Is mathematics mostly chaos or mostly order?

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.

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.

Large cardinal axioms are the paradigmatic example "invented" math.

> all internally consistent

Well, we hope.

What makes you consider it a "discovery" instead of a creation of us humans?

I am more on the side of seeing maths as a precision language we utilize and extend as needed, especially because it can describe physically non-existent things e.g. perfect circles.

This just steps one meta level higher. Yes, you can make your object of analysis the axioms and what they lead to and proof theory etc. But now you've just stepped back one level. What are the axioms that allow you to derive that "ZFC leads to Hahn–Banach paradoxes"? Is this claim True and discovered or is it in itself also simply dependent on some axioms and assumptions?

This is part of a broader meta-ization of culture. Philosophers are also much more reluctant to make truth claims in the last century compared to centuries ago. Everything they say is just "To a Hegelian, it is {such and such}. For Descartes, {x, y, z}." If you study theology, they don't teach with conviction that "Statement A". They will teach that Presbyterians believe X while the Anglicans think Y, and the Catholics think it's an irrelevant distinction. Of course when push comes to shove, you do realize that they do have truth claims, and moral claims that are non-negotiable but are shy to come forward with them and explicitly only talk in this "conditional" "if-then" way.

In fact many would argue that math is not too far from theology. People who were obsessed with math limits, like Gödel, were also highly interested in theology.

I guess physics is the closest to still making actual truth claims about reality, though it's also retreating to "we're just making useful mathematical models, we aren't saying that reality is this way or that way".

I rather think the discovered/invented thing is just semantics.

You can say that literally anything was "just discovered".

Thriller by Michael Jackson? Those particular ordering of sound waves always theoretically existed, MJ and various sound engineers just discovered them, they didn't create anything.

The cappucino? It's just a particular orderly collection of chemicals, such a collection always theoretically existed. Those baristas are explorers, discovering new latte art shapes, nothing creative there.

Cantor's diagonal argument? Yep, those numbers where just waiting to be discovered and written in that order.

And so on. The entire argument is meaningless, pointless philosophizing. Nobody wastes their time saying latte art was discovered rather than invented, but somehow when it comes to mathematics this is considered a deep and worthy discussion.

I like to think of axioms as "created" while the consequences (i.e. theorems) of said axioms are "discovered". You can't create logic consequences (conclusions) given a set of axioms, but you can certainly create the axioms (premises).

It's not clear to me why people think perfect geometries do not exist, they occur all the time in physics.

Of composite matter, sure, because it's composite in a certain sort of way, you do not get perfect circles. But the structure of macroscopic material does not exhaust the physically.

Even here, one could define some process (eg., gravitational) which drives matter towards being a perfect circle, because perfect circularity is a property of that process. This is, as a matter of fact, true of gravity -- if it weren't we'd observe violations of lorentz invariance, which we do not.

To me, the difference is: the way to __make__ music was invented, it was already there to be discovered.

Didn't mean to rub you the wrong way.

> Didn't mean to rub you the wrong way.

To me, it doesn't sound like you did. The parent comment of yours just stated, albeit bluntly, that the "invention" and "discovery" are fundamentally the same. Whether we use one or the other depends on how big the size of the space of the possibilities feels to us. Math has a very rigid and easily enumerable space of possibilities (strings of symbols), so we call it "discovery", while cooking has an enormous space of possibilities (countless pieces of meat and vegetables, each unique in its configuration of atoms, etc.), so we call it "invention".

When you invent a way to make music, did you really invent it? Or did you simply discover a particular configuration of atoms that can produce sound when handled in a particular way, that was already there in some platonic universe of ideals? Either way, the end result is the same. Nothing really changes.

Perfect in a single-body universe, perhaps, but the gravity field of a particle is perturbed by other nearby particles - where "nearby" is relative to precision desired - and therefore never a perfect sphere.

Or, to put it another way, so-called "perfect circles" exist in a real, 4-D, wibbly-wobbly gravity-distorted space, and are no longer perfect Cartesian circles.

They still only exist theoretically; not in practice.

Is TRIZ (and its great-great-grandchildren, LLM GenAI) invention, or discovery? https://en.wikipedia.org/wiki/TRIZ

> Math has a very rigid and easily enumerable space of possibilities (strings of symbols), so we call it "discovery", while cooking has an enormous space of possibilities (countless pieces of meat and vegetables, each unique in its configuration of atoms, etc.), so we call it "invention".

I think you've made a good point here.

Although, to nitpick a bit: Both spaces (cooking and maths) are infinite, and for most fields of math, uncountably infinite. The difference is in the numbers we are dealing with. For cooking, it's mixtures of trillions of molecules. For maths, it's usually in the order of thousands of symbols (although those ellipses do some infinitely heavy lifting!).

> Didn't mean to rub you the wrong way

Not at all, you sparked thought in an area I find fascinating (philosophy of maths). Albeit I find this specific topic a bit too commonly discussed relative to how important it is, but I'm still happy to talk about it and share my thoughts.

The overarching point still stands: our formal systems are just models built to describe the patterns we observe. In that sense, math “just is.” The fact that some models aren’t compatible with others doesn’t undermine that—it just shows they’re incomplete or context-dependent views into a larger structure.

Circularity is still a property of the process. One requires perfect circles to describe it.

It is also easy enough to construct circular state spaces, and the like.

The idea that what's real is simply the geometry of macroscopic visible matter, or even of matter alone, is a nonesense.

The world is "immanently abstract", and possess primeness, circularity, etc. in itself -- not as something merely imagined. This is obvious from the physical description of its evolution.

Irregularity, of this kind, is derivative of a geometrical reality. The irregular doesn't govern the irregular, if it did, there would be no structure whatsoever.

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.

No, you are wrong. 90% of Philosphy it's bullshit about giving a fake truth status depending of WHO said what. Meanwhile, Math and Science always put FACTS over personas.

Perfect circles just exist on probabily using 2pi for random directions and overall statistics.

When was the circle discovered? When it became essential to physics?

When it was essential to perception. Its necessary to have a model of a circle (, elipse...) in order to correctly parse (at least,) visual perception -- because space is inherently geometrical.