Numbers

[beero1000]

Luckily for all of us, physicists are much smarter than you are, and they actually thought through the consequences of the different possibilities. They figured out that the potential discretized spacetimes imply a violation of Lorentz invariance, which in turn implies consequence that would be observable with today's technology. Specifically, the divergence from predicted results in vacuum Cherenkov radiation, particle decay, gamma ray bursts, etc, would all show as consequences of there being a 'smallest possible volume of space'. We've checked dozens of these as precisely as is currently possible (and that is VERY precisely, down to well below the Planck scale), and the consequences do not appear. You are wrong.

Correct me if I'm wrong, but don't the presently measurable statistical consequences of discretization you're talking about scale with the size of the hypothetical spacetime quanta? So the nonappearance of these consequences can't ever rule out spacetime quanta, but can only put an upper bound on their size? For the sake of definiteness I used the Planck volume in my posts, and I take it that was way too big; but couldn't untermensche simply always postulate grains of unknown size far smaller than whatever the demonstrable upper bound is? This would make the spaceless interior of each quantum the Gap of the Gaps.

In a way, kind of, but in a sense it's also worse than just a single quantum 'god of the gaps' argument. You need two distinct steps, at least, and it's the second one that's harder to plausibly satisfy.

Discreteness is forbidden (even at arbitrarily small lattice sizes) in the standard model, due to the structure of the Lorentz group and how it would act on a discrete quantum spacetime lattice. So, the only way to show that space is discrete is first to show that the standard model is wrong. The negative experimental results for Lorentz violations show that we haven't found any issues with the standard model, so while you could 'god of the gaps' posit an issue outside our realm of measurement (IIRC, it's dozens of orders of magnitude smaller than the Planck scale, but that's at least a plausible experimental result), you're still facing the (worse) second issue -- that doesn't mean that spacetime is discrete, it just means that the standard model is wrong.

We'd need to then to replace it with a theory of discrete quantum spacetime and, importantly, there is no currently known discrete theory that could be expected to work to replace it. None of the ones that physicists have come up with in the last hundred years or so have survived experimental tests (if they even make any testable predictions at all). They just don't work -- and it's the discreteness itself that always seems to be the culprit.

So, in order for untermensche to be right, the single most supported theory of all time needs to be wrong, and then even if the standard model is wrong, it still needs to be replaced with a theory of discrete quantum spacetime when there are no solid theories of discrete quantum spacetime that are real contenders to replace it. The first is just an experimental result away, but the second needs a major theoretical breakthrough that has eluded the best physicists on the planet for a good long while.

 

[Bomb#20]

When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is this a possible scenario?

Observer 1's partition: |_____|_____|
Observer 2's partition: |___|___|___|​

Yes or no?

You are showing me some lines.

Not a real division of anything.

Sorry to be unclear. When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is it possible for it to be divided into different numbers of quanta for them?

 

[untermensche]

When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is this a possible scenario?

Observer 1's partition: |_____|_____|
Observer 2's partition: |___|___|___|​

Yes or no?

You are showing me some lines.

Not a real division of anything.

Sorry to be unclear. When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is it possible for it to be divided into different numbers of quanta for them?

That is ultimately an empirical question.

But if something is one quanta and indivisible it will be one quanta and indivisible for all observers.

But the question is: Will humans ever be able to accurately examine something so small?

 

[fromderinside]

I believe current theory leans to quanta dissipating. Process may be gradual. Parallel to gradual quantum dissipation is field amplitude decreases to zero.

Curious does something cease to exist if humans can't measure it?

 

[untermensche]

Can something make an infinitely small movement?

How far did it move?

 

[Speakpigeon]

Can something make an infinitely small movement?

How far did it move?

If it was an empirical question, then, clearly, we couldn't measure a smallest movement. But I will guess yours isn't an empirical question but a logical one.

Me, I can't see any logical problem with the notion that whatever distance you may want to pick something could move that much. For that to be physically possible, we would need a universe where the laws of nature would allow it, in particular that time and space should be continuous, and therefore infinitely divisible. Any movement would then be an infinite succession of infinitely small movements.

I don't think we could know that time and space would be continuous, so again, there's no empirical confirmation to be expected, but on the logical side of things, I think it works. I would even say that it's a rather trivial problem. It sort of befuddles me that you should be stumped by such a straightforward question.
EB

 

[untermensche]

If it was an empirical question..

I said measuring a quanta of space is an empirical question.

Understanding that there is no such thing as an infinitely small movement or the theoretically smallest movement is a logical question.

 

[bigfield]

the simplest function x ↦ x and the simplest applications (x ↦ x) (x ↦ x) = (x ↦ x).

Took me a little while to figure out the notation: the second (x ↦ x) on the left is the argument?

 

[Speakpigeon]

If it was an empirical question..

I said measuring a quanta of space is an empirical question.

Understanding that there is no such thing as an infinitely small movement or the theoretically smallest movement is a logical question.

No. You didn't say that in the post I was commenting on. You said something very different:

Can something make an infinitely small movement?

How far did it move?

Two questions. Empirical? Logical?

Or just you trying to fudge the issue by confusing these two options?

It's up to you to clarify but you're clearly unable to do that.

That measuring is an empirical question goes without saying, so you're not clarifying anything by insisting on that.

Same for understanding as a logical question in this case.

Never mind, I addressed both alternatives.

I just explained why I think you're wrong on the logical question and, no surprise, you have no argument on that.

And I explained under which conditions can something make an infinitely small movement, and that even then, we wouldn't be able to measure it.

You have no argument.
EB

 

[bigfield]

the simplest function x ↦ x and the simplest applications (x ↦ x) (x ↦ x) = (x ↦ x).

Took me a little while to figure out the notation: the second (x ↦ x) on the left is the argument?

NVM--Answered my own question by reading wikipedia.

 

[Bomb#20]

Sorry to be unclear. When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is it possible for it to be divided into different numbers of quanta for them?

That is ultimately an empirical question.

But if something is one quanta and indivisible it will be one quanta and indivisible for all observers.

In which case the observers can compare notes, and the one who measures it to be less foreshortened must be moving slower. From this they can work out an absolute rest frame. Relativity asserts no absolute rest frame exists; so if your postulate is right then relativity must be wrong. Until you provide an equally accurate alternative theory, given relativity's predictive success, claiming it's wrong is an extraordinary assertion. You need extraordinary evidence. Philosophical objections to a space time continuum don't qualify as extraordinary evidence.

But the question is: Will humans ever be able to accurately examine something so small?

Probably not; but then if a quantum is too small to measure, you're probably not going to be able to produce evidence that it exists.

 

[Phil Scott]

the simplest function x ↦ x and the simplest applications (x ↦ x) (x ↦ x) = (x ↦ x).

Took me a little while to figure out the notation: the second (x ↦ x) on the left is the argument?

NVM--Answered my own question by reading wikipedia.

Ah yeah, sorry if that wasn't clear. Normally, in maths, you write a function application as f(x) where the "f" is always the name of a function. Here, we're more flexible, and allow "f" to be anything that denotes a function, be it a name, a variable, a function literal, or a complex expression involving function literals.

 

[bigfield]

NVM--Answered my own question by reading wikipedia.

Ah yeah, sorry if that wasn't clear. Normally, in maths, you write a function application as f(x) where the "f" is always the name of a function. Here, we're more flexible, and allow "f" to be anything that denotes a function, be it a name, a variable, a function literal, or a complex expression involving function literals.

I'd never made the connection that the notation for a mathematical function is the same as the function definition/call syntax in many languages. Code syntax often seems weird but now I see that lambda functions, arrow functions and function calls are all inspired by mathematical notation.

 

[untermensche]

In which case the observers can compare notes, and the one who measures it to be less foreshortened must be moving slower. From this they can work out an absolute rest frame. Relativity asserts no absolute rest frame exists; so if your postulate is right then relativity must be wrong. Until you provide an equally accurate alternative theory, given relativity's predictive success, claiming it's wrong is an extraordinary assertion. You need extraordinary evidence. Philosophical objections to a space time continuum don't qualify as extraordinary evidence.

But the question is: Will humans ever be able to accurately examine something so small?

Probably not; but then if a quantum is too small to measure, you're probably not going to be able to produce evidence that it exists.

You have jumped to about ten bad conclusions.

We are talking about the nature of measurement.

All are subjective.

If we follow your logic we conclude nothing exists.

 

[Kharakov]

NVM--Answered my own question by reading wikipedia.

Ah yeah, sorry if that wasn't clear. Normally, in maths, you write a function application as f(x) where the "f" is always the name of a function. Here, we're more flexible, and allow "f" to be anything that denotes a function, be it a name, a variable, a function literal, or a complex expression involving function literals.

I'd never made the connection that the notation for a mathematical function is the same as the function definition/call syntax in many languages. Code syntax often seems weird but now I see that lambda functions, arrow functions and function calls are all inspired by mathematical notation.

Is this a more general way of looking at things, so you can understand functions under other syntactic/semantic environments?

I don't see its use, other than that. Reminds me of Google Translate's "interior language" that it uses to translate between one language and another.

 

[bigfield]

I'd never made the connection that the notation for a mathematical function is the same as the function definition/call syntax in many languages. Code syntax often seems weird but now I see that lambda functions, arrow functions and function calls are all inspired by mathematical notation.

Is this a more general way of looking at things, so you can understand functions under other syntactic/semantic environments?

I don't see its use, other than that. Reminds me of Google Translate's "interior language" that it uses to translate between one language and another.

I'm just making connections between languages that I'd previously learned in isolation from one another. It makes it easier to learn new things when I understand the interconnectedness of the stuff I already know.

 

[Phil Scott]

I'd never made the connection that the notation for a mathematical function is the same as the function definition/call syntax in many languages. Code syntax often seems weird but now I see that lambda functions, arrow functions and function calls are all inspired by mathematical notation.

Is this a more general way of looking at things, so you can understand functions under other syntactic/semantic environments?

I don't see its use, other than that. Reminds me of Google Translate's "interior language" that it uses to translate between one language and another.

I'm just making connections between languages that I'd previously learned in isolation from one another. It makes it easier to learn new things when I understand the interconnectedness of the stuff I already know.

It's more than a connection. Church was trying to come up with a new foundations for mathematics, and decided that set theory was inauthentic, and that the true primitive concept in maths was the function, not the set. His calculus turned out to contain a Turing complete fragment, and is still massively influential. Programming languages such as ML and Haskell are ultimately just extensions of Church's simple type theory, which is also still used as the basis for grand projects to formalise maths. In the actual semantics of lambda calculus, lambda abstractions are precisely mathematical functions.

None of this stuff is particularly coincidental. Folk like Church and Turing were mathematicians, after all.

 

[Kharakov]

I'm just making connections between languages that I'd previously learned in isolation from one another. It makes it easier to learn new things when I understand the interconnectedness of the stuff I already know.

It's more than a connection. Church was trying to come up with a new foundations for mathematics, and decided that set theory was inauthentic,

Inauthentic?

and that the true primitive concept in maths was the function, not the set.

What does it say about me that I like this?

 

[Phil Scott]

Inauthentic?

The exact word he used to describe Russell's type theory was "artificial", though it might be that he was specifically criticising the need for types to save higher-order logic, something he had to back down on when he used basically the same type theory to save lambda calculus. This was in his original paper introducing the calculus, which is ultimately pretty light on philosophising and heavy on the technical stuff. Hopefully, there's correspondence out there where he talks more about the motivations, but I confess to not having read much.

and that the true primitive concept in maths was the function, not the set.

What does it say about me that I like this?

Only good things!

 

[Speakpigeon]

and that the true primitive concept in maths was the function, not the set.

What does it say about me that I like this?

You're all set?

You're set in your ways?
EB