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Is Math/Science Absolute?

3 points by hegzploit ↗ HN
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.

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.

Kurt Gödel then showed that this was impossible and the number system needs to be inconsistent for it to be complete.

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.

What are your thoughts?

5 comments

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bediger4000
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".

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.

hegzploit
Thanks for your valuable insight, I'm yet to read more about these topics since I was just scratching off the surface and I find Mathematical Logic to be a very interesting area.

A lot of people have been recommending "Gödel, Escher, Bach" which I'll check out.

bediger4000
You might want to also look at Raymond Smullyan's "Beginner's Guide to Mathematical Logic", and "Beginner's Further Guide". I got hung up at the first order logic section, and the Axiomatic Logic is tough, but he includes a lot of topics and how they interrelate.
hegzploit
Thanks a lot for these recommendations, I'll get these on my Kindle as well.
al2o3cr 

    Kurt Gödel then showed that this was impossible
    and the number system needs to be inconsistent for it to
    be complete.
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.

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.