My skip manager started asking for weekly status reports so my manager started pulling Jira reports and feeding them to ChatGPT. It turns out that my skip manager was using Copilot to summarize those reports into basically what you could get directly from Jira.
I say this as someone who has been in deep learning for over a decade now: this is pretty wrong, both on the merits (data obviously lives on a manifold) and on its applications to deep learning (cf chris olah's blog as an example from 2014, which is linked in my post -- https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/). Embedding spaces are called 'spaces' for a reason. GANs, VAEs, contrastive losses -- all of these are about constructing vector manifolds that you can 'walk' to produce different kinds of data.
To prove that there is no value of k between 12 and 10^10 such that 2^k has all even digits, you only have to prove that there is an odd digit among the lowest X decimal digits for all 12 ≤ k ≤ 10^10.
The value of X necessary to prove this grows rather slowly compared to k. For example, the smallest power of 2 that doesn't have an odd digit in its last 16 digits is 2^12106. The smallest power of 2 that doesn't have an odd digit in its last 32 digits is 2^3789535319. So it makes sense to try increasingly large values of X until you are able to rule out all values of 2^k for k up to 10^10.
Here's a C++ program you can run to replicate this proof. It takes around 20 minutes to run, and can probably be optimized further, but it shows the principle: https://pastebin.com/DVK2JKdq
This reminds me of an ancient Chinese story from the Liezi, where a craftsman presents a robot to King Mu that can sing and dance. After the robot beckons to the kind's concubines, he orders the craftsman to be killed. The craftsman is terrified and deconstructs the robot, demonstrating to the king that it is simply a collection of inanimate items. The king is impressed and says "can it be that the skill of a man can be equal to that of the creator?" It's a great story that I discovered because it's an early instance comparing creativity and invention to divine power. Not sure if it has been translated but the text is here: https://ctext.org/dictionary.pl?if=en&id=37480
That link just goes to the whole book. For anyone curious: the relevant bit is on page 249, though some pages before and after provide interesting context.
"The same thing is affirmed by [long list] of Albertus Magnus; who, as the most expert, had made an entire man of the same metal[1], and had spent 30 years without any interruption in forming him under several Aspects and Constellations. [...] and being put and fastened together in the form of a Man, had the faculty to reveal to the said Albertus the solutions of all his principal difficulties. To which they add (that nothing be lost of the story of the Statue) that it was battered to pieces by St Thomas, merely because he could not bear its excess of prating. But to give a more rational account of this Androides of Albertus, as also of the miraculous heads, [...]"
[1] i.e., brass ("brazen heads" are mentioned earlier in the paragraph).
(I've modernized the spellings.)
So I think the Androides (I think this is intended as a Greek-looking singular title, not as an English plural; it's a translation of French "Androide") is meant to be a whole person, not just a talking head, although the book talks about it in the context of other things that were just talking heads.
The author declines to believe that Albertus actually made a statue that was able to talk rationally. The specific reasons he gives aren't super-convincing to a modern reader, but I suspect they're mostly rationalizations and his real reason for being unconvinced is just that the story doesn't sound plausible. (Plus, he wants to acquit Albertus of the charge of doing magic in the treating-with-the-powers-of-evil sense.)
He does say that statues able to make vaguely speech-like noises are surely possible "by the help of that part of Natural Magick which depends on the Mathematicks" :-).
