“Nature is pleased with simplicity. And nature is no dummy.”
― Isaac Newton

In part three of the series I gave an overview of this question to make it clear how little work has been done from my point of view. I feel I’m starting from scratch.
I propose our thoughts are limited to pieces of ideas that I call “oughts.” Oughts are elements of meaning, pieces of thoughts, that do not appear in our minds until we have a context for them. Oughts, by themselves, are are not enough to verbalize.
In this fourth and final (for now) installment in the Where Thoughts Come From series I’ll describe how thoughts might be formed using a basic model that starts simple and yields interesting results.
I have a couple of problems to address before I start, the first being that math turns peoples’ minds off. I hope that you can endure this description in order to see to the horizon at the end. I am going to go slowly and wave my hands. Think of numbers as another kind of words, and think of formulas as a kind of sentence. A mathematical statement is like any other kind of statement, it’s just that it rests on the logic of numbers rather than your experience.
The good thing about mathematical descriptions is that you can add complexity without sacrificing the clarity of what you started with. If things make numerical sense at every step, then the whole thing makes numerical sense. You can’t say the same with reason because extending one line of reasoning usually weakens another. This is also the downside of math: you’ll have a hard time adding new steps not because they don’t make sense, but because they make the whole story more difficult. In order to combat creeping complexity I’ll start as simple as possible and sacrifice some rigor in putting the ideas together.
Math
First, let me reprimand you: you must stop wallowing in the trauma of your childhood when it comes to math. Math was used as one of several paddles to make you feel stupid and inadequate. It’s necessary that we all be made to feel stupid and inadequate in order for us to obey authority and be eager to accept the model of material and spiritual scarcity that our society rests on.
Math is taught to kids as something that must be done exactly right, and then you’re given ideas you don’t understand and problems you can’t do. Of course you learn to hate math, it is used as coercion and belittlement, but that’s not what math is.
There is no “right” math, only math that’s put together right. You can no more be wrong at math than you can be wrong at blocks. Of course, your blocks might fall down, but that’s all in the course of learning. Similarly, there is no wrong math, only math that falls down.
We use math like we use blocks. We try to build things. Maybe they stand up, but that doesn’t mean they’re right, it means they’re consistent. Get over your math phobia, it’s all about the abuse of your childhood. Reclaim your right to build with numbers.
Oughts
We’re building a theory of oughts. Oughts are bits of feeling, elements of ideas that amount to inclinations. They are the bits of an image that mean little on their own. They are the red of a sports car, the flash of the sun in our eyes, or our first impression when we step into a room but have not yet seen what’s in it.
Each ought has certain positives and negatives. The positives lead us to see it more clearly and hold it in our mind to see what grows around it. The negatives repel us and cause a reflexive reaction to shrink it in magnitude and implication. Some oughts are all positive, like the suggestion of a kiss, others are all negative, like the suggestion that you’ve peed in your pants.
When oughts grow in strength they entrain similar oughts into existence and, through the process, an idea forms. Eventually, if this collection of oughts gains enough strength and has enough substance, it emerges as a thought, something with a vision with implications and perhaps a story.
Some oughts just fade away. They are not engaging, not because they’re repulsive but because they’re irrelevant, like my whole diatribe on mathematics which, for most people is irrelevant. You might reflect on it for a moment, but little is brought to mind.
Other oughts become inflamed: lust, hunger, anger, grief, and depression. There seem to be more negative triggers than positive and this is actually true: we are more vigilant of threats than we are expectant of rewards. Nevertheless, some ideas seem to grow to dominating proportions. We are motivated to do something or we would like to. This can lead to rash action, frustration, and regret if our thoughts get away from us.
Time
We’re told and we believe that time is continuous. Do you experience time continuously? Do you have a continuous memory of events? Does your sight grade continuously from one image to another?
Imagine the second hand, it moves in a graceful and continuous arc. Imagine yourself falling, you pitch forward in a graceful and continuous arc. Aside from few experiences such as these, which are more figments of your imagination than any sort of universal temporal experience, we feel time move in steps, waves, or pulses, like everything else in our bodies. The waves in your brain determine the resolution of your notion of separate events in time.
In this model, our experience of time is discrete, and our awareness proceeds in steps, much like the frames of a movie. By stringing them together we get the sense of continuous motion. Unlike a movie, however, the intervals between our steps can grow longer to slow time down, or occur more rapidly to give the impression of time speeding up.
Numbers
In the spirit of numbers that can say things, let’s say an ought has a strength that’s given by a number at any point in time. The strength is given by a number we’ll call x and the time by a number we’ll call n. So, x(n) is how strong this ought is at the time n.
Because we’re modeling time as the clocking of events that happen in steps, we can say that after the time given by “n” the next time is given by “n+1”. We don’t care what time n is or how much time has elapsed between times n and n+1. It could be anything and, in fact, it is whatever time we feel it to be. It’s just that things happen in steps, and first the strength of our ought is x(n) and at the next step the strength of our ought is x(n+1) and then x(n+2) and so on.
Here’s the first formula:
x(n+1) = x(n)
If that’s the case, the inkling we have at n persists unchanged at time n+1 and, if this keeps going, it stays the same forever.
In reality all of our thoughts degrade over time and sink from consciousness unless something keeps them afloat. They might be kept afloat externally, as is usually the case by a situation that persists, or internally by a process of reflection and rumination, which is also common but fades out fairly quickly. On the other hand, sometimes our thoughts do not fade out, and we’ll consider that too.
If our ought grows in time then it will eventually overwhelm our thinking. In this case, the power of the ought would grow and we could write x(n+1) = A·x(n) where A is a number greater than 1. We call A the growth rate. In this case, at every step the strength of the subsequent ought is greater than it was previously and, after enough steps, we are overwhelmed.
On the other hand, if A is less than one, then the ought degrades with each step until it eventually fades to a strength of zero. None of these three cases, A < 1, A = 1, A > 1 is particularly interesting. Also, my proposal is that oughts are just little pieces of thoughts and are not enough to have a full meaning no matter how loudly we hear them. We’ll need to combine the oughts, and we’ll do that shortly.
At the moment, we have just one ought and it either grows, stays the same, or shrinks over time. We can draw a graph of the change in the strength of the ought over one step in time as following one of three lines, as shown below. The strength of the ought at time n is given on the horizontal axis, and the strength of the ought at time n+1 is given on the vertical axis:
Figure 1: Different rates of growth. R=1 implies no growth.
To build an auditory analog, the strength given by x would be the volume of a tone and the passage of time would generate a beat. The tone would repeat at an even beat and either stay the same, diminish in volume to nothing if the growth rate was less than 1, or grow without limit if the growth rate was greater than 1.
We have many oughts that just fizzle out on the boundary of our consciousness. It’s only through their combination and reinforcement that they generate ideas that take up our attention.
Inhibition
Let’s next say that the power of an ought grows in time, that is that the A in the equation x(n+1) = A·x(n) should be greater than 1. However, we can’t let them run away and overwhelm us. There must be another force that balances and eventually stabilizes their growth.
Here is the next term in our equation: B·x(n)·x(n), otherwise written as B·x(n)^{2}. B is the inhibition term. The larger the value of x(n), the more negative the value of B·x(n)^{2}.
Adding our inhibition term gives the next instance of the equation for the power of the ought given by x(n+1) derived from the combination of growth and inhibition:
x(n+1) = A·x(n) – B·x(n)^{2}
Now we can perform a simple bit of magic of the kind that you can do more easily with math than with reason: we can simplify the equation without losing any generality.
We didn’t really care about the amount of time that elapsed between times n and n+1 as we can always add in that detail later. The time scale won’t change the general behavior of the evolution.
We also don’t care about the actual values of A and B. We don’t even care how different they are from any given value, we only care about their relative difference, which is their ratio: the ratio of the enlarging to the diminishing effect. Is the growth rate twice as large or half as large as the inhibition? That sort of thing.
We can simplify further. In order to get the general feel for the behavior described by this equation we can set A equal to B, setting the growth rate equal to the inhibition. This is simpler because now there’s only one extra variable determining the rate of growth.
This is a special case and we’ll need to recover the difference or the ratio of growth to inhibition at some point, but things are simpler and it is this special case that has been studied in detail. From here on we set A = B and replace them both with the letter R. The equation can then be written as:
x(n+1) = R·(x(n) – x(n)^{2})
The Logistic Equation
I think you’re still with me, and you may think we have not accomplished much. We have not said all that much, that is true, but our little equation is famous. It’s called the logistic equation and it describes how populations grow and shrink. It has two variables, x and n, and one parameter R.
From this modest equation you can derive the behavior of many of the systems that you’ll find on earth. It’s that powerful. If you don’t believe me watch the video, “This equation will change how you see the world (the logistic map)” which you’ll find on YouTube at: https://www.youtube.com/watch?v=ovJcsL7vyrk
Here I’ll give just a taste of it. The following graph is the extension of the first graph to include the effect of our inhibition term. The lines are now curved because the effect of the inhibition gets larger as the strength of the ought gets larger. I’ve also included the straight line that is the graph of x(n+1) = x(n) because it represents that static case where nothing changes over time.
Figure 2: Different rates of growth and inhibition.
The red dots are where locations at which the system first settles.
This is a dynamic situation. If you start with one value at a time n given by x(n) then you’ll give a different value the next time that will depend on where you started. And as time goes on, and you keep moving to the next value of x according to the equation, the resulting values will fall into various classes of behavior.
One possibility is that the strength of the ought will vanish. This happens when the amplification is less than 1, in which case the “amplification” term is not amplifying, it’s leading to a diminution. The additional inhibition terms makes the net strength only go to zero faster. The ultimate point toward which the strength of the ought moves over time is given by the red dot at zero.
When the amplification value exceeds the value 1, everything changes. Now the system moves to a balanced state that is not zero. Over time, the ought moves toward the red dot located at that point where the diagonal straight line crosses the parabolic curved line.
As the amplification increases the height of the parabola increases, and the location of this fixed value shown by the red dot also increases as it climbs up the diagonal line, but it does not increase indefinitely. What happens instead, when the value of R gets big enough, is that the system starts oscillating between values. What was one fixed point becomes two values that the system oscillates between. Then, as the amplification increases further, the system oscillates between four values, then eight values, and so forth continuing to double with larger values of R.
Return to the analogy of sound where the strength x is the volume of a note and the passage of time generates a beat. The volume of the note would repeat at an even beat and either grow louder or software to stabilize at a fixed volume, or the volume would cycle between louder and softer.
As the growth factor grew further, the note would alternate between four values: softer, soft, loud, louder, repeating this cycle indefinitely. With increasingly smaller changes in the strength of the growth factor, the number of distinct and regularly repeating volumes would continue to double. At a certain point the regularity would cease and the volume would change chaotically.
The Model of Thoughts
This is not yet my model as this is just a single ought. An ought is not an idea and is not enough to become one. An idea is a combination of oughts that reinforce each other or, in other cases, extinguish each other.
The simplest model is two oughts coupled to each other so that the action of one affects the other. If the strength of the first ought is x, then the strength of the second ought can be y. Separately, their strengths would evolve independently:
x(n+1) = A·x(n) – B·x(n)²
y(n+1) = A‘·y(n) – B‘·y(n)²
Where A can be different from A‘, and B can be different from B‘. These differences would clearly have an effect on how things developed, but the general shape of things will survive their simplification to B = B‘ = A = A‘ = R, that is to say we’ll set them all to the same value R.
Returning to the analogy of sounds, the addition of a second ought corresponds to the addition of a second tone. In this case, with each beat, there are two tones. Given the two separate equations, these two tones will behave independently, moving to find their own, separate equilibrium volume.
Now, I couple them together by putting the effect of one into the equation for the other. I make this addition using a multiplier “T,” which can be any value, and which I can lower to zero in order to recover two independent equations:
x(n+1) = R·(x(n) – x(n)²) + T·y(n)
y(n+1) = R·(y(n) – y(n)²) + T·x(n)
We can vary T between 0 and 1. When T = 0 we recover the two separate equations. The coupling between the values of x and y increases as the value of T gets larger.
Where the single logistic equation looked simple but generated astounding complexity, the coupled equations don’t look simple. I would like to say that now we have great insight into the structure of thought, but it rather seems that we’ve traded what was before opaque with what is now complex.
There is a vast amount of research on the features of the logistic equation and equations like it. There is little research on coupled logistic equations, and what research there is rarely goes beyond the coupling of two equations. I would like to couple many together, so I’m going to have to do this work myself.
One thing we can see is that the additional “T” term has the effect of raising the curves. With this additional term the strength of x(n+1) does not vanish when the previous strength x(n) vanishes. That is, the effect of the other ought continues to stimulate its partner and vice versa.
As a result, we expect to see some new behaviors, and some new thresholds of behavior, but we also expect to see many of the original behaviors. We expect to see the strengths at which the oughts stabilize to be higher, that is to say the effect of the stimulation will increase the magnitude toward which each ought settles. This is what we want: the effect of coupling oughts is to enhance each other.
Using the analogy of sound, we now have two notes in which the volume of one depends on the other. The volumes could stabilize at a constant value or they could cycle, or one could stabilize and the other could cycle. One might become very soft while the other became very loud.
If the combined strength of the oughts surpassed some threshold, then we expect thoughts, or a thought, to emerge in our minds. Over time these strengths will fade, as our models need to be amended to account for the fading of thoughts. Before that happens, we expect to see various new effects that represent new structures. In particular, we expect to see oscillations that are either synchronous or antisynchronous in the strength of the oughts.
In a 1998 paper in the journal Theoretical Population Biology (Kendall and Fox, 1998), the authors explore similar equations for two coupled logistic systems. I’ve highlighted the identical components of the two equations in blue and red lettering.
x(n+1) = (1T)·R·(x(n) – x(n)²) + T·R‘·(y(n) – y(n)²)
y(n+1) = (1T)·R‘·(y(n) – y(n)²) + T·R·(x(n) – x(n)²)
They use the coefficients T and (1T) so that when T varies from zero to 1 the equations are either fully coupled or uncoupled. The case of interest is in the middle of this range where the coupling between the two is weak.
At the simplest level, Kendall and Fox find the two parameters x and y either rise and fall together, or rise and fall in opposition. There are both stable and unstable configurations. In stable configurations, the system settles into unchanging values for x and y. In unstable configurations the values of x and y fluctuate chaotically.
Applications
I would like to suggest that when enough thoughts resonate in synchrony, we experience something like an emotion.
We’d expect fractal behaviors, as is typical of all of these structures. Similar patterns would emerge at higher rates of growth. This would be like the chords on a keyboard that can be transposed to higher and lower octaves.
As we add more oughts in the equation we add more notes in the sound analogy. All the notes are playing at each beat but their volumes differ. Those that stabilize at a single volume will continue at that volume, the volumes of those that cycle will rise and fall, while those that are unstable will change volume in a chaotic fashion seeming to appear and disappear at random. The behavior that prevails depends on the growth rates of the various oughts and how strongly they affect each other.
I will be interested to explore this further as it supports the claim I’ve made in the study of dreams that dreams have a fractal structure. That is, dream narratives tend to reiterate their themes so that positive dreams unfold into a splay of positive details of varying depth and size, and negative dreams unfold into a spectrum of negative details. This being one of the reasons that dreams have a strong effect on us, why we have a difficult time navigating them intentionally, and why we are protected from remembering them.
Finally, if we can identify oughts, then we might categorize and simplify different thought structures. We do that now to a rough degree, but we have no underlying theory and nothing explanatory. If we could identify which particular oughts contribute to particular states and behaviors, then we’d gain a general understanding of how our thoughts evolve.
All of this will have to wait for me to put these relatively simple equations on a computer and see what they generate. Most computation work focuses on the full range of outcomes with a focus on the most extreme behavior. In contrast, I am interested in the simplest behavior that persists as we make the system larger. Where others are looking at increasing detail, I’m looking at increasing aggregation.
I don’t expect to find anyone who’s done this work before. I also don’t expect to generate much interest in this idea from those who are already looking for different sorts of answers. One thing I’m sure of is that it will be some time before I have answers.
References
Kendall, B., Fox, G. (1998). Spatial Structure, Environmental Heterogeneity, and Population Dynamics: Analysis of the Coupled Logistic Map, Theoretical Population Biology 54, 1137. Retrieved from https://escholarship.org/uc/item/359696wj
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