Page 38 - GALIET PHYSICS BLOSSOMS I+

AXIOMS
(ii) Now explain what is meant by an axiomatic system. Discuss both the idea of a formal axiomatic system and how it is constructed, and also the “interpretation” of the axiomatic system.
An Axiomatic System
An axiomatic system is one where you start from a basic or intuitive set of propositions or axioms, which do not require proof or prior formal demonstration. The axioms do not need to be self-evident or true. We can think of them as reasoning starting points in a particular logical system. From a few axioms we can derive all theorems using the rules of inference or complex chains of deduction of the system. Brilliant Euclid was the first to create an axiomatic system for geometry for he aspired to derive all of geometry from a set of self-evident propositions. That is, he wished to have a logical system to deduce, with absolute certainty, the Truth or Falsehood of geometrical and arithmetical proposition.30 Quite a remarkable and extraordinary task! Euclid’s 5th axiom postulating that 2 parallel lines never meet, seemed so self-evidently true in reality, yet Euclid could not prove it from other axioms so it had to become an independent axiom from the first four. It wasn’t until 1829 with Lobachevski following on the work of Gauss (1824) and Bolyai (1832) that Euclid’s 5th axiom was denied, that is was false, leading to the development of Non-Euclidean Geometry. By denying the 5th axiom, these mathematicians, demonstrated “that we could get not just one, but an infinite variety of other non- Euclidean geometries simply by introducing other axioms in its place.”31 Even Einstein corroborated this by saying that Non-Euclidean geometry describes the universe.32 Interestingly, not even Kepler, Descartes, Newton, Laplace, Maxwell in the 16th – 19th centuries, ever questioned that Euclidean geometry did not describe the real world. Even Kant, realizing that our mind and senses have limitations in grasping space and time, never doubted that spatial geometry was Euclidean. Even something so self-evident in our realm of experience, was proven to be completely wrong!
30 Dr. Stamp. 31 Dr. Stamp. 32 Dr. Stamp.
Greek Mathematics. 14. Greek Mathematics. 14. Greek Mathematics. 14.
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