Democritus and his question

Kyriakos

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Democritus lived in the 5th century BC. He was one of the most celebrated philosophers of the era, and is mostly famous for his theory of 'atoms', namely that after some point in the divisions of something you can only get to having a vast void where infinitesimally smaller parts move around, which he termed as atoms. Atoms (Atoma) literally mean 'indivisible' in Greek, and the reason he developed this theory was to counteract the immediately previous one, by the Eleatic philosophers (such as Zeno and Parmenides), who argued that the divisions never end (and focused on what this would mean).

Democritus, like most other presocratic philosophers, also was interested in concurrent mathematics. It was a period shortly after the first Pythagorians. Democritus posed a question about geometry, which had obvious other consequences (including for his own theory of atoms). It was the following one:

-If one divides a perfect cone, using a plane, then at the two immediate points where the cone was divided you would have on each side two points in the outer surface, but would the point nearer to the base of the cone be countably larger than the point just one further away? (ie in the division location).
He noted that if the answer was 'Yes', that would mean the cone has a step form in its infinitesimal parts. If the answer would be 'No', then in those parts it would be indistinct as a progression in space (something between a cube and a cone, for example).

The question had the obvious link to atoms, since those infinitesimal parts of the cone were to be linked to atoms. In essense the question was whether you could get a visible progression by adding the smallest particle, just once. Mentally, we can always do that. But could it be the same in nature?
In the end the question is about the notion of a 'single point', in Geometry, and whether that notion is really in line ( ;) ) with the external, non-mental world of phenomena.
 
I read somewhere, maybe Velikovsky's World's in Collision or Santillana's Hamlet's Mill, that Democritus traveled to Egypt and Mesopotamia and upon his return said there are more planets than can be seen.
 
^Others of that early era travelled to Egypt and/or Mesopotamia, including Thales and (possibly) Pythagoras :)
It is said that Thales went to Mesopotamia and later provided the first math proof of what is now called 'the theorem of Thales', which supposedly was something used in Mesopotamia before, but without there being any formal proof that it is true (the theorem is that if the hypotenuse of a triangle is the diameter of a circle, then the meeting point of the two other sides on the periphery of that same circle creates always a right-angle).

Animated_illustration_of_thales_theorem.gif


Worth noting that (it seems) the Mesopotamians first came up with the idea to divide the circle in 360 parts. Not sure why, but it is nice that 360+360+180 (ie all angles outside those formed by a given triangle, counted as external to that specific triangle) equals 900 :) (also of note are the Vedic circles with 9 parts). Haven't looked up why they set 360 as the number of degrees, but it is of interest...
 
-If one divides a perfect cone, using a plane, then at the two immediate points where the cone was divided
If you do that you have either an unevenly truncated cone with an elliptical base, or a smaller cone with a circular base. Where do the two points come from? I think this refers to the surface area of the two separated parts.

You might get a different number of atoms, you mights not, I don't think it says anything about the notion of atoms themselves.
 
For the interest of being specific (and i suppose, but i am not sure, Democritus used this in his example too) let's say the cone has a perfectly circular base, and the difference in length (or non-difference) is to be counted in the two immediate levels in the plane (ie the one in the plane entering the cone in parallel to the circular base, and the level above it, ie just that bit further from that circular base).
I think it is a good question to highlight the use of the idea of 'single point', which may not be really something other than a needed axiom for the math-devised world, which is not allowing for 'real' examination of objects in the actual external world.

Important edit: BTW, this does not have to mean the idea of 'a single point' is a wrong idea. It likely would mean it is tied to human logic/thinking, and thus shows another gap between human perception and the 'reality' of the external world.
 
I think I see what you are getting at.

A mathematical point is an ideal, it bears little relation to the world we experience, in which everything is approximated. The 'size' of atomic sub atomic particles is representative of the fields of force around them and binding them together, not by the presence of any actual 'solid' matter.
 
I think I see what you are getting at.

A mathematical point is an ideal, it bears little relation to the world we experience, in which everything is approximated. The 'size' of atomic sub atomic particles is representative of the fields of force around them and binding them together, not by the presence of any actual 'solid' matter.

Yes, i think the above is very close to my own view of this issue. I think that the notion of 'a single point' is likely very deeply rooted in the dynamics of human perception- at least consciousness- (and by extension so is the notion of a Limit). But the external world likely is not featuring this parameter. But a human maybe should primarily be defined as an intellect by being that which can identify both a limit and an infinity, both as ideals (ie cannot sense them, but they are main axiomatic elements of his theories). (anyway, i wrote a short story about this theme a while back :D ).
 
I'm not sure where the question is going. A point is merely a coordinate, it can't have size. It is like saying north is larger than south.
 
I'm not sure where the question is going. A point is merely a coordinate, it can't have size. It is like saying north is larger than south.

And it is set as a 0-dimensional element, axiomatically. Without this could we really have any kind of Geometry?

And yet it seems to be already crucially distant from 'reality' in the external world (and maybe this is inevitable, cause we are humans and not the same as the phenomena we observe in that world).

In the above sense... maybe Pi (or any such number created in a different set of numbering systems) is not an element in the external phenomena either, but something hugely related to human consciousness. Afterall, a circle and a line are entirely ideal notions we have, and very 'simple', fundamental notions in human thinking as well (by now). So the pi times the diameter would also be very important for a human thinking system (but not having to be important for external phenomena, at least in a set manner).
 
maybe Pi (or any such number created in a different set of numbering systems) is not an element in the external phenomena either, but something hugely related to human consciousness.

Afterall, a circle and a line are entirely ideal notions we have, and very 'simple', fundamental notions in human thinking as well (by now).

Doubtful.

pi or x(pi) is going to pop up elsewhere around the universe, wherever intelligent creatures are computing the areas or circumfences of circles (or the volumes of spheres). A circle is a very basic shape, so it's not really something that's going to remain unique to humanity. If there are mathematicians out there somewhere, you can bet that they know about the line and the circle, unless they're still in caves and are making grunting sounds while they fight over a steak.. but then you couldn't really call them mathematicians.
 
Doubtful.

pi or x(pi) is going to pop up elsewhere around the universe, wherever intelligent creatures are computing the areas or circumfences of circles (or the volumes of spheres). A circle is a very basic shape, so it's not really something that's going to remain unique to humanity. If there are mathematicians out there somewhere, you can bet that they know about the line and the circle, unless they're still in caves and are making grunting sounds while they fight over a steak.. but then you couldn't really call them mathematicians.

I am not sure this has to be the case. A circle is a simple shape (for us, by now, in historic times), but why assume that 'the shape of which every point in its periphery is in equal distance from a center' has to be a shape known by any other intelligent being? For starters if you have an intelligent alien which is immobile, would that alien have senses enabling it to grasp empirically any forms which are distant in differing degrees from the alien observer?
 
I am not sure this has to be the case. A circle is a simple shape (for us, by now, in historic times), but why assume that 'the shape of which every point in its periphery is in equal distance from a center' has to be a shape known by any other intelligent being? For starters if you have an intelligent alien which is immobile, would that alien have senses enabling it to grasp empirically any forms which are distant in differing degrees from the alien observer?

Not by any other intelligent being, just by any other intelligent society capable of producing bona fide mathematicians :)

Circles appear in nature, for one.. not perfect circles, but gravity favours circular/spherical objects for instance. Any society attempting to work out the orbits of their planets is going to need to know about ovals and circles. That's just one example.
 
Not by any other intelligent being, just by any other intelligent society capable of producing bona fide mathematicians :)

Circles appear in nature, for one.. not perfect circles, but gravity favours circular/spherical objects for instance. Any society attempting to work out the orbits of their planets is going to need to know about ovals and circles. That's just one example.

What if they live in a planet with massively bigger 'gravity'-like forces? They would not be likely to focus on the sky or higher planes that much. I mean (wingless) ants quite likely don't have a sense of 'flying' :D By extension maybe some set parameters could disable even an intelligent species from having an idea of such planes.

Alternatively, maybe they would not view external (material) phenomena in a similar manner at all, due to more theoretical/vague differences in their own somatic-makeup.
 
Either way IMO an intelligent society wouldn't really be able to get far ahead in fields of architecture, mathematics, physics, and so on, without knowledge of the circle (or line). If they miss it, that's not a very good start or sign. I'm not sure if it could even happen, but if it did, I think it'd be rare.

Either way, as soon as you know about circles and how to calculate their areas and circumferences, you get pi. It's not a human thing, it's just the relationship between one thing to another. It might exist for them in a form we wouldn't recognize (different digits, different way of encoding numbers, different base, maybe a multiplier in front of the pi, etc.), but they'd have to know about it at that point.
 
^Of course if they have the notion of a circle and a line, it follows they will have (or potentially have) pi or a tied variant of it. I am not sure they would have the former, though..
 
You have a straight ruler or any other type of line. You secure one end to a piece of paper and rotate it.

Viola, you get a circle. There's no way a civilization would "miss" thinking up the circle, unless they're still living in caves. IMO.
 
You have a straight ruler or any other type of line. You secure one end to a piece of paper and rotate it.

Viola, you get a circle. There's no way a civilization would "miss" thinking up the circle, unless they're still living in caves. IMO.

You are thinking of humanoid aliens. What if the intelligent alien civ does not have sensory input of 'extensions' or other missing sensory basis compared to our own which led/enabled our species to bring forward those notions of set shapes in regards to distances from set points?
 
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