Let me start by saying that I don't understand most of Mueckenheim's arguments. Nevertheless, as I see it, one of the ideas that he has been promoting is important and may someday be viewed as a revolutionary idea.

The idea of "dark numbers" is closely related to the idea of observability, and if we start with the idea that the purpose of mathematics is to help us reason about the real (i.e. observable) world, then the idea of observability is essential.

Here's the idea that I have been promoting, that is closely related to "dark" numbers:

It's eminently reasonable to define mathematics to be a formalization of scientific reasoning. And then every abstract concept that is essential to scientific reasoning should be formalized and integrated into mathematics at the foundational level.

Science deals with an observable reality. And furthermore, there is close relationship between science and technology: technology is built upon scientific understandings, and technology is used to build laboratory instruments (e.g. microscopes) that let us peer more deeply into the observable reality that science seeks to understand.

For mathematics, the laboratory instrument is the computer. The computer helps us peer deeply into a world of computation, and mathematics can be defined as the study of the compututational phenomena we observe when we think of the computer as a microscope.

Part of our modern understanding of science is that our universe is expanding, which implies that there is an event horizon beyond which we can observe nothing at all. And this idea of an event horizon can be extented into mathemtics.

The laws of physics themselves put a theoretical limit on the power of computers, and it would be eminently reasonable to say that anything (such as numbers) that cannot be observed simply because the laws of physics forbid it, are "dark". And it should be noted that it would be very easy to formally define integers that are so large that they cannot be computed (for example, we could use Ackerman's function, or the Goodstein sequence)

And then, since we are viewing mathematics as a science wherein the abstract principles that are essential for scientific reasoning about the real, observable world should be formalized and integrated into mathematics, we must consider an "event horizon" to be part of mathematical reasoning, which, if I'm not mistaken, is what Mueckenheim's "dark" numbers are all about.

Anyways, to extend this idea further, we should recognize that scientific reasoning is not consistent with Cantor's notion of infinity, but it is consistent with Gauss' notion of infinity (in mathematics, infinity is nothing more that a figure of speech that mathematicians find very useful when reasoning about limits.) I used to think that that is part of Mueckenheim's position, but I'm not longer sure of that.

It's just speculation on my part, but I suspect that the reason that Mueckenheim's arguments are so hard to follow is that he imagines that he is proving to mathematicians that he is doing mathematics when he uses the language of Cantorian set theory to explain his ideas, even when those ideas are not compatible with the logic underlying Cantorian set theory.

David Petry wrote:
>
> Let me start by saying that I don't understand most of Mueckenheim's arguments. Nevertheless, as I see it, one of the ideas that he has been promoting is important and may someday be viewed as a revolutionary idea.
>
> The idea of "dark numbers" is closely related to the idea of observability, and if we start with the idea that the purpose of mathematics is to help us reason about the real (i.e. observable) world, then the idea of observability is essential.
>
> Here's the idea that I have been promoting, that is closely related to "dark" numbers:
>
> It's eminently reasonable to define mathematics to be a formalization of scientific reasoning.

No it isn't. It isn't eminently reasonable to define mathematics in any
way that limits its scope.

> And then every abstract concept that is essential to scientific reasoning should be formalized and integrated into mathematics at the foundational level.
>
> Science deals with an observable reality. And furthermore, there is close relationship between science and technology: technology is built upon scientific understandings, and technology is used to build laboratory instruments (e.g. microscopes) that let us peer more deeply into the observable reality that science seeks to understand.
>
> For mathematics, the laboratory instrument is the computer. The computer helps us peer deeply into a world of computation, and mathematics can be defined as the study of the compututational phenomena we observe when we think of the computer as a microscope.
>
> Part of our modern understanding of science is that our universe is expanding, which implies that there is an event horizon beyond which we can observe nothing at all. And this idea of an event horizon can be extented into mathemtics.
>
> The laws of physics themselves put a theoretical limit on the power of computers, and it would be eminently reasonable to say that anything (such as numbers) that cannot be observed simply because the laws of physics forbid it, are "dark". And it should be noted that it would be very easy to formally define integers that are so large that they cannot be computed (for example, we could use Ackerman's function, or the Goodstein sequence)
>
> And then, since we are viewing mathematics as a science wherein the abstract principles that are essential for scientific reasoning about the real, observable world should be formalized and integrated into mathematics, we must consider an "event horizon" to be part of mathematical reasoning, which, if I'm not mistaken, is what Mueckenheim's "dark" numbers are all about.
>
> Anyways, to extend this idea further, we should recognize that scientific reasoning is not consistent with Cantor's notion of infinity, but it is consistent with Gauss' notion of infinity (in mathematics, infinity is nothing more that a figure of speech that mathematicians find very useful when reasoning about limits.) I used to think that that is part of Mueckenheim's position, but I'm not longer sure of that.
>
> It's just speculation on my part, but I suspect that the reason that Mueckenheim's arguments are so hard to follow is that he imagines that he is proving to mathematicians that he is doing mathematics when he uses the language of Cantorian set theory to explain his ideas, even when those ideas are not compatible with the logic underlying Cantorian set theory.
>

--
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

On Sunday, June 6, 2021 at 9:03:36 AM UTC-7, Peter wrote:
> David Petry wrote:
> >
> > Let me start by saying that I don't understand most of Mueckenheim's arguments. Nevertheless, as I see it, one of the ideas that he has been promoting is important and may someday be viewed as a revolutionary idea.
> >
> > The idea of "dark numbers" is closely related to the idea of observability, and if we start with the idea that the purpose of mathematics is to help us reason about the real (i.e. observable) world, then the idea of observability is essential.
> >
> > Here's the idea that I have been promoting, that is closely related to "dark" numbers:
> >
> > It's eminently reasonable to define mathematics to be a formalization of scientific reasoning.
> No it isn't. It isn't eminently reasonable to define mathematics in any
> way that limits its scope.
> > And then every abstract concept that is essential to scientific reasoning should be formalized and integrated into mathematics at the foundational level.
> >
> > Science deals with an observable reality. And furthermore, there is close relationship between science and technology: technology is built upon scientific understandings, and technology is used to build laboratory instruments (e.g. microscopes) that let us peer more deeply into the observable reality that science seeks to understand.
> >
> > For mathematics, the laboratory instrument is the computer. The computer helps us peer deeply into a world of computation, and mathematics can be defined as the study of the compututational phenomena we observe when we think of the computer as a microscope.
> >
> > Part of our modern understanding of science is that our universe is expanding, which implies that there is an event horizon beyond which we can observe nothing at all. And this idea of an event horizon can be extented into mathemtics.
> >
> > The laws of physics themselves put a theoretical limit on the power of computers, and it would be eminently reasonable to say that anything (such as numbers) that cannot be observed simply because the laws of physics forbid it, are "dark". And it should be noted that it would be very easy to formally define integers that are so large that they cannot be computed (for example, we could use Ackerman's function, or the Goodstein sequence)
> >
> > And then, since we are viewing mathematics as a science wherein the abstract principles that are essential for scientific reasoning about the real, observable world should be formalized and integrated into mathematics, we must consider an "event horizon" to be part of mathematical reasoning, which, if I'm not mistaken, is what Mueckenheim's "dark" numbers are all about..
> >
> > Anyways, to extend this idea further, we should recognize that scientific reasoning is not consistent with Cantor's notion of infinity, but it is consistent with Gauss' notion of infinity (in mathematics, infinity is nothing more that a figure of speech that mathematicians find very useful when reasoning about limits.) I used to think that that is part of Mueckenheim's position, but I'm not longer sure of that.
> >
> > It's just speculation on my part, but I suspect that the reason that Mueckenheim's arguments are so hard to follow is that he imagines that he is proving to mathematicians that he is doing mathematics when he uses the language of Cantorian set theory to explain his ideas, even when those ideas are not compatible with the logic underlying Cantorian set theory.
> >
> --
> The world will little note, nor long remember what we say here
> Abraham Lincoln at Gettysburg

Zero is no quantity to see. It classifies as dark no quantity.
It's purpose is to set bases in quantity representation.

Mitchell Raemsch

On 6/6/2021 11:57 AM, David Petry wrote:

> The idea of "dark numbers" is closely related to the idea of
> observability, and if we start with the idea that the purpose
> of mathematics is to help us reason about the real
> (i.e. observable) world, then the idea of observability is
> essential.

It seems to me (JB) that if you (DP) succeed in your stated
goal of restricting the tools that math and physics have
available, no useful purpose will be served, and several
anti-useful purposes will be served.

| If you could soar high in the sky, as red kites often do
| in search of prey, and look down at the domain of all things
| known and yet to be known, you would see something very
| curious: a vast class of things that science has so far
| neglected. These things are central to our understanding of
| physical reality, both at the everyday level and at the level
| of the most fundamental phenomena in physics -- yet they have
| traditionally been regarded as impossible to incorporate into
| fundamental scientific explanations.They are facts not about
| what is -- the 'actual' -- but about what _could or could not be_
| In order to distinguish them from the actual, they are called
| _counterfactuals_
| -- "Prelude" (p. xv), _The Science of Can and Can't_
_a physicist's journey through the land of counterfactuals_
Chiara Marletto (2021)
https://www.amazon.com/Science-Can-Cant-Physicists-Counterfactuals/dp/0525521925

https://en.wikipedia.org/wiki/Constructor_theory
| | Constructor theory is a proposal for a new mode of explanation
| in fundamental physics, first sketched out by David Deutsch,
| a quantum physicist at the University of Oxford, in 2012.
| Constructor theory expresses physical laws exclusively in
| terms of what physical transformations, or tasks, are possible
| versus which are impossible, and why. By allowing such
| counterfactual statements into fundamental physics, it allows
| new physical laws to be expressed, for instance those of the
| constructor theory of information.

| Quantum mechanics and all other physical theories are claimed
| to be subsidiary theories, and quantum information becomes a
| special case of _superinformation_

I don't have anything to say about constructor theory -- yet --
beyond "This looks promising. Maybe you should look into it."
I am right now looking into it, myself. But, you know,
it _does_ look promising, and maybe you _should_ look into it.

On 6/6/2021 10:57 AM, David Petry wrote:
>
> Let me start by saying that I don't understand most of Mueckenheim's arguments. Nevertheless, as I see it, one of the ideas that he has been promoting is important and may someday be viewed as a revolutionary idea.

you do not understand, you call it revolutionary.

>
> The idea of "dark numbers" is closely related to the idea of observability, and if we start with the idea that the purpose of mathematics is to help us reason about the real (i.e. observable) world, then the idea of observability is essential.
>
> Here's the idea that I have been promoting, that is closely related to "dark" numbers:
>
> It's eminently reasonable to define mathematics to be a formalization of scientific reasoning.

no, it is not reasonable, it is a huge mistake. Math is stand alone.

> And then every abstract concept that is essential to scientific reasoning should be formalized and integrated into mathematics at the foundational level.

that is nonsense, and another mistake in reasoning.

>
> Science deals with an observable reality.

parts of it.

>And furthermore, there is close relationship between science and technology: technology is built upon scientific understandings, and technology is used to build laboratory instruments (e.g. microscopes) that let us peer more deeply into the observable reality that science seeks to understand.
>
> For mathematics, the laboratory instrument is the computer.

no. computer modeling, yes. but math in general, no. another serious
mistake.

3 mistakes, your out.

<snip crap>

>Let me start by saying that I don't understand most of Mueckenheim's arguments. Nevertheless, as I see it, one of the ideas that he has been promoting is important and may someday be viewed as a revolutionary idea.

Given ti is incoherent non-sense, no it won't.

>The idea of "dark numbers" is closely related to the idea of observability, and if we start with the idea that the purpose of mathematics is to help us reason about the real (i.e. observable) world, then the idea of observability is essential.

But that is not the purpose of it so it isn't essential.

>It's eminently reasonable to define mathematics to be a formalization of scientific reasoning. And then every abstract concept that is essential to scientific reasoning should be formalized and integrated into mathematics at the foundational level.

Not really.

>For mathematics, the laboratory instrument is the computer. The computer helps us peer deeply into a world of computation, and mathematics can be defined as the study of the compututational phenomena we observe when we think of the computer as a microscope.

Absolutely not, mathematics do not need a computer, never has, never will. It is a useful tool to reduce human work, but logic and reason is the tool of mathematics.

Mathematics deal with more than what is computatable.

The rest is irrelevant non-sense based in idiotic foundations I pointed out here

david...@gmail.com schrieb am Sonntag, 6. Juni 2021 um 17:57:07 UTC+2:

> The idea of "dark numbers" is closely related to the idea of observability, and if we start with the idea that the purpose of mathematics is to help us reason about the real (i.e. observable) world, then the idea of observability is essential.

So it is. Therefore there will always exist a comparatively small set of named numbers whereas most are inaccessible because of lack of ressources and time.

But that is not the only reason for dark numbers. This can be explained most easily by Zeno's paradox of Achilles and the tortoise:

Achilles and the tortoise run a race. The tortoise gets a start and the race begins (state 0). When Achilles reaches this point, the tortoise has advanced further already (state 1). When Achilles reaches that point, the tortoise has advanced again (state 2). And so on (state 3, 4, 5, ...). Since Achilles runs much faster than the tortoise, he will overtake (state omega), but only after infinitely many finitely indexed states of the described kind.. Their number must be completed. Otherwise Achilles will not overtake. But there must not be a last visible finitely indexed state. (The last state Achilles remembers have indices much smaller than omega.) This can only be realized by means of dark state.

In oder to describe reality, we have to accept that Achilles leaves the set of finitely indexed states when he gets to state omega. But every finite state we can name, has aleph_0 successors. They cannot be named but must be dark, independently of physical ressources.

Regards, WM