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On the very idea of a mind trail

Maths and Art

The first thing one sees when clicking on this website is a quote by the mathematician E. Frenkel which reads:

What if at school you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo and Picasso? Would that make you appreciate art? Would you want to learn more about it? Mathematics is just like art.

But the reader might wonder, “How?”, “How is the study of numbers and shapes, in all the abstractness they give rise to, possibly akin to art?”, after all, the quote just delivers a promise regarding the nature of mathematics.

In this shorter and introductory trail we will try giving a glimpse of just what is meant by the analogy between these two disciplines.

When one is unsure where to start, a good place is the beginning. All disciplines of the human spirit stem from the same place: our subjective experience of the world around us, and if we go back in time to the dawn of human thought, we will see that we have been playing with abstractions since our very first steps.

That is because thinking is often directed at something, you think about something, and when you single something out of the interconnected web of things that is reality you are engaged in abstraction from your experience; Humans have learned to play with those abstractions and something like language was born.

Art though does more than this, it takes these abstractions and injects feeling in it, pulls them out of their imaginative realm and renders them concrete: one both makes and enjoys art.

But mathematics, just like art, both stems from experience and above everything else is concerned with the making and enjoyment of compositions, as derived from the interconnections of their factors.

The notion of elegance is pervasive in mathematical practice: A proof can be elegant if it is concise, original, and provocative of novel ideas; but ideas themselves may be elegant from a conceptual point of view, and sometimes even from a purely visual point of view, as a quick search with the keywords “Julia sets” will show you.

This brings us much closer to an appropriate definition of mathematics, as not the “study of numbers and shapes” but “the study of patterns and structures”. As such it has an implicit tendency towards generalisation.

The example of non-planar geometries

The quintessential example of this fact is geometry, we started with shapes drawn in the mud, trying to identify truths about them and how these truths relate with one another. Eventually came along a mathematician by the name of Euclid who managed to synthesize this whole web of truths and pin its roots into only five of them: accepting these, then everything about planar geometry can be deduced through pure logic alone. Such unprovable ground truths are called axioms in mathematics, Euclid's are the following:

Which look straightforward enough. One thing people have tried to do for a long time though is deriving the last axiom from the other ones, since in some way it does not "feel" as fundamental as the others*.

[*The one given here is actually a logically equivalent statement

to the original 5th axiom, which was even wordier]

For the longest time though we were never sure of whether it was really an axiom, we did not doubt about its truth, only a madman would say that such there exists more than one such parallel line.

Then, two millennia later, a madman, by the name of Lobachevsky came along and did exactly what was thought to be nonsense, he changed the last axiom to

5) Given a line and a point external to this line there is an infinite number of lines passing through that point that are parallel to the first line.

And behold, what he obtained was not nonsense but geometry on a saddle, also called hyperbolic geometry.

This is a weird way to do geometry, and it is rather hard to gain an intuition for it. For this reason we will pass onto the other, more intuitive way to deny the 5th axiom. One thing should be mentioned before though, which is that there is a way this hyperbolic geometry can be nicely represented by diagrams like these:

They are called tilings in the hyperbolic plane, such constructions also somehow relate to black holes. Could make for an interesting future trail, would it not?

That being said, what is the other way to go against common sense? Introduce our second madman: the mathematician Riemann, who went ahead and claimed:

5) Given a line and a point external to this line there is no line passing through that point that is parallel to the first line

Now this is quite the claim; until one realises that we deal with this geometry every time we look at a globe, for this is simply geometry on a sphere, where the only way to properly define lines is to say they are great circles, akin to the equator. One might wonder whether such is a proper designation, couldn't circles of latitude be seen as lines?

Well, saying a curve is “straight” on a curved geometry is somewhat of a tricky concept, so what to do?

We try and reason what characterises lines in a domain we feel confident, like, planar geometry. There a line segment is the curve with minimum distance between two points, so let us do that here as well, let us run with it. If one keeps tracing such segments going further and further, then one indeed obtains a "great circle"

Think about planes, when mapping a route from Rome to New York (which are at the same latitude) it’s as if the plane “curved” north or south to get there, but that actually is the shortest path.

And there we have it, the best mathematicians are the ones not impeded by such things as common sense. They feel like they are discovering something almost akin to a world outside space and time.

The uses and nature of mathematics

What was noteworthy in the preceding section is the process of discovery, as well as the fact that something like spherical geometry became instrumental in devising Einstein’s theory of general relativity. This is part of a more general trend in 20th century physics, which is the realisation of the adequacy of pure mathematics to physical theorising.

Many still view this as borderline magic, as nicely shows in the title of a famous paper by E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences”. But if one regards mathematics as the study of abstract patterns, and if one believes that our universe does possess structure, then it is inevitable for some kinds of mathematics to give us precise ways to express, or more precisely model, that structure.

Another example of this is symmetry, which nowadays has a very important place in physics, and which is itself exemplified by another mathematical concept: that of a group, foreshadowing yet another future trail.

What is unreasonable then is mathematics itself. How did we get from counting sheep to the richness and variety of modern mathematics?

In the words of another mathematician, mathematics is but a divine madness of the human spirit, a slow process, building upon centuries of shared practice, and it gives the discipline the character of an abstract cosmos that is slowly being unveiled. Indeed, the greatest discoveries in mathematics take the form of newly found bridges between seemingly disconnected areas of this world. These form a great class of those beautiful conceptual ideas we discussed earlier, as one of the most beautiful aspects of mathematics is exactly its interconnectedness.

Aesthetics and the mapping of the mathematical world

This is in stark contrast with how mathematics is normally taught, even at university level, where one has different modules which possibly intersect only at a much later stage.

However, concealed in our initial quote lies a truth about learning: just like what is at play when contemplating a piece of art, the goal of education should lie in bringing about intuition, in immediate apprehension and appreciation of varieties of value, something possible in any discipline, something that could be qualified as aesthetic learning.

For mathematics this means solving exercises surely, the same way studies in anatomy makes one a better portraitist, but it also means staying faithful to the interconnectedness of various areas of mathematics, to that process of discovery outlined earlier, and to the general aesthetics or "feel" of the subject as a practice.

But how to do that? What structure would best serve this purpose?

In answering this, maps of this abstract cosmos and ways to navigate them were conjectured by the author.

The two dimensions of a sheet of paper are quickly diagnosed to be unable to contain all the information there is to pack. What about more abstract shapes? More dimensions? This is an entertaining exercise for those familiar with pure mathematics, the punchline though is that again and again arbitrary divisions loom everywhere.

What about a world with a local structure but lacking a global map? With each concept connected to other ones but without a bird’s eye view offering more than an abstract network. A magical doll house with each room connected to several other ones through a single door is a concrete way of imagining such an arrangement.

While better suited, this system still leaves things to be desired, for the same concept can be viewed in many ways and the same topic can be part of different discussions in a way not captured by discrete nuggets of information.

The concept of a mind trail

It is through different trains of thought then that the need for a more fluid concept comes to mind, something akin to a trail.

The discovery of mathematics through stories is, when complemented with practice, what is closest to the nature of the subject, intended with respect to those principles listed earlier.

A trail exemplifies what such "aesthetic" learning should be all about.

A trail is a trail, some people take more time, other people less.

Some people are there for the satisfaction found in completing it, others are there just to enjoy some nature.

A trail is flexible, it can be changed according to the whims of the human aesthetic sense, while having to respect the morphology of the mountains it goes through.

There is no right or wrong way to go about a trail, the view from the top and a novel perspective is all that is sought.

Then, for those who want to be more than casual hikers, practice becomes important, but practice is best done when one has mustered the curiosity necessary to tackle it wholeheartedly.

Creativity, Discovery and Romanticism

This leaves an open question though; how can mathematics be a creative endeavour of the human spirit and an abstract world to be discovered, at the same time?

In other words, do we invent, or do we discover mathematics?

The most correct answer is, without doubt, “both, and some more”.

We start from experience and create definitions and concepts that are the most natural to create meaningful novel structures which are not found directly in experience, soaring above the immediacy of perception. We discover new things in light of these definitions, logical necessity and intuition; we sometimes slightly change those starting blocks to conform to the nature of what is being found. One could say that on one hand such a process mimics invention in that what is theorised did not exist before, on the other it mimics discovery in that it relates in fixed ways to the rest of our knowledge once it is theorised.

What is beautiful about such a hybrid character is that in its very act of making, the logical rules of mathematics confer to the discipline a subtle haiku-like balance between freedom and restriction which makes up the essence of many creative endeavours. Indeed, no one would claim that the haiku master creates despite the constraints of his art form, rather one should say that he creates through them.

That is why, when walking, the attitude most closely resembling that of the mathematician by trade is that of the wanderer one often sees depicted in many romantic-era paintings: lost in a world born out of a unique perspective on this universe we inhabit, a world that is unfathomably rich yet one where he feels at home, a world where he is free to go one way or the other, along trails yet uncharted.

Which brings us full circle to where we started, to the artistic side of mathematics, while we wait for the opening of the next trail.


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