This has got me inspired again about my own history project. It is so easy nowadays to go back to original sources and unearth these hidden or forgotten narratives... and then return to tell these stories in a way that gives another point of view on our current day. Thankyou to the author of this article!
That's about how to solve problems, but to do research you need to find good problems, and that turns out to be more important, imo. How do you pick "major problem X" anyway? This strategy from Tao just leads to incremental results...
1. If Tao (a Fields medalist) is telling you a process to follow, a dismissal of "just leads to incremental results" requires more evidence than just a bald claim. IMO, he's giving away his working process for free.
2. If "all" you want is tenure at a research 1 institution in the us, there's a lot to be said for this process. Another ingredient is "pick an area where other people will be interested in the results."
PS I don't know what to make of your username, but perhaps you have more to say about picking a problem... in which case I'm sure many of us would love to hear it.
There certainly is a cultural division, but maybe there is no fundamental distinction to be made. Both art and science help us make sense of the world. They both tell stories, about an apparently objective reality. They both excite and inspire, and enable us to transcend a limited point-of-view.
If we take science and art to be the same greater thing, then I'd look at it in terms of constraints. Science could be defined as the subset of art that's about seeking maximally accurate representations of aspect of reality. In the same vein, math is a different subset, one that's maximizing self-consistency. Other types of art - i.e. the things we normally call art - have their own sets of constraints.
Math and science are thus special in the sense that their constraints - self-consistency and accurate representation of objective reality - give powerful, direct practical benefits. Other forms of art can have practical use too, e.g. by moving people, or refining their beliefs (or convincing them of lies in the interest of those who commissioned the art - that's advertising and propaganda).
Another way in which math and science are special is that their constraints are independent of humans. Where other forms of art are necessarily a function of how people perceive and feel, the output of science and mathematics is, in principle, universal. I don't know of any other form of art like this.
The answer to these questions is the Riemann hypothesis. John Baez just wrote a related paper about "motives" for beginners (undergrads?) [1]. It's worth a read even if you squint at the equations like i do, there's plenty of interesting commentary as well.
Yeah, it's a reference to how "category theory" generalizes "group theory". So if transformations in a group are called "symmetries" then you might call the transformations in a category "generalized symmetries" or "non-invertible symmetries" as in this article.
Erm the whole idea of a symmetry is that it is a group invariant. If they're using the term in some more general way, it would help if they said what the new way was. The article is otherwise almost completely uninformative. What on earth is a "generalized symmetry", especially one that still has something like a conservation law? Does it have applications in math as well as physics? E.g. topologists are always looking for new invariants.
I can understand that popularizations have to gloss over the math, but they usually at least identify the important points even if they don't get into the weeds of explaining the details. This seems like more of a sleight of hand.
Principle problem being that expert systems required meticulous inputs from domain experts, codified by skilled engineers. People don't have time or startup capital for actual expertise...
And AI requires the same thing, we just call them data scientists and ML engineers. Using linear-ish algebra instead of decision trees doesn't change the the fact that you need time and capital to hire experts.
The big difference is that data scientists only work on the model architecture and data sources, whereas expert systems need people who have expertise in the subject matter itself. One of the biggest changes from 'old AI' to modern ML is that we no longer try to use human domain knowledge as much, instead getting the model itself to see the same pattern from data.
Yes, but there is a whole field of artificial intelligence called unsupervised learning that tries to identify labels without pre-defined labels. At the extreme end there are no externally imposed / defined labels and artificial labels are determined by empirical clusters or some orthogonal data pattern or algorithm. Unsupervised learning is much less effective and not as mature as supervised learning. In the case of LLMs the label is "next words" and it's inferred from a corpus of text.
I'd say labels (for supervised ML) are fundamentally different from rules (for expert systems), because
- labels are easy to decide in many cases
- rules require humans to analyze patterns in the problem space
- labels only concern each data point individually
- rules generalize over a class of data points
Large language models are the thing the average joe in 2023 would call AI the most, and at the end of the day, if you go deep enough down the 500 billion parameters rabbit hole, it's just a "veryyyyyyy loooooong chain of if-then-else's" obtained after 10s of thousands of hours of computing time over basically all of the text generated by humans over 30 years of internet existence. I know it's not EXACTLY that, but it could be pretty much "recreated" using this metaphorical long chain.
Part of this I would attribute to the general collapse of western civilization that we seem to be witnessing. I think it helps to recognize this is what is happening.
As for exchanges, I would recommend Bitstamp [1]. It is the oldest crypto exchange & never been hacked (that i know of). The original owners sold out a few years ago, but it's still going rock-solid.
This applies to all goods and services. If everyone else is charging a fixed fee on top of a proportional one, and then one vendor advertises having no fixed fee, you can be certain that they're recouping it some other way - be it a steeper slope, non-linear pricing, or just a bunch of extra fees that will materialize after they lock you into a contract.
