Understanding Deeply

Diving into Set Theory


This excerpt is another continuation from my book which is temporarily called ‘On Theories of Everything’, read and enjoy… it is unedited and may contain simple to gross errors, readers beware.

I defy anyone who thinks he understands anything without set-theory. I still remember lightening-bolts striking the core of my being the day I started to understand set-theory and the foundations of mathematics. It’s a profound feeling, it colours every thought with a warm blanket of deeper knowing and one starts thinking like Neo in the Matrix. The world around you starts revealing it self, previously unattainable concepts come into grasp and slowly but surely you feel your consciousness rising.

I would like to says things only get better from that day onwards but as with everything in this world, the more you learn the more you start realising that you know very little. Things you previously thought you understand start exhibiting their true complexity and its an unsettling feeling when the familiar become a strange once again.

There is indeed some truth to the saying that ‘ignorance is bliss’.

Starting Out

Before we start on some actual set theory it will be beneficial if we quickly map-out what what learning it will hopefully give us.

Set-theory --> functions --> relations --> boolean-algebra
| | | |
V V V V
numbers calculus semiosis logic/gates/memory
| | | |
V V V V
space functors semantics knowledge-representation
| | | |
V V V V
tensors laws-of-nature deep-meaning creation

This obviously is not an exhaustive list and neither are the arrows in precise order. Its just a taster of what set-theory can offer us.

It is an amazing thing that most people don’t actually know what numbers are. Numbers actually underpin almost every important concept in our modern lives. Space is an abstraction over numbers. Tensors, which are a compact way of representing multi-dimensional spaces and the laws of nature, act on these spaces.

Calculus is the study of change and as I will argue later on in the book, Time is change. Understanding Time is one of our objectives.

Starting with relations we begin to see how things are fit together and moreover that relations are the foundation of meaning itself. How can one answer ‘what is the meaning of life?’ without a foundation on meaning itself.

With boolean algebra we learn to build the elements of computing. We learn to process information and control of the environment. We now use computing in almost every conceivable facet of our lives, precisely because computing is how to create on a massive scale.

With these building blocks we can then reach further, into language, art, ethics, history, into love and other ideas that we cherish.

The biggest idea in Set theory is that of Membership

It seems crazy at first to think that this little idea would have such extensive ramifications but as we will see later on, nature seems to be designed this way. Almost everywhere we look nature uses the very smallest number of axioms/ideas/assumptions and a generative process of multiplication with selective process of pruning to carve out gigantic structures of deep complexity. In fact the evolutionary algorithm is nothing but a specialisation on this idea.

A real problem with Set-theory (for us at least) is the fact that it is so simple.

Fortunately or not, it is paradoxical to think that the simpler an idea gets, the more difficult it becomes to understand. This is because our (everyday) world is actually quite complex! Our brains are trained from birth to deal with complexity. Keep this in mind if the below starts feeling unmanageable. Things are far easier than you are probably making it.

One last point before we start. You might get the feeling of

‘So what!?’

or

‘What’s the point of this contorted concept’

or something along those lines. I think you would be crazy if these ideas did not cross your mind. It’s natural. Try to ignore it.

You will also need the ability to suspend your skepticism for significant periods of time. It is import in Math (and other technical subjects) to be able to just go with the flow and accept what is being said. Only in this way does the information have enough time to get in your brain and form a coherent idea that can later be accepted or rejected by your inner scientist. Resist the temptation to stop at every juncture where you don’t understand or face confusion. Boldly go on.

All I can say right now is that it is through this “barely worth having idea” that almost all of our great modern advances have come and where our personal ‘deep understanding’ will sprout.

It’s usefulness will not become apparent until we have swallowed the pill whole. Now do you want to take the red pill or not?!

The idea is this:

Every Set is defined uniquely by its members.

Another way of saying this is that,

‘we can know something by its members/properties alone’

In most forms of Set-theory this definition is recursive. That is to say, the members of the Set are Sets themselves. Don’t concern yourself with this though. We will soon tackle what sets and members are.

One of the commonest mistakes to make here is to give a Set order. i.e. to think that the order in which the members are added matters. It does not matter in what order we identify the members.

Another common mistake is to think that can have more than one of the same member, it can’t, each member of the Set only counts once.

Examples will follow. For now just remember,

“No Order, No Duplication”

Again paradoxically the simplest Set we can consider is:

“the Set with no members”

otherwise called the,

“empty Set”

it is with this Set that we will build almost everything we know.

Why is a Set?

Well because any Set can be known by the members it is made of. By not being made of any members we can identify the first set as ‘the set with no members’.

representing sets

Describing sets continuously in English (or any normal spoken language) gets very verbose and highly tedious. So we’re going introduce some notation to make things easier. Just remember whatever notation we introduce maps back to English.

our notation.

= : is equal to or is the same as or is another word for

{ … } : is a set containing ...

: such that

∈ ,  : is a (member of) , is not (a member of)

 : the empty set

∀ : For all

 ,  : there exists, there doesn’t exist

 : and

 : or

Just keep these funny symbols and what the stand for in your head for a moment. It’s really not difficult and all will be explained.

We often represent a set with an opening { and a closing }, all the contents in the middle (separated by commas) are the contents of the set, for example:

{ green ball, red ball, blue ball } is a set containing a green, red and blue ball.

{ … } : is a set containing ….

but writing out sets explicitly like this every time is extremely irritating. So we might want to write it once and then call it by a name. Names can be any letter of the alphabet or a string of letters inside quotes, for example:

let X = { green ball, red ball, blue ball }

now we can use X anywhere we mean a set containing a green, red and blue ball. Better right!?

but X is not very descriptive so we will also allow “colored-balls” to be the name of the set.

“colored-balls” = { green ball, red ball, blue ball }

but what if our set is very large. Like the set of all children in America, writing out { Johna .D Cameron, Mary C. Cummins, …} would be very cumbersome (millions of entries) if not impossible.

We need something more powerful.

To do that we use declarative formulas. An example is easier to explain. Watch the transformations from english into set-theoretic notation.

a) the set of all children such that the child lives in America.

b) { all children such that the child lives in America}

c) { child | child lives in America and child is a young human}

d) Now let c = child, H= set of all young humans , A= set of all people in America, so that all the extra words are removed.

C = { c | c ∈ A c ∈ H}

it reads as: C is a Set such that it contains objects (c) who are members of the set of people living in America and are members of the set young Humans.

Now that’s starting to look like math!

But you may be wondering why we might go into the trouble of all this notation when it takes almost as much effort or more as the original sentence did?

The answer is that our example is a little contrived. In real mathematics we are mostly dealing with a small set of very well known mathematical objects. So a lot of effort is made to describe these objects in the first place, but once that is done we can use these objects all over the place with the impunity of a very terse syntax. That is powerful.

Let’s put this notation on the back burner for a little while and turn to why…

why sets are important

Sets are important because they allow us to simultaneously glue together and unglue concepts from each other. When we glue things together, we are saying, “it is important to consider together” and when we pry things apart, we are saying, “these things should be considered separately”

Turning back to our set C above, the set of all children in America. We are doing some gluing and ungluing at the same time. Firstly we are gluing together these children as one concept while simultaneous removing all members of our species who are not children and don’t live in America.

Why is this important? Well perhaps we want to implement a Health Policy for them. It would be very difficult administering these policies without such tools. How would you go about it without these simple concepts.

Another important and powerful result of this move is that we can consider objects as grouped / ungrouped without affecting the objects we are considering at all and we are free to consider things in anyway we like without the slightest fear that we are contaminating concepts coming from the layer before.

This is super powerful because we can start building concepts layer upon layer into mountainous towers only having to make sure that the current layer we are building is solid, in trust that the layers below are strong.

Not to belabour the point but with our example C above what we have done is taken to separate ideas:

  1. People who live in America
  2. Young Human Beings, i.e. Children, and
  3. created a third, “Children who live in America”

We have created a new category of objects that have meaning because of the way we carved them out. Like Michelangelo we can carve meaningful bodies out of what looks like just another block of stone.

This is a special insight because it tells us something non-intuitive. Most people think that we create meaning only by adding something, that its an additive process.

What set-theory tells us is that meaning is created by a sort of carving/removing process, by being selective from a larger canvas of possibilites.

Just as we pry colour out of white light by restricting frequencies. We pry meaning out of concepts by restricting their scope.

We do this carving with set-theory and arrive a Deeper Meaning.

Thanks for reading this. It is a first draft. I haven’t really thought about my analogies or tried very hard to improve things. Please let me know what you think.