Numbers

[Lion IRC]

Linking papers would be cool, I hope.

I would really love to link On the Infinite, but I can't find the damn thing. I know it's in "From Brouwer to Hilbert", but I have no idea why this paper from the 1920s is not freely available online in the 21st century. I almost want to go on a rant, but will save it for now.

https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html

DAVID HILBERT. Delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, in honor of Karl Weierstrass. Translated by Erna Putnam and Gerald J. Massey from Mathematische ...

 

[untermensche]

...Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments...

This guy gets it.

 

[Phil Scott]

Linking papers would be cool, I hope.

I would really love to link On the Infinite, but I can't find the damn thing. I know it's in "From Brouwer to Hilbert", but I have no idea why this paper from the 1920s is not freely available online in the 21st century. I almost want to go on a rant, but will save it for now.

https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html

DAVID HILBERT. Delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, in honor of Karl Weierstrass. Translated by Erna Putnam and Gerald J. Massey from Mathematische ...

I submit to your superior Google-fu.

Though now that I think about it, I was using DuckDuckGo.

 

[Kharakov]

I'm thinking Hilbert wasn't privy to modern science that indicates space, if it is granular, is so at scales a minimum of 10^-11 times smaller than the Planck length. This doesn't even include the time portion of the continuum.

 

[Bomb#20]

...Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments...

This guy gets it.

No, he demonstrably doesn't. Let's see that quote in context.

In addition to matter and electricity, there is one other entity in physics for which the law of conservation holds, viz., energy. But it has been established that even energy does not unconditionally admit of infinite divisibility. Planck has discovered quanta of energy.

Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments.

But what quanta of energy did Planck actually discover? Planck discovered energy comes in discrete multiples of Planck's Constant times the frequency. But that would only prove energy does not admit of infinite divisibility if it were the case that frequency does not admit of infinite divisibility -- hence Hilbert's critical insertion of the word "unconditionally". There had been no demonstration that frequency isn't infinitely divisible; 90 years on, there still hasn't. Hilbert jumped to conclusions based on a hasty generalization.

 

[Bomb#20]

https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html

DAVID HILBERT. Delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, in honor of Karl Weierstrass. Translated by Erna Putnam and Gerald J. Massey from Mathematische ...

Euclidean geometry necessarily leads to the postulate that space is infinite. Although euclidean geometry is indeed a consistent conceptual system, it does not thereby follow that euclidean geometry actually holds in reality. Whether or not real space is euclidean can be determined only through observation and experiment. The attempt to prove the infinity of space by pure speculation contains gross errors. From the fact that outside a certain portion of space there is always more space, it follows only that space is unbounded, not that it is infinite. Unboundedness and finiteness are compatible. In so-called elliptical geometry, mathematical investigation furnishes the natural model of a finite universe. Today the abandonment of euclidean geometry is no longer merely a mathematical or philosophical speculation but is suggested by considerations which originally had nothing to do with the question of the finiteness of the universe. Einstein has shown that euclidean geometry must be abandoned. On the basis of his gravitational theory, he deals with cosmological questions and shows that a finite universe is possible. Moreover, all the results of astronomy are perfectly compatible with the postulate that the universe is elliptical.

We have established that the universe is finite in two respects, i.e., as regards the infinitely small and the infinitely large.

That one is just asinine. From the fact that unboundedness and finiteness are compatible, the fact that elliptical geometry is the natural model of a finite universe, and the fact that Einstein has shown that euclidean geometry must be abandoned and a finite universe is possible, it does not follow that the universe is finite as regards the infinitely large. That's just a non-sequitur, an elementary logic error. From a demonstration that the universe might be finite, Hilbert jumps to the conclusion that it is finite. Elliptical geometry is not the only non-euclidean geometry; Einstein's discovery was equally compatible with hyperbolic geometry, the natural model of an infinite non-euclidean universe. (And the idea that David Hilbert of all people was unaware of hyperbolic geometry is beyond implausible.)

Moreover, astronomy progresses. "All the results of astronomy" are no longer perfectly compatible with the postulate that the universe is elliptical. The recently discovered acceleration of the expansion of the universe favors the hyperbolic hypothesis over the elliptical hypothesis.

 

[Speakpigeon]

I'm thinking Hilbert wasn't privy to modern science that indicates space, if it is granular, is so at scales a minimum of 10^-11 times smaller than the Planck length. This doesn't even include the time portion of the continuum.

Awesome.

I'd like to know "how some", too!

But don't bother, I won't get it.
EB

 

[Speakpigeon]

No, he demonstrably doesn't. Let's see that quote in context.

In addition to matter and electricity, there is one other entity in physics for which the law of conservation holds, viz., energy. But it has been established that even energy does not unconditionally admit of infinite divisibility. Planck has discovered quanta of energy.

Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments.

But what quanta of energy did Planck actually discover? Planck discovered energy comes in discrete multiples of Planck's Constant times the frequency. But that would only prove energy does not admit of infinite divisibility if it were the case that frequency does not admit of infinite divisibility -- hence Hilbert's critical insertion of the word "unconditionally". There had been no demonstration that frequency isn't infinitely divisible; 90 years on, there still hasn't. Hilbert jumped to conclusions based on a hasty generalization.

Right, but there I think a conventional infinite would suffice, i.e. unbounded, rather than some newly-minted notion like an actual infinity, with something like an actual point at the infinite.
EB

 

[untermensche]

No, he demonstrably doesn't. Let's see that quote in context.

In addition to matter and electricity, there is one other entity in physics for which the law of conservation holds, viz., energy. But it has been established that even energy does not unconditionally admit of infinite divisibility. Planck has discovered quanta of energy.

Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments.

But what quanta of energy did Planck actually discover? Planck discovered energy comes in discrete multiples of Planck's Constant times the frequency. But that would only prove energy does not admit of infinite divisibility if it were the case that frequency does not admit of infinite divisibility -- hence Hilbert's critical insertion of the word "unconditionally". There had been no demonstration that frequency isn't infinitely divisible; 90 years on, there still hasn't. Hilbert jumped to conclusions based on a hasty generalization.

What homogeneous continuum do you claim exists?

And what is your proof of it?

Sure he was limited by the knowledge of his day but his thinking is right on.

 

[Kharakov]

I'm thinking Hilbert wasn't privy to modern science that indicates space, if it is granular, is so at scales a minimum of 10^-11 times smaller than the Planck length. This doesn't even include the time portion of the continuum.

Awesome.

I'd like to know "how some", too!

But don't bother, I won't get it.
EB

lol. "How some" broke my brain to brain translator, we should talk past eachother for a bit.

 

[Bomb#20]

No, he demonstrably doesn't. Let's see that quote in context.

In addition to matter and electricity, there is one other entity in physics for which the law of conservation holds, viz., energy. But it has been established that even energy does not unconditionally admit of infinite divisibility. Planck has discovered quanta of energy.

Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments.

But what quanta of energy did Planck actually discover? Planck discovered energy comes in discrete multiples of Planck's Constant times the frequency. But that would only prove energy does not admit of infinite divisibility if it were the case that frequency does not admit of infinite divisibility -- hence Hilbert's critical insertion of the word "unconditionally". There had been no demonstration that frequency isn't infinitely divisible; 90 years on, there still hasn't. Hilbert jumped to conclusions based on a hasty generalization.

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. Physicists do not yet know what homogeneous continua exist.

Sure he was limited by the knowledge of his day but his thinking is right on.

This isn't a matter of his being limited by the knowledge of his day. He wasn't even current. His thinking was not "right on": I have just exhibited two obvious logic errors in his thinking. Unfortunately, brilliant people very often make elementary mistakes when they step outside their fields of expertise and try to draw conclusions about other people's fields of expertise. Hilbert was a great mathematician but he was not thinking competently about physics, even 1925 physics.

Note that this is not a claim that his conclusion is wrong; we don't know if the conclusion was wrong. This is a claim that his conclusion was unsupported because the reasoning he used to reach it was wrong.

 

[Kharakov]

We have established that the universe is finite in two respects, i.e., as regards the infinitely small and the infinitely large.

That one is just asinine.

I had a diatribe about Hilbert's claims being ignorant, and the paper being the product of a circle jerk session of like minded, rest on their Laurel and Hardee har har types, but it didn't make the cut. sigh.

Moreover, astronomy progresses. "All the results of astronomy" are no longer perfectly compatible with the postulate that the universe is elliptical. The recently discovered acceleration of the expansion of the universe favors the hyperbolic hypothesis over the elliptical hypothesis.

Not only that, but astronomical observations of distant GRBs have allegedly indicated that if space is granular, it is so at scales at least 10^11 times smaller than the Planck length.

It seems that looking outwards has shown 2 things about the universe. Although one could just as easily look inwards at the continuity of experience to know that the continuum is smooth (if one doesn't devote one's mind to splitting hairs, which leads to baldness, because follicle damage).

 

[Phil Scott]

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?

If you do, that's the finitist programme, and Hilbert put his money where his mouth was and started working on the concrete problems that would become that programme.

For me, I'll only judge this programme based on its fruits, not its motivating philosophical arguments. I couldn't care less at that point if they turn out to be asinine.

I feel the same way about Brouwer and intuitionism, which I've already largely judged to be awesome, so haven't felt much need to read the original philosophical motivation.

 

[Bomb#20]

But what quanta of energy did Planck actually discover? Planck discovered energy comes in discrete multiples of Planck's Constant times the frequency. But that would only prove energy does not admit of infinite divisibility if it were the case that frequency does not admit of infinite divisibility -- hence Hilbert's critical insertion of the word "unconditionally". There had been no demonstration that frequency isn't infinitely divisible; 90 years on, there still hasn't. Hilbert jumped to conclusions based on a hasty generalization.

Right, but there I think a conventional infinite would suffice, i.e. unbounded, rather than some newly-minted notion like an actual infinity, with something like an actual point at the infinite.
EB

I think you're using "unbounded" in a different sense from what Hilbert meant by it. He was talking about actual infinities even though he meant infinities that don't have a "point at the infinite". (See our other discussion.)

But when he talks of space being unbounded, what he means is it doesn't have a boundary. This is a hard concept for humans to visualize in three dimensions so it's customary to drop down to two dimensions for an analogy. Imagine that we're 2-D organisms living on the surface of a sphere. We could travel anywhere in our universe and never encounter a boundary -- we'd never bump into a wall we couldn't get past -- but since the space we live in curves back on itself in this scenario, if we go far enough in one direction we'd find ourselves back where we started. So the total amount of space is finite even though it's "unbounded" in the no-boundary sense -- the same way a person could go east on the Earth potentially forever even though its surface contains only 200 million square miles. The 2-D organisms could never leave the surface and view their spherical universe from outside, so they'd think it was infinite unless they made careful measurements and detected the curvature.

Hilbert believed that's how our universe works, only with one more dimension -- he believed the curvature Einstein discovered meant that space curves back on itself, it contains only finitely many cubic light years, and if we could go enough billion light years to the galactic north we'd come back to home from the galactic south. But he was jumping to conclusions. There's more than one way to curve, and space can curve without closing back on itself.

 

[beero1000]

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.

That actually makes a lot of sense. Someone of Hilbert's caliber wouldn't seriously make the mistakes of 'proof by example' that appear in that talk. Although, IIRC, this was right around the time he became seriously ill and his productivity declined.

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?

If you do, that's the finitist programme, and Hilbert put his money where his mouth was and started working on the concrete problems that would become that programme.

For me, I'll only judge this programme based on its fruits, not its motivating philosophical arguments. I couldn't care less at that point if they turn out to be asinine.

Hilbert's original finitist programme didn't survive after the incompleteness theorems came out showing it was impossible, right? I'm not very familiar with the results after that, was enough of it salvaged to count as successful 'fruits'?

I feel the same way about Brouwer and intuitionism, which I've already largely judged to be awesome, so haven't felt much need to read the original philosophical motivation.

I'd be interested in reading your opinions on constructivism. We should have a math subforum soon too!

 

[Phil Scott]

Hilbert's original finitist programme didn't survive after the incompleteness theorems came out showing it was impossible, right? I'm not very familiar with the results after that, was enough of it salvaged to count as successful 'fruits'?

The hope of a relative consistency proof of set theory within primitive recursive arithmetic is scuppered decisively by Goedel's second theorem, so you have to backslide from that lofty goal. But the line is that a surprising amount of mathematics still goes through finitistically. Here's a discussion about very recent results.

I'd be interested in reading your opinions on constructivism. We should have a math subforum soon too!

Would love to share them!

 

[Speakpigeon]

But what quanta of energy did Planck actually discover? Planck discovered energy comes in discrete multiples of Planck's Constant times the frequency. But that would only prove energy does not admit of infinite divisibility if it were the case that frequency does not admit of infinite divisibility -- hence Hilbert's critical insertion of the word "unconditionally". There had been no demonstration that frequency isn't infinitely divisible; 90 years on, there still hasn't. Hilbert jumped to conclusions based on a hasty generalization.

Right, but there I think a conventional infinite would suffice, i.e. unbounded, rather than some newly-minted notion like an actual infinity, with something like an actual point at the infinite.
EB

I think you're using "unbounded" in a different sense from what Hilbert meant by it. He was talking about actual infinities even though he meant infinities that don't have a "point at the infinite". (See our other discussion.)

But when he talks of space being unbounded, what he means is it doesn't have a boundary. This is a hard concept for humans to visualize in three dimensions so it's customary to drop down to two dimensions for an analogy. Imagine that we're 2-D organisms living on the surface of a sphere. We could travel anywhere in our universe and never encounter a boundary -- we'd never bump into a wall we couldn't get past -- but since the space we live in curves back on itself in this scenario, if we go far enough in one direction we'd find ourselves back where we started. So the total amount of space is finite even though it's "unbounded" in the no-boundary sense -- the same way a person could go east on the Earth potentially forever even though its surface contains only 200 million square miles. The 2-D organisms could never leave the surface and view their spherical universe from outside, so they'd think it was infinite unless they made careful measurements and detected the curvature.

Hilbert believed that's how our universe works, only with one more dimension -- he believed the curvature Einstein discovered meant that space curves back on itself, it contains only finitely many cubic light years, and if we could go enough billion light years to the galactic north we'd come back to home from the galactic south. But he was jumping to conclusions. There's more than one way to curve, and space can curve without closing back on itself.

Right, yes, I wasn't addressing Hilbert's views on infinity, and I also wasn't thinking outside a flat geometry, so "unbounded" only referred to the basic notion of infinity.

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

 

[beero1000]

Right, yes, I wasn't addressing Hilbert's views on infinity, and I also wasn't thinking outside a flat geometry, so "unbounded" only referred to the basic notion of infinity.

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

FYI, you can have a flat universe that is finite and has no boundary - think about Pac-Man's universe. There was a post on this a while back in the science forum where I went through possible candidates for 3D space. I can dig it up if you want.

 

[Speakpigeon]

Right, yes, I wasn't addressing Hilbert's views on infinity, and I also wasn't thinking outside a flat geometry, so "unbounded" only referred to the basic notion of infinity.

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

FYI, you can have a flat universe that is finite and has no boundary - think about Pac-Man's universe. There was a post on this a while back in the science forum where I went through possible candidates for 3D space. I can dig it up if you want.

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

 

[untermensche]

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.