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My question by itself is hard to ask and hard to answer. I don't mean to be ambiguous, but it is one of those philosophical questions that requires a bit of specification. Let me provide some context.

Axioms are often used in two ways. The older and original concept seems to me like a top-down description. You can work with logical systems like numbers or lines on planes all you want, but when you try to backtrack your deductions, you come across axioms that can not be traced further back, much like how atoms can be not split further. Axiosm are also often used to describe assertions that systems are built from, like building blocks. I call these bottom-up axioms. That there may be no distinction between the two types is a philosophical question for another thread.

Let me use Euclidean axioms as an example. Mathematicians thought they were top-down for hundreds of years until they realized that the first four can be used independently of the fifth. Then they realized that you can "play" with axioms and basically develop novel systems that may or may not correspond to real world things. A similar process with number systems lead to the develop of modern algebra. After trying to find the most fundamental statements underlying number systems, mathematicians realized you can basically mix and match properties of operations to create new systems. But how did these mathematicians become convinced that their axioms were "true" in the first place?

For further clarification, I have no doubt that the axioms, in whatever direction, used by mathematicians do indeed describe mathematical objects very well. But I am referring to top-down axioms in my question. With this additional information, maybe my initial question is easier to understand. How did/do mathematicians deduce that their axioms were/are sufficient in describing whatever abstract object that they are pointing to?

I believe that this answer boils down to experimentation. Assert the axioms and if anything "breaks" then add more axioms, roughly speaking. Further independent study of the axioms themselves can provide more intuitive confirmation. At some point, you become so sure of the axioms that it is like you might as well define objects by their axioms. Hence, this is why top-down axioms and bottom-up axioms are used interchangeably.

If you could direct to me to any relevant texts, that would be very welcome.

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  • $\begingroup$ Whether or not a mathematician views the axioms they work with as "true" is part of the Platonism vs Creationism debate in the philosophy of mathematics. But it doesn't really matter. At the end of the day, we find axioms by experimentation, see if they are useful by trying to prove things we want to prove in those axiomatic systems, and hope that the system isn't inconsistent (if it is sufficiently powerful). That's all math is. $\endgroup$ May 16 '21 at 2:30
  • $\begingroup$ The axioms of Euclidean geometry may well not be "true" in the physical world. Mathematicians simply introduce these axioms because they seem reasonable or interesting and then explore their consequences. You can view the axioms of Euclidean geometry as merely the definition of a "Euclidean plane". We are not claiming the axioms are "true" in the physical world or anything, we're just defining a Euclidean plane to be a set $S$ (whose elements we will call "points") together with a collection of subsets of $S$ ("lines") such that certain axioms are satisfied. $\endgroup$
    – littleO
    May 16 '21 at 2:34
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    $\begingroup$ (@DonThousand, to refer to an aspect of mathematical philosophy as "creationism" is colorful, and to a degree perhaps descriptive, but is loaded with too much dumb baggage to really convey your (sensible) message without some corruption...) $\endgroup$ May 16 '21 at 2:34
  • $\begingroup$ We may hope that formulas derived from the axioms of Euclidean geometry can be used to compute distances between locations in physical space (for example), but whether or not that is true can be answered only by experiments in the physical world. $\endgroup$
    – littleO
    May 16 '21 at 2:36
  • $\begingroup$ This was why geometers since Euclid tried to prove the fifth postulate from the others - because it was not a “clear” axiom. We now know you can’t. We also know if an axiom system is complex enough, we can’t prove whether it is consistent or not, much less whether it accurately describes what we are intending to describe. $\endgroup$ May 16 '21 at 2:49
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You weren't kidding about this being a difficult question to answer, however it's a really good question, and definitely a matter of philosophical debate. I'd definitely recommend reading Principia Mathematica, which covers a lot about this type of logical dilemma. Although I am extremely passionate about this area, I'll try to be as unbiased as I can.

Top-down

Concerning "top-down" axioms, as you put it, it seems to me that at least archaically, these were deduced through pattern recognition and intuition. Run enough test cases and you can more or less convince yourself that you've stumbled upon a tautology. This intuition is a mathematicians greatest friend, and simultaneously their arch nemesis. Often, it is impossible to prove things that may seem undoubtedly true, as in the case of the Collatz Conjecture. Hence, it would be rather immature to claim the Collatz Conjecture as an axiom (or genius, if you're crazy enough). Sometimes, our intuition fails us, and things we thought were true end up being inconsistent. Such as a lot of early set theory. Take the Axiom of unrestricted comprehension, which is very intuitive, but ultimately leads to paradoxes.

Bottom-up

"Bottom-up" axioms are more-so "crafted" to meet our expectations we obtain from our experiences. A good example of this are the axioms in Zermelo-Fraenkel set theory. Many were put in place mainly to prohibit inconsistencies such as Russell's paradox.

The balance

As a side note, when an axiom breaks, it is often better to remove or replace an axiom rather than add another. Mathematicians really want to minimize the amount of axioms, while maximizing what can be proven, on top of consistency. The more independent axioms you add, the more room you have for inconsistencies to arise.

How do we know what's true?

Some axioms can be proven to be true entirely by lower level logic, which may rely on even lower axioms. In which case, the immediate question is "but how do we know those lower level axioms are true?" and we're left with the same dilemma. On the contrary, it is possible to prove things without any axioms at all. These are called natural deduction systems, and instead of using axioms, the heavy lifting is done by "transformation rules". Again, one could ask "how do we know these transformation rules are consistent?" and we're left with the same problem.

All of this boils down to metamathematics, and the question of "how do we know anything is true?" Although we don't have a universally accepted answer, one thing we can be sure of is that axioms provide the backbone for being able to convince ourselves of some truth, at the least. That being said, It's more than okay to doubt axioms. Many are not universally agreed upon, such as the Axiom of choice. I think what I'm trying to get at here is that truth is not universal, but relative. Here's an example that I think sums the situation up well.

Is theorem $X$ true? We don't know, but we do know that $X$ is true if we assume that $Y$ is an axiom; and $Y$ is a really nice axiom.

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    $\begingroup$ And what about the mathematical truth "𝑋 is true if we assume that 𝑌 is an axiom"? Is that something which we can know to be true in a non-relative way?? $\endgroup$ May 16 '21 at 12:07
  • $\begingroup$ @PeterSmith Of course not, it's all relative. Or to be more precise, recursive. $\endgroup$
    – Graviton
    May 16 '21 at 22:56
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    $\begingroup$ Yes, I should have wrote "modify the axioms list" instead of "add more axioms." Thank you for your answer $\endgroup$ May 17 '21 at 0:40
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In some cases, it's possible to prove that every possible true formula (in a strictly defined language) about a structure can be proved from its axioms: Simply prove that the axioms let you reduce formulas down to an equivalent canonical form, then prove that the axioms determine whether or not these formulas in canonical form are true.

However, in the most interesting cases, like that of natural numbers or set theory, it is impossible (as proved by Gödel) to write down a complete axiom system. Mathematical Platonists believe that these abstract objects exist despite our inability to pin them down, and Mathematical Formalists do not.

I'd encourage you to look into how set theorists decide what extra axioms of set theory to accept. Since we don't have strong intuitions about really big sets, sometimes mutually incompatible axioms are proposed and worked with.

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