Extroverted introspection – There is nothing beautiful about Mathematics
An exploration of my relation to mathematics, where I make the case that it is not fair to claim that mathematics possesses any kind of beauty in itself.
I remember when I started out to study mathematics at the University level. (This was more than 10 years ago, before I took a long detour into other areas.) For the first time I was confronted with idea about the beauty of mathematics. Also, I distinctly remember the feeling of confusion.
– Excuse me, er, what do you mean by that?
An answer could be something like: the beauty inherit in a particularly simple proof, that can show something significant with unexpected simplicity.
And this confusion got worse when, over time, I could not really internalize any feeling of beauty. Keeping in mind that, when many describe an experience that is inexplicable to oneself, the shortcoming is usually ones own. Like you can lack the experience to fully enjoy an incredible movie, and later in life learn what escaped your notice the first time. And clearly, the beauty of mathematics is an experience shared by many. Just enter “the beauty of” into the Google search field, and check out the auto complete suggestions. At the time for the writing of this blog post mathematics is defending position number six, trying to catch up to life at number five.
Thus I begun to grow a cyst of inferiority complex when pondering why I am unable to see what is clearly there. Maybe I have not worked hard enough, or – more likely – I am just not smart enough. As it turns out I did lack some necessary experience. I did not see that something was wrong. Mathematics is not beautiful, instead, mathematics is elegant.
The concept beauty is highly a subjective one. When looking for beauty within any creative field – I hold the position that – it is foolish to go searching without intimately involving oneself. The ultimate judge of beauty is you, albeit it may take some time or even collaboration, before you can form a lasting opinion. In addition, note that no one can really be told that their experience of beauty is false, instead it is just not shared experience.
Here, you might object to my premise, since I just prohibited objections to anyone’s subjective feeling for beauty. But note that there is a distinction between something having a property in itself – mathematics is beautiful – and something invoking a subjective feeling – mathematics is beautiful, to me.
And, to me, beauty and maths becomes particularly difficult to join. In my subjective world of concepts and things (is that what Edmund Husserl called phenomenology?) beauty always have some degree of imperfection. But in mathematics, you strive for completion, the simplest lines of reasoning, and rigour in every detail. And to illustrate why perfection – in my mind – can eliminate beauty, allow me to make a crude comparison. Say you are a retoucher, working with a portrait, each small stroke making the photo a little bit better, until, at the end of your work you have arrived at a perfect picture of a exceptionally lifeless face. Instead, my subjective notion of beauty entails finding an interesting blend of necessary imperfections, and leaving them in place. Something I can not reconcile with mathematics.
In the place for beauty, as I have already mentioned, there is a much better noun we can recruit. Elegance. For something to be elegant is a more objective property, with a more well defined meaning, and one that I can reconcile with the mathematicians obsessive strive for complete, and completely rigorous, arguments. I grant that elegance is not an equally sexy and mysterious concept. So while visiting your local pub, if your goal is to convince the (objectively!) beautiful poet you just met to spend the night at your place, maybe the elegant nature of mathematics will not work to your benefit as a conversational topic. In that situation I will look the other way if you momentarily lapse into a discussion about the beautiful and sexy qualities of maths. But returning to work the morning after, I advice against reinforcing the impression that beauty is a general characteristic of mathematics. From a technical point of view I do not see how that can be right, and in the end you risk spooking the young and hopeful undergrad students (like I once experienced), and discourage them in their efforts.
As a physicist I like to apply my particular toolbox of thinking tools. One specific thought I have had on this topic is that perhaps beauty is a somewhat higher order emergent behaviour. And thus necessarily very hard to find in such a fundamental system of logic as mathematics. However, completing such an argument is not just a blog post, but instead requiring research at least comparable to a bachelor thesis. Any philosophy students out there? You are welcome! Just remember to send me a copy of your thesis.