What are the problems with finitism?

[mubashirmansoor]

Hello everyone,

What are the problems with finitism? Imean what would happen if there was a very large number (potentially infinite) instead of an absoloute infinity?

Do we really need an infinity?

Thankyou for you reply :)

 

[HallsofIvy]

Yes, we need infinity in order to be able to use calculus at all! Without infinity, you are restricted to "discrete mathematics" which, while it can solve many problems cannot solve them all, and is much clumsier than calculus for some of the problems it can solve.

It's not clear to me what you mean by "potentially infinite". If you are suggesting that there be some immensely large but finite largest number, that raises an obvious question- what do you get when you add 1 to that number?

Note that I am talking about mathematics here. This has nothing to say about whether there exist "infinite" numbers of things in the "real" world.

 

[Hurkyl]

I've moved this to the logic forum since this is dealing with foundational issues.

(Disclaimer: I have some familiarity with a particular kind of constructivism; I'm not sure precisely in what way it differs from finitism, or even if finitism refers to a specific philosophy)


I should point out that finitism has essentially nothing to do with whether or not you consider mathematical structures that contain an object called "infinity".


For example, a finitist would permit us to work with the positive rational numbers. The set of extended positive rational numbers consists of an additional point called $+\infty$; however, this set requires no more complicated than the set of rational numbers in the interval [0, 1]. So, a finitist would permit us to work with the set of positive rational numbers.


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.

 

[mubashirmansoor]

Yes, we need infinity in order to be able to use calculus at all!

why not using the immenesly large number instead? how can we prove that we are not doing so when dealing with infinities?

If you are suggesting that there be some immensely large but finite largest number, that raises an obvious question- what do you get when you add 1 to that number?

We can imagine a finite number line as a line segment or a closed geometical shape such as a circle. In the first case we would get nothing when 1 was added to the largest number, But in the case of a circle, we would get the smallest possible number or negative infinity (I mean potentially infinite)...

I'am sure what I've written above might sound a bunch of really useless and flawed sentences to you, but I just want to know if we can disapprove such an assumption or not... Just for fun :-)

 

[Office_Shredder]

We can imagine a finite number line as a line segment or a closed geometical shape such as a circle. In the first case we would get nothing when 1 was added to the largest number, But in the case of a circle, we would get the smallest possible number or negative infinity (I mean potentially infinite)...

So let's say the line segment was picked to be [0,1] (as clearly if it's [a.b] we can just shift it and divide to find an isomorphism between the two for addition and multiplication). Where would the number 1 appear on that line? Where would the number 2 appear on that line? etc. You quickly find that no matter how large a number you want to place on that line, it has to be at 0, as otherwise you either eventually reach 1 (representing infinity), and then your next number won't be represented (which means we haven't actually represented the real numbers)

 

[Hurkyl]

why not using the immenesly large number instead?

Why not use some number really close to '2', instead of actually using '2'?

I expect your answer to my question will also be applicable to your question.

how can we prove that we are not doing so when dealing with infinities?

By definition, the extended real number $+\infty$ is larger than every real number. Therefore, it's trivial to prove that $+\infty$ is not a real number.

 

[mubashirmansoor]

Why not use some number really close to '2', instead of actually using '2'?

I expect your answer to my question will also be applicable to your question.



By definition, the extended real number $+\infty$ is larger than every real number. Therefore, it's trivial to prove that $+\infty$ is not a real number.

Thankyou for your reply Hurkyl, but as you mentioned a logical outcome of the definition of infinity can be simply that infinity is NOT a real number.
As a result how can we place it in real number's number-line ? I think we cant, hence the numberline becomes finite and most probably reaccuring. such as this one:

..., -3, -2, -1, 0, 1, 2, 3,...,the largest number, a nil number, the smallest number,..., -3, -2, -1, 0, 1 , 2, 3 & so on. (a closed circle where zero is on the opposite side of the nill number)

This means when sketching an x-y curve, we'll be dealing with the surface of a sphere and that one of the poles of this sphere has the cordinates of (0,0) & the other pole (nil,nil).

This will solve the divison by zero problem because the answer would be nil...

Believe me I know what I've written is flawed but I couldn't really find a way to disaprove this closed numberline system... would you please help me to do so... Thankyou ver much once again. :)

 

[Kittel Knight]

As a result how can we place it in real number's number-line ? I think we cant, hence the numberline becomes finite and most probably reaccuring. such as this one:

..., -3, -2, -1, 0, 1, 2, 3,...,the largest number, a nil number, the smallest number,..., -3, -2, -1, 0, 1 , 2, 3 & so on. (a closed circle where zero is on the opposite side of the nill number)

This means when sketching an x-y curve, we'll be dealing with the surface of a sphere and that one of the poles of this sphere has the cordinates of (0,0) & the other pole (nil,nil).

Then N < N + 1000 is no more a valid expression!
:yuck:

Or maybe - who knows - you prefer to say N<N+1000 and N>N+1000 are both valid expressions!
:yuck: :yuck:

 

[Coin]

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.)

--- --- --- ---

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.

 

[Coin]

why not using the immenesly large number instead? how can we prove that we are not doing so when dealing with infinities?

I think we can show that there are situations where you get a meaningfully different answer when you use an infinity versus when you use "an arbitrarily large number". For example you can think about the limit of sin(x) as x goes to infinity. If you choose "a large number", you get an answer between 1 and -1. If you choose "an infinity", you do not get an answer at all.

(Of course maybe this isn't a very good example, since in the infinite case rather than getting "a different answer" you get no answer. It is the best I can think of on the spur of the moment, though.)

As a result how can we place it in real number's number-line ? I think we cant, hence the numberline becomes finite and most probably reaccuring. such as this one:

We don't put it in the number line, we put it slightly to the right of the number line. As Hurkyl said, if you have infinity then you aren't using reals, you're using "extended reals". What we have to do is define a new number line, containing all the reals PLUS exactly two extra points, infinity and minus-infinity, at either end like bookends.

 

[mubashirmansoor]

Then N < N + 1000 is no more a valid expression!
:yuck:

You are not right!

N+1000 = -infinity in closed numberline system & N is +infinity .

The logical outcome is; N > N+1000 because +infinity > -infinity

and remember that N is a constant not a variable.

 

[mubashirmansoor]

I think we can show that there are situations where you get a meaningfully different answer when you use an infinity versus when you use "an arbitrarily large number". For example you can think about the limit of sin(x) as x goes to infinity. If you choose "a large number", you get an answer between 1 and -1. If you choose "an infinity", you do not get an answer at all.

In fact this is a very good example and maybe all my problem is right here...

What I 've always thought about such an example; when trying to find the answer to sin(infinity), a set of possibilities between -1 & 1 are achieved. So we do get something... Is this assumption wrong ?

We don't put it in the number line, we put it slightly to the right of the number line. As Hurkyl said, if you have infinity then you aren't using reals, you're using "extended reals". What we have to do is define a new number line, containing all the reals PLUS exactly two extra points, infinity and minus-infinity, at either end like bookends.

Are simply two points at each end of the numberline enough?, I thought infinities would make the numberline endless, so there would no bookends at all...


Thankyou for your very interesting comments on finitism in post#9... :)

 

[Kittel Knight]

You are not right!

N+1000 = -infinity in closed numberline system & N is +infinity .

The logical outcome is; N > N+1000 because +infinity > -infinity

and remember that N is a constant not a variable.

Since you states that N > N+1000 , then N < N+1000 is no more a valid expression (as I had said) !
:yuck:

 

[CRGreathouse]

Are simply two points at each end of the numberline enough?, I thought infinities would make the numberline endless, so there would no bookends at all.

Here's the number line: holds up thumb and index finger about two inches apart. In the middle is zero; a little to the right of it is one; a little to the right of that is a trillion; a little to the right of that a googolplex; a bit further is BusyBeaver(10,000,000,000); a bit further is my index finger at infinity. Everything's there, what's the problem?

 

[Gib Z]

Here's the number line: holds up thumb and index finger about two inches apart. In the middle is zero; a little to the right of it is one; a little to the right of that is a trillion; a little to the right of that a googolplex; a bit further is BusyBeaver(10,000,000,000); a bit further is my index finger at infinity. Everything's there, what's the problem?

Most people would think you have a massively mutated hand =]

 

[mubashirmansoor]

Since you states that N > N+1000 , then N < N+1000 is no more a valid expression (as I had said) !
:yuck:

I'am talking about a closed number-line, which would make what I said true. But you expresions are true in an infinite number-line...

I thisnk you got mu point... :)

 

[mubashirmansoor]

Here's the number line: holds up thumb and index finger about two inches apart. In the middle is zero; a little to the right of it is one; a little to the right of that is a trillion; a little to the right of that a googolplex; a bit further is BusyBeaver(10,000,000,000); a bit further is my index finger at infinity. Everything's there, what's the problem?

Here is the problem; even extended reals are still finite even if it takes the whole milkyway galaxy to write the number...

In your example, the distance between your fingers is finite, while you have added an infinite notation within this finite lenght... Is this correct to do? if so how?