In contemporary mathematics, algebra and geometry intertwine in diverse and subtle ways. This paper distinguishes two fundamentally different modes of algebra--geometry relations: constitutive and representational. In algebraic geometry (AG), the algebraic structure is constitutive of the geometric object: the geometry exists as an intrinsic manifestation of algebra. In contrast, in algebraic topology (AT), the algebraic structures, such as chain complexes and (co)homology groups, are representational, serving as tools to analyze, classify, and measure geometric or topological properties rather than forming the object's ontology. We formalize criteria for distinguishing these modes, illustrate them with canonical examples from AG and AT, and extend the discussion to Derived Algebraic Geometry (DAG), which partially bridges these paradigms by internalizing higher homotopical data into algebraic structures, thereby enhancing constitutive features without fully erasing the distinction. This analysis provides a rigorous philosophical framework to evaluate the nature of algebra--geometry relations in modern mathematics.