The Square Circle

The round square and square circle (we will take these to be synonymous) are classic examples of concepts that allegedly cannot be instantiated. Contemporary philosophers likely associate the notion with Meinong on the basis of Russell's (1905) criticism of him, or with Quine's (1948) invocation of the round square cupola on Berkeley College in his criticism of Russell's previous acceptance of Meinong. Its use as an example of a non-entity per definition goes back at least to Aristotle; in the Categories we find him remarking, “A square is not more a circle than an oblong is, for neither admits the definition of circle” (Aristotle 1984, 19). Hobbes goes even further: “If a man should talk to me of a round quadrangle … I should not say he were in error, but that his words were without meaning, that is to say, absurd” (Hobbes 1839, 32–33). Other famous philosophers invoking the round square as an example of an impossibility include Bolzano (1972, 78), Kant (2004, 92–93), Berkeley (1735, 59), Leibniz (1966, 43), and Spinoza (1996, 7).

Let p be the sentence “There is no square circle.” Even to many contemporary philosophers, p seems to be not only true but necessarily, a priori, or even analytically true, just as many philosophers find it necessarily, a priori, or analytically true that water is H₂O, that Smith does not know that Jones owns a Ford or is in Barcelona, that nothing can be in two places at once, or that “cordate” does not mean the same as “renate.”

But start up any image manipulation program such as Paintbrush, GIMP, or Photoshop and draw a circle with a one-pixel radius, making sure that any antialiasing is off. The result will be something like figure 1, where the centre of each pixel is marked with an ×.

On one interpretation, this is the closest possible approximation to a circle of radius one on a square grid. But we can also see it in another way: a computer screen has its own type of geometry, in which the distance between points is like the Euclidean one, but rounded to the nearest integer.1 When the radius is a single pixel, the concepts of square and circle coincide.

This admittedly involves a lot of hand waving, such as assuming certain tacit interpretations of circularity and squareness. The obvious counterargument to the example is that these interpretations are wrong; squareness and circularity, correctly construed, are incompatible.

Now we can easily conceive of an interpretation of the constituent concepts that makes the impossibility a logical truth. But this only shows that the nonexistence of round squares does not contradict the definitions of roundness and squareness. Compare the parallel axiom: while the notions straight line and parallel do not contradict there being parallel straight lines that intersect, the existence of such intersections does not follow from them either.

So which concepts do circle and square entail? In order to minimise question begging, I will largely follow Euclid's original descriptions of his terms, and only invoke modern mathematical concepts when needed for precision or clarity. Like Euclid, we assume that we are working in what is nowadays called a topological 2-manifold, which means a space each part of which it is homeomorphic to some part of ℝ², that is, for each open subset of the space, there is a continuous and continuously invertible function from that subset to the real plane.2 Consider a subset X of that space, where each point a ∈ X corresponds uniquely to an ordered pair of real numbers (x_a, y_a). The usual definition of a circle is as follows.

Definition 1. A circle is the set of all points that have the same distance r from a point x, which is called its centre.

In mathematical terms, the distance between two points in X is determined by a function d: X × X → [0,∞), called a metric. A function from pairs of points of X to non-negative real numbers is a metric iff it satisfies the following three axioms for all points a, b, c of X:

1. d(a, b) = 0 iff a = b.
2. d(a, b) = d(b, a).
3. d(a, b) + d(b, c) ≥ d(a, c).

The first axiom says that the only thing at a distance 0 from something is the thing itself, and the second that the distance from a to b is the same as that from b to a. The last is known as the triangle inequality. It guarantees that taking a “detour” through a third point y will never give a shorter path than going directly from x to z. The most well-known metric is the Euclidean one, defined as

dEa, b=df. xb−xa2+yb−ya2

It is not the only one, as is also well known. For example, it is not the one that describes distances in an arbitrary plane in the actual physical space we live in. By definition 1, each metric on X gives rise to a specification of which sets of points are its circles. This means that circles exist in many spaces other than Euclidean ones, but that their properties in such spaces may differ from the ones we usually suppose.

What is a square, then? Definitions of the word “square” are unfortunately somewhat more complicated than those of “circle,” but a common, fairly straightforward one is the following:

Definition 2. A square is a quadrilateral with equal sides which meet at right angles.

This invokes the notions of quadrilateral and right angle, which I will define next. Beginning with “quadrilateral,” we have:

Definition 3. A quadrilateral is a closed figure consisting of four straight-line segments.

How do we define straight-line segment? Unfortunately, Euclid's definitions of a line as “breadthless length” and a straight line as “a line which lies evenly with the points on itself” (Euclid 1908, book 1, defs. 2 and 4) do not seem very enlightening. Archimedes is more useful here, although he frames his condition as an assumption rather than a definition: “Of all lines which have the same extremities the straight line is the least” (Archimedes 1897, 3). This allows us to characterise a straight line by its property of being the shortest distance between two points. Since distances are defined by a metric, this definition makes the concept of straight-line segment applicable to any metric space.3

To work out the formal details, it is more convenient to define lines in terms of another concept, namely, that of a path in X. By this we simply mean a continuous function p: [0, 1] → X, where [0, 1] is the closed unit interval. Let the parametrisation set P be the set of all n-tuples t = (t₁, … , t_n) of real numbers such that t₁ = 0, t_k < t_k+1 and t_n = 1, for all natural numbers n > 1, and let |t| be the number of elements of the tuple t. We define the length Λ(p) of a path p in X as

Λp=df.supt∈P⁡∑k=1t−1d(ptk, ptk+1)

This gives the length of a path p as the supremum (maximum) of the length of any piecewise linear approximation of p, as illustrated in figure 2. Since the length of a straight path between a and b is just d(a, b), this gives us a way to find the length of any path, or at least any sufficiently mathematically well-behaved one.

A curve segment or arc is the image of a path—what Euclid referred to as a line. For any arc A with endpoints (boundary points) a, b in a space X we say that a path p: [0, 1] → X is a parametrisation of A if p[0, 1] = A, p(0) = a, and p(1) = b. We let the length Λ(A) of an arc A be the minimal length of any path parametrising A—that is the shortest way to draw all of A.

The modern definition of a straight line (or arc) requires only that it is an arc that is of locally minimal length, that is, such that no small change in the arc's shape gives a shorter arc. This is captured in the modern notion of a geodesic. For the purposes of this note, however, we will not need to consider non-global minima, so it is enough for us to require that any arc A between a and b of minimal length is a straight line, as Archimedes did. Together with the triangle inequality this entails that Λ(A) = d(a, b) is a sufficient condition for A to be a straight line.

This completes our definition of “straight line.” The other concept we need for the definition of “quadrilateral” is that of closed figure, which, classically, means one which can be drawn without lifting the pen and which ends where it begins. In our terms, this is simply an arc whose endpoints coincide. Putting these characterisations into definition 3, we arrive at:

Definition 3´. A quadrilateral is the image of a path q such that q(0) = q(1), and there are t₁, t₂, t₃ ∈ ℝ such that 0 < t₁ < t₂ < t₃ < 1, such that the images q[0, t₁], q[t₁, t₂], q[t₂, t₃] and q[t₃, 1] are all straight line segments.

To use this to define squareness we also need to know what an angle is. This was a hotly debated topic in antiquity and is still the subject of some confusion among students, as witnessed by our tendency to talk of angles as having a dimension (for example, degrees or radians) that they mathematically do not have. We will follow tradition and define angles in terms of circular arcs:

Definition 4. A circular arc with centre x and radius r is an arc that is also a subset of a circle with centre x and radius r.

The following is then a standard definition of angle:

Definition 5. The angle between two straight line segments A, B intersecting at x is the least length of a circular arc C with centre x and endpoints on A, B, divided by its radius r.4

The reason for using the least length here is to fix the reference to the smaller angle between two intersecting lines, and not the larger one. This definition assigns a unique angle between two intersecting straight arcs in most spaces. It is based on the idea that an angle is simply a part of the unit circle spanning two lines, as in figure 3. Since the definition of an angle takes it to be a length divided by a length, it follows that it is dimensionless.

A right angle, according to Euclid (1908, book 1, def. 10), is one that obtains when one line stands upon another so that the standing line makes the same angle to either side with the line it is standing on. Since the sum of all angles around an intersection of lines is a full circle, we can express this as saying that a right angle at x is one that is one-quarter the circumference of a circle with centre at x, divided by its radius.

We have now finished the preparatory definitions and are ready to show that there are square circles. We do this by imposing the following metric d_C on X:

dCa,b=max⁡(xb−xa,yb−ya)

This is known as the Chebyshev distance, or the chessboard distance, since it describes the number of moves it takes to get a king from one square to another on a chessboard. Now let r be any real number and define the points a = (-r, r), b = (r, r), c = (r, -r), and d = (-r, -r), and let abcda, shown in figure 4, be the union of the straight lines ab, bc, cd, and da (these are uniquely determined, since we have assumed X to be homeomorphic to a subspace of ℝ²).

Given the metric we have imposed, the following is now provable simply by replacing defined terms with their definitions:

Theorem. abcda is a circle with radius r and a square with side 2r.

This proves that there are square circles, at least in the abstract, mathematical sense in which there are round circles, or square quadrilaterals.5 Rather than being an impossibility, the existence of a square circle implies no contradiction at all. In fact, if we use the standard set-theoretic interpretations of topological spaces, their existence is even provable in systems such as ZFC.

The actual mathematics involved in showing that there are square circles should not be news to a geometer educated after the late nineteenth century, even if their construction is not, as far as I know, something that is usually taught explicitly. Nevertheless, the round square remains a staple of philosophers’ impossibilia. A search in JSTOR's philosophy section resulted in 276 items mentioning “round square” or “square circle” published since the year 2000. Their non-existence appears as item [R4] of a list of “general intuitions” in the Stanford Encyclopedia of Philosophy's article “Intuition” (Pust 2016). In contrast, I have yet to see any contemporary philosopher invoking “the intersection point of two parallel lines” as a similar alleged impossibility: it is well known that the parallel postulate holds in some spaces and fails in others.

One should of course not jump to the conclusion that all these philosophers are unaware of the mathematical fact that there are round squares. Sometimes examples get carried along even when we know that they are not strictly true, or at least no longer have strong reasons to believe them. The pseudo-Aristotelian definition of man as a rational animal is an example of this. In that case, however, the presupposition that rationality is essentially human arguably held back theorising about rationality as well as about nonhuman behaviour. What we should ask, therefore, is if the existence of square circles is a mere mathematical curiosity, or whether recognising that the square circle implies no contradiction points to a deeper lesson about philosophical prejudices that we may have carried with us. I think that there are in fact several such lessons, but at least one of them concerns the stability and determinacy of our so-called everyday informal concepts. In the remainder of this article, it is this lesson that I want to use the example of the square circle to illustrate.

For concreteness, let us imagine a typical philosopher using the round square concept as an example of impossibility being presented with our proof of its existence. As with the pixelated round square, perhaps she would say something like “That is not at all what is meant by ‘round’ and ‘square’!” But what is meant, then? Our definitions of “circle” and “square” are the same as the ones that Euclid used (Euclid 1908, book 1, defs. 15 and 22). How do we determine their everyday meaning more precisely than that?

An appeal to Platonism would be of no help. I have shown that there are perfectly viable and natural concepts of roundness and squareness that imply no contradiction, so for a Platonist, such round squares should be as true universals as round circles or square squares. To deny this would be to hold that only certain geometric concepts correspond to universals, and besides calling for the question of how to decide which ones these are, it would exclude most contemporary mathematical objects from Platonic existence. Therefore invoking Platonism does not help the philosopher determine which of the purported universals (the Chebyshev round square or the Euclidean round square) is meant.

More useful, perhaps, would be an appeal to Kantian intuition. According to this view, concepts such as circlularity and squareness are features of our innate abilities to experience the world as subject to spatial organisation. Since these abilities are taken to be the same for all humans, one might postulate that the philosopher's use of “square” and “circle” would express these concepts, and no others.

The problem with this is that squareness and circularity are, first and foremost, geometric concepts, and as such the province of mathematics. Furthermore, as was made plain during the nineteenth century, the Kantian view does not at all capture what geometry is like. A valid argument for a geometrical thesis proceeds exactly like a valid argument for any other mathematical thesis, and never invokes spatial intuition.6 Furthermore, much of contemporary geometry concerns things about which most people, arguably, have no spatial intuitions at all.

Perhaps the philosopher only meant to say “When I have imagined a circle, I have never also imagined it as a square; in fact, I do not even think I can imagine a square circle.” This is hard to argue against, and it may also be true for many philosophers, at least if by imagine they mean something like form a mental image of. Of course, neither the impossibility nor the non-existence of round squares follows from such a remark. But it gives an indication of a possible relationship between the Kantian answer and the last one we will consider, namely, the physicalist one.

Most of our abilities to imagine shapes have likely developed or been learnt in close connection with our abilities to orient ourselves in physical space, especially around medium-size physical objects. It is therefore proper that we should also consider the answer that what is meant by “square” and “circle” should be actual physical squares or circles, or some kind of idealisation of them. This is right in the sense that there are physical things that are shaped in a roughly circular manner and physical things that are shaped in a roughly square manner, and we can often determine which of these the case is. Furthermore, as far as we have found, these classes of things do not overlap. But this does not even show that round squares are a physical impossibility, and there is good reason to believe that they are not.

The mathematical construction I have just given does not transfer directly to physical space, since the Chebyshev metric, which we used, is not invariant under rotational symmetry, and thus contradicts special (and even Galilean) relativity. But there are many other ways to construct square circles. According to general relativity, all objects that are under the effect of gravity alone travel in geodesics (that is, in straight lines). As an example, a satellite orbiting the earth will travel in what is both an ellipse and a straight line, at least if we disregard other forces. We can build a circular square using this fact.

Pick a suitable black hole H and let r be the radius of H's photon sphere—the distance at which light is bent sufficiently so that it travels in orbit. Suspend four glass panes A, B, C, and D at distance r from H so that each of them is equally far from the one coming before it and the one coming after it in sequence. Fire a light ray from pane A to pane B; then, because it travels in the photon sphere, the light ray will go through all the panes and arrive back at the origin, and thus make up a closed figure (see figure 5).

Since the light travels at a constant distance r from the point H, it makes up a circle with H as centre. And since light always travels in a straight path, each of the segments AB, BC, CD, and DA are equal straight lines, which furthermore meet one another at equal angles.7 The whole light ray ABCDA therefore constitutes a perfectly square circle, so square circles are at the very least a physical possibility. In fact, if we do not demand the panes of glass to “mark off” the sides of the square, it is even quite likely that there are square circles in the universe, at least if we take there to be physical geometrical objects at all. All it takes for one to exist is for some photon to be orbiting some black hole somewhere.

Taken together, these replies can be used to answer the rather obvious criticism of our thesis that what we mean when we talk about squares and circles are Euclidean squares and circles. As I mentioned, we have used Euclid's definitions (with the possible exception of straight line, where we followed Archimedes), so these are certainly Euclidean. What we have not assumed is the whole axiomatic system of Euclidean geometry, that is, the postulates and common notions together with the implicit axioms needed to make Euclid's treatment rigorous. But requiring all of these to be fulfilled for us to be able to talk about circles and squares would cripple geometry: they would make it applicable neither to intuition nor to the physical world, nor to most of mathematics. Euclidean geometry is a useful subject because much of the physical space immediately around us is almost Euclidean. But limiting concepts such as circle and square to this space alone would mean that they would become useless in many other spaces in which we would like to apply them.

An important point in our replies to the physicalist and the Kantian is that it is not the whole physical world that affects what our concepts are like. We typically begin to learn about circles and squares by ostension, and the examples of them we are given are medium-sized objects in a part of space close enough to be Euclidean that it is very hard to spot the difference. “There is no circular square with a radius between 0.001 m and 100 m in my vicinity” is a statement for which I have much stronger grounds than “There are no circular squares” or “There can be no circular squares.” We have a reasonable ability to identify circles and squares of the everyday kind, but not more exotic ones.

The main fact I would like to use the existence of square circles to illustrate is that even presumably “clear” or “precise” concepts such as squareness and circularity are only partially defined, or indeterminate, in important ways. A concept can, for our purposes, be minimally interpreted as an ability to classify certain objects as falling or not falling under it.8 Being abilities, they are evolved or acquired in specific contexts. In these contexts it is likely that we get significant levels of interpersonal agreement. But outside an informal concept's “natural habitat” the concept becomes vague, indeterminate, and sometimes simply inapplicable. This is the case with pre-formal versions of circle and square, whose ordinary use in classifying the shapes of middle-sized objects in a roughly Euclidean plane does not determine their interpretation or even their extension in other kinds of spaces.

One sign of indeterminateness in a concept is that significant disagreement about the concept's extension exists even among those that we have good reason to believe do grasp it. But even in cases where there is a large amount of interpersonal agreement when it comes to what falls under a concept, we cannot draw the conclusion that assertions using it or about it are objectively significant, rather than merely being an expression of the asserter's personal or socially conditioned prejudices. There is still rather large agreement that no circle is a square, and until the nineteenth century this agreement was almost complete. I have shown, however, that what was agreed on in this case is false.

What we need to do is to separate the typical cases of a concept, which are the ones that it is generally learnt from or has been evolved to capture, and what I will refer to as stable cases—ones where there is merely agreement among the members of a certain (possibly very large) group. Among the stable cases, we should furthermore distinguish between a concept's centre, which consists of those stable cases that are sufficiently similar to one or several of the typical cases, or are obtained by somehow interpolating between them, and the concept's periphery, whose members are stable but different from the typical ones. When considering the extension of a concept, it is therefore useful to think in terms of a taxonomy like the one in figure 6.

In the middle, we have a number of typical cases of what the concept applies to. These are the ones that a formal definition arguably cannot ignore while still claiming to be a definition of that concept; in the present case, for example, instantiations of squares and circles in Euclidean spaces. Outside this, certain populations A and B of users may have a large measure of agreement as to whether a certain thing falls under the concept or not. But agreement among a large group of users that something is true should not be conflated with it actually being true. For one thing, other populations may feel otherwise, and for another, we do not accept inferences from majority belief to truth in other areas, so why should we do so when it comes to concepts?

To further illustrate the difference between a concept's centre and its periphery, let us briefly look at a concrete model. One way to model a concept that brings out the classification scheme outlined here is as a perceptron.9 A perceptron is a type of artificial neural network with a particularly simple structure. Whereas connections in general neural networks can go between any neurons, a perceptron is organised into layers, and a connection goes from neuron a to neuron b iff a is in layer n and b is in layer n + 1. The first layer represents some kind of input—a pattern of signals—and the last gives an output. If we see a concept as an ability of classification, the inputs would be certain features that things can have—possibly to varying degrees—and the output would consist of two neurons: one that is activated if the input pattern corresponds to something falling under the concept, and one that activates if it does not (see figure 7).

Perceptrons, and various variants of them, are arguably the most important tool currently used in pattern recognition, and they also have the philosophical advantage of being somewhat more similar to the workings of our own brains than, for example, lists of necessary and sufficient conditions. What makes a perceptron represent the concept that it does is what weights it has attached to its connections; these can roughly be seen as the degree to which the firing of one neuron will cause the firing of another. A perceptron becomes useful (in our case, comes to represent a concept) by being trained. This may consist in feeding it various inputs and then modifying its weights if it gives the wrong verdict. How to modify such a network is far from obvious, but there are a few standard ways available, such as backpropagation.

All of the features of informal concepts that I have drawn attention to here can be exhibited in the perceptron model. Suppose that we have a collection C of perceptrons with a common structure (that is, the same number of layers and the same number of neurons in each layer) but with various weights on the connections. Suppose, furthermore, that we train these networks on the same, or largely the same, examples. These examples will then correspond to the typical cases of a concept, on which there will be a very high amount of interpersonal agreement. The stable cases, on the other hand, are merely those where a large majority of C give the same verdict, whether these were among the ones they were trained on or not.

Now, perceptrons would be of little value if it weren't the case that stable answers are often the right ones: if they could not correctly classify things they had not been explicitly trained to handle, they would not be useful for pattern recognition, for example. But there is a caveat. Perceptrons are generally quite good at interpolation but bad at extrapolation. More specifically, they often get it right when asked about things that are either similar to the typical cases, or cases whose features lie somewhere between those of two or more typical cases. When it comes to highly untypical cases, or cases whose features are more extreme than the typical ones, all bets are off (see, e.g., Barnard and Wessels 1992). The perceptrons in C then no longer give replies that are indicative of the truth, even if they should happen to agree with each other.

This is very similar to what we say holds for informal concepts. Circle is learnt in a certain environment, either ostensively or by giving descriptions or definitions that themselves rely on contextual factors, and this environment determines the typical cases. The centre of interpolated stable cases consists of those that are similar to typical cases, and here we still have reason to believe our verdicts. When it comes to the extrapolated stable cases of the periphery, however, we have no greater reason to believe our opinions about them than we have when it comes to cases where interpersonal agreement is entirely lacking. The agreement in extrapolated cases is simply an artefact of the conceptual apparatus, and it does not correspond to anything in the world. It is of the same kind as agreement among a group of astronomers who all observe an unknown object in the same location using the same telescope, which unfortunately happens to have a smudged lens.

That informal concepts generally make sense only when applied in situations close to the ones they have learnt in or have evolved in does not, of course, mean that we cannot extend them as we wish, and most of the time we arguably do not even notice that we do so. Thus, when the round square is taken as an example of an impossibility, the principle of charity makes us temporarily precisify the concepts round and square so that the example holds true. Likewise, there is nothing that stops us from stipulating that we will henceforward use the word “circle” to mean exactly the set of points of equal Euclidean distance from a point x. The downside to this, as I noted, is that we will no longer be able to talk about circles in many other spaces, such as the one we live in.

The conscious, systematic extension of geometric concepts is largely the business of mathematics. As so eminently illustrated by Lakatos (1976), mathematical concepts, like all informal concepts, are in general not crystallised, definite, or exact. Even when explicitly defined they tend to rest on other concepts in which vagueness or indeterminacy remains. The standard antidote to this—expression in a formal language—does not solve the problem completely. Even if we were to believe that some logical concepts have an “innate” absolute determinateness, very few theories or concepts can be reduced to these. Therefore we will always need undefined predicates whose interpretation will remain open outside the narrow confines in which they have been learnt or introduced.

I believe that recognising this properly has some importance for the methodology of philosophy. Among other things, it indicates that the practices of conceptual analysis and thought experiments may be far less useful than is sometimes presupposed. If only the centre of an informal concept is significant, it is meaningless to try to draw conclusions from opinions about what it would be applicable to in situations very different from these.

This throws doubt on, for example, arguments against utilitarianism that make use of extreme or uncommon situations, such as ones in which one can stem a riot only by convicting an innocent man, and makes questionable the value of, for example, discussions about the ethics of killing baby Hitler. It also suggests that we have no reason to think that our beliefs about most Gettier cases tell us anything useful about knowledge, since these involve events that are quite unlikely and also very much unlike typical uses of locutions such as “She knew that p.” And deriving a counterintuitive or even absurd consequence from a metaphysical thesis should not be taken to constitute a counterargument to that thesis, since metaphysics as a whole lies very far from our usual experience. After all, if even square and circle, in their informal versions, are so indeterminate that there is no fact of the matter about whether they are contradictory or not, how much more must this not hold of concepts such as knowledge, meaning, justice, rationality, thing, and reality?

This lesson is certainly not new; one can find it in Carnap (1950, chap. 1), Quine (1951), and Wittgenstein (1978), and in such contemporary critics of a priori conceptual analysis as Stich (1992) and Ramsey (1992). The case of the square circle simply provides it with a rather vivid illustration.