History of the mathematical notion of a single point?

[Kyriakos]

I am interested in this. I know that by Eukleid's Elements time (around 300 BC) the single point was axiomatically set as having no extention at all (ie 0 in a line, the simplest 1-dimensional form), and therefore axiomatically you can not get a line (nomatter how small) by adding up even an infinite number of single points.

This was not a random axiom, cause the lack of extention of a single point was an issue at least already 2 centuries prior to Eukleid, in the work of Democritos. For example Democritos posed the question as to what happens if one would divide the area of a cone, anywhere parallel to its base, by a single plane. Would that create two immediately equal or unequal (infinitesimally varrying) planes in the position of the cut?

If they are equal, then the cone would tend to become a cylinder, but if they are unequal this seems to mean that the cone has a step-progression in its infinitesimal differences in height.

Worth to also note that Democritos was mostly developing his own theory (of the last, smallest, particle-sized, 'atom' being the non-divided primary element of anything with mass, and the atoms existing in a huge (next to them) 'void'), as a reaction to the Eleatic philosophy, where the main argument was that there is no end to divisions to anything, which likely at least supports the notion of a 'single point' as non-extentable nomatter how many of them are added to one another.

Besides, one of the famous Paradoxa by Zeno of Elea, was that if you drop a mass of wheat to the floor, you will hear a sound, but you won't hear anything if you just drop a single bit of that wheat (ie the sound you hear cannot be sensed readily as the added sum of an infinite number of sub-parts, as in the case of the line and the single points).

 

[Thorgalaeg]

The concept of point is an intriguing one indeed. And not only in a theoretical or purely mathematic, but also in the physical world in the sense of fundamental particles like electrons or quarks having not size apparently (yeah, they are waves too and can be thought of as clouds of probability and such but are also particles). It seems there is not a volume there with something inside, so, are fundamental particles points in space with no size? In other worlds is there nothing "real" there after all? Only a mathematical concept like the point? It is difficult to accept it in a material universe. It is somewhat mindblowing. Thankfully some theories like superstrings add some lenght to particles, even if it is an incredibly small lenght.

 

[Kyriakos]

^+1

I firmly am of the view that the notion of the single point can be adapted so as to help expand the specifics in the resulting notion of an infinite progression. I mean if you have a point which is at the same time 0 (no extention) but also the specific place where two lines intersect solely (solely at least in the usual parameters set), then any minor alteration to the notion of a single point can usher in a myriad changes in the examination of limits and infinite sets.

 

[nc-1701]

I think the key here is the notion that a point does not "fill space" i.e. has measure zero. Which is what seems odd to us since we intuitively expect every construction to fill some amount of space.

As for your second post I honestly have no idea what you are talking about

 

[Souron]

The non-dimensionality of a point has some interesting mathmatical consequences, as first studied by the Greeks, but the basic idea of a location is pretty primitive. Like if we point to somewhere, we're not thinking of a spherical ball at that point, we're thinking of only a position, and the notion of being near it. It's sort of like talking about infinity; since time immemorial people have been imagining eternity and infinite things, but modern infinite set theory was not discovered till the late nineteenth century.

 

[Kyriakos]

^Thanks, i will try to check some of those, they seem very interesting (i was also told that in Riemannian synthesis of geometry with complex numbers, the zero and infinite tend to play a more ambiguous role).

Regarding the sphere, although i cannot claim it is having to be related, it is a nice bit of 'trivia' that the Eleatic philosophers (with the afforementioned infinite divisions of any space or other extentable quality) regarded 'the image of god' as a perfect sphere, with Parmenides in his work specifically claiming it is a sphere because it extends in the same manner towards all directions, not having any reason to be stopped anywhere or stop anywhere.
And while Xenophanes and Parmenides (and by obvious extension Zeno, Parmenides' student) viewed that cosmic sphere as infinitely divisible in its own existence, but having a border to something else - undefined as to what that would be; the sphere was just itself ultimately not infinite but part of something other), Melissos of Samos (the last Eleatic) argued that the god-sphere was without any border either.

While Melissos is regarded by Aristotle as being far less refined than Zeno, Democritos supposedly was in contact with him. It does seem that the early 5th century BC was very much centered in its philosophers to the notion of infinity and (on the other end) 'nothingness', and also 'void'.