# Keith’s Conundrums: The Cone and the Frustum

Last time, I asked if there’s anywhere else in the universe with a river of Coca-Cola. Alex B wrote to say that because of the likelihood of all combinations of basic stuffs elsewhere in the universe, there is somewhere else that has Coke.

I disagree, and there are some interesting metaphysical lessons here, independent of worries about infinity when it comes to combinatorics and probability that some might have. This is what is called “essentiality of origin.” If I make an exact duplicate, molecule for molecule, of a twenty dollar bill, it is merely a perfect forgery. Why? Because it is illegal in Canada for anything other than the official government source to produce currency. Similarly, if the place elsewhere in the cosmos is not under license from the company based out of Atlanta, it’s not Coke, but merely a “perfect bootleg copy”.

What does this teach us? One thing it seems to, something perhaps otherwise unexpected: it seems we have an example of a property that is both relational and essential. (There is some pre-1970s philosophy that asserts without argument that essential properties have to be intrinsic.) The second lesson is that we should be on the lookout for other things that also have essentiality of origin. S. Kripke in his famous Naming and Necessity thinks he has some, for example. And the discussion continuous over there; I’ll not bother with anything further on that line but encourage people to read this (in my view) flawed classic. A third lesson is that sometimes one doesn’t need to worry about infinity if the topic can be analyzed without it.

With that, I will move to this month’s conundrum.

The Cone and the Frustum

This one is old, attributed to Democritus, my favourite Presocratic philosopher.

Take a right cone, and slice it parallel to the circle that forms the base. Now one has a smaller cone (from the “top”) and a so-called frustum.

Now consider the two surfaces – the top of the frustum and the bottom of the small cone. Are these circles the same size or not?

If they are the same size, it seems that since we could have done the cut anywhere, the cone was in fact a cylinder. That’s not right, so let’s look at the other alternative: they have different sizes. But then one is larger than the other, presumably that of the frustum, and then there’s a circle, and another tiny bit of indent and the other circle in the original cone. But that’s not what we started with either – a sequence of circles with indents like that sounds like some sort of weird ziggurat. So that’s not it either. That seems to exhaust the possibilities, so what on earth is going on?

(Fields to consider: ?)

“… two surfaces – the top of the frustum and the bottom of the small cone..”

No, ONE and the same same surface mathematically–there’s not a top and a bottom.

So clearly, same size circle.

“..since we could have done the cut anywhere, the cone was in fact a cylinder..”

No idea what you’re talking about here??

What am I missing? I’m assuming (not necessary but best for visualizing) the big cone is sitting

on a table, so your cut(s )is (are) parallel to the table, i.e. horizontal.

Oh, by the way, Keith, those 3 papers by Oppenheimer and Zalta on the Ontological Argument

are undoubtedly the worst papers on formal logic ever written and published in 4 so-called scholarly

journals, the redundant one theological so “scholarly” is more than dubious –email me and

I’ll send you their extensive list of nonsense.

Assuming the cone’s slope is smooth, the two circles are not the same size. If they were at all points, the cone would be a cylinder, as you note.

The second possibility you raise is the correct way to think of it – the circle at the top of the frustrum is just a little bit larger (in the limiting case, by one molecule) than the circle at the bottom of the (new, smaller) cone. Just as no child seems to grow one day to the next but clearly does so one year to the next, our cone is constantly growing in circumference even if this growth is not evident from a casual glance. Increase the magnification of our observation sufficiently and the cone will, indeed, appear as if it was “some sort of weird ziggurat.” This way of thinking holds even if the increase in circumference is imperfectly smooth at higher levels of magnification.

Leslie