry. In this model of Euclidean geometry, a "point" is a pair (or triple) of real numbers, and a line can be "constructed" from "points" meaning that, whatever point of line you take, it has well-defined coordinates, or pair / triple of real numbers. Put it another way, whatever 2 intersecting lines you take, their common point has coordinates. From algebraic point of view this means that a system of linear equations with 2 independent variables and 2 equations (if considering bidimensional case) has one single solution in real numbers.

Again, Euclidean geometry constructed axiomatically is not something you can observe in real world. The Euclidean geometry constructed as 2- or 3-dimensional linear space is not something observable in real world. Depending on mathematical abstraction you choose, you get different views on a line as can / cannot be constructed of points.

P.S.

To give you more arguments of how different mathematical abstractions can heavily influence the perception of their results, here are some more constructions:

1. Non-Archimedean numbers, including real ones (https://en.wikipedia.org/wiki/Nonstandard_analysis). This construction with a hierarchies of infinities and infinitesimals does not satisfy classical axioms of 1-dimensional Euclidean geometry. But of course, some geometry can be constructed by analogy with linear algebra.

2. The notion of geometric duality:

• Point A ~ Line b
• A ∈ b ~ a ∋ B
• A ∉ b ~ a ∌ B
• a = BC ~ A = b ∩ c

and so on. In this correspondence, the notions of point and line are interchanged, so there is no way to establish "primary" notion and "derived" notion, which can be defined in terms of primary one. Or put it another way, any notion can be primary, the other can be defined in terms of it.

P.P.S.

If somebody considers that a mathematical construction by some axiomatic system (e.g. Hilbert's axiomatic system) is more fundamental that a mathematical construction modeled from something else (e.g. linear algebra as a model of Euclidean geometry), this is disproved by Löwenheim–Skolem theorem (https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem). It is true that real numbers satisfy axioms of 1-dimensional Euclidean geometry, or that 1-dimensional Euclidean geometry can be modeled by real numbers. The theorem states that there are other axiomatic systems, essentially different from Euclidean 1-dimensional, that can be modeled by real numbers, satisfying all their axioms. This means, the model is neither more fundamental than an axiomatic system, nor it is less fundamental mathematical object. So, if you want to detect in which way to consider Euclidean geometry: as an axiomatic system or as model of linear algebra, this theorem says: None, it is up to you.

Dear Alexandru

Thank you for this detailed and thoughtful exposition. I agree with your framing: mathematical constructs such as points, lines, and numbers are abstract entities, and their definitions depend on the formal system in use. However, your overview also — perhaps unintentionally — brings to light two deep foundational issues that I believe deserve clarification: circularity and duality.

1. On Circularity in Analytic Geometry

You describe how, in analytic geometry and linear algebra, the line is modeled using pairs or triples of real numbers — i.e., as a subset of R, R×R, or R×R×R. Yet R itself is introduced to represent the continuum — the very thing the line is supposed to instantiate.

In other words, the line is defined by R, but R is constructed in order to define the line.

This introduces a foundational circularity: the real line is assumed to be continuous because it consists of real numbers, and real numbers are justified because they yield a continuous line. Continuity is both the premise and the conclusion. This is not merely a philosophical nuance — it is a textbook case of conceptual circular reasoning. Continuity is not derived but presupposed.

From a constructive standpoint, this is even more problematic. If one accepts the principle — formalised in my work — that distinct dimensionless points cannot intersect (DDPDNI), then no line composed of such points can be continuous. From this angle, the classical real line is not only circular in its justification but also ontologically incoherent.

2. On Geometric Duality

You also mention the elegant projective-geometric notion of duality:

"Any notion can be primary; the other can be defined in terms of it."

This is certainly true at the formal level: point and line can be exchanged in incidence relations within projective systems. But this is a syntactic symmetry, not a constructive foundation.

Duality does not tell us how a line is generated from points — or vice versa — in a constructive or ontological sense. It treats both as primitives and focuses on their interchangeability within a system of axioms, but it avoids the question of generative priority.

In contrast, my framework is explicitly constructive and ontologically asymmetric:

• The point is the sole primitive.
• A bipoint is a directional, relational unit between two distinct points.
• A line is a discrete, quantized structure made of parallel, equipollent bipoints anchored at a common origin.

This construction prioritises generation over symmetry: points generate bipoints, bipoints generate dot-lines, and continuity never enters the system. Thus, while duality is valuable within symbolic systems, it does not resolve the foundational question of what comes first.

Conclusion

Modern frameworks such as Hilbert’s axioms or projective geometry offer symbolic elegance and logical flexibility. But they leave unresolved — and often obscure — the deeper issue:

How can a line be constructed without already assuming what it means to be a line?

To the best of my knowledge, no existing framework offers a constructive definition of the line that does not assume the existence of R. That is precisely the conceptual and foundational gap my work seeks to address.

With respect and appreciation for your careful contribution, Bertrand D. Thébault

Thank you Bertrand D J-F Thébault for your appreciation. I will certainly study your work more carefully.

I would like to clarify 2 points:

1. "In other words, the line is defined by R, but R is constructed in order to define the line."

This is true and is essentially the description of process of creation of some mathematical construction by human mind. But formally, there is no circularity in construction of R. It can exist without geometry / geometrical interpretation at all. The notion of "continuity" clearly sounds as a geometric property by usual human perception. But in analysis the very same notion / word is defined by Dedekind's sections. That s why I said "The set of real numbers R is continuous in a very specific mathematical understanding of this term".

Löwenheim–Skolem theorem is exactly about it. When a mathematician (say, Dedekind) develops some mathematical abstraction (e.g. R) he / she always has in mind some model of it (e.g. Euclidean line in his / her perception). However, after constructing the full set of axioms, which is never complete due to 1st Gödel's incompleteness theorem (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems), he / she creates a formal / axiomatic system corresponding not only to what it was intended for, but describing much more essentially different models, which may radically differ from initial one. For example, the axioms of R say nothing about geometrical continuity as you pointed out. We are free to choose the same (incomplete) set of axioms to model continuous Euclidean line or point-wise Euclidean line. Their properties are quite different, but all their axioms are the same.

2. "Any notion can be primary; the other can be defined in terms of it."

This is certainly true at the formal level: point and line can be exchanged in incidence relations within projective systems. But this is a syntactic symmetry, not a constructive foundation.

The projective duality can be actually used to construct something new. One example is my presentation "Space duality as instrument for construction of new geometries" (Poster Space duality as instrument for construction of new geometries

Dear Alexandru Popa,

Thank you once again for your thoughtful and generous reply.

You're absolutely right to point out that the construction of R is not formally circular in the logical sense: Dedekind’s cuts and other constructions of the real numbers can indeed proceed without explicit geometric interpretation. I also appreciate your reference to the Löwenheim–Skolem theorem — a valuable reminder of how axiomatic systems can generate a plurality of models, including unintended or non-geometric ones.

That said, my concern is not with explicit formal circularity, but with what we might call implicit conceptual circularity, especially as it pertains to post-Euclidean definitions of the line.

In practice, many constructions of the line — particularly those rooted in analytic geometry, Hilbertian axiomatics, or projective frameworks — rely on real numbers (or structures homeomorphic to R) to instantiate lines. While the algebraic or logical definition of R can be rigorously formulated, it is the implicit assumption of continuity that allows such a definition to function as a geometric foundation. The intuitive motivation, and often the semantic interpretation, is that R “fills” the line segment — thereby implicitly embedding a notion of geometric continuity into the foundational structure.

This is precisely where my theorem — DDPDNI (Distinct Dimensionless Points Do Not Intersect) — becomes relevant.

If one adopts the principle that dimensionless points cannot compose or intersect to form a continuum, then any definition of the line that relies on the real numbers (or similar constructions) must be re-evaluated. Even when continuity is not explicitly assumed in the axioms, it is often implicitly reintroduced through the semantic interpretation of R, or through limit-based constructs such as Dedekind cuts, Cauchy sequences, or topological closures.

To address this systematically, I intend to revise and extensively expand my current paper by analyzing all major post-Euclidean definitions of the line — including Hilbert’s axiomatic system, Dedekind’s cuts, Cauchy sequences, ZFC-based set-theoretic constructions, Tarski’s first-order geometry, analytic geometry, projective constructions, differential geometry, synthetic differential geometry, nonstandard analysis, and locale theory — through the diagnostic lens of the DDPDNI principle.

The goal is to distinguish which frameworks:

• Collapse, due to hidden assumptions of point-wise continuity;
• Can be reformulated symbolically or discretely;
• Or are already compatible, if reinterpreted within a point-free or relational ontology.

I am also interested in your work on space duality as an instrument for generating new geometries, and would be glad to see how it is impacted by the DDPDNI theorem within a constructive approach.

With sincere appreciation for your insights and engagement, Bertrand D. Thébault

Alexandru Popa

I have a conservative alternative to your notion of geometric duality:

• Point A ~ Line b
• A ∈ b ~ a ∋ B
• A ∉ b ~ a ∌ B
• a = BC ~ A = b ∩ c

The idea is to use constructive deprojection, I have used in the follwoing article for the stretched point euclidian line in this article:

Article Distinguishability conservation imposes rational cartesian coordinates

Dear Bertrand D J-F Thébault, in my research I only used duality to justify introduction of new properties and relations. These new geometric constructions, of course, I introduced in a systematic rigorously way using the model of homogeneous spaces described by me.

You can find this in my PhD thesis "Analytic Geometry of Homogeneous Spaces" (Preprint Analytic Geometry of Homogeneous Spaces

One example of new relation is unconnectable points (section 2.1.4, no line contains both), which is dual to parallel lines (have no common point). There are several possibilities for points unconnectability.

Another property, which may be interesting for your research of structure of line is points separability on a line (section 3.2.1). Again, there are several cases in different geometries: points can be non-separable, weak separable or strong separable.

Dear Dr Alexandru Popa

Thanks for pointing out the relevant sections in your PhD thesis. Excellent you had already identified these difficulties.

Each time I came to geometry was to solve problems elsewhere:

1) irrational versus rational 2)

Quantum physics turned into Classic Physics with basic 3D geometry

Deleted research item The research item mentioned here has been deleted

But I don't really do geometry like you (no PhD just truth seeking)

So keep in touch for next move (3)

Best regards

Bertrand

Dir, Dont waste your valuable time by thinking this. Its very easy to grasp and simple to understand if you try to draw a line a circle with pencil or pen. In realistic view line is continuum of points. If you are not drawing it and just thinking its equation only, then its abstract of that. But you shall really draw a line by joining infintely many points. Just frame that as a definition. A line is set of all points which satisfy the condition y= mx+c.

Dont waste your valuable time for these types of silly matters

Hi Nishad T M

Philosophical Argument: On the Nature of Lines—Material Intuition versus Mathematical Continuity

Premise 1: Physical intuition suggests that when we draw a line with a pencil, we create a continuous entity.

Premise 2: Upon closer examination, any physical pencil line is composed of atoms—discrete, separable units—so the “line” is fundamentally a collection of distinct points, not a true continuum.

Premise 3: Modern mathematics (from the late 19th century onward) models a straight line in the plane as the set of all points (x,y)(x, y)(x,y) satisfying a linear equation such as y=mx+cy = mx + cy=mx+c, where the coordinates are taken from the set of real numbers. This relies on the real number system, itself constructed as a dense, ordered continuum in the sense of Dedekind, Cantor, and later set theorists.

Premise 4 (my DDPDNI critique): However, according to my DDPDNI theorem (“Distinct Dimensionless Points Do Not Intersect”), such a definition has no genuine geometrical background, because a collection of dimensionless, non-intersecting points cannot generate a continuous line in geometry.

Premise 5: Physical analogies (like the pencil line) are helpful for intuition, but they do not constitute rigorous mathematical definitions.

Conclusion: Thus, both the physical intuition behind a pencil line and the standard set-theoretic conception of the real line as a point-set reveal foundational difficulties in offering a fully satisfactory geometric account of continuity. This is the motivation behind my paper—perhaps not so “silly” after all.

Out of curiosity, how would you define a pencil drawing in mathematics?

Best regards,

Bertrand D J-F Thébault