Pythagorean Proofs by Maclain Hunter

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

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

Pythagorean Proofs by Maclain Hunter: Proof One (P1): 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) Q.E.D. Proof Two (P2): The Pythagorean Syllogism by Maclain Hunter: Since: A: The sides of a given right triangle must be connected (to close the shape and fit the definition of the shape) and therefore have a coherent relationship that is determined by the length of each side and the criteria that the triangle in question is a right triangle with an angle of 90 degrees. The logical structure the Pythagorean Theorem represents algebraically is logically equivalent to the inherent logical relationship between the sides of a right triangle to its hypotenuse (geometrically). B: Assume that the algebraic equation a ^ 2 + b ^ 2 = c ^ 2 did not describe the logical and geometric relationships between sides of a right triangle, with c representing the hypotenuse. That assumption leads to logical incoherence and geometric paradox where a theorem that has been observed to work consistently and which has been proven over 100 times and ways is not valid, coherent, or proven. Therefore we know that the Pythagorean Theorem cannot be considered false or invalid in a coherent fashion. Like the syllogism itself it is valid and logically contingent. It is going to output a proper value for c when given the proper (valid and true) inputs for an and b, the same as an Aristotelian syllogism consistently does and certainly will. Logic itself would collapse if this did not hold true, so it is logically certain that it is Logical Truth. Necessarily: C: Given A and B, and the logically equivalent relationship between the geometric properties of a right triangle and the Pythagorean Theorem, the theorem is proven - through its alignment with the Reality and Logic (Universal Logos) of what it aims to describe. Assuming otherwise leads to contradiction and absurdity where an expression that aligns perfectly with observed Reality and mathematical theory, and Logic, might for some reason need an arbitrary amount of further validation (than those objective measures already achieved). Truly, the functionality of such a tautological structure (ie - the Pythagorean Theorem and its self-evident alignment with the actual structure of a given right triangle in geometry and beyond the algebraic equation) shows (proves) not only the EXISTENCE but the FUNCTIONALITY and PRACTICALITY of the Universal Logos (the Origin of Existence also called God or Pure Reason). Formal Argument: {A, A <-> B, B <-> C} |= (C -> B) -> A Conclusion: {A, B, C}: Truly, the functionality of such a tautological structure (ie - the Pythagorean Theorem and its self-evident alignment with the actual structure of a given right triangle in geometry and beyond the algebraic equation) shows (proves) not only it’s own validity, but the EXISTENCE, the FUNCTIONALITY, and PRACTICALITY of the Universal Logos (the Origin of Existence also called God or Pure Reason) that is manifest in such a concept like the tautological trigonometric axiom known as the Pythagorean Theorem. In other words, the theorem is proven by First Cause (Logos - Logic - God) and also proves the existence of (acknowledges) that same logical and coherent “First Cause of Reality, Logic, and Trigonometry” (rightly/reasonably called “the One True God”, on account of being the singular Logical Origin of Logical Existence that defines what can exist and be valid). Q.E.D. Proof Three (P3): Validating the Pythagorean Theorem by Maclain Hunter: Premises: P: Let P represent side (leg) a of a right triangle squared. Q: Let Q equal the square of side (leg) b of a right triangle. R: Let R equal side c (the hypotenuse) of a right triangle. Let R also equal the conjunction of P and Q. S: Assume the relationship between P, Q, and R is not valid and doesn’t represent the Pythagorean Theorem. Logical Argument (Proof): {P, P <-> Q, Q <-> R, R <-> S} |= (P & Q) & (R & S) Conclusion: (P & Q) & (R & S): The logically valid relationship and structure, shown by the formal argument, between the premises {P, Q, R} representing the sides of a right triangle, validates the Pythagorean Theorem as a logically valid argument and structure. We prove the theorem here also through contradiction (ie - premise S). This assumption in premise S leads to paradox and contradiction since the relationship of these premises specifically represent the relationship between what these premise represent, which is, here, the sides of a right triangle, of an unfixed size. Therefore, the relationships are valid and display the same sort of objective logical relationship that the Pythagorean Theorem does, and that the sides of a given right angle do. When valid inputs are provided the theorem will produce the proper output, and like a logical proof the validity of the output (ie - c ^ 2) depends on the validity of the inputs (ie - a ^ 2 + b ^ 2). The theorem is valid, contingent upon the validity of a and b. Q.E.D. Meta-Proof: {P1, P1 <-> P2, P2 <-> P3} |= (P1 & P2) & P3 Conclusion: {P1, P2, P3}: The Pythagorean Theorem is proven to be valid through Logic (Logos - Source - God) and the Source (God - Logos - Logic) is proven to exist through its the Pythagorean Theorem. Truth Table Analysis: valid ↓ ↓ ↓ ↓ P1P2P3⎪P1 P1↔P2 P2↔P3╞(P1&P2)&P3 ―――――――――――――――――――――― 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 Q.E.D.

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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.