0 reactions
How are axioms “agreed upon”?
I hope this question is stated well enough for an answer, but this question has nagged me more and more as I’ve studied areas that rely more on formal logic. When learning calculus in high school we have a pretty good idea of what the topics introduced mean (such as derivatives being the slope of the tangent line of a function and integrals representing area under a section of a function). But when you get to real analysis, all these results are permitted only though use of the axioms of set theory and some definitions (like what a real number is). There’s nothing remotely close to geometry in these, yet we’re able to interpret our deductions are somehow related to the real world. This feels like way too much of a coincidence.
I get that we don’t have some magic tablet with ZF axioms inscribed in them telling us that these are the correct axioms for set theory and that we’re free to create other theories where we have different axioms. I’m wondering why the axioms of ZF set theory and any other theories are agreed to be what they are. Do they all intend to achieve something? I’m assuming ZF aims to recreate the real world as closely as possible, after all the axioms are extremely intuitive and match our intuitive understanding of certain concepts like equality. It seems that we could add a ton of axioms that come from “common sense” yet we only choose a couple—is there somewhat of an agreement to try and keep theories as minimal as possible? I could go on much longer but I will keep it at that.
This is definitely a very loaded question and more philosophical than mathematical but it’s beginning to irk me a lot and I thought I’d throw out the question to see if anyone has any good insight on the topic. Thanks for any help.
#science
I hope this question is stated well enough for an answer, but this question has nagged me more and more as I’ve studied areas that rely more on formal logic. When learning calculus in high school we have a pretty good idea of what the topics introduced mean (such as derivatives being the slope of the tangent line of a function and integrals representing area under a section of a function). But when you get to real analysis, all these results are permitted only though use of the axioms of set theory and some definitions (like what a real number is). There’s nothing remotely close to geometry in these, yet we’re able to interpret our deductions are somehow related to the real world. This feels like way too much of a coincidence.
I get that we don’t have some magic tablet with ZF axioms inscribed in them telling us that these are the correct axioms for set theory and that we’re free to create other theories where we have different axioms. I’m wondering why the axioms of ZF set theory and any other theories are agreed to be what they are. Do they all intend to achieve something? I’m assuming ZF aims to recreate the real world as closely as possible, after all the axioms are extremely intuitive and match our intuitive understanding of certain concepts like equality. It seems that we could add a ton of axioms that come from “common sense” yet we only choose a couple—is there somewhat of an agreement to try and keep theories as minimal as possible? I could go on much longer but I will keep it at that.
This is definitely a very loaded question and more philosophical than mathematical but it’s beginning to irk me a lot and I thought I’d throw out the question to see if anyone has any good insight on the topic. Thanks for any help.
#science
| 19,000 mlx | |
| 19 | |
| 0 |