As mentioned in a note that I wrote earlier on the philosophy of science, reality is nothing but a set of predictive models we’ve deemed useful for predictions.

I like to imagine true reality as some kind of a fuzzy blob that doesn’t have clear boundaries and access to which is forever out of our reach. The neat categories of reality (such as atoms, molecules, people) and so on exist in our mind as useful models to help navigate in that fuzzy blob of true reality. To put it a bit harshly, the reality that we perceive is an evolved illusion created by the mind to help us survive.

Since true reality is forever out of our reach and yet we have the illusion of understanding the world, the question of what does understanding something even mean becomes important.

The genesis of “aha”

The “aha” moment that comes with understanding something is a bit of a mystery. When we find ourselves dealing with an unfamiliar situation, there’s a fog of confusion. Then after putting in some effort, the fog lifts and we get a nice, warm feeling of understanding.

I think this positive feeling of “aha” is a similar motivator as the feeling of happiness upon eating a cookie or the feeling of orgasm after having sex. These positive feelings were evolved to motivate us to do something that’s beneficial for our survival and/or reproduction.

In case of understanding, what’s clearly beneficial to us is to be able to predict the dynamics of the unfamiliar situation or object so that we can exert control over it for our own benefit.

So, that is what understanding boils down to ultimately:

Understanding something is building a (physical, mathematical or mentally simulated) mechanistic model of it.

In fact, we get a deep understanding or aha moment, when we’re able to play in our mind with the model of the object to be understood and find that our simulated model’s outputs match with what we observe in the object out there in reality.

Ways of understanding things

There are at least 3 ways of building a model of the thing to be understood:

Mathematical (or code)

Physical

Mentally simulated/imagined

An imagined model is actually simulated physics in the head. For example, when we’re trying to understand how the moon remains in orbit around the earth, we invoke in our mind a rudimentary physical model of reality where forces act on objects and they make the object do things. An unmotivated thinker can stop at that but a more motivated person may try to imagine additional constraints such as the impact of Sun and/or other constraints.

But there’s a limit to what quality and fidelity of physical world simulation we can run in our heads. Our brain wouldn’t notice inconsistencies that silently emerge in our imaginations which cannot exist in the real world. This limits the usefulness of mentally simulated models of understanding. This mode of understanding is only capable of very simple (linear) models with few categories of inputs/outputs.

Of course, most of our everyday understanding is via mental simulations only. When our moms scold us to not keep food outside, we mentally simulate the effect of keeping the food outside and link it to its attraction of ants, flies or bugs. This is a simple (linear) model that serves a useful purpose (prevents food spoilage) and everyday language is sufficient for that.

Supremacy of mathematics in model-building

However, everyday language fails when we need precision. Coming to the previous example again, what if our interest is not just in whether the food kept outside will get spoiled but also in how it will get spoiled, when will it get spoiled, and where will the spoilage start?

As the scope of what we desire from the model increases, so does the limitations of using everyday language to describe it. This is because our everyday language never evolved for precision and our capacity for mental imagery remains limited because there was no point of spending so much energy expanding that capacity when limited capacity would do. These constraints for precision necessitate the use of mathematics

And precision is just one of the benefits of mathematics. As detailed in Why fundamental reality is describable using mathematics, the internal logical consistency of mathematics forces your built model to be internally consistent as well. And because the physical reality (as far as we know) is internally consistent, mathematical models produce more accurate (and hence more useful) models than everyday language.

A motivated knowledge seeker is attracted to mathematical model building because the precise outputs of the model help in verification of the accuracy of the model. For example, if one says things like “world is a simulation” or “consciousness is a field”, it’s difficult to tease out their consequences and without that, there is no way to verify if they model reality or not. Compare this to a mathematical model which says if 2 atoms come in a superposition, they start demonstrating a precise deviation from the precise predictions of quantum mechanics. Now this can be verified and hence confidence in such a model can be accurately calibrated.

As a thumb rule, then,

the more motivated a learner is towards understanding something, the more s/he should start by trying to build a mathematical model of it.

Physical models as ground-truths

Mathematics can help us build rigorous and precise models about the thing that we want to understand. But, ultimately, since mathematics is logic made precise, it can model both: actual reality that exists or a reality that is logically possible but doesn’t exist.

To find out whether our model actually corresponds to physical reality, we need to translate the mathematical model to a real world physical model or experiment. Doing so exposes our mathematical model to real world interactions, which if turn out to be dominant and unaccounted for, can make the mathematical model less useful (or entirely useless).

So, in this way, the true test of understanding is to build a physical model of what we want to understand and check if predictions from our mathematical model actually correspond to how the physical model behaves.

Hierarchy of models as the basis of understanding

The models that we build for understanding themselves have to be understood. For example, the model “food outside → spoilage” is built upon the primitives of our understanding of the words “food”, “outside”, “spoilage” and the notions of time and casualty. What do we understand these words in terms of?

I think the buck stops at our ingrained concepts that evolution has hardwired into us. These include sensory qualities such as color, smells, etc., concepts such as animals, people, plants, sky and also notions of casualty. Consider these as the building blocks of understanding. Learning, then, constitutes assembling them in order to understand an unfamiliar situation. (e.g. spoilage is when edible apple becomes inedible over time). We continue to build more and more complex models based on the (simpler) models we’ve already built.

Mathematics works the same way. We start with the concept of sets or addition and build our way to topology or group theory. But the building block of these higher concepts is actually what we understand intuitively (thanks to hardwiring by evolution). For example, 1+1=2 is understood in terms easy to imagine mental simulation of putting an apple and another apple together on the table. We just know it has to be true. Similarly, an empty set is modeled as an empty container. We know an empty container cannot be a filled container simultaneously and no further proof is necessary.

Complex systems reveal limits of understanding

The best model of a thing is thing itself. But we can’t understand a thing unless we’re able to make a much simpler model of it that while being simple, still captures the essential dynamics of the thing.

It’s not guaranteed that in reality we will always find situations/systems that can be described by models much simpler than the thing itself. Complex systems such as a biological cell or Earth’s climate are perfect examples of systems that resist simplification.

They resist simplification because they have many important components that interact in all sorts of ways leading to dynamics that can’t be captured via simple models. It’s like there are a thousand details and all of them are important.

To capture the dynamics of complex systems, we need complex models and hence machine learned models such as deep neural networks work great in such situations. But since the model itself is so complex (e.g. GPT3 which is a model of English has 170+ billion parameters), we can’t claim to have an understanding even when the model works well.

The take home message is that (all of) reality doesn’t guarantee its distillation to simpler models. Whatever we can distill, we can claim to have understood. What we can’t, we can’t (even if we’re able to use computers to make predictions).

## Understanding as hierarchical model building

Crossposted from anotein my knowledge garden.As mentioned in a note that I wrote earlier on the philosophy of science,

reality is nothing but a set of predictive models we’ve deemed useful for predictions.I like to imagine true reality as some kind of a fuzzy blob that doesn’t have clear boundaries and access to which is forever out of our reach. The neat categories of reality (such as atoms, molecules, people) and so on exist in our mind as

useful modelsto help navigate in that fuzzy blob of true reality. To put it a bit harshly, the reality that we perceive is an evolved illusion created by the mind to help us survive.Since true reality is forever out of our reach and yet we have the illusion of understanding the world, the question of what does understanding something even mean becomes important.

## The genesis of “aha”

The “aha” moment that comes with understanding something is a bit of a mystery. When we find ourselves dealing with an unfamiliar situation, there’s a fog of confusion. Then after putting in some effort, the fog lifts and we get a nice, warm feeling of understanding.

I think this positive feeling of “aha” is a similar motivator as the feeling of happiness upon eating a cookie or the feeling of orgasm after having sex. These positive feelings were evolved to motivate us to do something that’s beneficial for our survival and/or reproduction.

In case of understanding,

what’s clearly beneficial to us is to be able to predict the dynamics of the unfamiliar situation or object so that we can exert control over it for our own benefit.So, that is what understanding boils down to ultimately:

In fact, we get a deep understanding or aha moment, when we’re able to play in our mind with the model of the object to be understood and find that our simulated model’s outputs match with what we observe in the object out there in reality.

## Ways of understanding things

There are at least 3 ways of building a model of the thing to be understood:

Mathematical (or code)

Physical

Mentally simulated/imagined

An imagined model is actually simulated physics in the head. For example, when we’re trying to understand how the moon remains in orbit around the earth, we invoke in our mind a rudimentary physical model of reality where forces act on objects and they make the object do things. An unmotivated thinker can stop at that but a more motivated person may try to imagine additional constraints such as the impact of Sun and/or other constraints.

But there’s a limit to what quality and fidelity of physical world simulation we can run in our heads. Our brain wouldn’t notice inconsistencies that silently emerge in our imaginations which cannot exist in the real world. This limits the usefulness of mentally simulated models of understanding. This mode of understanding is only capable of very simple (linear) models with few categories of inputs/outputs.

Of course, most of our everyday understanding is via mental simulations only. When our moms scold us to not keep food outside, we mentally simulate the effect of keeping the food outside and link it to its attraction of ants, flies or bugs. This is a simple (linear) model that serves a useful purpose (prevents food spoilage) and everyday language is sufficient for that.

## Supremacy of mathematics in model-building

However, everyday language fails when we need precision. Coming to the previous example again, what if our interest is not just in whether the food kept outside will get spoiled but also in how it will get spoiled, when will it get spoiled, and where will the spoilage start?

As the scope of what we desire from the model increases, so does the limitations of using everyday language to describe it. This is because our everyday language never evolved for precision and our capacity for mental imagery remains limited because there was no point of spending so much energy expanding that capacity when limited capacity would do. These constraints for precision necessitate the use of mathematics

And precision is just one of the benefits of mathematics. As detailed in Why fundamental reality is describable using mathematics, the internal logical consistency of mathematics forces your built model to be internally consistent as well. And because the physical reality (as far as we know) is internally consistent, mathematical models produce more accurate (and hence more useful) models than everyday language.

A motivated knowledge seeker is attracted to mathematical model building because the precise outputs of the model help in verification of the accuracy of the model. For example, if one says things like “world is a simulation” or “consciousness is a field”, it’s difficult to tease out their consequences and without that, there is no way to verify if they model reality or not. Compare this to a mathematical model which says if 2 atoms come in a superposition, they start demonstrating a precise deviation from the precise predictions of quantum mechanics. Now this can be verified and hence confidence in such a model can be accurately calibrated.

As a thumb rule, then,

## Physical models as ground-truths

Mathematics can help us build rigorous and precise models about the thing that we want to understand. But, ultimately, since mathematics is logic made precise, it can model both: actual reality that exists or a reality that is logically possible but doesn’t exist.

To find out whether our model actually corresponds to physical reality, we need to translate the mathematical model to a real world physical model or experiment. Doing so exposes our mathematical model to real world interactions, which if turn out to be dominant and unaccounted for, can make the mathematical model less useful (or entirely useless).

So, in this way, the true test of understanding is to build a physical model of what we want to understand and check if predictions from our mathematical model actually correspond to how the physical model behaves.

## Hierarchy of models as the basis of understanding

The models that we build for understanding themselves have to be understood. For example, the model “food outside → spoilage” is built upon the primitives of our understanding of the words “food”, “outside”, “spoilage” and the notions of time and casualty. What do we understand these words in terms of?

I think the buck stops at our ingrained concepts that evolution has hardwired into us. These include sensory qualities such as color, smells, etc., concepts such as animals, people, plants, sky and also notions of casualty. Consider these as the building blocks of understanding. Learning, then, constitutes assembling them in order to understand an unfamiliar situation. (e.g. spoilage is when edible apple becomes inedible over time). We continue to build more and more complex models based on the (simpler) models we’ve already built.

Mathematics works the same way. We start with the concept of sets or addition and build our way to topology or group theory. But the building block of these higher concepts is actually what we understand intuitively (thanks to hardwiring by evolution). For example, 1+1=2 is understood in terms easy to imagine mental simulation of putting an apple and another apple together on the table. We just

knowit has to be true. Similarly, an empty set is modeled as an empty container. We know an empty container cannot be a filled container simultaneously and no further proof is necessary.## Complex systems reveal limits of understanding

The best model of a thing is thing itself. But we can’t understand a thing unless we’re able to make a much simpler model of it that while being simple, still captures the essential dynamics of the thing.

It’s not guaranteed that in reality we will always find situations/systems that can be described by models much simpler than the thing itself. Complex systems such as a biological cell or Earth’s climate are perfect examples of systems that resist simplification.

They resist simplification because they have many important components that interact in all sorts of ways leading to dynamics that can’t be captured via simple models. It’s like there are a thousand details and all of them are important.

To capture the dynamics of complex systems, we need complex models and hence machine learned models such as deep neural networks work great in such situations. But since the model itself is so complex (e.g. GPT3 which is a model of English has 170+ billion parameters), we can’t claim to have an understanding even when the model works well.

The take home message is that

(all of) reality doesn’t guarantee its distillation to simpler models.Whatever we can distill, we can claim to have understood. What we can’t, we can’t (even if we’re able to use computers to make predictions).