Program Particulars: Math, Purpose, Truth

Program Particulars

*Times indicated refer to web version of audio

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(01:52–03:56) Music Element

"The Multiples of One" from Awakening, performed by Joseph Curiale


» Enlarge the image Kurt Gödel and Albert Einstein regularly walked together in Princeton, New Jersey while working at the Institute for Advanced Study.

Kurt Gödel and Albert Einstein regularly walked together in Princeton, New Jersey while working at the Institute for Advanced Study.

(02:11) Kurt Gödel

Kurt Gödel (1906–1978) — one of TIME Magazine's 100 most important people of the 20th century — was an Austrian mathematician and logician who is best known for his "incompleteness theorems." In 1931, he published a scientific paper describing mathematical theorems that explored limitations in set theory, a core branch of mathematics that describes how objects can be grouped together in "sets" that are defined by rules known as "axioms." One example of a set is a group including all whole numbers, "natural numbers" such as 1, 2, 3, and so on. Gödel formally proved that mathematical axioms capable of describing such a set could indeed be highly accurate and internally consistent, yet fail to predict every possible number in the set.

To better understand this idea, an analogy can be drawn between the "incompleteness theorems" and a philosophical problem called the liar paradox, or Epimenides' paradox. In Levin's A Madman Dreams of Turing Machines, mathematicians and philosophers of 1930s Vienna reflect on this paradox, wondering if they can ever believe someone who says, "I am lying." That statement, by itself, is grammatically correct, but logically incoherent.

The "incompleteness theorems" explored a similar paradox in mathematical set theory. In the same way that language can be consistent but incoherent, the implications of Gödel's theorems were that mathematical theory itself could never fully encompass the totality of abstract mathematical truth. The idea shocked a generation of thinkers that believed an overarching mathematical formalism did exist and could describe all mathematical truth.

Kurt Gödel shared these ideas with fellow mathematicians and philosophers in Vienna, many of whom were of Jewish ancestry. As a result, Gödel had difficulty maintaining his academic tenure in the anti-Semitic climate of newly formed Greater Germany. He left for the US in 1940, settling in Princeton, New Jersey. At Princeton's Institute for Advanced Study, he became lifelong friends with Albert Einstein. Later in life, though, Gödel succumbed to paranoia. Refusing to eat for fear of being poisoned, he starved himself. He was admitted to a Princeton hospital in a severely malnourished state, and died there at the age of 72.

» Enlarge the image Alan Turing stands beside the Ferranti Mark 1 Console, the world's first commercially available general-purpose computer.

Alan Turing stands beside the Ferranti Mark 1 Console, the world's first commercially available general-purpose computer.

(02:12) Alan Turing

Alan Turing (1912-1954) — also one of TIME Magazine's 100 most important people of the 20th century — was a British mathematician and code-breaker who worked, along with others, to decipher coded German army transmissions during World War II. Their efforts helped the British anticipate and counter German attacks on supply ships headed for British shores.

Turing was recruited for this task after writing a 1937 paper that imagined a machine capable of computing numbers based on instructions it was fed by a human operator. This imaginary machine, later nicknamed the Turing machine, could essentially be made to think the way a human could. This idea helped lay the groundwork for the modern computer, though many other scientists contributed significantly to its development as well.

Turing continued his research during the post-war period, but died before the age of mass computerization. In 1952, he admitted to a homosexual relationship with an acquaintance. Turing was arrested and charged with indecent behavior, then subject to chemical castration. The ensuing hormone treatment radically altered his body, causing obesity, impotence, and the development of feminine secondary sexual characteristics, including breasts. In the wake of these transformations, Alan Turing succumbed to depression and committed suicide at the age of 41 by eating a poisoned apple.

(03:29) Cosmology

Cosmology, particularly in reference to physics and astronomy, is the mathematical study of the nature of the universe, its origins, formation, size, and behavior. The Big Bang Theory is perhaps the best known expression of the work of cosmologists.

(06:28) Quantum Mechanics

Quantum mechanics is a core branch of physics that studies the behavior of energy and matter on the subatomic scale. There, behavior of matter is entirely different and often unpredictable compared to the behavior of matter as we experience it. A handy introduction to quantum mechanics and its relevance to our understanding of reality can be found in the PBS series The Elegant Universe.

The elements of randomness and unpredictability in the field of quantum mechanics bothered Einstein, who viewed the universe as following strict rules. The universe does follow the rules Einstein described, but not at the subatomic level of quantum mechanics. Einstein responded to the randomness of quantum mechanics by proclaiming, "God does not play dice with the universe." Niels Bohr, another famous physicist and contemporary of Einstein's, replied, "Who is Einstein to tell the Lord what to do?"

(06:49) The Vienna Circle, and Wittgenstein's Tractatus

The German philosopher Moritz Schlick (1882-1936) founded the Vienna Circle (Der Wiener Kreis_, a small group of intellectuals who convened between 1929 and 1938. Other members included Kurt Gödel, sociologist and philosopher Otto Neurath, and mathematician Olga Hahn-Neurath.

The group advanced the ideas of logical positivism, believing that one could only verify factual knowledge through direct observation. At its core, logical positivism says that a statement can only be true if there is a finite set of steps that can verify its truth — God, numbers, and many other entities were eliminated because they could not be proven in a finite number of steps.

Ludwig Wittgenstein's Tractatus Logico-Philosophicus was a foundational book for the Vienna Circle. The German philosopher's work is composed of short declarations such as "The world is all that is the case" and "What we cannot speak about we must pass over in silence." These black-and-white pronouncements generated fervent discussion in the Vienna Circle, a group that Wittgenstein himself was not aware of until he met Moritz Schlick.

In the years leading up to the Second World War, many members of the Vienna Circle left Austria and Germany for Britain and the United States. Some, like Moritz Schlick, remained and continued their lives as teachers, academics, scientists, and researchers. Schlick taught at the University of Vienna, where he was murdered in 1936 by an anti-Semitic student who mistakenly believed him to be Jewish.

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(07:25–07:55) Music Element

"Chaconne In G Major" from Moondog (The Viking of Sixth Avenue), performed by Moondog


(07:25) Reading from Levin's Book

The extended passage of the reading in the program is excerpted from Janna Levin's book, A Madman Dreams of Turing Machines:

On every previous Thursday, Kurt has been a silent spectator. Tonight he looks from one person to another as he waits for the right opportunity. His temperature fluctuates while openings come and go until he throws out a question he knows they have asked themselves a thousand times. "How do you recognize a fact of the world?" Moritz laughs, but not rudely, and nods, which loosens his hair only marginally from its proper place before he stops himself, slightly sorry for his reaction as he takes in Kurt's serious expression. "It is a fair question," he confesses. "How do I verify a fact of the world?" Such a simple question. He cannot even answer this simple question. Despite his proclamations, Moritz knows something is wrong. No matter how disciplined he is in his adherence to logic, he cannot make sense of a method to verify fact of the world. He comes to ever narrowing definitions that magically take him farther from clarity. Spirals of rational thinking thread him closer to understanding only to unravel disappointingly far afield. As Moritz reaches for his coffee, his motion is very slow, and it seems so even to him, the perspective telescopic. His fingers surround the cup and feel the heat. He pulls the cup to his face as sees the dark liquid. "How do I know this cup exists? I don't," he admits to himself. "I don't." Being honest he can be sure only he sees. He can be sure only he touches. He watches Olga pull on a mammoth cigar. She has a calm about her, always at ease. The smoke drifts in curly plumes sifting through her lashes. She doesn't seem to mind and even tends to hold the burning cinder vertically and uncomfortably close to her eyes. Her hair is collected loosely at the nape of her neck, a rumpled frame for her big and broad broken eyes. She cannot see the cigar. What does it mean for her to say there is a cigar between her fingers? The meaning of the statement is that she feels the tobacco-stuffed wrapping. She tastes the juice and smoke. She senses it. Does it exist. That is not a meaningful question. But what really arrests Moritz, what keeps his fingers in a frozen clutch around the cup, coffee suspended near his chin, is this question: Does Olga exist? He hangs there for what seems like a very long while. The conversation stalls, suspended along with the coffee. "Olga?" "Yes, Moritz. I'm here." She reaches over and hooks his thumb with her forefinger. The rest of her fingers scramble over to clasp his hand. But all Moritz concedes is that he can feel what he has learned to describe as pressure on what he believes to be his hand.

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(09:21–09:51) Music Element

"Chaconne In G Major" from Moondog (The Viking of Sixth Avenue), performed by Moondog


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(14:54–16:12) Music Element

"Seraphic Light" from Stellar Regions, performed by John Coltrane


"On Truth and Beauty"
The science magazine Seed held a salon with acclaimed fiction writer Jonathan Lethem and physicist and novelist Janna Levin in March 2007. They discuss the importance of truth in their art and the impurity of metaphor — and therein lies elegance and beauty.

(16:01) Second Reading from Levin's Book

The extended passage of the reading in the program is excerpted from Janna Levin's book, A Madman Dreams of Turing Machines:

I have tried to stay out of these stories but I am here too. I am standing on a street in a city. On my way to a train. I can see someone standing in a window several stories high in an ornate building overlooking the park. If I were up in that building, I'd look out of the window too and wonder who was down on the ground and how we got here and why and I would admit what I have already admitted a thousand times: We are not much. We are confused and brilliant and stupid, lost clumps of living ash. On this street at this hour is an old woman in a huge coat. I overtake her easily as she makes her way down the avenue. With the old fashioned streetlights and the backdrop of the park, it could be another decade altogether. She reminds me of Olga if she had grown so old, but Olga died in the Netherlands at the age of fifty-five after a kidney operation. A woman and her dog walk toward me and move to pass on either side. She lowers the dog's leash so that I can easily step over and for a fraction of a second while my foot is suspended in the air, I'm a little girl about to skip rope and she's my friend swinging the jump rope handle. Then she and the dog are gone, along with the sensation. In the park, over the low wall, there are two girls playing in the grass. Giants looming over their toys, monstrously out of proportion. They're holding hands and spinning, leaning farther and farther back until their fingers rope together, chubby flesh and bone enmeshed. What do I see? Angular momentum around their center. A principle of physics in their motion. A girlish memory of grass-stained knees. I keep walking and recede from the girls' easy confidence in the world's mechanisms. I believe they exist, even if my knowledge of them can only be imperfect, a crude sketch of their billions of vibrating atoms. I believe this to be true. The girls are not a product of my imagination and I am not a product of yours. There are just some things we can't strictly prove. So let's take it to be true, this is me. I am on an orbit through the universe that crosses the paths of some girls, a teenager, a dog, an old woman. Maybe I should list more significant events that shaped my relationship to truth but that story would be a lie too. I could have written this book entirely differently, but then again maybe this book is the only way it could be, and these are the only choices I could have made. This is me, an unreal composite, maybe part liar, maybe not free.

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(17:25–17:55) Music Element

"Seraphic Light" from Stellar Regions, performed by John Coltrane


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(22:35–24:20) Music Element

"The Secret Fluid of Dusk" from The Sad Machinery of Spring, performed by Tin Hat


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(28:42–30:12) Music Element

"Focus on Sanity" from The Shape of Jazz to Come, performed by Ornette Coleman


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(30:44–31:03) Music Element

"The Land of Perpetual Sleep" from The Sad Machinery of Spring, performed by Tin Hat


(31:53) Extra Dimensions, Multi-Dimensional Spaces

In attempting to reconcile different theories of physics — primarily the probabilistic nature of quantum mechanics and the deterministic nature of Einstein's general relativity — physicists have searched for a so-called "theory of everything." This Holy Grail of physics would describe all physically observable phenomenon in one elegant mathematical model.

Theoretical physics today have a number of models that are close to unifying the two different views of the universe. Among these theories is superstring theory, in which the fundamental building blocks of the universe are not particles, but tiny loops of energy known as strings.

The mathematics of superstring theory is exceedingly complex, even for today's most gifted physicists. The mathematical gymnastics behind superstring theory have lead scientists to imagine that the universe is composed of more than the three physical dimensions of height, width, and depth, and the dimension of time. Instead, competing versions of superstring theory claim that the universe is composed of between 10 and 26 dimensions. Physicist Brian Greene's The Elegant Universe explores the implications of superstring theory and describes how these extra dimensions would function.

(32:20) Copernicus

Polish astronomer Nicolaus Copernicus (1473-1543) was the first astronomer to mathematically prove that the Earth revolved around the Sun. His 1543 work On the Revolutions of the Heavenly Spheres caused mild controversy in its day. However, the later support of Copernicus' work by Italian astronomer Galileo Galilei (1564-1642), in defiance of Christian doctrine, publicized the theory and led to Galileo's lifelong imprisonment and house arrest under the Roman Inquisition.

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(38:35–39:18) Music Element

"Lonely Woman" from The Shape of Jazz to Come, performed by Ornette Coleman


(39:06) Third Reading from Levin's Book

The extended passage of the reading in the program is excerpted from Janna Levin's book, A Madman Dreams of Turing Machines:

I approach a man and a woman on a park bench. His glasses, her handbag, her umbrella. They're as still as stone. They're in front of me and then, within a few strides, I pass the green-painted bench with its concrete feet. I turn to watch them but they are swept along by time, unable to stop, dragged away along with the street and the wind. I see them everywhere, my two mad treasures. An elegant man in a hat with an able, stocky woman at his side. A solitary black-haired boy with a peculiar stride. 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. They are here in our minds, Turing's luminescent gems, Gödel's Platonic forms. There are no social hierarchies to scale. No racial barriers. Given to us along with our brains. Built into the structure of our thoughts — no bullying into blind faith, no threats of eternal damnation — just honesty, truth, and reason. I am here in the middle of an unfinished story. I used to believe that one day I would come to some kind of conclusion, some calming resolution, and the restlessness would end. But that will never happen. Even now, I am moving toward a train. My heart is thumping. My lungs are working. There is a man, a woman, a bench, the glasses, the smooth hair, an umbrella. We are all caught in the stream of a complicated legacy — a proof of the limits of human reason, a proof of our boundlessness. A declaration that we were down here on this crowded, lonely planet, a declaration that we mattered, we living clumps of ash, that each of us was once somebody, that we strove for what we could never have, that we could admit as much. That was us — funny and lousy and great all at once.

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(41:12–41:43) Music Element

"Lonely Woman" from The Shape of Jazz to Come, performed by Ornette Coleman


Finding Rings of Dark Matter
NASA and the European Space Agency observed dark matter in the constellation Pisces using the Hubble Space Telescope. An astronomer explains dark matter, what they saw, and how they found it.

(41:50) Dark Matter

Most scientists believe that most of the universe is made up of dark matter and dark energy, an unknown substance that neither reflects nor emits light. A theory of dark matter was suggested in the 1930s as a way to account for gravitational forces bending light without observable matter such as stars or galaxies. In early 2007, computational models predicting dark matter were realized after the Hubble Space Telescope recorded the collapsing of dark matter in the Pisces constellation.

Trying to account for this "empty" space in a static universe (the prevailing idea of the day), Albert Einstein created a cosmological constant to gel with his theory of general relativity. As Steven Weinberg writes in Physics Today (pdf):

"In thinking of Einstein's mistakes, one immediately recalls what Einstein (in a conversation with George Gamow) called the biggest blunder he had made in his life: the introduction of the cosmological constant. After Einstein had completed the formulation of his theory of space, time, and gravitation—the general theory of relativity—he turned in 1917 to a consideration of the spacetime structure of the whole universe. He then encountered a problem. Einstein was assuming that, when suitably averaged over many stars, the universe is uniform and essentially static, but the equations of general relativity did not seem to allow a time-independent solution for a universe with a uniform distribution of matter. So Einstein modified his equations, by including a new term involving a quantity that he called the cosmological constant. Then it was discovered that the universe is not static, but expanding. Einstein came to regret that he had needlessly mutilated his original theory. It may also have bothered him that he had missed predicting the expansion of the universe."

But Einstein may have been too hasty in his dismissal of his calculation. Some scientists think that his cosmological constant might be the best explanation for the constant expansion of the universe.

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(45:30–45:58) Music Element

"Configurations" from Stellar Regions, performed by John Coltrane


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(47:33–48:05) Music Element

"Viking 1" from Moondog (The Viking of Sixth Avenue), performed by Moondog


(47:51) Fourth Reading from Levin's Book

The extended passage of the reading in the program is excerpted from Janna Levin's book, A Madman Dreams of Turing Machines:

Here I am, in New York City. It is the twenty-first century. This is as good a place, this time as good a time, as any. I am stepping off the curb. The subway entrance is just across the street. Big green orbs signify that the downtown entrance is open. Artificial light competing against the sun. There are children in the park; a woman with silver hair and a long, old coat; a couple on a bench. A yellow plague of taxis infests the streets. I walk behind some and in front of others. We all move along fixed trajectories, following our prescribed arcs. The plague disappears behind the stone wall as the old steps take me down into the subway. I am under the city. In this stink of an underground, miscellaneous liquids pour from the streets off the grating and onto the tracks. A train comes. The car is empty. I can see my eye in the window of the C train. A big black eye. I'm looking our and outside looking in. I stare into the opaque disk at a biological machine. There is no ending. I've tried to invent one but it was a lie and I don't want to be a liar. This story will end where it began, in the middle. A triangle or a circle. A closed loop with three points. A wayfaring chronicle searching for a treasure buried in the woods, on the streets, in books, on empty trains. Craving an amulet, a jewel, a reason, a purpose, a truth. I can almost see it on the periphery, just where they said it would be, glistening at me from the far edges of every angle I search.

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(49:09-50:15) Music Element

"Viking 1" from Moondog (The Viking of Sixth Avenue), performed by Moondog


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(50:13–50:15) Music Element

"The Secret Fluid of Dusk" from The Sad Machinery of Spring, performed by Tin Hat


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is an astrophysicist and writer. She has contributed to an understanding of black holes, the cosmology of extra dimensions, and gravitational waves in the shape of spacetime. She is the author of A Madman Dreams of Turing Machines, which won the PEN/Bingham prize.

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