A Mathematical Theory of Creativity

I was listening to In Utero by Nirvana last night, and I was reminded by how great they were as a band. That album in particular has a really unique, and unusual sound, and is harmonically quite strange. I was also reminded of the fact that Cobain is certainly not the best guitarist in the world. Nonetheless, his songs are legitimately interesting. You could say that this is the result of working well with what you have, and in some sense, this must be true, because this is exactly what he’s doing. It turns out that we can think more rigorously about this distinction between creative brain power, and technical competence, using computer theory.

For example, if I give you a set of notes that are all in the same key, it’s trivial to construct a melody that fits them, and machines can certainly do this. If, however, I give you a set of notes that are not in the same key, then constructing a listenable melody using those notes is non-trivial, and I’m not sure that a machine can do this. This is exactly what Nirvana’s music does, which is to take a set of disparate key signatures, and connect them with listenable melodies that your average listener probably wouldn’t second guess at all. That is, their music generally sounds normal, but harmonically, it’s actually quite interesting. The same is certainly true of Chris Cornell‘s music as well, in that he also makes use of what is effectively multi-tonal music, in a manner that isn’t obvious in the absence of conscious reflection.

Creativity in the Arts

Taking it to the next level, we can think of the rules of art as constraints:

You’ve got a set of what are effectively rules of harmony, a set of expectations in your audience, which impose some more rules, and a space you populate with notes. So any final product can be viewed as a collection of notes arranged in that space, which either satisfy the rules, or don’t. What artists like Cobain, Erik Satie, and others do is to create simple, but non-obvious solutions to the problems of art. In their case, the real work is not necessarily in the performance of the piece, but is instead in the contemplation beforehand that produced the piece.

But this type of brain power is certainly not unique to art, since we can imagine mathematical constraints that together pose a difficult problem, that nonetheless has a low complexity solution. That is, the solution to a computationally hard problem could itself have a low Kolmogorov complexity. There are real problems like this, such as Diophantine equations, which as a class are non-computable. We can easily state the answer to a Diophantine equation, since it’s just a set of integers. We can also quickly confirm that it is in fact the correct solution, since that requires only basic arithmetic. As a result, the solution to any Diophantine equation will have a trivial Kolmogorov complexity, and confirming its accuracy is also computationally trivial. Nonetheless, because Diophantine equations are non-computable as a class, any given problem could require infinite time to solve, or, a non-Turing Equivalent machine.

So, we know these types of problems exist in mathematics, where the complexity of the solution to a problem is low, but the computational power required to find that solution is extremely high, and possibly infinite (at least on a UTM). My opinion is that great artists are capable of exactly this type of thinking, where a superficially impossible aesthetic problem is solved by a particular piece. I think Prokofiev takes this approach to the extreme, producing works of art that are not only non-obvious, but also technically complex, in that you really have to be a technically competent musician to play one of his pieces. As a general matter, I would say that Prokofiev’s music solves complex aesthetic problems in non-obvious ways, using complex solutions. In contrast, I would say that Cobain and Satie solve complex aesthetic problems, using simple solutions.

Creativity in Mathematics

Nature itself imposes constraints that distinguish between true and false claims about mathematics. This is an abstract way of thinking, but it is necessarily the case. Just consider a simple example: “all even numbers are divisible by two”, versus, “all odd numbers are divisible by two”. If you have a collection of objects that is even in number, you will find that it can in fact be divided into two equally sized collections of objects. In contrast, if you have a collection of objects that is odd in number, you will find that it cannot be divided into two equally sized collections of objects. This seems trivial, but it’s actually quite profound: it shows that theorems of numbers operate as primordial rules of our reality, that are arguably even more fundamental than the laws of physics, since there is absolutely no possibility that they will ever be refined.

As a result, we can think of mathematicians as solving problems imposed by the most fundamental constraints of reality itself, in turn generating claims about mathematics that are demonstrably true in our external world. The most interesting theorems of mathematics are the ones that seem to follow only from these primordial rules, where you’re forced to wonder how it is that the mathematician in question ever conjured such a result. In contrast, the least interesting theorems of mathematics are those that can be derived mechanistically from known theorems. For example, if I tell you that A is false, you know mechanistically that (not A) is true, but that’s certainly not interesting.

Stated differently, if you begin with a known result, and proceed mechanistically by evaluating all possible consequences of that result, whatever inferences you arrive at, are, frankly, boring, because it doesn’t require any creativity to do that – a Turing Machine can do that, and much faster than a human being can. In contrast, if someone identifies a new and correct theorem of mathematics, that arguably doesn’t follow from any known result, you’ll be surprised and amazed, precisely because it’s not clear how people do these things. In fact, even if you yourself do this all the time, it’s still surprising when someone else does it.

A few examples of theorems that I think are of this type are Kuratowski’s Theorem, Ramsey Theory generally, and of course, the Graph Minor Theorem. Taking Kuratowski’s Theorem as an example, imagine a machine trying to solve this problem, even after it’s already been stated. A truly creative mathematician will begin by asking interesting questions in the first place, thereby stating interesting problems, and it’s not clear that even this initial step is computable. For example, in the case of Kuratowski’s Theorem, why would there necessarily be a finite set of forbidden sub graphs? You’d have to first state this question, in order to prove that it is the case. But even if we skip this step, it’s just not realistic to assume that a Turing Machine could winnow down the set of forbidden sub-graphs to a finite set, which is exactly what Kuratowski’s theorem achieves. In particular, determining whether two graphs are isomorphic is a hard problem, and that’s just a small part of what’s necessary to solve a problem like Kuratowski’s theorem, as a practical matter.

A Mathematical Theory of Creativity

All of this allows us to distill precisely what makes great mathematicians, and great artists, so impressive, in mathematical terms:

They begin with an enormous, or possibly infinite, number of possibilities, and somehow present you with a correct one. This suggests that true creativity is the power to solve computationally hard problems, and possibly, non-computable problems.

I say this all the time, but it’s pretty obvious, to me at least, that some human beings are simply not equivalent to a Turing Machine, but are instead, categorically superior. (Note that a reasonably intelligent human being is always at least as powerful as a Turing Machine). I think a good place to look for this kind of brain power is in musicians and architects, since both are forced to solve complex technical constraints in a manner that is more than just functional, since they must also satisfy aesthetic criteria. Obviously, mathematicians and scientists would be a great place to look for this type of brain power as well, but regrettably, I get the sense that this bloodline has been massacred.

Which makes you wonder –

Why is it that so many of our finest musicians seem to end up dying long before their time?

Perhaps I’m not the only person aware of the connections between creativity in the arts and in mathematics, and maybe others don’t like the economic and military risks posed by too many creative people, giving them an incentive to ensure that we don’t live long enough to have too many children.


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