Richard Carter uses his writing journal practice to "think out loud" to himself. Often, laying out extended arguments helps people to refine and reshape their thinking as they're better able to see potential holes or missing pieces of arguments. It's the same sort of mechanism which is at work in rubber duck debugging of computer code: by explaining a process one is more easily able to see the missing pieces, errors, or problems with the process at hand.

Carter's separate note taking and writing journal practice being used as a thought space or writing workshop of sorts is very similar to the process seen in my preliminary studies of Henry David Thoreau's work in which he kept commonplace books and separate (writing) journals which show evidence of his trying ideas on for size and working them before committing them to his published works.

This definition can be used to demonstrate why the following function is continuous:

\(f: [0,2\pi) \to S^1\) where \(f(\phi)= (\cos\phi, \sin\phi)\) and \(S^1\) is the unit circle in the cartesian coordinate plane \(\mathbb{R}^2\).

Intuition

The preimage of open (in codomain) is open (in domain). Roughly, anything "close" in the codomain must have come from something "close" in the domain. Otherwise, stuff got split apart (think gaps, holes, jumps) on the way from our domain to our codomain.

Formalism

For some \(f: X \to Y\), for any open set \(V \in \tau_Y\), there exists some open set \(U \in \tau_X\) so that it's image under \(f\) is \(V\). In math, \(\forall V \in \tau_Y, \exists U \in \tau_X \text{ s.t. } f(U) = V\)

Demonstration

So for \(f: [0,2\pi) \to S^1\), we can see that \([0,2\pi)\) is open under the subspace topology. Why? Let's start with a different example.

Claim 1: \(U_S=[0,1) \cup (2,2\pi)\) is open in \(S = [0,2\pi)\)

We need to show that \(U_S = S \cap U_X\) for some \(U_X \in \mathbb{R}\). So we can take whatever open set that overlaps with our subspace to generate \(U_S\text{.}\)

proof 1

Consider \(U_X = (-1,1) \cup (2, 2\pi)\) and its intersection with \(S = [0, 2\pi)\). The overlap of \(U_X\) with \(S\) is precisely \(U_S\). That is,

$$ \begin{align} S \cap U_X &= [0, 2\pi) \cap U_X \ &= [0, 2\pi) \cap \bigl( (-1,1) \cup (2,2\pi) \bigr) \ &= \bigl( [0, 2\pi) \cap (-1,1) \bigr) \cup \bigl( [0,2\pi) \cap (2,2\pi)\bigr) \ &= [0, 1) \cup (2, 2\pi) \ &= U_S \end{align} $$

Notation of a random process that has outcomes

The "universal set" aka "sample space" of all possible outcomes is sometimes denoted by \(U\), \(S\), or \(\Omega\): https://en.wikipedia.org/wiki/Sample_space

Probability theory & measure theory

From what I recall, the notation, \(\Omega\), was mainly used in higher-level grad courses on probability theory. ie, when trying to frame things in probability theory as a special case of measure theory things/ideas/processes. eg, a probability space, \((\cal{F}, \Omega, P)\) where \(\cal{F}\) is a \(\sigma\text{-field}\) aka \(\sigma\text{-algebra}\) and \(P\) is a probability density function on any element of \(\cal{F}\) and \(P(\Omega)=1.\)

Somehow, the definition of a sigma-field captures the notion of what we want out of something that's measurable, but it's unclear to me why so let's see where writing through this takes me.

Working through why a sigma-algebra yields a coherent notion of measureable

A sigma-algebra \(\cal{F}\) on a set \(\Omega\) is defined somewhat close to the definition of a topology \(\tau\) on some space \(X\). They're both collections of sub-collections of the set/space of reference (ie, \(\tau \sub 2^X\) and \(\cal{F} \sub 2^\Omega\)). Also, they're both defined to contain their underlying set/space (ie, \(X \in \tau\) and \(\Omega \in \cal{F}\)).

Additionally, they both contain the empty set but for (maybe) different reasons, definitionally. For a topology, it's simply defined to contain both the whole space and the empty set (ie, \(X \in \tau\) and \(\empty \in \tau\)). In a sigma-algebra's case, it's defined to be closed under complements, so since \(\Omega \in \cal{F}\) the complement must also be in \(\cal{F}\)... but the complement of the universal set \(\Omega\) is the empty set, so \(\empty \in \cal{F}\).

I think this might be where the similarity ends, since a topology need not be closed under complements (but probably has a special property when it is, although I'm not sure what; oh wait, the complement of open is closed in topology, so it'd be clopen! Not sure what this would really entail though 🤷‍♀️). Moreover, a topology is closed under arbitrary unions (which includes uncountable), but a sigma-algebra is closed under countable unions. Hmm... Maybe this restriction to countable unions is what gives a coherent notion of being measurable? I suspect it also has to do with Banach-Tarski paradox. ie, cutting a sphere into 5 pieces and rearranging in a clever way so that you get 2 sphere's that each have the volume of the original sphere; I mean, WTF, if 1 sphere's volume equals the volume of 2 sphere's, then we're definitely not able to measure stuff any more.

And now I'm starting to vaguely recall that this what sigma-fields essentially outlaw/ban from being possible. It's also related to something important in measure theory called a Lebeque measure, although I'm not really sure what that is (something about doing a Riemann integral but picking the partition on the y-axis/codomain instead of on the x-axis/domain, maybe?)

And with that, I think I've got some intuition about how fundamental sigma-algebras are to letting us handle probability and uncertainty.

Back to probability theory

So then events like \(E_1\) and \(E_2\) that are elements of the set of sub-collections, \(\cal{F}\), of the possibility space \(\Omega\). Like, maybe \(\Omega\) is the set of all possible outcomes of rolling 2 dice, but \(E_1\) could be a simple event (ie, just one outcome like rolling a 2) while \(E_2\) could be a compound(?) event (ie, more than one, like rolling an even number). Notably, \(E_1\) & \(E_2\) are NOT elements of the sample space \(\Omega\); they're elements of the powerset of our possibility space (ie, the set of all possible subsets of \(\Omega\) denoted by \(2^\Omega\)). So maybe this explains why the "closed under complements" is needed; if you roll a 2, you should also be able to NOT roll a 2. And the property that a sigma-algebra must "contain the whole space" might be what's needed to give rise to a notion of a complete measure (conjecture about complete measures: everything in the measurable space can be assigned a value where that part of the measurable space does, in fact, represent some constitutive part of the whole).

But what about these "random events"?

Ah, so that's where random variables come into play (and probably why in probability theory they prefer to use \(\Omega\) for the sample space instead of \(X\) like a base space in topology). There's a function, that is, a mapping from outcomes of this "random event" (eg, a role of 2 dice) to a space in which we can associate (ie, assign) a sense of distance (ie, our sigma-algebra). What confuses me is that we see things like "\(P(X=x)\)" which we interpret as "probability that our random variable, \(X\), ends up being some particular outcome \(x\)." But it's also said that \(X\) is a real-valued function, ie, takes some arbitrary elements (eg, events like rolling an even number) and assigns them a real number (ie, some \(x \in \mathbb{R}\)).

Aha! I think I recall the missing link: the notation "\(X=x\)" is really a shorthand for "\(X(\omega)=x\)" where \(\omega \in \cal{F}\). But something that still feels unreconciled is that our probability metric, \(P\), is just taking some real value to another real value... So which one is our sigma-algebra, the inputs of \(P\) or the inputs of \(X\)? 🤔 Hmm... Well, I guess it has the be the set of elements that \(X\) is mapping into \(\mathbb{R}\) since \(X\text{'s}\) input is a small omega \(\omega\) (which is probably an element of big omega \(\Omega\) based on the conventions of small notation being elements of big notation), so \(X\text{'s}\) domain much be the sigma-algrebra?

Let's try to generate a plausible example of this in action... Maybe something with an inequality like "\(X\ge 1\)". Okay, yeah, how about \(X\) is a random variable for the random process of how long it takes a customer to get through a grocery line. So \(X\) is mapping the elements of our sigma-algebra (ie, what customers actually end up experiencing in the real world) into a subset of the reals, namely \([0,\infty)\) because their time in line could be 0 minutes or infinite minutes (geesh, 😬 what a life that would be, huh?). Okay, so then I can ask a question like "What's the probability that \(X\) takes on a value greater than or equal to 1 minute?" which I think translates to "\(P\left(X(\omega)\ge 1\right)\)" which is really attempting to model this whole "random event" of "What's gonna happen to a particular person on average?"

So this makes me wonder... Is this fact that \(X\) can model this "random event" (at all) what people mean when they say something is a stochastic model? That there's a probability distribution it generates which affords us some way of dealing with navigating the uncertainty of the "random event"? If so, then sigma-algebras seem to serve as a kind of gateway and/or foundation into specific cognitive practices (ie, learning to think & reason probabilistically) that affords us a way out of being overwhelmed by our anxiety or fear and can help us reclaim some agency and autonomy in situations with uncertainty.

D. J. Levitin, The organized mind: thinking straight in the age of information overload. New York, N.Y: Dutton, 2014. #books/wanttoread

A general truism in my experience, but I'm curious what else Levitin has to say on this subject.

Thinking in solitary can increase one's susceptibility to confirmation bias. Thinking in groups can mitigate this.

How might keeping one's notes in public potentially help fight against these cognitive biases?

Is having a "conversation in the margins" with an author using annotation tools like Hypothes.is a way to help mitigate this sort of cognitive bias?

At the far end of the spectrum how do we prevent this social thinking from becoming groupthink, or the practice of thinking or making decisions as a group in a way that discourages creativity or individual responsibility?

I wonder a bit about applying behavioral economics or areas like System 1/System 2 of D. Kahneman and A. Tversky to social media as well. Some (a majority?) use Twitter as an immediate knee-jerk reaction to content they're reading and interacting with in a very System 1 sense while others use longer form writing and analysis seen in the blogosphere to create System 2 sort of social thinking.

This naturally needs to be cross referenced in peoples' time and abilities to consume these things and the reactions and dopamine responses they provoke. Most people are apt to read the shorter form writing because it's easier and takes less time and effort compared with longer form writing which requires far more cognitive load and time expenditure.

search engines usually can surface that faster (less cognitive load than recalling what and where you store something) than you retrieve it in your second brain (abundance info, do can always retrieve from external source in a JIT fashion)

some thoughts when skimming through stream-of-consciousness journals like these

if I want to absorb the information and "learn" faster, then reading faster or summarising the text is not the solution, because a text is already a compressed lossy encoded form of the initial thought. to decode it further and transfer it into my head would risk too much missing bits of information.