Steven Strogatz, the popular applied mathematician and educator, recently tweeted a link to a paper on the question of whether mathematics is something that was invented or discovered:

Is mathematics discovered or invented? That is THE QUESTION. Smart thoughts on it from a leading mathematician: https://t.co/QX2vjWYb1c

— Steven Strogatz (@stevenstrogatz) July 2, 2017

 

The author of that paper, Barry Mazur, highlights the importance of the subjective experience of doing math in addressing this question. As someone who works in computational neuroscience, I wouldn’t fancy myself a mathematician and so I can’t speak to that subjective experience. However I can say that working in a more applied area still leads one to the question. In fact, it’s something we discuss at length in Unsupervised Thinking Episode 13: The Unreasonable Effectiveness of Mathematics.

It’s been awhile since we recorded that episode and its something that has been on my mind again lately, so I’ve decided to take to the blog to write a quick summary of my thoughts. Mazur also gives in that article a list of do’s and don’ts for people trying to write about this topic. I don’t believe I run afoul of any of those in what follows (certainly not the one about citing fMRI results! yikes), but I suppose there is a chance that I am reducing the question to a non-argument. But here goes:

I think the idea of mathematics as a language is a reasonable place to start. Now when it comes to natural languages, like English or Chinese, I don’t believe there is any argument about whether these languages are invented or discovered. While it may have been a messy, distributed, collective invention through evolution, these languages are ‘invented’ nonetheless.  The disorder around the invention of natural languages means that they are not particularly well designed. They are full of ambiguities, redundancies, and exceptions to rules. But they still do a passable job of allowing us to communicate about ourselves and the world.

We cannot, however, do much work with natural languages. That is, we can’t generally take sentences purely as symbols and rearrange them according to abstract rules to make new sentences that are of much value. Therefore, we cannot discover things via natural language. We can use natural language to describe things that we have discovered in the world via other means, but the gap between the language and what it describes is such that its not of much use on its own.

With mathematics, however, that gap is essentially non-existent. Pure mathematicians work with mathematical objects. They use the language to discover things, essentially, about the language itself. This gets trippy however–and leads to these kinds of philosophical questions–when we realize that those symbolic manipulations can be of use to, and lead to discoveries, in the real world. Essentially, math is a rather successful abstraction of the real world in which to work.

But is this ability of math due to the fact that it is a “discovered” entity, or just that it is a well-designed one? There are other languages that are well-designed and can do actual work: computer programming languages. Different programming languages are different ways of abstracting physical changes in hardware and they are successful spaces in which to do many logical tasks. But you’d be hard-pressed to find someone having an argument about whether programming languages are invented or not. We know that humans have come up with programming languages–and indeed many different types–to meet certain requirements of what they wanted to get done and how.

The design of programming languages, however, is in many ways far less constrained than the process that has lead to our current mathematics. An individual programming language needs to be self-consistent and meet certain design requirements decided by the person or people who are making it. It does not have to, for example, be consistent with all other programming languages–languages that have been created for other purposes.

In mathematics, however, we do not allow inconsistencies across different branches, even if those different branches are designed to tackle different problems. It is not the case, for example, that multiplication doesn’t have to be distributive in geometry. I think a strong argument can also be made that the development of mathematics has been influenced heavily by a desire for elegance and simplicity, and by what is useful (and in this way is actually influenced by whether it successfully explains the world). If programming languages were held to a similar constraint, what could have developed is a single form of abstraction that is used to do many different things. We may then have asked if “programming language” (singular) was a discovered entity.

So essentially, what my argument comes down to is the idea that what we call mathematics is a system that has resulted from a large amount of constraints to address a variety of topics. Put this way, it sounds like a solution to an engineering problem, i.e. something we would say is invented. The caveat, however (and where I am potentially turning this into a non-problem), is that what we usually refer to as “discovering” can also be thought of as finding the one, solitary solution to a problem. For example, when scientists “discovered” the structure of DNA, what they really did was find the one solution that was consistent with all the data. If there were more than one solution that were equally consistent, the debate would still be ongoing. So, to say that the mathematical system that we have now is something that was discovered, is to say that we believe that it is the only possible system that could satisfy the constraints. Perhaps that is reasonable, but I find that that formulation is not what most people mean when they talk about math as a discovery. Therefore, I think I (for now) fall on the side of invention.

 

Meta-caveat: I am in no way wedded to this argument and would love to hear feedback! Especially from mathematicians that have the subjective experience of which Mazur speaks.