Coffee cups and donuts

by Neil Rickert

There’s a saying among mathematicians, that a topologist is someone who cannot tell the difference between a coffee cup and a donut.  I’ll discuss that in this post, and I’ll suggest implications beyond mathematics.

Usually, when we say this, we are thinking of the donut and the coffee cup as two-dimensional surfaces.  Once we go to the three-dimensional objects, nobody denies that the donut has a soft and spongy texture which makes it clearly different from a coffee cup.

Topology

Let’s start with a brief rundown on what is topology.  It is a branch of mathematics where we discuss ideas such as continuity, convergence, etc.  A classic example of convergence is with the sequence 0.9, 0.99, 0.999, …  We can see that the sequence gets closer and closer to 1, and we say that it converges to 1.  So topology has something to do with the geometric ideas of getting closer.  But it does so without needing a notion of metric (or distance).

A key notion in topology, is that of homeomorphism.  A homeomorphism is, quite simply, a continuous one to one mapping between two spaces (or objects or surfaces).  And, because the mapping is continuous, it preserves all of the topological relations.

It turns out that the donut and the coffee cup are homeomorphic (as two-dimensional surfaces).  The donut, as a surface, is sometimes called a torus.  And, at other times, it is described as a sphere with one handle.  All spheres with one handle are topologically equivalent, in the sense that they are homeomorphic.

If you should come across a coffee cup with two handles, that would be homeomorphic to a sphere with two handles, but would not be homemorphic to a donut.

What about shape?

When we look at a coffee cup and at a donut, they seem easily distinguishable to us on account of their different shapes.  Our notion of shape includes that of curvature.  Shape and curvature are metric properties.  We define them in terms of distance.  For example, we usually define curvature in terms of the radius of the circle that fits best.  And radius is a distance measurement.

What seems to clearly distinguish the coffee cup from the donut, turns out to be due to metric properties.  But metrics are defined by human conventions.  And different conventions are possible.

Take any metric (distance measuring function) appropriate for measuring distances on the surface of a donut.  Because there is a homeomorphism between the donut and the coffee cup, we can use this homeomorphism (or continuous map) to define a corresponding metric on the surface of the coffee cup.  The rough idea is that to find the distance between two points on the coffee cup, we map those to the corresponding points on the donut, and measure the distance on the donut.

Given a metric property such as shape, on a donut surface, we can use this corresponding metric on the coffee cup to see that the same property exists there.  This illustrated the relation between metric properties and the metrics under which we determine those properties.

Intrinsic vs. conventional

To sum this up, there are intrinsic properties of a sphere with one handle (or a donut or a coffee cup) which distinguish it from a sphere (one without handles).  But what distinguishes a donut from a coffee cup is not an intrinsic property of those two-dimensional surfaces.  Rather, it is a conventional property, a property introduced by our conventions — in this case, by our measuring conventions.

While people might sometimes say that conventions are arbitrary, they are not completely arbitrary.  We adopt conventions for their usefulness.  Our measuring conventions are adopted for their pragmatic value.

What I see as an important philosophical point, is that our knowledge of the world is very much tied to our adopting pragmatic conventions.  We cannot explain human cognition on the basis of logic alone.  We must also take into account the ability of a cognitive agent to make pragmatic judgments, and to adopt pragmatic conventions.

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