This marks the 20th post on this page - thank you all for continuing to read the random articles I put here. It’s very much appreciated and glad we have a small audience.
Let’s continue exploring the theory of reflexivity from post 19 - it’s been a topic of interest for the last few days.
Fallibility and Reflexivity
To review from the last post, let’s look at the definitions of fallibility and reflexivity once more.
i) Fallibility - in situations that have thinking participants, the participant’s view of the world never perfectly corresponds to the actual state of affairs (i.e., distorted views)
ii) Reflexivity - the imperfect views can effect the situation to which they relate through the actions of the participants (i.e., views influencing the situation)
For example, if investors believe that markets are efficient, that will change the way they invest, which will in turn change the nature of the markets in which they are participating (although, not necessarily making them more efficient… likely the opposite!).
It’s also worth noting the difference between these concepts when applied to the natural world vs social world.
In social systems, fallible humans are not merely scientific observers but also active participants in the systems themselves. This is what makes social systems reflexive… while natural systems are not reflexive… because they do not have thinking participants.
Put differently, thinking is part of the reality that people have to think about. That’s reflexivity (in other words, self reinforcing). Your views shape reality, reality leads to refined views, which further shapes reality, etc. That’s circularity in social systems.
We briefly mentioned subjective and objective reality in the last post.
Subjective reality includes the participants thinking.
Objective reality deals with the actual state of affairs.
Here is an example of subjective reality - remember the blue black / white gold dress issue from a few years ago?
The dress can only be one set of colors and not both. But people view this reality differently (we have a distorted view).
Alternatively, take the below picture. Do you see a rabbit or a duck?
Hopefully these are fun examples that drive the point home. We all view things differently. These views then shape our actions. Our actions effect the world, which further refines our (distorted) views.
There can only be one objective reality - the actual state of affairs. Although there can be many subjective realities based on participant’s views, and these subjective realities can diverge or converge to objective reality.
****A side note on subjectivity - when you observe something, you don’t observe the other side of it. For instance, imagine looking at the moon. You see one side of the moon, but in observing that side, you are unable to observe the other side. We can never get a full picture of the actual state of affairs. We have a fundamental limitation of grasping the whole.
Take AI as a practical example. You might think AI is purely objective, but it’s not. AI requires humans to train and teach AI. Those humans have distorted views and biases and they may be unaware that they are acting upon those views when training AI.****
The conversion or diversion process mentioned above is what’s referred to as the negative and positive feedback looks. Positive feedback further separates views from reality. Negative feedback converges views and reality to an equilibrium.
In dealing with other people, we can aim to understand their subjective reality.
How do they see the world?
How can their actions influence the world?
Based on your estimate of their subjective reality, what actions are they likely to take?
Back to reflexivity, but applied to markets.
Markets react to participant’s expectations.
And those perceptions influence prices, tending to validate themselves in a self-reinforcing process until some unpredictable event jolts expectations.
This is how financial markets function - a self reinforcing, reflexive process between expectations, prices, and fundamentals.
High expectations leads to higher prices. Higher prices can actually improve fundamentals, which further validates the high expectations, driving prices further higher, etc. etc. this process happens until expectations can be validated no further / a climax is reached and the reflexive process moves in the other direction.
As you can infer, equilibriums don’t exist in markets. Instead you have reflexive processes as described above. Meaning, thinking and actions of market participants will effect market behavior.
The market will in turn influence the fundamentals and shape new expectations… in a continuing reflexive process.
I know this is repetitive - I’m trying to restate the idea several times in different ways to convey the concepts clearly.
Uncertainty, Imperfect Understanding and Mathematics
In the last post, I mentioned that “we act on the basis of imperfect understanding.” Let’s explore this idea.
To put it differently, we can never perfectly grasp reality (this is the concept behind fallibility).
Something remarkable happened in 1931 to cement this idea - known as Gödel’s Incompleteness Theorem.
Before we jump into Gödel’s discovery, let’s briefly review basic mathematics as it will help us frame the building blocks to the Incompleteness Theorem.
Mathematics has to start somewhere, and that starting point is known as an axiom. Axioms are essentially widely agreed upon mathematical assumptions - the seed that allows mathematics to grow and discover new ideas.
Example of an axiom - you have two distinct points on the same plane. There is a distinct line that passes through the 2 points.
Another axiom is Euclid’s 5th Postulate. If you have a line L and a point P not on L, there is exactly one line through point P that is parallel to L.
These are the building blocks to math. Postulates, mentioned above, are statements that are assumed to be true without proof. We’ll come back to this.
So, math entails formal systems that starts with a set of axioms. The axioms to be used depend on the observer - the one who chooses the axioms.
“Who is watching the watcher?”
You have a formal system with axioms. Then you have rules of inference (logical rules). If A is true and A implies B, then B is true (Aristotle, also said A is A. Ayn Rand mentioned this many times).
Axioms - accepted as true statements
Logical Inference - way to produce new statements
Theorems - name of the new statements that have been deduced
How far can you go? How many statements can you create?
But, you need to be consistent. Consistency in the sense that the system is not self contradictory.
Back to Gödel. Gödel says that this procedure (starting with axioms and using logical inference to derive theorems) will never prove all statements true.
This is an incredibly important point and was a groundbreaking discovery in 1931.
If you have a sufficiently formal system, if it’s consistent, there will be a true statement that cannot be derived from the process of deriving theorems from axioms. This is Gödel’s Incompleteness Theorem.
Mathematics
Back to the fun. We learned above that there are true statements that don’t have proofs. These are postulates but also unknowable statements that we have yet discovered and will not be provable (this is the Incompleteness Theorem). To recount, it says that the number of mathematical truths is always greater than proofs.
This is important because it supports our discussion around imperfect understanding. If we cannot prove every true statement, we can never perfectly understand the world. Knowledge will never be perfect, so there is constantly room for improvement and further understanding (improving upon our imperfect understanding of the world). Further, with reflexivity, the [social] world changes as our views change, so there is no finite perspective we are working towards. Knowledge is infinite due to reflexivity. We must constantly adapt and refine our understanding of the world.
As mentioned above, if you have a true statement that can’t be proved, add this as a new axiom and that will expand what you can prove within mathematics.
Pre-Gödel, common belief was that there will be a set of axioms in which you can deduce all sets of truths within mathematics.
But Gödel argued otherwise.
Gödel’s Incompleteness Theorem
Without getting too complex (mainly because it’s beyond my understanding!), Gödel’s system turns every statement into a number. Think of this similar to binary code on computers. For the statement to be provable, it’s code number must be divisible by the axiom’s.
Now, suppose we have a statement: “This statement cannot be proved from the axioms”
Since Gödel turns every statement into a code number, he can then turn it into an equation. And every equation must be true or false, so let’s explore the two options.
False - assuming the above statement is false, that means the statement is provable from the axioms.
But….
A provable statement must be true
We started with something we thought was false, but now we’ve deduced it’s true —> therefore we have a contradiction.
But, assuming math is consistent, we can’t have contradictions (also, hats off to you Ayn Rand - contradictions don’t exist, check your premise), so the statement must be true
Within a system, we have found a true statement which cannot be proven true within the system.
Put simply, since there are more true statements than there are proofs, we will never know everything with certainty. There will always be true statements which cannot be proved.
This leads to i) an imperfect understanding of the world and ii) uncertainty. Remember, uncertainty in this case relates to our understanding and interpretation of the world.
More posts to come on these topics as I spend more time with Soros’ ideas… and potentially continue down the math theory rabbit hole.
+++
Until next time.
John Galt