If you’re looking for a whole new perspective on the value of mathematics, Stanford University’s Keith Devlin shall provide. With his wonderfully lilting English (Yorkshire?) accent and as sharp of a mind as you can imagine, he compares mathematical equations to sonnets and says that what most of us learn in school doesn’t begin to convey what mathematics is. That technology may free more of us to discover the wonder of mathematical thinking — as a reflection of the inner world of our minds.
What an absolutely splendid way to usher out the day, thanks to explore-blog:
This is amazing: A mesmerizing Beach Boys vocals inspired by the physics of church bells from Alexander Chen, who wrote code to draw a circle for each note of the song using a mathematical relationship between a the circumference of a circular surface and pitch. A fine addition to these synesthetic visualizations of music.
~Trent Gilliss, senior editor
Mathematics, Purpose, and Truth: The World Feels More Spacious
by Krista Tippett, host
I picked up Janna Levin’s novel off a table at a bookstore, drawn to it initially perhaps because we had just completed our show with Paul Collins and Jennifer Elder on autism. Mathematician Alan Turing — known as the father of modern computing — is one of the autistic personalities who was mentioned in that interview. I was immediately taken by Janna Levin’s lush prose and the alluring, provocative ideas that she brings to life through human stories in space and time.
A Madman Dreams of Turing Machines sounds depths I had never considered before, between mathematical truths and great existential questions. It does so by probing the parallel lives and ideas of Turing and another pivotal 20th-century mathematician, Kurt Gödel. Turing’s discoveries were made possible in part by Gödel, who shook the worlds of mathematics, philosophy, and logic in 1931 with his “incompleteness theorems.” They demonstrated that some mathematical truths can never be proven. Or, as Gödel says in Janna Levin’s novel, “Mathematics is perfect. But it is not complete. To see some truths you must stand outside and look in.” This held unsettling scientific and human implications; it posited hard limits to what we can ever logically, definitively know.
Janna Levin is an atheist, if we care to categorize her. And while that simple fact informs our conversation along with her exquisite intelligence and her mathematical training, we cover territory that can’t be bounded by such definitions. Janna Levin’s most certain “faith” is in the conviction that we can agree on basic realities described by mathematics — that 1 plus 1 will always equal 2. Putting God into that equation, or barring God from it, is not her concern. Yet this conversation is a beautiful example of the deep complementarity of religious and scientific questions, if not of answers. The ideas and questions Janna Levin lives and breathes open my mind to new ways of wondering about purpose, meaning, and ultimate reality.
There is much in her thought that I struggle to comprehend and will continue to ponder. I’m intrigued, at the same time, by echoes with the wisdom of ordinary life. Gödel’s idea that there are some truths we can only see at an angle — by standing outside, looking in — is a fact even in the work I do, of speaking of faith. The deepest truths are usually impossible to see and articulate straight on.
And I feel a kindred pull to Janna Levin’s delight and passion in the great narrative of the world and humanity, epitomized in these lines from her book that we read in the show:
"I am looking on benches and streets, in logic and code. I am looking in the form of truth stripped to the bone. Truth that lives independently of us, that exists out there in the world. Hard and unsentimental. I am ready to accept truth no matter how alarming it turns out to be. Even if it proves incompleteness and the limits of human reason. Even if it proves we are not free."
Of all the ideas Janna Levin presents, the most provocative and disturbing, perhaps, is her doubt that there is free will in human existence at all. She cannot be sure that we are not utterly determined by brilliant principles of physics and biology. Yet she cleaves more fiercely in the face of this belief to the reality of her love of her children and her hopes and dreams for them. She sees “evidence of our purpose” in figures like Gödel and Turing, even though they did not the find the clarity in life that they wrested from mathematics on all our behalf.
Paradoxically, perhaps, the world feels more spacious to me after this conversation with Janna Levin — even, to use her words, if it suggests incompleteness and the limits of human reason and faith; even if it suggests we are not free. She possesses a quality that keeps me interviewing scientists as often as a I can — a delight in beauty, a comfort with mystery, a limitless ambition for one’s grandest ideas combined with a humility about them that many religious people could learn from.
"I don’t believe that math and nature respond to democracy. Just because very clever people have rejected the role of the infinite, their collective opinions, however weighty, won’t persuade mother nature to alter her ways. Nature is never wrong."
—Janna Levin from How the Universe Got Its Spots
Photo by Agustin Ruiz (Taken with instagram)
Symbols of Power: Adinkras and the Nature of Reality
by S. James Gates
Physicists have long sought to describe the universe in terms of equations. Now, James Gates explains how research on a class of geometric symbols known as adinkras could lead to fresh insights into the theory of supersymmetry — and perhaps even the very nature of reality.
Complex ideas, complex shapes Adinkras — geometric objects that encode mathematical relationships between supersymmetric particles — are named after symbols that represent wise sayings in West African culture. This adinkra is called “nea onnim no sua a, ohu,” which translates as “he who does not know can become knowledgeable through learning.”
In the land of theoretical physics, equations have always been king. Indeed, it would probably be fair to caricature theoretical physicists as members of a company called “Equations-R-Us”, since we tend to view new equations as markers of progress.
The modern era of equation prediction began with Maxwell in 1861, continued through the development of Einstein’s equations of general relativity in 1916, and reached its first peak in the 1920s with the Schrödinger and Dirac equations. Then a second, postwar surge saw the development of equations describing the strong force and the electroweak force, culminating in the creation of the Standard Model of particle physics in about 1973. The equations trend continues today, with the ongoing struggle to create comprehensive equations to describe superstring theory. This effort — which aims to incorporate the force of gravity into physical models in a way that the Standard Model does not — marks the extant boundary of a long tradition.
Yet equations are not the only story. To an extent, geometrical representations of physical theories have also been useful when correctly applied. The most famous incorrect geometrical representation in physics is probably Johannes Kepler’s model of planetary orbits; initially, Kepler believed the orbits could be described by five regular polygons successively embedded within each other, but he abandoned this proposition when more accurate data became available.
A less well known but much more successful example of geometry applied to physics is Murray Gell-Mann’s “eightfold way”, which is a means of organizing subatomic particles. This organization has an underlying explanation using triangles with quarks located at the vertices.
For the past five years, I and a group of my colleagues (including Charles Doran, Michael Faux, Tristan Hubsch, Kevin Iga, Greg Landweber and others) have been following the geometric-physics path pioneered by Kepler and Gell-Mann. The geometric objects that interest us are not triangles or octagons, but more complicated figures known as “adinkras”, a name Faux suggested.
The word “adinkra” is of West African etymology, and it originally referred to visual symbols created by the Akan people of Ghana and the Gyamen of Côte d’Ivoire to represent concepts or aphorisms. However, the mathematical adinkras we study are really only linked to those African symbols by name. Even so, it must be acknowledged that, like their forebears, mathematical adinkras also represent concepts that are difficult to express in words. Most intriguingly, they may even contain hints of something more profound — including the idea that our universe could be a computer simulation, as in the Matrix films.
Imagination Is More Important Than Knowledge
by Krista Tippett, host
I interviewed James Gates once before, a few years ago, when we were creating our show on Einstein’s ethics. We talked then about Einstein’s little-remembered passion for racial equality. James Gates spent part of his childhood in segregated schools — experiences he does not take for granted now that he is a preeminent, African-American physicist. But what I was so taken by in that conversation years ago was how he explained Einstein’s social activism in terms of the values and virtues of scientific pursuit. He spoke of empathy as a potential byproduct of the process of discovery. A scientist’s “What if…” questions can evolve into human “What if…” questions.
James Gates’ capacity to share both from his humanity and his life in science strikes me again, and comes through even more forcefully during our more recent conversation in “Uncovering the Codes for Reality.” This time, I spoke with him about his particular passions. He is a string theorist, with a special emphasis on supersymmetry — a quality in the universe which, if demonstrated, might help support string theory as a way to reconcile the greatest puzzle modern physics has tried to solve since Einstein. Simply put, the universe seems to follow different rules at the highest and the smallest levels of reality. String theory imagines that deeper than atoms, deeper than electrons, behind quarks, all of reality is brought into being by filaments of energy. These “strings” might span the whole of reality, and possibly explain why gravity behaves so differently from varying vantage points. Some leading string theorists posit that there are at least eleven dimensions — far more than the three or four dimensions we are equipped to experience.
That is about how far I comprehend the idea behind string theory. The lovely thing about a conversation with James Gates is that my incomprehension does not matter. He gives me much to chew on, and be enriched by.
For starters, he is just the latest voice — others include the astrophysicist Mario Livio, and the astronomers Guy Consolmagno and George Coyne — to let me in to the secrets and power of science’s language of mathematics. He calls mathematics a kind of sixth sense — an organ of “extrasensory perception” — for scientists. By way of mathematics, scientists perceived and described the atom years before microscopes sophisticated enough to view them could be invented. Now, with mathematics, he and his colleagues are tracing clues and cosmic hints that may never be provable with our five senses — but that may shift our very sense of the nature of reality.
One of the things James Gates and some of his colleagues have “seen,” for example, are underlying codes embedded in the cosmos — error-correcting codes, like those that drive computer programs. (Full disclosure: he’s a fan of The Matrix — so am I — and we hear a little bit of that iconic movie in our one-hour podcast.) This is just one of many observations he makes that raises questions, he says, that physics alone can neither answer nor probe.
He is also working on an interesting frontier of expanding science’s own imagination about mathematical equations in describing reality. He and his colleagues have recently employed something called adinkras, visual symbols that may be able to unlock truths that equations alone cannot capture, just as there are truths that only poetry can convey.
There’s also a lot of fodder for one of my fascinations with the realm of science — the creative, playful, even spiritual act of naming things, especially in physics: beauty quarks and anti-beauty quarks, sizzling black holes, and superstrings, for example. The term adinkras, which comes from West Africa tradition and connotes pictures having hidden meaning, carries on this tradition.
James Gates’ own delight is infectious and illuminating, as much when he is letting us in on mysteries of the cosmos as when he shares the human lessons of his life in science. I’ll leave you with this, for example, as an enticement. When I asked him what he thought of Einstein’s statement that “imagination is more important than knowledge,” he said he had puzzled over this for many years:
"For a long time in my life, imagination was the world of play. It was reading about astronauts, and monsters, and traveling in galaxies, all of that kind of stuff, invaders from outer space on earth. That was all in the world of the imagination. On the other hand, reality is all about us. And it’s constraining, and it can be painful. But the knowledge we gain is critical for our species to survive.
So how could it be that play is more important than knowledge? It took me years to figure out an answer. And the answer turns out [to be] rather strange… Imagination is more important than knowledge because imagination turns out to be the vehicle by which we increase knowledge. And so, if you don’t have imagination, you’re not going to get more knowledgeable.”
Being Comfortable in the Presence of Mystery
by Krista Tippett, host
Mario Livio speaks with Brian Greene (photo: ©The Philoctetes Center for the Multidisciplinary Study of the Imagination/Flickr)
When I first picked up Mario Livio’s book Is God a Mathematician? I knew I wanted to speak with him. Given that title, it is perhaps surprising to learn that he is not himself a religious man. But in his science, he is working on frontiers of discovery where questions far outpace answers — exploring the nature of neutron stars, white dwarfs, dark energy, the search for intelligent life in other galaxies.
In vivid detail and with passionate articulation, he reinforces a sense that has come through in many of my conversations with scientists these past years. That is, in contrast to the nineteenth- and twentieth-century Western, cultural confidence that science was on the verge of explaining most everything, our cutting-edge, twenty-first-century discoveries are yielding ever more fantastic mysteries. The real science of the present, Mario Livio says, is far more interesting than science fiction could ever be.
For example, the fact that the universe is expanding rather than contracting is new knowledge. That has led to the discovery of what is called, for lack of precise understanding, “dark energy,” which is accelerating this expansion. This utterly unexplained substance is now thought to comprise something like 70 percent of the universe. Likewise, the Hubble telescope has helped humanity gain intricate new detail on the unimaginable vastness of the cosmos and the relative insignificance of the space we take up in it. At the same time — and this is one of Livio’s intriguing mysteries — this new knowledge and perspective also shine a new kind of light on the inordinate power of the human mind.
Livio’s question “Is God a mathematician?” is actually an ancient and unfolding question about the uncanny “omnipresence and omnipotent powers” of mathematics as experienced by science and philosophy across the ages. The question itself, as Livio says, is as rich to ponder as any of its possible answers. And so is the fact, behind it, that our minds give rise to mathematical principles, which are then found to have what physicist Eugene Wigner called “an unreasonable effectiveness” in describing the universe.
Livio also picks up on an intriguing theme left dangling in my lovely conversation in 2010 with the Vatican astronomers Guy Consolmagno and George Coyne — the enduring question of whether mathematical truths, laws of nature, are discovered or invented. Livio unapologetically offers his conclusion that there is no either/or answer possible here — that mathematics is both invented and discovered. That is to say, as he tells it, scientists habitually “invent” formulations and theories with no practical application, which generations or centuries later are found to describe fundamental aspects of reality. Even mathematical ideas that are at first invented yield real discoveries that are relevant, true, and wholly unexpected.
I was also interested to learn, as I went into this conversation, that when Mario Livio is not doing science he is a lover of art. “Beauty” is a word that recurs across my cumulative conversation with scientists, and Mario Livio infuses that word with his own evident passion. He is not quite sure, when I press, what that might have to do with his simultaneous passion for art. And yet there is something intriguing — mysterious even — about his description of how echoing allusions from science and art come to him effortlessly in his writing.
And in the backdrop of our conversation, images from the Hubble Space Telescope have brought a lavish beauty of the cosmos into ordinary modern eyes and imaginations. One senses that of all the accomplishments in which he has played a part, Mario Livio is most proud of this one. For him, science is a part of culture — like literature, like the arts. And he wants the rest of us, whether we speak his mother tongue of mathematics or not, to experience it that way too. This conversation brings me farther forward on this path.
I kept thinking, as I spoke with Mario Livio, of Einstein’s references to the reverence for beauty and open sense of wonder that Einstein saw as a common root experience of true science, true religion, and true art. His use of the word “God,” Mario Livio tells me, is similar to Einstein’s grasp for the word “God” as a synonym for the workings of the cosmos. I am struck once again with the capacity of modern scientists to be more comfortable with the presence of mystery, and bolder in articulating its reality than many who are traditionally religious.
The Fundamental Tension between Stories and Statistics
by Shubha Bala, associate producer
My favorite dog-earred, page-stained book growing up was The Phantom Tollbooth. I must have read over 40 times about Milo’s quest through the Kingdom of Wisdom to reconcile the rulers of Dictionopolis, the lover of words, and Digitopolis, the lover of numbers. The conclusion of this book, and of John Allen Paulos’ recent post in The New York Times, is that both language and math should reign equally.
Paulos, a mathematician and professor, argues that while narratives and statistics play important roles, people approach them both with different mindsets:
"Despite the naturalness of these notions, however, there is a tension between stories and statistics, and one under-appreciated contrast between them is simply the mindset with which we approach them. In listening to stories we tend to suspend disbelief in order to be entertained, whereas in evaluating statistics we generally have an opposite inclination to suspend belief in order not to be beguiled."
He goes on to demonstrate this tension by citing examples of statistical errors that are completely natural in storytelling, like the conjunction fallacy.
Journalism, to me, seems to be the attempt to reconcile that tension by finding common space between the data and the narratives. Do you think there is an inherent difference in how we mentally approach statistics and stories? Or is it a tension which can be bridged?
Image above: The dodecahedron, from the children’s book “The Phantom Tollbooth,” has 12 faces each showing a different emotion. (illustration by Jules Feiffer)
Mathematics in Sunflowers
Shubha Bala, associate producer
This week’s show with astrophysicist Mario Livio explores, amongst other things, how math is implicated in the nature of the world. The Nobel physicist and mathematician Eugene Wigner, who wrote "The Unreasonable Effectiveness of Mathematics in the Natural Science," argued that math is so successful in predicting events in physics that it could not be a coincidence. Even on our previous show, "Asteroids, Stars, and the Love of God," the astronomers pointed out the complexity in declaring whether math is discovered or invented.
While producing these interviews, I happened upon the video above. The visualization helped me by filling in some of the specific examples in nature that mathematicians can easily visualize on a daily basis. It shows how three mathematical concepts, including the golden ratio, translate into simple objects in nature.
What I really love is the about page, which deconstructs how the Fibonacci series and golden ratio translate into the spiral of a shell, and the spirals within a sunflower. When listening to Livio, what examples of math explaining the cosmos came to mind for you?