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... 

AI economics

 "Money" is just a proxy for resources. 

"Jobs" is just a proxy for labor resources.

The resources and labor exist whether money or jobs exist or not. It is the distribution of said resources that necessitates a system of money and jobs. 

We should not confuse the measurement of something with the thing itself. Measurement stands only as proxy and when the concept fails, we should fall back on the ground truth.

For instance, for many, the fear of AI is based around the fact that it will automate away many jobs. But the focus on "jobs" as a measure of economic health and prosperity only holds under certain circumstances. And said circumstances may not hold under an AI-driven economy. The same human resources exist whether AI is adopted or not. So the real problem, which people inherently understand but don't necessarily express correctly, is that during the transition period AI will make certain skills less valuable to others, and there will be cost for re-skilling people, while at the same time the rest of society will benefit from lower cost. Also relevant is the risk of monopoly as AI companies subsume entire industries.

Relatinoship dynamics and "optionality"

Incomplete, just notes:

 https://www.youtube.com/watch?v=lad3sX6cPJo

"facts"
- women leave men at higher rates (2x)

- men mature slower

- men seek younger women regardless of their own age

- men live shorter lives

- age gap M > F

Consider: "viable dating age" for women vs men -- women have shorter timeframes, hence denser "time in a relationship"?

Optionality? Or is the real problem that younger men are competing against older men for a smaller pool of younger women? 

What about the fact that women leave more often, initiating breakups? Simply, competition
Yes, evo bio is a part of it. no, that doesn't make women (or men) "superficial" and self-serving. Women seek a partner who will support/love and is mature. That's all modeling on the micro-scale though. Given a dating time distribution, it could describe any kind of micro-level situation. The same picture happens even if men break off the relationship but have trouble finding the right person due to competition. 

How about we enhance our model to talk about the economics of optionality? Do women really have more options? Supply vs demand? 

In general, my goals are:

-Model the econometrics of dating such that it may possible explain popular observed phenomena such as "men are terrible" and "women have optionality" as either secondary inevitabilities of the mathematics, or more primarily motivated by the mathematics by some causal mechanism.  

    - Do women actually have optionality? In what sense?

    - Are men incentivized to be more awful? Are they? Or is there an inevitable reason why that sense would arise/become popular? 

    - Does this model give us any other insights? Explain other phenomena? (Crazy cat ladies, e.g. women left behind by the dating market, their number, divorce rates, etc)

https://medium.com/the-renaissance-economist/how-an-economist-sees-the-dating-market-eb3afe0c9e35

Another math fb comment

 

My response: