“Always listen to the experts. They’ll tell you what can’t be done and why. Then do it.” ~Robert Heinlein
Sitting at my desk vigorously coding, and the office starts to come alive. A friend of mine stops by to say hello we chat for a bit, we exchange some experiences with the technologies we are working with and we both move on with our day. Or at least he did, i unfortunately got derailed. During our discussion we were talking about Reflection vs. Serialization and which one was faster and under which circumstances. Somehow, not quite sure how, it brought back some bad memories from my undergrad at Randolph Macon College. I was a Math/Computer Science double major and constantly finding myself doubting the things i was learning. During my times at Macon (RMC) there were 3 main things that i couldn’t accept. The fact the PI never repeats, The Halting Problem, and People abusing Infinity. There is no way for me to fully explain my grumbles with these theories/proofs but i will try, often unsuccessfully, to explain whats going on in my head.
Even though it is bad form, feel free to skip this one, I will start with the theory that i have the least stable argument against: how does PI not repeat itself? I find it hard to believe that the “most famous” irrational number Π does not repeat itself. I guess my issues stem from the fact that this number is used all throughout science and engineering fields yet can’t be fully calculated. This number can accurately describe the ratio of a circles circumference to diameter. This number is also the ratio of the circles area to the square of its radius. Yes i am currently just throwing to very simple well known formulas at you. I guess my unfounded believes are based on the fact that we just haven’t been able to calculate with enough precision PI and that the proof of PI’s irrationality is done through “reductio ad absurdum.” This method of proof, which i have major issues with, is proof by contradiction. You start with the assumption that PI is irrational and then magically deduce that oh wait… since we can’t prove it is.. it is false.
Alan Touring “proved” that it is not possible to design a program that can not “Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input”. Yet again i have issues with a proof that is proven by assuming it is true, and the deduced that it is not possible. An outline of the proof can be found here Wikipedia Halting. They start with the assumption that they have a program, i will call Super-P, that can determine given ANY input and program it can determine if it will halt. Well they throw a monkey wrench at Super-Z and pass it itself. Their conclusion is no matter what the program determines you will get into an infinite loop and thus your program isn’t really a true slayer of the halting problem. My issues stem from the first assumption. We assumed that our program can determine for any input if a program will halt. With that assumption the program that we are describing can also determine if itself halts. They set the proof up to fail. I think it is very easy to prove something wrong then prove it true.. and i am still waiting on a halting problem proof that i believe…
Last but not least Infinity. Question is simple… how is:
2∞ > ∞
So the idea is 2 * infinity is greater than infinity. Aneurysm forms just thinking about it. We have a number that represents something that doesn’t end. If it doesn’t end then how is 2*Infinity greater? You can’t ever calculate 2* Infinity because in order to start the calculation it would have to end. I.e.
99999999999999999999999999999999999999999999….. (-> Infinity)
Alright… so where do you start? We multiply starting at the right and move left? Now i know this is a theoretical concept… but to me… Infinity is Infinity there is no such thing as N*Infinity. Above wasn’t a technical proof… but maybe it could be. If you prove u can’t multiple infinity, i guess that would prove that 2∞ > ∞ can’t be true… thus absurdum is hit.