Euclid — "The postulates are not self-evident, but they are necessary for the development …"
The postulates are not self-evident, but they are necessary for the development of geometry.
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More from Euclid
"To inscribe a regular hexagon in a given circle."
"The laws of nature are but the mathematical thoughts of God."
"A straight line is that which lies evenly between its extreme points."
"The properties of figures are derived from their definitions and postulates."
"If a straight line be cut into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal…"
Details
Implied understanding from the structure of 'Elements', though not a direct quote.
Date: c. 300 BCE
WisdomVerification
Unverifiable
Found in 1 providers: grok
1 source checked
Understanding this quote
What it means
Some foundations cannot be proven or made intuitively obvious — they must be accepted as starting points before any larger structure can be built. Their value is not that they feel naturally true, but that without them no coherent logical framework can follow. This separates two distinct qualities: whether something is self-evident, and whether it is necessary. Necessary wins — the system must begin somewhere.
Relevance to Euclid
Euclid's Elements opens with five postulates, the most contested being the fifth — the parallel postulate — which ancient readers found far less obvious than the other four. Euclid carefully distinguished postulates from 'common notions,' signaling he knew they carried different epistemic weight. His willingness to build an entire logical edifice on admittedly non-obvious assumptions reflects his commitment to rigor and intellectual honesty over false certainty.
The era
Around 300 BCE in Alexandria, Greek philosophers actively debated the foundations of knowledge. Aristotle had recently argued all reasoning must start from unprovable first principles. Euclid worked at the newly established Library of Alexandria under Ptolemy I, systematizing centuries of scattered Greek mathematics. In a culture obsessed with rational demonstration, deciding which assumptions to accept without proof was a genuine philosophical challenge, not a technicality.
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