Proof Through Abstraction? Hellenize Everything!
The Greeks gave us Proof through Abstraction, or [and] so I thought! Sans Cartesian doubt, the presence of doubt doesn't presuppose a thinking self; cogito ergo sum. In a pseudo-Existentialism, dystopian-"ish" reality, the thinking vessel [self] carries a bifurcating companion, replicating, scaling at will and without final cause. If we can all agree, layman and fellow philosophers alike, that the regression toward the mean, in this case, represents a trajectory for the conversation that is semantic in its origin and futile in its discovery. Let us instead focus on abstraction.
How do you proof through abstraction? It seems that all the proofs I've seen have been mathematical proofs in which you have to first derive something from what was already present. The argument becomes much more complex because, with abstraction, it's more of a puzzle.
Abstraction is actually a key part of mathematics. For example, in geometry, there are two different things that we call angles. The first one is the tangent and the other one is perpendicular. If you ask someone how they would prove this, they will probably answer with the tangent. This is true but, if you ask them how to actually prove that the tangent is equal to the perpendicular, they will be at a loss because it just doesn't make any sense.
But, when you take the tangent and divide it into two parts, you get a right and a left. You then have to prove that the left is bigger than the right. If the tangent was not real, you could do this by taking the hypotenuse and dividing it into two parts. This gives you the two opposite sides and you have to show that these two sides are equal to each other.
The more abstract the idea, the easier it is to prove. In geometry, if you want to prove that the tangent is equal to the perpendicular, then you have to take the tangent into two parts and split the hypotenuse into two parts and then divide these two parts into the right and left. With abstraction, it doesn't really matter what the tangent or hypotenuse is, you just have to take whatever abstract thing you want and divide it.
So, when I say abstract, I mean abstract in a way that it doesn't really make sense. To prove something, you have to first find a concrete proof that it is true and then apply your abstract proof to that concrete proof. This process is called proving it isn't so much that you're proving that the abstract idea is true but that you are proving that the abstract idea isn't true.
So what is an abstract proof? Well, it can be anything that you can take and use to prove something else. For example, if you want to prove that the square root of 9 is greater than nine, take that number and divide it into two parts. One of which is the hypotenuse and the other is the divided part.
Now, the first part is the abstract part and the second part is the concrete part. By abstracting, you can now prove that the first part is smaller than the second part. If you were to divide it into two parts again, you will get the exact same results you got last time but you will just have to take the hypotenuse and divide it into two parts.
So, when I say abstract proof, I mean a way of proving something without actually finding any concrete proof to back it up. So, you can use abstraction and prove something and then use your abstract proof to prove that it is true.
What it means to abstract the idea is, that is, you look at something in the abstract and try to figure out what it is, and how it works. That way, when you come to prove the abstract idea, you don't need to go and find any concrete proof to support it.
If you want to prove something is true, then when you abstract the abstract idea, you can look at it and figure out exactly what it is and how it works. The only thing you have to do when your abstract is to divide it into two parts and then divide those parts into the other parts. Once you've done that, you have the proof that proves you can use to prove that the abstract idea is true. and then use it to prove that it is true.
I hope you understand that when I use the word abstract, I'm not simply saying that it is impossible to prove something is true or to make it true. It's just that abstract proof is a different approach to proofs. It doesn't necessarily involve concrete proof but when it does, the proof can be much stronger.