23,904 karma · joined September 6, 2012
Most people, if they haven't studied mathematics in university, would need to learn hundreds of concepts just to get to the leaves of the tree, and many of these concepts are truly difficult to understand, requiring weeks of study and practice.
That's not going to happen. Mathematics isn't just unfamiliar, it's truly difficult to understand. You have to put in a lot of work to understand each concept and the concepts build upon each other to form a vast tower of abstractions that has been growing for thousands of years. Just as there is no royal road to geometry, there is no elevator to the top of the tower.
Pronouns like you/me/he/she/they/them are context dependent in everyday English writing but they're only ambiguous when the context is unclear, otherwise most people have no trouble dealing with them at all!
Z[i], the Gaussian integers, is the subring (of C) generated by Z union {i} where i is the imaginary unit in C, the complex numbers. The Gaussian integers correspond to the integer grid-points of the complex plane, if you want to visualize them.
People genuinely struggle to think verbally or visually once we extend beyond 3 dimensions and start talking about infinite-dimensional constructs, uncountable sets, and so on...
First of all, no, mathematics would be far less approachable if it did that. Most of the Greek letters used in mathematics don't have a universal meaning, they're context-specific and defined by convention or just prior to use.
Second of all, mathematics is optimized for hand calculation on paper, not long-term programming and code maintenance. Writing out long names over and over on a whiteboard gets tiring extremely quickly, so mathematicians prefer to stick to single-letter symbols.
I hadn't read or heard anything from him in recent years, though I had heard he had a falling out with Leo Laporte. He might have reconciled more recently but I am not sure. I know Leo will genuinely mourn him, as they were friends and colleagues for many years. I really miss him now too!
I don't think so. There are a ton of varieties of banana that are grown from seed. Humans have cultivated them and selectively bred them from seed for thousands of years, just like every other plant. The seedless bananas that dominate western grocery stores are 1 variety, Cavendish, that have been endlessly cloned from cuttings. They weren't the first seedless variety commercially available, they were preceded by Gros Michel. But either way, travel to southeast Asia and you'll find a lot more varieties on sale.
China also benefits massively through all of the commodity hardware they make. Sure, they aren't competitive in the chip arms race, but they dominate all the component and power electronics markets. Think about all the power supplies and other commodity components needed to build a gigawatt-scale data centre.
Furthermore, to elaborate on my point above: building new power plants in tandem with data centres that demand GWs of additional power does nothing to address the needs of the grid itself. The grid is not built to handle all that new power, and the consumer electricity demand is not there anyway.
It's like comparing a railway line from a mine to a smelter with a city's road network.
It's like the tulip bubble of the 17th century [1]. Having a bunch of money tied up in useless tulip bulbs didn't do anything productive after the collapse.
No, it actually means something. We have access to refrigeration, climate control, near-limitless computation, entertainment, transportation, knowledge, communication, and medicine. All of this stuff would BLOW THE MINDS of medieval kings. Even if you're living in a 1BR apartment, working at Walmart and struggling to pay rent, you enjoy many luxuries Charlemagne could scarcely dream of.
Go to the store, grab a pineapple off the shelf, take it home, and eat it. Then read about the great lengths [1] the wealthy people of the past went to in order to try to grow pineapples in Europe, and how obsessed the culture became with this fruit.
Stockfish never settles down to an exact move, unless it sees a forced mate sequence. You could leave it running indefinitely, but the exponential blowup will soon grind you to a halt.
What new versions of Stockfish buy you are optimizations and better heuristics. An older version of Stockfish can still beat a new one, if given enough of a compute advantage.
Hence my library mention. Humans have been doing this for millennia: orienting ourselves within a library (the physical kind, full of books) and calling upon its information resources as needed to accomplish tasks (research). Ultimately, it's all just one big cache hierarchy. Your short term memory, your long term memory, the book in your hands, the desk at the library, the nearby shelves, the card catalogue, the stacks, the inter-library loan system.
To manage it all, we humans have developed our abilities for abstraction. When we build clean, tight abstractions we reduce our cognitive load. Perhaps the best abstraction we've built so far is the TCP/IP and web stack. We don't need to care at all about the hardware details of a server in order to talk to it. It's such a powerful and airtight abstraction that we take it for granted.
I'd like to hear from more people who have spent a lot of time building with LLMs, because so far what people are saying is that these models do not have the ability to reason about and build the kind of marvellous abstractions us humans have built.
Look at all the libraries full of books we've built. It's useful for more than mere training sets.
We have examples of AI producing theorems but there is no evidence that AI will be able to find all of the practical applications of mathematics, at least not any time soon.
As I understand it, most of the theorem proving work done by AI today consists entirely of “glue code” style work: combining a bunch of different known results to prove a new result. This is great for those who desire completeness in the mathematical project but it’s often outside the areas of interest for mathematicians who are pursuing “big idea” problems whose solutions likely require development of entire new branches of mathematics. One famous example of such was Fermat’s Last Theorem.
I have some sad news for you. 99% of the work mathematicians do has no immediate application, nor even an obvious path toward application in the near future. You mentioned cryptography, so for an example consider number theory: no apparent practical applications, going back thousands of years to the time of Euclid and earlier.
It’s been religion, philosophy, and recreation that have provided the motivations to study mathematics all these years, not applications. Applications have almost always followed long after the development of the pure mathematical theory. For number theory, that was the development of cryptography during WW2, millennia after the ancients laid those foundations.
Most unfortunately, it’s the truth value and the understanding which drive applications of mathematics, not the proof work itself. If the AI revolution decapitates the institution of mathematics which produces the understanding, and is unable to replace it, then the applications will cease as well.
Unfortunately, if you don’t lift heavy (or if you use electrical stimulation that’s mild enough to sleep) then you’re not going to put your muscles into hypertrophy, so you won’t gain muscle mass either.
Having a grid with no baseload generation and only storage is going to spell disaster during extended cold+calm periods. Rolling blackouts when it’s -30C outside…
There is. "We want to escape" is a very different viewpoint from "we want to liberate the masses."
Freeing yourself from the social media is definitely doable. Depending on how firmly engaged you are at the moment, it can vary in difficulty between fait accompli and moderately challenging. It's obviously possible for anyone to do themselves.
Liberating the masses? Morpheus said it best:
"The Matrix is a system, Neo. That system is our enemy. But when you're inside, you look around, what do you see? Businessmen, teachers, lawyers, carpenters. The very minds of the people we are trying to save. But until we do, these people are still a part of that system and that makes them our enemy. You have to understand, most of these people are not ready to be unplugged. And many of them are so inured, so hopelessly dependent on the system, that they will fight to protect it."