Squeeze theorem

To quote from “The girl next door”
The first lesson of politics is “Always know whether the squeeze is worth the juice”. Now i was trying to finally make a genuine effort at understanding Central Limit theorem. Throughout my life(30 years), i have always been suspicious whenever statistics goes beyond the mean, median, mode, SD and Variance. (i.e to say, whenever any stat goes above first and second moments). Part of it because i never really learnt or rather never paid enough attention to convince myself of the theorems involved in reasoning with distributions. Anyways, i figured Central limit theorem would be a good place to start and in learning by teaching am summarizing what i’ve learnt so far.

It started off as i came across this post on HN and going through comments and critique realized the demo is more of a special case and while i did get that specific example(and sure of what CLT says) am still unsure of why Central limit theorem is true or how one formulate it in math terms. It is important for me to understand those, if i am ever to be able to question someone claiming some implication of CLT. Anyway, i came across the squeeze theorem in one of the HN comments and since it seems it’s part of the proof for CLT, I ended up reading and here’s the result of that.

Anyway, enough story. Let’s go onwards. So here goes straight from the wikipedia page:

There are three functions f,g,h defined over a limit l.
a is a limit point.
f,g,h may not be defined at a, since it is the limit point.

g(x) leq f(x) leq h(x)

lim_{x to a} g(x) = lim_{x to a} h(x) = L

To be proved:
lim_{x to a} f(x) = L



I’ll try and clarify what is a limit as mathematically defined, and hopefully without equations,but words only.
Well, according to wikipedia page, limit of a function f(x) means that the function f(x) can be made as close to a value (say L),
by making x sufficiently close to c.

Or to write out the equation
lim_{x to c}f(x) = L

Consciousness — some dreamy hand-wavy theorizing

I am not a professional scientist or an expert in any of the areas, I refer to in the following article. I have picked up a couple of unrelated degrees in my life so far, but don’t consider myself an expert in any of them. I consider the rest of the article as something, I would have written if I had been interviewed for that (pop-sci) book “What I believe, but cannot prove”.

This is an attempt to summarize a bunch of my thoughts and ideas on consciousness.
This definitely is a biased summary, as abhay asked me to write a post and with a set deadline, my brain was scampering around and recalled the most recent thoughts.
A few basic assumptions*, I tend to make are:
1.Consciousness is a emergent phenomenon, that arises out of essentially materialistic universe.
2.Only known/agreed fundamentals of universe are dictated by physics as Space, Time, Energy, Mass.
3.While our neuronal activity may not be sufficient to explain all the phenomenology of consciousness,
they definitely are a necessary condition for enabling consciousness.**
4. Space-time continuum and the extended Space-Time-Energy-Mass Continuum is true, and will be proven sometime in the future.

A couple of basic hypothesis I would like to add is:
1. Attention is a core, integral, fundamental part of our universe on par with Space, Time, Energy and Mass.
2. Attention is part of a continuum of Space-Time-Energy-Mass-Attention Continuum
3. Big bang is the event when this continuum evolved/broke apart enough to these distinct components.

For the rest of the writing, I would use the terms consciousness and attention rather inter-changeably, because that’s how I consider it to be.
Do note that, I haven’t defined attention, because despite the standard definiton, I am not sure, that fits in with what am building here.
Instead, I’ll try to provide as many specific examples as I can think of and let you infer or construct a definition, that fits.

Some evidence*** I would like to sketch out are:
1.Heisenberg’s uncertainty principle:
i.e: It is impossible to know both position and mass of a particle to the exact precision allowed by the equipments’ capability.
This has been interpreted in many ways, and discussed rather extensively in some philosophical circles and popular science books.
The main point,I wish to make being once you think of the measuring equipments as exhibitng some characteristics of attention, it’s easy to interpret this as evidence for attention being a part of the universe and being converted into other forms.

One corollary of this is

Am picking just the terms I am familiar with. But this principle, that all human beings/living beings are part of a whole has been proposed in quite a few mythologies and stories spread around the world. Now, it could be argued that, this really is just a bias that led to this kind of independantly originated similar myths, but am at the other end.I’ll just list out a few variants, I’ve come across here, “Treat unto others”, “Gaia theory”
3.Causality and Probabilistic Graphical models:
Judea Pearls’ work on causality is fairly known.But I only have the basic ideas from reading easy text and avoiding the core math part.Based on a few of the sketches, I have indeed read, I would say, quite a few of our currently used models of causality are suspect, which I believe is the biggest argument for a peer review process, rather than a made-up gold standard template for scientific experimentation and conclusions.
But I digress, my core argument being that, if and once you agree that highly engineered, technical equipments(like the LHC) possess some level of attention/consciousness, then your probability graph immediately adds in complexity. To put it mildly, the possible number of agents goes up so dramatically, that trying to assign strong probability to any event as being the cause of another becomes a unrealistic computation problem.

4.Speed of light constant:
That mysterious rule that Speed of light is a maximum limit constant that cannot be exceeded.Also the corollary, about exchanging information.

Some predictions I would like to make are:
1.Sometime in the distant future, physicists will start acknowledging and forming theories around the agency of their equipments and how to make experiments that account for those, but still add value/evidence to the original hypothesis.
2.Neuroscience,technology, and our understanding of ethics will progress to levels, that enable us to more actively experiment on consciousness by enabling us to set/change/modify variables involved in affecting it.

Few corollaries, supposing those hypothesis are true:
1.Consciousness/Attention has been increasing in the universe, since Big bang.
2.It’s possible to convert from consciousness to mass/energy/space/time. (this means some superpowers like that of Vista, clockblocker, etc. in The Worm web serial novel)
3.The orders of magnitude of Consciousness contained in the form of mass/space/time is humongous. (Just think how much Energy is released by converting mass into Energy.of the order of 10^^12 times our known measure)
4.Our current measures of Consciousness/attention are too crude and they underestimate by incredible numbers.
5.In terms of life as a optimization problem viewpoint, we are currently woefully stuck in local optima points. We are just simply ignoring what is atleast 1 extra dimension, by not acknowleding it. (Not to imply we are utilizing the others we acknowledge optimally)

*— Some of these are just plain and simple biases I’ve accrued over 30 years of my life, and not necessarily proven science facts.
** — It follows that, I don’t consider bacteria, viruses, DNA/RNA etc. as conscious. I remember Harward declared animals are conscious, (though not sure they included microbes or not) and I agree with that. Am open to being convinced otherwise though.They do have a few basic reactions to environmental changes, so to take a weaker position, I would say, the bacteria and virii have a very tiny/micro/minute level of consciousness.
*** — evidence as suggested by law scenes in movies and serials, rather than from hard sciences.I could probably argue, there’s probabilistic graphical models based strong evidence, but I don’t have the time or effort to put a good one. This includes narrative, story-based evidences.