The Cathedral of Logic
Mathematics — the great language of words. Everything we have today, we have thanks to mathematics. While logic is our first attempt to formalize the way we reason and think, it is mathematics that took us to the next level and helped us achieve such great technological prowess. But it all started with words. Or more precisely, with sounds.
Back in 1959, Eugene Paul Wigner, theoretical physicist and Nobel Prize winner, delivered a lecture at New York University in which he spoke about “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In the lecture, he argued that many mathematical concepts that initially appeared to be just abstract ideas were later found to perfectly describe and predict real-world phenomena. He echoed Gellileo’s centuries old assertion that “the universe is written in math”. In the said lecture, he cited a few great examples. Although it wasn’t mentioned in that lecture, my favorite example is probably that of a “Black hole”. In 1915, Albert Einstein published his equations for General theory of Relativity. Only few months later, another German scientist, Karl Schwarzschild, used those same equations to infer a solution that described an extremely compact point where gravity is so strong that even light cannot escape it. For almost half a century, most scientists, including Albert Einstein himself, thought that these were just abstract mathematical ideas — something fun to play with on paper, but not a real physical phenomena. But they were mistaken. Astronomers soon started finding these weird objects in space, and these objects that perfectly matched the “extremely compact point” proposed by Karl Schwarzschild all those decades ago.
Mathematician Richard Courant once said that Mathematics ”as an expression of the human mind ,reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality”.
Indeed, Mathematics is our Cathedral of logic. But every cathedral has a shadow.
Incomplete, Undecided, and Unfolding
Kurt Gödel’s Incompleteness Theorems, published in 1931, established a fundamental limit to what mathematics can prove. The theorems demonstrate that any formal system of logic that is both consistent and robust enough to encompass arithmetic must, by its very nature, be incomplete. This means that within one such system, there will always be mathematical truths that cannot be logically proven. Before Gödel, mathematicians believed that math is complete in a sense that “every true thing can be proven and every provable thing is true”. They thought that if a fact is true, they would be able to prove it by using math and logic. But as it turns out, that is not the case.
A few years after Gödel’s seminal work, Alan Turing — the British mathematician and computer scientist, decided to tackle the so-called “Halting problem”, and soon proved that there cannot ever be a computer powerful enough to solve all possible problems. This means that even if we create a perfect super-computer that can solve even the hardest mathematical problems, there will still be some problems that such a computer wouldn’t be able to solve.
Logic is the “rules of the game,” and Gödel and Turing proved that for any game complex enough to be interesting, you can always describe a move that the rules of that game can’t handle.
Back in 2015, a group of scientists (Cubitt, Perez-Garcia, and Wolf) took this idea one step further and proved that some properties of quantum systems are fundamentally unpredictable. This implies that even if we know all the details about some material, we cannot mathematically predict its macroscopic properties (like whether the material is a superconductor or not).
As you can see, the world we live in is mathematically incomplete, computationally unsolvable, and realistically unpredictable.
There are fundamental limits to what we can know. We can obviously know (and we do know) a lot, but certainly not all. So no matter how hard we try, some mystery will always remain. In fact, it appears that mystery itself is a fundamental property of the universe.
And I think this is a good thing because it allows for things such as free will and complexity. A predictable world is a boring world. Everything we see and interact with is one constantly emerging outcome. Isaac Newton saw our world as one grand mechanical clock. Later on, some scientists began viewing it as an evolving organism. But since our universe is already full of mechanical clocks and squishy organisms, maybe we could say that
the universe is all there is combined into something radically different than the sum of its parts — some surprising and astonishing but ultimately indiscernible pattern emerging from the constantly expanding enthropy.
“Aren’t we all?”, I hear you asking. Well, that’s a wonderful question. To the best of our knowledge, humans are the most complex beings in the universe. Our brain alone is more complex than anything we have ever seen. Homo sapiens is the only species considered capable of complex, abstract thought, cooperative culture, and advanced technology. What makes us distinctively different from all the other animals is this unique combination of certain key characteristics such as: bipedalism, large brains, opposable thumbs, very efficient perspiration system, and few others. Indeed, there’s nothing quite like us. But life wasn’t always this good. Earthquakes, volcano eruptions, predators, diseases, and other vims of nature challenged our claim to supremacy at every step of the way. It took us approximately 200,000 years to get where we are today.
However, it wasn’t the human brain that got us this far — it was human brains.
The first quarter of the 21st century is behind us. Since January, 2000, we have survived: terrorist attacks, recession, and even the great plague of our times. But let’s not ‘bs’ ourselves. This isn’t the age of heroes. We didn’t kill no dragon. It’s the generations before us who are the true heroes. They survived all the famine, all the plagues, all the mass migrations, and all the doomsdays. They witnessed all the great empires fall one by one, leaving behind just ruins and memories that slowly faded into legends. If the age of heroes is behind us,, how should we name our age? Maybe we can only name ages once they pass.
Many have argued that history is merely a work of fiction written by the victors. Thomas Carlyle may have put it best when he said that:
“the history of the world is but the biography of great men”.
I would argue that there are 2 histories. First there’s history as a systematic study of (primarily) human past. But there’s also what I call the real history, a series of true events that, in large part, might have been forgotten but have ultimately shaped who we are as a society today. This real, largely forgotten history, is the true history of our supremacy.
There are some 80 billion neurons inside the human brain. These neurons form about 100 trillion connections (synapses). Multiply this by (estimated) 117 billion men and women throughout our real history, and what do you get? You get to be alive here and now. You get the privilege of quality education, healthcare, hot showers, PS5, and BMW M4 Competition
Logos
In the beginning was the Word (John 1:1)
Logic teaches us that there are some things we can never do — but art nudges us to still give it a try. Each one of us is a system who lives inside of a system that is a part of a larger system. We live in a society which is a part of the greater system we call nature. We are bound by laws of nature and laws of our society. But in our minds, we are unbounded — we are free. We have two languages to express anything our minds can deduce or conjure. We have math, for the matters related to logic and numbers; and poetry, for matters related to life and relationships with other people. But how did it all start?
Everything started with words, or, if we go even further back, with sounds.
At the beginning (when we were still more apes than humans) there were only certain sounds we could produce. Maybe to attract someone’s attention in order to show them something or warn them. Unlike other Apes, our unique anatomy allowed us to produce a much wider range of sounds. Soon, we began combining these sounds into more complex sounds (tokens), and finally, we started uttering complete words—a fact that changed everything. Words spread both spatially and generationally, and slowly evolved into language. It is probably language that allowed us to impose our supremacy over all the other apes. What made it so powerful is the fact that it allowed us to communicate better. The key to our supremacy was communication, but what made communication so powerful was words.
Words are the foundation from which we built language. From there, by observing the natural order, we inferred logic, which then allowed us to formalize it even further into math and geometry.
If we look closely, we’ll see that language and math are of the same substance.
Logic was the inflection point. What made logic possible was imitation. We are good at noticing the casual order in nature. We observed nature and we inferred a few universal truths from these observations which we then formalized into rules. Words are the prime substance, and language is the prime skill. We can use language for all sorts of different purposes. We can sharpen it and use it for science — to describe and dissect what we perceive. Or we can dip it into the can of our emotions and use it to create art. Either way, words are very powerful.
I have found that there are 2 types of powerful words. I call them — “key” words and “spark” words.
Key words are very high frequency words. Quiet and discreet like a dog whistle, not everyone can hear them, but those with delicate enough ears can. These words unlock something in people; they’re like the missing pieces of a puzzle. When Einstein said E=MC², most people couldn’t care less. Even today, most people still don’t.
Spark words, on the other hand, are very low frequency. Loud and rash, like a bullhorn, everyone can hear them. These words have power to move not just individuals, but also masses. The right combination of such words, delivered with the right rhythm can fire up the crowd, like a hot spark touching gasoline. When Churchill said, “We shall fight on the beaches”, or when Brian Johnosn said, “Thunderstruck”, masses moved (and to this day, they still move when AC/DC does their thing).
Both the soldiers on the beaches of Normandy and the scientists at Los Alamos were moved by words. And even divided by the ocean, they have danced to the same song. That is what powerful words can do when they are arranged in just the right order—they can move people, both mentally and physically.
The Invitation
We live at the most extraordinary time in human history. Not because we have all the answers — we don’t (and scientists proved we never will) — but rather because we now have the tools that are unlike anything we had before. Everything we ever achieved will converge by the end of the decade (±3 years). Throughout our long history, we were always looking everywhere around us for inspiration. Nature always had something to show us. And just like children, we were always eager to copy and imitate. This time we turned inward for inspiration. We somehow managed to capture in code what nature had captured in flesh. This is the beginning of something new. We aren’t the only intelligent entities on the planet anymore. This new intelligence didn’t come from out of space, as we were expecting—it came from within. And its modest beginnings were similar to ours in a way.
If 50 years from now you ask a robot how it all began, it might just say — “It all began with a word”.
We first formalized the external world into logic and math. Now we’re formalizing our mental faculties with AI. And just like Gödel showed that math cannot complete itself, AI won’t complete us either — but the attempt will surely change everything. As you are reading this, the gears of history are turning and grinding faster than ever before. Many great challenges lie ahead, but also countless new possibilities.
We are standing on a precipice — the point of big convergence. I invite you all to engage with what’s coming. AI is not just another disruptive technology — AI is the first general purpose technology, and, as such, it has the potential to completely revolutionize the way we work and live. We need to face what’s coming with an open mind — but also with great caution.
150 years have passed since the first telephone call; 133 years since the first radio; 98 years since the first television; and some 37 years since Tim Berners-Lee invented the World Wide Web. All these technologies fundamentally transformed our world, and they all made our lives easier. The rate of technological development is accelerating. Artificial intelligence will accelerate it even more. We all must do our part to ensure that this phase transition unfolds as smoothly as possible.
So don’t just be a witness — take action. And don’t just wait for the future — the future is now.
Originally posted on Medium by Tom Nikola
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