The Axiom of Geometry — Full Supplied Copy

Authored by Maclain Hunter · first-party-authored · authored-foundational-record

Stored 4P source diagnostic: clear · finding no_violation_detected · admission admitted_provisionally. This intake result is separate from the theorem checks of the exact propositional sequent.

The Axiom of Geometry by Maclain Hunter Lines, angled/curved or not, derived from Logic (Logos - Logical Coherence), form planes, and (higher dimensional) shapes that can be rationally categorized by structure (ie - a circle can be seen as a line with a consistent curvature that is the reciprocal of the radius). These shapes are consistently defined by the interconnections, angles, and/or number (finite or infinite) of lines/sides, chartable as finite points (that can themselves be grouped into sets), existing with temporal independence and logical coherence superseding spatial realities and logically and geometrically contingent spacetime. Proof of Axiom by Contradiction: Assume for a moment the negation of the Axiom of Geometry proposed above. This leads to logical contradiction since it would negate the necessary logical grounding of geometry and geometrically contingent concepts like shape, space, and spacetime. Likewise, “linear time” presupposes the line, as well as geometry and the Logos (Logic - Logic’s Foundational Coherence). It may satisfy some incoherent notions of geometry to base geometry in itself or some other concept (besides Logic), but it does not satisfy Logic, which requires, to be satisfied, that all things including the Cosmos, the physical world, and (tautological) Logic itself must find their Logical Foundation (Origin - Ground - Cause) in Pure Logic (Logos - Reason - Logic). Assumption of the negation would also mean arbitrarily negating the potential for lines and/or points to be represented as planes, and multidimensional spaces, branching timelines, or complex objects (and vice versa) and (set theoretical or other) sets. Such positions are logically untenable, especially in the face of the coherent alternative offered by the proposed Axiom of Geometry. To say that geometric construction cannot be achieved through the Axiom is also an untenable position as the Axiom derives all planes, shapes, and points, and indeed all aspects of geometry consistently from Logic, allowing for the Law of Identity (understood as the Principle of Coherence, as Logic, and as “the Logos” or “Foundational Coherence”) to build and serve as the basis of universally applicable and logically consistent geometry and geometries. Axiomatic Construction: Point: A point is the most finite and particular of geometric constructions, the quantum level of geometry that all integrated aspects of geometry intrinsically encompass as a matter of logical and geometric form. Geometry without a point (literally) would be geometry without a point (literally and figuratively), as it would contradict Logic (Logos). This quantum unit of geometry, in the context of geometric reasoning has no measurable size or length, and takes up no measurable or meaningful space, much like the quantum unit(s), contingent upon geometry (and the concept of “quantum geometry”), in physics. Line: A line is a measurable length drawn between two or more points. The points must be completely aligned and coherent from one perspective to allow for the construction of a single perfectly straight line through them, though a line (not necessarily straight) can be drawn to connect any number of finite points, regardless of position, alignment, or distance. Lines of this sort, particularly the lines between two points that form the sides (or boundaries) of more complex shapes, defined by the number of sides and angles, and the exact proportions and measurements of those sides and angles, starting with the triangle as the minimal feasible shape (made from lines) if the lines are straight and starting with the circle (or ovals, or the like), potentially constructed from a single curved line curving at an angle that equals the reciprocal of the radius (for a perfect circle). From more complex shapes and geometric objects or planes are not only possible, but logically follow (ie - rectangles and polygons). Circle: Traditionally a circle is constructed by first placing a finite point and then choosing a radius, then taking the radius squared times pi, but this is an inconsistency in how we construct shapes upon further reflection, as we do not traditionally begin drawing triangles, rectangles, polygons, or even imperfect circles from a central point, which also adds an unnecessary extra step that fails Occam’s Razor and demands for parsimony. The more consistent approach would be (and is) to start with a single point and to, for a perfect circle, to start drawing a line with a perfectly consistent curve to it. Any amount of consistent curve will result in a perfect circle with less drastic curves resulting in larger circles and more extreme curvatures resulting in smaller circles (but never feasibly a finite point as that would contradict and ignore the construction of the circle using and requiring the measurable (not quantum) line made from the connection of two points. In the case of the circle the start and end points of the line do and must align/cohere, as the alternative leads to contradiction and a circle that fails to be a perfect and closed loop, like Logic itself (which the perfect circle is a fantastic even if not exclusive representation of, even in traditional cultures where things like the ‘Ouroboros” or “Tail-Devourer” is rightly considered a coherent geometric representation of the Divine Logos underlying all realities including shape/form and substance/meaning). Plane: A plane is formed when lines intersect and create a closed two-dimensional space, like a tile, linear or curved, and complex shapes can enjoy many sides with a number of planes. The type of plane is defined by the length and curvature of the intersecting lines that form it. Polygon: Polygons begin with triangles and extend infinitely to infinitely complex shapes tab we don’t necessarily have established names for. A polygon is defined by the number of sides and planes it has and by the angles and curvature (or linearity) of these features. A basic polygon is constructed by starting with a point, and drawing lines to other points (at least three) until a closed shape is formed and the line returns to the coherent start/end point(s). Angle: An angle is formed by intersecting lines (derived from logically derived points and coordinates). The way the lines intersect determines the angle with a right angle being the result of perpendicular intersection, and without obtuse or acute angles being the result of a lack of perfectly perpendicular intersection. Space: A space is a geometric construct and container or area in which other geometric and logical objects can exist, move, be moved, and in general act or interact with each other. Dimensions of space from a given/fixed standpoint include height, width, depth, and time. The Shape of All Shapes: This refers to the ability of a shape of infinite complexity, with an infinite number of points and lines (curved and/or straight) to encompass all other geometric constructs, including all points, lines, planes, shapes, and spaces that can be conceived of. These are shapes and structures that logically follow from the coherent geometries inherent in the Logical Structure of Reality (Logos - Logic), that are rational and not arbitrary, even while dynamic. This works in the same manner that a triangle can be made from drawing a line from corner (point) to corner (point) in a rectangle, and that a point can be found within even a circle drawn from a single point with a single line, or a rectangle and a square can be made within a pentagon by adding an internal line between two points on either side of one of the five points that define the pentagon, and so on (until infinite complexity and all shapes are combined and coherently integrated. This helps us to understand the geometric structure of the Cosmos (the Logical System of All Logical Systems) itself, with this “Shape of All Shapes” reflecting the “System of Systems” (Logos). Within that integrative structure exists all finite points and all quanta, whether we are speaking about the Cosmos or the Shape of Shapes. The relationship of the finite point to the Shape of Shapes is analogous to the relationship of subatomic (quantum) particles and the physical world or Cosmos in physics. Some have confused the infinite nature of the Universe (the Shape of Shapes) as meaning that the Universe has no shape. Its not “no shape”, it is a logically grounded infinite shape, that dynamically sustains itself and builds/realizes the infinite possibilities latent within Logic (God). The actual Reality is already and eternally complete. Only in the finite/temporal realities within spacetime, deep within the Completed Reality, is it “still developing” or does it appear to start with a “big bang”. In Truth the Universe comes from the Absolute Truth (Pure Logic - Logical Completeness - Full Coherence), and in Logic from Pure Logic (Tautological Reality - Absolute Truth). Understanding Existence chronologically rather than logically and as a matter of Logical Order (Logos) can lead to confusion, as you’ll be thinking it came from the “beginning” of the chronology, but no. Time encompasses beginning and end, as shape includes all shape, and that is then encompasses by things like Geometry (where spaces and lines including the concepts needed for a timeline come from), and Logic (the All-Encompassing True Origin that holds all existences and knowledge). Modal Proof of the Axiom of Geometry: P: Necessarily there must be a coherent axiom, that is a pure reflection of the Logos (the Logical Structure of Reality - Pure Logic), in the context of geometry, as the alternative is not coherent and leads to contradiction and “coherent geometry” becomes impossible. Q: Necessarily the coherent Axiom of Geometry can only be a single axiom, as above, in order to be singularly coherent. Multiple axioms, as found in less coherent traditional sets of geometric axioms (ie - Euclidean, Hilbert’s, Riemannian, etc.), cannot amount in the strictest logical sense (or even a relatively lax one) to a “coherent axiom” - which must be a unified rule to not be contradictory to the Logos—the Unified Rule of the Cosmos—which must manifest in geometric context, as it has to lead to spacetime and geometry, for the Universe to have and for humanity to find a coherent Axiom of Geometry like the one above. R: Necessarily Logic (Logos) supersedes geometry and coherent geometric axioms and systems (logically speaking). The alternative cannot be true since that leads to logical contradiction which makes a Logos (Universal Logic) coherent geometric axiom a rational necessity and imperative, that humanity has been incoherent objectively for failing to discover before now and before me (Maclain Hunter - the logician). S: Necessarily the contradiction of the previous premises and the assumption of ~(P, Q, R) lead to logical contradiction, otherwise Logic itself fails and fails to be supreme, as logically Logic is (and must be to succeed and work rationally without arbitrary function or needing contingent/illogical derivation). To assume the negation of the previous premises (P, Q, and R) means rejecting that geometry and geometrically contingent concepts like spacetime and physical reality need a/the Logical Foundation (Logos), which is a logically untenable position to hold. {P, Q, R, S} |= (P & Q) <-> (R & S) Conclusion: {P, Q, R, S}: The proposed Axiom of Geometry offers a coherent basis for geometry and the construction of spaces and shapes, as well as for (geometrically contingent) concepts of spacetime and physical reality, that is desperately needed in human academics and which is necessary and a fact of Reality - that Reality as we know it absolutely cannot exist without. Truth Table Analysis: valid ↓ ↓ ↓ ↓ ↓ PQRS⎪P Q R S╞(P&Q)↔(R&S) ――――――――――――――――――――――――― TTTT⎪T T T T TTT T TTT TTTF⎪T T T F TTT F TFF TTFT⎪T T F T TTT F FFT TTFF⎪T T F F TTT F FFF TFTT⎪T F T T TFF F TTT TFTF⎪T F T F TFF T TFF TFFT⎪T F F T TFF T FFT TFFF⎪T F F F TFF T FFF FTTT⎪F T T T FFT F TTT FTTF⎪F T T F FFT T TFF FTFT⎪F T F T FFT T FFT FTFF⎪F T F F FFT T FFF FFTT⎪F F T T FFF F TTT FFTF⎪F F T F FFF T TFF FFFT⎪F F F T FFF T FFT FFFF⎪F F F F FFF T FFF Q.E.D. Reinventing the Wheel: There are few aspects of shape construction using the Axiom of Geometry and Logic applied to geometry that introduce a notable change to (and critique of the inconsistency of) the traditional methods of shape construction as we see with the perfect circle. Therefore I will elaborate on it, and I will introduce the concept of the “arc unit” (au) to help us understand curvature in geometry without explicit use of pi and the headaches that pi and attempts to manually calculate or approximate pi can and often do cause in practical application. If you accept the geometric construction of the circle as traditionally conceived then you either end up with a circle with a radius line going to the point in the middle construction started from, or you have to add two or more extra steps compared to proper and consistent construction style from a point and with lines. Otherwise, you must introduce the concept of erasing lines or having imaginary (indrawn but perceived) lines. It adds steps and inconsistency precisely because you have to start with a point and a radius line, then complete the circle based on an extremely complex calculation using pi (an inelegant option even for calculators and computers that have deep approximation of pi), or introduce a mechanical tool like a compass randomly and arbitrarily into the construction of shapes that is not used anywhere else, for polygons or the like, which is again a form of inconsistent workaround for subpar construction methods. From there you then need to go back and erase the radius (line) and the center point to end up with just a perfect circle and no extraneous lines or points. To be more efficient and consistent we can simply start with a point and follow a consistent curve of any grade back around to the starting point to make a circle of a size that is always perfect and with inverse proportions between how aggressive the curve of the line is and how large the circle is, with a gentle curve leading to a larger circle and an extreme curve leading to a smaller circle. The other option (in traditional construction) that still leads to abnormalities would be to have undrawn placeholder lines that can be used to get from the center point and radius (line) to the drawn perimeter of the circle alone. This introduces a complicated and unnecessary abstract aspect into shape construction that is inconsistent with the methods used for other shapes, which is logically inconsistent, which reduces self-evidence of shape construction, and which being more complex than my alternative also fails Occam’s Razor If a consistent curve has a 1:1 ratio in its movement along the x and y axis’ of a graph then it will loop back to its starting point creating a circle that is 2 units in diameter with a radius of 1. Then the area of that circle will be the circumference divided by two times the radius of 1. The constant for deriving the circumference need not be pi (necessarily and in particular), as it can be replaced in effect by a special, easier to work with circumference measuring unit, defined as, “the length of a perfectly consistent arc of 1 linear unit along the x axis of a grid, and 1 linear unit along the y axis of that graph”. Such a proposed unit is called, in my system of geometry, an “arc unit” or “au” for short. That way a circle of those proportions (a radius of 1 linear unit) can be drawn, where the arc of each quadrant is “1 au” (one arc unit) and the whole circumference is simply “4 au” in length. This makes solving for the area a calculation of 4 au divided by 2 times 1, for an area in this case of 2 square arc units. It drastically simplifies sphere construction and the construction of arched planes to use my methods, in the sense that a sphere can be made by starting from a point, moving 1x au (which will ultimately be the size of one quadrant of one curved plane’s boundary in a given sphere) and making a point before continuing the line another 1x au (for a total of three points with a 2x au line across the three equidistant points). We do this to make the foundational boundary of a spherical plane. Then we cover the sphere with a consistently curved plane (measured in au), the center of which follows the track of a perpendicular line from the center point out of the three points of our boundary, halfway along the first 2x au line. That plane, in terms of circumference in the center, is 4x au in length. We cover this space with such a curved plane, closing the sphere. This means the sphere has a height, width, and depth of 2x linear units (each) while the circumference in any direction or from any starting point on the sphere is 4x au, making the total area of the sphere a simple calculation of squaring the area of a two dimensional circle of the same proportions and radius, or: 4x au / 2 * x ^ 2 What is convenient also is that it doesn’t mean we are doing away with what pi represents, just the complexity of working with it directly, and for any case where we need to convert from arc units (au) to pi based calculations, we can use the equations below: For a circle with a dynamic (unfixed) radius: 1 au * r = πr ^ 2 / 4 For a circle with a fixed radius of 1: 1 au = π * 1 ^ 2 / 4 This fundamentally changes the way we have approached circle construction, curved planes, and non-linear and complex/hybrid shapes (that may have some linear and some curved lines and planes), freeing us from inconsistent shape construction and the often burdensome use of pi, and from the rigid linearity of Euclidean Geometry. Specific Units and Quantum Coherence: If you think about it, we would not typically think of the straight line as some twist in the default, with the default being the curved line, though with a round earth and planetary bodies being (arguably) more common than lengthy perfectly straight lines in nature it could actually make a lot of sense and be well grounded in Reality’s Logic (Logos) to do so. Yet, we do essentially that with making straight lines the default and curved lines something that we measure rather arbitrarily in units designed for measurement of perfectly straight lines, that allow for convenience in measuring straight lines, but which present headaches in curved lines (just as using arc units to measure a straight line would present). That’s why it makes more sense to have units tailored to each. From there the two can be linked through a simple coordinate system that accounts for the start and end position of the line rather than the topological length of the curve, which also allows for us to meaningfully work with lines with complex curves. It is like how we could know that a point twelve aerial miles upstream on a river is exactly how far we have to go in aerial miles to get to that point, no matter how complex the river’s bends are and what the actual length of the river, which is not necessarily meaningful for plotting our trip or understanding in a practical sense how far we must go from point a to point b. That coordinate (point) based approach can also help us understand various phenomena in physics, particularly in quantum mechanics, where patterns and interactions that appear random or unequal actually have profound forms of coherence and simplicity (and/or identity) underlying the emergence of complexity. If the whole thing that matters is that the start point (a) is at 0 on the x-axis and 0 on the y-axis (0, 0), and the end point (b) is (1, 1), then the actual length of the line drawn between them is trivial and is representative of the logical connection between the points in any case. This allows for all paths between an and b to be equal in their fundamental outcome and (point or quanta charted) logical structure - allowing for coherence to underly surface level chaos in the “quantum mapping”. That serves as a necessary bridge between Foundational Coherence (God - Logic - Logos), also understood as “Universal Order”, and both complexity and chaos (or incoherence and evil/incorrectness when measured against Logic). In this sense we can geometrically, or through aspects of the Philosophy of Geometry, solve the “problem of evil”, as incoherence between natural differences in the characteristics of finite shapes and geometrical structures/concepts can be shown to resolve through the Axiom of Geometry, “geometric universals”, and, of course, Pure Logic (Logos - A Vision and Reality of Absolute Unity). From that perspective of “non-duality” (oneness) and non-contradiction (Logic), the gaps that exist in common understandings of physics, between things like Quantum Mechanics (QM) and Relativity fade and even incoherence, mystery, and a lack of understanding can be understood as having been weaved into a perfectly logical (and geometrically/physically manifest) metaphysical and universal (non-local, and not merely metaphysical) Logic - the Logos. One of the “quirks” the underlying coordinate system leads to is that as a result of being a map of “points”, per the consistent approach and definitions used, the total system, even while containing an infinite number of points, collectively takes up no space and no time. It is possible to collapse into one coherent geometric object that requires no further explanation or justification, logically, mathematically, geometrically, physically, scientifically, legally, or otherwise. This helps us understand why there is necessarily, even from the perspective of Geometry alone, something like the “quantum superposition”. It is because that is how geometry works, and how “quanta” work, both physically and geometrically. They are what are used to construct and realize theoretically feasible complexity (and “advanced contingent concepts” - which Logic logically leads to) latent within the Natural Order (Logos). These quantum structures transcend, by definition, the spatial and temporal constraints of the spacetime they construct. It can’t be any other way, as the assumption of the negation of my statements leads to absurdity, inconsistency, a lack of Logic based proper “Logical Order”, and logical contradiction. These quanta exist naturally and logically beyond spacetime, and thus the study of Quantum Mechanics is not strictly physical science in the sense of being constrained to spacetime and objects that have geometric shape and measurable physical weight or energy. It is very much so an empirical observation of metaphysical structures - empirically measurable with logical and scientific tools. In summarizing this section of the broader Axiom of Geometry, to establish specific units for specific measurements, and then increasingly or directly universal measurements, and methods conversion for translation between them, is logically consistent and coherent. It is something humanity has (in some cases) overlooked, like the historical absence of formal recognition of the intrinsic and naturally existing concept of the arc unit. This concept is not arbitrary or merely “imaginary” (meaning “only existing in the finite mind, or in a way contingent upon the finite mind” in this context). It is an aspect of Reality’s own self-construction and Logos-based rationalization (realization) - a concept that is just “there”, which amounts to understanding curves for what they are instead of what they are compared to a straight line (which is “apples to oranges”), that was merely awaiting a name and discovery when I came to it through logical reasoning and applying Logic to problem solve and tackle challenges in the field. The Redundancy of Proof for the Pythagorean Theorem by Maclain Hunter: Premises: Let P represent Logic and the Logical Order inherent in deriving trigonometry (a) from the Pythagorean Theorem (b), deriving the Pythagorean Theorem from syllogistic logical reasoning (lower order formal logic), and then deriving lower order syllogistic logic (c) from the Highest Order of Logic (Universal Logic) also called the Logos (d) or “Universal Logos” (containing all logoi and finite reasoning). Let Q represent proof, as well as the contradiction inherent in the assumption that the validity and real-world success of trigonometry is not demonstration of the validity of the Pythagorean Theorem. Let R represent the statement “the Pythagorean Theorem is valid and true, as demonstrated by its continuous practical and accurate function |= the Pythagorean Theorem is valid and true, as demonstrated by its continuous practical and accurate function”. Let R also represent the contradiction that arises from assuming that the Pythagorean Theorem is not inherently “proven” (aligned with Logos demonstrably) by being a valid geometric application of algebra in syllogistic form. Assuming the negation of the set of premises {P, Q, R} leads to logical contradiction, therefore the set of premises is true. Thus the reasoning holds that: {P, P <-> Q, Q <-> R} |= (P & Q) & R Conclusion: {P, Q, R}: The Pythagorean Theorem is already self evident and is proven by its own valid and true identity, which is what any proof or logics must achieve (in any case) - so they are objectively consistent with Logic being Logic. It is absurd to think the Pythagorean Theorem lacks Coherence (Logic) and thus “lacks proof”. It is self-evident in the same way a syllogism with true premises is. The Pythagorean Theorem can, in fact, only leas to a “wrong answer” when the values that are used are incorrect and don’t actually match the triangle at hand - which is not a flaw in the theorem but in the use of it. The Logical Order of concepts a, b, c, and d, representing trigonometry, the Pythagorean Theorem, syllogisms, and Logic (Universal Logos) is as follows: {a, a <-> b, b <-> c, c <-> d} |= (d -> c) -> (b -> a) Truth Table Analysis: valid ↓ ↓ ↓ ↓ PQR⎪P P↔Q Q↔R╞(P&Q)&R ―――――――――――――――――――――― TTT⎪T TTT TTT TTT TT TTF⎪T TTT TFF TTT FF TFT⎪T TFF FFT TFF FT TFF⎪T TFF FTF TFF FF FTT⎪F FFT TTT FFT FT FTF⎪F FFT TFF FFT FF FFT⎪F FTF FFT FFF FT FFF⎪F FTF FTF FFF FF Additional Note: If you don’t accept that the algebra of the Pythagorean Theorem is proven to not be invalid and arbitrary by its proper description of concepts in geometry, that have been shown to align, then identifying “proof” (alignment with Universal Logos) is surely not your forte. It is like thinking “2+2=4” needs external validation, or that the Law of Identity (which is what it is consistent with itself) needs external law. It’s not so. Like the syllogism is a universally valid form of proving C based on A and B the Pythagorean Theorem is a more specific application of this same kind of Logic, tailored to the problem of figuring out a tangent based on sine and cosine, or any combination of those, using two (out of the sine, cosine, and tangent) to prove the remaining/unknown one. Truly, to deny the Pythagorean Theorem which follows and gives example of valid syllogistic form would be to defy and undermine Logic in perhaps its most classical and sound form. Thus the Pythagorean Theorem can also be proven through contradiction by assuming that it is false and not possible to logically proof, and by inspecting the absurdity and negative implications for other aspects of Logic (like syllogistic form) that then arise - consistent with that line of reasoning. It would make sense if Pythagoras had understood this and that it is why proof of the theorem was not such a priority for him. Q.E.D.

Lean 4 verification and timestamps

The machine formalization date is separate from the original authored work. See the verification record for exact checked scope and any unresolved parts.

Logic-centered continuity

This authored proof is connected to Logic, Logos, God, and the self-evident Ground of Reality, and to the distinct systems 4P Engine, MCB Core, 4P-MCB, and the source-first Canon. Formal proof, empirical evidence, and system-generated records remain separately typed.