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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-Eucli

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.

vintermann 1y ago
> all internally consistent Well, we hope.
bonoboTP 1y ago
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".
- anthk 1y ago
  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.
cheesecompiler 1y ago
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.
nexttk 1y ago
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.

vintermann 1y ago

> all internally consistent Well, we hope.

bonoboTP 1y ago

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".

anthk 1y ago
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.

cheesecompiler 1y ago

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.

nexttk 1y ago

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.