When we say mathematics is rigorous, exactly what do we mean?

There are no assumptions.

here's an attempt: rigorous = can straightforwardly be converted into a valid statement in formal logic

that's just the first thing to come to mind; go ahead and nitpick/criticize (for example: what counts as "straightforward"?)

>>8149002

>wat is an axiom?

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If you believe that you can do pattern matching on a finite number of strings, then there shouldn't be a problem.

Theory fragment of numbers.
* Language:
1. "0" ... a term
2. "n", "m" are term variables
3. if "n" denotes a term, then "Sn" denotes a term.
4. brackets (, ) as you'd expect
* Axioms:
n+Sm may be rewritten to S(n+m) (plus axiom)
n+0 may be rewritten to n (zero axiom)

so

SSS0+SS0 may be rewritten to S(SSS0+S0) by the plus axiom
S(SSS0+S0) may be rewritten to S(S(SSS0+0)) by the plus axiom
S(SSS0+S0) may be rewritten to S(S(SSS0)) by the zero axiom
so we saw that SSS0+SS0 may be rewritten as SSSSS0
Here SS0 may be abbreviated as "2", SSS0 as "3" and SSSSS0 as "5" and thus we have that 2+3 may be rewritten as 5.

Here's one framework to set up formal theories and derive theorems:
https://en.wikipedia.org/wiki/Hilbert_system

>>8149024
Axioms are not mathematics.'Mathematics' is done on top of axioms. Mathematicians do not create axioms for their daily routine, they choose axioms so that they can do mathematics.

And as it turns out, this is so true that we only take the axioms that serve an innovative purpose.

You could take any system and take away axioms and put in new ones but is that doing mathematics?

Can you even publish a paper where you just propose an axiom? No, if you are going to do something like this you propose an axiom and then do mathematics on top of it to see what happens.

>>8149014
Gödel's incompleteness theorem defeats that, though.

>>8149029
One should note that it's not like everyday mathematicians think of axioms much, or think of their theories as written down in terms of formal axioms. The validation of basic systems, such as that of numbers or plane geometry, or metric geometry, or linear algebra - this validity is never questioned and the inference rules are just common knowledge and done as a routine.
The idea of strictly formalizing stuff, or rather to actually do it, came only after 1880 with Frege, Hilbert, then Russel and friends.
For most, it's reassuring that much can be done formally, but the advantage from formalizing stuff is mostly one for computer science and purists.

>>8149031
No. Just because in the language you set up there are statements that can be written down but not proven or disproven .
.. doesn't mean you can't formalize some theory you have in mind.

It will be just incomplete, but it's not like the unformalized theory was more whole before.

>>8148814
Funny, this is how I always get whenever anyone tries to explain anything to me. I do great but I can't work with people because I never understand what they're saying unless I actually work it out myself
Is anyone else like that ?

>>8149131
nah, you're just a cunt

also, in order to be considered intelligent, you have to be able to understand what people are saying without them having to construct their sentences with rigorous logic

if you can't understand them, it's because you choose not to or are autistic

>>8148814
>since I can easily keep asking for further assumptions
That's the reason there are axioms (the popular ones right now are ZFC) and undefined terms (set, belong). The rest comes from them.

The language used is fairly precise too, compare with natural languages like english that are shock full of ambiguities, with something like the formal languages in math that are so unambiguous they can be parsed by a computer.

>>8149298
belong?
If you mean "[math]\in[/math]", this is exactly what's implicitly defined in e.g. the ZFC axioms

>>8148814
It means that every statement either logically follows from a previous statement or from the axioms.

>>8149744
But to define the [math]\in[/math] operator in ZFC, you already need a naive notion of sets and membership.
This is like how ZFC is defined using logic, but to define logic you need to talk about a set of symbols to form your language, so there is always going to be some assumed naive theory that your rigorous set theory is defined in.