Numbers

[untermensche]

I had a category theory mate who was pretty up on his history of maths, and I once asked him about Hilbert's paper. He suggested that there was folklore that Hilbert was straight up trolling.

The point I'd throw in for untermensche is this: Hilbert was expressing scepticism on the infinite, but he wasn't merely relegating it to a mathematical construction. He flat out thought the idea had an unreality which meant its place in abstract mathematics was also severely compromised. Accordingly, the only way that the infinite could retain any place in mathematics was if Hilbert could establish that it was a mere convenience, and so ultimately superfluous. That comes with a positive proof: Hilbert was tasked to show rigorously that the infinite could be, in principle, banished from mathematics whilst retaining all of the theorems of real mathematics.

Do you agree with this, untermensche?..

I have no opinions on the use of infinity in mathematics.

My only objection, and it has been my constant objection, is when the concept of infinity is applied to anything real.

The second you try to apply infinity to anything real you immediately encounter insoluble absurdities.

But I thought you wanted to discuss the philosophy of mathematics. Try to get to the bottom of the concept of numbers.

I asked: What is "one"?

You gave some modern understanding but "one" existed long before that understanding.

So what was "one" when it first became a concept? "One" existed and was used a long time before the understanding of it you provided.

I know this is thought experiment and cannot be answered definitely but it is actual philosophy as opposed to mere mathematical conjectures. Not to diss mathematical conjectures but they are not philosophy.

 

[Bomb#20]

Still, there isn't any straightforward relation, and we certainly ordinarily don't consider any such relation, between this basic notion of infinity as applied on the one hand to space, and at least to the future if not necessarily to the past, and on the other hand to the kind of infinity we think of when we talk of infinite divisibility and the continuity we ordinarily attribute to space and time.
EB

That's right. It's curious, but the way we normally think about them, "the infinitely small" is a bigger infinity than "the infinitely large". When we think of the infinitely large, we typically think of the stars going on forever -- that after N light years, a photon can always go another light year. This would mean the universe goes on in space and time for a "countable" infinity of light years, a number mathematicians call "aleph null". But the number of distinct points in even one centimeter of an infinitely divisible continuum is "two to the power of aleph null", an amount that Cantor proved was more than aleph null.

 

[Bomb#20]

What homogeneous continuum do you claim exists?

I don't claim any homogeneous continuum exists. Unlike you and Hilbert, I limit my claims to what the evidence supports.

We are allowed to think and use our rational capacities.

We can rationally examine this notion of a real infinity.

We can look at something like space and ask is it possible to have the smallest amount of space?

What volume of space could we have where theoretically we could not have a smaller volume of space?

What would that smallest theoretical volume be?

This requires thinking, I warn you.

You and I have been through this before; so far you've declined to think about the problem using your rational capacities.

The rational case for a homogeneous continuum is simple: we have a well-tested theory called "relativity" that accounts for all our measurements; and it implies there can't possibly be a smallest nonzero volume of space. This is because distance is relative to motion, and distance is foreshortened in the direction of travel. So if we suppose there is a smallest volume, and we have a particular region of space that has that volume, that same region has a smaller volume from the point of view of an observer moving past it quickly than from the point of view of an observer moving past it slowly. No matter how small a region is, there is always another frame of reference in which it's smaller. Therefore, no minimum volume, other than zero. QED. Therefore, if, nonetheless, the universe really does have a minimum volume, it follows that Einstein's theory of relativity must be wrong. Therefore, if there's a minimum volume, then there must be an alternative competing theory that's at least as good as relativity at accounting for all our measurements.

Well then, if you insist that space must have a minimum volume, present that alternative competing theory. Show us a mathematical model of space -- one that matches our measurements -- in which distance doesn't foreshorten in the direction of travel. Put up or shut up.

Here, I'll even get you started. We actually already have an alternative competing theory of space, that matches our measurements, in which distance doesn't foreshorten in the direction of travel. It's called "Lorentz Ether Theory". Unfortunately for your purpose, LET doesn't have a minimum volume either. But if you want to try your hand at constructing a theory of granular spacetime, that's probably the best place to start.

 

[untermensche]

We are allowed to think and use our rational capacities.

We can rationally examine this notion of a real infinity.

We can look at something like space and ask is it possible to have the smallest amount of space?

What volume of space could we have where theoretically we could not have a smaller volume of space?

What would that smallest theoretical volume be?

This requires thinking, I warn you.

You and I have been through this before; so far you've declined to think about the problem using your rational capacities.

The rational case for a homogeneous continuum is simple: we have a well-tested theory called "relativity" that accounts for all our measurements; and it implies there can't possibly be a smallest nonzero volume of space. This is because distance is relative to motion, and distance is foreshortened in the direction of travel. So if we suppose there is a smallest volume, and we have a particular region of space that has that volume, that same region has a smaller volume from the point of view of an observer moving past it quickly than from the point of view of an observer moving past it slowly. No matter how small a region is, there is always another frame of reference in which it's smaller. Therefore, no minimum volume, other than zero. QED. Therefore, if, nonetheless, the universe really does have a minimum volume, it follows that Einstein's theory of relativity must be wrong. Therefore, if there's a minimum volume, then there must be an alternative competing theory that's at least as good as relativity at accounting for all our measurements.

Well then, if you insist that space must have a minimum volume, present that alternative competing theory. Show us a mathematical model of space -- one that matches our measurements -- in which distance doesn't foreshorten in the direction of travel. Put up or shut up.

Here, I'll even get you started. We actually already have an alternative competing theory of space, that matches our measurements, in which distance doesn't foreshorten in the direction of travel. It's called "Lorentz Ether Theory". Unfortunately for your purpose, LET doesn't have a minimum volume either. But if you want to try your hand at constructing a theory of granular spacetime, that's probably the best place to start.

If you totally ignore all my questions you are just wasting my time.

I am saying there is no such thing as the smallest theoretically possible volume. It is not a rational concept.

You confuse my arguments about movement with my points here.

To have any movement it must be a movement greater than zero. Any movement greater than zero is a finite movement.

So to have movement you must make some first finite movement. You cannot make an infinitely small movement. There is no such thing.

 

[Juma]

The second you try to apply infinity to anything real you immediately encounter insoluble absurdities.

Please give example of such insoluble absurdities...

 

[untermensche]

The second you try to apply infinity to anything real you immediately encounter insoluble absurdities.

Please give example of such insoluble absurdities...

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

 

[Speakpigeon]

I see the Pac-UBot is now eating posts in the Numbers thread.
EB

 

[Juma]

The second you try to apply infinity to anything real you immediately encounter insoluble absurdities.

Please give example of such insoluble absurdities...

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

”You should realize” isnt valid argument here.
Show that a infinite number of things wouldnt fit in an infinite universe.

 

[untermensche]

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

”You should realize” isnt valid argument here.
Show that a infinite number of things wouldnt fit in an infinite universe.

If the number of objects are not finite and occupy space there is no volume that can contain them.

No matter how much space you had it would be completely full and there would still be more objects.

You would always need more and more space without end. And no matter how much space was filled there would always be more objects. Infinitely more objects.

 

[bilby]

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

”You should realize” isnt valid argument here.
Show that a infinite number of things wouldnt fit in an infinite universe.

It would if you start from the assumption that the universe is finite, because you are too dumb to remember the basic premises of a brief argument for as long as it takes to read it.

With a pig-headed refusal to consider the arguments as presented, you can prove anything that's even vaguely part of your dogma.

 

[Phil Scott]

Clever.

Not exactly how we ordinarily think of an infinite space, but I can see it works. And I remember programming something like the Pac-Man's space myself and this definitely feels somewhat similar to a curved space, or more exactly a cube.
EB

The two dimensional case is the study of surfaces and topologists exhibit examples often by thinking about pacman worlds. More specifically, they start with a square and then say what the rules are on hitting a boundary. In Pacman's world, the left and right sides wrap around. You can embed this three dimensionally as Pacman moving around the outer surface of a ring. There would be some distortion due to the curvature, but it's not something you would notice at small enough scales.

Now in Asteroids, the world wraps in top and bottom as well. You embed this in three dimensions as flying along the surface of a doughnut.

Doughnut world truly lacks any edges. You can fly anywhere on the surface without hitting the boundary. This is conceptually true even in the 2d embedding. We might think there is a boundary at the edge of the screen, but you can always translate and bring that boundary to the centre. It is the edge of the screen which is artificial, rather than a real part of the world (and in Asteroids, the world is continually translated to keep the player in the centre of the screen).

You can do more interesting wrappings too: take Pacman again, and put a cool twist on the horizontal wrapping: have it so when you cross either side of the screen, you flip Pacman's y coordinate. Pacman's world now embeds on a band with a twist in it: a Moebius band.

What if we do the same for Asteroids: keep the vertical behaviour but flip the y when crossing the left and right edges? Now the world has no 3d embedding. It can only be realised in four dimensions. You might have already heard of this bizarre object.

What is common in these cases is that, at small enough scales, they start looking like flat planes. All manifestations of this idea turn out to be fully classifiable. I haven't studied the 3d case of surfaces (manifolds), where locally the world looks like ordinary 3d space, but am told it is the same there and would enjoy reading beero's link.

 

[untermensche]

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

”You should realize” isnt valid argument here.
Show that a infinite number of things wouldnt fit in an infinite universe.

It would if you start from the assumption that the universe is finite, because you are too dumb to remember the basic premises of a brief argument for as long as it takes to read it.

With a pig-headed refusal to consider the arguments as presented, you can prove anything that's even vaguely part of your dogma.

What volume would hold infinite objects that take up space?

And I want a real world volume, not an imaginary concept attached to the word volume.

I can remember my point was about trying to attach infinity to something real.

Can we attach the idea of infinite objects to some real world volume? If so what volume would that be?

If all we can say is infinity can fit into infinity we are admitting the absurdity of trying to attach infinite objects to reality.

 

[Phil Scott]

But I thought you wanted to discuss the philosophy of mathematics.

According to the link you provided, I have been. Please acknowledge this.

I asked: What is "one"?

You gave some modern understanding but "one" existed long before that understanding

Lots of notions existed in folk understanding before our modern understanding. The folk understanding often turns out to be wrong.

My answer to your question isn't worth anything, nor are any other idiosyncratic philosophys on anonymous internet forums, especially when it is the philosophy of subjects that the poster has no knowledge of, interest in and thus stake. I would rather discuss the ideas of better people.

If you find modern accounts too modern, we could try Husserl. He was a mathematically trained contemporary of Frege who, like Frege, initially wrote about the concept of number, but I believe in a way which led to his prescientific phenomenology.

 

[Speakpigeon]

Still, there isn't any straightforward relation, and we certainly ordinarily don't consider any such relation, between this basic notion of infinity as applied on the one hand to space, and at least to the future if not necessarily to the past, and on the other hand to the kind of infinity we think of when we talk of infinite divisibility and the continuity we ordinarily attribute to space and time.
EB

That's right. It's curious, but the way we normally think about them, "the infinitely small" is a bigger infinity than "the infinitely large". When we think of the infinitely large, we typically think of the stars going on forever -- that after N light years, a photon can always go another light year. This would mean the universe goes on in space and time for a "countable" infinity of light years, a number mathematicians call "aleph null". But the number of distinct points in even one centimeter of an infinitely divisible continuum is "two to the power of aleph null", an amount that Cantor proved was more than aleph null.

Do you have any suggestion to explain that the basic arithmetic system humans have initially developed, broadly considering that of the Ancient Greeks, should so tightly fit with their prior intuition about continuity? The Greek were obviously still struggling to understand the idea of the continuity of space and time, even though they already had the rational numbers, at least some irrational numbers, such as the square root of 2, and at least one transcendental with π, that even the Babylonians and Ancient Egyptians knew of (obviously as approximations)?

It seems an interesting coincidence to me that humans seem to have had the intuition of the continuity of space and time even before they could develop, crucially independently from that intuition, a basic, unsophisticated, arithmetic that only much later, broadly in the 19th century, could be shown to snug tightly into that intuition with the first formal expression of the notion of limit. And it's also only much later that more sophisticated mathematicians started to go beyond our basic intuitions, with for example the Complex numbers. And yet, I don't know that anybody has ever imagined any non-Real number that would nonetheless be conceived of as properly belonging to a continuous interval between two integers for example. As if we can't go beyond our basic intuition about the continuity of space and time. I believe it should be possible to invent such a number in the abstract (as for the Complex) even though our intuition seems to say such a number couldn't possibly represent any real point in space or time. Is it just that mathematicians are stopped by their very human intuition, or is it really impossible to invent that kind of number? The later would be a real shocker to me.
EB

 

[untermensche]

But I thought you wanted to discuss the philosophy of mathematics.

According to the link you provided, I have been. Please acknowledge this.

I asked: What is "one"?

You gave some modern understanding but "one" existed long before that understanding

Lots of notions existed in folk understanding before our modern understanding. The folk understanding often turns out to be wrong.

My answer to your question isn't worth anything, nor are any other idiosyncratic philosophys on anonymous internet forums, especially when it is the philosophy of subjects that the poster has no knowledge of, interest in and thus stake. I would rather discuss the ideas of better people.

If you find modern accounts too modern, we could try Husserl. He was a mathematically trained contemporary of Frege who, like Frege, initially wrote about the concept of number, but I believe in a way which led to his prescientific phenomenology.

Better people?

What is that?

Are you claiming people weren't using "one" properly until some arbitrary scheme existed? Are you saying they could not make full use of it?

It looks like you are refusing to even take a philosophical peek at the concept of number.

You are no philosopher. Maybe you learned some things anybody could learn.

But anybody with Google knows that too.

But having Google won't help one think.

 

[bilby]

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

”You should realize” isnt valid argument here.
Show that a infinite number of things wouldnt fit in an infinite universe.

It would if you start from the assumption that the universe is finite, because you are too dumb to remember the basic premises of a brief argument for as long as it takes to read it.

With a pig-headed refusal to consider the arguments as presented, you can prove anything that's even vaguely part of your dogma.

What volume would hold infinite objects that take up space?

It's a mystery

 

[untermensche]

”You should realize” isnt valid argument here.
Show that a infinite number of things wouldnt fit in an infinite universe.

It would if you start from the assumption that the universe is finite, because you are too dumb to remember the basic premises of a brief argument for as long as it takes to read it.

With a pig-headed refusal to consider the arguments as presented, you can prove anything that's even vaguely part of your dogma.

What volume would hold infinite objects that take up space?

It's a mystery

No. It's an absurdity.

If all you can do is invoke one imaginary concept to explain another you haven't related anything to reality.

 

[bilby]

It would if you start from the assumption that the universe is finite, because you are too dumb to remember the basic premises of a brief argument for as long as it takes to read it.

With a pig-headed refusal to consider the arguments as presented, you can prove anything that's even vaguely part of your dogma.

What volume would hold infinite objects that take up space?

It's a mystery

No. It's an absurdity.

If all you can do is invoke one imaginary concept to explain another you haven't related anything to reality.

I related it to YOUR claim. Are you finally admitting to the absurdity of your claims?

The second you imagine having an infinite number of something real that takes up space you should realize that they would not fit in infinite universes. Which means they could not fit anywhere in any way.

Instant absurdity.

Instant absurdity is something of a hallmark of your posts; Along with very short term selective amnesia.

 

[untermensche]

Anybody with a brain can see you did not address one point I made.

Are you claiming you have empirical evidence that infinite marbles can fit into infinite universes? Or is it just an opinion you pulled from your ass?

Again.

My claim is when you try to apply infinity to the real world you instantly have an absurdity.

Immediately having to invoke imaginary concepts is the definition of absurdity.

 

[Phil Scott]

It seems an interesting coincidence to me that humans seem to have had the intuition of the continuity of space and time even before they could develop, crucially independently from that intuition, a basic, unsophisticated, arithmetic that only much later, broadly in the 19th century, could be shown to snug tightly into that intuition with the first formal expression of the notion of limit. And it's also only much later that more sophisticated mathematicians started to go beyond our basic intuitions, with for example the Complex numbers. And yet, I don't know that anybody has ever imagined any non-Real number that would nonetheless be conceived of as properly belonging to a continuous interval between two integers for example. As if we can't go beyond our basic intuition about the continuity of space and time. I believe it should be possible to invent such a number in the abstract (as for the Complex) even though our intuition seems to say such a number couldn't possibly represent any real point in space or time. Is it just that mathematicians are stopped by their very human intuition, or is it really impossible to invent that kind of number? The later would be a real shocker to me.
EB

You can expand the continuum using the hyperreals and the surreals (don't know so much about the latter, but am consistently told they are mind-blowingly cool). The hyperreals consist both of non-real infinite numbers, and non-real numbers that are the sum of an ordinary real with an infinitesimal part. They can be used to do calculus without using limits. Where a standard analyst would take a limit, a hyperreal analyst will just drop the infinitesimal part and see what real is left. I think historians broadly agree that the hyperreal approach is more closely aligned with how Newton understood calculus. The standard approach is more aligned with the Greeks.

The hyperreals, in one "construction", arise by looking at the first-order axioms of the reals and saying: I can always consistently say there is a number bigger than any given numeral, so I must be able to consistently say that there is a number bigger than all of them. This gives you an infinite number, and since we're still looking at the axioms for reals, we note that we've still got all the normal algebraic rules and, in particular, we can take the reciprocal of our infinite number and get an infinitesimal.

The surreals, as I understood, arise by taking the basic idea that Dedekind came up with in constructing the reals and then going completely insane with it.