I have zero doubt that symbology is connected to "feel". There's an entire field of academic research on that connection (semiotics). I don't, however, see what "feel" has to do with spatiality. The various layers of abstraction are all dimensionless.

Why insist they're separate layers? Because it seems like they are. Remember the sand grain? It conveys information, but when grouped with a bunch of other grains, the collection conveys even more information. If a finger writes a message in that sand -- a higher level of abstraction -- we get even more information.


Are you're asking if it's the representation -- the symbols -- that gives meaning to an otherwise meaningless hunk of information? I don't think so. Symbols give information "concreteness", that's true, but there are so many examples where the choice of symbols makes no difference whatsoever, that it's difficult to think that the symbol is what matters. For instance, I don't get different meaning from the base-2 representation of a number versus its base-10 representation.

That's not to say that the symbols chosen can't introduce yet another level of abstraction that carries its own level of information. We see this all the time in marketing materials, where things like the choice of font makes a significant difference. But I see this is a meta-layer, not the fundamental thing.

Insist that they’re “separate layers” of what though? More 0D information effectively in the same geo-dimensionless burlap bag of 0-length points?

Re: symbols, I wasn’t making distinction between WHICH symbols, just the fact that 2D (minimum) symbols are utterly necessary, and they too are 0D information(!)

 

Let's run with this for a moment. Every piece of information, including numbers are 0D, yes?

No, not "0D". Information is un-dimensional -- there is no geometry to information. "0D" implies a geometry, e.g., the dimensions of a point.

There are no lines of any length when everything is 0D length, right?

Since you don't seem to grok the state hierarchy, let's try a more concrete approach. What color is the money in your bank account? As you know, your bank does not have a vault of money with your name on it. The money in your account is information, and information has no color.

If you withdraw $20, they hand you a green piece of paper -- a symbol -- that represents some amount of money. Relative to the U.S. dollar, the paper represents a magnitude of twenty. Relative to the euro, the paper represents a magnitude of about 17.8. Depending on whom you ask, the piece of paper has a different currency value, and these values exist in an nth order abstraction level of "money". At this level, we can compare magnitudes of money, but we can't compare magnitudes of length -- money has no length.

We can zoom in some and consider the $20 bill as a discrete physical thing. At this abstraction level, we can compare the piece of paper to other discrete physical things, such as rulers. Using an agreed upon reference system, we find that the piece of paper has a length of 156 mm. If we measure a $1 bill, we find that it has the same magnitude of length. At this level of zoom, the $20 bill and the $1 bill have the same magnitude.

If we zoom way in, just above the ground floor -- the level of fundamental physics -- it becomes much more difficult to talk about the $20 bill as a discrete thing. This is the subatomic level where our macroscopic notions of length and position fail to be meaningful. There are informational degrees of freedom, which we can compare, but there's not enough states-of-states structure to do much else. Put it this way: We can encode messages in sand, because sand has plenty of states-of-states structure. It's much harder, though still possible, to write messages in a grain of sand. But we can't write messages on the ground level floor.

So, where are "lines"? They belong to nth-order abstractions, which have plenty of states-of-states structure for us to compare their magnitudes. Lines do not exist at the ground floor, but neither does money or $20 bills or even the concept of length.

 

How do you figure they are not results?

Anything derived from a process can be termed results. ℝ elements are the results of multiple-level abstract arithmetic computation beyond counting that results in non-integers.

And this here is entirely what I’ve been saying since day 1 in Wiki:

“A model for the real number system consists of a set ℝ, two distinct elements 0 and 1 of ℝ, two binary operations + and × on ℝ (called addition and multiplication, respectively), and a binary relation ≤ on ℝ.”

“A model using the two most elemental numbers, 0 and 1 represented as binary bit elements (compact unary). No advanced frameworks necessary. All non-ℕ numbers and sets are the results of computation on the integers.“ — L. Kronecker II

Even sqrt(-1).

ℝ is not a geometric line unless mapped, sure—but it is an implied conceptual one, if it has any kind of mappable sequentiality.

I’ve been lurking. And there are two repeated posts that bother me.

One, the confusion with the number of digits in a representation with the finite properties of a number. An infinite number of digits does not imply that the number is not finite. Wrong, wrong, wrong, to paraphrase Gomer Pyle.

Secondly, the dependence of your thoughts on a binary number system. Numbers can be represented with an infinite number of bases. The base of the representation does not change the “value” of a number.

 

Since you don't seem to grok the state hierarchy, let's try a more concrete approach. What color is the money in your bank account? As you know, your bank does not have a vault of money with your name on it. The money in your account is information, and information has no color.

If you withdraw $20, they hand you a green piece of paper -- a symbol -- that represents some amount of money. Relative to the U.S. dollar, the paper represents a magnitude of twenty. Relative to the euro, the paper represents a magnitude of about 17.8. Depending on whom you ask, the piece of paper has a different currency value, and these values exist in an nth order abstraction level of "money". At this level, we can compare magnitudes of money, but we can't compare magnitudes of length -- money has no length.

I grokked that before. My issue is, all of that is drawing a strange partiality between representation and information, as an information processor that only knows what information is, though. Representation vs. the information itself is the same to the information processor. Yes?

I have no problem with “state hierarchy.” I have a problem with non-dimensional information being described with something you insist has different “weight” than information itself, in a physical brain doing so, when all you have at your disposal as a brain-based machine is non-dimensional bits using information you got from a physical space you readily describe as having dimension, information about which came from non dimensional photon bits.

 

I’ve been lurking. And there are two repeated posts that bother me.

One, the confusion with the number of digits in a representation with the finite properties of a number. An infinite number of digits does not imply that the number is not finite. Wrong, wrong, wrong, to paraphrase Gomer Pyle.

Secondly, the dependence of your thoughts on a binary number system. Numbers can be represented with an infinite number of bases. The base of the representation does not change the “value” of a number.

Thanks for your post!

One: the number of digits in a number describes how finite it is when employed. It takes more digits to describe this property. To claim a number is finite that has essentially a “dial-in knob” of finitude, should not be semantically labeled “finite” in my estimation. 3 has no “dial”. Pi does.

Two: numbers can be represented with any base, sure. The base does not change the value, correct. But notice you are using “representation” as a form of information independent of information itself. If you are just a brain or machine (which is being discussed here), all is information.

 

See how this rubs you as a starting point (bare in mind, some of that is probably explanatory text included within the axioms for now):

Axiom 1: All information, conveyed by humans using principally words, has no geometric dimension, and is classified by a human as signal or noise.

Axiom 2: Information exists as a phenomenon independently of how it is represented using visual or audible symbols. Representation, though information itself, is being defined as meta-information here for clarity, which is geometrically 1D (point) or 2D (lines). Meta-information is the basis of knowing what information is, because it cannot be cognized or conveyed without it. Meta-information has shape.

Examples of meta-information include audible words and written geometric symbols (glyphs). Physical objects themselves can be used as well. Meta-information implies dimension.

Axiom 3: Physical space is not information and is the space humans are born into.

Axiom 4: A thing is an object in physical space, and is also not information. It is described using meta-information to define its informational qualia properties (size, color, texture, sound etc.). Visible light, through photons each having non-dimensional information per photon, is the principal carrier of information concerning physical space and objects therein.

Axiom 5: The human brain only stores and computes with non-dimensional information. Per axiom 2, 2D meta-information can not exist within it without being converted to a non-dimensional state.

Axiom 6: A human uses words to describe things in physical space. One does not know what information means until it is mapped to things in physical space, as learned by every healthy human child.

Axiom 7: By axiom 6, knowing and meaning exists only when information is mapped to things.

Axiom 8: Dimensional meta-information must be a metaphysically innate phenomenon mapping physical objects to non-dimensional information within the human being and not itself residing in the non-dimensional information-carrying brain.

Theorem:
If non-dimensional information per axiom 1 is being used by dimensional meta-information to describe things using informational tokens of length, width, and height, and one does not know what these terms mean until mapped to physical space by way of light carrying non-dimensional information, physical space and things in it must have dimension as an innate property.

 

Until one makes a delineation between information, the "information about the information" (its "representation") and ("physical space" — "something else"), the pursuit is futile.

I have. The states-of-states hierarchy delineates information based on its level in the hierarchy.

So if I said "such and such is not REAL" but was using the term with respect to physical space, where EVERYONE ELSE inherently uses the term effortlessly and intrinsically, well then "she doesn't understand what she's talking about." No, she understands fully. The mathematics discipline does not understand this clearly, and theories can be built with misnamed things that are "self-consistent" but can also be ontologically blind. Why does that matter? It matters if you believe math's primary PURPOSE is to describe the world in which we live.

You assume "everyone else's" use of "REAL" will effortlessly hold up to scrutiny, but that is clearly not the case. Most people don't give it any thought at all. The history of philosophy is littered with serious thinkers unfruitfully debating the nature of "REAL", and yet you expect some commonsense, unanalyzed notion of "REAL" to be the rigid mast upon which to hoist your ontological sail. Good luck with that!

As for the purpose of mathematics, I cannot speak. But I don know that for most professional mathematics, math is its own purpose. This has been the case for hundreds of years. By and large, mathematicians do not care about its applicability to the world in which we live. G.H. Hardy is the exemplar. Physicists have learned that wordly insights can be gained from exploring the non-worldly creations of mathematicians, but for most mathematicians, such insights are merely a curiosity.

You also have said Euclid's "point" definition sucks. These are considered staples in mathematics, so you are not at odds with the above.

I am certainly at odds with the above, and it has nothing to do with Euclid. (BTW, it's pretty annoying when you proclaim what is or isn't a staple in contemporary mathematics. You don't have enough experience with contemporary mathematics to make such claims. I barely do.)

So when you make a QED out of certain terminology, and it's not necessarily ontologically applicable, the QED is NOT the full story. It's "self-contained consistent," but it is NOT the full scope.

When I write a proof in this thread, I do so to show the necessity of my conclusion, or the impossibility of yours. In other words, I'm not just writing proofs to write proofs, I'm using them to resolve our disagreements. I well know that a mathematical or logical proof is not the "full scope", but when you introduce a formal concept -- such as ℝ -- into the discussion, I will analyze it in the most appropriate manner, i.e., within the formal system in which it is defined.

You seem to have the idea that you can pick and choose "parts" of a formal system, use it as a foundation for your ontology, and discard the rest of the formal stuff that necessarily comes with it. It doesn't work like that. The instant you introduce Pi or the unboundedness of ℕ, you've brought in the entirety of their corresponding theories, because those things do not have any MEANING without the theory.

For example, what if space is NOT bent at all, and gravity is NOT an effect ontologically.

More "what if" questions. What's the point? There is an endless stream of "what if" questions -- they add nothing to the conversation. "What if space were a toroid?" "What if space were a 1D line?" "What if space were only in our minds?" If you think a hypothesis is worth exploring, then explore it!

GR doesn't say that space is "bent"; it says that a geometrical model of the universe is necessarily a curved geometry. There is a sh!t ton of scientific research that corroborates this model, so if you disagree with it, you have a LOT of work to do to explain all the phenomena accounted for by GR that Newton's theory cannot explain. It's not enough to ask "What if?" -- you have to offer an argument in order for me to consider questioning my belief in GR. Otherwise, there's no point in bringing up GR.

I have issues with the use of the term "infinite." To call pi a "finite number" mathematically, even if "true," is a "REAL" criminal "point" in full effect.

I understand that your conception of Pi gives you cognitive dissonance. As someone who used to have a similar dissonance, I am suggesting that the way out is to learn more math.

In the context of mathematical frameworks, "pi" is a "finite number, a point on a number line." Sure, got it. I understand all about it. But I can grok instantly that this is linguistically akin to having a child who was learning violin, and she can play one Mozart minuet and one Bach in two keys memorized, and the parent just went ahead and bragged about her being a "musician."

I don't understand the analogy. The parent prematurely calls the child a musician, a fairly normal behavior of boastful parents. What does that have to do with Pi? In any case, I'm hoping that you have revealed a crucial problem with your internal conception of Pi. Your use of "linguistically" suggests that you might be trying to conceive of Pi and ℝ and such within a natural language. This is a doomed plan of attack!

Natural languages (such as English) are woefully unsuited for expressing mathematical concepts. Sure, we can translate basic ideas fairly easily, but as the mathematical structures of study start stacking up on each other, it becomes increasingly difficult to "plainly say" what a mathematical object is. For example, I don't believe I could explain what a "sheaf" or a "tangent vector" is in plain English. In the definitions of each, there are too many references to other mathematical objects to find adequate English representation. Yet, mention "sheaf" to any algebraic geometer and she'll instantly know exactly what you're talking about, as if you had mentioned "car" or "dog".

I acknowledge that it's weird and even uncomfortable at first to divorce mathematical ideas from linguistic constructs, but it's an essential step toward understanding the highly abstract concepts in math. I thought I should mention that.

As for the rest of your rant, give me some meat. Don't just tell me that ℝ and such are bullsh!t, show me with logic.

 

The moment you try to describe, using words, a "hierarchy" or "nested form" to non-dimensional states (information), you are using dimension-indicating meta-information to do it. You want to say states exist within states, but you can't do that without Dimensionalizer v1.0 running on your desktop.

My abiding issue since page 4.

 

How do you figure they are not results?

Anything derived from a process can be termed results. ℝ elements are the results of multiple-level abstract arithmetic computation beyond counting that results in non-integers.

The digits of a real number are derived from a process. The result of that process is a sequence of digits. The number represented by those digits is independent of both the process and the digits.

If, as you say, a number is the process (or its result), then each number would have a unique process (or result). But very clearly this is not the case. There are innumerable ways to express any number, each with its own different process.

Therefore, a number is neither a process nor a result.

 

As young children, we first see, feel, and draw shapes in physical space before there is any further abstract informational characterization of any kind.

So? Are you arguing that young children are better at characterizing shapes than mathematics?

 

Insist that they’re “separate layers” of what though? More 0D information effectively in the same geo-dimensionless burlap bag of 0-length points?

Information is not "0D". If you insist on categorizing it as such, then you need to provide a complete geometry for information. You can't pick one part of geometry -- dimension -- apply it to non-geometrical things and expect anyone to know what you're talking about.

I don't think you understand how empty your "0D" protests are to me. I don't have any idea of what wrong you're trying to right until you're more precise with your language.

 

Information is not "0D". If you insist on categorizing it as such, then you need to provide a complete geometry for information. You can't pick one part of geometry -- dimension -- apply it to non-geometrical things and expect anyone to know what you're talking about.

I don't think you understand how empty your "0D" protests are to me. I don't have any idea of what wrong you're trying to right until you're more precise with your language.

Yes, you’re correct, bot 0D... I changed it to say “no dimension” not “0 dimension.”

 

Thanks for your post!

One: the number of digits in a number describes how finite it is when employed. It takes more digits to describe this property. To claim a number is finite that has essentially a “dial-in knob” of finitude, should not be semantically labeled “finite” in my estimation. 3 has no “dial”. Pi does.

Two: numbers can be represented with any base, sure. The base does not change the value, correct. But notice you are using “representation” as a form of information independent of information itself. If you are just a brain or machine (which is being discussed here), all is information.

What is a “dial-in knob”? Your response does not make any sense without a definition of that phrase. “3” and “π” are both finite. I don’t understand your distinction.

 

The digits of a real number are derived from a process. The result of that process is a sequence of digits. The number represented by those digits is independent of both the process and the digits.

If, as you say, a number is the process (or its result), then each number would have a unique process (or result). But very clearly this is not the case. There are innumerable ways to express any number, each with its own different process.

Therefore, a number is neither a process nor a result.

I like this, but bare in mind this is incredibly rigid, and everyday use is entirely devoid of such very precise understanding even in higher circles (confirmed). Even most computer scientists and mathematicians would say computers compute using the numbers 0 and 1 without blinking. “0 and 1” stand for logic states, and since computers compute with these things, they yield digits having the same symbology in base 2 which overlap with numbers 0 and 1 empirically.

 

What is a “dial-in knob”? Your response does not make any sense without a definition of that phrase. “3” and “π” are both finite. I don’t understand your distinction.

I was speaking colloquially to illustrate.

3 cannot be expressed digitally with varying degrees of finitude. Pi can.

Someone doing work around the house might make a calculation using:

3.1459 for pi.

Someone at a fabrication facility might make a calculation using:

3.1459265 for pi.

NASA might need:

3.14159265358979 for pi.

All are considered gradations of finiteness for the same “pi”, you can “dial in” with more digits. The difference is how much “resolution” or granularity required for the application. The number 3 does not share that capacity.

This is why, like L. Kronecker, I only acknowledge integers as true numbers, and all else, including pi, are numeric expressions derived from integer computation.

 

I was speaking colloquially to illustrate.

3 cannot be expressed digitally with varying degrees of finitude. Pi can.

Someone doing work around the house might make a calculation using:

3.1459 for pi.

Someone at a fabrication facility might make a calculation using:

3.1459265 for pi.

NASA might need:

3.14159265358979 for pi.

All are considered gradations of finiteness for the same “pi”, you can “dial in” with more digits. The difference is how much “resolution” or granularity required for the application.

This is why, like L. Kronecker, I only acknowledge integers as true numbers, and all else, including pi, are numeric expressions derived from integer computation.

And this illustrates my primary issue with your arguments. You interchange representation with value. They are not interchangeable. π can only be represented by an approximation. In Engineering, this is sufficient. But by using a representation of an approximation, does not change that π is a single point on a line.

 

I was speaking colloquially to illustrate.

3 cannot be expressed digitally with varying degrees of finitude. Pi can.

Someone doing work around the house might make a calculation using:

3.1459 for pi.

Someone at a fabrication facility might make a calculation using:

3.1459265 for pi.

NASA might need:

3.14159265358979 for pi.

All are considered gradations of finiteness for the same “pi”, you can “dial in” with more digits. The difference is how much “resolution” or granularity required for the application. The number 3 does not share that capacity.

This is why, like L. Kronecker, I only acknowledge integers as true numbers, and all else, including pi, are numeric expressions derived from integer computation.

You might acknowledge integers as true numbers, but if you aren’t wrong, you’re at least on the fringe of understanding.

FYI, my credentials are BSAM Cum Laude

 

And this illustrates my primary issue with your arguments. You interchange representation with value. They are not interchangeable. π can only be represented by an approximation. In Engineering, this is sufficient. But by using a representation of an approximation, does not change that π is a single point on a line.

The reason I exchange value with representation is because, ontologically, a non-dimensional-information processor (machine or computer) makes no distinction. Only a living human being seems to. And this is the thrust of my argument, that the very distinction between information and its representation is invalid unless a mechanism can be pinpointed to do so. No material information-processor does such, so the information (value) and its representation (dimensional symbology) remains non-dimensional and there is no hardware level value/representation distinction to it between strings of non-descript bits or unary “u-its” used to describe it.

Also, pi is not a single point on a line unless the construction of the line can be delineated via this value/representation differentiation, and geometry created out of what is non-dimensional phenomenon by default.

 

You might acknowledge integers as true numbers, but if you aren’t wrong, you’re at least on the fringe of understanding.

FYI, my credentials are BSAM Cum Laude

I actually only acknowledge the numbers “0 and 1” as foundational, and the rest are shortcut meta-informational ways of grouping them. These base-2 numbers {0, 1} double as logic states {0, 1}, and every binary computer is empirically able to do any numerical calculation using this bijection. Every element of ℝ is derivable using this overlap in definition, and then we assign nth order symbologies to these strings of binary bits and dub them n-base concatenated integers.

 

There is no intellectual justification to use the tool of information to insist information (value) is separate from its representation, unless representation is separate from information. If it’s not separate, there IS no distinction, as is the case in a computational machine. QED.

Because representation is the tool used to convey information, and is the foundation of meaning connected to information, a representation mechanism must exist independent of information itself.

This mechanism must possess dimension, since all representation is described by the human as having such (lines, circles, glyphs, etc.). Therefore, the concept of “form and shape” cannot have been learned from non-dimensional bits from photons imparted from light. The ability to cognize or construct a meaningful 2D or 3D spatial image out of non-dimensional information from photons is not a function of any n quantity of information, but an innate higher dimensional meta-information processing capacity of a living human.

It’s not possible to define meta-information as having been learned via light carrying non-dimensional information about objects. The implication here is that information is non-dimensional points, and representation is spatially dimensional, because there is no knowing what representation is until one sees and interacts with dimensional objects in physical space to even ascribe such properties.