I have finally found a simple explanation, consisting of nothing more than two words, of why I hate theoretical mathematics:
Banach-Tarski.
You see, these two crackheads mathematicians, Stefan Banach and Alfred Tarski proposed, and supposedly proved, the theorem that given a solid ball in three-dimensional space, you can take that ball, split it up into a finite number of non-overlapping pieces, and then reassemble those pieces in a different way (using only translation and rotation on the pieces) to create two identical copies of the original ball.
Yes. Two.
Yeah, I know. Conservation of mass gets violated, right off the bat, and then it goes downhill from there.
But, you see, the trick is that they are talking about mathematical spheres, not physical spheres, and specifically mathematical spheres that are infinitely divisable, with infinite density. Yeah. Because that makes sense (and, likewise, violates the conservation of mass right off the bat all over again, but in a mathematically-acceptable way… *sigh*).
Ever since I was a child, I have always searched for practical applications of various sciences that were taught to me over the years… For applied sciences, like physics and chemistry, that was never terribly difficult, but once we started branching into theoretical sciences and maths, my eyes start to glaze over and my brain runs out of connections to the real world. Even worse is when those mathematics involve spaces or dimensions that are distinctly non-euclidean, and instead carefully crafted to ensure that whatever theory we are talking about suddenly, coincidentally works. Oh, it will never work in this lame three-dimensional universe we inhabit, but once you start adding dimensions, or changing the way things interact, or redefining basic ideas, then obviously it will work!
*headdesk*
Hell, who am I kidding – my mind broke in my probability class (a very applied course, especially at my school) when my professor proved, in a wholly understandable way, that there is a 0.00% probability of a car’s front left wheel stopping at any given point, yet it stops all the time.
It is probably just as well I went the “engineering” route… But marrying a discrete mathematician, that might have a somewhat significant impact on my sanity in the future. And yes, my introduction to the insanity theorem of Banach’s and Tarski’s was her fault – she was looking for anagrams of their combined names. “A absinth rack” works, in more ways than one, despite the grammatical and spelling shortcomings.
My favorite was Gabriel’s horn. Finite volume, but infinte area.
Good friend of mine is a microbiologist. The shit he works with you can’t even SEE, let alone handle. That’s too removed for me.
Hell the Mrs. is a CHEMIST. She talks about all these crazy machines, that I KNOW how they work, but let’s face it with stuff as small as a benzine ring, or a hydrocarbon chain, it might as well be magic.
This sort of goofy intellectual masturbation I have zero patience for.
Reputo: Ok, that one hurts my head too. Especially with the “painter’s paradox” explanation.
Weer’d: Even those kinds of things I can abstract-ize to a point where I can understand it… Little tiny machines, and whatnot. But cloning a ball without the introduction of additional materials or energy? That breaks my noggin.
Rule #1: Weird things happen at infinity.
For instance, there’s the same number of even positive integers as there are positive integers.
But how can that be, the even positive integers are only half of all positive integers?
Well, yes. In any *finite* set of them. But remember Rule #1. Weird $h1t man, Weird $h1t.
Two sets have the same number of things in them if you can find a 1-to-1 relationship between each and every thing in each set.
Set “adult” (Cat, Dog, Man, Woman) and Set “Child” (Kitten, Puppy, Boy, Girl) are equally sized sets because you can link Kitten to Cat, Puppy to Dog, Boy to Man and Girl to Woman (You can link them other ways too, but it’s only necessary to find 1). There’s no item *unused* and there’s no item *reused*.
Question: Do the set of positive integers and the set of even positive integers (a subset of positive integers) have the same size?
Intuitively we would say no. The set of positive even integers is 1/2 of the positive integers. However, finding a 1-to-1 relationship between the positive integers and positive even integers is easy: take any number in the positive integers and double it. No integer is left out and none are reused so therefor they have the exact same size.
The same thing can be done for the odds. So you can take the set of all positive integers and split it into two groups, the evens and the odds, each of which has the exact same size as the original.
Told ya. Weird $h1t man.
And don’t even start with fractals and non-integer dimensions
See, it is that kind of made-up nonsense that drives a concrete-thought guy like me right up the wall.
“Sure, it does not work when you talk about a rational number, but once we throw ‘infinity’ into the mix, everything works out just fine!”
“Well can you demonstrate it?”
“Can you find me something of infinite density?”
“No.”
“Well then of course not! But it sure sounds neat, does it not!”
*headdesk*
I know, I know, a lot of the things that our current society is built on is based on mathematics that cannot really be “demonstrated”, but a lot of the explanations for that math seem a lot more like workarounds than logical progressions (at least to an applied engineer like me).
See, it is that kind of made-up nonsense that drives a concrete-thought guy like me right up the wall.
Yeah, that was my mind blowing moment akin to your probability class. OK, yeah, I understand it, but WTF?!
“Well then of course not! But it sure sounds neat, does it not!”
Well, nothing in the real world has a length (or width) some multiple of the sqrt(-1), but when you put’em in fractals, they do make some really pretty pictures.
Now that is definitely true… some of the most beautiful abstract art I have stumbled across on the cortex actually stemmed from graphical representatives of seriously jacked-up math.
I guess consuming copious quantities of crack while scribbling numbers and characters on a sheet of paper does have its uses after all…