201–210 of 439 posts
Wow, from the 2015 article: [A journal reviewer of his famous paper says]: "you should be careful. This guy posted a paper once, and it was wrong. He never published it, but he didn’t take it down, either.’ ” The reader meant a paper that Zhang posted on the Web site arxiv.org, where mathematicians often post results before submitting them to a journal, in order to have them seen quickly. Zhang posted a paper in 2007…
Math used to be a young person game, but it now requires so much knowledge just to get to the frontiers of human knowledge, not to speak of making a dent into uncharted territory, that results are being obtained later and later in life. When mathematicians have had time to accrue sufficient knowledge while still being sharp enough to make the intellectual leap. The sad part is that as the trend continues we may reach…
Over time I think we need to either create a GAI or enhance ourselves(via genetic engineering, brain-computer-interface, eternal youth or other ways) to overcome this fundamental human limitations.
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Patent clerk (at least back then) is a more skilled job than it sounds, as it's someone that actually reviews the patents rather than just a general office dogsbody, so you have to understand the technology and to some extent the science in those patents. It's not totally unrelated to being a scientist and definitely a less surprising early career job than delivery driver.
A patent clerk now is a technical job as well. It’s a flaw in peoples thinking that “clerk” implies something low rank. If someone tells you they were a “clerk” for a Supreme Court judge don’t be surprised when later in life they have an impressive legal career
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I suspect computers can also help us get deeper. Stuff like computer algebra systems. Maybe some CAS-assisted work gets us into feedback loops allowing us to go indefinitely, as in a technological singularity. But the "shallow" part is also quite wide. You can teach people what you've learned forever, for instance.
do you have experience with CAS? i would love to learn how to use CAS to write proof more effectively
I don't know what field you are in so it may or may not be helpful to you.
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To add to your comment, quaternions predate rotational matrix operators by a considerable amount (1843 vs not exactly clear, ~1900 with Peano or ~1920 with Weyl), despite quaternions being much more challenging to manipulate. There are definitely simpler ways to view the same things. There was a cottage industry of exotic hypercomplex numbers that disappeared when linear algebra matured to eclipse them. In fact, Maxw…
I am somewhat unconvinced that quaternions are more challenging, or at least a worse way to think about the issue : https://eater.net/quaternions And speaking of Maxwell's Equation"s" : http://www.av8n.com/physics/maxwell-ga.htm#sec-preview
With some familiarity with linear algebra, it's easy to derive the formula for constructing a rotation matrix. You just have to think about what the operation does to the axes. The derivation for quaternion rotation is far more abstract, by virtue of the operation we actually care about involving a sandwich of multiplications with unclear 4 dimensional meaning. There's no hyperspheres with a rotation matrix.
Augmenting your space to handle not just rotations & scaling, but translations is easy for matrices, just requires a homogeneous coordinate and you get 4x4 matrices with intuitive columns.
Augmenting quaternions to handle translations requires the 8 dimensional dual-quaternions.
I definitely like geometric algebra, it's a very nice continuation of topics in linear algebra and makes it clear why things like normals behave differently from standard vectors. But I don't use it every day. I use standard linear algebra every day.
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>Math used to be a young person game, but it now requires so much knowledge just to get to the frontiers of human knowledge, not to speak of making a dent into uncharted territory >The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient Hmm. I had never thought of it like this. Is it possible for human knowledge to become so advanced in…
Here is a short story that covers that idea: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
I've read it, lost it, and have been searching for it for a few years now.
Can someone explain the decimal constants that are used throughout the proof? For example, on page 52. It's rare to see these kinds of numbers used in mathematical proofs, but I'm sure they were chosen for good reasons.
Unfortunately, nobody can explain anything like this right now. The paper was posted today, is 111 pages long, and it will likely take even professional mathematicians around a year to understand / check it completely.