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Numbers: Unity And Division’s Bridge

January 26, 2008

Through a more thorough understanding of numbers, I’ve came to solve other riddles of the mind I’ve been thinking about for years.  Whether it be what happens when we die, to infinity, to what is a thought, I’ve found numbers to be a strange bridge linking all these things together.

Unity and Division have always been the most complex notions for me to understand.  You can view all of space and time as a series of  infinitesimally small instants, or as a static whole.   The objects of ordinary everyday experience, whether a table, a book, or a computer, can all be viewed as a whole, or as the sum of their corresponding parts.  Also, a person can be viewed many different ways, such as a whole of their past actions, their present instant (their bodily flesh), or as their projected future (free will).

A person’s past can be broke up into into a series of their life events, or, if you decide to disregard a person’s past, for whatever purpose you’re trying to convey, you can say they are nothing but a free will trapped in a body for the present.

I started a journal entry the other day, which got really technical and hairy really fast, talking about realtivity and the nature of motion, space, and time.   All of that is heavily based on unity and division.

I’ve been heavily studying the fundamental theorem of calculus lately, trying to understand it better.  It basically says you can take an infinite number of infinitely small changes, add them all together, and come back to the whole.  It’s rather dull if you do not understand what a number is, as you think, “Why do I care about some area under a curve?  Or the length of some curve segment?”  But numbers have a tight relationship to things (a relationship we’ll soon get to).  They are the bridge between division and unity.  I find it so fascinating because it is a method of reassembling something which has been divided up (especially when it’s infinitely divided up).  (They call this process ‘integration’).

You may wonder how unity and division have anything to do with life after death.  I’ll tell you.  Next time you get your hair-cut, keep the cut hair, and place it outside where you can watch it rot into dirt again.  Then, watch the atoms and matter, which were once your hair, become a flower, or grass, or some other plant.  Watch it be consumed by bugs, and become the bug.  (The same could be said of the food we eat.  It was once a bagel, then a chewed up bagel in the stomach, then broken down into all kinds of stuff in the body, then it eventually becomes the person who ate the bagel, and the bagel no longer exists.)

Now we get to the ultimate discussion:  What is a person?  We see a person, and we think of them as their body, but I hate such thoughts.  Chop off a person’s leg, and watch it rot outside and become bugs and plants.  That leg is not them, it is nothing but matter.  Their hands, their eyes, or any of their body for that matter, is not them.

This same thought experiment can be applied to any conception of anything within the mind (“what is a thought”).  Say you have a small wooden home made out of boards.  At first you start off with a full house.  Remove one of the boards.  Is it still a house?  You say, “Yes, it’s a house”.  I remove another board.  Is it still a house?  You say, “Yes, it’s a house”.  Finally I remove lots of the boards, it starts to lean sideways, and is going to fall over.  Is it still a house?  Then you might start to think, “I’m not sure anymore.  It is moreso a pile of boards now.”  Then you take the other pile of boards I’d been removing from the house and I say, “Are these the house?”  And you say, “I don’t know.  They WERE the house.  Are they now?  I don’t know.  They are a pile of boards.”

Basically the question always being asked is “how much must something deviate from something before it can still be classified as the object we’re after.”  Life in the womb is no different.  How far developed does the baby have to be before it’s finally classified as “alive”.

I was talking to Greg the other day on the phone regarding fractions.  Take a book for instance.  You can view the book as a whole, or we can represent the book as a series of parts (fraction).  But things become interesting when you think of a book as a fraction.  Say someone was to come up to me and say, “Jason, I want 1/4 of that book”.  I remember in math class we’d do all kinds of arthithmetic and calculations involving fractions and long decimal numbers.  I really never thought about what that meant though.  What does 1/4 of a book even mean?

I could divide the book into 4 vertical columns.  I could also divide the book into 4 horizonal columns.  I could divide the depth of the book into 4 columns.  I could divide it into 4 triangles, or all kinds of geometric shapes.  All of these are geometic divisions of the book, but we need not divide the book this way.   I could divide the book by the pages within it.  If the book has 1000 pages, I could rip out the pages and 250 pages for each division.  Then again, I need not divide it this way either.  I could look at the table of contents and say it has 4 main chapters.  I could rip out each corresponding chapter and count that as a division of the book.

Now you have to ask yourself the question, why did you divide the book up to begin with?  Maybe you’re trying to hide it from someone as it contains valuable information.  You take one part of the book and store it in this location, and scatter the other points to different locations throughout the world.  Maybe you’re designing a course curriculum and need to divide the content up for your classes.  Or maybe you’re slightly crazy and like cutting up books into pieces.   Whatever your fancy, how you divide the whole into its parts is up to you.

I used to think of human reality in many different ways.  At one time I viewed a person as the sum of their past actions, at another time their current beliefs and projects they’re purusing out of life (what they value), at other times as a purely empricistic point of view and viewed them simply as their body.

When I began to think about fractions and numbers (thanks to Bertrand Russell’s Principles of Mathematics), I began to see that I can view human reality any way I want, based on what purposes the view is designed to achieve.

If you’re looking at a person from a psycho-anlalytical point of view, a person is the sum of their past events and decisions, because you want to see what past painful events and other reinforcements have come to shape the personality and character of the person in question.  (I’m not advocating determinism, I’m simply saying past events do have an influence on a person).  Then again, when I would think of a person in the existential view (as what they value – a pure free will of infinite potential), I would be trying to show the capabilities humans have to achieve things they want out of life, not to confine yourself to the present or your past, or how things look, but believe in yourself, to watch out for things that can enslave the mind, and never run from your own freedom.  This includes things like confidence in decision making, and the consequences of indecision.  Then there’s the empiricistic point of view, where you view a person as their body.  If you’re doing any type of science work, and trying to predict how an illness spreads through the body, you must view a person in these terms to be effective in treating the illness.

So each mindset has its purpose based on what you’re trying to achieve.  Every division of anything into a set of corresponding parts is done so to achieve some purpose and the divisions are ultimately human.  It’s not a matter of whether a person IS their body, or IS their free will, or IS their past events – it’s based on what purpose you’re trying to achieve.

(Then again, that point could be argued that the relationships all exist prior to us being revealed so by reality.  But either way, what does it matter?  Does it help us either way, saying all these things “existed” beforehand, or whether such relationships are human and exist in our minds?  I see no real tangible difference.)

Bertrand Russell defined a number as a class of similar classes (cardinal numbers).  A number is a symbol, representing a unity of a diverse group of things, based on a shared common property, which is the number.  When properly understood, numbers are the bridge between unity and diversity.  What a crazy idea.  I’m still pondering this.  No wonder such a genius spent his life as a mathematician.

But I can’t say I’m digesting it easily.  There’s really so much to think about.  Identity is such a complex notion.  Take for example: 1 = 1.  That’s an insanely complex proposition.  How can two different things ever be the same?  Take Wittgenstein’s comments on the subject:

(Tractatus 5.5302)
“[Bertrand] Russell’s definition of ‘=’ is inadequate, because according to it we cannot say that two objects have all their properties in common.  (Even if this proposition is never correct, it still has sense.)

(Tractatus 5.5303)
“Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.”

It’s no wonder people get confused when wondering about death, and wondering if they survive.  They think about their body rotting, and questions of identity start to arise.  But take a look at this simple example.  1 = 1 is hard enough to figure out!  But the same key which unlocks the mystery of 1 = 1 is the same key to understanding life after death.  Something to think about  🙂

My only comments on the subject really are these.  When writing computer software code in Pascal, or C++, whenever you create your own custom ‘types’ (classes), if you are to take two instances of that class, and say that they are ‘equal’, you have to define what you’re wanting to show by the equality.

Say you write one software code class “Dog”, which stores information related to a dog.  Then you have another software class “Bobcat” which has all of its information.

So if you were to write the following code:

if (Dog1 = Bobcat1) then
begin
// Do this code
end

You have to write the code as to whether you consider those objects “equal” or not.  Why must you do this?  Is a Bobcat “equal” to a Dog?

Well it depends on what you want.  If by equal you want to mean they both have tails, then both bobcats and dogs have tails.  If by equal you want to mean they both have teeth, then that’s true, they both have teeth.  If by equal you want to test whether they both purr, you will find only the Bobcat does this, and the equality is false.

Could this possibly be the same thing?  Saying 1 = 1 is worthless?  Is it some arbitrary procedure where I define what I want ‘=’ to mean?  After all the number 1 is a container class of all collections of 1 term (object).  It really amounts to kind of the same dilemma doesn’t it?  The only difference is it compares not only 1 Dog with 1 Bobcat, but 1 anything with any other thing.  What kind of procedure would this entail?  I cringe at the thought of writing such a body of code.  It has to be more simplistic than this.

I think the key to solving this problem is to once again, come to the purpose as to why the ‘=’ symbol was added to mathematics.  This has to do with some sort of complex relationship system between symbols.  Like in Algebra.  The ‘=’ symbol is used for when you modify one side of the equation, you have to do proper modifications to the other side as well.  Have to keep things balanced.  You can also put one system on one side of the equals, and another equation on the other side, and do all kinds of testing.

The equals symbol doesn’t always make sense in mathematics.  Take this for instance:

X^2 + 1 = 0

Sure it’s a normal looking equation, but no number multiplied by itself ever gives you -1.  Well, taking all real numbers into account, and not getting into the ‘imaginary’ numbers (another confusing topic).

So far I’ve been pondering similarity (identity, equality, whatever name you want to slap on it).  I think another key to this dilemma is to define what ‘truth’ is to mean.  I was talking to Leon about this the other day on MSN.

Regardless of what people may think, truth is a mindset.  Truth is the hardest concept in the world to define out.  Religious people for instance, they view truth as revelation from God, or scriptures in their holy book.  All other forms of truth, whatever they may be, are secondary.  Scientists view truth as empirical data, which is testing and observation – probability.

However if truth is defined in terms of some experiment performed, and the probability of a certain outcome, then what are we to say of all the events we haven’t done our probability calculation for?  What about all the galaxies and other things going on in the universe which we’re not seeing?  There’s been no experiments done on the events going on out there.

Subtle mysteries, I must say, very subtle.

Philosophers (such as Aristotle, John Locke, David Hume, Immanuel Kant, Descartes etc) invented the scientific mindset to achieve certain purposes.  Basically they wanted people to put their faith in what they saw instead of clinging to all the religious superstition surrounding them.  That was the general movement of thought, anyway.

Truth was invented because in life we have to make decisions.  People oftentimes fear what decision to make sometimes, because they don’t know what to do.  They used to pray to God to tell them what to do, and society as a whole would do this.  Philosophers soon found this harmful as people were doing actions which seemed very unwise to people experienced and have seen that things simply don’t work that way.  The heavenly realms haven’t provided information to mankind to tell us what actions to do.  Well, they might have but we haven’t listened, but that’s a huge topic.  Anyways, soon the idea of “experience” as the guide to making decisions, and looking at what happened when other people tried the same thing became the idea of what truth and wisdom.

Soon the idea rose up that if 999 out of 1000 people who have tried this doing it this way have succeeded, “chances” are you will to.  Hence was birthed the concept of probability and the rational man making decisions based on what is most highly probable.  This is the closest they have come to a definition of certainty and security in this life.

So you see, truth is a mindset as to what you put your faith in.  Everything, and I mean everything, about this entire life is about what do you value?  What do you put your trust in?  What do you think will bring you happiness and what are you pursuing?  Truth is just one of man’s creations to try to get closer to whatever “wisdom” may be.  Toward avoiding pitfalls and pain, and succeeding in life.

So to say that 1 = 1 is “true” is… well, I don’t know.  I don’t know what exactly you’d be trying to say.  Truth originated from other dilemmas, but I don’t doubt there’s definitely something being said in mathematics when you crunch down an equation and get X = whatever.

This situation reminds of me statistics.  You can create the most beautiful statistics model for the prediction that some event will happen.  The weather channel tells you, “It’s 95% probable that it will rain today.”  Then you go outside and there’s not a cloud in sight.  What does it mean?

It’s just a calculation.  The value in the meaning of the numbers is in whether after you write down these mathematical symbols you can predict something before it happens.

If you’re an engineer, you start to see consistent relationships between all the “physics” calculations you do, and what actually takes place when you go to build the bridge.   The value of the “truth” of the mathematical relationship is in whether when you do these numbers and write down these symbols, a consistency exists between this “mathematics/physics” and what happens as you go to build the bridge.

There’s a weird sort of consistency between these mathematical symbols and operations and physical reality.  It’s amazing the power they hold.

We saw by Bertrand Russell’s definition of a number that he roots his conception of them entirely on similarity.  The number “1” is a unity class containting all classes with one “term” (object).  The number “2” is a unity class containing all classes with two “terms”.  And overall, a number is a class containing similar classes.  (All contained classes having a common property, which is the number – we represent this common property with our numerical symbols).

So when calculating things with numbers, you’re showing some common relationship which exists between all the objects those classes contain.  Numbers are in a sense reality.  They’re unity classes containing the objects of reality.  They’re big baskets full of the objects of reality.  But they’re less tangible and more generic than ordinary objects we see everyday.  They’re a common property our mind can perceive in the items around us.  (Once again, or it could be a relationship which exists in reality – I really can’t say.  Mind or reality – what purpose does it serve?)  When I see 1 book and 1 orange, I simply perceive it as 1, and they both have this in common.  My own mind in a sense singles these objects out through a negation sort of thing, and I’m left with just those.

But that leaves me with another kind of weird question.  I don’t think I’ve ever perceived one-million people or one-million books.  Put me in a room with bookshelves upon bookshelves of books, and say the count of the books in that room was one-million.  I couldn’t look at you, just by gazing around the room and say “Ah, I perceive there’s a million books in here.”  I’d have no idea the count of the books.  It’d be too many for me to directly grasp just by looking. I don’t even know how many books I have here in this room, but I know it’s less than a million.

Which gets into the concept of counting, a complex notion indeed.  But I feel like that’s getting away from the subject of similarity and equality, which is what we’re discussing.

I guess I’ll now summarize.  Objects are both diverse and whole depending on what properties you’re looking for in the object.  You’ll see whatever it is you’re looking for.  Numbers bridge the gap between unity and diversity, which makes them particularly interesting.  Also the concept of equality in mathematics holds the secret to life after death.  Understanding what stays the same through change, and whatever this consistency is is likely the entire subject of knowledge (epistemology).  Knowledge is often acquired by observing change, then reassembling the changes and forming a model.  An ideal example of this is integration in Calculus and the Fundamental Theoream of Calculus.  It’s an ideal assembling of perfect infinitely small changes with no gaps back into a whole.  I suppose you could say integration in Calculus is strangly reassembling all the possible changes of any object in reality and brings you to a unity.

For an example of how powerful this stuff is.  Take Issac Newton’s laws of motion for example.  You can simply start with the acceleration of an object (simply observing change in movement).  Create whatever unit you want and label this variable G.

So a = acceleration as a function of time, v = velocity as a function of time, and p = the position of the object as a function of time.  v(0) = initial velocity of the object.  p(0) = initial position of the object.  t = time.  G stands for gravity pulling the object.

a(t) = -G
v(t) = Integral(a(t)) = -Gt + v(0)
p(t) = Integral(v(t)) = (-G/2)*t^2 + v(0)t + p(0)

When I write ‘Integral’ that’s where the magic happens.  Reassembles the infinite changes (the acceleration) and gives me a perfect equation which gives me the speed.  I do the same thing with speed, and I get a perfect mathematical relationship telling me the position of this object at all times.

I perceive one relationship in the movement of the objects, then I’m able to infer all kinds of others.

Use similar methods and next thing you know you’re doing everything from ballistics, to calculating the orbits of planets.  No doubt this stuff has value.

Anyways, time for bed.

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