This shit blew my motherfucking mind. Yes, math that makes you curse like a sailor. I absolutely hated it at first and insisted that it was completelyThis shit blew my motherfucking mind. Yes, math that makes you curse like a sailor. I absolutely hated it at first and insisted that it was completely stupid and made absolutely no sense. "How can parallel lines continue to get closer to each other for infinity?? GODDAMMIT!!?!?" And this was in a classroom with people much calmer and more mathematically inclined than myself.
One day while venting about how little sense this kind of math made to me, someone somehow calmed me down by basically explaining that...well, actually now I can't even remember what they said, but somehow it soothed my confusion and totally indignant outrage.
I still don't understand things like this. Higher level mathematics (and by extension much of particle physics, namely quantum mechanics) is often so outrageously counter-intuitive that one has to wonder if those for whom these topics are bread and butter are just spinning an epic and elaborate joke. Part of me somewhat likes to imagine that behind closed doors it's all cackles and guffaws at mathematics and physics conferences.
I still don't understand Lobachevsky's wild mathematical models, but I now more or less dumbly marvel at them rather than get frothy with incredulous rage.
P.S. Cantor's ideas about multiple infinities also leave my jaw agape and the basic logic modules of my brain inflamed....more
I began writing a short story about Alan Turing last year. Despite a lengthy scribbled outline it remains a stunted opening gambit. After reading JannI began writing a short story about Alan Turing last year. Despite a lengthy scribbled outline it remains a stunted opening gambit. After reading Janna Levin's A Madman Dreams of Turing Machines I really feel far less of a need to finish what I started, because she basically captured what I'd kept confined in my head, off the page. I still might finish it one day, but after reading David Leavitt's beautiful Turing biography (The Man Who Knew Too Much: Alan Turing and the Invention of the Computer) and this incredible historical fiction of Levin's I feel like they've jointly completed what I wanted to see carried out: a sensitive, detailed, intellectually astute and "literary" portrait of this far too underappreciated genius and his tragic decline.
This is the historically-informed story of two 20th century intellectual giants, Alan Turing and Kurt Gödel. Other real life figures make supporting appearances such as Wittgenstein and Otto Neurath. There are also very brief and well-placed metafictional entries and minute allusions that bring the author into the fold in a narrator-as-character manner, as can be seen in the very (non-)beginning of the book:
"There is no beginning. 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.
At one apex is a paranoid lunatic, at another is a lonesome outcast: Kurt Gödel, the greatest logician of many centuries; and Alan Turing, the brilliant code breaker and mathematician. Their genius is a testament to our worth, an antidote to insignificance; and their bounteous flaws are luckless but seemingly natural complements, as though greatness can be doled out only with an equal measure of weakness."
The connection between mental illness and artistic and intellectual greatness is a long established cliché by this point and is probably far too often overstated via confirmation bias. There’s a fantastic documentary called Dangerous Knowledge which focuses on four mathematicians and/or scientists who all grappled with hugely complex and difficult issues like the nature of the deepest structures of reality, infinity, human consciousness, free will v. determinism, etc, and all ended up killing themselves. Turing and Gödel are two of the four. There’s an implication that it was their theories and obsessive intellectual aspirations that drove them to commit suicide, which I think is a rather flawed notion considering the facts and other plausible explanations. However, it does make for compelling narrative to peer into the lives of tortured geniuses consumed by their own big brains or whatever, and is an excellent sounding board for thinking about the pursuit of knowledge and its various costs and benefits. In any case, these are fascinating stories, and Turing’s in particular I find the most captivating and tragic.
Alan Turing's influence is felt hugely in the realm of computer science, cryptology, Artificial Intelligence and mathematical logic more generally. He’s often credited as one of the single most important influences on the development of the modern computer—without Turing we may not be having this exchange of information right now. He also played a hugely instrumental role in cracking the German Naval Enigma Code in WWII with his tireless cryptology work and innovations in the field which allowed for a far more rapid decoding of the German transmissions that were quite literally matters of life or death. After the war he was arrested for admitting to having homosexual relationships to the police after he reported being burgled by a casual fling—arrested and prosecuted by the very same government he’d served and protected. Instead of going to prison he was chemically castrated. The regimen of huge doses of estrogen caused him to gain weight and grow breasts, fall into a chemical depression, and ultimately end his life by eating a cyanide-glazed apple, mimicking one of his favorite films, Snow White. Turing was persecuted to death. The British government has the blood of a genius (who saved them from further Axis-led destruction) on their hands. Only as recently as 2009 has the British government issued an official apology for this incident that occurred in 1952 (however the same government has rejected the proposal to posthumously pardon Turing of his "crimes"). It took the Roman Catholic Church 359 years to finally officially apologize for persecuting Galileo for positing that the Earth revolves around the sun, so perhaps this is the sign of a kind of progress, but it still all feels far too little, far too late.
Kurt Gödel was a mathematician and logician (the distinction between the two starts to break down at a certain point) who famously constructed his Incompleteness Theorems, which I still have trouble explaining, because I’m a dummy when it comes to mathematics and formal logic (Gödel's Incompleteness Theorem For Dummies). I am truly ignorant about mathematical matters, but I can appreciate from my perch of acknowledged ignorance the allure of "the sanctity and purity of mathematics, the profound truth so completely immune to human stains." Gödel was also an absolute loon. He held a deep paranoid fear of being poisoned and as such rarely ate anything and only enough to keep his skeletal frame alive, and had elaborate rituals involving his wife’s cooking. He was also a member of the famed Vienna Circle, a group of intellectuals who met weekly to discuss the tenets of their new unifying idealistic philosophy of Logical Positivism. Gödel, along with Wittgenstein, each in their own ways, aided in dismantling this group with their unorthodox ideas. There are great sections in this book where Gödel's proposed notions of “Incompleteness” cause a great uproar amongst those seeking complete unifying theories to knit all of reality together. Gödel lived much longer than Turing, but ultimately died by starving himself to death both out of his paranoid notions of being poisoned and for other sad and errant reasons: there’s a passage in the book where Gödel delusionally claims that his refusal to eat is a proof of his free will, something he desperately wanted to believe in, along with the existence of an afterlife.
Turing and Gödel never met, but they were certainly aware of each other’s work and so the only way they collide in the book is in mentioning one another’s ideas.
Janna Levin is a physicist with a concentration in philosophy (her primary professional focus is cosmology but she had formal focus on philosophy as well, and I think it shows) yet on the stylistic level her writing is fantastic and surely shames huge numbers of authors who’ve workshopped their way through MFAs and maybe even published for years and years, while narrowly focused on literary fiction and nothing else. Janna Levin churns out steadily captivating prose that soars richly and exultantly without succumbing to a plummeted decadence; regularly supplanting ho-hum descriptions with a strikingly vivid lyricism through the conjuring of unusual imago-sensory crossbreeds that dance across the neural pathways with pleasantly assured aplomb.
The book is thoroughly researched as the notes provided at the back of the book further prove. It’s intellectually dexterous in its portrayals of these brilliant and flawed figures. The subjects (and the sort of human beings most tightly latched upon them) that are classically conceived of as cold and cerebral and arrogantly cocksure are sensitively imbued with the squirming life and heat of fallibility, frailty, confusion, and the portrayal of the true scientific spirit, where truth is provisional, and self-doubt and self-interrogation are constant companions.
While this is a book of heady ideas, it’s also a humanizing ode. The sections on Turing especially tugged the heartstrings. He was an odd but deeply sympathetic person. There are gripping descriptions of London being bombed by the German Luftwaffe, of Alan’s loneliness and tragic loss of his one true love as a schoolboy, and multiple gorgeous sections about the interconnectivity of things that just need to be read to be felt.
Both Turing and Gödel chased after the Truth with great fervor accompanied by great doubt. This classic yearning for the Truth of All Truths is maybe something many can easily set aside as not worth wasting time over when there is a more pressing desire for the Pursuit of Happiness on offer. I myself have often done this and will continue to do it. Hitting a wall where I no longer hunger for deep abstract truths about the nature of consciousness or reality or death, etc. But the desire never fully cools either. Also, even if one doesn't care at all about such cliché or high-minded foolishness, everyone knows what it’s like to yearn strongly for something Ideal, be it Romantic Love or the Perfect Career or the Perfect Artistic Creation and so on. As Olga Neurath says to Gödel about her and her husband finally accepting his Incompleteness Theorem:
"Your incompleteness theorem was hard for him to accept. It was hard for all of us, for every mathematician alive. But then Moritz always knew that it did not matter what he believed. What matters is the truth. And somehow you found it hidden where none of us could see. We all came to realize that mathematics is still flawless—no paradoxes, contradictions—just some truths that cannot be proven. Not so bad. We can live with that. He could live with that. [...] I myself worried from the start. Kurt, you worried us. It was hard for us for a time, to be sure. If not even arithmetic is complete, then what could we hope for from our philosophies, from our sciences, from the very things that were to be our salvation? The buoys that we clung—perhaps, I would admit now, with too much desperation—were taken away. [...] And here we are again with our hopes being crushed. I used to believe that when I was older I would come to some kind of conclusion, some calming resolution, and then the restlessness would end. I would know something definitive and questions would fade. But that will never happen. [...] We wanted to construct complete worldviews, complete and consistent theories and philosophies, perfect solutions where everything could find its place. But we cannot. The girls I hear playing in the park when I walk to the institute, our neighbor the old woman who will die soon, our own circle, we all prize a resolution, a gratifying ending, completeness and unity, but we are surrounded by incompleteness.
So I think that reading about the pursuit of Truth can still be moving and redemptive and nourishing for those who do not currently or never have really put much value on it. And then the journey becomes more valuable than the destination, as the ol' cliché reminds us....more
Alan is five years old and taking a bite out of an apple for the first time. Human life is rich with such firsts, as we well know and make known withAlan is five years old and taking a bite out of an apple for the first time. Human life is rich with such firsts, as we well know and make known with our various rituals and markings, preservations and engravings. First tooth. First step. First word. First day of school. First kiss. But many firsts go uncelebrated, unmarked, fail to be photographed or scrapbooked, and countless sums pass by human sensors unknown, even to those who personally bear them. No one—neither parents nor Alan or otherwise—could’ve realized the somber significance of this as he happily tore the meat of the commonly blossomed fruit away from its seed-laden axis, his jaw working the tart mouthful into a swallowable sweetness and repeated in between the beaming smiles and clear dancing eyes of satisfaction.
The verdant freshness of childhood innocence harmoniously converges with the surrounding organic hustle and bustle of firsts smattering the flora and fauna. The bite’s unknowable meaning radiates silently amid the pastoral scene of the Turing family picnic. The not all too frequently unclouded light of the Scottish sun warmly contributes to the gorgeous weather and blankets the feelings of time-halting serenity and familial love that mingle and gently swirl about this moment, in this clime, in this fraction of a fraction of the world.
Palpable glimmers of Alan’s remarkable intelligence perched upon the early signposts. His mind grasped the landscape of the idyllic family picnic as not merely a series of pleasant impressions bleeding into one another and lapping at his mind-as-center-of-it-all, but as composed of distinct pathways and trajectories, not exactly upon an actual visualized grid—as might some cartoonish version of a mathematics genius as a child—but as things that adhere to deeper principles of space and movement.
Urged by an offhand remark about how nice it would be to have some honey with their biscuits, he traced the flight patterns of the nearby bees and intuitively calculated them into a hunch that, if followed, could bring their sweet secretions into the already plentiful spread that his family’d been enjoying. Within minutes he’d scampered to the central hub of the hive, exhilarated not as much by the possibility of snatching up a syrupy comb or two, but by the series of gratifying clicks within the mind of being able to anticipate and accurately predict the workings of the world—a complex feeling of exerting power and mastery, tempered by the simultaneously humbling sense that his own workings as a person could likewise be anticipated, predicted and uncovered, and as such fall into place with their own satisfying clicks. He both looked out upon the world and felt himself to be its kin, all of course in ways that a child, and most people beyond childhood, could not articulate.
Alan had a conductor car toy that he rolled around so regularly that, within a year since the Christmas he’d received it, the model vessel and even the miniature conductor himself had worn away significantly—human hands and entropy. When a wheel had fallen off, his impulse was to dig a shallow grave for the toy in the backyard. His parents discovered this and thought that he’d tried to hide the evidence of misusing their gift, but in fact his unusual manner of thinking had given him the idea that a proper burial would allow for some sort of magical rejuvenation of the object, causing it to rise from the soil, Phoenixlike, reborn anew. In general, children are prone to magical thinking, yes, but his particular form of it was unique—a vision of causality quite fenced off from others—a self-contained logic churning within the young boy’s mind that was perhaps a sign of revolutionary leaps to come. ...more
This was an extremely difficult book for me. I understood so little of it on my own but luckily was reading it in school where I had people around meThis was an extremely difficult book for me. I understood so little of it on my own but luckily was reading it in school where I had people around me who were more mathematically fluent than myself to explain what the hell was going on. It was fascinating and frustrating and the basic ideas I gleaned from it were worth the headaches...more