Every Number Is Infinite: Squaring the Circle


Hello my fellow gnostics and Thelemites!

It's been a long Hiatus, since the last issue of Nu Gnosis Magazine was published in 2018.Unfortunately due to other projects I had to halt this project indefinitely.  But now since I've reclaimed a fraction of the time that I had, I decided to revive Nu Gnosis in spirit by continuing with this blog.  Simply sharing my thoughts and those of my gnostic partners in crime as they come up.  

"If the doors of perception were cleansed every thing would appear to man as it is, Infinite.  For man has closed himself up, till he sees all things thro' narrow chinks of his cavern."

Every Number is Infinite

My latest thoughts have been focused line 4. in Chapter 1 of Liber Al vel Legis:

"Every number is infinite; there is no difference." 

This line often struck me as odd and counter intuitive, yet it never haunted my mind for an answer.  It just felt right, in my heart there was a feeling of 'knowing it to be true'  but this is something I would never say publicly.  My mind and my pride would not allow it.  

Mainly because I had no sensible way of articulating why I thought it was right, other than some specious notions I had of Zeno's paradox.  Which as you will see later on, isn't too far from the mark once we explore this question further.  

The bottom line is that I never felt a personal need to explore this question further.  It seemed as true as the fact that 'I love to play drums', or that 'I am irritated by people who say obvious and irrelevant information'.  There is a direct intuition, an inner experience of this truth that cannot be readily articulated or necessarily proven, but whose truth was immediately apparent to my reflective consciousness. 

This was the case until a few days ago, where I stumbled upon a video about Aristotle's Wheel Paradox taken from his text: 'Mechanica'.  In short this is a problem where one has a model wheel with smaller wheel drawn closer to it's center as in the image below:

The paradox can be discerned by marking both the inner and outer circle by a radial line proceeding from the center of the disc to the the outer circumference and rolling out the circle to measure it:

If the larger outer circle rolls one revolution without any slips until the radial line that it started at, it will have made a line the length of it's circumference.  This can be seen in the image above by observing the line between P and Q.   So far this makes logical sense and can be demonstrated with any wheel even that of a toy car.  

Albeit, things begin to get weird when we examine the smaller inner circle in the image above.  For simplicity sake we will say that the circumference of the outer circle is 2X while that of the inner circle is 1x.  How can it be that a smaller circle and a larger circle produce a line the same distance in length while both turning only one revolution each? 

On the surface there is a logical paradox here, nevertheless there are two plausible solutions:

The first explanation is that outer circumference of the wheel moves much faster than the inner smaller circumference since the outer wheel has a greater distance to travel in the same moment of time.  However, this doesn't adequately explain how the smaller circle is able to travel greater than it's own circumference in one revolution.  This problem dissipates once we understand that while the outer circle and it's circumference is rolling, the inner circle cannot: it must be both sliding and rolling in one revolution.  

This must be the case because physically if you cut out the smaller wheel and rolled it along it would travel a much shorter distance.  Half the distance of 2x in this example.  It's when the smaller circle is affixed to the larger wheel that we get the illusion of it surpassing the limits of it's circumference: in actuality it is sliding. 

The second explanation is that all the points of the wheel have in fact traveled a smaller distance but with identical displacements as seen in the diagram below:

When Galileo reproduced this experiment using polygons instead of circles the skips of the inner polygon are readily apparent, especially when each is coated with paint and simultaneously rolled along two surfaces.  

This lead Galileo to the conclusion that the circle must be akin to a polygon with an infinite number of sides making an infinite number of skips.  If this is so, using the reference to Liber Al, each number, (the circumference) of each circle is paradoxically infinite.  

How is this possible? Well we will continue to look further.  But going back to Al, and with the tentative assumption that this is the case, we see that each circle is infinite in quantity, but finite in quality in relationship (relative) to others. 

Today many see Galileo as being partially right, but the inner circle is not actually skipping or slipping since we can still roll it along: rather it must be sliding.  

In spite of this insight, modern mathematicians still hold this to be unsatisfactory since there is still a one to one correspondence between the two different sized circles as the outer rolls along in one continuous motion.  So we are still at the beginning of the problem!  But at least now we have distilled the question to a more relevant one:  

'How can two lines of unequal length correspond?'

Still further:

'What does it mean to be equal?'

We can approach this question by unrolling the circumferences of the two circles into two 
differently sized lines:




We can begin by making a one to one correspondence between the two lines by drawing a triangle:
Each vertical line represents a point of contact between the inner circumference 1X and the outer circumference 2X.  So this is one way of illustrating the one to one correspondence between the inner circle and the outer circle.  No matter where you place the line, (the point of contact) it will always connect the inner circle with the outer circle. 

We can proceed to fill in the circle or triangle with lines indefinitely making complete contact between the two different lengths of line or two different sizes of circle.  Each sized line or circle being a number, in this respect we see the assertion quit clearly that says in Al:  '...there is no difference...'

'...there is no difference'

So using the common sense 'correspondence' approach to equality, we see that there is no difference in the sums of each line in respects that they have a seamless one to one correspondence.  

So the socialists may run in horror or gun me down as equality in quantity seems to be inherent in all things, the only inequality would be in the quality of the things themselves, at least in nature.  (I will save my political economic thoughts for another day  :-)

The other option here is to consider that a one to one correspondence does not always mean two things are equal.  A two inch line might correspond perfectly to a one inch line as shown above, but does not necessarily mean they are equal.  At least not in all respects.  

The implications of this deepen further when we consider that numbers themselves are represented on a line...

But to explore these implications we need to understand one flaw in this last consideration of correspondence between lines.  While distinct objects like marbles can be split up into discreet numerical packages; circles and lines cannot because they are continuous.  Sound familiar?

"... that men speak not of Thee as One but as None; and let them speak not of thee at all, since thou art continuous!" Al ch.1 l.27

Numbers themselves however, are very different in quality than discreet objects such as apples or marbles.  Whole positive numbers 12345... are discreet because there are spaces between them, so they are best represented as a dot rather than a line, yet there is an infinity of them.  

The same can be said about negative whole numbers, they are also an infinity and they are an equal infinity to that of positive numbers, meaning they both correspond perfectly and have an infinite supply.  It's rather strange to say an 'equal infinity' here, but my math geek friends assure me that this is the correct way of saying it!

But these numbers are artificial and incomplete and do not compose a true number line.  To do that we would need to add all the other numbers: fractions, decimals, irrational numbers, transcendental numbers ect  Every conceivable number needs to be added to fill in the gaps to create what is referred to as 'the real number line'.

Now the real number line is completely different in quality: it is continuous like the line and circle in our paradox.  The real number line is a continuous stream flowing from negative infinity to positive infinity.  

On the number line there will always be a specific number no matter where you point your finger.  If you make a segment of 0 to +1 and from 1 to +1000 0000.

And just like the circle and triangle examples from before you can draw a perfect correspondence between these two different numbers.  

This leads to the conclusion that there are an equal amount of numbers between the number 0 to 1 as there are between 1 and 1000 000.  In this respect there truly is 'no difference' as stated in Liber Al. 

Furthermore, and to really nail this coffin shut, we must now concede that there is in fact the same quantity of numbers between 0 to 1 as there are between zero and infinity.  

Thus every number is infinite.

For a concrete example we might ask: what is the number after zero?  

We could say: .01

But why not: .001, or .0001, or .00000001?  

It doesn't end. 

This also leads us to two types of infinities: a countable infinity and an uncountable infinity.  Much of this is but the beginnings of the bizarre and groundbreaking work of the controversial mathematician George Cantor 1845, who created Set Theory which is now an accepted and fundamental theory in modern mathematics. 

So it's not such a new theory.  But what I like about the circle paradox, is the with it we can see how this statement in liber Al vel Legis is true and how it is reflected in a concrete day to day phenomenon. 

What would be of further interest is to examine how this fact is relevant for the human being as a whole entity, and the soul so called.  What does this mean about people?  

What does this mean about equality?

And perhaps most relevant, what might the implications of this be when orienting one's self in a meaningful way in relationship to being, the universe, and life as a whole?  

One answer might be just to be and understand that you and the world you live in is perfect just the way it is, and to recall that:  

"If the doors of perception were cleansed every thing would appear to man as it is, Infinite."

NOTE: The author Ryhan Higgins Orshalev is an initiate of the A.'. A.'.
If you would like ot learn more about the A.'. A.'. and mentorship
please visit:  www.astronargon.org

To apply for membership:  ananta231@gmail.com

If you would like to learn more about Cantor and the Infinity Paradox, you might enjoy this video: