Quoting a relatively famous mathematician, linear algebra is one of the few branches of math we've really truly understood. It's very, very well behaved
kogasa
joined 1 year ago
Lived in Iowa for a few years, there were a few authentic Mexican places, just not as many as Americanized ones.
Yes, with Iosevka font
Google it? Axiomatic definition, dedekind cuts, cauchy sequences are the 3 typical ones and are provably equivalent.
I'm fully aware of the definitions. I didn't say the definition of irrationals was wrong. I said the definition of the reals is wrong. The statement about quantum mechanics is so vague as to be meaningless.
That is not a definition of the real numbers, quantum physics says no such thing, and even if it did the conclusion is wrong
Being suitable for human consumption doesn't mean it's not also suitable for playing a role in a more efficient food chain
Stokes' theorem. Almost the same thing as the high school one. It generalizes the fundamental theorem of calculus to arbitrary smooth manifolds. In the case that M is the interval [a, x] and ω is the differential 1-form f(t)dt on M, one has dω = f'(t)dt and ∂M is the oriented tuple {+x, -a}. Integrating f(t)dt over a finite set of oriented points is the same as evaluating at each point and summing, with negatively-oriented points getting a negative sign. Then Stokes' theorem as written says that f(x) - f(a) = integral from a to x of f'(t) dt.
Going to almost certainly be less than 1. Moving further up the food chain results in energy losses. Those fish are going to use energy for their own body and such
For sure, which is why I said "another food source would be needed." I had in mind something like the wild-caught fish being processed into something useful as part of a more efficient food chain, e.g. combined with efficiently-farmed plant material.
Moreover there’s high mortality rates inside of fish farms for fish themselves.
I don't have any context on the other pros and cons of fish farming, so definitely not arguing whether they're a net positive or not.
Your first two paragraphs seem to rail against a philosophical conclusion made by the authors by virtue of carrying out the Turing test. Something like "this is evidence of machine consciousness" for example. I don't really get the impression that any such claim was made, or that more education in epistemology would have changed anything.
In a world where GPT4 exists, the question of whether one person can be fooled by one chatbot in one conversation is long since uninteresting. The question of whether specific models can achieve statistically significant success is maybe a bit more compelling, not because it's some kind of breakthrough but because it makes a generalized claim.
Re: your edit, Turing explicitly puts forth the imitation game scenario as a practicable proxy for the question of machine intelligence, "can machines think?". He directly argues that this scenario is indeed a reasonable proxy for that question. His argument, as he admits, is not a strongly held conviction or rigorous argument, but "recitations tending to produce belief," insofar as they are hard to rebut, or their rebuttals tend to be flawed. The whole paper was to poke at the apparent differences between (a futuristic) machine intelligence and human intelligence. In this way, the Turing test is indeed a measure of intelligence. It's not to say that a machine passing the test is somehow in possession of a human-like mind or has reached a significant milestone of intelligence.
The assumption is that the size decreases geometrically, which is reasonable for this kind of self similarity. You can't just say "less than harmonic" though, I mean 1/(2n) is "slower".