Wednesday, November 17, 2004

If Squares Were Circles

In the Semantics class I took last semester, we were taught to analyze counterfactuals in terms of possible worlds, so that the counterfactual "If X were true then Y would be" (for false 'X') is true iff the following truth condition is met:

(w) ('X' is true in w and w is otherwise similar to the actual world -> 'Y' is true in w)

Simply put, the counterfactual is true iff 'Y' is true in all those close possible worlds where 'X' is true.

Now, one problem with this analysis is that it implies that any counterfactual with a necessarily false antecedent will be vacuously true. (Consider: there won't be any falsifying cases where 'X' is true but 'Y' isn't, because 'X' is never true.) But it doesn't quite seem right to say it's true that "if there were a largest prime number then Kerry would be president".

Incidentally, a fun way to deal with this could be to extend our ontology to impossible worlds, and proceeding in an analogous way to that described in the linked-to post (see also here). But that's not really what this post is about.

Perhaps counterfactuals involving necessary falsehoods aren't usually meant as counterfactuals at all. (The above example sounds very unnatural, after all. I can only imagine someone saying it for rhetorical effect, with the implicature "stop wishing for the impossible, it's time to move on!") The notion of putting apparent 'counterfactuals' to factual use is discussed at Siris and Mixing Memory. Anyway, I just want to discuss a particular example I find interesting, which was originally brought up in our Semantics tutorials. The question is, which (if either) of the following two sentences is true?

(1) If squares were circles then cubes would be spheres.
(2) If squares were circles then cubes would be cylinders.

Despite appearances, these statements probably shouldn't be interpreted as counterfactuals. For one thing, according to the previously mentioned 'truth conditions', they'd both be vacuously true. We don't want that, as on any sensible reading they are clearly incompatible! (Though we could just reject the earlier analysis as a flawed or incomplete semantic theory of counterfactuals.)

They seem to me to just be a fancy way of expressing a sort of 'ratio' or comparison. We're effectively being asked: "square is to cube as circle is to ____?"

So what's the answer?

I think almost everyone in our class chose 'sphere'. From a mathematical point of view though, 'cylinder' is probably the better option. It all depends, of course, on how you get from square to cube. The simplest construction is to just take the 2-dimensional object and 'raise' it through the third dimension. Layer a whole bunch of squares atop each other (just pretend they have some small amount of 'thickness' to them) and you eventually get a cube. Do the same with circles and you get a cylinder.

So why do most of use choose 'sphere'? I'd guess it's because we weren't really thinking about how to (geometrically) construct a cube out of squares. Instead, we were comparing our geometric concepts, looking for salient similarities. Now, a cube is the most 'regular' 3-d analog of a square (all sides being of equal length), and something similar can be said of spheres in relation to circles: a sphere has constant radius, just like circles do, whereas the length from the centre of a cylinder to its surface will vary depending on which point of the surface you choose.

For another point in favour of 'sphere', consider working backwards, 'compressing' a 3-d object into its 2-d silhouette. Note that a cube is such that from every side it looks like a square (if looked at front-on, that is - I exclude 'diagonal' or rotated views). And of course the silhouette of a sphere is always a circle (no matter how you rotate it - even better!). But a cylinder? From the wrong side it looks more like a rectangle.

So, overall, I would say that "square is to cube as circle is to sphere". Nevertheless, I think (2) above is true, and (1) is false. So I have to abandon my 'ratio' analysis of the counterfactuals (huh, I didn't see that coming until I actually wrote this post!).

I think the difference is that the 'ratio' question invites us to compare concepts, and the concept of a 'sphere' is the circular analog of our 'cube' concept in a way that 'cylinder' is not. However, the original counterfactual formulations seem more objective, somehow. They're less concerned with our own subjective understanding of geometric concepts, and are instead more concerned with geometric reality - and that means constructions. (Despite our mistaken shortcut of comparing our concepts instead.) If you applied the general "build a cube" method using a circle for your base template instead of a square, the end result would be a cylinder, rather than a sphere. Hence, if squares were circles, then cubes would be cylinders.

Maybe it's a genuine counterfactual after all?


  1. at its root the question is like
    if humans were frogs what would horses be?
    Obviously since the first point is not true (humans are not frogs) the function does not exist. So that in itself alows the potential for a true contradiction in our attempt to solve the problem because if there was a contradiction it would not matter to the universe.

    However the transformation itself seems to be insufficiently defined. the things required are for example "assuming the most simple transformation" (which is usually implicit in such questions and why after 12 most people would say 3 4 5 6 not 4 8 16 32) and probably some more detail as to exactly what "simple" means.

    If we take it one level higher I would say Circle because I think point, line, plane, cube, moving cube in that order. AND I would think if a "what would become a sphere if a circle could not?"
    triangular pyramids squares and spheres and such seem to be the "key" objects that should be accounted for in the model - BUT a less scientific mind might easily come up with a more practical solution involving any other shape with a circle profile depending on their experiences in life. 

    Posted by geniusNZ

  2. Mathematically, I would say that "spheres" is a better option. The generalized n-dimensional definition of a sphere is the set of all points in n-space that are equidistant from a given point, which makes circles the 2-dimensional spheres. The generalized definition of an n-dimensional cube is more complicated, but it makes squares the 2-dimensional cubes.  

    Posted by Blar

  3. Sphere is better, because the applying the transformation of circle to cylinder, to the square, can lead to many rectangular solids rectangular solids and a cube is only one such rectangular solid. A square going to a cube requires the height having the same size and the length and width. A sphere will also have all three dimensions the same size. A cylinder *may* have a height equal to the diameter, but there are an infinite number of cylinders having a height not equal to the diameter.

    So in conclusion I'm saying that while cylinder can be correct due to the method of transformation, sphere is a better answer as it has a greater correlation in the attributes of the two dimensional and three dimensional objects. 

    Posted by Brian (Shadowfoot)

  4. Brian - you're right that rather than the generic "cylinder" it should be specified that the shape we have in mind has height equal to the circle's diameter. Once this is clarified, would you still think sphere is the better answer?

    Blar - the "generalized n-dimensional definitions" you bring up are interesting. It reminds me of the point I made about the 'regularity' of the shapes, but put more rigorously (and thus more convincingly!). So thanks for that.

    But I still wonder if that's more a point about the concepts rather than actual constructions in geometric reality. If we took a geometric space, and turned all the squares there into circles, what would the old cubes look like?

    Note that if we cut through a cube with a horizontal plane, the intersection is always a square of constant size. If we turn all those squares into circles, then the resulting shape is a cylinder.

    So we would get cylinders IF we only changed squares in the horizontal plane into circles. This is basically what I was doing with my talk of 'raising' a (horizontal) 2-d object into the third (vertical) dimension. That yields cylinders as a nice, clean answer.

    If you try to change all the squares, in all dimensions, into circles, that is an impossible task. (Changing one distorts the shapes in the other dimensions so that they are no longer squares.) Just try taking a cube, and change every face into a circle - it would be like trying to merge three orthogonal cylinders together, which I don't think can be done whilst retaining their 'circularity'.

    I agree that insofar as our mathematical concepts are concerned, 'sphere' is the 3-d analog of 'circle'. But if we take the counterfactual seriously in its request that we consider what would happen "if squares were [turned into] circles", I can show how cubes would turn into cylinders (according to one interpretation). I cannot see how they would turn into spheres.

    But if anyone has a construction to prove me wrong, I'd love to hear it!

    Oh, hand on... Working backwards, if we take a sphere, slice it into circles (of varying diameter), and turn each circle into a square, we might end up with something like a 3-d 'diamond', or a cube resting on its corner? I'm having trouble visualising this though. Perhaps someone else can offer a second opinion on what this resultant shape would be? (If it is indeed a cube, then 'sphere' would appear to be just as good an answer to the original question after all.) 

    Posted by Richard

  5. Oops, that should be "hang on", of course!

    Back to the semantic side of all this... do you guys think that (1) and (2) should be understood as genuine counterfactuals? Or are they just a sort of analogy or metaphor (or something), phrased like counterfactuals for rhetorical effect? 

    Posted by Richard

  6. I think it would be a lantern with two circles as the skeleton and the "skin" pulled very tight around it . The top and bottom of the lantern are the smallest squares. 

    Posted by genius

  7. and you have a good point regarding that being the best (but not only answer)... but few people will think of it because it is not one of the standard shapes. 

    Posted by Anonymous

  8. You could think about transforming a square into a circle as puffing out the sides of the square until they're all the same distance from the center as the corners are. Each point only moves outward radially (from the center), and there's a fairly simple mathematical description for where a point ends up based on where it starts.

    In 3-D, puffing out a cube with a similar transformation (moving each point outward radially and using a similar equation with one more variable to account for the additional dimension) would turn it into a sphere where each point ended up at the same distance from the center as the corners originally were.

    (Your "working backwards" construction would not give a cube, though. A cube has 8 corners where three faces meet, but you would only get 2 corners where more than 2 faces meet - one at the north pole and one at the south pole.)

    The main problem with saying that the cube would be a cylinder is that you lose symmetry. Circles are more symmetric than squares, so it's weird to say "If squares were circles then cubes would not be symmetric with respect to 90-degree rotations," but that's what would happen if cubes were cylinders. If you choose a cylinder, then then the square that was on the side of the cube would become a rectangle that was puffed out into another dimension, like a speed bump, while the squares on the top and bottom become circles. If you choose the sphere, though, then the square that was on the side of cube would become a somewhat deformed circle that was puffed out into another dimension like a mound, as would all the other faces of the cube. 

    Posted by Blar

  9. Yeah, I was a bit concerned about the 'rectangular' aspect of cylinders, as you may have picked up from my earlier comments (about how it only works if we change only the squares in the horizontal plane into circles). I just didn't see any better alternative. But your "puffing out" idea would seem to work quite nicely. 

    Posted by Richard

  10. I will accept :

    Squares are to Circles as Cubes are to Spheres.

    Squares are not to circles as cubes are to cylinders.

    But ignoring the linguistic murder of claiming that squares actually are circles in some alternate world, it doesn't follow that cubes are spheres. What are they claiming? That an alternate physics allows squares and circles to be indiscernable? Without a description of said geometry, you might not be able to know.

    A square might be considered "a 2 dimensional object all of whose edges are the same length" while a circle is "a 2 dimensional object where every point on its edge is equidistant from its center"

    Projecting those definitions into 3 dimensions gives squares and spheres, and no cylinders. 

    Posted by Tennessee Leeuwenburg

  11. I don't know about the spheres, but I think that cylinders are the wrong way to go. Here's why:
    First, we need to figure out how to interpret the antecedent of your counterfactual. When we talk about circles, spheres, cubes and such, are we talking about regions of space or their occupants? Let's interpret it to be about occupants, just to make the visualising easier. And when you say that squares are circles, do you mean all squares? Surely not - otherwise, the issue of cubes versus spheres wouldn't come up - we'd be left with dust (i.e., a 3D version of Cantor dust - I think the process of changing each square we find into a circle, from largest squares to smallest, would give us a process analogous to the process used to create cantor dust). So let's limit it, to catch what I take to be the spirit of what you're after: if all squares with a certain sidelength were actually circles with diameters equal to that sidelength, what would a cube with that sidelength be?
    Since I've made this about occupants rather than space, I have no trouble interpreting your original statements as counterfactuals (although I wouldn't mind it anyway - I lack the intuition that there's something wrong with vacuously true counterfactuals with necessarily false antecedents). The antecedent of your counterfactual is possibly true, if we interpret it as describing a world which is like ours except that every instance of a square (with a certain sidelength) in our world corresponds to an instance of a circle (with a diameter equal to the square's sidelength) in that world.
    To show why I think cylinders don't work: suppose I want to do a demonstration, so I construct a cutting machine that, when given an object, would cause all squares (say, with sidelength 2) in that object to become unit circles. Suppose that for my first demonstration, I gave it a cube with sides of length 2. As expected, it turns the cube on one of its faces (let's say, with the x and z axis flat), and cuts down on the y axis, causing all of the squares on the x-z planes to become circles. We are left with a cylinder. However, the machine does not shut down as we expected it to: it detects another square! As you mentioned, the cylinder has a rectangular aspect. If we look down on it from the front (with the z axis coming directly toward us) or from the side (with the x axis coming directly toward us) we will see a square shape. And the square will have sidelengths of 2, since as you pointed out, the cylinder will have a height, width and depth identical to the diameter of the circle. This square is on the list of things to be cut, because it is a two dimensional entity with sidelengths of 2, even though it is embedded in our cylinder. So it must become a circle. To be economical, the machine uses the same unit-circle cutter to get rid of it. (Let's say it cuts across on the x axis.) This cutting doesn't effect the square that exists on the x-y plane, though, so the machine will have to repeat the process (cutting in on the z axis) to get rid of that last square.
    Of course, the resultant shape is not a sphere. It's a weird puffy thing . . . I have no idea if it has a name. I drew it out a couple times, then tried cutting an apple in the way described so I could see it (I'm not good at visualising shapes). I think that this shape shows us how cubes might look in the world you described. Though the method of cutting I described is just one way to meet the conditions of your antecedent (well, for an eternalist, cutting will be unhelpful in causing all the squares in a world to be circles, because if the squares exist at t then at every time they exist in virtue of it still being the case that they exist at t. But we can ignore that for the moment!). We could imagine a different, more precise machine that will only alter squares (with the right sidelengths). The first cut would be the same, because we're dealing with a bunch of squares of sidelength 2 stacked on top of one another. But the second cut would alter only the 2D bit of the cylinder that bisects it on the y-z plane. This square (and the last one, on the x-y plane) would be changed into a unit circle, but nothing around it would be. I'm sure there isn't a name for this object . . .
    This might show how the counterfactual and ratio ways of reading your statements differ: on the counterfactual account, it doesn't matter which process we use to alter the squares, as long as we can apply it and get a world in which the antecedent is true. On the ratio account, we have to come up with a process that causes squares to become cubes, figure out what the relevant similarities and dissimilarities between squares and circles are, and then apply an analogous process to the circles. I'm not even going to begin that . . .
    Thanks for the fun post! 

    Posted by Shieva


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