But we haven't defined it.

Here's my first iteration of an attempt. An information processor that recognizes associations as being associations has awareness of those associations. Recognition only means carrying state about other states. For example, the computer that measures CPU temperature to control its fan keeps state on the conditions of the fan and CPU temps. It recognizes the dependence between the two, and so has awareness of the association between CPU temperature and fan speed.

I define SELF as the concept that is formed from the awareness of the distinction between internal and external states. We describe an information processor with a notion of SELF as having consciousness.

What if it's as weird as the "continuum" — and there might be a "continuum" element to the being that has nothing to do with n associated states (Cue: "soul").

I haven't seen any evidence to suggest that a continuum-like "soul" is necessary to explain consciousness. In contrast, there is a ton of evidence pointing to a mechanistic explanation of life.

ℝ as a set, for example, is really a continuum. Any interval in the continuum is 0% of the rest of it, but yet we can point to an interval, as though we can "rationalize" or "discretize" a portion as "one componental thing," but it's not a fraction of the rest of it.

Incorrect. Any non-empty interval of ℝ, no matter how small (by Lebesgue measure), is 100% of ℝ. That's what makes it a continuum. We can map every point in [0.0001, 0.0002] to the entire real line, or to the entire Euclidean plane, or to the entire 1024-dimensional Euclidean hyperplane.

 

The set of ℕ represented in base-1 unary is rendered as a set of infinite marks.

"A set of infinite marks" is a confusing way to say it. Each mark in ℕ is finite. We just can't list them all, because there's no largest natural number.

The set of ℝ as represented in base-1 unary is also rendered as a set of infinite marks.

Incorrect. Almost all numbers in ℝ cannot be written down in any base. This is not because they are infinite (they are all finite), or because they have non-terminating representations, rather they literally have no representation. They are not computable.

Any sequence of numerals that you can write down (or compute on an infinite memory computer) can be mapped to a unique number in ℕ. This includes the representations for the irrational numbers like root-2. Indeed, take every number in ℚ (which includes every number in ℕ) and adjoin \( \sqrt{2} \) to it. Then, \[ |\mathbb{Q} \cup \{\sqrt{2}\}| = |\mathbb{N}| \] In other words, any set of computable numbers is countable. ℝ is special because it is uncountable. Furthermore, it's self-evident that any number representation in any base is computable -- if we can write it by hand, then an algorithm exists for its representation. Likewise, if a number is computable, then we can write a representation of it by hand in some base. The set of all these computable/writeable numbers is countable. Therefore, most numbers in ℝ cannot be written down or computed.

This is no different than the difference between discrete vs. continuous mathematics. They are not directly comparable.

How so? In signal processing, we compare discrete vs continuous math all the time. The Fourier transform can surely be compared with the DFT; differential equations with difference equations; etc.

 

Ok, from here on out, “presence or absence” means “high or low”.

That's fine, as long as we don't ignore that there's more than just symbol swapping happening here. "Presence or absence" implies a discrete domain, something is either there or it's not. In contrast, the levels corresponding to "high or low" are not discrete, they're a range of values. This is an important distinction because you claim that discrete presence or absence is fundamental, but computers most definitely do not work with discrete voltage levels. Another way to say this is that every digital circuit is made up of analog components.

Do computers only use logic 0 and logic 1, or is there a logic 2, a logic 5?

It's not this simple, as there are several layers of abstraction and particular details for what you mean by "logic". At the formal system level, we have two-valued propositional logic, three-valued ternary logic, and, more generally, n-valued multi-valued logic. This formal level defines the possible states that can be given to a variable in a model. In propositional logic, variables are replaced with one of two values: "T" or "F". Ternary logic includes another state, such as "undetermined" or "don't care".

At the circuit level, logic gates implement the logical operations defined by whatever formal system the designer chose. A binary operation is called binary because it has two operands (inputs), not because it has two possible values. We can, for instance, have binary operations in ternary logic. A ternary operation has three operands; again, we can have ternary operations with any number of logic values.

The familiar boolean gates (AND, XOR, etc.) are implementations of the binary boolean operations. Here, the "binary" refers to the number of inputs and the "boolean" refers to the number of possible logic states.

Do computers use only binary gates? Absolutely not. Computers are filled with unary, binary, ternary, and all manner of n-ary inputs. What about the formal logic, do computers use only boolean? Nope. For example, computers use tri-state gates at the interface to a bus, which have a high-impedance third logic state to disconnect the gate from the bus. There are also many multi-level communications protocols (like Ethernet) that use n-state logic (5-state, 9-state, 16-state, etc.).

 

That's fine, as long as we don't ignore that there's more than just symbol swapping happening here. "Presence or absence" implies a discrete domain, something is either there or it's not. In contrast, the levels corresponding to "high or low" are not discrete, they're a range of values. This is an important distinction because you claim that discrete presence or absence is fundamental, but computers most definitely do not work with discrete voltage levels. Another way to say this is that every digital circuit is made up of analog components.


It's not this simple, as there are several layers of abstraction and particular details for what you mean by "logic". At the formal system level, we have two-valued propositional logic, three-valued ternary logic, and, more generally, n-valued multi-valued logic. This formal level defines the possible states that can be given to a variable in a model. In propositional logic, variables are replaced with one of two values: "T" or "F". Ternary logic includes another state, such as "undetermined" or "don't care".

At the circuit level, logic gates implement the logical operations defined by whatever formal system the designer chose. A binary operation is called binary because it has two operands (inputs), not because it has two possible values. We can, for instance, have binary operations in ternary logic. A ternary operation has three operands; again, we can have ternary operations with any number of logic values.

The familiar boolean gates (AND, XOR, etc.) are implementations of the binary boolean operations. Here, the "binary" refers to the number of inputs and the "boolean" refers to the number of possible logic states.

Do computers use only binary gates? Absolutely not. Computers are filled with unary, binary, ternary, and all manner of n-ary inputs. What about the formal logic, do computers use only boolean? Nope. For example, computers use tri-state gates at the interface to a bus, which have a high-impedance third logic state to disconnect the gate from the bus. There are also many multi-level communications protocols (like Ethernet) that use n-state logic (5-state, 9-state, 16-state, etc.).

Ok, great clarification.

So now, in propositional logic, variables are replaced with one of two values: "T" or "F". IC's might use multiple inputs, but in the end, after all is said and done, a logic high or low is on each output pin, correct?

Assuming the "unknown" element of ternary logic being basically "an answer in waiting", would you not agree fundamentally that logic and computation boils down to True or False?

 

Unary isn’t really considered a base. It’s not base 1, because 0 alone can’t be used. It’s not base 2 because that involves 0 and is binary.

You think because it can’t be written down by a finite brain it’s not a number? R is a number set with all numbers and numeric computations in it, essentially. All the numbers in N can’t be written down! It is infinite! You think because segments of R are infinite that there’s “more infinite.”

Every single number in R can be manifested in its original representation of unary, or it’s not a number! Why the partiality to base 10? Anything other than unary or binary is a higher abstraction of convenience only. Code and computation is built on logic high and low.

What does this question look like in 1820 before sets were invented?

The semantic problem with the word countable:

Countable is being used as synonym for nameable. But the problem with this is that it doesn't confer the extent of the nameability. It also doesn't confer the sense of whether or not the emphasis is on the nameability of the subject OR rather the inherent potential limitations of the person naming! You're a fellow wordsmith, so I'm sure you grok this. Talk about lack of specificity!

That which is infinite is unbounded, and therefore is NOT finite and NOT terminated. It is IMPOSSIBLE to name or count all of an unbounded, infinite set by the proper definition. There is no point in calling the set infinite otherwise. Infinite DENOTES unboundability. Otherwise it's like saying it's "infinite until it's not". Naming IS a discretizing operation no matter how you cut it. N+1 is NOT infinity. N+1 IS "the next unknown number needing a name in the sequence."

Fact is, if we stayed with unary and had no number sets, this entire hellish semantic clusterf*ck would completely disappear. If we have an infinite continuum of unary bits, any one section can be cordoned off as nameable and countable if we want, or unameable/uncountable if we want. Every computation result is found somewhere in the continuum, and if it can be conceived as something knowable, it is somewhere in the continuum represented as a "fraction of it." And that fraction ALSO contains ALL OF THE CONTINUUM as you said above!!

And as I'm saying this, this so speaks to what I was saying about numbers being nothing more than "cordoned off portions of an infinite continuum". That's exactly what they are. They are "dicrete names for infinite portions of a continuum!"

Every time I hear the word "countable" as including the "extent of countability" a young kitten is trampled somewhere.

Because it's already in use, I will defer to its use. But it's obtuse AF. There are "numbers for the purpose of counting" and then there are computational results based on them. Computation is done with NUMBERS. If your result is not a number, it's not a computation. If it can't be known right now, does not mean it's NOT in the continuum. It's just unknown.

Just because your result can't be known, doesn't mean it can't be computed.

We cannot exactly compute where something will be in the universe, doesn't mean there's not "countable" numbers to represent it. We just don't know the true extent of the variables required to do it.

Agreed on this?

 

Incorrect. Any non-empty interval of ℝ, no matter how small (by Lebesgue measure), is 100% of ℝ. That's what makes it a continuum. We can map every point in [0.0001, 0.0002] to the entire real line, or to the entire Euclidean plane, or to the entire 1024-dimensional Euclidean hyperplane.

That is 100% correct above, of course. I was referring to a different angle.

The problem is, ℝ is an intra-number continuum, where as ℕ is only an upper bounds continuum, and ℤ is an upper and lower bounds continuum. Truly, everything other than ℝ is the set of perfumed bullshit.

One cannot compare ℝ to ℕ, because ℕ is a set of discrete elements where ℝ is not really a set of nameable components — it's a continuum that includes ℕ as part of its unbroken continuum. You cannot say one is less than another because you can't compare discrete sequences of nominal elements to continuous infinity!

Dammit! This is really the most cluster-f*cked semantic thing ever devised, right out the bowels of Dante's Inferno!

 

The One True Number Set

There's a reason why most mathematicians ignore the number sets.

Not off to a good start, lol. I can't think of a strong enough adjective to emphasize how wrong that sentence is. I don't know of a single mathematician that "ignores the number sets". Where did you get such a wacky idea?

The only thing I can think of is that perhaps you misinterpreted my earlier comment that many mathematicians ignore the transfinite details of set theory. If so, please believe me that just because the average mathematician probably doesn't spend much time thinking about ZFC, does not in any way, shape, or form imply that they ignore number sets.

It's because they don't really exist as legitimate mathematical elements.

Again, I don't mean to be insulting, but it is preposterous to claim that number sets are not legitimate mathematical objects. If we could somehow erase the notion of number sets from everyone's mind while they slept tonight, the first thing they'd do (after making coffee) is re-define the number sets. I mean, all of the canonical algebraic structures -- groups, rings, modules, fields -- are based on number sets. Open sets are essentially the currency of topology. Analysis is literally all about ℝ and ℂ. You can't solve differential, algebraic, or arithmetic equations without number sets, and enormous amounts of mathematics has come from the theories of finding these solutions.

On the most elementary of levels, any modern computer does any imaginable computation and logic evaluation using the classical bit and no number sets. Because this is true, there is an overlap in utility between the classical bit, having logic state 1 and 0 and the numbers 1 and 0 as the basis of all computation.

When a computer does a computation or logic evaluation, it uses neither 1 nor 0. It is very important that we recognize the layers of abstraction.

A boolean logic system is an abstract concept that uses two values {F,T}, which are not numbers. A circuit is a physical device that can be designed to behave as if it were evaluating a logical operation -- i.e., we interpret the result as a logical operation -- but the circuit itself can only alter the flow of electricity.

A number system is an abstract concept that includes a set of elements (numbers) and arithmetic operations. Neither the numbers nor the operations have a physical form. In order to conceptually manipulate them, we express numbers and arithmetic operations using symbols in some definite representation. Again, we can design a circuit to behave as if it is manipulating symbols, but that is entirely our interpretation; a circuit can only manipulate electricity.

The way to keep all of this clear is to consider the domain of discourse used in each abstraction. For boolean logic systems, the domain of discourse is {F, T}. For number systems, the domain of discourse is a set of numbers. For circuits, the domain of discourse is electricity.

In a computer, 1 and 0 stand for logic states HIGH or TRUE, and LOW or FALSE.

Inside of a computer are various voltages. There are no numbers inside of a computer.

A certain amount of voltage present on a line stands for logic 1, and a certain contrastive amount stands for logic 0.

A range of voltage values is taken to represent logic HIGH or TRUE (1 is a number, not a logic level), another range of voltage values is taken to represent LOW or FALSE.

The transistors in IC's let voltage flow in a given direction or not. It is a binary phenomenon.

No. A transistor is an analog device with a range of operating modes depending on how it is biased. In its linear mode, a transistor is an amplifier. When biased into saturation, a transistor behaves like an analog switch.

In any form of physical substrate, given sufficient complexity of arrangement of power, lines, switches and 1 or more timing devices, any form of logical evaluation and numeric computation is empirically proven to be achieved with just these two elementary states, which double as elementary "building-block" numbers.

"Empirically proven" is a category error. You can empirically demonstrate a phenomenon, or you can prove a theorem in a formal system. But you can't mix the two.

There has long been a debate concerning the cardinality of different number sets being larger or smaller. But this debate is easily settled and all confusion removed when we reduce all number sets, namely N,W,Z,Q,I,R,C to their basest of forms, and that is the "unary" base, involving just the number 1. All numbers and computation can be done with the number 1. It is not efficient, but it is elementary, and necessary to disabuse confusion.

You are being disingenuous: the sizes of these sets are not a debate in mathematics. Only a tiny fraction of mathematicians have a problem with it. I wholeheartedly agree with you that all computations can be represented in base-1, though I would say that every computable number can be represented in base-1.

BTW, there are far, far more number sets than just those listed.

How many values are in each of the 7 number sets when every term is represented by strings of the number 1? Infinite. What is their cardinality? Infinite.

Using "infinite" as a number is definitely non-standard. We don't even need to use the word "infinity".

What's the cardinality of ℕ, ℤ, and ℚ? \( \aleph_0 \)
What's the cardinality of ℝ and ℂ? \( 2^{\aleph_0} \)
How are they compared? \( \aleph_0 < 2^{\aleph_0} \)

From this perspective, it is easy to see how preposterous it is in calling the set N < R, once the positional illusion of base-10 is removed. It is due to the use of higher bases that the illusion of comparability where there is none persists. Any such manufactured sets are 100% equal in cardinality, because they are subsets of NUMBERS: no computer would argue, and most working mathematicians tacitly agree in their everyday dismissal of the 7 sets.

All number bases are positional. The fact that N < R has nothing to do with representations, it is a property of the numbers themselves. You seem to be confusing the notion of computable numbers.

Once leveled by unary representation, all sets therefore implicitly disappear entirely, and we are left with one set, called NUMBERS in the unary base. This set has a most curious property: it is both countable AND uncountable at the same time.

Crazy talk! How can I be able to count something and not be able to count it? You're flying off into the crackpot weeds.

The same concept can be understood in the analysis of a Euclidean line with end points A and B. How many points are in between each end point? Infinite. But one can pick an interval p and q in between A and B and create another line from it. How many points are in between new line p and q? Infinite.

That's because Euclidean geometry is built on top of ℝ, where any interval equals the whole. Notice that if you build a geometry on ℕ this is no longer true!

One can label a stick of length 6 vs. length 3, and in unary that is IIIIII vs. III. Which stick is bigger than the other? Sure, 6 comes "after" 3's ordinality in the NUMBERS set, but with a set cardinality of INFINITY, does that mean the stick is "bigger?" No.

You can't tell which of the two sticks looks bigger?

So a stick of 6 inches is "bigger in magnitude" than a stick of 3 inches only because we've defined an inch in terms of something else. But why is attributing an inch different than a cm or any other thing that we've measured in terms of a number we've assigned to it? This is a grand illusion!

More crazy talk. Carve equally spaced notches on each of the two sticks. Count the number of notches on stick, then the other. The stick with more notches is bigger! You're way over-complicating a simple thing.

 

Ok, great clarification.

So now, in propositional logic, variables are replaced with one of two values: "T" or "F". IC's might use multiple inputs, but in the end, after all is said and done, a logic high or low is on each output pin, correct?

(Just wondering, why italicize entire sentences? They make it hard to quote.)
The domain of ICs is electricity, not logic. I know that we casually say, e.g., "set the pin to logic high", but we need to be precise and conceptually clear.

Assuming the "unknown" element of ternary logic being basically "an answer in waiting", would you not agree fundamentally that logic and computation boils down to True or False?

Not at all. "Unknown" is just as valid a logic state as "true" or "false" or "don't care" or whatever. "Unknown" doesn't need to be later replaced by "true" or whatever. One might even argue that any logic system with an "unknown" state is more accurately representative than one without.

 

I'll try to not italicize as much. I use it as "emphasis" indicator.

The number sets exist as a means of doing mathematics, of course. But their very existence and absence of proper delineation with respect to the underlying order of how numbers work underneath it all is causing confusion and insanity. I am not alone in this. Kronecker, Poincare, Wittgenstein, and many others felt this way. Cantor died in an infirmary. The man lost his mind. There are SERIOUS problems semantically going on that need to be addressed. Even the word "countable", as I addressed above, is FLAWED, period. If you think "REAL" is flawed as a word by the great Rene Des Cartes, all I can say is, it's open season on every term. Lol.

Current math and logic know positively NOTHING about many, many elements of existence, and there is a lot of convolution I sense that isn't based in... wait for it... REALITY (ughhh!) Lol.

The domain of ICs is electricity, not logic. I get it, and I know you are a formal systems juggernaut. But level with me a sec and realize I'm trying to find core drivers and relationships between things, so though they are "category errors," let's assume on some level the categories do not exist to the rigidity level they have just temporarily.

A computer has a power source. That power source is high, and it is then "filtered" through the electronics to mimic T/F logic. In the end, it is the voltage itself that is the origin of the beginning of computation. Propositional logic's worth is in deriving T or F ascription to a computation, no? Unknown means you do not have enough data to answer it, or it's not a valid question.

 

Countable is being used as synonym for nameable.

Though "countable" does speak to nameability, it does so only in the sense of computability. The defining characteristic of something being countable is how it relates to ℕ.

That which is infinite is unbounded, and therefore is NOT finite and NOT terminated. It is IMPOSSIBLE to name or count all of an unbounded, infinite set by the proper definition.

This is important. Countable means only this: a set is countable if and only its cardinality is less than or equal to ℕ. That's it.

The root "count" in countable comes from "counting numbers", which is the name we give to ℕ. Countable literally means "cardinality comparable to ℕ". Don't think of cardinality as "size", just think of it as what it is: a way to express the number of elements in a set.

We never have to use the words "infinite" or "infinity" when discussing these things.

We cannot exactly compute where something will be in the universe, doesn't mean there's not "countable" numbers to represent it. We just don't know the true extent of the variables required to do it.

Agreed on this?

Nope. Quantum physics tells us that even the idea of something having an exact location is wrong. The issue isn't that we can't compute it, the issue is that "well-defined position" is not physically meaningful.

 

One cannot compare ℝ to ℕ, because ℕ is a set of discrete elements where ℝ is not really a set of nameable components — it's a continuum that includes ℕ as part of its unbroken continuum. You cannot say one is less than another because you can't compare discrete sequences of nominal elements to continuous infinity!

ℝ is defined as a set of elements. ℕ is also defined as a set of elements. Categorically speaking, they are the same types of things, therefore we can 100% compare them. Apples to apples, sets to sets.

Let me put it another way. You agree that ℕ is contained within ℝ. You agree that ℝ contains a bunch of other stuff, too. Isn't it clear that ℝ is bigger?

If I put all of my shoes in a box, and put the box inside of a closet that has a bunch of other stuff, isn't it clear that the box is smaller than the closet?

Dammit! This is really the most cluster-f*cked semantic thing ever devised, right out the bowels of Dante's Inferno!

ℝ is f*cking maddening. I warned you about the dragons!

 

"The size of these sets is undisputed." I don't think so. "N" is not countable forever. N+1 is not infinity. There are sets of various forms of horse-sh*t in what is "considered as" mathematics, and I'm not alone in saying this.

Also, John Von Neumann, as you know is considered one of the smartest mathematicians who ever lived. The man believed the brain was binary. I intuit the same thing — I just have to often invoke other intellects so I don't appear "crazy" if I can't describe it as formally as he. "Kronecker" and "Poincare" and others are the same deal. Not name-dropping, just showing it's not a "crazy, isolated thought from an individual who can't invoke the languages of formal systems to describe it."

 

The domain of ICs is electricity, not logic. I get it, and I know you are a formal systems juggernaut. But level with me a sec and realize I'm trying to find core drivers and relationships between things, so though they are "category errors," let's assume on some level the categories do not exist to the rigidity level they have just temporarily.

The only reason I'm so anal about this category stuff is because I know firsthand how much confusion occurs when the categories of abstraction aren't kept straight. We're trying to form a reasonable theory of the universe, life, consciousness using words and ideas from several different types of abstract systems that are, frankly, confusing as hell on their own.

I'm all for finding connections between these things (I believe they're all indeed connected), but we have to be very careful as we do it, or else we will end up with a rat's nest of conceptual spaghetti in a haystack of foggy confusion soup.

A computer has a power source. That power source is high, and it is then "filtered" through the electronics to mimic T/F logic.

Yeah pretty much, though I wouldn't say "mimic T/F logic", rather we interpret the resulting physical state as if it were T/F logic.

Propositional logic's worth is in deriving T or F ascription to a computation, no?

No, the computations themselves never have a T or F value associated with them. (We design the circuits such that every computation is valid.) The primary worth of logic to a computer is in control. As a very simple example, a computer might have an LED to indicate that it's in sleep mode. To drive the LED, the sleep mode circuitry might use an AND gate with one of its input tied to voltage HIGH, and the other tied to the sleep mode state. If the output of the AND gate is tied to the LED driver, then when the sleep mode becomes active, both inputs to the AND gate go HIGH and the LED shines.

Unknown means you do not have enough data to answer it, or it's not a valid question.

Yup, and that's a perfectly reasonable answer. Another simple example: a computer might have a circuit that turns on a GREEN light when it determines that its CPU temp is normal, a RED light when it determines its CPU temp is too high, and a YELLOW light when it doesn't know (temp sensor broken). We could design this with a few boolean logic gates, or with a single three-level gate.

 

"The size of these sets is undisputed." I don't think so. "N" is not countable forever. N+1 is not infinity. There are sets of various forms of horse-sh*t in what is "considered as" mathematics, and I'm not alone in saying this.

Go to math stackexchange and search on cardinality of number sets. I predict you'll find thousands of posts explaining the subtleties at various levels, and zero or almost zero disputing |N| < |R|.

I can't stress how much easier it is to think about this stuff if we simply don't use words like "infinity" and "forever". The cardinality of N is aleph-naught. That's no harder to swallow than "the value of x is B". The cardinality of R is two to the aleph-naught. "The value of y is 2^B."

|N| < |R| is just "x < y".

 

A computer has a power source. That power source is high, and it is then "filtered" through the electronics to mimic T/F logic.

Yeah pretty much, though I wouldn't say "mimic T/F logic", rather we interpret the resulting physical state as if it were T/F logic.

Ok, but the essence of ourselves as "state machines" is that T/F is what matters to *us* as humans. I don't care how many transistors, wires, voltage levels, etc. In the end, whether classical or quantum, T/F is the basis of propositional logic, the basis of human reason, the basis of computers and their behavior as "computers." They "compute T/F" outputs. We amalgamate those output strings into higher abstractions.

Yes?

 

Go to math stackexchange and search on cardinality of number sets. I predict you'll find thousands of posts explaining the subtleties at various levels, and zero or almost zero disputing |N| < |R|.

I can't stress how much easier it is to think about this stuff if we simply don't use words like "infinity" and "forever". The cardinality of N is aleph-naught. That's no harder to swallow than "the value of x is B". The cardinality of R is two to the aleph-naught. "The value of y is 2^B."

|N| < |R| is just "x < y".

This is EXACTLY the problem. The reason for this "cardinality nonsense" is found in the reason Hawking cut off the rest of Kronecker's quote to make his book title fit this bias, which I fix below:




Aleph-naught is devil-speak for "over-complication to obfuscate the truth." God is found in the details, and so is the devil. Speaking metaphorically.

"N < R" is Cantor's secretary at Bellevue talking.

If both are reduced to unary 1's, it's easy to see there are infinite 1's in both. That would be Pythagoras "crazy-speak." He thought all numbers were born from 1 as I do. ;--)

The continuum of N can only be seen in one direction in base 10. The continuum of R can be seen in both directions AND intra-value. The presence of continuum in both sets is sufficient to call them both equal in cardinality. It's all an illusion.

The reason R is maddening is because of this nutbag partiality and need to see it "in light of a subset of itself!". Numbers are numbers! Continuum is continuum. Sqrt. 2 is NOT a number. It is a computation. It just happens to relate in PART to other components of a geometric figure, that TOO is an obfuscation.

In order to blaze new trails and find core connections, we have to question all existing presumptions, no matter how popular... and we cannot necessarily invoke consistency checkpoints from existing systems in the process!

 

The only reason I'm so anal about this category stuff is because I know firsthand how much confusion occurs when the categories of abstraction aren't kept straight. We're trying to form a reasonable theory of the universe, life, consciousness using words and ideas from several different types of abstract systems that are, frankly, confusing as hell on their own.

I'm all for finding connections between these things (I believe they're all indeed connected), but we have to be very careful as we do it, or else we will end up with a rat's nest of conceptual spaghetti in a haystack of foggy confusion soup.

100% agreed! But that is a massive problem... because what if there are either errors or inadequacies in a given system? It's as if we need to build a semantic eco-system from scratch and THEN work it into existing formalities.

 

So I'm convinced you have neither job, house, nor family and you operate from a self-made hot air balloon over the arctic somewhere, working for multiple governments, wearing nothing but eskimo boots, a white t-shirt that says "F*ck Rene" while brandishing an old Tandy that you retrofitted with prototype RISC chips hardwired with a SCSI cable through self-surgery to your cortex. Along with creating your own ride-able Roomba from IC's that you actually self-manufactured, you also taught yourself surgery through website tutorials. Though you go by Javier on here, your real name is "Nathan," and you are the lead character in the documentary-as-film "Ex Machina." Meantime I'm over here on an island you somehow are constantly circling over, and I'm desperately trying to semaphore you to land within 20 miles on an abandoned light house. Every once in a while, while you cycle around, there are like 3 signals that give me hope you'll land. My arms are about to fall off from fatigue, and your balloon is about to deflate at some geometric rate that is undoubtedly consistent with some Hilbert-describable n-tuple metric you created. I often feel pigeonholed trying to fit my thinking into old rap tunes from the 1990's and wondering if Vanilla Ice could lend a hand to that end. "STOP, collaborate and LISTEN! Jen is back with a brand new edition! HILBERT grabs a hold of me TIGHTLY..."

For those tuning in, I have absolutely no pretense to the fact that I'm crazy AF when it comes to all of this. We're talking "crazy daughter-of-a-b*tch." I could study formal systems and learn various nomenclatures and such, but then they'd influence my thinking too much, and I couldn't maintain my "lawyer/apologist" tabula rasa insistence. That's why we have a good balance here. I'm unencumbered, and you're, well... not unencumbered, but in the most positive of lights.

So we have to find some kind of balance there... it's 100% true we have to be rigid. But if we build a new eco-system, as long as we're lexically self-consistent within it, we're good. And then we can attempt to connect it to the existing systems through your lexical Parallels™ Academic Edition netware.

You have to somehow disengage the "category errors" element while we work on this, because I'm rendered one big blue screen of death with 404 emblems made out of old ANSI characters otherwise.

 

... We need to connect the bit with the number with the continuum with the 3D-1D form business.

I propose we start a new number set to begin, called I.

This number set is unary and infinite. Any interval created has infinite @‘s in between them.

T. Rasa—

1) What is a number in this set?

 

Another pot on the stove:

Mathematics is, among other things, about the relationship between numbers. Why does your brain insist on any relationship at all?