It's just math
I remember when I was at work and a few of the guys there had this saying. Whenever you were discussing programming and you would say, "I don't want to do X because it would be so hard to program," they would reply, It's just code. I always thought that was pretty funny, and I am starting to say that here. Now, that's not the same as what I am writing about right now, but I guess it's related in a tangential way....
In Calculus III the other day Dr. Anderson mentioned the floor function. I don't know when I learned about it, but I have known about it for a while. It's almost exactly what it sounds like, you take a number and then the answer is the lowest integer of the number. I don't know if that describes it right. How's this:
floor(1.5)=1
floor(1)=1
floor(-1.5)=-2
Make sense? Well, one of the guys in the class said, "You can put stuff like that in a mathematical equation?" I remember thinking something to that effect when I learned something similar. The floor, the ceiling, the unit step function, heck even modulus, they all seem so arbitrary. Not like math where you just pump it out. But that is beside the point. It's just math. It's similar to functions that are defined partially by their input (the name escapes me). They seem so arbitrary, but again, it's just math. What gets really interesting is when you start mixing in Turing Machines; again, just math. But a Turing Machine is the lowest common denominator for computers. You can do some special things with hardware these days, but anything a computer can do, so can a Turing Machine.
I think that the reason these things throw us off is that they are not "single" step functions that can be done arithmetically. Modulus, for example, requires that one uses long division. And the unit step function is little more than an if-statement. And obviously a Turing Machine could be arbitrarily many steps. It could even go on forever! But even so, It's just math.
I do want to note though, that although all of these things are math, it's probably not canonical to put Turing Machines into equations.
In Calculus III the other day Dr. Anderson mentioned the floor function. I don't know when I learned about it, but I have known about it for a while. It's almost exactly what it sounds like, you take a number and then the answer is the lowest integer of the number. I don't know if that describes it right. How's this:
floor(1.5)=1
floor(1)=1
floor(-1.5)=-2
Make sense? Well, one of the guys in the class said, "You can put stuff like that in a mathematical equation?" I remember thinking something to that effect when I learned something similar. The floor, the ceiling, the unit step function, heck even modulus, they all seem so arbitrary. Not like math where you just pump it out. But that is beside the point. It's just math. It's similar to functions that are defined partially by their input (the name escapes me). They seem so arbitrary, but again, it's just math. What gets really interesting is when you start mixing in Turing Machines; again, just math. But a Turing Machine is the lowest common denominator for computers. You can do some special things with hardware these days, but anything a computer can do, so can a Turing Machine.
I think that the reason these things throw us off is that they are not "single" step functions that can be done arithmetically. Modulus, for example, requires that one uses long division. And the unit step function is little more than an if-statement. And obviously a Turing Machine could be arbitrarily many steps. It could even go on forever! But even so, It's just math.
I do want to note though, that although all of these things are math, it's probably not canonical to put Turing Machines into equations.