Eckard,
You said you have trust only in the definition of a point as having no extension. In my opinion, this may be one example of "Your Physics Suffers from Unwarranted Interpretations", the title of your essay.
In my 2013 Essay, I briefly discussed how your definition came about. Let me copy and paste for your convenience the relevant portion, with bold and italics to highlight the important things to note:
"Divisibility played a key role in apprehending the objects of geometry and arriving at a best definition of them. If all that has magnitude must be divisible, then only a zero magnitude cannot be divided and this will be the fundamental unit of geometry. One other argument attributed to Simplicius, goes as follows. If a body, having 3-dimensions is one dimension away from surfaces, then surfaces will have 2-dimensions. If a line is one dimension away from surfaces, then lines will have 1-dimension. By this logic then, the point would be one dimension away from a line and thus be of 0-dimension [1], p.157.
Right from the early era the definitions have been contentious. The Platonic view was that things in the physical world are imperfect replicas of things in a perfect realm and should be taken as such. On this basis, it meant nothing for a line to have no breadth, even though nobody had seen such a line in reality... The Aristotelian view on the other hand appeared more inclined to getting descriptions as practical as possible to the reality we can actually behold. Like the Pythagoreans who were of the view that points were extended objects and called them monads to differentiate them from the dimensionless object of the Platonic school, Proclus and Aristotle also felt that points must really exist and had the attribute of position, but they were unclear at what magnitude the point could then be defined as the limit of divisibility.
Although commonly portrayed as the arrowhead of the view that points were of zero dimension, Plato himself is quoted as somewhat disclaiming this. In Metaphysics, book I, part 9, paragraph 14, Aristotle tells us, "...Plato even used to object to this class of things as being a geometrical fiction". Instead, Plato is said to prefer that points be referred to as the 'beginning of lines' or as 'indivisible lines'. This defense was however denied him as Aristotle counters that if that were so, then the same argument and logic that makes lines exist must then equally prove that points also exist.
For a fuller account of these arguments, other ancient definitions, criticism by commentators and modern views, see [1], pp.155-157 and [2], pp.76-80. Both also quote Aristotle's Metaphysics and Physics frequently. In summary, the fact as to whether a point is an extended object or a zero dimensional idea has had to be postulated, i.e. has had to be an act of faith and not from evidence. In our thinking, to accommodate the contending views, Euclid restricts himself to a least contentious, middle-of-the-road definition, but not as explicit in its detail, i.e. all are agreed that the fundamental unit of geometry would have no part. A point having no magnitude and of zero dimension would have no part. Likewise, an infinitesimal magnitude not further divisible into parts of itself would also satisfy Euclid's definition 'that of which there is no part'. This ambiguity may however have implications for the foundation of our physical theories, space being all pervasive".
References
[1] Heath, T.L., The Thirteen Books of Euclid's Elements. (3 vols.). New York: Dover Publications (1956).
[2] Proclus, A Commentary on the First Book of Euclid's Elements, translated by Morrow, G.R., Princeton: Princeton University Press (1992).
So Eckard, since you are following the Platonic interpretation may I ask:
1. Is your point a geometric fiction or is it real?
2. If it is real, can something that is of zero dimension exist? If so, how and where? Can it occupy a position?
3. If it is not real, can something that is not real be a part of what is real? A line is real, at least in 1D (even though my line is 3D). Can an unreal point form a part of a real line which Euclid calls the extremity?
4. When you divide a line into segments, each of those segments must by Euclid's definition have points as their extremity. From this does your point constitute lines?
5. Finally, you talk of the continuum. In most cases, the linear continuum consists of an infinity of points. In your exchange with Edwin above, you said "Akinbo Ojo suggests splitting pebble-like non-zero dimensional numbers". No, I didn't say so. A non-zero dimensional point cannot be split. It satisfies Euclid's definition and cannot be split into parts. It is the smallest indivisible extension. That is why, Dedekind, Peirce and others have been worrying: "how then can we cut a line, if it consists of points, finite or infinite, since at every incidence for cutting a pint is located therein. My hypothesis suggests a solution. I never knew of Peirce till I read your essay. Thanks. I will read Giovanni Prisinzano views after this.
Best regards and thanks for the conversation
Akinbo
