# Mathematics

Andrés Villaveces

Freedom

It is not easy to locate the precise moment in which I decided to study mathematics: to dedicate a good part of my efforts not only to understanding but to generating new mathematics. There is no exact time that I can pinpoint. Naturally, like several of us at Refous College, he appreciated the beauty of certain topology demonstrations, or the geometry seen in terms of groups of plane transformations or dubis and dirks in the parquet. But in those final years of high school - when you make decisions that end up affecting the rest of your life - I also liked many other things. Ancient languages, philosophy. Literature. Music naturally. I particularly liked writing (and voraciously reading how much literature there was out there); He participated somewhat more tangentially in the school's orchestra and choir. And at home I lived among many subjects of science, philosophy and history of science, epistemology: my father was a chemist who gradually followed his structural concerns and came down a tortuous path to certain very mathematical ways of thinking about chemistry.

Perhaps (I cautiously say so!) It was a then dimly intuited sense of extreme freedom that led me to decide to study mathematics. Alongside many characteristics of the discipline, there is something a little paradoxical, fascinatingly paradoxical, in the juxtaposition between the highest rigor and almost absolute freedom. Music achieves a similar freedom, although with other types of parameters. Mathematics (that other music, Monsieur Jeangros once told me) sometimes achieves that freedom: spaces of as many dimensions as one manages to work, infinity of different infinities, principles of reflection of conjunctival universes that show how the large universe it can reflect, infinitely perhaps, in small universes, as multiple ways of being “continuous” (for a function or a process) as topologies can imagine, spaces so strange that sometimes they don't even have points (but they have open ones). At the same time, mathematics sets a standard of rigor that few other disciplines, perhaps no other, have. That is paradoxical. Scary and beautiful, like the challenge of climbing a great mountain wall, or launching yourself into the musical world of great works. Freedom, true freedom, scares, demands and reveals one's weaknesses. Mathematics brings all that.

Creativity

The human being creates music. Create literature. Create photographs, pictures, poems. Perhaps initially it does not seem so clear that this is the case with science: the force of gravity is not created, it is discovered and studied, and if one is in a good mood, one can understand it. Likewise with elementary particles of physics, or the chemical structure or DNA chains in biology. But in mathematics ... is it created or discovered? Again, mathematics has a privileged role here. There is no immediate answer to that question; there is no consensus. Great thinkers have argued both that it is created and that it is discovered. One can invent theories in logic, to later notice that they explained something that one knew on the other hand. Creation or invention?

Even at the level of elementary axiomatic geometry, the question remains: do we discover the lines, the parallelism ... or do we invent versions of these in order to understand them and (as Monsieur Jeangros did at school) we call them with strange names (dubis, self-crystallinity) to understand them better?

In any case, the mathematical activity, whether at the moment of understanding complex phenomena (infinity / infinities, the structure underlying quantum, the structure of world music, the fundamental symmetry underlying quasicrystals, to name a few examples that I like) that is to say, when inventing theories (or strange examples or topologies) is at the crossroads of human creativity.

Staying creative is never easy: it requires (as those who write poetry or novels perfectly, those who compose or invent theatrical scripts, those who make movies) a very rare combination of very hard work, which sometimes does not seem very creative on the surface, with a certain sensitivity to not let go of the creative moment when one finally generates it (or arrives). Make variations on themes from other composers, variations on one's own work, invent different ways of approaching… formal exercises many times, but suddenly, like when a cloud dissipates and a ray of light illuminates the snowy summit of some mountain and the perfect photo remains (even if it is only in the mind of those who see it), they allow you to create something new. That rare and difficult moment is part of what keeps activity for years in something like mathematics.

Look

The word theory, central in mathematics, has the Greek root theōros, the one who looks, the viewer. Theoria originally meant show, what is seen, what is seen. Mathematics is also a way of looking, a way of contemplating.

The part of the math I ended up working on is called model theory - "models" are structures (for example, reals with sum, product, and order or a parquet and its dubis or some Hilbert space in physics quantum - they can be infinitely varied structures). The "theory" side could actually be written in the plural: we have as many theories as we can imagine sets of formulas, sets of sentences. It is a way of seeing the world ... or at least a way of looking, of contemplating certain structures that are there, before our eyes that are almost always covered.

Being able to contemplate the world with the lens (or prism) of mathematics, which one patiently polishes and must reconstruct every day, is a very peculiar human activity.

Curiously, learning to contemplate mathematically ends up teaching one a certain resistance, a certain way of contemplating at the same time happily enthusiastic and seriously doubtful.

The opening of the day, the dawn of the first light, coexists with the sunset, with the resistant and hard twilight. Alba and twilight are constantly intertwined in mathematics.