Boole's work stems directly from trying to delineate the Laws of Thought upon which mathematics is made, and it informs the work of Shannon, et. al. and the entire information age is built on that book and Boolean Algebra, and unless I can truly see practical application otherwise beyond that book, everything else is just Boolesh*t. Nothing more is needed than that, and to me is just convoluting the matter...

Except that Boole's work, though historically important, isn't actually foundational (see Frege for that). What we call "Boolean algebra" is not the haphazard symbol mashing that George Boole wrote about; we associate his name with it as an honorific, but the truth is Boole didn't know what he was doing. He was an early-adopter and simply didn't have the requisite mathematical sophistication to properly treat the subject.

An analogy: George Boole is to logic as the Wright brothers are to jet airplanes. Sure, Wilbur and Orville were historically important to show that it could be done, but no one references the 1903 Wright Flyer when building 747s. Similarly, Boole's contribution was to show that logical propositions could be manipulated algebraically. He wasn't the first to think of the idea, but he was one of the first to publish a palatable demonstration of it. However, the important work -- the foundations upon which we've built pretty much everything -- happened later.

I say all this to nudge you away from Laws of Thought, as I would had you decided to base your theory of medicine on the writings of Hippocrates.

 

1) Original axiom — any material substrate, including the human brain is a bit-processing substrate of n bits.

I would word this differently, but the idea I think is the same. How about:

Anything that can change state (distinct configurations) can process information. A state with \( 2^n \) possible configurations can process n bits of information.

I prefer "state configurations" as it doesn't rely on the potentially infinite-recursion semantics of "substrate".

2) How does the substrate define a number if not by bits?

Reworded: How do states encode numbers if not by bits?

I'd say that the natural numbers can be directly mapped to states. Then, counting is associated with state transitions. Then, the notion of unbounded counting (INFINITE) is simply "starting over" when every possible state transition has been performed.

Let's illustrate this with a toy model, a system with 32 possible state configurations (a 5-bit "brain"). The brain can uses three of its bits to associate the states {000, 001, 010, ...,, 111} with the first seven natural numbers. Counting to seven is simply the state transitions \( 000 \to 001 \to 010 \cdots \to 111 \). The brain has two extra bits, which we can think of as memory. So, once the transition to state 111 happens, the brain "remembers" that 7 was its last number and starts the transitions from 000 to 111 again. Now it has counted up to fifteen, using three "counter bits" and one "memory bit". Unbounded is the notion that with more bits, the counting process can continue.

I think something comparable happens when humans count. We eventually run out of "counter memory" and forget the last number we reached, but we have enough memory to store the idea that if we had more memory, we could count higher. This doesn't take any more memory (doesn't have more information) than the idea that with an unlimited amount of memory, we wouldn't have to stop counting.

3) What is _INFINITY to the substrate in terms of those bits?

The various concepts in _INFINITY are extra-model, though they can certainly be expressed in terms of states -- they are, after all, almost invariably expressed in terms of sets, and we can map sets with states. But that would be both superfluous and exhausting.

 

Questions and issues above I will address, btw, but I feel we periodically we need to defrag the drive of the discourse and get back to the initial axiom, because every sector on the drive is a portal to a RAID 5 of disks.

 

I would word this differently, but the idea I think is the same. How about:

Anything that can change state (distinct configurations) can process information. A state with \( 2^n \) possible configurations can process n bits of information.

I prefer "state configurations" as it doesn't rely on the potentially infinite-recursion semantics of "substrate".

Yes, but because defining the difference between something and nothing is part of this, and these may have 5D definitions, I feel we need a hard physical, natural world palpability here... i.e., any observable “substance” that can do this. Because the point is to start off with a proof there is something more than this going on in the brain. That’s why I liked “substrate” because it makes it tangible(?)

Reworded: How do states encode numbers if not by bits?

I'd say that the natural numbers can be directly mapped to states. Then, counting is associated with state transitions. Then, the notion of unbounded counting (INFINITE) is simply "starting over" when every possible state transition has been performed.

“Encode” speaks to representation. Define speaks to what is. I was careful to use define here for that reason, because _DEFINE is not in the domain of the bits themselves, only digital representation and storage (E.g., smoke signals and morse code are different than what they’re encoding). If you’re cool with define not being part of the brain’s functionality purview, we can use that as part of the “not in the brain” springboard.

The various concepts in _INFINITY are extra-model, though they can certainly be expressed in terms of states -- they are, after all, almost invariably expressed in terms of sets, and we can map sets with states. But that would be both superfluous and exhausting.

“Extra-model” in this case is “extra-brain." The thrust here is to prove the brain is insufficient as an exclusive starting point to create a “theory for everything.” Same issue here, it goes back to _DEFINE.

 

Yes, but because defining the difference between something and nothing is part of this, and these may have 5D definitions, I feel we need a hard physical, natural world palpability here... i.e., any observable “substance” that can do this. Because the point is to start off with a proof there is something more than this going on in the brain. That’s why I liked “substrate” because it makes it tangible(?)

But even the concept of "tangible" is spongy, soft, and fraught with implicit conceptions. The tips of your fingers are not touching the keyboard, at least not in the way we usually think of "touch". To leave the model for a second, the reason my butt doesn't fall through the seat is because the electrons surrounding the atoms that comprise my butt are repulsed by the electrons surrounding the atoms that comprise the chair. At no point do these electrons touch -- in fact, they stay as far away from each as physically possible. This "staying away" is what we feel as "touching", but it's certainly not contact in the conventional sense of the word.

So, what is a substrate or substance? At a closer zoom level, the things that we think of as solid objects are mostly empty space. If we try to pinpoint a defining boundary, we'll find that the closer we zoom in, the fuzzier the boundary becomes. At some point, any notion of boundary completely dissolves. So, what is the substance? Physics tells us that the substance is quantum fields, and everything we experience is a result of quantum field interactions. So, unless we want to go down the long winding road of quantum physics, we should avoid reference to notions of physical "substance".

Tangible is a fourth (or nineteenth) order concept. Let's stick with first or second order concepts for now?

“Encode” speaks to representation. Define speaks to what is. I was careful to use define here for that reason, because _DEFINE is not in the domain of the bits themselves, only digital representation and storage (I.e., smoke signals and morse code are different than what they’re encoding). If you’re cool with define not being part of the brain’s functionality purview, we can use that as part of the “not in the brain” springboard.

A state is a particular configuration of a thing, not the thing itself. We never experience the thing in itself (Kant's ding an sich), we only experience a perception of the thing. I would describe "perception" as the process of mixing the state of the thing we're perceiving with the state of our brain.

A "thing" itself is unknowable; therefore, it's reasonable to leave "thing" out of the model entirely and focus on states. All we're left with is encodings: state configurations that represent unknowable, undefinable things. We can cogently talk about states; we can't cogently talk about unknowable things.

“Extra-model” in this case is “extra-brain”. The thrust here is to prove the brain is insufficient as an exclusive starting point to create a “theory for everything.” Same issue here, it goes back to _DEFINE.

As I've been using the term, "extra-model" means outside our model of information (which, I think, is a fine starting place for a ToE). I think we definitely have the goal of including brains in the model, but also including, say, computer CPUs. I wouldn't call CPUs "extra-model".

I call _INFINITY extra-model because it encompasses highly technical, convoluted topics that feel outside the scope (at least for now, heh) of this discourse. Parsimony is never fun, but it's essential to good model building.

 

But even the concept of "tangible" is spongy, soft, and fraught with implicit conceptions. The tips of your fingers are not touching the keyboard, at least not in the way we usually think of "touch". To leave the model for a second, the reason my butt doesn't fall through the seat is because the electrons surrounding the atoms that comprise my butt are repulsed by the electrons surrounding the atoms that comprise the chair. At no point do these electrons touch -- in fact, they stay as far away from each as physically possible. This "staying away" is what we feel as "touching", but it's certainly not contact in the conventional sense of the word.

So, what is a substrate or substance? At a closer zoom level, the things that we think of as solid objects are mostly empty space. If we try to pinpoint a defining boundary, we'll find that the closer we zoom in, the fuzzier the boundary becomes. At some point, any notion of boundary completely dissolves. So, what is the substance? Physics tells us that the substance is quantum fields, and everything we experience is a result of quantum field interactions. So, unless we want to go down the long winding road of quantum physics, we should avoid reference to notions of physical "substance".

Tangible is a fourth (or nineteenth) order concept. Let's stick with first or second order concepts for now?

Agreed concerning "tangibility" (I was using that in the sense of abstract definability).

Without a word like "substance," definining the opening axiom with the use of "material" becomes problematic. What is the brain made of?" The goal is to use the brain as the premise, since this is our naturalist starting point. This is essential. No matter how we cut the cake, if we zoom to the quantum level, we're still dealing with a "thing" made out of "things" — quarks or what have you. If a brain is on a table and we're discussing it, we are making a differentiation between "not there" vs. "there" — what is this distinguishing element? A "thing" must have an "existential boundary" to it. The model must be anchored to the ontology.

When discussing the n-bit processing of a thing, what term could be used outside substance? A brain is a substance. It's made of smaller substances. You seriously don't think "substance" works?

A state is a particular configuration of a thing, not the thing itself. We never experience the thing in itself (Kant's ding an sich), we only experience a perception of the thing. I would describe "perception" as the process of mixing the state of the thing we're perceiving with the state of our brain.

A "thing" itself is unknowable; therefore, it's reasonable to leave "thing" out of the model entirely and focus on states. All we're left with is encodings: state configurations that represent unknowable, undefinable things. We can cogently talk about states; we can't cogently talk about unknowable things.

Everything above is abstract thought from philosophers, though. There are all sorts of theories and branches to those theories. "We never experience the thing in itself" is Kant's view. There is no proof to this yet. It has some truth potentially, but the point to this model is to not reference existing philosophical notions, and instead, build from the brutally observable, logical inference, ground-up.

I argue we *can* talk cogently about what we "think" is currently unknowable — and that's part of the point here. Once we can prove the brain is NOT where axioms, definability, sense, and reason is happening (to my knowledge, such a proof has NOT been built based on hard obvious deduction), we can build a framework of discourse in a new logic-derived (5D) meta-space.

As I've been using the term, "extra-model" means outside our model of information (which, I think, is a fine starting place for a ToE). I think we definitely have the goal of including brains in the model, but also including, say, computer CPUs. I wouldn't call CPUs "extra-model".
As I've been using the term, "extra-model" means outside our model of information (which, I think, is a fine starting place for a ToE). I think we definitely have the goal of including brains in the model, but also including, say, computer CPUs. I wouldn't call CPUs "extra-model".

I call _INFINITY extra-model because it encompasses highly technical, convoluted topics that feel outside the scope (at least for now, heh) of this discourse. Parsimony is never fun, but it's essential to good model building.

_INFINITY is extra-model? I thought INFINITY was extra model, and _INFINITY was intra-model, and we're defining it as "not a number" as a baseline property?

The parsimony element is utterly KEY to this! The core opening issue here is cordoning everything off from excess noise so we can infer that something "meta" within the being has to be identified and triangulated.

We start with the foundation of the theoretical "limits of material substance" as a discrete bit-processing medium. You wanted n→∞ originally. We agree now that ∞ is not a number, and so even this speaks to the limits of physicality.

What is the theoretical limit to the storage capacity of bits in the material plane?

 

Except that Boole's work, though historically important, isn't actually foundational (see Frege for that). What we call "Boolean algebra" is not the haphazard symbol mashing that George Boole wrote about; we associate his name with it as an honorific, but the truth is Boole didn't know what he was doing. He was an early-adopter and simply didn't have the requisite mathematical sophistication to properly treat the subject.

An analogy: George Boole is to logic as the Wright brothers are to jet airplanes. Sure, Wilbur and Orville were historically important to show that it could be done, but no one references the 1903 Wright Flyer when building 747s. Similarly, Boole's contribution was to show that logical propositions could be manipulated algebraically. He wasn't the first to think of the idea, but he was one of the first to publish a palatable demonstration of it. However, the important work -- the foundations upon which we've built pretty much everything -- happened later.

I say all this to nudge you away from Laws of Thought, as I would had you decided to base your theory of medicine on the writings of Hippocrates.

So I have one thing to say to the above with all due respect and in all levity:



Not to open up a jar of Medusa a la Vodka, but I have to personally say the above is objectively not correct from my analysis and experience — beware the DEBKAC below.

Say what you want, but Claude Shannon built the entire information age on Boole, not Frege. Marginalizing Boole here is to marginalize Shannon's explicit deference and incorporation of his crystallization of algebraic notation in his seminal thesis A Symbolic Analysis of Relay and Switching Circuits that is the foundation of computer science (and dare I say is counter to your philosophical bias against equating numbers and bits).

On page 9 of the thesis:
"We are now in a position to demonstrate the equivalence of this calculus [above] with certain elementary parts of the calculus of propositions. The algebra of logic (1), (2), (3) originated by George Boole, is a symbolic method of investigating logical relationships."

Per Wiki:
"In 1932, Shannon entered the University of Michigan, where he was introduced to the work of George Boole. He graduated in 1936 with two bachelor's degrees: one in electrical engineering and the other in mathematics. In 1936, Shannon began his graduate studies in electrical engineering at MIT, where he worked on Vannevar Bush's differential analyzer, an early analog computer.[8] While studying the complicated ad hoc circuits of this analyzer, Shannon designed switching circuits based on Boole's concepts [NOT Frege's]."

His thesis that changed the entire world is not some sort of "Hippocratic Oath-esque honorific lip-service nod" to Boole. It's the very basis of it. And it's the basis of my approach here as well. I'm sure Shannon was well aware of all of the other names in the metamathematics sphere, and Boole was his choice, and certainly not by some "ignorance" of the other logician figures in the field.

On a personal note: In the preface to the Frege's Begriffsschrift, Frege attacked what he considered to be the false assimilation of logical functions to algebraic operations — and frankly, this is utterly nonsensical in the face of what Shannon literally did with it—because our ability to type to each other on these Boolean-algebra based circuits is proof. Technology today is not built on Fregean precepts, it's built on Boolean ones, and the notion Boole set forth in equating numbers, bits and algebraicizing them is pivotal, and in turn is just as utterly pivotal in furthering a model for reason based on further analysis of precepts he set forth in Laws of Thought.

Schroder, a leading logician in Germany at the time, remarked that Frege:

....had not built further on the work undertaken by Boole and himself. "I consider it a shortcoming," he wrote, "that the book is presented in too isolated a manner and not only seeks no serious connection with achievements that have been made in essentially similar directions (namely those of Boole), but even disregards them entirely." He was convinced that Frege's system "does not differ essentially from Boole's formula language," adding: "With regard to its major content, the Begriffsschrift could actually be considered a transcription of the Boolean formula language. With regard to its appearance, though, the former is different beyond recognition—and not to its advantage." While Schroder granted that Boolean algebra was defective when it came to expressing existential judgments, he saw no major achievement in Frege's quantifier notation. "Frege correctly lays down stipulations that permit him to express such judgments precisely," he wrote, but "the analogous modification or extension can easily be achieved in Boolean notation as well."

That said, there can be positively no nudging of me away from Boole's foundational work in analyzing reason, and:

Let 1 = "A universe of thinkable thoughts", and 0 its complement

...here as the basis of this model, which is to equate numbers, logic, and implicit geometric form. Turing and Church did it on paper, and Shannon's work is the hard, applied, empirical, technological proof that logic states ARE numbers that can be used to do arithmetic on them. 1 is a number and logic state. 0 is a number and a logic state. They are the foundational BITS. We can add them and create unique "supra" logic states called NUMBERS. If they weren't NUMBERS, we couldn't compute with them!! You are not escaping this, or I'm pulling out another CPU and sending you back to 2090 to Cyberdyne for repair!! Lol.

We don't have anything more to use than bits here with the BRAIN as our basis.

"RED" is not on the left side. RED is a 394th order symbology, a UNIVERSE of knowledge, and belongs on the RIGHT side of Boole's Law here. You think the likes of RED can go on the left because you refuse to equate bits with numbers. You might as well put dog-in-light there, when it belongs on the right side! The "dog in the light” is on the right side of this identity, where “a universe of thinkable thoughts” implies knowable geometric elements.

Each number on the left side is being equated to what we need to work on defining further on the RIGHT side, namely _INFINITY and its connection to geometric form and sense (here we can invoke Frege a bit!).

This foundational identity is a LAW of human reason, the very foundation of computer science, and we need to probe the right side of it, not marginalize its creator or its value in any way, shape or form. Because if you reject it, you and I don’t exist enough to have this conversation any more than Mario and Luigi do.



</END DEBKAC RANT>

Now back to your originally scheduled replies...

 

A "thing" itself is unknowable; therefore, it's reasonable to leave "thing" out of the model entirely and focus on states. All we're left with is encodings: state configurations that represent unknowable, undefinable things. We can cogently talk about states; we can't cogently talk about unknowable things.

Also, if this is true, we should it pack up right now then, because the "model" and everything in it is unknowable—a string of nondescript logic states—because we're just DOS prompts with hair and glasses, who have no right to discuss the bit-driven pixels that make up the glyphs that represent us on a screen.

Again, it goes back to the right side of that Boolean Law: If there is no "indivisible THING" on the right side of that identity, there is no such thing as knowledge, and no model to "know" as a "thing" — and maintaining this philosophical bias at this point in the game is the very definition of a religion.

We most certainly can theorize about things, by invoking _INFINITY as the basis of a continuous geometric form in a 5D space, in an originating metaspace within the being. We'd be triangulating, deducing, and actually observing the inner 3D objects through thought experiment approach. This is my entire thrust since day 1 of this.

 

So, here’s what I propose:

1) using bare reasoning power
2) no use of existing number theory outside basic counting and fractions
3) you can count and infer
4) you have access to basic set symbology and relations
5) access to basic boolean algebra and arithmetic
6) higher calculus not necessary until later

Example rough sketch:

Let n represent the max bit utilization and binary state processing limit of the natural world, in particular the human brain substance.

Human brain is a bit-processing substrate of max n-bit length (n does not go to infinity). Brain is incapable of “knowing any form” due to discrete bit processing limitations that have no geometric correlation. It can only encode bits and process their states which are encodings of knowledge, which implies observable forms and their relations as the basis of “knowing what is.” There is no native geometric form in itself or outside itself that it can know exists. CT scans can prove no form exists as it allegedly describes them. No physical device can augment this or endow or be endowed with this capacity.

Let _INFINITY be defined as:

1) the universe exceeding n, which cannot be represented in the brain substance.
2) the property of unboundedness
3) the uncountable number of bits between any x and y that represents an addressable, boundable manifestation of itself

Form is defined as any _INFINITE 2D or 3D indivisible shape. While n bits can create shape in natural world, no number of n-bits can endow a physical substance the ability to know it exists; it is a property of the mind to interpolate the bits of a form in the natural world to the first order one in the mind so as to _KNOW it exists.

If form exists, it therefore must exist in a 5D colocated space. In its space are native tokens that comprise knowability, definability, congenital axioms and _INFINITY with respect to an observer and an external space (_REALITY), which is the basis of knowing scientific truth.

_KNOW is the innate human property of living human that is the universe of _INFINITE geometric, thinkable thoughts outside a human‘s brains and its bits.

_SENSE is the base empirical instrument to observe, compute and base instrument to _KNOW.

_OBSERVER is an extension of space and is “that which knows”

Token “dog” is unknowable by brain substrate, but knowable by _OBSERVER. One must use internal empirical observation to define dog before it can know what one is outside itself.

“Dog” is first defined as an unbroken _INFINITE line comprising its native definition first as a 2D _INFINITE sprite, or a 3D figure.

.....Etc.

The congenital, native scientific axioms of the mind now become REAL, discussable scientific things, and not Harry Potter! The 5D space becomes a triangulated infinite-bit phenomenon where forms exist, and they are knowable via the first order internal “senses.” See where I’m going here if we can build a proof to exit the brain as the starting point??

 

Without a word like "substance," definining the opening axiom with the use of "material" becomes problematic.

Words like "material" and "substance" are bursting with both philosophical and physical preconceptions. You wanted to start at the beginning, a tabula rasa; we can't toss in "substance" and expect it to be meaningful. So, though we could spend days trying to figure out the best way to clean-room define "substance", I suggest we take the expedient path and just drop it from the model. Notice that my replacement "state" is far more generic and flexible. We want the foundational tokens in our model to be generic, flexible, and as free from philosophical-physical-cultural baggage as possible.

What is the brain made of?" The goal is to use the brain as the premise, since this is our naturalist starting point. This is essential. No matter how we cut the cake, if we zoom to the quantum level, we're still dealing with a "thing" made out of "things" — quarks or what have you. If a brain is on a table and we're discussing it, we are making a differentiation between "not there" vs. "there" — what is this distinguishing element? A "thing" must have an "existential boundary" to it. The model must be anchored to the ontology.

Why do we need to care what things are made of? What changes in the model if today we believe that everything is made of "glorp", but tomorrow we find out that everything is made of "tse-doh"?

I'm confused about the "brain as premise" tact. Given the enormous complexity and mystery-shrouded aspects of the brain, shouldn't we build up to it? Starting with human brains as a given feels like the Wright brothers starting with the space shuttle as a given. We're still at the stage of trying to explain kites and you invoke helicopters.

As for the "thing" vs "not thing" distinction, personally I think it's an illusion, a byproduct of the specific evolutionary pathway most organisms have followed. Had we been much smaller, or had more electromagnetic sensitivity, we'd experience a very different world. Accordingly, I would not describe "thing" vs "not thing" as a universal fact. If your ontology assumes it, it must do so explicitly and with the expectation that the assumption needs to be defended.

When discussing the n-bit processing of a thing, what term could be used outside substance? A brain is a substance. It's made of smaller substances. You seriously don't think "substance" works?

I quite seriously dislike "substance". I have suggested "state" as a viable alternative, a first-order abstraction that does not depend on omniscient-like powers to know the fundamental building blocks. Whatever you're thinking of when you use "substance", surely you believe that it has what we can describe as states, yes? You might even say that the state of the substance -- its particular configuration -- is what determines the behavior or properties of the substance. So, if the substance's state is the important aspect -- the thing that influences its environment (and us!) -- then let's focus on the states.

This also aligns with your desire for "thing" vs "not thing", as we can associate "thing" with one state and "not thing" with some other state. I'm cool with that because "state" is a fluid enough concept to allow for the possibility of a single universal state, while still allowing us to focus and explore its various sub-states. Imagine all the traffic lights in NYC; the green/yellow/red status of each light is encapsulated in a very large, frequently changing state. At any time, we can focus on some smaller sub-state -- say, Brooklyn's traffic lights, or a particular block in the Bronx -- and cogently talk about the sub-state as if it were an independent "thing". Make sense?

Everything above is abstract thought from philosophers, though. There are all sorts of theories and branches to those theories. "We never experience the thing in itself" is Kant's view. There is no proof to this yet. It has some truth potentially, but the point to this model is to not reference existing philosophical notions, and instead, build from the brutally observable, logical inference, ground-up.

But when I say "we can't know the thing itself", I'm not invoking all of Kant's metaphysics, nor am I invoking all of quantum physics -- though both Kant and quantum physics agree with me. It's an epistemological stance, and -- if we're being intellectually honest -- it should be the default stance until shown otherwise. It's a huge assumption to say that we can know what a thing is beyond our perception of the thing. I prefer to leave the huge assumptions out.

I argue we *can* talk cogently about what we "think" is currently unknowable — and that's part of the point here. Once we can prove the brain is NOT where axioms, definability, sense, and reason is happening (to my knowledge, such a proof has NOT been built based on hard obvious deduction), we can build a framework of discourse in a new logic-derived (5D) meta-space.

Inevitably, every ontology must assume an epistemology. I'm a hard-line "we can't know what actually is" kinda guy. The only cogent thing we can say about that which we don't know is to acknowledge that we don't know it. Anything else is no better than religions arguing among themselves whether god likes pancakes or waffles for breakfast.

_INFINITY is extra-model? I thought INFINITY was extra model, and _INFINITY was intra-model, and we're defining it as "not a number" as a baseline property?

I've been using the convention that an underscore prefix signifies "outside of language", as in compiler extensions or directives (e.g., "__attribute__" in gcc). Thus, _INFINITY encompasses all the varied notions about \( \infty \) that are the purview of advanced mathematics (and, hence, irrelevant to us). Examples of _INFINITY are: transfinite numbers, the hyperreals, sets of infinite sets, etc.

In contrast, INFINITY is the in-model token I use to signify the "common sense" version of \( \infty \) as an unbounded process. If we want to allow the natural numbers into the model (and I think we do, as counting is a canonical process), we need INFINITY in the model.

The parsimony element is utterly KEY to this! The core opening issue here is cordoning everything off from excess noise so we can infer that something "meta" within the being has to be identified and triangulated.

Then we must surely cordon off "substance", as that it is pure noise.

We start with the foundation of the theoretical "limits of material substance" as a discrete bit-processing medium. You wanted n→∞ originally. We agree now that ∞ is not a number, and so even this speaks to the limits of physicality.

In English, the string of symbols \( n \to \infty \) means "Let n approach infinity". In other words, the \( \infty \) symbol in the string isn't acting as a number, rather, the arrow symbol is saying "let n grow without bound". This is INFINITY, the simple concept that we're welcoming into the model.

What is the theoretical limit to the storage capacity of bits in the material plane?

What does "material plane" mean? If we reword this as, "What is the theoretical limit to storage capacity for the obvservable universe?", it turns out that we can actually estimate the answer. If you start with the Bekenstein bound, which sets the information density limit at \( 10^{69} \) bits per square meter (trying to squeeze another bit in will lead to a black hole, and black holes -- it weirdly turns out -- saturate information density; they are the universe's best possible hard drive), and if you work out various Planck-scale factors and the size of the observable universe, you get something on the order of \( 10^{122} \) bits as the maximum number of bits in any physically possible computation.

 

I've been using the underscore prefix to denote "That which is not yet defined in the model." I.e., INFINITY is our everyday discourse to discuss it. But _INFINITY will be THE crystallized token in the model.

I hope from the most recent posts it's more obvious, but the reason I want to start with the brain is because it is the very cockpit of the airplane in the scientific airport. It's the very oscilloscope we're using to analyze the circuit. We don't need to know how it works to start from the premise that it is limited in no ways different to any other physical substance with respect to the limit of what it's able to do, and that is process a discrete and LIMITED number of bits — but also not "know" what these bits represent (true geometric form).

I theorize there is a co-locational 5D substance in the being responsible for engendering true infinite forms that are the basis of axioms and reason itself and the "right side" of Boole's identity. "The dog in the light." In order to prove it exists, it needs to be deductively inferred to a place of asymptotic certainty by saying "all that this is is NOT found in the physical brain and responsible for the mechanicsm of reason is THIS." So where is it? It has to be between one's ears, but it cannot be perceived directly by the physical senses. It is perceived by internal senses. You can "see" the dog in the light in the mind and also "see the dog" outside the mind in the natural world. It is the one in the mind that is the reference for the one in the real world. We make a distinction between "in thought" and "out of thought" for that reason.

And though you don't like "substance," the problem I see with "state" is that it is not speaking to an observable actuality. The brain, the laptop, the tree. These are all referential tokens that have pinpointability.

The brain is THE starting point for all naturalist modern science. There is nothing more than the physical chemicals making it up. To get OUT of that starting point so that we can start having "sense" of what "bits represent," we have to build a model starting from the *inside out* which is what Boole began to do when he laid out the laws of thought and algebraic notation for logic that undergirds math. To do this, the brain cannot a considered an abstract "state". It has to be defined as an observable "thing" that has limits.

"Material plane" = physical world = natural world = all that is not this extra-brain "mystery notion"

 

So I have one thing to say to the above with all due respect and in all levity:

LOL!

Say what you want, but Claude Shannon built the entire information age on Boole, not Frege. Marginalizing Boole here is to marginalize Shannon's explicit deference and incorporation of his crystallization of algebraic notation in his seminal thesis A Symbolic Analysis of Relay and Switching Circuits that is the foundation of computer science (and dare I say is counter to your philosophical bias against equating numbers and bits).

Shannon is one of my personal heroes, so I'm pretty sure I'm not marginalizing his work. In any case, Shannon assuredly did not open a copy of Boole's texts while doing his own work. Shannon was a first-rate mathematician and would have been annoyed (at best) at Boole's haphazard treatment of the subject. By the time Shannon was writing his thesis, propositional logic (what Boole had tried to define nearly a century earlier) had become a mathematically mature subject. Shannon would have had available to him dozens of proper mathematical treatments of propositional logic and the more general field of boolean structures (algebras, rings, etc.).

I mentioned Frege only to contrast the quality of his work vs Boole's, in the context of both being "founding fathers" of logic.

On page 9 of the thesis:
"We are now in a position to demonstrate the equivalence of this calculus [above] with certain elementary parts of the calculus of propositions. The algebra of logic (1), (2), (3) originated by George Boole, is a symbolic method of investigating logical relationships."

C'mon, that's thesis filler, a throwaway sentence to give "historical context" to his work. I looked up his thesis to be sure. I also noticed that his references do not include Boole's work.

His thesis that changed the entire world is not some sort of "Hippocratic Oath-esque honorific lip-service nod" to Boole. It's the very basis of it. And it's the basis of my approach here as well. I'm sure Shannon was well aware of all of the other names in the metamathematics sphere, and Boole was his choice, and certainly not by some "ignorance" of the other logician figures in the field.

As I said, Shannon's a hero to me. His most important work was not his thesis on relays as logic blocks -- that idea was in the air and would have certainly been seized upon had Shannon never existed. What made Shannon the father of computer science was his work a decade later on the foundations of information theory.

As Dijkstra said, computer science has as much to do with computers as astronomy does with telescopes. Building circuits of relays to do logic is a good start for building practical computers, but computers are just tools. This is why we don't call William Shockley (credited with inventing the transistor) the father of computer science. The really interesting sh!t, the stuff of computer science, is information.

On a personal note: In the preface to the Frege's Begriffsschrift, Frege attacked what he considered to be the false assimilation of logical functions to algebraic operations — and frankly, this is utterly nonsensical in the face of what Shannon literally did with it—because our ability to type to each other on these Boolean-algebra based circuits is proof. Technology today is not built on Fregean precepts, it's built on Boolean ones, and the notion Boole set forth in equating numbers, bits and algebraicizing them is pivotal, and in turn is just as utterly pivotal in furthering a model for reason based on further analysis of precepts he set forth in Laws of Thought.

When I was in school (the second time), I went on a boolean structures kick. In addition to my formal studies on various logic theories, I devoured everything I could find on boolean algebras, boolean rings, boolean lattices. I even did what I thought was original research on boolean circuit minimization using boolean rings (turned out that a couple of researchers in Hong Kong had scooped me a decade prior).

I'm pretty familiar with all things boolean, and I can tell you that Boole's "algebra" is not a boolean algebra. (In fact, there's a well-known essay by Hailperin with that very title.) Unfortunately, the nomenclature is very confusing, so it's important for us to note the distinction between three different things that all have similar names:

• Boole's logic, the particular system of symbolic propositional logic espoused in Laws of Thought
• boolean algebra, a type of algebra over sets (there are an infinite number of boolean algebras)
• Boolean algebra (sometimes prefixed with the), is the particular example of a boolean algebra we tend to use when reasoning about computers

Before we had any of these notions, we had propositional logic. The rules of propositional logic are almost as old as humanity. Leibniz was probably the first to cogently write about mechanizing propositional logic, which is what Boole set out to do a century later. His resulting system -- what I call "Boole's logic" -- allows for propositional statements to be expressed arithmetically. This was significant as a seed for what would come later, but Boole's system itself was semi-formal and not rigorous enough to be foundational.

Mathematicians and logicians fixed up Boole's work, plugging in the holes, spelling out the definitions, nurturing the seed of an idea into a well-rooted tree. Some of the branches of this tree are what we now call boolean algebras, which are well-defined mathematical structures \( (S, \wedge, \vee, \neg) \) over some set \( S \) and whose operators share certain axioms. A particularly well-known example is \( \mathcal B = (\{0, 1\}, \wedge, \vee, \neg) \), which is what people usually think of when they hear "Boolean algebra".

Note that \( \mathcal B \) is neither unique nor privileged among boolean algebras; it just happens to be a convenient choice for reasoning about on/off circuits. Another example of a boolean algebra is \( \mathcal P = (\mathcal P(X), \wedge, \vee, \neg) \), where \( X \) is some finite set and \( \mathcal{P}(X) \) is its powerset. Another boolean algebra is \( \mathbb{D}_n = (\mathbb{D}_n, \text{gcd}, \text{lcm}, \neg) \), where \( n \) is a square-free integer and \( \mathbb{D}_n \) is its set of divisors.

Obviously, each of \( \mathcal{B}, \mathcal{P} \), and \( \mathbb{D}_n \) seem quite different in flavor, but they are all equally suited to expressing the calculus of propositional logic. That's the heart of what boolean means, and it's something that George Boole had no way of understanding. Perhaps you have a sense now of why I resist giving Boole credit for something he didn't even comprehend.

Let 1 = "A universe of thinkable thoughts", and 0 its complement

Can you give me a cite to where Boole actually said this? If I google

boole "a universe of thinkable thoughts"

I get this thread as the first hit.

I can't help but wonder if old Boole wasn't talking about "universe of discourse", which actually has a precise mathematical meaning, i.e., the set of all elements in the "universe" being considered. For example, if we're talking about unary binary functions, the universe of discourse is \( \{0, 1\} \). The complement to the universal set is the empty set \( \emptyset = \{\} \), which is not zero (as zero belongs to the universe of discourse).

...here as the basis of this model, which is to equate numbers, logic, and implicit geometric form. Turing and Church did it on paper, and Shannon's work is the hard, applied, empirical, technological proof that logic states ARE numbers that can be used to do arithmetic on them.

Erm, what proof that logic states are numbers? Show me, please.

 

Also, if this is true, we should it pack up right now then, because the "model" and everything in it is unknowable—a string of nondescript logic states—because we're just DOS prompts with hair and glasses, who have no right to discuss the bit-driven pixels that make up the glyphs that represent us on a screen.

Maybe so! But it seems to me that the only way to find out that we are not just DOS prompts is to think about it. If, in the end, it's DOS prompts all the way down, then so be it. But we gotta try, right?

 

So, here’s what I propose:

1) using bare reasoning power
2) no use of existing number theory outside basic counting and fractions
3) you can count and infer
4) you have access to basic set symbology and relations
5) access to basic boolean algebra and arithmetic
6) higher calculus not necessary until later

I'm good with this.

Let n represent the max bit utilization and binary state processing limit of the natural world, in particular the human brain substance.

You just listed the six restrictions and then proceed to toss in "brain". How about instead of brains we consider information processors, which can easily be described using your six restrictions, and then see if we can build our way up to brain-like capabilities?

Brain is incapable of “knowing any form” due to discrete bit processing limitations that have no geometric correlation.

What precludes geometric forms from being represented by finite, discrete bits?

Let _INFINITY be defined as:

1) the universe exceeding n, which cannot be represented in the brain substance.

To borrow compiler terminology, "the universe exceeding n" is a type error. The symbol n stands for a quantity, a count that can presumably be mapped to a natural number. By "universe" do you mean all quantities greater than n? If so, let's not overload "universe", which typically means everything including things that are less than n and things that can't be compared to n.

2) the property of unboundedness

This is INFINITY. I prefer to keep _INFINITY as an extra-model token, for those instances when we need to refer to any of the various advanced mathematical notions of infinity. Otherwise, we'll need __INFINITY and perhaps ___INFINITY, and it all gets typographically and lexically confusing.

3) the uncountable number of bits between any x and y that represents an addressable, boundable manifestation of itself

No idea what this is trying to say.

Form is defined as any _INFINITE 2D or 3D indivisible shape. While n bits can create shape in natural world, no number of n-bits can endow a physical substance the ability to know it exists; it is a property of the mind to interpolate the bits of a form in the natural world to the first order one in the mine so as to _KNOW it exists.

If an indivisible thing is infinite (in any sense), then by definition that thing is everywhere. This can't possibly be what you mean, so try again? Also, you invoked "mind" and "_KNOW". How would you characterize what those tokens represent using your guide of six?

 

I've been using the underscore prefix to denote "That which is not yet defined in the model." I.e., INFINITY is our everyday discourse to discuss it. But _INFINITY will be THE crystallized token in the model.

From my perspective, we crystallize a token by introducing (defining) it in the model. INFINITY already exists within our model; it is the token that characterizes unboundedness. Things outside the model aren't usable inside the model, hence the underscore prefix for "out of band" communication. I'm in no way leading up to defining _INFINITY, as I've been using it, within the model. Maybe we need a new token to express whatever it is you're leading up to?

I hope from the most recent posts it's more obvious, but the reason I want to start with the brain is because it is the very cockpit of the airplane in the scientific airport. It's the very oscilloscope we're using to analyze the circuit. We don't need to know how it works to start from the premise that it is limited in no ways different to any other physical substance with respect to the limit of what it's able to do, and that is process a discrete and LIMITED number of bits — but also not "know" what these bits represent (true geometric form).

I agree that the brain is an information processor, similar in kind to any other. Which is why it's a good idea to start with information processor, a much more distilled and general notion than "brain", which brings with it a truckload of extraneous baggage that we'd have to navigate.

If the information processing aspect is important to your theory, then it should apply to any general information processor, not just brains.

I theorize there is a co-locational 5D substance in the being responsible for engendering true infinite forms that are the basis of axioms and reason itself and the "right side" of Boole's identity.

In the context of our model, this statement can't be parsed, much less assigned a truth value. I get the sense that the tabula rasa was only meant to apply to me.

"The dog in the light." In order to prove it exists, it needs to be deductively inferred to a place of asymptotic certainty by saying "all that this is is NOT found in the physical brain and responsible for the mechanicsm of reason is THIS." So where is it? It has to be between one's ears, but it cannot be perceived directly by the physical senses. It is perceived by internal senses. You can "see" the dog in the light in the mind and also "see the dog" outside the mind in the natural world. It is the one in the mind that is the reference for the one in the real world. We make a distinction between "in thought" and "out of thought" for that reason.

We seem to have abandoned the model completely and are back to dogs and light. Maybe this is just an interlude? In any case, I say that the dog is not in the light, any more than the dog is in the brain of the perceiver. The dog may not even exist -- it could be a projection, or a hallucination. What matters is that information (presumably about a dog) has been transferred via light to the brain. The information, as a state configuration, exists in the light and it exists in the brain.

And though you don't like "substance," the problem I see with "state" is that it is not speaking to an observable actuality. The brain, the laptop, the tree. These are all referential tokens that have pinpointability.

With all due respect, WTF is an "observable actuality"? As I see it, a state is as observable as a laptop or tree. Indeed, I can associate the laptop and tree with particular states. That "state" is more abstract than "tree" is a bonus, not a bug; we can pinpoint states with ease. We can be exactly as precise or general as needed when pinpointing states, which is not the case for laptops or trees.

The brain is THE starting point for all naturalist modern science. There is nothing more than the physical chemicals making it up. To get OUT of that starting point so that we can start having "sense" of what "bits represent," we have to build a model starting from the *inside out* which is what Boole began to do when he laid out the laws of thought and algebraic notation for logic that undergirds math. To do this, the brain cannot a considered an abstract "state". It has to be defined as an observable "thing" that has limits.

The brain is the starting point for neurosciences, sure, but other than that? It's pretty rare to find mention of brains in, say, physics or quantum chemistry. Science is filled with deep, rich theories that have no concept of "brain"; I don't see why we should have to be different. Further, you're assuming that the possibility that the brain is nothing more than chemical reactions is a hypothesis from which we must escape. I'd prefer not to assume anything and see where we get. Unless you can show that it is logically impossible for thought to be just chemical reactions -- and I very much doubt anyone can do that -- we have to take it as a serious possibility.

 

Can you give me a cite to where Boole actually said this? If I google I get this thread as the first hit.

It's somewhere! I know I read it. Will have to find it. Point is, he defined 1 on one side, and equated it to some kind of "reality" on the right that was not a number or logic state.

Erm, what proof that logic states are numbers? Show me, please.

I don't have an official logic proof, but a hash-out:

It is incongruous to add TRUE or FALSE. Yet, boolean algebra is doing this, no?

How can you "subtract" or "add" states of "true" and "false" and arrive at strings that are the base 2 representation of numbers?

1 + 0 = 1
can be written
1 OR 0 = 1

...because 1 is both the elemental quantity and also a denotation of the "Truth that you have the elemental quantity." If this overlap didn't exist, you couldn't use logic-state processors (computers) to "compute" new sequences of logic states that also happen to be the base 2 elementary nomenclature of numbers.

If we didn't have the luxury of shortcut squibbles or glyphs to represent numbers, we would use base 2 to represent them. The logic string 010111 can be "added" to 01010110 (which is incongruous — adding is only done with numbers!) to yield 01101101 — but if you didn't have the numeric glyphs of "23" + "86", you would think it was normal that it was "109". It's an illusion that "109" is what that "number" is called. It's a higher order symbology for the basest way of denoting a number.

1 represents "the truth something is present" AND the something itself.
0 represents "the truth nothing is present" AND the nothing itself.
"OR" works as a representation of addition AND logic evaluation BECAUSE of this relationship. I have SOMETHING (1) OR I have SOMETHING (1) = I have SOMETHING. I have SOMETHING (1) OR I have NOTHING (0) = I have SOMETHING. I have NOTHING (0) OR I have NOTHING (0) = I have NOTHING.

This allows us to interchange quantities with logic states to yield more logic states that also double as the most elemental way of denoting quantities in a digital system.

 

When I was in school (the second time), I went on a boolean structures kick. In addition to my formal studies on various logic theories, I devoured everything I could find on boolean algebras, boolean rings, boolean lattices. I even did what I thought was original research on boolean circuit minimization using boolean rings (turned out that a couple of researchers in Hong Kong had scooped me a decade prior).

I'm pretty familiar with all things boolean, and I can tell you that Boole's "algebra" is not a boolean algebra. (In fact, there's a well-known essay by Hailperin with that very title.) Unfortunately, the nomenclature is very confusing, so it's important for us to note the distinction between three different things that all have similar names:

• Boole's logic, the particular system of symbolic propositional logic espoused in Laws of Thought
• boolean algebra, a type of algebra over sets (there are an infinite number of boolean algebras)
• Boolean algebra (sometimes prefixed with the), is the particular example of a boolean algebra we tend to use when reasoning about computers

Before we had any of these notions, we had propositional logic. The rules of propositional logic are almost as old as humanity. Leibniz was probably the first to cogently write about mechanizing propositional logic, which is what Boole set out to do a century later. His resulting system -- what I call "Boole's logic" -- allows for propositional statements to be expressed arithmetically. This was significant as a seed for what would come later, but Boole's system itself was semi-formal and not rigorous enough to be foundational.

Mathematicians and logicians fixed up Boole's work, plugging in the holes, spelling out the definitions, nurturing the seed of an idea into a well-rooted tree. Some of the branches of this tree are what we now call boolean algebras, which are well-defined mathematical structures \( (S, \wedge, \vee, \neg) \) over some set \( S \) and whose operators share certain axioms. A particularly well-known example is \( \mathcal B = (\{0, 1\}, \wedge, \vee, \neg) \), which is what people usually think of when they hear "Boolean algebra".

Note that \( \mathcal B \) is neither unique nor privileged among boolean algebras; it just happens to be a convenient choice for reasoning about on/off circuits. Another example of a boolean algebra is \( \mathcal P = (\mathcal P(X), \wedge, \vee, \neg) \), where \( X \) is some finite set and \( \mathcal{P}(X) \) is its powerset. Another boolean algebra is \( \mathbb{D}_n = (\mathbb{D}_n, \text{gcd}, \text{lcm}, \neg) \), where \( n \) is a square-free integer and \( \mathbb{D}_n \) is its set of divisors.

Obviously, each of \( \mathcal{B}, \mathcal{P} \), and \( \mathbb{D}_n \) seem quite different in flavor, but they are all equally suited to expressing the calculus of propositional logic. That's the heart of what boolean means, and it's something that George Boole had no way of understanding. Perhaps you have a sense now of why I resist giving Boole credit for something he didn't even comprehend.

Of course! I have a sense, all right. "School..." he says. "Hong Kong..." he says. I don't buy a word of any of the above! I think you ARE Boole (iDead® module insert)! And you have some kind of function running called play_along()....

With that in mind — so you're Ken Jennings, and you're playing against Watson. Watson is a piece of software running on hardware. Discrete 0's and 1's incredibly well organized.

What are "you?" (serious question to gauge where your baseline reference is)

 

From my perspective, we crystallize a token by introducing (defining) it in the model. INFINITY already exists within our model; it is the token that characterizes unboundedness. Things outside the model aren't usable inside the model, hence the underscore prefix for "out of band" communication. I'm in no way leading up to defining _INFINITY, as I've been using it, within the model. Maybe we need a new token to express whatever it is you're leading up to?

Ok, got it — we have INFINITY in the model defined as a token of meta-numeric "unboundedness." If it's not a bit in the information processor, what is it?

I agree that the brain is an information processor, similar in kind to any other. Which is why it's a good idea to start with information processor, a much more distilled and general notion than "brain", which brings with it a truckload of extraneous baggage that we'd have to navigate.

If the information processing aspect is important to your theory, then it should apply to any general information processor, not just brains.

"Information processor" some reason doesn't feel quite 100%, but let's roll with it, because it's probably sufficient.

In the context of our model, this statement can't be parsed, much less assigned a truth value. I get the sense that the tabula rasa was only meant to apply to me.

No, it certainly applies. I was just projectile spir-balling there to convey my intuition on the matter. There is no truth value to it.

We seem to have abandoned the model completely and are back to dogs and light. Maybe this is just an interlude? In any case, I say that the dog is not in the light, any more than the dog is in the brain of the perceiver. The dog may not even exist -- it could be a projection, or a hallucination. What matters is that information (presumably about a dog) has been transferred via light to the brain. The information, as a state configuration, exists in the light and it exists in the brain.

Lol I was bringing into focus the ultimate point here. One of the very earliest questions that spawned this was, "where is the dog in the light." Problem is, in a bit processor with no one bit truly connected to any other without the user making "sense" of it, "dogs, "lights," or any other object, or existence itself is not anywhere to be found.

With all due respect, WTF is an "observable actuality"? As I see it, a state is as observable as a laptop or tree. Indeed, I can associate the laptop and tree with particular states. That "state" is more abstract than "tree" is a bonus, not a bug; we can pinpoint states with ease. We can be exactly as precise or general as needed when pinpointing states, which is not the case for laptops or trees.

Hold the phone! Tree = 01100110101101010101111010101010101010110 to you, being a brain. NO one 1 or 0 is concerned with any other! Where is the tree again? You can't see any "tree" anywhere in that. There's neither tree nor the word "tree" to identify the tree!

Observed actuality = something you can empirically sense directly like a "brain" or "tree", NOT a binary representation thereof. "State" is merely something the brain can "accomplish" and no number of strung-together concatenated bytes of 0's and 1's changes that. You can't see the state change as easily as just saying "brain."

The brain is the starting point for neurosciences, sure, but other than that? It's pretty rare to find mention of brains in, say, physics or quantum chemistry. Science is filled with deep, rich theories that have no concept of "brain"; I don't see why we should have to be different. Further, you're assuming that the possibility that the brain is nothing more than chemical reactions is a hypothesis from which we must escape. I'd prefer not to assume anything and see where we get. Unless you can show that it is logically impossible for thought to be just chemical reactions -- and I very much doubt anyone can do that -- we have to take it as a serious possibility.

See above for the problem here. Tell me exactly where you see "Tree" with all of its properties in a few billion 0's and 1's that have only an end-user connection to each other. A true ToE must first fully delineate this problem before it goes outward. If one wants to be productive with a DAW and only knowing how to hit the record button and having no analysis tools or VST's to work on the captured waves, how much can he really do? The DAW is the brain, our starting reference point... (so I don't care what anyone else is doing—this is a novel approach to the problem).

No one bit is associated with any other bit when attributing a "noun" marker to it to discuss it. This is all zeroes and ones to a “you” that does not exist. “Where” is the demarcator that knows the difference, the binary is just representative.

 

It's somewhere! I know I read it. Will have to find it. Point is, he defined 1 on one side, and equated it to some kind of "reality" on the right that was not a number or logic state.




I don't have an official logic proof, but a hash-out:

It is incongruous to add TRUE or FALSE. Yet, boolean algebra is doing this, no?

How can you "subtract" or "add" states of "true" and "false" and arrive at strings that are the base 2 representation of numbers?

1 + 0 = 1
can be written
1 OR 0 = 1

...because 1 is both the elemental quantity and also a denotation of the "Truth that you have the elemental quantity." If this overlap didn't exist, you couldn't use logic-state processors (computers) to "compute" new sequences of logic states that also happen to be the base 2 elementary nomenclature of numbers.

If we didn't have the luxury of shortcut squibbles or glyphs to represent numbers, we would use base 2 to represent them. The logic string 010111 can be "added" to 01010110 (which is incongruous — adding is only done with numbers!) to yield 01101101 — but if you didn't have the numeric glyphs of "23" + "86", you would think it was normal that it was "109". It's an illusion that "109" is what that "number" is called. It's a higher order symbology for the basest way of denoting a number.

1 represents "the truth something is present" AND the something itself.
0 represents "the truth nothing is present" AND the nothing itself.
"OR" works as a representation of addition AND logic evaluation BECAUSE of this relationship. I have SOMETHING (1) OR I have SOMETHING (1) = I have SOMETHING. I have SOMETHING (1) OR I have NOTHING (0) = I have SOMETHING. I have NOTHING (0) OR I have NOTHING (0) = I have NOTHING.

This allows us to interchange quantities with logic states to yield more logic states that also double as the most elemental way of denoting quantities in a digital system.

More crystallization:
The true nature of ℕ is base 2. Any other base for ℕ is an abstract, short-cut base for easier reference and involves non-numeric, aggregative, axiomatic glyphs. E.g., "Nine" and "9" is the shortcut, conglomerating English symbology for the ACTUAL 5-digit number 01001. That's nine's TRUE, elemental, simplest form—a concatenation of 5 independent base 2 ℕ numbers and binary logic states. (Additionally, a lightbulb for me: Kronecker is 100% right. Because computers can represent all the reals using base-2 ℕ, I'd hazard ℝ is really just a subset of ℕ using a kind of "fractional isomorph" of ℕ delimited by the radix and concatenation of the base 2 mantissa to the integer/characteristic portion of the number). In addition, a computer can represent every arithmetic operator and the signed state as a form of addition. All is done with the intersection of base-2 ℕ and binary logic.

The magic here is also linguistic in nature: the specific intersection of the definition of "number" and "logic state" lies at the linguistic intersection between "disjunction" and "addition operator" — i.e., the seamlessly synonymous use of "OR" and "+" as Mr. Boole laid out in Laws of Thought (and yes, that's what I believe Shannon's essential takeaway was with him).

In the example use of 1 + 0 = 1
LOGIC: Augend and addend are implicitly summing via disjunctive logic evaluation
MATH: Augend and addend are explicitly summing via additive arithmetic

CONCLUSION: Logic states and numbers are two sides of the same ontological coin, utilizing the same axiom-denoting “+“ glyph.

100% identical operation, as proven in hardware first by Mr. Shannon.

 

Unless you can show that it is logically impossible for thought to be just chemical reactions -- and I very much doubt anyone can do that -- we have to take it as a serious possibility.

Oh, but this is the very point!! I 150% intend to prove this using logic and raw reasoning power with you here on this digital restaurant napkin, and you can do the magic of crystallizing (The word that keeps on giving) it using proper propositional logic symbology. I am thoroughly intuitively convinced that reasoning is NOT happening whatsoever in the brain, and is the very point of scientific constipation that has led to other spurious dead-end lifeless religions theories.

Positively essential to this is the starting point of the brain as the point of analysis, because it is here that "thoughts are thought to reside." This is why I'm hellbent on surrounding the brain with a pride of logical lions that haven't eaten in weeks, where one is forced to deductively abandon it as the place where reason and thought begins. If we can pull this off, it will change science forever. Lofty goal, eh?

What precludes geometric forms from being represented by finite, discrete bits?

Nothing. REPRESENTED is the operative term there. Representation of a geometric form in the brain is not the form itself (hearkening back to my very original question, "Where is the cube, as described, with 8 hard corners, in the grey matter?") This will be part of the proof. If Siri captures a pic of a cube with the camera and identifies it as a cube, does Siri KNOW what the cube is, and does Siri have an archetypal definition of it that involves 8 hard corners? Nope. Machine "sees" cube, wave hits machine from cube, wave is digitized to 01100110111011010101101... the number is put through gates to match against stored numbers a living USER has programmed to represent a "real cube," and if it finds a match of the same 01100110111011010101101... it says "YES! It's a cube!" But what it's really only saying is "I've matched 01100110111011010101101 to a stored 01100110111011010101101, I have no idea what a cube IS". The problem of consciousness and self-awareness is concerned with the to-be-modeled-and-triangulated "knowing and reasoning about what IS," altogether separate from "digital representation and computing happening in a material substrate.”

For this reason, I'd like to use "Any Natural World Information Processor" if you're cool with that, because it's entirely intrinsic to the proof that we prove the limits of the natural world machine in order to triangulate the "5D GPU."

And yes, I'm well aware that computer science is allll about information theory, not actual computers.