Hurkyl said:
The main thing that finitism forces upon us is algorithmic representations of objects. The finitist would not allow us to work with the real numbers as we normally do; he would insist that we instead work with, for example, algorithms that produce rational approximations to any degree of precision we request.
I think the modern term for this would be "computable numbers".
So here's my take on this:
As far as I know the main thing limiting finitism is that it more or less died out in the 1920s! A lot of the logical tools we might use to approach the subject of finitism in an intelligent way weren't available to the people who originally approached finitism as a philosophical subject; and by the time we were well-equipped to talk about what can be finitely computed or represented, nobody cared about finitism anymore.
So the original finitists-- like Kronecker, the most well-noted finitist, who famously was said to have denied the reality of the square root of two-- aren't really much of a guide in telling us what "finitism" means. Our perspective on things has changed so much that it's difficult to meaningfully evaluate anything the finitists said. (The square root of two
is a "computable number", since we can compute any arbitrarily close approximation to it we like, but I think the idea is that Kronecker would have claimed that the square root of two has no reality, only our approximations to it. But who's to say? Kronecker died a good forty years before anybody got down on paper a really rigorous of what it means to "compute" something.)
Meanwhile in the modern age we have a lot of tools at our disposal for talking about calculations which are in some sense finite. This is, if you think about it, a major part of what computer science theory is all about! Computation theory allows us to identify the class of strings (strings of symbols-- "numbers", if you will) which can be calculated in finite time, and those strings which we can't calculate in finite time but which we can create a finite
specification for an infinite-time calculation (in other words, a computer program), and we can even talk about things which can't be finitely specified by any computer program at all. We can even go one step further than this-- we can define a sort of "hypercomputation", where you're allowed to perform an infinite number of steps all at once. (The interesting thing is that once we've done this, it's possible to identify things that not even the hypercomputer can do).
The thing is though, now that we know all these neat things about what you can potentially do in a finitist system, we don't really have any interest anymore in the philosophical ideas that comprise finitism itself. CS people are interested in what can and can't be finitely computed for
practical reasons-- we've built these computers, and we want to know what they can and can't do. Worrying about whether or not the infinity-infected things that the computers can't work with are "real", or whether they're "needed", doesn't really help you-- in fact, it's downright counterproductive, since a lot of things computers do involve approximating non-finitist things like real numbers, and it's kind of difficult to talk about such things without at some level assuming the things you're approximating exist. Once you have a clear idea of what is and isn't computable-- and therefore what can practically "exist" in the finitist universe-- then the jump between finitist and not-finitist just seems like just another gap between two entries in a big heirarchy of complexity classes. And you'd feel kind of silly trying to pretend that that one gap is much more important than any of the others, especially when you've got Mean Old Mr. Goedel and his friends looking over your shoulder and reminding you that even when you limit yourself to the integers then your abilities are still incomplete in a very deep philosophical sense. (There are things which can't be "finitely computed" but don't themselves have anything to do with infinities, like the busy beaver functions.)
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Hm, I hope I didn't get kind of carried away here... but, one last thing:
Aside from all this if you look at that Tegmark paper that was linked in the "beyond the standard model" forum awhile back, he does propose sort of a modern sort of finitism. Tegmark has this philosophical paper where he tries to talk about the question of "can our universe be exactly described by a mathematical system?" and then toward the end goes one step further and asks "can our universe be exactly described by finite computations?" This is more interesting to me. I think the problem with finitism is, it's just not a useful distinction to make-- who
cares which things in math are and aren't finitely computable, as long as the infinite things are mathematically rigorous? It's all just symbols anyway. The question Tegmark raises is more interesting-- are the things
in physics finitely computable? This seems like a much more meaningful question.
