“If a cut were made through a cone parallel to its base, how should we conceive of the two opposing surfaces which the cut has produced - as equal or as unequal? If they are unequal, that would imply that a cone is composed of many breaks and protrusions like steps. On the other hand if they are equal, that would imply that two adjacent intersecting planes are equal, which would mean that the cone, being made up of equal rather than unequal circles, must have the same appearance as a cylinder; which is utterly absurd.” Democritus, The Presocratics, Philip Wheelwright
My opinion says there is no such thing, even mathematically, as a smooth sided cone, dare I say. Do you agree?
There’s certainly no cone in our reality that is perfectly smooth. But when you start talking about things that ‘exist mathematically,’ you’re treading in very abstract waters – waters where the term ‘exist’ may not mean what you think it means, or indeed may not even be meaningful at all.
Mixing the conceptual with the physical is dangerous territory.
The conceptual cone, when divided, has conceptually the exact same (ie 2 dimensional) surfaces at the divide. The cut is a physical impossibility, being merely an infinitely thin divide.
If you could sum an infinite number of infinitely thin surfaces, and knowing nothing else, you could not predict the result. It might be conical. It might be expanding or shrinking. It might be anything. The mathematics doesn’t tell you enough because you didn’t provide enough information. And that sum might be merely an infinitesimal slice.
When dealing with infinite or infinitesimal things, one must be careful of their logic.
I agree with the last two posts. These kinds of problems are meant to be solved in their own sphere (not that imagining a cone inside a sphere is the solution).
It’s not like trying to solve the paradoxes of square circles, for example, because a square circle is, on the face of it, an incoherent concept. But the concept of a cone isn’t. We can both visualize a cone and conceptualize one, and in both those exercises, it seems that what we are dealing with should be possible and consistent mathematically. So when Democritus finds a paradox, we are pressed to figure out what went wrong (and it makes more sense to assume, as a starting point, that it is Democritus and the way he phrased the problem that went wrong somehow).
So if I were to take a stab at the problem, I would say that the circles are neither equal nor unequal–at least not in a straight forward fashion. I would say the circles are unequal in a not-so-straight-forward fashion–namely, in the sense that the difference between the two is no greater than an infinitesimal (where by “infinitesimal” I mean different by the smallest possible amount–0.000…0001 where the “…” represents an infinite series of zeroes). You could say infinitesimals are impossible anyway, but I don’t think so in the sphere of pure mathematics. I mean, geometric points are impossible if you think about it–what object actually has no extension in any dimension–but we don’t consider it an impossibility mathematically. An infinitesimal difference is just based on a geometrical point–a difference the size of a geometrical point.
So the two circles are unequal in this not-so-straight-forward sense–unequal infinitesimally. And this makes a difference to Democritus’s reasoning. In a sense, you could say that this difference implies a series of steps, but since the height of these steps is also infinitesimally small (that just follows from the thickness of a circle or any 2D form), stacking them on top of each other just results in a smooth surface for the sides of the cone. It’s similar to how you could describe a circle as a ring of infinitesimally short lines connected end-to-end. If they are infinitesimally short, and the angles that subtend between them infinitesimally less than straight, then they will constitute a smoothly curved arch.
In the thinking of the day the atomist, system is often regarded as essentially Democritus’
The atoms i.e. atomos or atomon, were ‘indivisible,’ were infinite in number and various in size and shape, and perfectly solid, with no internal gaps.
Consequently in Democritus’ thinking, on the cone issue, there seems to be no allowance for an a infinitely thin section of of a circle cut from a cone. In his thinking, the stacking up of different diameter circles in order to make a cone would be step-like because the single atom that made up each circle would have a vertical dimension of at least one atom thick.
That was a practical way of thinking rather than a abstract way, in that did not lend itself to an infinitely thin concept.
Perhaps with our greater technical facilities, if were were to look at a cone with an electron microscope, we would fine that Democritus was correct.
Of course we would find he was correct. We already know that physical objects are made of atoms. We don’t need to look at cones under a microscope. But this misses the point entirely. We’re talking about geometric objects, not physical ones. Geometric objects are not made of atoms.
Democritus thought that a cut always removed material. A mathematical ‘cut’ does not remove material. It is really a logical separation.
The same kind of thinking may be applied to cutting a line segment into two. Democritus would think that after the cut, the sum of the lengths of the two segments would not total the length of the original.
BTW, you can logically separate physical objects, not just geometric ones.
I guess there is a solution to the cone paradox. Suppose i intersect a geometrical Line to one of its points. Suppose then that I take away the one segment from the other. One might say that the ending point of the First segment is the same with the starting point of the other segment. Although we treat it as two distinct points, it is actually the same point in two different positions (you can find that view - at least in antiquity). The same goes for the cone. Democritus speak of two distinct circles. But what if we claim that they are not two distinct circles but rather the same circle in two different positions? If we do so the problem dissapears. This is because the problem exists only if we have two distinct circles. Maybe this sounds odd but in that way the problem dissapears.
I’m being sincere.