I was reading in my free-time for the past couple months about the history of Mathematical Logic and how Euclid's attempt to axiomatize Geometry turned out to be just a way of interpreting objects and by changing the fifth axiom we get a completely different type of Geometry which yield their very different interpretations.<p>And then after, Hilbert attempted to axiomatize Euclid's geometry by proposing 20 axioms and he took it a bit further by constructing an analogue of his geometry within the Cartesian coordinates which has elevated the problem to that of arithmetic itself.<p>Kurt Gödel then showed that this was impossible and the number system needs to be inconsistent for it to be complete.<p>This made me think as if most sciences are statements in their own formal language in which stuff is modeled after, and is true with respect to that formal language.<p>What are your thoughts?
Math, maybe, despite your very valuable readings and insight. By "maybe I mean "inside some axiomatic system (like ZFC, or lambda calculus) math is absolute".<p>Science definitely isn't absolute. It's based on observation, hypothesis, experiment and verification. A new observation can be made at any time. Most theories have some things that don't fit, like Mercury's orbit didn't fit Newtonian mechanics. Right now we've got observations of galaxy-sized objects that don't quite fit Einsteinian Relativity. There's probably other things that we know that just don't fit. So, no absolutes there.
<p><pre><code> Kurt Gödel then showed that this was impossible
and the number system needs to be inconsistent for it to
be complete.
</code></pre>
IMO this is favoring the wrong end of Gödel's argument: a "complete" number system sounds like something desirable, but it's equivalent to one where "1 = 0" is a true statement and literally anything can be proved.<p>The converse part is usually more important: given any consistent system of mathematical logic, it is always possible to produce a theorem in that system that the system cannot prove to be EITHER true or false.