Although the term “exact sciences” has fallen into disuse, we tend to think of mathematical concepts and their corresponding terminology as precise and immutable. Don’t we still use Pythagoras’ theorem or Euclid’s postulates as they were formulated more than two thousand years ago? But this is not always the case, and in fact one of the most interesting and instructive aspects of the history of science is the way in which the names and/or definitions of some mathematical objects are modified (it is very significant, in this sense, that geometry began as the “measuring of the earth”, as its name indicates). In the words of the master Martin Gardner: “Usually, the process is as follows: objects are given a name x and are roughly defined, according to usage and intuition. Then someone discovers an exceptional object that fits the definition, but which no one thinks of when an object is called x. Then a new and more precise definition is proposed, which covers or excludes this exceptional object. The new definition remains in effect until new exceptions appear, in which case the definition must be revised again, and this process can continue indefinitely” (Penrose mosaics and coded hatches1990).
As we saw last week, something as familiar and seemingly simple as a curve is not easy to define precisely, and the numerous attempts by readers to give an unambiguous and comprehensive definition attest to this (see comments by What is a curve?). In fact, as Adelaida López pointed out, in mathematics we do not usually talk about curves “in general”, because in many cases it is necessary to specify what type or concept of curve we are referring to.
And, needless to say, curves are not an isolated case. Terms and concepts as seemingly clear as number, set, dimension or infinity are equally slippery and can give rise to disturbing paradoxes (as you will discover if, following Einstein, you try to explain them to your grandmother, real or imaginary).
Heaps and sets
The sorites paradox, or the heap paradox, which we have discussed on more than one occasion, is largely due to the ambiguity of the term itself, which has to do with quantity but is not quantifiable, so that, at first glance, sets, which are like piles with no pretensions of abundance, would seem immune to paradoxes. However, as Bertrand Russell demonstrated at the beginning of the 20th century (developing an idea of Cantor’s own, a detail that is often omitted), a merely intuitive notion of set (what is yours, astute reader?) leads to contradictions such as the following: let us call typical those sets that do not contain themselves and atypical those that contain themselves (for example, a pile/set of chickpeas is not a chickpea, while a set/heap of piles is a heap).
What will be the set of typical sets, which we will call T? If T is typical, it must be included in T, and therefore it is included in itself, and therefore it is atypical…
At the same time that Russell, with his barber’s paradox (the most popular version of the paradox of typical and atypical sets), was blowing up Frege’s logicist project, the French mathematicians Gaston Julia and Pierre Fatou They developed their theory on the iteration of complex functions, which gives rise to “monstrous” sets (read fractals), such as Julia’s own or the well-known Mandelbrot, whose intricate graphic representations are of a breathtaking beauty. But that is another article. Or several.
Balls and conics
Flashlights usually project a more or less defined cone of light, and a conic is the intersection of a plane and a cone, so the ball is not even necessary (see last paragraph of last week), as you can easily see by focusing on a wall at a short distance: facing you, a circle of light will form, by tilting the flashlight slightly the circle will become an ellipse, tilt it a little more and you will get a parabola… If you put a ball in the cone of light, a cone of shadow will form with which you can obtain more defined “Chinese” conics.
And speaking of balls and conics, the typical image of the tennis hawk’s eye looks like an ellipse. Is it? Why? Shouldn’t it be a circle, since it represents the intersection of a sphere (the ball) and a plane (the court)?
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