Common notions or axioms - Philosophical Concept | Alexandria
Common Notions, often interchanged with “axioms,” are the self-evident truths that form the bedrock of logical and mathematical reasoning. But are these notions truly "common," universally understood, or simply assumptions cleverly disguised? These foundational principles dictate how we perceive equality, quantity, and spatial relationships, yet their origins and implications are far from simple.
The earliest recorded articulation of what we now recognize as Common Notions appears in Euclid's Elements, circa 300 BCE. These weren't mere abstract philosophical musings; they were the operational rules for geometry, inscribed within the context of the Hellenistic world – a period of intense intellectual ferment sparked by Alexander the Great's conquests, where mathematical rigor was simultaneously a tool for understanding the cosmos and constructing empires. Euclid’s Common Notions, such as “Things which equal the same thing also equal one another,” were the unchallenged keystones upon which he built his geometrical edifice.
Over centuries, the interpretation and application of Common Notions have undergone a subtle, yet significant, transformation. What began as self-evident truths within a specific mathematical context gradually became generalized principles applicable across various scientific and philosophical domains. Thinkers like Isaac Newton implicitly relied upon them in formulating his laws of motion, assuming a universality that Euclid perhaps hadn't explicitly claimed. Yet, the 20th century saw challenges to inherent certainty, as non-Euclidean geometries questioned the absolute nature of spatial axioms, prompting a re-evaluation of what constitutes a "common" or necessary truth. Intriguingly, Gödel’s incompleteness theorems cast shadows on the completeness of axiomatic systems themselves, hinting at inherent limitations to our efforts to capture all truths through finite sets of foundational principles.
Today, Common Notions, or their functional equivalents, continue to underpin not only mathematics and logic but also computer science and artificial intelligence. They are the silent assumptions coded into algorithms and the unspoken rules governing how machines "think." But what if our "common" notions aren't so common after all? What if the very foundations of our logical systems are built upon culturally biased or inherently limited assumptions? Are we building thinking machines in our own image or can we transcend the limitations of human thinking by questioning these most basic of assumptions?