Okay, that adds a crucial layer of intellectual honesty and rigor to the proposed Appendix. Instead of just justifying π and φ, it should first undertake a broader survey of candidate “natural constants” and then argue *why*, within the specific context and axioms of infomatics, π and φ emerge as the most fundamental or appropriate choices for structuring the informational field I, while acknowledging the roles other constants like *e* or √2 might play. **Revised Plan for Appendix: “Natural Constants, Mathematical Frameworks, and the Π-φ Basis for Infomatics”** 1. **Critique of Conventional Mathematics (as before):** - Limitations of Base-10/Decimal System (historical context, linearity, approximation errors). - Problematic Nature of Zero (placeholder vs. nothingness, singularities). - Critique of Artificial Constructs (Negative numbers as operations, questioning ontological status of ‘i’). - Need for a system reflecting continuity and fundamental relationships. 2. **Survey of Candidate Natural Constants:** - **What constitutes a “natural constant”?** Define criteria: dimensionless (ideally), arising from fundamental geometric, topological, or dynamic principles, scale-invariant, not dependent on arbitrary units or specific physical systems. - **Candidate: π (Pi):** Discuss its origin in circular geometry ($C/d$). Role in cycles, oscillations, waves, phase, rotational symmetry. Ubiquity in physics. Status: Strong candidate for governing cyclical dynamics and phase relationships in I. - **Candidate: φ (Phi, Golden Ratio):** Discuss its origin in proportion and recursion ($(a+b)/a = a/b$, Fibonacci). Role in optimal growth, scaling, self-similarity, efficient packing, potentially stability/minimal action principles. Appearance in diverse natural systems (biology, potentially physics). Status: Strong candidate for governing scaling between resolutions (ε), recursive dynamics, and potentially contrast (κ) or repetition (ρ) scaling in I. - **Candidate: *e* (Euler’s Number):** Discuss its origin in continuous growth/compound interest ($lim (1+1/n)^n$). Role as the base of the natural logarithm, fundamental to exponential growth and decay, rates of change, complex analysis ($e^{i\pi}=-1$). Status: Clearly fundamental in describing *rates* of continuous processes and linking cyclical (π) and exponential aspects. While essential for describing *solutions* to dynamic equations within I, infomatics hypothesizes it might be *derived* from or secondary to the *structural* roles of π and φ, rather than defining the fundamental geometry itself. Its dimensionality in $e^x$depends on $x$. - **Candidate: √2 (Pythagorean Constant):** Discuss its origin in diagonal geometry ($a^2+b^2=c^2$). Role in basic spatial relationships, potentially in transformations (e.g., rotations by π/4). Status: Represents a fundamental geometric relationship, but perhaps less universal or dynamic than π (cycles) or φ (scaling/recursion) for structuring the *entire* informational field I across all scales and processes. Might appear in specific geometric contexts within I, but less likely to be a primary governing constant for overall dynamics or structure compared to π and φ. - **(Optional) Other Candidates?:** Briefly consider others if relevant (e.g., Feigenbaum constants related to chaos/bifurcation?), but likely focus on the main contenders. 3. **Justification for Prioritizing π and φ within Infomatics:** - **Direct Link to Infomatics Axioms:** Argue that π (cycles/phase) and φ (scaling/recursion/proportion) map most directly onto the core posited dynamics and structures of the informational field I as conceived by infomatics–a continuous field supporting sequences (τ, related to π) across resolutions (ε, related to φ) with varying contrast (κ, potentially scaled by φ) and correlations (m, potentially involving π/φ ratios). - **Foundation for Geometric Description:** Argue that π and φ provide the most fundamental basis for a *geometric* description of the informational continuum, focusing on inherent structure and relationship rather than just rates of change (*e*). - **Potential for Derivation:** Reiterate the goal of infomatics to potentially *derive* other constants (like the effective values of G, c, ℏ, maybe even *e* in certain contexts) from π, φ, and the informational variables, positioning π and φ as the most foundational layer. 4. **Towards a Π-φ Natural Notation (as before):** - Sketch the idea of a system using π and φ directly. - Emphasize ratios and relationships over linear counting. - Connect to transcending decimal/floating-point limits and artificial scales like Planck length. **Implications for Section 2 (Axiom 3):** - Section 2 can now introduce Axiom 3 by stating Infomatics posits fundamental geometric constants structure I. - It can briefly introduce π and φ as the *primary* candidates chosen by infomatics for governing cycles and scaling respectively. - It should then explicitly **reference the Appendix** for a detailed discussion of *why* π and φ are prioritized over other potential natural constants like *e* or √2, and for the full critique of conventional mathematics. This approach makes the choice of π and φ seem less arbitrary by acknowledging other constants but providing a reasoned argument, grounded in the specific axioms and goals of infomatics, for prioritizing these two. It also strengthens the Appendix by making it a more comprehensive analysis of natural constants in relation to the framework. Does this revised plan for the Appendix and its connection back to Section 2 seem like the right way to address the justification for π and φ rigorously and transparently?