Tag Archives: Poetry

Kurt Gödel

As I peruse through biographies of the lives of philosophers, scientists, mathematicians, and other researchers, I find myself fascinated. I wonder how their hometowns, education backgrounds, and people they’ve met throughout their lives influenced their success in their work. In investigating what it means to be a genius and what it takes to produce amazing work, I still wonder about how people interacted with scientist Albert Einstein, mathematician Bertrand Russell, logician Kurt Gödel, and philosopher Ludwig Wittgenstein and what they thought of their greatness. With the distance and objectivity I have as I read about these famous minds of history, I appreciate the way history becomes more of a gradient of many events that give rise to settings, culture, and ideas themselves.

Einstein’s job as a patent clerk set the stage for his rumination of space and time. As he struggled to find a place in the world, he thought about the universe itself and what sort of theories would describe it. Russell bore wealthy apparel that represented his aristocratic lifestyle. It gave him an authoritative feel to his advice and commentary on society. As radios and newspapers covered his work, people revered him as a celebrity, especially how he emphasized scientists to engage with the public. Mathematician Alfred Whitehead and Russell’s Principia Mathematica laid the foundation for mathematics and computer science of the 1930s and beyond. Even Wittgenstein’s friendly demeanor earned him friends with other philosophers. Gödel made similar contributions to mathematics and philosophy. His story, though, was not marked with such popularity.

While Gödel studied at Princeton University alongside Einstein, he contrasted the physicist’s casual attire with his formal white jacket. He appeared distanced and detached, as though he were observing everything in the universe around him. Among Gödel’s contributions include his incompleteness theorems and an argument for God’s existence. The first incompleteness theorem let rule-based systems like those of Principia Mathematica prove arithmetic truths though they’ll always remain incomplete. The second incompleteness theorem prevent arithmetic theories from being proved consistent within those theories. At the time of their publication in the 1930s, scholars found it appealing there were truths they couldn’t prove. Gödel’s formalisms of this idea even appealed to scholars who searched for romantic beauty in their work. These incompleteness theorems have been referenced throughout mathematics, computer science, and logic and even in more distant disciplines such as aesthetics and neuroscience. This beauty of logic carried through Gödel’s discussions, lectures, and publications as other intellectuals admired his work. Even professor of cognitive science Douglas Hofstadter would write on the themes of mathematics and symmetry encompassing the theorems in Gödel, Escher, Bach. In the book, Hofstadter described the connections between music, art, and logic through a “strange loop.” Hofstadter argued this “strange loop” gives rise to consciousness, and, as the neurons of the brain respond to what the body perceives, they give rise to the entity itself that creates those perceptions about the world. This creates a “strange loop.”As he inspired discussions in philosophy, literature, science, and theology, the poet Hans Magnus Enzensberger wrote “Home to Gödel” and musician Hans Werner Henze’s “Second Violin Concerto was based of the setting of this poem. He might not have been the most popular celebrity among the general public like Einstein and Russell were, but Gödel surely had his fans among the intellectuals.

In this excerpt from “Home to Gödel,” Enzensberger describes the incompleteness conclusions and how fascinating Gödel’s thought process may be.

     In any sufficiently rich system
     including the present mire
     statements are possible
     which can neither be proved
     nor refuted within the system. 

     Those are the statements
     to grasp, and pull!

Gödel himself was nowhere near as outspoken as his fame might suggest he was. Born to a Luthern family in Moravia, he was shy and often nervous around others. As he identified as Austrian, he spoke German rather than Czech. He excelled in school and studied mathematics at the University of Vienna. There, he met with philosophers of the Vienna Circle, a group that supported logical positivism, a form of philosophy that used logic to address philosophical issues. They used this “scientific outlook” as a way to explain truths in science without using other forces such as mysticism. As they discussed what about mathematics truly existed in the universe, they formed their arguments and engaged in debates between each other. Gödel himself agreed with many of their views, but his belief in God and his belief in Platonism differentiated himself. This Platonistic view held that mathematical truths were discovered as though the mathematician were an explorer in a cave, rather the Constructionist view, which is that of an inventor building a machine. While a Platonist might believe mathematical objects, such as numbers and symbols, are real, Constructivists would be inclined to argue those objects are only works of fiction humans have created.

When Gödel received his doctorate in 1930, his dissertation showed first-order logic is complete. This meant every logical truth using first-order predicate logic could be proved in rule-based systems. It was after this he would perform work on the incompleteness theorems. Logician Jaakko Hintikka wrote Gödel’s announcement of the incompleteness conclusion at the Königsberg Conference on Epistemology of the Exact Scientists was the most important moment in 20th-century logic and possibly in all of logic. When Gödel announced it, however, his audience remained silent. Only Hungarian mathematician John von Neumann realized its important implications at the time. When Gödel published the proof, he lectured at Princeton University and rose to popularity. Later, theoretical physicist Roger Penrose argued that Gödel’s theorem shows that “Strong AI” doesn’t hold true. Human minds are not computers, and the artificial intelligence will never completely replicate them. Gödel himself didn’t believe his incompleteness conclusion proved Platonism true, but it did refute the anti-Platonist view that, in mathematics, truth and provability are the same thing.

Gödel would suffer from poor health physically mentally through his remaining work. When he married dancer Adele Nimbursky, he moved to Princeton University where he remained until his death. He applied for U.S. citizenship and, while examining the propositions of the constitution, discovered how the U.S. could legally turn into a dictatorship. Similarly, as Gödel studied the field equations of Einstein’s general theory of relativity, he imagined a world in which time could move backwards. His perfectionism lead him to only publish a few more papers that described his Platonist views, including “Russell’s Mathematical Logic” and What’s is Cantor’s continuum problem?” But the logician’s health would get the best of him. As he grew weak and disturbed by psychological effects, he grew a distrust towards everyone except his wife. While he made appointments with doctors and wouldn’t attend them and refused medicine, he was hospitalized after refusing to eat in the 1970s.  When he died in 1978, the certificate read that the cause of death was “malnutrition and inanition caused by personality disturbance.”

Other logicians such as Austrian philosopher Ludwig Wittgenstein engaged in these debates as well. When he met with Russell during a meeting of the Moral Science Club at Cambridge University in the 1940s, some say Wittgenstein threatened philosopher Karl Popper with a fireplace poker during a debate. Regardless, Wittgenstein’s peers described him with emotion ranging from hatred to adoration. Indeed, Wittgenstein sought to use philosophy to grant solutions to the “puzzles” of contemporary life, and his seminal work Tractatus Logico-Philosophicus (Latin for “Logico-Philosophical Treatise”) sought to answer these issues philosophy faced. Such an ambitious goal showed how Wittgenstein believed philosophy to represent the integrity and truth of life that everyone seeks to achieve. The work’s central thesis was that propositions, or statements of the world, were pictures of reality. According to philosopher Norman Malcolm, Wittgenstein found this idea when he came across a newspaper using a map to describe the location of an automobile crash. The map pictured reality the same way propositions do. Philosopher Georg H. Von Wright described the beauty of Tractatus in its simple, static sentences similar to one of Wittgenstein’s architectural projects. Wittgenstein himself built a mansion using utmost precision down to the smallest details in Vienna for one of his sisters and later built a sculpture in the studio of his friend sculptor Drobil. Wright described the perfection and elegance of the finished work contrasted Wittgenstein’s dynamic, searching personality.

Wittgenstein himself sought this intellectual comfort in all of his work. As he drew inspiration from Russian writer Leo Tolstoy and his book The Gospel in Brief, he found solace after witnessing a tremendous amount of suffering against the Russians during World War I. Wittgenstein’s cousin economist Friedrich Hayek once described him so engrossed in a detective novel he wouldn’t speak to him. When he did finish, though, he engaged in lively discussion of the Russians in Vienna. The experience had shaken Wittgenstein fundamentally as a person and left him disillusioned with some of his views of society. Still, Wittgenstein’s enthusiasm for intellectual discussion showed how he lectured for long periods of time as professor of philosophy at Cambridge University. He dressed informally in a way that students found approachable even in his longwinded explanations. He did this without getting tired and even cut into his student’s lunch breaks.

The challenges logicians and other philosophers face today may still leave us speechless, as they did during the Königsberg Conference. Still, the beauty of logic permeates through whatever theorem, equation, or idea it prove. The lives of these minds reveal the nuances and arguments that carried through the decades, and they’ll remain relevant through the future of mathematics, physics, and philosophy.

The aesthetic value of Umberto Boccioni’s “Dynamism of a Cyclist” shaped the Futurist movement. It was an admiration of science and technology. The constant movement of the perceived world reveals how our minds conceive such speed and motion.

Beauty is truth, truth beauty, — that is all 
Ye know on earth, and all ye need to know – John Keats, “Ode to a Grecian Urn”

To some, beauty may be only in the eye of the beholder. But if there were an objective basis for it, researchers have only begun to uncover what it is and what it means. To find a philosophical and scientific basis of art, we can study how our brains respond to aesthetics itself. The humanities and sciences seek different approaches. The humanities are speculative while science is empirical. But both can understand art. We may compare both methods to reveal how our brains process art, aesthetics, and, in some ways, morality. There remains debate among philosophers about Kant’s theories of aesthetics. We can reveal the connection of beauty with epistemology and ethics – from neuroscience, too. The potential for this area of research could hold benefits for art-based therapeutic treatments. It may also help determine art’s relationships to morality and justice and the neuroscientific basis for what it means to be human.

We need to describe the limits of the ways our bodies respond to works of art such as paintings or pieces of music. I cite beauty as one method of how art moves us on physical and intellectual levels as a way of determining this bodily response. This comparison has its limits because a bodily movement doesn’t guarantee one finds a work beautiful. Perceiving a work of art as beautiful comes from a harmony among our faculties of cognition. It includes our senses such as auditory and visual parts of the nervous system. Like my previous posts on symmetry and rhythm, this harmony is “purposive without purpose”, according to Kant. This means that the harmony of art has a purpose intended by the artist. But, as we experience the artwork, they should only affect us as if they had purposes even while we don’t find the purpose. This is what makes a work of art “beautiful” and explains how we perceive pleasure from it. 

Using beauty to show aesthetic value has its limits. If neuroaesthetics were to gauge beauty of artworks, it should use objective, universalizable methods. We must be careful to universalize our own subjective experience in a way that all people experience. Beauty itself can vary between cultures and groups of people. Art can move the individual without being beautiful, as argued by scientists Bevil R. Conway and Alexander Rehding. There are ways to avoid these issues, though. While the subjective experience itself varies, it has universalizable, objective methods of arguments. Two people may dispute that one work of painting is more beautiful than another. But their methods of deduction rely on objective, nuanced, justified arguments. This could be arguing that there is a certain balance of colors or that the rhythmic symmetry of painting in one is superior to that of another. It’s within these methods of reasoning that beauty emerges, not the subjective eye of the beholder. These methods of reasoning could help neuroaesthetics find a neuroscientific, empirical basis for art. It also explains how we draw moral and ethical judgements from works of art. We may have different movements. Some of us express more shock of pupils dilating at the sight of a sensationalist news headline. But our methods of reasoning provide a fundamental, shared way of communicating. Through this, we draw meaning from such an artistic move. Indeed, scholars have found success in these methods of inquiry. Neuroscientist V.S. Ramachandran and philosopher William Hirstein studied practical rules that connect features of artwork to beauty itself. They described “eight laws of artistic expression.” They include peak shift principle, which is the stronger response to stronger representations of desirable features. The scientists also described grouping, that is, discriminating a figure from the background. The other laws include problem solving, the way we favor the detection of objects through struggle. One may also use symmetry, or complementary objects revealing more information easily.

Kant went deeper in characterizing aesthetic judgments in such ways to address these problems. For Kant, deep aesthetic encounters are states of “disinterested interest.” This means that our judgements of beauty create pleasure, not the other way around. Many have attacked this claim. German philosopher Friedrich Nietzsche and psychologist Sigmund Freud believed art must relate to will. Political theorist Karl Marx believed all art must be political. The philosophical notion of expressionism views art as a subjective response to the world around us. Still, Kant’s ideas describe a method of how aesthetics is much more concerned with our judgements. We form these judgements in response to art alongside the bodily response we experience. As Kant theorized, art is argument, criticism, and persuasion. 

This makes the question of determining neural correlates all the more confusing. Neuroscience, as it seems this way, is far too limited in capturing art. In fact, it might be the case that art itself sets the foundation for understanding neuroscience, rather than the other way around. We describe the beauty and elegance of mathematics, physics, and physiology of the processes of the nervous system. From there, those features can be used to explain phenomena that emerge from them. 

Kant’s notion of the sublime provides a way of determining a connection between aesthetics and morality. Before the eighteenth century, philosophers claimed the sublime was beauty that left us awe-struck in such a way we couldn’t even measure it. Kant later wrote that sublime even goes beyond form and purpose in a way that we can speak of morality in its terms. The sublime power of Beethoven’s music could describe how it moves audiences to a type of ecstasy. This is beyond the comprehension of their senses. It incites feelings of wonder and romanticism that can only music itself can explain. The sublime also relates to expressions of fear, uncertainty, and terror in art. Kant grounded his moral theory in reason. He described a “moral culture” emerges when our moral faculties of reason create the limits of experiencing the sublime.

To find the neural correlates of art, we probe our senses. Particularly, understanding the sublime gives us greater meaning. I finish with a discussion of the sublime among philosophers Amir Aczel and Sandra Shapshay, English professors Paul Fry and Alan Richardson, and conductor James Judd. In the YouTube video The Sublime Experience, they discussed how the sublime through all forms of art and religion provide the grounds for differentiating between pleasure. The scholars debated notions of beauty, greatness, picturesque, infinity and other features of art. These ideas relate to our arguments in religion and philosophy. 

Zebra finches use a “critic” in the brain to differentiate between the rhythm of songs of other birds and, through this, learn songs.

Like the ebb and flow of the ocean, 

A rhythm emerges from the pen,

I capture it, imagine it,

before it disappears. 

Appropriate rhythm in writing means making sense of the relation between words and phrases. Stress, repetition, fluctuation, rhyme, meter, pattern, juxtaposition, and harmony all come together. These form the aesthetic and intellectual properties of rhythm. For a philosopher studying semantics or neuroscientist uncovering our nature, rhythm poses challenges. I’ve written on the subject with respect to symmetry. Let’s delve into rhythm’s secrets philosophically, mathematically, and scientifically.

Through much of my scientific writing, I pay close attention to how lengthy my sentences are. Too many long sentences at once can lose the reader in monotonous, cumbersome passages. I especially fall into this trap with description and exposition.  Long sentences bore the reader. Short, astute sentences can feel abrasive and clumsy. We alternate between the brevity of Twitter to the nuance of academic prose. it’s easy to succumb to habits and forget about the appropriate rhythm with which to write. Rhythm is both something we plan in advance and  re-evaluate through reflection and speculation.

In the realm of aesthetics, philosophers have debated the role of rhythm since the Classical era. In Book III of Plato’s The Republic, Socrates clarified that rhythm and meter are what separate poetry from pure prose. Pre-Medieval philosopher St. Augustine developed a theory of aesthetics based on ideas of rhythm in De Musica. In congruence with the theologian’s religious beliefs, God is the origin of rhythm. We discover these mathematical truths, pre-determined by God, of rhythm. It’s like how Plato believed humans collectively remembered them.

Emerson’s poem “Merlin” showed the use of rhyme and meter to create rhythm. Particularly the lines that moved back and forth between his own sensations and the way to craft meters of poetry from them showed this. Emerson in this section, showed how the rhyme fit so naturally that it seems like part of human prose. Socrates’ idea of rhyme and meter separate prose from poetry  lets Emerson use this distilled rhythm.

Thy trivial harp will never please

Or fill my craving ear;

Its chords should ring as blows the breeze,

Free, peremptory, clear.

No jingling serenader’s art,

Nor tinkle of piano strings,

Can make the wild blood start

In its mystic springs.

We teach ourselves rules and tips on creating great writing to take into account the effect of the rhythm on the piece. To imagine and care for these aspects of the reader gives the writing a property only observed at a scale larger than individual words. Rhythm comes from how words interact with each other, yet remains limited by our conventions of writing. It emerges when you take a step back from your computer screen and look at the whole picture. Rhythm is this frequency. Time itself limits these perceptions as we read. As such, it reveals deeper features of our subjective perceptions, such as the stress, intonation, and tempo of speech itself.  Yet we speak of it as something deeper than the combined intrinsic content of words themselves. In this sense, it’s like an emergent phenomena, much like evolution selecting certain genetic traits.

What makes these six clave patterns fundamental are that they reveal maximally even rhythms and maximum sum of pairwise distances between the points as vertices on a tetrahedron.

In a scientific context, we rely on our empirical observations of rhythm to determine higher truths of rhythm.  Canadian computer scientist Godfried Toussaint finds this connection between shape and musical rhythm. Due to the evenness of six musical clave patterns, they have mathematical significance. That a mathematical algorithm could generate music raises questions about what is music. It raises questions as what sort of mathematical or empirical technique governs what “good” rhythm sounds like.

Placing the six intervals in histogram form reveal patterns among themselves that may dictate the nature of rhythm as a whole.

We can find fundamental features among strings of numbers such that the patterns of these features give rise to Euclidean rhythms. These are rhythms created by Euclidean distance, or the straight line distance between the points when arranged in a circle. The numbers show the span between the beginning of successive notes and, thus, represent the simplest way to represent rhythm. This needs much more empirical evidence before showings its truth in all music. Toussaint’s claim that Euclidean rhythms that are reverse Euclidean strings appear to have a much wider appeal. French mathematician Jean-Paul Allouche showed these strings of numbers are like combinations of words. Other mathematicians and computer scientists have developed Euclidean rhythms from Euclidean strings. Toussaint argues that the Euclidean algorithm finds the greatest common divisor of two numbers. It can generate rhythm timelines by using the two numbers as an input to the Euclidean algorithm. The two numbers would dictate the beginning of each note in the rhythm and the span between notes. 

In the field of cognitive neuroscience, we can study the ways humans and other organisms produce and  test rhythms. Computational neuroscientists Kenji Doya and Terrence J. Sejnowski discovered a “critic” within the zebra finch brain. It lets the finch to differentiate between songs. NMDA receptors, a specific method of chemical signaling in nerve cells, activate to let bird to learn the “correct” song. Scientists Philipp Norton and Constance Scharff found patterns between the nerve and muscle cells. They corresponded with elements of notes and duration of the notes themselves. This is like Toussaint’s study of the beginning of each note determining rhythm. It includes the span between them, both fundamental components of rhythm. This research holds value for finding similar discoveries in the human basis of rhythm as well.

Much the same way a poet translates nature into word, a scientist would find quantifiable metrics of music. They may raise questions for musicology, geometry, and, with enough empirical evidence, neuroscience. Now take a deep breath and let the waves beat upon the seashore. Detect the rhythm and move along like before. 

Sources

Doya, Kenji & Terrence J. Sejnowski (1999). The New Cognitive Neurosciences. II. MIT Press. pp. 469–482.

G. T. Toussaint, “The Euclidean algorithm generates traditional musical rhythms“, Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science, Banff, Alberta, Canada, July 31 to August 3, 2005, pp. 47–56.

Norton, Philipp & Constance Scharff (2016). “Bird Song Metronomics”: Isochronous Organization of Zebra Finch Song Rhythm. Frontiers in Neuroscience. 

No! I won’t! I won’t write a poem! 

You can’t make me! Nor will I succumb to my desires. No, no, no…

I’m a researcher. That’s right. I seek knowledge and certainty. I seek soundness and completeness. I seek objective truths. 

For I see the world in black and white. Atop a ship in a sea of gray,

In absolutes, in truth and beauty I can describe the world.

Still, the mighty roar of the foggy ocean, surrounds me on all sides, 

through its cloudy mist light cannot penetrate. I fear what lies beneath the surface.

Einstein was the wisest man alive, as science gives us answers,

or Aristotle, a thinker like no other, with philosophy, more questions, 

I search for land, refuge from an infinite sea. I won’t read Coleridge or Whitman or Thoreau. 

I’ll remain willfully blind to what can’t be described or learned.

I choose not to forsake my judgement in rhetoric and logic,

lest I should become overpowered by my desires within.

I won’t. I’ll lock the chest and throw away the key.

You won’t get a poem out of me.