Proof by contradiction

yeah, I made a boner there. the argument's still the same though. sometime's it's easier to understand why p --> !!q (q can't be false).
it's also frequently the case that proofs by construction are difficult to understand and end ip just being memorized, even if they're algebraically useful.

those are two different things

No. You're describing contrapositive, first of all, and odds are your "proofs by contradiction" are not actually using the contradiction in any meaningful way and are just sloppily written.

>>it's also frequently the case that proofs by construction are difficult to understand and end ip just being memorized, even if they're algebraically useful.
Even if difficult, they are helpful for understanding as well and the strategy needed for construction can be useful elsewhere.

>OP disagrees with Euclid's parallel lines and the entirety of On Conics

Ah, proof by contradiction, otherwise known as reductio ad absurdum, something that Apollonius wielded quite well. You would do well to respect it sir, it helps to rigorously establish all aspects of a proof.

For instance, if you ever have an instance where a ratio has to reach a certain equality (like greaterthan/equal to or lessthan/equal to) you will fundamentally have a situation where if the ratio is greater than or less than the opposing ratio (usually the parameter to the diameter in the case of conic sections) then NOTHING is true. For instance, when the perpendicular squared is to the radius squared of the base in the same ratio as the diameter to the parameter of a given hyperbola, you can construct the cone with the hyperbola inside it. If it is less, then you can construct two of them in the same cone. If it is greater than it, then nothing. That's right. Nothing.

How do you prove THAT? As opposed to proving that state simply just exists because it has to, you must first suppose that it WOULD be possible to construct a cone with these dimensions with a hyperbola inside it. So you would go about it like you would the other two situations and just make an erroneous assumption somewhere. That's how these work, something has to fundamentally be wrong in order for us to proceed with the proof. But usually the mathematicians are intelligent about it. They'll use the same situation as set up for the initial case with the one hyperbola.

And even in the other two cases where there were hyperbolas, Apollonius is rigorous. He uses proof by contradiction to show that there is no way there is a THIRD hyperbola in the case where there is two, and there is no way there is a SECOND in the case where there is one. Sometimes the reductio ad absurdums are easy, one liners. Sometimes they can go on for a page or two simply because of how rigorous proving a mathematical impossibility is.

All of these are constructive arguments, not proofs by contradiction...

Proof by contradiction is a reductio ad absurdum. I always notice when a reductio ad absurdum is used, friend. I am very familiar with it and how it works. It is absolutely essential to mathematics.

In that post I mentioned a proof, VI. 32 to be precise, in which reductio ad absurdum is used three times.

Suppose P, derive absurdity, therefore not P.
Suppose not P, derive absurdity, therefore P.

Both reductio ad absurdum, one of them by contradiction, other just a proof of negation...

>confuses intuitionistic with intuitive
wew lad
en.wikipedia.org/wiki/Intuitionistic_logic

>Suppose P, derive absurdity, therefore not P.
Are you saying this is the proof by contradiction? I would agree.

This is how most reductio ad absurdums are used by mathematicians.

Let me try to rephrase this. A reductio absurdum used by a mathematician is typically

True proof:
Suppose P, therefore C, prove this exists
Reductio ad absurdum:
Suppose X, therefore C, prove this exists, is absurd.

The way I've worded it is important, because in order to prove something is absurd, you must assume everything you've assumed in the first case. In your example, the first think you asserted is only part of how a reductio ad absurdum is utilized by mathematicians. There are other ways, as I've stated. Typically the true construction proofs are not reductios at all as you've pointed out, but once we breach the territory of having to explain that the situation where NO hyperbolas could be constructed is possible it's not as simple as just assuming P, then deriving the absurdity.

But the funny thing about this is that even if it were, your overall point is incorrect. In this case either they use my example of a reductio ad absurdum or they use yours. Either way you slice it it's NOT a proof of negation.

>>Suppose P, derive absurdity, therefore not P.
>Are you saying this is the proof by contradiction? I would agree.
I left it as exercise for you to figure out which one is which--well, that's the proof of negation.

Rest of your post is mostly retarded notions of logic and proof theory--what do you mean by "true proof?" How is "suppose P, derive absurdity, therefore not P" not a proof of negation?

Anyway, take a look at this: math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/

Look the point is, you're wrong, kid.

Here's why,

True proof:
Suppose P, therefore C, prove this exists
Reductio ad absurdum:
Suppose X, therefore C, prove this exists, is absurd.

That's right. It's a copy/paste, because you failed to read my fucking post.

Okay then lets see, you should have realized that the 'therefore C' is static. Unchanging. In other words, you are incorrect in what a reductio ad absurdum is, because there is no standard to balance it against. You were correct that it would be suppose NOT P, because it's definitely supposing different things, but the absurdity is derived AFTER you've stated what is the therefore.

You should think about the concepts you're trying to type before you type them out of your stupid fucking skull, brainlet. Kill yourself before you shit up this board anymore, neither of those examples you gave me were valid.

Your garbage is an incoherent mess. What the fuck does a "true proof" mean? Your view of what is reductio ad absurdum is stupid (it's same in form as your "true proof" except you just used X as a symbol instead of P and you tacked "is absurd" at the end). If someone needs to end his worthless life here it's you.

Great argument.

Look, you're not entirely wrong, it's just that we got sucked into an argument about reductio ad absurdum.

Think about superposition arguments, they are somewhat modeled after what you were talking about. Superposition arguments in geometrical propositions are the closest you'll get to a proof by contradiction, because to be quite honest with you the limiting factor of proof by contradiction is that it's binary.

There are many situations where there is not one of two options. I just have a bent against people that exclusively copy/paste wikipedia to help them win arguments.

where the fuck did these logicians come from?

>where the fuck did these mathematicians come from?

ftfy, not all mathematicians are obsessed with plugging in numbers and derivatives. Some might actually be interested in the fundamental philosophy of it.

>proof by induction

>Assuming LEM

>
>Your garbage is an incoherent mess. What the fuck does a "true proof" mean? Your view of what is reductio ad absurdum is stupid (it's same in form as your "true proof" except you just used X as a symbol instead of P and you tacked "is absurd" at the end). If someone needs to end his worthless life here it's you.
Holy shit just fucking kill yourself you worthless autist