User:Average/Math
Of course. Where else can you p0wn nearly everyone on the planet, with hardly an effort? Optimal.
There are four (and only four) driving values for math -- two of which are foundations (things which are demanded) and two which are ideals (things which are hoped or striven for):
- concision: is it stated syntactically and grammatically correct?
- precision: is it logically accurate?
- parsimony: does it abide with other parts of the field?
- elegance: is it as beautiful as possible?
Practicality is a hidden force that drives some mathematics (often driving it away from elegance), as is what I call being "gangbusters" — trying to get as big or better than ever (driving it often away from parsimony). But these are neither principles nor ideals.
Purity if not a value, is the substrate of math. The only other force is compatibility, with Existence itself, from whence derives all the useful axioms of algebra, geometry, and math; otherwise, there is no a priori cause for them to be selected, is there?
Let it be known that math has been a little sloppy on certain things. Unless it's a transcendental function, the result of a function should not contain more information than it`s input. If a result of an equation is sqrt(2), then the amount of information is about 2 decimal digits.
Aphorisms:
- Simple functions (x2-x) can have complex graphs and simple graphs (the bottom half of a circle touching the X-axis at x=0) can have complex or no functions.
- For the expression: -x = 1/x, the solutions are +/-i.
- STUB Differentiate between different types of functions. For now, I'll call them normal and inverted. Beyond this are transcendental functions, that transcend syntactical boundaries. The log10 function is an inverted function. You can't describe the function using only arithmetical syntax and tokens. Takes it`s inverse, though, and you have a normal function: 10x. Sqrt is another inverted function and doesn't use the syntax of arithmetic (same, surprisingly as factorial (!). E.g. factorial(0) -- why = 1?). These all belong outside of domain Q. But transcendental functions go beyond the well-maintained boundaries of the written word. Conjecture (check): Loge x (or ln x) and it`s inverse en (n in Q) are transcendental functions giving a legitimate irrational answer from finite-information rational source or through some difficulty need irrationals out of a rational input. See Law of the Eternal and/or Mark Janssen for more info. Think on it...
- Functions must behave well their mapping from input to output when working with the above. This means that they cannot output more information than what went into it. (OR: in what case is it okay? From R to Q?)If they take rationals, they should output only rationals. They can't give an infinite series of significant digits. The square root gives out more information, but it's not a normal function. If they take reals, be perplexed if they return rationals, but it is perfectly acceptible. Even the factorial, for example, gives out more information than went in. How? Because it is using non-arithmetic symbols! It may be acceptible, however, as it is merely agorithmically getting more infomation.
- This is strange. It seems domain R is for kooks.
- Reals are distinct philosophical domain from rationals with a different purpose, so check any and all interactions, like fractional exponents on reals.
- Rationals and Geometrics are and must be exact. Reals and Complex might be exact if you started from the realm of ideals. Mandelbrot sets with rational co-efficients or geometric figures with the reals.
- Extra credit: Is i = {} (the empty set)?
