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Correspondences and Mathematical Functions

13:24 / 19.03.04

Many people posit that understanding correspondences between systems is a way to understand or define archetypal energies in terms of the physical/human-understood world. These corresponding groups defined are the expression of a particular archetype.

Not knowing much about math, this is kind of a stab in the dark for me... but, wouldn't it be beneficial to define correspondences in regards to, not only whole numbers (as has traditionally been associated with correspondences), but with mathematical functions as well?

Here's where I'm coming from -- If you take a series of numbers (1,2,3...ad infinitum) and run it through a mathematical function (for example Sqrt(1), Sqrt(2), Sqrt(3), ... ad infinitum), the result would be a series of numbers who were all very different, but held on innate similarity, which would be the function(archetype) used to calculate it. The example of square root may be a poor one, since one could calculate a cube root, etc. So, this could all be grouped under the function of x root y.

Other examples could include Sin(x), Cos(x), Tan(x), ArcSin(x), ArcCos(x), ArcTan(x)... all Trigonometric Functions. As well, there are Transcendental Functions, which I can't really remember which ones they are.

I'm no mathematical genius here, so if there is a flaw in this logic, please feel free to steer me in the right direction.

14:10 / 19.03.04

Basicly what you are suggesting, is that functions are to numbers as archetypes are to human understanding? I'll admidt, I see the pattern there. Its suggestive of the view that all existence is of numbers, made of numbers, can be expressed as numbers.

But I tend to think that notable relations are a more useful expression of this idea. For example, 1/0=infinity. Does this not express the eastern mystical idea that dissolution (0) of the Ego (I or 1) leads one to transcendental experience (infinity)?

Or wrap your head around my favorite of the lot:

e^(-PI*i) = 1

3 irrational numbers, allied in a particular way, make 1. I look at this and wonder then, what do Pi, i, and e all represent escotericly?

Fetch goes on quite a bit about Pi in the Isian numerics threads. I see it as represenative of the fractle nature of the universe, infinity, more detail always the closer you look. Infinite variation. Motion.

i is the projection of math onto a new non-spatial (imaginary) dimentional plane. Numbers that can't exist in normal space, but which can represent relations within it. The transcendent dimention. The dimention of spirit and magick, perhaps?

e, well, the derivitive (rate of change) of e^x is e^x. Thus, e represents a cyclic nature. Consciousness as a system which includes itself, forever cycling and changing yet remaining consistently itself. I see e thus as sentience, consciousness, the self aware system that drives intelligence as we know it. The balance poinit of perfect recursion.

So this relation suggests that when one seeks the root (negative powers are roots, like x^-2 is the same as saying square root of x) of eternal detail upon the magickal plane within the conscious self, unity is achieved. The process of seeking God brings about the experience of God, suggesting again that the quest for enlightenment is itself the enlightened state rather than a means to reach it. Dynamism, all remains in motion. Life is change and motion. The meaning of life is to ask what the meaning of life is.

See what sort of trouble thinking about numbers spits out of my head?

16:18 / 19.03.04

I think that you've just (re)invented the mathematical notion of sets.

I forget the exact notation but you could describe using it: "The set of all positive natural numbers" 1,2,3,4,5...

or your example is the set which is (a) more than zero and (b) when squared the result is in the set of natural numbers.

I seem to recall using it in my undergrad days for clearly and distinctly specifying servitors. Seems so long ago now...

I think that thinking of functions as archetypes is interesting. Not sure if it'd work or not but off of the top of my terribly geeky head:

(for clarity) 'lambda' is to 'any/unknown funtion' as 'x' is to 'any/unknown number'

Ain: zero or [] - The empty set. Nothing, but with the posibility of something.

Kether: 1 or 'x'. Something from nothing. (Sphere 1 - existance). or possibly the set of ALL values.

Chockmah: lambda(x) - a function, something acting on something else. More than one thing required to exist (Sphere 2 - action). Or maybe the set of all functions.

Binah: y = lambda(x) - something being acted on to producing something else. A function acting on a value (or set) to produce a value (or set), or in classic terms, a bloke acting on a woman to produce a kid. (Sphere 3 - reaction). or maybe the set of all functions APPLIED to the set of all values.

It gets much harder to come up with anything beyond the supernals. Great. Another thing to lose sleep over...

08:35 / 20.03.04

A quick note: it is probably important to keep in mind what the domain and range of the functions you're working with are when doing this sort of thing.

The domain is the set of numbers that a function is defined for--basically, the numbers that it is 'legal' to use. So if you've got 1/x, x=0 is excluded from the domain but all other numbers are okay. Or, you can't take the even root of a negative number, unless you're allowing complex solutions in your problem. Inverse sine functions require you to specify where you're looking, or else you'll get duplicate values.

The range is the possible output values. Your sqrt series would be {squrt(1), sqrt(2), ...}. The usual square root function, which allows inputs other than just integers, has a range from 0 to infinity. Sine and cosine range from -1 to 1.

I wish I had more to say--finals have temporarily suspended my ability to be clever. But, noting domain and range could surely have magical uses. If a function isn't defined for some interval, you could define it for your own purposes... the inverse trig functions might be useful if you don't define a domain, and so exploit their 'one-to-many' property... basic trig functions would naturally be good for periodic behavior, or for their 'many-to-one' properties... so on and so forth.