The Back-and-Forth of “Being” and “Making”: From Euclid to the Present — Using Realism × Structuralism Lead

The Back-and-Forth of “Being” and “Making”: From Euclid to the Present — Using Realism × Structuralism Lead This piece offers a map for consciously shuttling between “realism” (what is) and “structuralism” (what we can make/relate), using the most familiar classic—Euclidean geometry—as a worked example. The upshot is simple: Euclid’s system is a three-layer device—definitions + postulates (permissions to construct) + common notions (axioms)—that lets us separate existence from operation and then recombine them in proofs. It is an early hybrid manual for “seeing what is” and “doing what can be done.” 1) Becoming the puppeteer, not the puppet — philosophy as practice Realism adopts what already exists as its starting point; structuralism begins from how to construct or relate. Our aim is not to be enslaved by either stance, but to switch and compose them as the situation demands—in short, to be the puppeteer (operator), not the puppet (a fixed stance). This practical posture resonates with post-structuralism in the humanities and with the Buddhist Madhyamaka (emptiness, two truths, skillful means). 2) Euclid as a hybrid 2.1 The three-layer architecture Definitions: supply the vocabulary and target objects. e.g., point = that which has no parts, line = length without breadth, surface = that which has length and breadth only. These are realist in flavor: descriptions of “what is.” Postulates (permissions to construct): declare what operations are allowed. e.g., draw a straight line from any point to any point; produce a finite straight line continuously in a straight line; describe a circle with any center and radius. These are structural/constructive. Common notions (axioms): rules for universal inference and invariants. e.g., things equal to the same thing are equal to one another; if equals are added to equals, the wholes are equal; the whole is greater than the part. Together they create a loop where “being” (definitions, axioms) and “making” (postulates) feed into each other to drive proofs. 2.2 Two tones already inside the definitions Realist-style definitions: #1 point, #2 line, #5 surface, #13 boundary, #14 figure—attempts to state what something is. Structural-style definitions: #8 plane angle (the inclination of two lines), #15 circle (points equidistant from a center), #23 parallels (lines that do not meet even when produced indefinitely)—concepts fixed by relations and constraints. → Even at the definitional level, qualities (being) and relations/operations (structure) are interwoven. 2.3 The famed Fifth Postulate (parallels) as a hinge The fifth postulate is historically tricky. In a modern equivalent (Playfair’s axiom), through a point not on a line there is exactly one line parallel to the given line. Here, a property (parallelism) and an operative implication (permission of exactly one construction) interlock—a visible duality of being and operation. 3) After Euclid: three sharpened programs Formalism (Hilbert): absorbs constructive intuition into existence axioms; starts from undefined terms + axioms to secure rigor. (His quip: if the system runs, we could replace “point/line/plane” by “table/chair/beer mug.”) Mathematical structuralism (Bourbaki): treats objects up to isomorphism; the structure—relations—is primary across fields. Intuitionism/constructivism (Brouwer et al.): to exist is to be constructible; prefers witness-bearing proofs over non-constructive existence. (Foundations aside: Gödel and Cohen show—via the constructible universe and forcing—that the Continuum Hypothesis and the Axiom of Choice are independent of ZF. Public moral: axioms themselves can be chosen/engineered; a very structural viewpoint.) 4) Mini case studies (to make it tangible) Straightedge–compass constructions: the impossibility of angle trisection or cube duplication is not about clumsy hands; it reflects algebraic constraints (field extensions). Operational limits expose structure. Two kinds of existence proofs: Non-constructive (e.g., via Choice or Zorn’s lemma) gives a map (“it exists”) but no recipe; constructive proofs provide algorithms or convergent procedures, hence implementability—at the price of explicit cost. → In practice we decide which to grasp first—the map or the recipe—depending on the task. 5) Bridging to clinic, research, and design (three steps) Speed up hypotheses with the courage to treat as if it exists (realist stance) and draw a quick overall map. Land interventions by checking can we actually construct it? (constructive stance). Make the shuttle a habit: after results, update either the premises (axioms) or the operations (postulates) as needed. The mantra remains: separate to diagnose, mix to operate. 6) Appendix: Book I, ultra-brief crib Common Notions (Axioms) 1 Things equal to the same thing are equal to one another. 2 If equals are added to equals, the wholes are equal. 3 If equals are subtracted from equals, the remainders are equal. 4 Things which coincide are equal (congruence). 5 The whole is greater than the part. Postulates (Permissions to Construct) 1 To draw a straight line from any point to any point. 2 To produce a finite straight line continuously in a straight line. 3 To describe a circle with any center and radius. 4 All right angles are equal to one another (note: Euclid does not use degree measure “90°”). 5 Parallel Postulate (often given in Playfair’s form: through a point not on a line there is exactly one parallel to the line). (For the Definitions 1–23, see the earlier list; quote inline only what you need in the main text.) 7) Takeaway Euclid’s geometry is a three-layer engine: being × making × invariants. Modern programs sharpen the two faces (existence / construction) in different ways. In real work, map (non-constructive) and recipe (constructive) should be cycled. Philosophically, don’t be a puppet of one stance; be the puppeteer who selects and composes stances as tools. Ancient text, modern craft: build the reflex to go back and forth between “being” and “making.” It makes thinking nimbler—and moves clinic, research, and design forward.