Two Essays on the Archimedean versus Non-Archimedean Debate
Abstract
Part 1 : For more than two millennia, ever since Euclid's geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus by Newton, we may actually have empirical evidence that time, and thus space as well, are not Archimedean. Part 2 : There is an insufficient awareness about the rich and complex structure of various totally ordered scalar fields obtained through the ultrapower construction. This rich and complex structure comes from the presence of infinitesimals in such fields, presence which leads to the fact that such fields are non-Archimedean. Here, with the concept of walkable world, which has highly intuitive and pragmatic geometric meaning, the mentioned rich and complex structure is illustrated. The issues presented have relevance for what are usually called the "infinities in physics".