←back to thread

Is mathematics mostly chaos or mostly order?

(www.quantamagazine.org)
116 points baruchel | 1 comments | 20 Jun 25 15:21 UTC | HN request time: 0.218s | source
Show context
dgfitz ◴[24 Jun 25 06:04 UTC] No.44363259[source]▶
I’ve always considered math is something that is discovered, neither chaotic or orderly, it just… is. Really brilliant people make new discoveries, but they were there the whole time waiting to be found.

This article seems to kind of dance around yet agree with the discovery thing, but in an indirect way.

Math is just math. Music is just music. Even seemingly-random musical notes played in a “song” has a rational explanation relative to the instrument. It isn’t the fault of music that a song might sound chaotic, it’s just music. Bad music maybe. This analogy can break down quickly, but in my head it makes sense.

Disclaimer - the most advanced math classes I’ve taken: calc3/linear/diffeq.

replies(3): >>44363838 #>>44363896 #>>44364735 #
isaacfrond ◴[24 Jun 25 07:57 UTC] No.44363838[source]▶
Mathematics isn't monolithic—it depends heavily on the axioms you choose. Change the axioms, and the theorems change. ZFC, ZF¬C, intuitionistic logic, non-Euclidean geometry—each yields a different “math,” all internally consistent. So it’s not right to say math “just is” in some absolute sense. We’re not just discovering math; we’re exploring the consequences of chosen assumptions.

For instance:

Under Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), every set can be well-ordered, but we do get the Hahn–Banach paradoxes.

Under ZF without Choice, analysis as we know it no longer holds.

In constructive mathematics, which avoids the law of the excluded middle, many classical theorems lose their usual formulations or proofs.

Non-Euclidean geometries arise from altering the parallel postulate. Within their own axioms, they are as internally consistent and "natural" as Euclidean geometry. Do non-intersecting lines exist in this universe? I've no idea.

replies(4): >>44364077 #>>44365083 #>>44367719 #>>44368386 #
1. nexttk ◴[24 Jun 25 17:08 UTC] No.44368386[source]▶
About the axioms, not really. Axiom sets is mostly there just as a 'short hand' to quickly describe a context we're talking about, but ultimately you could just do away with them. E.g. if we let A be the set of axioms from some theory (e.g. set theory, number theory etc.) and you have a mathematical statement of the form X => Y within that theory, you could just as well consider the statement "A ^ X => Y" in the purely formal system without any axioms at all, then it is purely a logical question (essentially, if X => Y is a theorem within theory A) and more objectively true than "X => Y" which would be theory-independent.