Rest

 I hope that everybody in the world gets their infinite moment of respite today. 

Skolem's paradox and the chauvinism of models + category theory, Plato's ideals...

I see Skolem's paradox as only paradoxical to the extent that I expect our understanding of reality to be objective and absolute. The corollary of this arguably chauvinistic [this usage is kind of inflammatory but it fits to what I'm thinking/feeling] view is that "subsets" of said reality can be understood, by extension, in much the same way the larger reality is understood. 

I'll elaborate with an example. Take the usual topology of the unit square with no boundary. Topologically this is isomorphic to R^2 even as it is a subset of R^2 itself! [I need more examples here. It kind of gets in the way of the analogy that the unit square is isomorphic to R^2 itself. I would've wanted it to have its own "life" to drive the point home] 

Containment and subsets

Containment, in the various contexts in which we use it, is rather nominal, often one-dimensional, superficial. Imagine the imprisoned philosopher: the body is bound, but the mind cannot be.

[This is super verbose but whatever] We are tempted into a notion of a hierarchy, that things which "contain" other things are "above" the things which are "contained". But now we are shown that when we make the ruler itself smaller along with the space, we get complexity that may not only be lateral to the larger space, but even more complex, and the notion dissolves. We like to use the phrase "dumb as a rock", imagining that the rock has very simplistic, if any, thoughts. However, this rock, which in our understanding of the world is a static, relatively uninteresting object, lives as rich an inner life as any human being. We can understand this from our current understanding as follows: to take the perspective of the rock, we begin as humans, then slowly "limit" our concepts, make our ruler smaller, our thoughts simpler. In other words: become dumber. The dumber we become, the more profound everything else is. Once we have "become the rock", the critical realization is this: "dumber" is a relative term. In fact, we only see that we have become "dumber" because we are stuck in the frame of our own human minds, not the rock's, and that we have not really "become the rock". We are drawing an arbitrary mapping between the rock's mind to ours, measuring its experience only in terms of ours. But the rock's experience, as unknowable as it is, is just as profound and valid as ours, incomparably so until we decide to draw arbitrary comparisons. Not to say that drawing comparisons isn't fun, but we should recognize our experiences for what they are: experiences. When measuring other experiences against our own, we should not be surprised when we get strange results, or even when the other experience deigns to measure ours against theirs. For surprise comes from the incorrect expectation that other experiencers measure themselves against our own as well, which will almost never be the case. It's prudent of us, then, to consider when we might err in this way, not only as it relates to the cardinality of sets but even more mundane things (mundane, from the perspective of one who finds counting stuff to be interesting).

Looking beyond the sensors

Whenever I feel I know something, or have some hunch, I like to short-circuit it by thinking of myself as a being with unreliable sensors. I certainly feel that it's paradoxical that a countable model of set theory includes an uncountable set. But what exactly do we mean by countability? What image-based ideas and analogies have we attached to the formal notion of countability, what experiences do we have as children that model these ideas [Here, I'm thinking about how we count as kids. Think Sesame Street] ? For another: I certainly feel that it's warmer in this room -- but what else besides actual temperature might be causing me to feel that way? 

Looking upon a "countable" universe from an "uncountable" one, we cannot imagine that living within such a universe, we could speak of the uncountable. Ernst Zermelo, for at least a decade and a half, refused to acknowledge countable models of set theory, even citing Skolem's work as an intrusion of relativism in mathematics in a note titled "Relativism in Set Theory and the So-Called Theorem of Skolem" in 1937. Yet the rest of the mathematical community moved forward, forced to re-evaluate its thinking around countability and uncountability. In the language of the sensors analogy above, Zermelo represents someone who lives in the world their sensors show them -- a first-order world of seeming objectivity. 

In this "deconstruction" of countability, it might help to think about the original, natural notion of counting itself. Why do we define countability the way we do, and how does it originate from our physical experiences? Mathematics as a formal system need not be beholden to any human's "objective" mental model of reality. As it were, countability vs uncountability isn't just grains of sand vs water anymore. By letting go and reckoning with the superficiality of our own understanding, we seem to gain a kind of clarity akin to what Plato's subjects in his Allegory might have gained.*  

Category theory : mathematical ideas :: mathematics : reality

[Need a better heading here. Concept is that category theory deals with ideas, the "stuff of isomorphism" which is shared by things that are isomorphic to each other, which lends itself to the Platonic idea of.... well, Ideas/Forms, which correspond to the "stuff of isomorphism", and everything else just being a representor/flawed manifestation/implementation of such. The thing is, mathematics already does this for reality, so there's two levels here. categories -> rest of math -> reality]

Set theory likes to talk about sets properly containing other sets. In category theory, the picture is different: there are two objects in their own right, and we choose to relate one to the other with some monomorphism, or from the set perspective, an inclusion map of underlying sets. In set-based mathematics we might say that as sets, although one is contained in the other, there is structure that is not necessarily contained, and thus the inclusion map doesn't imply much about the relationship between these two structures. However, from the categorical perspective, none of this needs to be said in the first place -- these are two separate objects to begin with, and arbitrarily mapping one to the other isn't necessarily profound or meaningful. Category theory treats the structure of objects as first-class citizens, their innate nature which they share with other objects of the same isomorphic type. Rather, category theory doesn't even treat them as separate objects to begin with, seemingly directly dealing with the "central spirit" of things rather than how they are implemented. To take it back to reality: it doesn't matter, in the end, how something is implemented. One object can be "inside" of another object, like a pebble, which is a portion of our world, yet still as equally complex. Or it can even be the same "thing" implemented in two different media, like waves of sound or electromagnetic fields (or even our own reality -- implemented as a simulation or not). The important thing is the "idea" of it, what Plato called Forms. 

*Sort of a bastardized verison of Plato's Allegory of the Cave can be made with Zermelo as the tied prisoner, believing the world ought to be like the shadows made on the cave wall, that this was "objectivity". If Zermelo were to be freed, he would see that the "smaller" set-theoretic universes he saw (and hence, rejected the existence of) were only projected shadows of what are perfect Forms in their own right, only seeming small from his limited perspective, not by virtue of the Forms themselves. 

Note on usage: Forms = Ideas here. 

Life lessons from mathematicians: L. E. J. Brouwer

 In later years, he became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting. Dutch mathematician and historian of mathematics Bartel Leendert van der Waerden attended lectures given by Brouwer in later years, and commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on — and only on — the foundations of his intuitionism. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy."

          L. E. J. Brouwer - Wikipedia 

I think the lesson here is about "tunneling". In Brower's case he was so convinced of a worldview that he felt that everything must be filtered through this lens. He lived his philosophy, which is admirable in some sense. 

I often fell, and still fall, into the trap of becoming so engaged with something to the extent that I forget everything else: sleep, work, even other people or pursuits I cherish and value, if I would remember them. Often at school and work I'd become obsessed with some puzzle, question, or thought, usually of philosophical or mathematical nature, and squander what little time I had. It's hard to look up when you believe that you're on the verge of some breakthrough, even if you've been looping the same thought for hours. Once I do look up, I realize there will always be thoughts, and there will always be things to keep you occupied. Good ideas and apparent "breakthroughs", it seems, are a dime a dozen. It seems it's better to keep your head up: it's the only way you can look forward. 

Some thoughts

There were a few times in my life where I felt like I was really on my own. On one hand there was that realization that I had to provide for myself and not starve, and while this brought on uncertainty and a little fear, it also gave me a refreshing sense of freedom, autonomy, self-determination. It was exciting. 

It also made me realize how "safe" I play it usually. Yes, I have obligations, responsibilities, and constraints (rather, just really strong incentives and ties to family). But I also shy away from spending, shy away from risk, shy away from enjoying life because, well, guilt and considerations, even things like "it's bad for the environment". Most of all, "being a good person". This Youtuber really hit it home for me:  "Smart" Financial Decisions Create Deadbeats in their 20s (youtube.com)

He talks about how many people have been provided for by their parents their whole life, encouraged to go into "safe" careers. Many times, these "safe" careers tended to become oversaturated, and economic outcomes were not as expected. I heard stories about kids who study hard their whole life, get into tech, but live lonely lives because they aren't very well-rounded. Why don't we just educated people to be flexible, open-minded, well-read, sociable, and critical-thinking? I don't understand this fixation on "jobs" and "careers". 

Jobs and careers are only an economic proxy to society's needs and opportunities. And these opportunities are everywhere if you just look. This is because society pursues new opportunities based on the skills it currently has. And individuals make up society. Data science didn't come out of a vaccuum, it came about because we had the technical know-how of statisticians and computer scientists, and really good computers.  This means that it doesn't really matter so much that you develop the "hot" skills, because as part of society, you impact the "skill portfolio" of society and you, as an individual, have a say in what direction society goes and what opportunities it should exploit.

And you don't have to be an entrepreneur to have a say where society goes, because you have a say in where you go. See, entrepreneurs are simply the extreme end of opportunity discovery and exploitation -- they convert implicit opportunities into net new explicit functions (often formalized in terms of "jobs"). But as they say, there is nothing new under the sun -- a lot of jobs already do exist to some degree* -- it's just that they're on the tail end of the distribution and often not marketed in obvious ways. Finding the right spot for you is a matter of navigating the "search space", attaining skills, attaining industry/related knowledge so that you know where and how to look -- e.g. Networking with the right people? Communities? Even search engine keywords said communities might expose? 

So yeah, this compartmentalization of societal functions in "jobs"? It's not so clean-cut, and at the atomic first principles level it's never been about jobs, it's been about individuals making up a society and collaborating to... do stuff. Be productive, fill needs, express themselves, whatever humans want to do. Maybe we gravitated towards jobs because it's structure, it's nice and makes things simpler. And that's fine. 

Being well rounded you make it so that your passions and skills can fit anywhere. There's a case for specialization if you're already sure of what you're going to do but often a breadth-first-search approach is more successful in this kind of search space. 

*If we're going to be nonsensically technical here, we can also think of entrepreneurs as not necessarily creating anything new, but rather selecting things from the hypothetical search space of possibilities and real-izing them into society. And to a lesser extreme you could also entrepreneurs are simply the inevitable product of their times, as we all are. 

Society is a transcendental function (almost), and we are trying to factor it as a polynomial because we are human and finite. Jobs, careers -- these are lower order terms.

Twin Twister 2 - An English axiom of choice

When we are given a statement like "At least K out of N objects are <property 1>" (1), we might say something like "Then the other ones must be <property 2>" (2). When we say "other ones", we are selecting, presumably, a set "N\K" with size N-K. The problem is, this implies that there is some choice of K to begin with, when really all we gave was some statistic regarding the whole N-set. 

However, there are times when (2) is true no matter what our choice of the K-set is, as long as the K-set satisfies <property 1>, e.g. in the Twin Twister problem. 

But what are the times when it does matter? 

And in the Twin Twister problem, we have probabilities, with the statement "Then the other ones must be <property 2> with probability P." How does this change the problem? 

Is the probability computed over a set of outcomes spanning the different possible choices of K? If yes, how should those different choices be weighted? If not, is there something contextual that provides a canonical choice of K (e.g. say the vet knows something)?  Given that we find a way to choose K, is there some identifying factor that allows us to compute P over an outcome space that holds K fixed (e.g. identifying code on the lambs, and the outcomes all have this invariant -- thus turning the probability to 1/2)?

A general problem is this: A non-probabilistic, 0th order sentence S has a singular interpretation in a single context, but in the scope of hypothetical realities and possibilities, we might run into a situation where there is no "canonical" interpretation over them all -- i.e. an "atom" or "object" in a sentence may not necessarily correspond to a singularly identifiable "object" in all of the possibilities. There may even be multiple levels to this: e.g. there are multiple choices of K, and multiple possibilities of gender assignment. Unless the problem is conditioned so that we restrict ourselves to the outcome space where the lambs are identified based on genetic code, we run into a "choice function" scenario. 

Twin Twister

A puzzle from The Guardian:  

A 17th century farmer observes that one of his sheep is pregnant. As all famers know, lambs arrive as non-identical twins, each with a 50-50 chance of being male or female. The local vet has an Elizabethan ultrasound machine and finds out the genders of the lambs: “Is it true that at least one of them will be male?” asks the farmer. “Yes, it is true” replies the vet.

“In that case,” the farmer says, “the other one will most likely be female”. Is the farmer correct?

Solution Yes!

There is a 2/3 chance that one of the lambs will be female. If we know that at least one lamb is male, then the possible pairings of the first and second lamb are male-male, male-female and female-male, and each of these pairings is equally likely. There will be a female in two of the three scenarios, hence the 2/3 probability.

Did you solve it? Art thou smarter than Shakespeare? | Mathematics | The Guardian

I can see the confusion here. When the farmer says "the other one will most likely be female", we may model that situation as the following:

As one of the lambs are male, take this one, call it lamb 1. What is the probability that the "other" lamb, call it lamb 2, is female? 

Also consider this variation:

We have 100 lambs. The doctor now says, "Lambs 1 and 3-100 are male". What you don't know is that the doctor is assigning the numbers after the fact -- if there were 50 lambs he would have said "Lambs 1 and 3-51 are male", strategically leaving out 2 and filling in the rest as male. What is the probability that "lamb 2" is female? 

In either case, it really depends on what we are referring to when we discuss "lamb 2", i.e. the farmer's "other lamb". Suppose the doctor knows the genders of both lambs. Saying "the other one" implies the existence of the "first one". Suppose that the doctor assigns "first one" to the first male lamb he has scanned -- he scans them in order. Say that the lambs also are born with unique genetic codes derived deterministically from the mother, regardless of whether they are male or female: call them lamb A and lamb B. As it happens, lamb B was the first to be scanned and was male, hence this is now referred to as "the first one". Therefore, lamb A is now the "other one". What is the probability that the "other one", lamb A, is female? Of course, the probability that lamb A is female is 50%. 

So it really depends on what the farmer means by "the other one". What happens when there is no genetic code invariant across the different possibilities, with which we could interpret an association with the farmer's "other one"? If the farmer's emphasis is on the "other" part of "other one", then perhaps 2/3 is correct. If the farmer's emphasis is on the "one" part of "other one"... then, it gets difficult. Identification across multiple realities requires something to identify by... if that doesn't exist, isn't the identification purely local to this particular reality? And what is the farmer's reality? It depends -- it could be M-M, M-F, or F-M, which then reduces down to the computation* below, assuming that the farmer chooses on a "first one" completely random 50-50 chance. 

If we were to interpret this way: that "the other will be female" is "most likely" correct, i.e. correct in 2/3 of the circumstances, then his statement is true. 

Or we could argue that the lamb is either a male or female, and isn't "most likely" anything, so he's correct in 2/3 cases and wrong in 1/3 but no one would take this interpretation... 

We should think about what went through the farmer's head when he said that. How can he choose which one to make the "other one"? 

Maybe like this:

1. There are two lambs. 

2. As we now know one of these two lambs is male, select one male lamb from the set... randomly. 

3. Label this select lamb "first". Label the other, "other". 

4. The probability that "other lamb" is female is 2/3. 

I don't really like this because I know for sure the farmer wasn't really thinking about all this, and I'm sure the farmer didn't consciously make a random choice. Though what did happen is that he did "select out" a male. But... how??

Take this example:

There's 16 candy bars. As it happens, by weight I can tell there's at least 15 Hershey bars in there. You say, "Ah, the other one's most likely a Dark Chocolate Hershey Bar, isn't it?" As it so happens all 16 are Hershey Bars, and there are 2 Dark Chocolate Hershey Bars. So... what?

What if it's like this: with 50-50 chance it's either Hershey or a Butterfinger. And if it is a Hershey, it could either be a Dark Chocolate Hershey bar or a Milk Chocolate one, with 50-50 chance again. 

What if I told you there's at least 2 Hershey bars in there? What do we mean by "the other ones"? Especially if there were actually 4? By "selecting out" 13, aren't we selecting out 2 Hershey bars as well? 

How can we make using "the other one(s)" totally nonsensical? Suppose Hershey bars come with a code. Your assorted bag of candy, the probability of the ratio of Hershey bars to Butterfingers is actually determined by the included codes written on the Hershey bars. Say that for bags containing only a Hershey bar with the code "A", there's only Butterfingers for the rest of them, and for bags containing Hershey bar "B", there's Snickers. Only thing is: there's always a code "A" Hershey bar. 

Given that there is at least 1 Hershey bar in this bag, how do we interpret the following statement: "the other candy bars are probably Butterfingers"? What if we "choose" Hershey bar A as the "chosen" bar, and the rest as the "other ones"? What if we "choose" Hershey bar B, then we would be wrong..? 

...Is there any scenario where we could "get them" for using "other ones" without thought? Where the truth of an "other ones" statement depends on the choice of said "other ones", where the context is a quantitative problem?

*Computation: in the M-M case, the farmer might have chosen either of the two males as a "first one", hence we have M-M-1 and M-M-2 each with probability 1/6. In the M-F case, we have M-F-1 with probability 1/3, and in the F-M case, we have F-M-2 with 1/3. The "other one" is F with probability 1/3 + 1/3 = 2/3, and M with probability 1/3. 

Potential hampered by... air quality?

 Some days I just feel like I have the potential to do great things and make a real impact, help lots and lots of people, but I'm just being held back. The usual scapegoat is air quality. But I'm not sure if that's the real reason. 

I've read stuff about what CO2 can hurt cognitive performance, how VOCs can cause bad health outcomes, how chronic sinusitis can cause depression, and so on. But what is it really? Am I using this bad air quality as an excuse, in much the same way I "have" to clean my desk before I can start working? Am I self-handicapping, to protect myself from failure and difficulty? I've seen people's lives ruined by obsession over things like mold and other invisible threats. Am I one of them? 

I just feel subpar most of the time these days and I don't know if it's coming from something physiological, or something else...