Philosophical Implications of g(x) = (3/4)x² Representation

1. Formalism and the Nature of Mathematical Truth

Representing g(x) in first-order logic emphasizes the formal structure of mathematics. It shows that mathematical truths can be derived from axioms and rules of inference without appealing to intuition or physical interpretation. The transformation from f(x) to g(x) is purely syntactic, a manipulation of symbols governed by logic.

Philosophical implication: This supports the formalists’ view that mathematics is a game of symbol manipulation, where the meaning of g(x) arises solely from its place in a deductive system.

2. Abstraction vs. Experience

The function g(x) = (3/4)x2 describes a geometric transformation—a vertical scaling—but in first-order logic this transformation is encoded purely symbolically, without any visual or experiential content.

Philosophical implication: This highlights the tension between abstract representation and phenomenological experience. Logic captures structure but omits the subjective “feel” of seeing a curve stretch, mirroring debates about qualia in the philosophy of mind.

3. Model Theory and Ontological Commitment

By expressing the function in first-order logic, we commit to a model—a domain of real numbers, operations like multiplication and inversion, and interpretations for f and g.

Philosophical implication: This raises questions about ontological realism. Are these mathematical objects “real,” or are they useful fictions within a formal system? The representation remains agnostic, aligning with model-theoretic semantics where truth is relative to a structure.

4. Reductionism and Compositionality

The proof shows that g(x) is composed from simpler operations: multiplication, inversion, and function application. This reflects a reductionist view of knowledge—that complex phenomena can be broken down into atomic operations.

Philosophical implication: Understanding is seen as compositional. We grasp wholes by understanding parts—but this also invites critique: does such reduction miss something essential? Is the “whole” more than the sum of its logical parts?

5. Epistemology: Knowing Through Proof

The line-by-line derivation of g(x) = (3/4)f(x) shows how knowledge is constructed through deductive reasoning. It’s not just that the function behaves a certain way—it's that we can prove it does.

Philosophical implication: This reflects a rationalist epistemology, where knowledge arises from reason rather than empirical observation. The graph may suggest the scaling visually, but the proof secures it conceptually.

---

Summary of Themes

Theme	Implication
Formalism	Truth is derived from syntax, not intuition
Abstraction vs. Experience	Logic omits subjective experience
Ontology	Mathematical objects exist within models, not necessarily “out there”
Reductionism	Understanding through decomposition and structure
Epistemology	Proof as a pathway to knowledge beyond empirical observation