The Calculus of Thinking
part 1

A scaleable framework for thinking creatively and for thinking for yourself
A scalable framework for learning, problem solivng, thinking creatively and thinking for yourself. Neil Keleher.


The Calculus of Thinking, part 1

An article on the FS blog suggests that no skill is more valuable (and harder to come by) than the skill to think critically.

This skill is particularly important when dealing with problems.

I'll suggest here that thinking is only part of what you need in order to solve problems.

Schools may or may not teach the art of thinking. What I'll attempt to do here is show how thinking, and its complement can actually make learning (and "understanding") easier. And that can be important in an age where machine learning is getting a lot of attention.

Why learn to think? Why learn its compliment and how they work together? Because they can make the process of learning more enjoyable. And while we may not always think critically, sometimes it's nice to turn our thinking machines off for a while. And that's where the compliment to thinking comes into play!

Thinking for yourself

The FS article, "How to Think", relies on a section in a lecture given by William Dereseiwicz as its basis on how to think.

Dereseweicz defines thinking as "concentrating on one thing long enough to develop an idea about it".

He calls this "thinking for yourself".

When we can think for ourselves, we are less dependent on others. But that also means that we can work better, and more effectively with others, if we choose to.

The compliment to thinking

I'd define thinking as the process of "defining ideas".

It's a bit more than that. In part it is the process of dividing big ideas into smaller ideas and through that seeing how the smaller ideas relate.

But more importantly, when using thinking as part of the problem solving process, and even as part of the learning process, something else is required. I'll refer to this for now as "the compliment to thinking".

Thinking offers only one point of view

Saying that thinking, and the ability to think critically, is all that we need to solve problems is roughly the equivalent of looking at light and seeing that it has particle-like properties and thus modeling light as if it only acts as a particle.

Or it's roughly the equivalent of looking at light and seeing that is has wave-like properties and thus modeling light as if it only acts like a wave.

Breakthroughs occur when we learn to see how two seemingly contradictory views are simply different ways of viewing the same thing. This can be especially true when we can easily switch between these two points of view.

Thinking, at least in the initial stages of learning, tends to offer only one of those points of view.

A background in thinking and problem solving

So what's my background in thinking and problem solving?

I spent 5 years as an armourer in the British Army.

I learned how the various weapons that I had to keep in working order worked so that I could fix them and maintain them.

Part of learning each weapon involved taking it apart and putting it back together again, repeatedly (but not mindlessly).

Another part involved learning what happened inside each weapon when it was operated in different modes (as well as what could happen if the weapon being operated had some fault, say a bulge in the barrel or worse, a partially blocked barrel).

After leaving the army, I spent 5 years studying systems design engineering at the University of Waterloo.

During a co-op work term at university, I re-designed and built a testing device for irradiation controllers. I had to learn how the irradiators and their controllers worked in order to build an effective test device.

I also helped build a custom Harley from a mix of new and second hand parts. There were problems getting second hand parts (some modified) and new parts working together, and so I learned how these parts were supposed to work as part of the problem solving process.

That too involved taking things apart and putting them back together again as well as trying to understand how those parts were supposed to work together when the bike was working.

Degree in hand, I spent about three years working with different types of systems in the engineering world. I learned the systems through the experience of disassembling and re-assembling them as well as testing them and using them.

I learned these systems well enough that I could talk clients through fault finding procedures over the phone without requiring a manual.

In one case, I knew the system well enough that when a client requested a novel change, something we'd never done before, I was able to come up with a solution in a flash of inspiration.

We did test my solution first. It worked, was cheaper than all other suggested solutions and so we implemented it.

This came about because I understood the system. I'd spent time thinking about it and working with it.

Figuring out problems without asking the internet

I'm a yoga teacher these days. My understanding of the human body, mine in particular, is a mix of thinking and the direct experience of my own body.

But I've also spent a lot of time teaching body awareness or "Intelligent movement" in yoga classes at the various gyms I teach at.

I basically teach non-athletes. But I've successfully taught students who often only see me once a week or less, how to better feel and control their body. In the process I've gained a better understanding of how the body works, in particular when trying to teach people who can't activate particular muscles.

Based on my experience with my own body and also teaching others, I'm able to come up with new exercises or figure out how to work on particular parts of the body without having to consult "expert" groups on facebook or reddit.

I figure it out by myself.

Indexing Chinese characters effectively

Another of my projects is the devising of a lookup system for Chinese characters that doesn't require pre-knowledge of a characters pronounciation or radical in order to find a character easily.

This all started when I first moved to Taiwan and was trying to learn to read Chinese characters.

The problem was, the lookup process was tedious and error prone. And so I started building my own database to make lookup easier. If I looked up a character in my paper dictionary, I then entered its "index number" in my database. This then made it easier to go to the entry of any character that I'd already looked up.

As my database got larger, it became apparent that I needed an effective and easy to use lookup and indexing system, and so I ended up developing one.

This involved a lot more than thinking. It also required repeated efforts to get things right. That meant repeated bouts of thinking, and then trying and noticing the results of trying.

Thinking helps us to "model" systems

Dereseweicz defines thinking as: "Concentrating on one thing long enough to develop an idea about it."

In general, the thing we are concentrating on when solving problems is some sort of system.

What we are trying to do when we think is develop a useful model of the system that we are thinking about. This is a mental-model, one that exists in our consciousness.

The better the model, the better our understanding of the system it represents. And the better our chances of solving whatever problem we are trying to deal with. Or, the easier it is to come up with our own "ideas" with respect to the system in question.

So when talking about building a model in our consciousness what I am trying to say is that when we focus on one thing, one system, long enough, what we are trying to do is build a model of that system.

Thinking for yourself means being able to test your ideas

Note here that the FS blog talks a lot about mental models.

Their idea of mental models is slightly different than the one I am using here.

What I am describing is a general approach to thinking about and modelling any system, one that uses limits to clearly define, ideas, relationships and systems so that we can use them as general building blocks and assemblies in our "mental models".

An important part of this "thinking for yourself" process includes being able to test our mental-models so that we can determine for ourselves whether they are useful or not.

The better we are at testing our models, the less dependent we are on others.

Also note, above where I said "clearly define". When we have clearly defined ideas, relationships and systems, that makes the testing process so much easier because we know what we are testing for.

Defining a system

A system is any collection of things that affect each other or are inter-related.

I'm using a very general definition here so that the word "system" can include subjects that we might be studying at some sort of educational establishment whether it be English, History, Maths (or as some would say, "mathematics") etc.

The idea of a system can also apply to things like weapon systems, motorbikes, computer systems, the human body, the Chinese writing system, apps, programming languages etc.

It can also apply to ourselves, our mind, our body, etc.

I should also suggest here that the smallest possible system is a "relationship".

A relationship is made up of two connected things that affect each other.

Note that we can look at a relationship from two possible points of view. We can look at the two things that a relationship connects. We can also look at the "change" that is transmitted or shared between these two things. Alternate words for change include information and energy.

I'm defining things like systems and relationships because they can be used as basic building blocks for thinking critically or clearly.

I'm also introducing concepts like information and energy because these offer us another way of looking at the thing we are modelling.

Note that I use the term energy in the same way it tends to be thought of as in physics, as an equivalent to mass.

What word can we use to describe the things that are connected in a system or relationship?

The word I use is idea.

Defining an idea (or recognizing it)

An idea is anything that is clearly defined or easily recognizable.

An idea can also be thought of as the potential for change.

Ideas can be parts of relationships or systems. They can also represent (or contain) relationships or systems.

When working on a problem, an important question you can ask yourself is: "What is the change that I am trying to create?" or "What is the change that this system is supposed to create?"

What's the Big Idea?

Creating mental models via Relationships and Ideas

Ideas, relationships and systems are the basic building blocks for building mental models.

When thinking about a system or analyzing it, we can define ideas and relationships to help us break down the system and model it.

The process of problem solving, which generally includes the process of learning, is to break down a system into smaller "digestable" components so that we can learn them bit by bit.

We then assemble our mental model from these component parts.

A general point that can make the digestion process easier is understanding the limits of our short term memory.

If we work within the limits of short term memory, assembling our mental models can be a lot less time consuming and a lot less frustrating.

Problem solving is a learning process

Problem solving is a learning process. You have to learn something in order to solve a problem with it.

When I was in the army, they taught us how the various rifles worked ahead of time. As a result we could get on with fault finding and fixing.

Working on building that custom harley, when we ran into problems, because I didn't have a sufficient mental model, I had to learn the various systems on the fly.

And sometimes what that meant was pulling the bike apart, making a change, then putting it back together again and testing the change. Was the problem fixed? If not, then we repeated the process.

The learning process is a process of switching between experience and thinking.

(And if you think that it should be the other way around, that's fine also. The important point here is that we have two different but complementary views!)

When we have detailed mental models, this problem solving process often becomes easier because we can do the assembly and disassembly in our minds.

But generally, to get to that stage, we need experience.

Note here that one nice thing when working with new systems is that if we know where the problem is, we can focus on learning the part of the system that contains the problem.

So for example, working with the harley, when we figured out that the front brake wasn't working, we didn't look at the engine. We looked at the front brake system, in particular the reservoir.

How we understand

What is understanding?

It is having a model in our consciousness, a representation that is constructed from the dual points of view of thinking and experiencing.

As an example of this, as part of our armourer training, I learned the component parts of a rifle. I learned them by repeatedly pulling a rifle apart and then reassembling it. There was a set sequence to this process. As a result, I not only knew the parts of the rifle, I knew how they related to each other.

When we knew the parts we then learned the sequence of events that occured within the rifle in different modes of operation. Assuming a loaded rifle, we learned what happened when it was in safe mode. And then we learned what happened when it was in fire mode. Following the sequence of events within the rifle, we noted how each part, each idea, transmitted change to the next part.

One way to think of the events that occur within a system when it is operating is as a "story". A system may have one possible story or many. Note that when looking at a relationship, the equivalent ot a story could be thought of as a signal or as a change.

Note the two different views.

Applying this process to ourselves, we get direct the experiential view by focusing on what we are doing at the time. This can be something as simple as focusing on the person we are with and giving them our undivided attention.

And that means not thinking about what we want to say while they are talking. It means noticing the words and the contents of the words that they are delivering.

In physical activity it can involve focusing on what is going on around ourselves. And/or it can involve focusing on what is going on within our physical body.

That can mean focusing on the sensations generated by floor contact, by muscle activation, connective tissue tension. And rather than just noticing what is happening, we respond based on the idea of whatever it is that we are doing.

The thinking and analyzing can occur in retrospect, after the activity or after the conversation.

Learning while making Instagram videos

Recently I've started doing videos for instagram. Part of my process at present, is to record a segment of me doing an exercise. Then I review the video immediately after I've filmed it.

When I first started doing this it was to try and fix a shoulder imbalance. I'd try to use a particular muscle control technique while exercising, and then review to see how successful or unsuccessful my attempt was. Then I'd try another technique.

While I was doing the exercise I didn't worry about whether or not I was balanced, I focused on whatever technique was my focus. Afterwards I'd review the video to see if the technique helped or not.

Also, while doing the exercises, I didn't have to think about how to activate a particular muscle. I simply focused on activating that muscle, feeling it activate, and then relaxing it and feeling it relax.

Note that each unsuccessful attempt could be viewed as a mistake.

Thus I was making mistake after mistake after mistake.

But it could also be viewed as experience. I was having an experience and then reviewing it. I was learning what didn't work. And the cost was a minute each time of filming, plus another minute or so to review. In the process I practiced different control techniques so the time wasn't wasted.

Plus the time to review also gave me a bit of a rest. It also gave me time to think of what to focus on next.

Why we have the ability to think

Thinking and experiencing are two parts of a process. Together they can be used to build models within our consciousness.

When we make the periods of thinking and experiencing short and clearly defined, we can build up our model piece by piece using a modular approach.

Plus, this can make it easier to work within the limits of our short term memory.

Going back to my attempts at instagraming, while I was making mistakes, they didn't cost a lot.

Also, each time I shot a new video, I clearly defined for myself what it was that I was trying to do. Then I focused on doing it.

And this is a taste of what thinking is about. It is the tool, the machinery that we use to come up with clear definitions. Depending on what we are working with, thinking can be used to recognize pre-existing differences. Or it can be used to create differences, to define them.

Taking a break from thinking by flowing

Note that a special quality of thinking, one that we can easily recognize, is that it does take effort. It can be tiring. And one reason that lots of people turn to yoga is to give them a rest from thinking.

Yoga poses offer one way for people to get into the experiential mind set. And generally, with classes like flow classes, it gets them into this mindset by giving them challenging poses to do one after the other.

The interesting thing is, we can get into the experiential mind state by doing math.

When I was in the army, I started studying A level maths via correspondence course because I'd left school without graduating. When I'd learned a particular technique well enough, I'd do a practice question, check my answers redo if necessary, then do the next question.

Time flew by because I wasn't thinking. I was simply doing. I was taking in data, the question, the conscious model that I'd built would work on each part of the question in the background and I'd write the answer.

Even though I was doing math, I wasn't thinking.

As another example of this, I used to practice writing the answers to simply math questions as quickly as possible. Given 100 questions, I would try to write the answers in less than 2 minutes. I didn't have time to think. I'd see a question, and while I was writing the answer my eyes would be on the next question.

After finishing, I'd check my answers, and most of them were right. The ones that were wrong were questions that I had insufficient experience with. I hadn't memorized the questions and their answers. I didn't yet have a model of them in my consciousness.

Note that many of you will recognize this as flow state or being in the flow, or being present or being in the zone. They are all names for the same state of being. The name itself doesn't matter. It's helpful when teaching or marketing but the name only points to the actual state.

I like the Japanese term for it:
No mind,
or as I like to think of it:
Know mind

A lot of what the FS author and William Deresiewicz call thinking could better be labelled as contemplation or a different type of flow. And I've experienced this a lot also.

Rhythm can lead to a different type of flow experience

Once while in Kuala Lumpur with my mum and sister a possible solution to a problem I'd been having with a particular tai ji sword movement popped into my head as I was wondering back to the hotel from Starbucks. So as soon as I got back I tried the movement using an umbrella.

Note that the idea occured to me based on what I already knew, the sword form, but also based on a particular move that I'd been repeating with the intent to get it to flow more smoothly.

The sartori occured while I was walking, a rhythmic action, and in a daydreaming state of mind.

In her book "the Artists Way" Julia Cameron talks about the benefits of activities with a rhythmic component, activities like shaving, walking, for getting into this sartori inducing frame of mind.

This frame of mind is not thinking!

Getting back to my sword form sartori, the problem was clearly defined. I didn't name the problem or describe it. I simply new it from the experience of doing the form over and over again. I knew the problem because I'd clearly defined where this particular movement began and ended. It started with the sword vertical and me on one foot. It ended with the lifted foot on the floor and the sword doing a full circle. (My memory of it now, some 14 years later, is a bit rusty!)

And that is a lot of what "good" thinking is about. It's about clearly defining what we are working on.

Note that "clearly defining" something doens't mean that we name it or use words to describe it, though we can when required. It means knowing for ourselves what we are trying to do. A clear definition, or something that is clearly defined, or definable, is something that we can see and take in at a glance. If we've defined it sufficiently well, then further thinking is no longer required. We can get on with doing.

The power of thought allows us to break things down

With complex systems, say a tai ji sword form, it can be hard to clearly define the whole form all at once. So, part of what thinking allows us to do is to break things down. We can use thinking to break long sequences down into short sequences. We can use it to break down big systems into smaller systems.

We can then learn these systems, bit-by-digestable-bit as the sum of their parts.

How small do we go?

As small as is useful. And small enough that we can hold a clear definition within our short term memory.

Published: 2021 08 27
Defining ideas, relationships (and change) for better understanding, problem solving and experiences

Articles by date

2021 11 12

The overlooked costs of poor indexing
Indexing methods; The benefits of good indexing; Why it takes time to save time

2021 10 18

An intro to better Mental Models, part 1
Mental models and their uses; 4 types of mental model; Simple building blocks for mental models

2021 09 16

Modularizing habits
Why we have habits and how we can change them and use them

2021 08 27

The Calculus of Thinking part 1
A scaleable framework for thinking creatively and for thinking for yourself

2021 08 25

Do you feel lucky?
Systemizing luck via patterns, models and good old "understanding"

2021 08 24

Learning to Understand
Becoming more self-reliant, while learning to think less (by pre-thinking)

2021 08 19

Overcoming frustration through habits
Frustration isn't always avoidable, but it can be minimized through habits

2021 08 18

A Good Death is its own Reward
Turning the thinking mind off

2021 05 03

Creating Space
To get in the flow a basic principle is to look for the space to flow through rather than at the things that prevent flow

2021 03 14

Working from first principles
Two points of view for understanding any system

2021 01 24

Relationships as a context for change
The [relationship] as a general building block for reasoning from first principles

2021 01 22

Muscle Control
A first principles approach to learning to feel and control your body

2021 01 20

Ideas as First Principle building blocks
The qualities of ideas that make them useful for working from first principles

2021 01 19

First principles
The art of modelling for function rather than form

2021 01 07

Building intuition
Why working from first principles is more than just understanding component parts (but also component relationships)

2020 12 03

Ideas as Units of Meaning
And as the Potential for Change

2020 10 05

Creating an easy-lookup indexing system for Chinese Characters
The importance of indexing in general

2020 09 19

Learning to understand complex systems in terms of ideas, relationships and change
Plus side trips down memory lane and how the method of loci relates to understanding

2020 09 18

Information, energy and the idea of change
Why it makes sense that information could have mass

2020 09 15

Indexing, context and understanding
How effective indexing makes it easier to find things and can lead to better understanding via the method of loci

2020 09 01

Right and wrong versus better possibilities
sometimes you just have to make a decission

2020 09 01

How to make decision making easier
Understanding short term memory (so that you can work effectively within its limits)

2020 09 01

A calculus for learning your body
The basics of "learning to understand"

2020 07 31

Learning Chinese by reading It
How to say "peed all over the toilet seat" in Chinese

2019 07 22

About Neil Keleher
Simplifying chaos

2019 07 21

Rewriting Our Operating Systems
Becoming Better at Being ourselves

2019 07 21

Being Present, What it Means
and How to Get There

2019 07 19

Being Present, a Non-Critical (but critical) State of Mind
(That's often more fun!)

2019 07 12

Basic Principles: Ideas as Units of Meaning
And as the Potential for Change

2019 04 24

What is zero parallax?
How to account for viewing error to measure change, create change and to understand

2019 04 24

Zero Parallax
Tools for learning to understand

2019 04 24

Flexible thinking(Formerly "Learning to Understand")
A First Principles approach to understanding systems from two points of view by using components and stories