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


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.


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.)


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.


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 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.


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.

You have good points above, I will consider that further... I suppose some of the issue is trying to connect math with ontology.

In the case of pi, my thinking is ontologically based with respect to no machine making a distinction between value and representation, as you pointed out above with the 2D array being seeing as a 1D stream of bits (and to the hardware, no-D). So to say there is a “distinction” between info and its representation, when representation IS info, is precisely why I say pi is in state of dynamic flux, chasing higher degrees of finitude, vs. the integer portion of “3” which is static.

 

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?

The "partiality" (do you mean distinction?) comes because the information conveyed by each level of the hierarchy is different from the other levels. You say that an information processor only knows information as if all information is the same. It's not!

We don't need human brains to see this. The network connection on your computer conveys different levels of information through different levels of state hierarchy. At a low physical level, the information conveyed causes various circuits in your computer to change state. This is an electrical language that only has meaning at this particular level. At a higher level, the CPU interprets these circuit states as a signal that follows a communications protocol. This is a transmission language, with its own syntax and semantics that apply only to this level. At an even higher level, the CPU decodes the transmitted message and interprets it in some other language, e.g., as ASCII characters to be printed on a screen, or an HTTP redirect command to fetch another link.

You can argue that none of that would be possible without programming, and I wouldn't disagree in the slightest. But you can't argue that the computer is "impartial" to the various levels of information. If the computer couldn't distinguish between representations on multiple levels, they'd be useless. Certainly, the internet would not be possible!

The universe allows for multiple levels of information, the states-of-states phenomenon. Information processors leverage this capability and organize information hierarchically. What causes biological processors to do this is anyone's guess. I suspect that it has something to do with the universe being maximally stable in a high entropy configuration, but that's speculation. What's indisputable is that information processors work by distinguishing between levels of information. This is as true for a computer as it is for a DNA molecule as it is for human brain.

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 . . .

I don't know what you mean by "weight". An Ethernet frame conveys information, different from the IP datagram contained within the Ethernet frame, different from the TCP packet contained within the IP datagram. States of states.

 

One: the number of digits in a number describes how finite it is when employed.

When you employ numbers, does 4 make more than 3?

So how finite is 1/3 again?

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.

A representation has its own level of information, distinct from the thing it represents. Surely you agree that the string "dog" includes information that is completely distinct from the thing it represents. Concepts such as "English word", "three-letter word", "monosyllabic", etc. apply to the representation strictly as a representation, and not in any way to the thing being represented.

 

The "partiality" (do you mean distinction?) comes because the information conveyed by each level of the hierarchy is different from the other levels. You say that an information processor only knows information as if all information is the same. It's not!

We don't need human brains to see this. The network connection on your computer conveys different levels of information through different levels of state hierarchy. At a low physical level, the information conveyed causes various circuits in your computer to change state. This is an electrical language that only has meaning at this particular level. At a higher level, the CPU interprets these circuit states as a signal that follows a communications protocol. This is a transmission language, with its own syntax and semantics that apply only to this level. At an even higher level, the CPU decodes the transmitted message and interprets it in some other language, e.g., as ASCII characters to be printed on a screen, or an HTTP redirect command to fetch another link.

You can argue that none of that would be possible without programming, and I wouldn't disagree in the slightest. But you can't argue that the computer is "impartial" to the various levels of information. If the computer couldn't distinguish between representations on multiple levels, they'd be useless. Certainly, the internet would not be possible!

The universe allows for multiple levels of information, the states-of-states phenomenon. Information processors leverage this capability and organize information hierarchically. What causes biological processors to do this is anyone's guess. I suspect that it has something to do with the universe being maximally stable in a high entropy configuration, but that's speculation. What's indisputable is that information processors work by distinguishing between levels of information. This is as true for a computer as it is for a DNA molecule as it is for human brain.


I don't know what you mean by "weight". An Ethernet frame conveys information, different from the IP datagram contained within the Ethernet frame, different from the TCP packet contained within the IP datagram. States of states.

See my more recent prior posts on this theme of "no-D information" vs. "dimensional information."

No amount of dimensionless information yields dimension (QED all day for that).

We insist information is separate from its representation because either dimensionality exists as some stand-alone phenomenon, or everything has no dimension.

 

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

It's weird as hell to start a ToE with humans (of all things) as part of the axioms. Plus, I disagree that information is classified as signal or noise -- it's not only arbitrary, it's a false dichotomy. In my job we actually treat noise as signal.

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.

Yuck. Representations and geometry are completely independent concepts. Information, of any kind, is shapeless. Shapes can convey information, but information has no 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.

Certain glyphs may remind of us geometrical shapes, but glyphs are not geometrical shapes. It's a category error to think otherwise. A geometrical shape is an object of a geometry, which is a formal mathematical system. Glyphs are things we draw, and we can't draw geometric shapes. We can draw circle-like shapes, but we can't draw a circle.

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

In other words, physical space is physical space, which is not information. Great, so what's physical space again?

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.

You're not anywhere near the ground floor here. What are things made of? What determines whether a thing is red or blue? If the size of color of the thing changes, is it still the same thing?

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.

Per axiom 2, yuck.

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.

I'm presuming then that only unhealthy human children know what love and justice mean, as those cannot be mapped to anything in physical space.

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

So "confusion" has no meaning, gotcha.

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.

Do we experience metaphysical phenomena in the physical world? If yes, how? If not, then how are we experiencing it?

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.

Given the axioms, I accept the theorem. But I reject the axioms.

 

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.

Prove it. Show me how it is logically necessary for me to include geometric dimension in my description of a hierarchy of states.

 

When you employ numbers, does 4 make more than 3?

So how finite is 1/3 again?

4 is more than 3, and both are finite.

1/3 is .333... and is not in a state of "escalating finitude". 3/10 is the same as 33/100 is the same as 333/1000.... Finite number.

Pi is in a state of dynamic, escalating, finitude — it has different levels that can be "dialed in" based on where that finitude is "rationalized" for calculation. Biiiig diff in my estimation.

A representation has its own level of information, distinct from the thing it represents. Surely you agree that the string "dog" includes information that is completely distinct from the thing it represents. Concepts such as "English word", "three-letter word", "monosyllabic", etc. apply to the representation strictly as a representation, and not in any way to the thing being represented.

But the point is, "dog" is a "non-dimensional" informational utterance representing something that is NOT information, which must rationally be a spatial thing with dimension — we learned the very word in order to map it to a spatial thing. We can now use "dog-like" to describe things with it. It reeks with dimension. You can't know what a dog "is" with NO dimension. Tell me one "representation" element that isn't based on some dimensional spatial thing you learned. "English word" (English == place == England — it gets its meaning from the dimensional physical place of England), "three-letter word" ("letters" are minimally 2D pictographic things), etc.

Every representative "meta-information" element you would attribute some kind of dimension to, since you have to ascribe SOME kind of dimension in physical space to talk about it. Words themselves are largely tokens of dimensional spatiality ascribed to the physical space we are born into.

 

Prove it. Show me how it is logically necessary for me to include geometric dimension in my description of a hierarchy of states.

If you didn't know what anything was in physical space where each thing is minimally 2D, you couldn't describe hierarchy to anyone else! It would have no meaning. "Hierarchy" reeks of positionality.

From the web: What is a hierarchy? "A hierarchy is an arrangement of items in which the items are represented as being "above", "below", or "at the same level as" one another."

Geometry all day long. You can't draw two points without one being above, below, behind another point. Then you draw a line between them.

 

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.

Sigh. Yes, precision of language is "rigid". Everyday use of language is sloppy and vague. It's sloppy to use "0" and "1" as symbols for logic states, and even sloppier to say that a computer computes with these symbols.

Do you want to have an intellectually rigorous conversation, or would you prefer we chat about everyday things in everyday language?

 

It's weird as hell to start a ToE with humans (of all things) as part of the axioms. Plus, I disagree that information is classified as signal or noise -- it's not only arbitrary, it's a false dichotomy. In my job we actually treat noise as signal.


Yuck. Representations and geometry are completely independent concepts. Information, of any kind, is shapeless. Shapes can convey information, but information has no shape.


Certain glyphs may remind of us geometrical shapes, but glyphs are not geometrical shapes. It's a category error to think otherwise. A geometrical shape is an object of a geometry, which is a formal mathematical system. Glyphs are things we draw, and we can't draw geometric shapes. We can draw circle-like shapes, but we can't draw a circle.


In other words, physical space is physical space, which is not information. Great, so what's physical space again?


You're not anywhere near the ground floor here. What are things made of? What determines whether a thing is red or blue? If the size of color of the thing changes, is it still the same thing?


Per axiom 2, yuck.


I'm presuming then that only unhealthy human children know what love and justice mean, as those cannot be mapped to anything in physical space.


So "confusion" has no meaning, gotcha.


Do we experience metaphysical phenomena in the physical world? If yes, how? If not, then how are we experiencing it?


Given the axioms, I accept the theorem. But I reject the axioms.

LOL!

But what are shapes to you if all is no-D information??? Shapes ARE dimensional!

 

Sigh. Yes, precision of language is "rigid". Everyday use of language is sloppy and vague. It's sloppy to use "0" and "1" as symbols for logic states, and even sloppier to say that a computer computes with these symbols.

Do you want to have an intellectually rigorous conversation, or would you prefer we chat about everyday things in everyday language?

Absolutely intellectually rigorous, of course — we're building a ToE here. I'm just saying, that what we "think" is rigorous might need re-evaluation at points lest we "over-rigidize," because some of these things we are BUILDING a definition for, and building that definition might require "universal innate use" as one of its metrics of rigidity.

 

The reason I exchange value with representation is because, ontologically, a non-dimensional-information processor (machine or computer) makes no distinction.

Nonsense. How do you think a compiler works? Every variable is a representation for a value, and the compiler most certainly knows the difference. Indeed, a C compiler understands that there are multiple levels of representation -- pointers to pointers to values, etc. -- and works with them flawlessly.

 

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.

[Emphasis mine] Go ahead and show me the binary string that encodes Chaitin's constant.

 

In my job we actually treat noise as signal.

Other than being information and mathematics itself as humanoid flesh and walking among us, I'm very curious what job Skynet programmed you with when you came through the time displacement wormhole from 2090 — i.e., I'm very curious what do you do with signal vs. noise for your job (if your circuits will permit disclosure)? ;--)

 

Nonsense. How do you think a compiler works? Every variable is a representation for a value, and the compiler most certainly knows the difference. Indeed, a C compiler understands that there are multiple levels of representation -- pointers to pointers to values, etc. -- and works with them flawlessly.

Nope, all of them are remaining as discrete switches in a machine that does not "know" anything, because no one is home to know. “Know” is alll about dimension, at least provisionally. You’re discussing a computer that itself requires dimension to even begin describing. You ascribe artificial connection between non-dimensional states that are, well, non-dimensional. As any non-dimensional information processor, you don't even know the compiler exists.

I defy you to find any one person that will agree with you above. If you do, I will pay you cash.

 

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.

What do you mean by "separate"? Consider a heap of sand; now consider a grain of sand in the heap. Is the grain "separate" from the heap?

What's different about the grain and the heap is the level of abstraction that's used to consider them. And what makes these levels of abstraction possible is the states-of-states phenomenon. The grain has its own state, the heap has a different state. But the heap's state is not just the sum of the states of each grain of sand. By virtue of being a collection of states, the state of the heap has extra degrees of freedom -- it can carry more information than just the sum of individual grains could convey. States of states.

Like the grain and the heap, a representation exists on a higher level of abstraction than the thing it represents. Consequently, a representation has more degrees of freedom than the thing it represents. An information processor can use these extra degrees of freedom to convey more information (e.g., choice of font in a message), or it can use the extra degrees of freedom to hold multiple, equivalent representations (as in choice of base for a number).

The universe provides the degrees of freedom at each level of the states-of-states hierarchy. If nothing else, an information processor can distinguish levels by simply counting degrees of freedom.

 

What do you mean by "separate"? Consider a heap of sand; now consider a grain of sand in the heap. Is the grain "separate" from the heap?

What's different about the grain and the heap is the level of abstraction that's used to consider them. And what makes these levels of abstraction possible is the states-of-states phenomenon. The grain has its own state, the heap has a different state. But the heap's state is not just the sum of the states of each grain of sand. By virtue of being a collection of states, the state of the heap has extra degrees of freedom -- it can carry more information than just the sum of individual grains could convey. States of states.

Like the grain and the heap, a representation exists on a higher level of abstraction than the thing it represents. Consequently, a representation has more degrees of freedom than the thing it represents. An information processor can use these extra degrees of freedom to convey more information (e.g., choice of font in a message), or it can use the extra degrees of freedom to hold multiple, equivalent representations (as in choice of base for a number).

The universe provides the degrees of freedom at each level of the states-of-states hierarchy. If nothing else, an information processor can distinguish levels by simply counting degrees of freedom.

Sand == word describing mystery object in physical space — neither of which is defined! Same for “heap” thereof!

You cannot describe sand without ascribing at least provisional dimension to it using “dimensionless” information!

“There is no such thing as a Euclidean line without metaphysicality!” Said Euclid, Pythagenstein, and Newton! I think these dudes know what they’re talking about!

“All your dimensional base are belong to us. Please insert $.25 dimensional quarters in this dimensional arcade machine to describe such a phenomenon without dimension.” ;—)

 

In the case of pi, my thinking is ontologically based with respect to no machine making a distinction between value and representation, as you pointed out above with the 2D array being seeing as a 1D stream of bits (and to the hardware, no-D).

The entire point of my 2D matrix example was to show that information processors can interpret data using any dimensions they wish. Hopefully, after my latest few points, you can see the distinction.

 

No amount of dimensionless information yields dimension (QED all day for that).

Demonstrably false, which, ironically, was the entire point of what I was saying in the screenshot you took. The compiler treats "maxtrix[1][2][3][4]" as being a 4-dimensional structure, even though it's not physically laid out in four dimensions.

 

Demonstrably false, which, ironically, was the entire point of what I was saying in the screenshot you took. The compiler treats "maxtrix[1][2][3][4]" as being a 4-dimensional structure, even though it's not physically laid out in four dimensions.

What compiler? The one running on the dimensional object in physical space, neither of which we have defined? The one a load of non-dimensional photon bits sent to your informational brain is using words to reflect the dimensionality of?