When Einstein Walked with Godel

Sharing my learnings from the book, When Einstein Walked with Godel by Jim Holt

When Einstein Walked with Godel by Jim Holt

Does time exist? What is infinity? Why do mirrors reverse left and right but not up and down? In this scintillating collection, Holt explores the human mind, the cosmos, and the thinkers who’ve tried to encompass the latter with the former. With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth. 

Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot. Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U.S. Constitution contained a terrible contradiction—and whether the universe truly has a future.

• In 1905, while still holding down a day job at a patent office in Switzerland, Einstein published four short papers that forever changed how we understand the world
  • he showed that light comes in discrete particles, later dubbed photons.
  • he finally proved that atoms were real and showed how to calculate the seemingly random way they move around in gas or liquids.
  • he introduced his theory of relativity, completely upending our notions about space and time.
  • he coined his famous formula E=mc2, illuminating the relationship between mass and energy. 
• Einstein regarded the implications of quantum theory as too “spooky” to be true, alienating himself from much of the scientific community at the time. And so he spent his days going on long, solitary walks around the Princeton campus. After a while though, he found an unlikely walking companion: the much younger Kurt Gödel. The genius logician Gödel was highly regarded for his incompleteness theorems, with which he had shown that no logical system is 100 percent airtight. With his ideas, Gödel challenged the notion that humans could ever achieve something like absolute knowledge. 
• Gödel also shared Einstein’s skepticism of quantum mechanics. And on the matter of time, Gödel took Einstein’s famous relativity theory even further than its originator.
• Gödel took Einstein’s idea even further. Trying to solve Einstein’s equations, he came to the conclusion that the universe wasn’t expanding, but rotating. In such a rotating universe, it’s theoretically imaginable that someone could travel back in time. If time travel is mathematically possible, Gödel concluded, then time itself doesn’t exist at all. 
• Illusion or not, Gödel’s own time on Earth met a tragic end. After Einstein’s death in 1955, the brilliant mathematician grew more and more secluded. He also became increasingly paranoid; convinced that someone was trying to poison him, he finally stopped eating altogether. In January of 1978, he died of self-starvation at Princeton Hospital. 
• In the 1980s, the neuroscientist Stanislas Dehaene was studying a French man who’d suffered an injury to the back left half of his brain. As a result, the man had extreme difficulties processing numbers. Dehaene concluded that our brain must have a special “number sense” that allows us to roughly estimate and add together objects in our environment. Later, he even located the precise brain cells that enable us to do this.
• By now, researchers believe that a dysfunctional “number sense” may express itself as dyscalculia, a difficulty in processing numbers that is similar to dyslexia. 
• In the nineteenth century, German mathematician Bernhard Riemann discovered a mathematical function that may hold the key to understanding the “music of the primes.” The Riemann zeta conjecture holds that the zeros of the Riemann zeta function can help us predict the distribution of all prime numbers. But so far, no one has been able to conclusively prove its verity.
• What is math?
  • Applied mathematics, for instance, uses numbers, functions, and calculations to solve real-world problems
  • pure mathematics is not really about finding practical solutions to real world problems. It’s more about manipulating numbers, functions, and calculations on their own to find interesting connections between them.
• famous philosopher Ludwig Wittgenstein, have thought that pure math is basically just a logic game played with numbers. But others, including “Platonists” like Einstein and Gödel, have held that even abstract mathematics has a basis in the real nature of our world.
• mathematicians are not just looking for any kind of proof though – they’re looking for beautiful ones. Some mathematicians go so far as to consider beauty the ultimate measure of a theory. For them, beauty usually means a mix of simplicity, strangeness, and inevitability. 
• In his famous 1940 book A Mathematician’s Apology, British mathematician G. H. Hardy argued that pure mathematics is more like art than science – it’s not about solving problems, it’s about painting a compelling picture using the language of mathematics. 
• In the late 1970s, the Polish-French-American mathematician Benoit Mandelbrot introduced the idea that certain rough physical structures are “self-similar,” or “fractal.” This holds for many structures in our universe, such as clouds, networks of blood vessels, and clusters of galaxies. Later, when he worked for IBM in New York, Mandelbrot used some of the very first computers to show how complex fractal patterns can arise from very simple formulas – often producing stunning, geometric digital drawings. 
• What many people consider infinity comes in two versions.
  • there’s infinitely big. When scientists explore this type of infinity, their work often approaches mysticism. Some Russian mathematicians of the 1920s, for instance, conflated their mathematical explorations of infinity with theological explorations of God. 
  • there’s also the idea of the infinitesimally small, or infinitesimal – and it’s no less puzzling than the infinitely big. In mathematics, for example, each number is divisible into infinitely many fractions. 
• In the seventeenth century, French mathematician Blaise Pascal shortly revived the idea of the infinitesimal, using it to calculate the areas under curvilinear forms – that is, forms characterized by curved lines and shapes. Isaac Newton, too, made use of the infinitesimal when he calculated the exact orbit of a planet around the sun. But Newton agreed with his contemporary Gottfried Wilhelm Leibniz that infinitely small quantities were nothing but “well-founded fictions” in mathematics – useful, but not really real. 
• was only in the 1960s that mathematician Abraham Robinson was able to rehabilitate the image of the infinitesimal. Robinson was able to draw on Gödel’s completeness theorem – the counterpart to his incompleteness theorems – to show that even if we can’t be sure of the reality of the infinitesimal, we can still be sure of the logical consistency of the calculations that involve it. That, Robinson argued, should be enough to satisfy the standards of modern mathematics.
• The history of science is littered with strange and untimely deaths. But perhaps none is as mysterious as that of Alan Turing, the father of modern computer science. Turing died in June 1954, apparently from eating an apple laced with cyanide. The death was ruled a suicide, but many people are not convinced. 
• Turing got his PhD in mathematics at Princeton in 1937. During his time at Princeton, he conceived the blueprint for the modern computer. Turing’s abstract computing devices were dubbed “Turing machines.”
• When he cracked the Nazi code, Turing put his theoretical ideas about computation to the test. The German Enigma machine used a complex system of rotating wheels to encrypt each letter in such a way that only a specially programmed receiving machine could decipher it. But Turing found that by analyzing certain repeat phrases the Germans used, such as “weather,” he could detect logical consistencies in the way the Enigma coded letters. To help with his work, he built a counter-coding machine called the Bombe, which is now considered a milestone in the history of computing. 
• Einstein’s general theory of relativity went a long way in explaining the fundamental interactions of gravity, mass, space, and time. And a little later, quantum theory helped explain how subatomic particles behave at the micro-level – apparently in a much more random and uncertain fashion than huge physical forces. Both general relativity theory and quantum mechanics seem to be true. But for a while, no one was able to bring them together. Until string theory came along. 
• String theory main idea is this: the smallest elements that make up our world aren’t discrete particles, but tiny, vibrating “strings of energy.” Different vibrations produce different natural phenomena.
• Scientists today pretty much agree on how the universe started – with the big bang, 13.82 billion years ago. They also agree that the universe has been expanding. What they don’t agree on is how it will all end. 
• they’ve come up with three competing theories on how the universe will end
  • the big chill assumes that the expansion will go on forever, until all particles of the universe are spread so far apart that life as we know it becomes practically impossible. 
  • big crunch proposes that the expansion of the universe will eventually stop. The universe will then begin to collapse back onto itself.
  • the big crack-up is based on the notion that the universe isn’t just expanding, but expanding faster and faster. In ten to twenty billion years, protons will eventually decay, and matter will dissolve entirely