Happy New Year! People like to ponder grand themes each time the Earth completes another orbit around the Sun, so let’s give that a try.
Maria Mannone is a musician who studies the relation between mathematics, music and the visual arts. We met at a conference on The Philosophy and Physics of Noether’s Theorems. Later she decided to interview me for the blog Math is in the Air. There’s a version in English and one in Italian.
She let me reprint the interview here… so with no further ado, here it is!
MM: You are one of the pioneers in using the internet and blogs for scientific education, with ‘This Week’s Finds.’ Which words would you use to feed the enthusiasm of young minds towards abstract mathematics?
JB: It seems only certain people are drawn to mathematics, and that’s fine: there are many wonderful things in life and there’s no need for everyone explore all of them. Mathematics seems to attract people who enjoy patterns, who enjoy precision, and who don’t want to remember lists of arbitrary facts, like the names of all 206 bones in the human body. In math everything has a reason and you can understand it, so you don’t really need to remember much. At first it may seem like there’s a lot to remember – for examples, lists of trig identities. But as you go deeper into math, and understand more, everything becomes simpler. These days I don’t bother to remember more than a couple of trigonometric identities; if I ever need them I can figure them out.
But the really surprising thing is that as you go deeper and deeper into mathematics, it keeps revealing more beauty, and more mysteries. You enter new worlds full of profound questions that are quite hard to explain to nonmathematicians. As the Fields medalist Maryam Mirzakhani said, “The beauty of mathematics only shows itself to more patient followers.”
MM: I love the reference to patterns, and the beauty to find. Thus, we can say that mathematical beauty is not ‘all out there’ as the beauty of a flower can be. Or, that some beautiful geometry present in nature can give a hint or can embody some mathematical beauty, but people have to work hard to find more of it—at least they have to learn how to look at things, and thus, how to mathematically think of them.
In the common opinion, a rose, or a water lily is beautiful (and it is!), but a bone is not ‘beautiful’ per se. Personally, each time I find patterns, regularities, hierarchical structures, I get excited and things seem to be at least mathematically interesting. I would like to ask you how would you relate the beauty in the natural world, both visible and ‘to discover,’ and the beauty of math. I’m wondering if they should be considered as two separate sets with occasional, random intersections, or as two displays of a generalized ‘beauty,’ as two different perspectives. Or, maybe, if the first can guide our search into math, or if math can teach us ‘how to look at things and finding beauty.’
JB: I think all forms of beauty are closely connected, and I think almost anything can be beautiful if it’s not the result of someone being heedless to their environment or deliberately hurtful.
It’s not surprising that flowers are very easy to find beautiful, since they evolved precisely to be attractive. Not to humans, at first, but to pollinators like birds and bees. It’s imaginable that what attracts those animals would not be attractive to us. But in fact there’s enough commonality that we enjoy flowers too! And then we bred them to please us even more; many of them are now symbiotic with us.
Something like a bone only becomes beautiful if you examine it carefully and think about how complex it is and how admirably it carries out its function.
Bones are initially scary or ‘disgusting’ because when they’re doing their job they are hidden: we usually see them only when an animal is seriously injured or dead. So, you have to go past that instinctive reaction—which by the way serves a useful purpose—to see the beauty in a bone.
Mathematics is somewhere between a rose and a bone. Underlying all of nature there are mathematical patterns – but normally they are hidden from view, like bones in a body. Perhaps to some people they seem harsh or even disgusting when first revealed, but in fact they are extremely elegant. Even those who love mathematics find its patterns austere at first—but as we explore it more deeply, we see they connect in complicated delicate patterns that put the petals of a rose to shame.
MM: Thus, there seems to be an intimate dialogue between nature, both visible and hidden, and mathematical thinking. About nature and environment: in your Twitter image, there is a sketch of you as a superhero saving the planet, with the mathematical symbol ‘There is one and only one’ applied to our planet Earth. Can you tell the readers something about the way you combine your research in mathematics with your engagement for the environment?
Also, it is often said that beauty will save the world. Do you think that mathematical beauty can save the world?
JB: I mainly think of beauty—in all its forms—-as a reason why the world is worth saving. But we are very primitive when it comes to the economics of beauty. Paintings can sell for hundreds of millions of dollars, and we have a market for them. But nobody attaches any value to this critically endangered frog, Atelopus varius:
To my mind it’s more beautiful and precious than any painting. Not the individual, of course, but the species, which has taken millions of years to evolve. We are busy destroying species like this as if they were worthless trash. Our descendants, if we have any, will probably think we were barbaric idiots.
But I digress! I switched from pure mathematics and highly theoretical physics to more practical concerns around 2010, when I spent two years at the Centre for Quantum Technologies, in Singapore. I was very lucky that the director encouraged me to think about whatever I wanted. I was wanting a change in direction, and I soon realized that mathematicians, like everyone else, need to think about global warming and what we can do about it: it’s the crisis of our time. I spent some time learning the basics of climate science and working on some projects connected to that. It became clear that to do anything about global warming we need new ideas in politics and economics. Unfortunately, I’m not especially good at those things. So I decided to do something I can actually do, namely to get mathematicians to turn their attention from math inspired by the physics of the microworld—for example string theory—toward math inspired by the visible world around us: biology, ecology, engineering, economics and the like. I’m hoping that mathematicians can solve some problems by thinking more abstractly than anyone else can.
So to finally answer your last question: I’m not sure the beauty of mathematics can save the world, but its beauty is closely connected to clear thinking, and we really need clear thinking.
MM: Yes, in a certain sense, despite culture, technology, and thousands of years of human history, people are quite primitive when it comes to evaluating beauty as detached from the economy.
You brought up an important point: the research focus of mathematicians. This is a tricky point because young researchers are kind of split between following new ideas and projects, and the search for funds, that often leads them to join existing projects or just well-funded areas and to put aside their more ‘visionary’ ideas. What would be your suggestion to find a balance?
JB: I don’t know if I can give advice here: I’ve never needed to search for funds, I get paid to teach calculus and other courses, so I always just do the best research I can. That’s already quite hard—I could talk all day about that!
I suppose if you’re struggling for funds you have to fight to remember your dreams, and try to work your way into a situation where you can pursue these dreams. I imagine this is also true for any entrepreneur with a visionary idea. Academics struggling to get grants really aren’t all that different from executives in a large corporation trying to get funding for their projects.
MM: My last question is about the theme of peace, very important to the Baez family. Many innovations are related to the military. Do you think that the needed clear thinking you mentioned, can first of all come from times, themes, and ideas of peace?
JB: We are currently in a struggle that’s much bigger, and more inspiring, than any war between human tribes. We’re struggling to come to terms with the Anthropocene: the epoch where the Earth’s ecosystems and even geology are being transformed by humans. We are used to treating our impact on nature as negligible. This is no longer true! The Arctic is rapidly melting:
And since 1970, the abundance of many vertebrate species worldwide has dropped 60%. You can see it in this chart prepared by the Worldwide Wildlife Fund:
If this were a war, and these were humans dying, this would be the worst war the world has ever seen! But these changes will not merely affect other species; they are starting to hit us too. We need to wake up. We will either deliberately change our civilization, quite quickly, or we will watch as our cities burn and drown. Isn’t it better to use that intelligence we humans love to boast about, and take action?
MM: Thank you Professor, I hope these words will enlighten many people.