a consideration
No possible geometry can be created, only discovered and realized.
An “impossible geometry” is a shape.
“Im-possible” is that which cannot have potency or potential; the possible is potent and potential as such, even if not located (“brought into locality vs nonlocality”) in immediate facticity.
No pure algebra is possible, only imaginable.
Possible algebra is geometric, and thus is discovered and realized.
Impossible algebra isn’t geometric, and so is created and not discovered.
A possible geometry exists even if it hasn’t been discovered or realized.
A possible algebra is geometric.
All math that exists is geometric. So in this sense “all math is discovered not created,” but unfortunately we can “imagine” things to be math that aren’t (in not existing), like pure algebra. So as long as we are not including “pure algebra,” then math is discovered not created (suggesting that what we include in the category of “math” impacts how we answer the question).
Geometry is discovered.
Algebra is created.
Algebra exists because we can imagine geometry as smaller units, which can help us “do” geometry, but only if we don’t forget to “bring algebra back up” into geometry. If we don’t, we end up in reductionism. We end up in imagination and confused.
Algebra is not “bad,” only a map of a territory of geometry. It is useful. The mistake is confusing use with “standalone existence” (self-sustaining). The existence of algebra is “found in” geometry. If geometry is left behind, algebra will be in reference to a realm of imagination. It could still be useful (perhaps finding formulas that one day will be applicable), it could still be aesthetically pleasing (in being maximally elegant), but it will not be real unless and until it comes (back) into geometry (say because someone finds applicability).
Geometry is the test of algebra: it is a means of falsifying if a given algebraic premise is created or discovered. It is possible for any algebra to be brought into geometry, and hence the real existence of a mathematical notion is testable not simply ponderable.
The test of the real existence of a math is not its self-consistency or lack of contradiction, but if it can be brought “up” into another dimension. Since all possible geometry exists, then any algebra that can be brought into geometry exists. Even if it (then) doesn’t come into texture, color, smell…it still exists; it’s just not realized.
How does geometry exist? Geometry is the form according to which phenomena “unfold” as themselves through time. “Shapes” are geometries that are spatial but not temporal, while “forms” are geometries that are spatial and temporal “Shapes” don’t exist but are created and can be useful to the degree they help us approach “forms.”
“Forms” exist as spatial-temporal geometries, and all math that exists is of form. Math that cannot be “formed” is created.
Math that can be “formed” is “(be)coming” (to allude to the magnificent “Fre(Q) Theory” of Alex Ebert), and so entails movement, potency, and realization into and as actuality. Hence, forms exist and form.
Forms self-(feed/form). They are like wheels within wheels. They are like love that moves the sun and other stars.
To the degree a thing aligns with its form through discovery and realization — versus un-align with it toward “shape” via imagination and creation — is to the degree a thing is true, good, and beautiful as itself, which is an unfolding.
Geometric creation unfolds, while algebraic creation is imagined moving. Literature happens.
Form sublates shape (sequentially in thought, “always already” in life), which means form entails shape but is not reducible to it. So too time sublates space, as geometry sublates algebra; subjects, objects. So it goes with all properties and epistemologies: form sublates imagination (for example) into itself, making it “imagination that can be realized” (which is real compared to algebraic “imagination that can’t be realized”). Nothing of shape is “bad,” only incomplete and unrealized if not brought into geometry. What doesn’t exist is privation — it is simply mis-ordered — alluding to St. Augustine. “All things are good.”
If things didn’t have imagination, they couldn’t imagine “a (future) geometry” which could bring about vertical causation upon them, and so things would not be “forms” but “shapes” and hence stagnant and at “a flat equilibrium.” But where there is imagination, there is the risk of treating math as real that isn’t (like “pure algebra”) — but this risk is necessary and also affords the possibility of choice and meaning.
Where there are forms, there is choice and meaning. Forms necessitate freedom, at least in the sense that we must be able to mistake “forms” as “shapes.” Because we can err, we are free.
“Formal freedom” is when we freely align with the form. Because this isn’t necessary, the choice can be more meaningful to us, in that it can manifest to us as a non-necessary potential that nevertheless sustains itself “as if” it was necessary. This is miraculous. It is beautiful. Beauty is found in a potential that doesn’t have to be and yet is as if it must be.
Forms are possible because we are free to mistake them as “shapes,” and because we can mistake forms as shapes, we are free. Freedom and form are the essential relationship at the root of all things — the simplest of all possible relations — from which all complexities can flow.
Forms freely flow, for flows freely form — existing, thus.
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