Ex falso quodlibet

[Anonymous No.15250612]

Retard take here: Why are axioms just assumed to be true? We literally take them for granted. What are we gonna do if one of them actually turns out to be utter bullshit?

[Anonymous No.15250623]

>>15250612
We just stop using that axiom.

[Anonymous No.15250646]

>>15250623
one does not simply 'stop' using an axiom.

im quite sure whatever it was it would have extensive implications

[Anonymous No.15250662]

>>15250623
what if this exact axiom is used to deduct fundamental mathematical concepts? If your assumptions turn out to be false every consequent conclusion drawn must be considered invalid

[Anonymous No.15250686]

The only way an axiom could be false is if it contradicts other axioms in that same system of Mathematics, in which case you'd just disregard that axiom and all of its implications, as they would contradict all the implications of the other axiom.

[Anonymous No.15250709]

>>15250612
This has happened. For a long time, it was assumed that all the postulates of Euclid were true. Then somebody asked the question "What if the fifth postulate isn't true?" and that opened the door to all of non-euclidean geometry. The fifth postulate went from being something that was accepted as universally true to being a statement that's only true in the special case of flat geometry.

[Anonymous No.15250733]

>>15250612
Unless an axioms condtradict each other, you can never prove one to be false. If you disagree, try to present us with a set of non-contradictory axioms and a proof that makes one of the false

[Anonymous No.15250736]

>>15250709
But no one realized "the fifth postulate is bullshit", they just thought "hey what happens if we leave it out and see where things go" a.k.a. they just used a new, smaller axiom set

[Anonymous No.15250813]

>>15250736
Yeah …what's your point?

[Anonymous No.15250858]

>>15250612
>Why are axioms just assumed to be true?
Axioms in the sense you're talking about are taken to be true because their falsity leads to incoherence. People will often bring up non-Euclidean geometry as a counter-example, but they fail to grasp that Euclid's parallel postulate is specifically an axiom of the geometry that Euclid concerned himself with, and it is indeed incoherent to talk about Euclidean geometry without the parallel postulate.

[Anonymous No.15250860]

>>15250736
Yeah …what's your point?

[Anonymous No.15250877]

>>15250709
>the fifth postulate went from being something that was accepted as universally true to being a statement that's only true in the special case of flat geometry.
The possibility of the 5th postulate being false for Euclidean geometry is as inconceivable and incoherent now as it was back in Euclid's time. What's changed is that Euclidean geometry is no longer the only geometrical context.

[Anonymous No.15250885]

>>15250709
>>15250736
What if Cantor was wrong and infinity doesn't exist?

[Anonymous No.15250924]

>>15250662
So you're asking if we found out something like the induction axiom was false? Like if Peano Arithmetic were inconsistent? Those kind of consequences are difficult to imagine, since it would mean a great majority of human intuition is incorrect about math.

In any case. We could remove the axiom and still have Robinson Arithmetic. But pretty much 99% of math would have to be discarded.

[Anonymous No.15251014]

>>15250612
there was that one Principia Mathematica book series, by Bertrand Russell, that may explain this shit to you.

[Anonymous No.15251079]

>>15250612
axioms live in their own little world. if they don't contradict each other, they can be used to prove all sorts of things about them.
But what if those axioms don't reflect the real world? Now you have the problem of mathematics anon. Maths is built upon axioms that haven't proven to be true of the real world. People might say that they can't be proven because of infinite regress. There might be something that can be done about this problem. But it hasn't been solved yet.

[Anonymous No.15251088]

>>15251079
There is nothing to be "solved" about axioms, retard. They are taken to be true because there is no such thing as "true knowledge". Therefore, someone basically drew an arbitrary line somewhere and said "this is good enough, let's just assume these are true" because otherwise, what else can you do? So, you operate under the assumption that the axioms are true and see what you can build off of that. They are not universal guarantees.

[Anonymous No.15251096]

>>15251088
i didn't say axioms would be solved. i said the problem of if axioms can be proven true without infinite regress hasn't been solved so I don't know where you got that from.
Next, there is such thing as true knowledge. If there is a ball in my hand, that's true. If you subscribe to continental philosophy, you can disagree but I don't care about that sort of wishy washy philosophy in any way. Just because there isn't a way to have the foundations of mathematics free from infinite regress now, doesn't mean there can't be one in the future, unless you're of the sort that thinks everything to know about the foundations is already known.

[Anonymous No.15251104]

>>15250612
It's more subtle than assuming they're true. There's a fine distinction.
What we're doing is saying CONDITIONAL on the axiom being true, here's all the cool shit we can prove. We suppose that they're true, but only for the sake of seeing what comes out of it.
We don't actually assume that they're actually true, we just spend a lot of time looking at the consequences, conditional on the axioms.

Now why would we do that? Why not pick different axioms?
Because the math you do with those axioms is really useful, and the math you do with other axioms happens to be a lot less relevant. That's it. That's all there is to it.
We looked, and the other way around just happens to be kind of irrelevant.

[Anonymous No.15251143]

>>15251104
we base them around human intuition basically.
We assume that 1 + 1 = 2;
that 1 + (-1) = 0,
and so forth, and generalize them.

These axioms make sense to pick, since they fit quite concretely with how we view the world, and how we would typically count.

That's my take.

[Anonymous No.15251144]

>>15250885
perhaps

[Anonymous No.15251153]

>>15251143
-- and with that little shit, you can create this huge logical framework, that can be used either to expand this formal framework, or to apply it into our real world... as it is often effective at describing different shit...

[Anonymous No.15251154]

>>15251143
Can't really argue with that.
My only nitpick is that if we found a 'better' set of axioms, which are less intuitive and fit less well how we view the world, but are actually better at solving real problems, then physicists would be using those.
We've started with those because they match our intuition, but at this point math is far enough along that people routinely work on extremely unintuitive things anyways. So intuition isn't really the main factor in what mathematicians chose to study.

[Anonymous No.15251169]

>>15250612
As for ex falso, there a systems which don't adopt it, but 1. it falls out from other sensible logical principle, and 2. it's justified in arithmetic, where 0=1 in light of x+0=x implies you can prove equality for any two given numbers, proving all concrete equations.
Thirdly, or rather an expansion on 1, very weak logical axioms grant you that a contradiction prove every negation (in which case double negation elimination implies ex falso).
I say this as someone who's not eager to adopt ex falso. This is me steelmanning it.

[Anonymous No.15251175]

>>15250612
Everything humans talk about happens inside the confines of a model.
If you visualize a model to represent a square inside which everything plays out, then the axioms are the "walls" of the square (there is also a nice parallel to unit vectors here, but I digress).
If an axiom falls, the model fails. You just use a different model.

The fact that you didn't know you were operating within fundamentally manmade models when you use logic, language and various subset models of these, suggests you are a turbo-NPC who has never thought about the foundations of stuff.

[Anonymous No.15251881]

>>15251096
Yes. I know what you were saying. If you could comprehend what I wrote, that would be obvious from context. But clearly you are too unintelligent for this conversation given that you have such concerns about axioms in the first place.