>>967
>The graph is not an approximation
>magnitudes can only be apprehended by geometry
I could argue that every graph is an approximation because it's impossible to represent idealized mathematical structures in the "real world" (just as it's impossible to determine the exact decimal expression of Pi), but that would not address your argument, although it's tangentially related to my hypothesis because the "real" (phenomenal) world is just an imperfect derivative of an idealized (noumenal) world, which is not unitary/singular and uniform (a set of all possible sets, whose contents or borders would be reducible to a common, uniformly expressible ur-element (even at infinitesimal level) or extensible to any notion of totality, even in infinity). Nor is it the causative agent (source) itself, but just a focal point of limited projection dependent on fundamentally multiple (plural, hence why poly-theism), irreducible, continuously/transcendentally synergic entities for manifestation, those beings a-priori qualitatively and quantitatively transcending any of the potential limitations of such an alleged super-set at their (inner) source, but logically, not consistently at their external manifestation due to limits involved with the manifestation itself and potential interference of other such beings. The existence of such Gods being implied (but not wholly expressed) with certain mathematical properties such as those of transcendental numbers. Or further:
<A complex number is said to be hypertranscendental if it is not the value at an algebraic point of a function which is the solution of an algebraic differential equation with coefficients in Z[r] and with algebraic initial conditions.
<The term is related to transcendental numbers, which are numbers which are not a solution of a non-zero polynomial equation with rational coefficients. The number e is transcendental but not hypertranscendental, as it can be generated from the solution to the differential equation
You mentioned previously how without spatial relations (or implicitly, without convergent limits), it would be impossible to go from point A to point B. But if Monad is a dimensionless point where no distances exist, nor is any (inner) movement possible, yet, represents an ur-element that contains all the topological configurations of itself as potential (anything that could ever be constructed from such a singular, irreducible point, or what would be your definition of a "god" or ur-parent, which in reality is just a sort of deus ex machina used to advance the plot in someone's favor), then graphs are not exactly the expression of the ultimate reality. Your idea of One is simply Monad extended over the boundless space, its actualization or extension.
>algebraic expression of an object
I'd like to introduce you to algebraic independence (pdf related)
<It should be possible to show that if α is an algebraic number with 0<|α|<1, then the set { ƒω(α): 0<ω<1 } contains uncountably many algebraically independent numbers.
>What is meant by x² + y² = a², where a is the radius and x and y are, respectively, the opposite and adjacent sides of a right triangle with hypotenuse a, is that regardless of the angular position of the radius (i.e. the legs of the triangle)
>We are not able to completely render all digits of π because the number system cannot represent magnitudes
You cannot square a circle. If you refer to the power of a point and law of cosines, it would still depend on an irreducible, implicit center-point (monad) to define a circle (constructed from the algebraic perspective of monadic limits and structures), but it would not fully capture the transcendental properties of Pi, only bind them in its own (limited) frame of reference. In other words, you are transposing monadic logic over entities possibly exceeding its limits (from their own frame of reference). Which is normal since your entire consciousness is based on cultural discovery of these mnemonic rules and relations, their dialectical properties etc. making it very difficult to see through them. Your language, identity, sense of "power" (in relation to your entelechy and generated environment), knowledge of the Forms, are all rooted in this schema. I'm not trying to disprove that x² + y² = a² here, I am challenging the notion of the universal point itself. For if there exist other, irreducible and non-repeatable elements and properties, it could alter the root of the structure itself. And under the right circumstances, ancient number systems hold more truth than the newer ones, because they were based on certain metaphysical truths.
>In theory, yes.
Since you like graphs, there would be another way to 'reconstruct' the circle. Imagine splitting its circumference in infinite parts. No matter how small, each part would contain the curvature (information) needed to complete the circle. This would not be the case if you reduced it to the aforementioned point (it would need to be constructed by various techniques and use of other, identical points of reference), yet, a circle is supposed to be a continuous ring of points? Or an infinitely long line. Do you understand the perspective that I'm trying to infer here? A point would produce the same results independently of perspective, yet, an infinitesimal part of the circumference of the circle could be rotated in different directions. While the value of the circumference may not change (I won't go into potential paradoxes here although I can envision a possibility of non-reciprocal results due to what you term 'indefinite' series or a potential randomness), would it still be the same circle?
>Except we do know the underlying patterns of irrational numbers, even if the decimal expansion is non-repeating, viz. continuous fractions. It's only random if you limit yourself to the decimal expansion.
Does that apply to all transcendental numbers, hypertranscendental numbers etc.?
<The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern
Even in generalized continuous fractions, wouldn't the irreducible values of ratios eventually reach infinity? Continuous fractions are just a representation, they do not actually solve anything (nor provide the actual ratio). Any alleged 'architect of the universe' would need to know exact values to be able to fully utilize them, otherwise it can only approximate. And that's without considering various functions and operations involving these numbers, from which infinite complexity could be derived. This doesn't actually refute any of my central points.
>That's not infinity. For something to truly be infinite, it must not be comprehended or bounded by anything (including a sequence); otherwise, it's just indefinite (as I argue irrational numbers to be).
That's an acceptable definition of infinite nowadays, giving us a classification of different types of infinities. Regardless, "indefinite" may qualitatively be an order of magnitude greater than "infinite" (with the exception of 'absolute infinity', the existence of which I found doubtful, but rather, find the closest representation of in the non-unifying number Zero), making arguing about semantics superficial. Especially if we consider the starting parameters of any chaotic, or self-organizing system that could be exponentially developed to some type of infinity, where in one case the sequence would be cyclically repeatable, and in another not .... Sounds quite like the concept of Manvataras, doesn't it?
>0.9999... is just another way of signifying the number one
And what is the basis for this "signification"? I have already mentioned it, but feel free to give us an alternate answer. Your argument comes down to: "1+1 is just a signification of the number 2", trying to deflect without adding anything to discussion
>because 0.3333... is not converging to anything; 0.3333... is just the decimal expansion of 1/3, and it would be non-repeating in another number system like duodecimal
Is duodecimal a more base and systematically more simple system than the decimal?
>Wrong, circles aren't eccentric; they cannot be lines.
Maybe it needs a more adequate explanation
https://yourquickinfo.com/can-a-circle-be-infinite/
<As the center gets farther and farther away, the radius gets larger, and the curvature gets smaller. When the “circle is centered at infinity” the curvature drops to zero, and the edge becomes a straight line. Old school topologists get very excited about this stuff.