What does it mean for something to be "logically possible"?

[untermensche]

..even though UM is only repeating himself ad nauseam.

It takes a lot of repeating to get some to understand.

Some still do not understand.

I'm sure somebody will again claim an infinite series can have an end. Can have a last element.

I can imagine infinity but I can't construct it.

I don't think you can.

All one can do is imagine something progressing towards some horizon. Imagine a progression too long to see it's end. You can't imagine an infinite amount of anything.

One can in theory say a finite line has infinite dimensionless elements.

But you can't imagine a dimensionless element.

 

[Speakpigeon]

I can imagine infinity but I can't construct it.

I don't think you can.

All one can do is imagine something progressing towards some horizon. Imagine a progression too long to see it's end. You can't imagine an infinite amount of anything.

One can in theory say a finite line has infinite dimensionless elements.

But you can't imagine a dimensionless element.

That's right. You have a point there. It's true I can't imagine infinity or any infinite number of things.

I really meant, "I can conceive of infinity".

I guess most people can conceive of infinity in the sense that they can have a concept of infinity that doesn't seem self-contradictory, or absurd, to them.


Whereas we're perfectly able to imagine a finite set of numbers, for example.

Wait, no! Not even that. We can certainly imagine any relatively small set of numbers but certainly not any very large set (most people at least can't). Yet, that doesn't stop us accepting that very large sets of things do exist for real. We cannot see atoms making up our own body but still we accept that it is made up of an extraordinary large number of atoms each too small to see we the naked eye, a number we can't imagine (although we could imagine the figure representing this number).

Do you not accept that the figure "436.376.800.227.546.569.001.438.665.123.582.982.688.435.776" as meaningful and therefore to represent potentially the number of some set of things. Yet, you could not possibly count those things from the first to the last. Your life wouldn't be long enough. If you could see all of them together you wouldn't be able to see each of them individually. Like our own galaxy at night. We see a few of its stars with the naked eye, we could see potentially perhaps a few thousand at best, but "Astronomers estimate there are about 100 thousand million stars in the Milky Way alone" (Wiki). So, do you not believe that the Milky Way is made up of all those stars?
EB

 

[Phil Scott]

And no one can show how to generate any infinite set...

All data that constructivists get their hands on is finite. Though we would say that some data, such as the natural numbers, are (potentially) infinite. That's mostly to say that for any natural, I can find one that's bigger, though the difference in the logic between constructivism and finitism is more subtle.

So, Constructivists are closet-Realists about infinity. They want their constructivist cake and eat it to get the extase of infinity.

If you say so.

Finitists are stricter, and it is always impossible possible to interpret finitist discourse as being about a world in which there is only a finite amount of data available.

It is truly fascinating that the mathematical ship of fools should split on the reef of our performing abilities. I can do this but I can't do that. I can imagine infinity but I can't construct it.

The fundamental nature of this problem may also explain why this thread is going on and on and on, even though UM is only repeating himself ad nauseam.

I'm trying to bring closure here, and relief.
EB

Thanks for correcting the typo.

On the subject of finitism, there has been some interesting metatheoretical results in recent years saying that a larger amount of mathematics than previously thought is finitistic. My own preference would be for a finitistic mathematics, but I haven't figured out how to write a theorem prover for it.

 

[Speakpigeon]

On the subject of finitism, there has been some interesting metatheoretical results in recent years saying that a larger amount of mathematics than previously thought is finitistic. My own preference would be for a finitistic mathematics, but I haven't figured out how to write a theorem prover for it.

I'm more broad Church, each one doing very different things that may nonetheless have things in common. So, why not have one mathematics of infinity alongside another mathematics of finitude? That's what we already have I guess. And I'm sure most mathematicians understand how these two tracks relate to each other though they differ quite a lot.

We're not all the same. I mean, our minds are not all the same. They differ in so many different ways, some subtle, some not so subtle. So, in the end, the question has to be whether we all agree on the logic of what people say even if we don't like what they say. I'm sure most people would.
EB

 

[untermensche]

Wait, no! Not even that. We can certainly imagine any relatively small set of numbers but certainly not any very large set (most people at least can't). Yet, that doesn't stop us accepting that very large sets of things do exist for real. We cannot see atoms making up our own body but still we accept that it is made up of an extraordinary large number of atoms each too small to see we the naked eye, a number we can't imagine (although we could imagine the figure representing this number).

We accept that the atoms that make up our body are finite because our body takes up a finite albeit an ever changing volume of space.

If an atom takes up any space an infinite number of them would not fit in the universe. They would not fit in infinite universes.

They could not "fit" anywhere.

 

[Phil Scott]

I'm more broad Church, each one doing very different things that may nonetheless have things in common. So, why not have one mathematics of infinity alongside another mathematics of finitude? That's what we already have I guess. And I'm sure most mathematicians understand how these two tracks relate to each other though they differ quite a lot.

In my experience, most mathematicians have no idea how the logics differ, and become routinely confused. It's an absolutely standard confusion among mathematicians that constructivists disallow proof-by-contradiction, for example. I was once asked to explain how one can prove the irrationality of root 2 constructively, by someone who couldn't see the plain fact that the standard proof is constructive.

As for finitism, this is even less well-understood by contemporary mathematicians. Classical mathematics reigns so supreme that it is basically how modern mathematicians think day-to-day. It's certainly how I was trained to think in my degree.

It is always nice to have a plurality of views, but we certainly don't have one today. And maybe we don't want one. Had Hilbert's programme not been scuppered by Goedel, I suspect we'd all be finitists now, even as we thought day-to-day like classical set theorists. This was Hilbert's idea: classical set theory is mostly nonsense, but it might be a very very useful nonsense that provides all sorts of cool metatheoretical shortcuts for real finitistic mathematics. Hilbert was wrong here, but maybe there are other metatheoretical shortcuts we can find that save some sort of finitist programme. I'm not personally sure what they look like.

My own discontent with constructivism is that it is still too easy to prove constructive theorems that are basically meaningless, since they talk about situations that require several universe lifetimes to work through. Finitism is a bit saner here, since all finitistic results can be interpreted right now on any computing device, from a humble abacus to a supercomputer.

 

[ryan]

You said that if something has an order, then it has a beginning. But there is no beginning to the integers.

Of course there is.

The positive integers begin a 1 and the negative integers begin at -1.

Okay, but I wasn't talking about the negative integers, and you know that.

Is there a beginning to the set of odd integers?

 

[Kharakov]

They're all odd to untermensche.

 

[Kharakov]

We accept that the atoms that make up our body are finite because our body takes up a finite albeit an ever changing volume of space.

If an atom takes up any space an infinite number of them would not fit in the universe. They would not fit in infinite universes.

They could not "fit" anywhere.

Umm, no, an infinite number of them would fit into an infinite volume.

 

[untermensche]

Of course there is.

The positive integers begin a 1 and the negative integers begin at -1.

Okay, but I wasn't talking about the negative integers, and you know that.

Is there a beginning to the set of odd integers?

Again, there is the set of the positive odd integers and the set of the negative.

Two infinite series.

ALL infinite series have a first element.

To have a series you must define a first element.

- - - Updated - - -

We accept that the atoms that make up our body are finite because our body takes up a finite albeit an ever changing volume of space.

If an atom takes up any space an infinite number of them would not fit in the universe. They would not fit in infinite universes.

They could not "fit" anywhere.

Umm, no, an infinite number of them would fit into an infinite volume.

No volume would ever be enough.

There is no volume large enough that would be enough.

There is no such thing as "infinite" volume. Infinite is not an amount.

There can only be a volume increasing without end. But no matter how big that volume increased it would never contain infinite elements that take up space.

 

[Kharakov]

Actually, a finite hypervolume that contains an infinite volume would be enough.

You could have a 2 x 2 x 2 x 2 meter hypercube that encloses a hyper-M-set fractal with a boundary volume of infinity, even though it has a hypervolume of less than 2 meters^4 (16 hypercubic meters).

 

[untermensche]

No you can't and you can't show me such a thing.

 

[fromderinside]

Right.

But not because it doesn't exist.

It's because you can't sense it, or, you can't understand the math.

Oh.

Pow!

So fallen.

 

[Kharakov]

The simplest object that I can think of for you to visualize this property of continuous space is the 4d analog of the Menger Sponge. I doubt you'll understand it, but....

The  Menger Sponge is an object with a volume that's ~0. It's boundary area is infinite.

In the case of a hypersponge, the boundary volume is infinite (instead of the boundary area being infinite), while the hypervolume of the object is infinitesimal like the sponge's volume is infinitesimal.


So, if spacetime consists of more dimensions than 3 + time, we know for a fact that continuous spacetime of >4 dimensions can enclose infinite volumes in finite hypervolumes.

 

[untermensche]

Right.

But not because it doesn't exist.

It's because you can't sense it, or, you can't understand the math.

Oh.

Pow!

So fallen.

I know people talk of many fantastic things. Infinite angels dancing on pins.

But what can they actually prove?

 

[beero1000]

This has only been gone over about 50 times.

The series of the positive integers begins at 1.

The series of the negative integers begins at -1. It is the same exact series as the positive integers with a negative sign in front of the number.

But none of this has any application here.

None of this applies to time or infinite time.

You said that if something has an order, then it has a beginning. But there is no beginning to the integers.

He's clearly wrong, even though he'll never admit it; twisting himself into pretzels trying to split them into the positive and negative integers (and forgetting zero!) so that he can have his 'beginning'.

 

[George S]

Right.

But not because it doesn't exist.

It's because you can't sense it, or, you can't understand the math.

Oh.

Pow!

So fallen.

I know people talk of many fantastic things. Infinite angels dancing on pins.

But what can they actually prove?

What can anyone actually prove?

The truth or falsity of certain mathematical propositions can be proved from the axioms of math.

The scientific method is a matter of disproof, not proof. We chip away, removing the false; what remains is what we know so far.

Within the world of mathematics I can, in a sense, count to infinity.

Consider a line segment _________________________. Mark the left end with 0. Go halfway to the end at put a 1 there. Go halfway to the end from 1 and put a 2 there. In general from any n go halfway to the end and put n+1. This marks the number line with all the integers. It takes the entire infinity of them to reach the end.

Now start a pencil moving at a constant speed along the line from 0 to the end, counting mentally as you go. When the pencil reaches the end you have mentally counted to infinity.

The mental counting goes really, really fast near the end.

That line segment (a pin perhaps?) is a finite length containing a "countable" infinity of points. (A countable infinity can be put into 1-1 correspondence with the natural numbers.)

 

[untermensche]

Within the world of mathematics I can, in a sense, count to infinity.

No you cannot.

Nobody believes that.

Consider a line segment _________________________. Mark the left end with 0. Go halfway to the end at put a 1 there. Go halfway to the end from 1 and put a 2 there. In general from any n go halfway to the end and put n+1. This marks the number line with all the integers. It takes the entire infinity of them to reach the end.

You can do that for a lifetime and not get any closer to the end than when you started.

Now start a pencil moving at a constant speed along the line from 0 to the end, counting mentally as you go. When the pencil reaches the end you have mentally counted to infinity.

You can imagine a pencil moving your entire life and it will be no closer to moving an infinite distance than when you started.

 

[bilby]

You said that if something has an order, then it has a beginning. But there is no beginning to the integers.

Of course there is.

Nope.

The positive integers begin a 1 and the negative integers begin at -1.

Ryan is talking about the Integers. Not about two subsets of the integers; About the entire set.

ALL infinite series have a beginning.

Nope.

NONE have an end.

Wrong yet again.

The set of rational numbers between 1 and 2 has a beginning and an end, and is infinite.

The set of integers has neither beginning nor end (and is also infinite).

 

[bilby]

Repeating false nonsense about infinite sequences isn't going to change that time itself doesn't have a beginning, nor the fact that infinite sequences can have ends and beginnings.

That is true.

That is why I only speak the truth.

Oh get the fuck over yourself. You don't even KNOW some very obvious 'truths', yet you pontificate about things you are utterly wrong about as though you were infallible. You are totally out of your depth, and a modicum of humility might just allow you to get off your high horse long enough to learn something that would save you from such embarrassing displays of stupidity.

Infinite time in the past is impossible.

Just not in any way you can demonstrate.

Infinite time never passes. It cannot be in the past.

It is not possible to crawl out of a hole infinitely deep.

The you should stop digging.

ALL infinite series have a beginning. You can't show me one without one.

The integers.

NO infinite series has an end. You can't show me one with one.

The rational numbers between 1 and 2

If you show me an infinite series you think ends that only means you don't know the difference between it's beginning and end.

The rational numbers between 1 and 2 are infinite, and are bounded at both ends.