I think that’s why the author put “vector” in quotes. I kind of imagine it as an ephemeral, infinite list where for some real, when we use that real value as an index into our “vector”/function, we get the output value as the item in this infinite, ephemeral list.
I think the only thing that matters is that the indices have an ordering (which the reals obviously do) and they aren’t irrational (i.e. they have a finite precision).
Imagine you have a real number, say, e.g. 2.4. What stops us from using that as an index into an infinite, infinitely resizable list? 2.4^2 = 5.76. Depending on how fine-grained your application requires you could say 2.41 (=5.8081) is the next index OR 2.5 (=6.25) is the next index we look at or care about.
A vector is always a vector -- an element of something that satisfies the axioms of a vector space. The author starts with the example of R^n, which is a very particular vector space that is finite-dimensional and comes with a "canonical" basis (0,...,1,...,0). In general, a basis will always exist for any vector space (using the axiom of choice), but there is no need to fix it, unless you do some calculations. The analogy with R^n is the only reason the "indices" are mentioned, and I think this only creates more confusion.
> and they aren’t irrational (i.e. they have a finite precision)
No, if you want only rational "indices", then your vector space has a countable basis. Interesting vector spaces in analysis are uncountably infinite dimensional. (And for this reason the usual notion of a basis is not very useful in this context.)
> and they aren’t irrational (i.e. they have a finite precision).
I'm not sure if I'm misunderstanding what you mean by 'finite precision' but the ordinary meaning of those words would seem to limit it to rational numbers?
In practice you're always computing with finite precision. (Even computing with symbolic expressions is just a preliminary step to what's ultimately a numerical result with finite precision.) The whole point of real numbers is to abstract away from detailed considerations of precision, and figure out what happens if you only ever care about putting satisfactory bounds on the output and are willing to bound your input to the extent required.
The idea of R is that it allows you to reason about things like "I need more than X input precision to achieve Y bound on my output". Just sticking with naïve computation in Q does not suffice for that.
When an alarming number of friends (all under 40 years old) from the same small neighborhood in my hometown were diagnosed with leukemia I started to look into the superfund site nearby. The pond that is connected to the stream that supplies the municipal wells in the area was still disgusting (with visible oily residue on the surface) nearly 15 years after the company, Congoleum, stopped operations and the plant was demolished. Soil testing some years earlier revealed benzene, which has been linked to AML.
This is one of the most impressive demos I've ever seen. Truly incredible. I have been dreaming about working on something like this for a while. Open to help?
I've been building https://heylangley.com as a solo side project for a little while now. The initial MVP took about 2 months, though I have not had much time to market it. It's pretty much identical to this. I also applied to YC for W23 and was rejected.
Not sure what else to say beyond this: if you have a project that you've been wanting to start, believe in yourself and build it!
This reminds me of a time when I was meditating in my dimly lit office after work some years ago. For context, I had been meditating daily for 20-30 minutes quite consistently at the time.
During this session, about 15 minutes in I had the sensation of "seeing with my eyes closed". I could see very clearly my workspace, desk, monitors, keyboard, etc. It was one of the most remarkable experiences of my life.
What's the distinction between vivid mental imagery and a closed-eye hallucination? I have aphantasia too (what I used to call "the opposite of a photographic memory") and it always struck me as odd that seeing imagery with closed eyes is considered normal. We put people in mental institutions for hearing voices in their head, but we get a pass if we're hallucinating a visual scene.
My theory is that it's a spectrum from "I only see images when photons hit my eyes" to "I see god right here in front of me." and somewhere on that line is the optimal amount of visual hallucination. It certainly serves some evolutionary advantage.
There's a distinction between delirium, being in a delusional state, where you can't tell the difference between fantasy and reality, which is pathological, vs the normal everyday action of replaying a sensory memory in your head, or imagining sensory input in order to explore an idea. Delusional states may or may not involve a sensory component. It's not a spectrum from one to the other.
The term "hallucination" is tricky because the usual connotation is that it is part of a delusion, whereas some people can generate "closed-eye hallucinations" (or whatever you want to call it) as an alternative kind of mental imagery.
For me personally, the distinction between ordinary "mental imagery" vs "closed-eye hallucinations" that I generate under conscious control is that they are subjectively quite different. The former feel like they are in a different "mental domain" than visual input, and I can perceive this imagery with my eyes open. Another commenter in this thread called this a third frame buffer in their head, separate from their eyes. The latter kind of imagery feels like it is coming from my retina, but it's very dim, and I can only see it with my eyes closed, or in a completely dark room. Apparently, visual input from my retina interferes with this second kind of imagery. And it's this kind of imagery that the OP discusses. You can train yourself to experience it. I've managed to improve both kinds of mental imagery by practicing.
Just bootstrap a second fusion power plant with the first, then continue on, similar to how compilers for a language can be written in the language itself.
Bootstrapping the power plant isn't the problem. The economics of the power plant are the problem, naming producing a worthwhile surplus of power, after accounting for all the power needed by the plant itself.
