A Toy Model Of Epistemology

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 Chaitin 2 composite2

We now close out what I started in the post-structuralism and Hinduism post. There was described the seemingly eternal tension between the wild Dionysian spirit of the continuum (Brahman) and the Apollonian desire to tame the continuum (Maya). Last time, we framed this “essential tension” as that between what we can grasp in our minds and what we cannot grasp in our minds. Here I offer a mathematical view of how the graspable and ungraspable forever attempt to eat each other. That is, I offer a toy model of epistemology.


 

Table of Contents:
Part 1 : Structuralism, Functionalism, Post-structuralism & Hinduism
Part 2 : The Graspable and the Ungraspable
Part 3 : A Toy Model Of Epistemology


 

To a man with a hammer, everything looks like a nail.
Unknown

Everything Moves, Especially Ideas
Last time we divided all of which we are aware into two categories: the graspable, G, and ungraspable, U. We generalized G and U in Chaitin’s terms where the G is that knowledge that can be contained by a “theory” whose bit count is much, much less than the data the theory describes. U is that knowledge where there is no theory, and our description of the knowledge has about the same bit count as the data that is being described.

We now wish to take these ideas and embed them in a dynamical model. Why? Because knowledge is not static. This is the problem with the mathematician’s interpretation of Plato’s realm of pure thought; they seem to incorrectly assume it is static. They assume all knowing (or at least mathematical truths) stands eternally revealed as if a bunch of statues were lining the walkways of some 18th century European walking garden.

Sorry, it’s just not like that. All the vistas of the Mental plane are in constant motion. Even those mental vistas that appear to be static are just moving very slowly relative to the others. Plato’s “realm of ideas” is just the mental plane. Even the ultimate knowledge at the Adi plane, or what Yogi’s call the “asmita” level of the gunas, is not static and eternal. At this level is the Logos, the Divine Plan, in its entire insanely complex, infinitely simple unspeakable beauty. But even this moves. As I discussed in Experience: everything relative moves.

Knowledge we can grasp and knowledge we will never grasp both move. They move in many ways. The dynamic model I now discuss captures only one way they move. But it is an important movement because it describes how U and G interact. They interact thus: U always tries to engulf G and G always tries to cage U and thereby convert it to G.

Grasping the Ungraspable and Losing It Again
We are confronted with U, with the ungraspable, in our experience: time, space, consciousness, the continuum, and so on. We seek to understand it, and thereby convert it to the graspable; to G.  Codifying and formalizing knowledge is the act of converting U to G. Successful formalism, in whatever field of understanding—mathematics, science, linguistics, social science, yes, even in religious terms, and so on—all represent the process of converting U into G. We convert what was previously ungraspable into the graspable.

However, no matter what formal system we devise, whether as seemingly tight as Zermelo–Fraenkel set theory, or as loose as the Ten Commandments, the ungraspable rears its ugly head and seeks to engulf any formalisms we invent (or discover, depending on one’s proclivities). The idea of U engulfing G is the idea that the more we know, the more new questions arise.

For example, classical physics ran into serious problems with light and energies at atomic scales. Quantum mechanics was invented because it solved those problems. But in doing so, it opened up whole new cans of worms. Now some people worry about the double slit experiment, nonlocality, and the collapse of the wave packet. We are defeated as soon as we succeed because U always impinges on and seeks to engulf G.

This recalls Newton’s quote: “…I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

John F. Kennedy said it in a less poetic manner: “The greater our knowledge increases the more our ignorance unfolds.”

Socrates perhaps said it most succinctly: “I know that I know nothing”.

The Competition Between U and G
Thus, there is ever a dynamic interplay between those ungraspable things of which we are aware and our need and desire to convert them to the graspable. We can express this very simply by saying: G seeks to destroy U, but U seeks to destroy G. By “destroy” I mean that G converts U to G, which ends up getting rid of, or destroying U. Similarly, U seeks to blur out, to make G fade away, which again, gets rid of G.

Is there a way to express the dynamic interplay of U and G? Yes there is. There are many such ways in fact. It is a standard idea in dynamics to mathematically describe the competition between any two factors. Such relationship is called a “mutual antagonism”. The resulting math gives rise to a circumstance called “winner take all”.

As usual, we can draw a diagram to illustrate a mutually antagonistic relationship:

UvsG
We can also write an equation that describes the “competition” between U and G:

Eq1
This is an old fashioned ordinary differential equation (ODE). It is called an ODE “system” because there are two equations making up the system. It is called a “coupled” ODE system because the 1st U equation contains G, and the 2nd G equation contains U.  It is the exact same equation I used to describe whether a biological cell lives or dies after being injured. This system of coupled ODEs is my hammer.

It doesn’t matter for the sake of this post to understand the math in any depth. What it basically says is: As G gets bigger, U gets smaller, and as U gets bigger, G gets smaller. The ODE system decides, in any instance of a competition between U and G, which one will win the competition.

How does it determine who wins and who loses? The answer is pretty simple. The equation has three parameters: n, and the two h terms: hU and hG.

The n tells how strong U and G interact with each other. The higher n is, the more they interact. If n is a low number, U and G are only loosely bound to each other; one would say they are loosely “coupled”.  However, it is a thing of diminishing returns, and after about n = 25, it doesn’t matter if you increase n; U and G are coupled about as tight as they will ever be.

h” stands for “thresHold”. U and G each have their own thresholds: hU for U and hG for G. The numerical value of each threshold tells at what point each begins to engulf the other. The number value of the threshold tells who will win the competition. There are only three possibilities:

  1. hU > hGU is greater than G and U will win
  2. hU = hGU equals G and it is a tie
  3. hU < hGU is less than G and G will win

Qualitatively, that is all one really needs to know about how this equation works. The larger n is, the stronger U and G interact.   Who wins depends on the value of the thresholds as listed above.

The equation is perfectly general. It can apply to any two things that mutually try to destroy each other. In this sense, it is the equation of Yin and Yang. Because the equation can be interpreted in this fashion, it has very wide applicability to describing the things in nature. Using it as I am here to describe the relationship between ungraspable (U) and graspable (G) knowledge is only one possible application. As I said, I have also used it to explain the interplay between the forces of life and death after a cell is injured.

Solving the System
There is a whole mechanical procedure for solving ODEs in general; several of them actually. They are programmed into Matlab and accessible to the masses. We need not concern ourselves with any of that. What we do need concern ourselves with is the following.

I made a big old deal in What Is Science? about the gunas, about how they map to our understanding of dynamics. Here we need to focus on tamas: lethargy, lack of motion, darkness, stillness. These are the terms Hindus associate with tamas. In dynamics, there are what are called “fixed points”. These are points, precisely defined by the ODE system and the specific numbers you plug in for the parameter, where the system stops. Whatever change the ODE is supposed to model, it stops changing at the fixed points.

The exact definition of the fixed points for the above ODE system is: Eq2The rates of change equal zero: tamas; stillness.

This is ironic in a sense. ODEs describe change, but they also describe when the system no longer changes. This is all a very big deal in mathematics. Poincaré was probably the first to recognize the importance of the fixed points. We can say, with respect to an ODE, that determining the fixed points “solves” the ODE. This is what we want to do now.

The Solutions of the System
What is really amazing about the ODE system above is that sometimes it has one answer, in terms of fixed points. But sometimes it has two answers, in terms of fixed points. These are called “monostable” and “bistable”, respectively.

With respect to our ODE system, the monostable solutions indicate unambiguously that either U wins or G wins. There is never any case where U and G tie that can manifest in reality. I won’t try to explain that here but will say this much. The only time U ties with G is at infinity, a fact itself which is ungraspable. So, given our model, we could say that when U ties with G that U actually wins! (that is kind of a weird joke but I won’t try to explain it).

The bistable solutions are the interesting ones. These have two answers. For one answer, U wins and for the other answer G wins. It is a very strange situation where the system is such that both have the possibility to win.

How do we express whether U wins (monostable), G wins (monostable), or both win (bistable)? We make a plot, a graph. We plot the 3 parameters against each other, in a 3D space. Then, when the combination of the 3 parameters is such that both U and G win (e.g. is bistable), we plot a point in the 3D space. Recall the three parameters:

hU – the threshold at which U begins to eat G
hG
– the threshold at which G begins to eat U
n
– how strongly U and G interact with each other

The 3D space has the axis as follows:

x-axis = the number value for hG
y
-axis = the number value for hU
z
-axis = the number value for n

So, to express the answers, we plot a point where the 3 number values, when plugged into the ODE system above, give the bistable answer where both U and G win.

And here, my Friends, is what that looks like:

n=1to20

All bistable solutions are contained inside of the blue volume that kind of looks like a boat.

How To Read The Graph
You can see that the resulting graph on the left makes something that looks like a right-leaning pear. Here we are looking down on the plot so it is only 2 dimensional. The middle and right plots now tilt into the 3D space. As we tilt, you can see it makes a funky shaped something-or-another. It kind of reminds me of a boat a little bit.

How to you interpret this shape?

For any numbers falling inside of the boundary of the pear (or boat), you get bistable answers.

Everything outside the boundary is monostable. Points to the upper left are where U wins. Points to the lower right are where G wins.

If anyone wishes to see this thing in 3D, I made an animation which you can see here, and from which the following picture is taken:

nAAAA-0276

Yeah, yeah, yeah…I was going for the 1980’s look with the checker board floor….!

Just so all three of my Readers know, in all my reading about differential equations, I have never seen anyone or any source illustrate the general solution to the above ODE system in this form. To my knowledge, the above pics are the first in this regard. So, you are getting some real, original results here.

What Does This Mean?
There is an idea in math and physics called a “toy model”. A toy model is a math model that is designed to illustrate some point, knowing full well that it is not meant to be thought of as “reality”. The value of a toy model is that it simplifies things, or isolates some one feature of a thing, that the toy model highlights and makes plain and obvious.

This is what the ODE system and the solutions illustrated above are: they are a toy model of epistemology. The model is very simple, unrealistically simple, as toy models are supposed to be.

Recall where we started: U is meant to represent a Chaitin construction where the theory is about as big as its output. G is meant to represent a Chaitin construction where the theory is much smaller, in bit size, than the output. Although I used the idea of “theory” previously, Chaitin is really talking about computer programs. So, let’s shift to this language. U is a computer program where the bit size of the program is about equal to the bit size of its output. G is a computer program where the bit size of the program is much less than the bit size of its output.

Each point in the 3D space is a pair of U and G programs and how they “interact” with each other. So, the whole 3D space contains all possible outcomes of the interaction of any pair of all possible G or U type programs.

It is very abstract perhaps, but that is how math is.

What does it mean? We can interpret each point in the space as follows: There will be some U programs that can be “explained away” by G. This is what it means for G to “win”. On the other hand, there will be some U programs that can never be explained by a G program, and that is what causes U to “win”. In both of these cases, by “win” we mean the monostable solutions.

However, as we can see by the pear or boat shaped volume in the plots above, there is a whole region of this space of interacting U and G programs where both U and G win. This is the region of bistability bounded by the boat-shape in the 3D space.
What does it mean for both U and G to win?

The short answer is: I don’t know.

[Oh! First a startled silence from the audience, then people start booing and throwing tomatoes!!].

Wait, wait, wait: I should say, I don’t know for sure. But of course, I can speculate about what it may mean.

It suggests a condition where it is possible to take some ungraspable aspect of experience and codify it as knowledge, but at the same time, it is possible for the ungraspable aspect to nullify the codified knowledge. That is to say: it is some aspect of nature that can never be fully grasped, but that we can have some degree of understanding of, like the quantum mechanics example above.

There the issue was light and energy. We had an initial coding (“program/theory”)—Maxwell’s Equations—that seemed to work. But then there were problems, and a new coding (“program/theory”) was needed: quantum mechanics. But this gives rise to many new problems and clearly indicates that there will be in the future a new coding (“program/theory”) that will solve the riddles of quantum mechanics, but in the process generate new riddles.

That is, we come to Hegel’s idea of “thesis, antithesis, synthesis”, what is also called the Hegelian dialectic.

So…what we have done is take Chaitin’s general idea of a “theory”, expressed it as a U or G “theory”, embedded this in a dynamical model, and bingo! Out seems to pop Hegel’s ideas!

I think that is pretty cool, actually.

What The…?
Let’s end this coming back to Leibniz. Leibniz used to love to talk about all the possible worlds that God could have made, and wondered why God made this specific world within which we actually exist. Leibniz came to the conclusion that our world must be the “best of all possible worlds” out of the infinite possible worlds God could have created. Our world must be the world with maximum richness with the minimum of rules governing it.

That’s how Leibniz saw it.

I would submit that the boat-shaped space we get from solving our little toy model is, in some sense, this best of all possible worlds that Leibniz used to wonder about. It is governed by one rule: the ODE system above applied to ungraspable and graspable knowledge. From this rule only three types of worlds are possible:

  • Worlds in which nothing can ever be known (U wins always)
  • Worlds in which everything can be known with 100% certainty (G always wins)
  • Worlds in which U and G constantly eat away at each other in an eternal give and take and both always exist.

We obviously live in the 3rd type of world. I’ll let you draw your own conclusions about what’s best or not.

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2 thoughts on “A Toy Model Of Epistemology

  1. PeterJ

    Some of this is over my head, and maybe it’s a bit out on a limb, but there seems to be something in it. Would it be right to say that in mathematics G wins, but in the foundations of analysis U wins?

    • Hi Peter

      Aww, the whole thing is so vague, I really don’t know what it means (to be honest)!

      But math is probably the quintessential example of what this toy model can address. Math is a very clear cut example of the ever-present tension between the graspable (i.e. integers, discreet algorithms) and the ungraspable (the continuum, real numbers).

      Really, math is an example of some kind of equilibrium between U and G. It shifts from time to time, but the tension is always present, and always quite obvious. Again, one can fall back on Weyl’s insight that math exists at this border. He calls it “freedom to construct the possible” in a field of necessity demanded by logic and math itself.

      Who knows, maybe math as we understand it is as close as we humans can get to the equilibrium point between U and G, which would be the repellor states, and not the attractor states.

      Take care, Sir, and thanks for commenting!

      Don

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