renyicircle
Mathematicians often reuse the same notation for different things where it's clear (for the mathematicians) what the implied meaning is. That means however that non-mathematicians can get confused.
The plus here is the operation that we call "the sum of complex numbers", where the numbers are a and bi. We can work towards this by first defining complex numbers as a tuple of two real numbers with particular sum and multiplication laws for these tuples. So (a, b) + (c, d) = (a + b, c + d) where the left side plus is for complex numbers and the right side plus is for real numbers. Similarly for multiplication, (a, b) * (c, d) = (ac - bd, bc + ad).
We associate (1,0) with the real number 1, and similarly for tuples of the form (a,0). The association is what allows us to see the real numbers as a subset of the complex numbers rather than being some unrelated set. It makes sense because the set of tuples of the form (a,0) with the multiplication laws of complex numbers behaves the same way as the real numbers and is in 1:1 correspondence with them. This association is used every time we write (a,0) as a(1,0) or just "a". We also write (0,b) as b(0,1) since (b,0)*(0,1) = (0,b), from the multiplication law. A general tuple (a, b) then can be written as (a,0) + (0,b) or, as what we've said is the same thing, a + b(0,1). Now we have simplified notation for all of these things except (0,1). It follows from the multiplication law that (0,1)*(0,1) = -(1,0) = -1. The complex number whose square is -1 is denoted as i. All these simplifications allow you to write (a,b) as a + bi.
> More fundamentally what's addition and multiplication? Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?
I'm not personally aware of any concept of continuity between these operations. In abstract algebra, operations are fundamental to an algebraic structure. The field of real numbers or the field of complex numbers are algebraic structures defined by their elements, the binary operation called addition, the binary operation called multiplication, and the laws that these operations satisfy. For example, both are commutative and associative, but the distribution law shows that they're different: a*(b+c) = a*b + a*c, but not a+b*c = (a+b)*(a+c). Also, a+0 = a, but a*1 = a, so the operations have different special elements that "fix" the other element when an operation is applied. It may be interesting to think what a "half-way operator" would have to be like with respect to these laws.
>What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line?
The idea of complex numbers and the more general idea of field extensions, which describes the process of obtaining complex numbers from reals, came from the study of polynomials. Polynomials can only have terms of integer degree. They satisfy many nice properties and quite rich mathematical structures arise from them which would not be possible with general non-integer degrees. Second degree polynomials with real coefficients, like x^2 + 1, are the lowest-degree polynomials such that they may not have real roots, and x^2 + 1 is the simplest example, but any other polynomial can be used to produce the complex numbers as well, if has a factor like x^2 + a. This is where the 2 comes from. It's not significant from a real line perspective (2 is just a number among the infinite reals) but it's significant from an algebraic one.
https://www.danfoss.com/en/service-and-support/fix-and-troub...
They highlight connection through a firewall, mention secure remote access methods, monitoring for data breaches, strong access controls, etc. Reasonable, but these are just recommendations, so the stores could have followed all of them or none at all.
An article about Danfoss vulnerabilities mentioned in a different comment (https://claroty.com/team82/research/freeze-the-controller-de...) says
> we searched publicly available internet-wide scanning platforms, including Shodan and Censys, for signatures associated with its management interface. It didn't take long to find what we were looking for. Numerous controllers [2765] were directly accessible from the public Internet.
So none at all is a possibility.
I wonder if the incremental efficiency gains from this outweigh the damages. That doesn't even sound like a bad idea, unless of course nobody stops to think "but what if someone hacks this".
In which ways does the situation where "most Russians turn on Ukraine" differ from the current reality?
> will math academia be big enough and accessible enough for anyone with sheer passion (despite not being the brightest mind in the field) to pursue a career in, or will it inevitably shrink such that it will only really be accessible to the brightest minds?
Could someone with a recent math PhD experience tell me if it this hasn't been so for years? Competition is incredibly high because people have to compete with every "brightest mind" from all around the world. PhDs are overproduced relative to the number of permanent positions. It's not a question that you only start to think about after you've seen LLMs do mathematics, and if that's the case the student should be made fully aware of what he's getting into. I don't want to call the original post "dishonest" but I wish this was addressed here without the LLM angle. However, it was written by a tenured professor, so he did say the only thing he realistically could.
I'm curious to know if the withdrawal was due to an actual mathematician looking at the papers and noticing the errors, or they ran a model on these to proofread, which would not be the first time, presumably, since they would have surely done that before publishing. Both options have interesting implications.
Well, let them have these. They'll play around with open problems which generate media hype and then they might run out and move on to something else, because "AI came up with a problem and solved it in 3 days" won't have the same effects as "AI solved a problem in 3 days that humans couldn't solve in 100 years".
Regardless of that, what happens long-term if AI gets good enough to solve all the hard open problems?
Millenium problems may give AI companies some temporary hype, but advancing research in some narrow directions to find new problems is probably not worth it to them. In addition, current problems have clear, human-defined acceptance criteria. Problems that are both found and solved by AI will be opaque enough not to generate interest in anyone unless they are found to have immediate applications. Mathematicians will somewhat distrust 1M+ LOC Lean formalizations as well as any original AI output in the form of creating a bunch of new concepts. There will surely be heated discussions around that.
Afterwards, mathematicians take some time to catch up with the solutions to existing problems, use AI to understand them and map out further directions. Problem solving by AI companies slows down since there are no cool problems left. They move on, mathematicians get back to whatever they were doing before, now with the help of AI.
With every mathematician having a strong mathematical AI model, a new problem arises. At any point of the current mathematical frontier, a mathematician can use AI to explore a new direction at a high speed. If all mathematicians do that, each will have their very advanced understanding of their direction until they fail to catch up the AI, and nobody knows whose direction is the one that others should check up with. Many possible connections with other branches and explored directions will be lost, unless there's a unified map of all these explorations. It will be hard to create this map, because while exploring one direction may have a moderate cost for a mathematician, having an AI do an overview of all other explorations to find some commonalities and synthesize the findings into a unified theory will take serious amounts of resources only available to AI companies, which they might be reluctant to spend.
Individual mathematicians' explorations will slow down to human level, too, with their unique findings being hard to publish because nobody will care and the mathematical community wants to do things that other mathematicians can appreciate.
In the end, the mathematicians' work does not change that drastically, unless AI gets good enough to one-shot some new crazy branch of mathematics that reveals something with high real-world impact. But that's a discussion for another time.
I really don't get the hate for mathematical notation here, do you have specific examples? Notation is a necessary tool to express complex ideas compactly so that others can understand them. It allows you to carry out technical proofs that confirm things that are "obvious". Then it uses previously introduced concepts to build new concepts on top of them. This keeps mathematics interconnected, there's value in explaining a concept in terms of already known things rather than just vibing it.
> makes itself intentionally hard
It doesn't make itself intentionally hard, it is just hard. You can simplify some expositions and make concepts easier to understand intuitively but it won't necessarily give you the skills to actually work with them in mathematical practice (which is the point of mathematics as a discipline).
I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.