Because of the unreal level of abstraction. ;--) This is like a 938GL language. Like that statement is 300% zoom, and I need like minimally 1200% just to understand what you're talking about.

Complexity is a result of abstraction. The three physical rocks represent some low level of abstraction and complexity. Their information is processed by a brain, adding another level of abstraction and complexity. Concepts, like "number 3" are formed, which are an even higher level of abstraction and complexity.

Abstraction (and so complexity) are part of the deal. If you want a particular zoom level, give me an example so I can try to match it.

A machine is based on some kind of instructive, sequential calculation schema utilizing a highly tuned ratio of external stimulus and stored stimulus that an agent programs. Concepts like "it was formed" over time is to me like saying, "Well, over time, my camera on my laptop just began developing circuitry that started doing face recognition, and the code appeared in the stack that is able to organize all of the stimulus into just the right neuronal flip flops. And then one day I found it video-conferencing on its own with another computer in Bejing. What's the big deal?"

You seem to be forgetting that we are machines. And, indeed, we were formed over time. I have an ultrasonic picture of my daughter when she was a much simpler machine than she is now.

Can we talk about the differentiator without invoking one word invoking incredibly abstract terms like "evolve" or "chance" for a moment? It is the very Differenza ® SDK that I'm interested in understanding. You know, like all of that code that's in the 6 interrogatives that is parsing the data nad storing it, and treating continuous phenomena as discrete in a very specific manner?

Not sure what you're asking for. You want example of the software that my brain runs? Sadly, we haven't figured it out to that kind of level of detail yet.

 

Side with them on what? Leibniz believed in infinitesimals, which are way[/] outside the realm of integers.

You know how you're anal about terms™? Well, here we go: Man has created terms. To me? "There are not integers, there are not REALS, there are not transfinites." There are bits (which one MIGHT call an integer — it's a difficult issue). 100% proof? We build numbers, systems, and interrelations using bits.

It seems pretty arbitrary to suggest that sets are "man-made" and numbers are "built-in". They're both abstractions. Moreso, we can define numbers in terms of sets, but we can't go the other way. It seems cogent to say that sets are more fundamental in some way. Indeed, math is built on top of logic, and logic and sets go hand in hand.

Concatenated bits are foundational. Everything else is a conceptualization differentiator.

Who is "boundarizing" infinity?

Transfinitization is in essence turning infinity into comparative-sized quantities. There IS no size to an infinite set. It is sizeless, or it is not infinite! You can't walk to school OR carry your lunch.

 

You're confusing concepts. What makes binary logic binary is that there are only (not "at least") two different logic states. Conditional branching is an entirely different notion. To wit, you can have three or seven or however many conditional branches you want, irrespective of the number of acceptable logic states. This is so because we can use relation operators, such as "equals to" or "greater than", that evaluate to one of the logical states.

Remember, any single if/else conditional statement has an equivalent form as two if statements:

if ( x == a ) then doA();
else doB();

is equivalent to

if ( x == a ) then doA();
if ( x != a ) then doB();

What would conditional branching look like in a unary computer? A bunch of if statements:

if ( x == a ) then doA();
if ( x == b ) then doB();
if ( x == c ) then doC();


You're right, it's utterly irrefutable that we can represent any number or calculation in base-2. But the gigantic, blinking-neon point that you're missing is that it is utterly irrefutably true for any base. There is nothing special about base-2.

Ok, terminology issue alert—I just realized. THIS is another good window to open on the desktop here:

I'm presuming I already proved my intuition as I'm speaking, terminology-wise, when I say "base 2," a direct "intersection" between "TRUE/FALSE" binary and "base 2" numbers. I completely understand from a numeric perspective we can use any base. In practice, we literally can use the same symbols. 0 = false and 0 = zero; 1 = true and 1 = one. In the end, you can take 0 and 1 logic states and pop them into a binary converter table and find any kind of information depicting geometric 2D and even 3D on screens/holographic projectors.

In line with my ultimate "foundations aim here," I want to write a proof of this, and I am 95000% positive it is truth.

I.e., is it not true, that essentially we have binary systems using logic TRUE and FALSE and in the end, these bits containing one element of the set {TRUE, FALSE} are the MOTHER set of all numbers, information and calculation as proven in the hardware and software? "True" meaning switch high, "false" meaning switch low.

Assuming your unary element is actually itself some kind of strange abstraction (based in my other post, I show you need binary states to determine the truth or falsehood of whether or not you have a certain "number of @'s"... you can't even measure the number of bits you can work with in the system without becoming a cave man and grunting for each @ — 1^n style. Lol).

 

You know how you're anal about terms™? Well, here we go: Man has created terms. To me? "There are not integers, there are not REALS, there are not transfinites." There are bits (which one MIGHT call an integer — it's a difficult issue). 100% proof? We build numbers, systems, and interrelations using bits.

As I see it, bits are as abstract as numbers. We use bits to measure information. Being discrete, bits allow us to count information, like we count sheep. There is a bijection between sequences of bits and counting numbers, so they are isomorphic in at least one category.

When we explore the logical consequences of our abstraction of counting numbers, we end up with all the other stuff, like arithmetic and non-counting numbers, I wouldn't call counting numbers more fundamental than all the other stuff, rather, counting numbers are just the lowest hanging fruit. The other stuff was there all along, it's just harder to find. Find where? In the information provided by the universe. I don't think mathematical objects are physical objects; I believe the mathematical objects we conceive of reflect the way the universe is.

Concatenated bits are foundational. Everything else is a conceptualization differentiator.

I would say, the set of states and transformations that we call the universe are foundational. Our concepts come from trying organize the resulting information flow.

Transfinitization is in essence turning infinity into comparative-sized quantities. There IS no size to an infinite set. It is sizeless, or it is not infinite! You can't walk to school OR carry your lunch.

Ah, I now understand, thanks. And I agree that it is unintuitive to think of infinities as having different "sizes". But I'd suggest that this un-intuition is a byproduct of a loose definition of "size".

When normal people mention size in the course of normal conversation, they invariably are referring to a geometric notion of size, such as length or volume. Mathematically speaking, you need a sh!t ton of structure to be able to define geometric concepts such as lengths, areas, and volumes. For example, you need a linear space over a field (like ℝ) with a suitable inner product. Each of those implies a whole bunch of mathematical machinery.

Sets, on the other hand, have almost no mathematical structure. Consequently, there are no geometric notions in set theory; it's basically just counting. And counting the elements in a set tells us the set's cardinality, which is the only notion of "size" in set theory. If set A has three apples, and set B has three galaxies, both sets have the same "size" (cardinality) in set theory. We cannot infer anything about the length or volume of a set, because such ideas don't have any meaning in set theory.

Of course, we still have the problem of counting an uncountable set, and this is what Cantor, et al, made rigorous with transfinites. But this is strictly the domain of set theory.

So, with a suitable measure we can say, without any hint of contradiction, that, the interval [1, 2] ∈ ℝ has half the length (half the size) of the interval [1, 4] ∈ ℝ. But, in set theory, we say that the cardinality of [1, 2] equals the cardinality of [1, 4]. If we don't agree in the truth of that statement, then we get a bunch of inconsistencies elsewhere. The key to keep it intuitive is to remember that "size" in terms of cardinality is very different from "size" in terms of length.

 

I.e., is it not true, that essentially we have binary systems using logic TRUE and FALSE and in the end, these bits containing one element of the set {TRUE, FALSE} are the MOTHER set of all numbers, information and calculation as proven in the hardware and software? "True" meaning switch high, "false" meaning switch low.

That's an ambiguous statement -- what does "MOTHER set of ..." mean precisely?

It's easy to get lost in the many levels of abstraction that we're discussing here. There's a distinction between formal systems, systems of logic, and models of arithmetic. Remember way back when, when I spelled out the difference between boolean logic (where we can do TRUE/FALSE statements) and boolean rings (where can do binary arithmetic)? That was touching on this.

A formal system is the base level of reasoning -- it sets the language, grammar, and rules of the system. Formal systems are purely syntactic. There is huge body of research on formal systems at this base level.

A logic system is a particular type of formal system -- i.e., it has a language, grammar, and rules of inference -- that comes equipped with a set of axioms and a specific form of semantics that we call truth values. Different logic systems come with their own set of axioms and semantics. Propositional (boolean) logic is just one possible model; it is in no way fundamental or privileged.

Mathematical systems are another type of axiomatic formal system. They may come equipped with semantics (boolean rings) or without (ring theory). It's not cogent to say that a mathematical system is based on a logic system -- they're different kinds of formal systems, with different kinds of axioms, elements, etc. Rather, we can sometimes (definitely not always) express mathematical statements as logical statements. But just because a statement is a theorem in a mathematical system does not make it a theorem in a logical system. The distinction needs to be clear.

With all of that in mind, hopefully you can better see that TRUE/FALSE -- which is just one particular semantic for one particular logic system -- is not in any way universal to all formal systems.

Assuming your unary element is actually itself some kind of strange abstraction (based in my other post, I show you need binary states to determine the truth or falsehood of whether or not you have a certain "number of @'s"... you can't even measure the number of bits you can work with in the system without becoming a cave man and grunting for each @ — 1^n style. Lol).

Let x = "@@@";
funtion f = { GRUNT; }

if ( x == @ ) then f();
if ( x == @@ ) then f(f());
if ( x == @@@ ) then f(f(f()));

Never said unary was an efficient representation.

 

Question... using this proof you created as the backbone where we essentially use the "bit" (rather than the u-it, which I think is basically a "special needs" bit), I'd like to know if we can propositionalize my thoughts on this matter and follow your proof... again, tabula rasa to the max. We don't care about existing "formal systems." We're building something new here based on the most basic observations (my text follows below the bold-text existing proof):

Axiom 1: Information is a measurable quantity.
Axiom 2: A state is a particular configuration or arrangement of measurable quantities.
Axiom 3: A process is a mapping between states. Some process P transforms state A to state B according to some rule.

As a measurable quantity, information can be processed. Examples of information processes are transferring and storage, wherein state is copied and saved. Let I represent some particular configuration of information. To transfer and store I, then, means to configure the storage state S to that of I: \[ S \to P(I) \to I \] Axiom 4: A bit is a discrete unit of information; we measure information by counting bits.

A convenient representation for bits is sequences of 1s and 0s; we call such sequences bit strings. There are precisely two possible 1-bit bit strings: "1" and "0".

The information capacity of a state is the amount of information that can be stored in the state. This is equivalent to the count of possible configurations of the state.

Lemma 1: There is a one-to-one correspondence between any state of n possible configurations and a bit string of log2(n) bits.
Proof: A string of n-bits can represent \( 2^n \) different configurations. Taking the base-2 logarithm of both sides gives us the lemma. QED.

Theorem 1: The information capacity of a state is given by the count of bits in its corresponding bit string.
Proof: Using Lemma 1, map the state to a k-bit bit string. Then, by axiom 4, the state has k bits of information. QED.

Essentially, there are special amalgamations of k-bit bit strings. These amalgamations humans call "numbers" or "names," but these are arbitrary tokens, and are the basis of human knowledge and reasoning.

Perhaps we create a novel "binary logic" set called, I dunno — "O," to comprise exactly 2 discrete logic states represented by symbols 0 and 1: {0, 1}.

Then we need another set called, I dunno, "I" to represent the set of all discrete and continuous information in existence, outside of the concept of unbounded infinity, including all existing number sets. Any waves and any discrete quantities coming into a state processor, are digitized and done ON this set by way of k-bit bit strings from set O.

Because O is essentially the foundation of I, the proof is that all other numbers, sets and systems and their derivatives are carried out by representing I "via" O.

And because O is 1 and 0 representing two logic states, the same symbols seamlessly parallel the base-2 integers 1 and 0, and this is how we can effectively use logic states as base-2 integers to do all computation on any discrete or continuous info by this mystery overlap!

 

That's an ambiguous statement -- what does "MOTHER set of ..." mean precisely?

It's easy to get lost in the many levels of abstraction that we're discussing here. There's a distinction between formal systems, systems of logic, and models of arithmetic. Remember way back when, when I spelled out the difference between boolean logic (where we can do TRUE/FALSE statements) and boolean rings (where can do binary arithmetic)? That was touching on this.

A formal system is the base level of reasoning -- it sets the language, grammar, and rules of the system. Formal systems are purely syntactic. There is huge body of research on formal systems at this base level.

A logic system is a particular type of formal system -- i.e., it has a language, grammar, and rules of inference -- that comes equipped with a set of axioms and a specific form of semantics that we call truth values. Different logic systems come with their own set of axioms and semantics. Propositional (boolean) logic is just one possible model; it is in no way fundamental or privileged.

Mathematical systems are another type of axiomatic formal system. They may come equipped with semantics (boolean rings) or without (ring theory). It's not cogent to say that a mathematical system is based on a logic system -- they're different kinds of formal systems, with different kinds of axioms, elements, etc. Rather, we can sometimes (definitely not always) express mathematical statements as logical statements. But just because a statement is a theorem in a mathematical system does not make it a theorem in a logical system. The distinction needs to be clear.

With all of that in mind, hopefully you can better see that TRUE/FALSE -- which is just one particular semantic for one particular logic system -- is not in any way universal to all formal systems.


Let x = "@@@";
funtion f = { GRUNT; }

if ( x == @ ) then f();
if ( x == @@ ) then f(f());
if ( x == @@@ ) then f(f(f()));

Never said unary was an efficient representation.

But they're all tacit backend XOR essentially. It's @, @@, @@@, and one needs to say "@" is TRUE while @@ and @@@ are FALSE in order to get the result we need.

Again, tabula rasa-fication. We don't care about any other system.

I know there are ALLL sorts of abstractions. But I want to find the "MOTHER" one, which is using the simplest number of terms as a direct reflection of the hardware and the connection between binary and the most elemental of informatic representation (this can't be overstated). We have at our disposable a physical substrate and electricity that is high or low.

 

Just an appendage to #866 above:

It’s possible if we set up two sets:

O: {0,1} (logic states)
L: {0,1} (base-2 numbers)

Basically we can prove a 1-to-1 correspondence between both sets in terms of the tabula-rasa most elementary computation rules. Everything from a Univac to a 593-core PC to a D-wise quantum computer is using these implied sets.

 

Side with them on what? Leibniz believed in infinitesimals, which are way[/] outside the realm of integers. Gauss, the greatest of them all, invented non-Euclidean geometry. Cauchy formalized the notion of a limiting process; it is because of Cauchy sequences that we can define ℝ. Kronecker needed to get laid. I don't know Konig is. I don't think Wittgenstein knew much about math.

The post-Cantoral math world.

As Poincaré, who didn't believe in it as well, elegantly said:

"Mathematics is the art of giving the same name to different things."

 

Apparently, the author is considered a crackpot by professional mathematicians. Anyway, in his "proof" there is at least one glaring flaw: he assumes en enumeration of ℚ that preserves the order of ℚ, which cannot be.

I got news for ya. This entire thread is the inadvertent initial by-laws draft for the "International Association of Crackpot Professionals," and two contributors to the thread may or may not be founding members.

 

What does infinity have to do with positive/negative or attraction/repulsion? It's crazy-talk to suggest that 1 and 0, pillars of finitude, are flavors of infinity.

The Atlantic Ocean called... it said it’s having some trouble routing a bill for cremation services from the funeral home to your new address?



Agreed, as are most initial intuitional conjectures. It's also crazy-talk to think there's a spirit or soul, Newton. We live in a crazy world. Our friend Newton—balancing a compass on the raised hairs of his chest at just the right cosines, and attempting to chart the path of Mercury through 3 sidereal Nakshatras—all while suspended from a bungie cord in his candle-lit shower, might just hazard such a statement. In my estimation, we should arrive there after this proof is laid down.

 

You're confusing concepts. What makes binary logic binary is that there are only (not "at least") two different logic states. Conditional branching is an entirely different notion. To wit, you can have three or seven or however many conditional branches you want, irrespective of the number of acceptable logic states. This is so because we can use relation operators, such as "equals to" or "greater than", that evaluate to one of the logical states.

Not from a place of considering a mechanical substrate with voltage high or low. Using two logic states, we've built the entire information age. All complex conditions, NAND's and XOR's are all built from two basic states. A mathematical statement is a TRUE one or a FALSE one in the end.

 

Essentially, there are special amalgamations of k-bit bit strings. These amalgamations humans call "numbers" or "names," but these are arbitrary tokens, and are the basis of human knowledge and reasoning.

What does "special amalgamation" mean within the theory?

Then we need another set called, I dunno, "I" to represent the set of all discrete and continuous information in existence, outside of the concept of unbounded infinity, including all existing number sets. Any waves and any discrete quantities coming into a state processor, are digitized and done ON this set by way of k-bit bit strings from set O.

What do "continuous" and "waves" mean within the theory?

Tabula rasa means we have to define these things.

And because O is 1 and 0 representing two logic states, the same symbols seamlessly parallel the base-2 integers 1 and 0, and this is how we can effectively use logic states as base-2 integers to do all computation on any discrete or continuous info by this mystery overlap!

Just so we're clear, the set {0,1} you're describing is not a subset of the integers ℤ = { ..., -2, -1, 0, 1, 2, ... }. Taken as a number field, {0,1} is called GF(2), the Galois field of two elements. These elements are not integers. Proof: The statement "1 + 1 = 0" is true in GF(2) but not in ℤ, therefore the "1" of GF(2) cannot be an integer.

Again, we are in a conceptual maze of twisty little passages, all seemingly alike. We must be very clear about exactly what we're describing.

 

But they're all tacit backend XOR essentially. It's @, @@, @@@, and one needs to say "@" is TRUE while @@ and @@@ are FALSE in order to get the result we need.

No, we do not need XOR. Nor do we need a notion of FALSE.

Consider the following code:

let x = @@@;

if ( x == @@ ) then f();
if ( x == @@@ ) then g();

The assignment requires us to store the value of @@@. We'll say that any voltage between 0.8 V and 1.2 V represents "@". Our unary computer has RAM, which works like this: the first cell can hold a single "@". The nth cell can hold a single "@" if and only if the (n -1)th cell is holding a "@".

So, the "let x = @@@" assigns "@" to the first three cells in RAM. The fourth cell has no "@"; it is in an undefined state.

The fist IF statement guards the f() function from executing unless the conditional expression evaluates to "@". To test if x equals "@@", the hardware first uses an op-amp to sum the values of all the cells in RAM; this is tied to the reference input of a comparator. The other input is tied to the output of an op-amp that sums the value of "@@". Since the comparison fails (@@ < @@@), the comparator outputs an undefined voltage and the IF expression fails.

In the next IF statement, the same thing happens again. This time the comparator outputs "@" and the g() function is executed.

Obviously, this is not by any stretch an ideal way to compute, but the point is to show that computation is possible without XOR or a FALSE state.

Here's the thing that bothers me, though. If you understood that physical computation is just manipulation of representational objects, then we wouldn't need this tangent. You clearly recognize that we can "add numbers" in any base. I use scare quotes because we're not really adding numbers, we're just manipulating symbols (representations). When adding base-10 representations, we use different rules than when adding base-2 representations. You get that, right? No matter what base we use, we always get the same number (represented in that base). Whether we do "2 + 3 = 5" or "010 + 011 = 101", we used different manipulations but got the same number.

For some reason, you seem to have no problem understanding that concept when it's applied to numerals (symbolic representations), but fail to see that it's precisely the same thing when applied to voltages (physical representations). Adding two voltages can be done in unary (base-1), binary (base-2), ternary (base-3), 42-ary (base-42). It makes no difference to the math whether we represent numbers with symbols or with voltages or with rocks.

Once you grok that representation doesn't matter -- no matter what the medium -- then it's plain-as-day obvious that general computation can take any form: unary, binary, analog, whatever. And then it becomes perfectly clear that {0, 1}, XOR, TRUE/FALSE, and all the rest are not fundamental.

I know there are ALLL sorts of abstractions. But I want to find the "MOTHER" one, which is using the simplest number of terms as a direct reflection of the hardware and the connection between binary and the most elemental of informatic representation (this can't be overstated). We have at our disposable a physical substrate and electricity that is high or low.

An obvious criterion for "the simplest number of terms" is the least amount of terms, in which case base-1 takes the cake. But I wouldn't call base-1 the "MOTHER" of anything -- they're all on equal footing, one base is not more special than the other.

Again, to my original point, mathematics -- the language of computation -- is a formal system, distinct from logic systems. We often cast mathematical proofs in the language of logic, but the math itself is not a logical system (like boolean logic). They have entirely different kinds of domains, axioms, and theorems. I'm confident that if you get that clear you'll stop trying to shoehorn math and computation into a boolean framework.

 

Just an appendage to #866 above:

It’s possible if we set up two sets:

O: {0,1} (logic states)
L: {0,1} (base-2 numbers)

Basically we can prove a 1-to-1 correspondence between both sets in terms of the tabula-rasa most elementary computation rules. Everything from a Univac to a 593-core PC to a D-wise quantum computer is using these implied sets.

Your last example is a very interesting counter-example. Whereas a bit can only be in one of two distinct states, a qubit can be in a complex-valued superposition of states. In other words, there is no 1-to-1 correspondence between a qubit and the set {0,1}. Therefore, {0,1} cannot be fundamental. QED.

 

What does "special amalgamation" mean within the theory?


What do "continuous" and "waves" mean within the theory?

Tabula rasa means we have to define these things.


Just so we're clear, the set {0,1} you're describing is not a subset of the integers ℤ = { ..., -2, -1, 0, 1, 2, ... }. Taken as a number field, {0,1} is called GF(2), the Galois field of two elements. These elements are not integers. Proof: The statement "1 + 1 = 0" is true in GF(2) but not in ℤ, therefore the "1" of GF(2) cannot be an integer.

Again, we are in a conceptual maze of twisty little passages, all seemingly alike. We must be very clear about exactly what we're describing.

If you mention the likes of Galois, ℤ or similar one more time, I’m coming to your house and will sit Indian-style on your lawn chanting “REALITY, REALITY, REALITY” over and over while wearing a custom $2000 silicon Kronecker mask.


Rasa. Remember what we agreed to earlier. You’re in safe mode, it’s the year 1200 AD. You’ve never heard of any of these things!

 

No, we do not need XOR. Nor do we need a notion of FALSE.

Consider the following code:

let x = @@@;

if ( x == @@ ) then f();
if ( x == @@@ ) then g();

The assignment requires us to store the value of @@@. We'll say that any voltage between 0.8 V and 1.2 V represents "@". Our unary computer has RAM, which works like this: the first cell can hold a single "@". The nth cell can hold a single "@" if and only if the (n -1)th cell is holding a "@".

So, the "let x = @@@" assigns "@" to the first three cells in RAM. The fourth cell has no "@"; it is in an undefined state.

The fist IF statement guards the f() function from executing unless the conditional expression evaluates to "@". To test if x equals "@@", the hardware first uses an op-amp to sum the values of all the cells in RAM; this is tied to the reference input of a comparator. The other input is tied to the output of an op-amp that sums the value of "@@". Since the comparison fails (@@ < @@@), the comparator outputs an undefined voltage and the IF expression fails.

In the next IF statement, the same thing happens again. This time the comparator outputs "@" and the g() function is executed.

Obviously, this is not by any stretch an ideal way to compute, but the point is to show that computation is possible without XOR or a FALSE state.

Here's the thing that bothers me, though. If you understood that physical computation is just manipulation of representational objects, then we wouldn't need this tangent. You clearly recognize that we can "add numbers" in any base. I use scare quotes because we're not really adding numbers, we're just manipulating symbols (representations). When adding base-10 representations, we use different rules than when adding base-2 representations. You get that, right? No matter what base we use, we always get the same number (represented in that base). Whether we do "2 + 3 = 5" or "010 + 011 = 101", we used different manipulations but got the same number.

For some reason, you seem to have no problem understanding that concept when it's applied to numerals (symbolic representations), but fail to see that it's precisely the same thing when applied to voltages (physical representations). Adding two voltages can be done in unary (base-1), binary (base-2), ternary (base-3), 42-ary (base-42). It makes no difference to the math whether we represent numbers with symbols or with voltages or with rocks.

Once you grok that representation doesn't matter -- no matter what the medium -- then it's plain-as-day obvious that general computation can take any form: unary, binary, analog, whatever. And then it becomes perfectly clear that {0, 1}, XOR, TRUE/FALSE, and all the rest are not fundamental.


An obvious criterion for "the simplest number of terms" is the least amount of terms, in which case base-1 takes the cake. But I wouldn't call base-1 the "MOTHER" of anything -- they're all on equal footing, one base is not more special than the other.

Again, to my original point, mathematics -- the language of computation -- is a formal system, distinct from logic systems. We often cast mathematical proofs in the language of logic, but the math itself is not a logical system (like boolean logic). They have entirely different kinds of domains, axioms, and theorems. I'm confident that if you get that clear you'll stop trying to shoehorn math and computation into a boolean framework.

Again, none of it exists, lol. We must write from what we agreed to earlier! We have no knowledge of frameworks!
We call it “base whatever” AFTER. We have 2 sets all computers work with. Assume qubits are nested bits. They are still digital computers, just more bits per space.

You wrote up that lovely proof tabula RASA. same here!

You agree to it, then you default back to Banachian transgendered ordinals of the 4th order of a Timbuckthree. I'm not trying to put any of that down(!) But you agreed to simple reasoning, simple inference, simple arithmetic, for a REASON that you share with me, or you wouldn't have agreed it was a good starting point. ;--)

 

You say that you begin with the tenet "There is Information "as your starting point, which essentially means, "that's all there is": there is no such thing as a separate "knowable element" aside from it. It's all states, there are no absolutes.

The problem is, tabula rasa, you (we as humans) don't behave or think that way all day long.

If any fellow reasonable human being asks you, "Is your name Javier?" you don't say, "Well, according to Cantor, at this very vector in time, I'm 8 levels deep in a 2D array, connected to a Hilbert space that I'll get back to you on." Even that statement's MEANING to you is a "true" statement or "false"! (it's TRUE I'm 8-levels deep..., or FALSE).

You say "TRUE" or "FALSE." If you DON'T, no one, including yourself know ANYTHING MORE about "Javier" until I say "It is TRUE he exists independent of all additional information." If I say, "Do you have family?" You say "TRUE" or YES. I defy you to try out the "information-only" stance and not say any one of them exist independent of the information you use to describe them on some basic level. ;--) "Dad, am I your daughter?" "I'll get back to you honey. Objective truth on this matter is not my belief." In fact, ONE CLOCK CYCLE of delay on that, and you could literally damage her psychologically or cause her great pain, because all information is based on absolutes of *whether or not it's true they exist* to talk qualitatively further about them! This is why I wrote those two stories, to illustrate the essential ludicrousness of not having any basic TRUTH stance concerning things happening in existence.

Would you like to talk about Eddie Van Halen and his guitar playing abstractions before we both decide it is TRUE he exists apart from us? "It's false he exists. I will not entertain additional information about him."

You make the statement "There is information" as your starting point. But what you're really saying "It is TRUE there is information." If you say "False" you can't make any further statements about it. Even the basis of conjecture is binary. QED. A question is, "Does X exist?" The answer is YES or NO. In addition, when you say "It is TRUE there is information, you're also saying it is TRUE you exist to ask the question, and TRUE that light exists to give it to me." If you say FALSE to any of them, there is no additional information to know or compute upon.

Further, you need to assume it is TRUE that light EXISTS independent of information, or you can't know anything more. You can't say "maybe" it exists, or "maybe information exists." LIGHT exists in order for you to KNOW and COMPUTE. You must start from a binary place of YES or NO. QED.

100% proof that reason is first binary, as in "true or false" until you qualify further:

You cannot know what anything is until you output a true or false about whether or not it exists. QED.

This is why tabula rasa means we don't invoke any higher abstractions built on the foundation that all computational systems use. We have nothing more than voltage on the line or not on the line. The voltage exists or it does not.

You can't say BASE-384 until you say IT IS TRUE it exists from a binary place. There are two options before you go further with any form of scientific inquiry.

A qubit is nothing more than a state processor that is evaluating more states per second than a classical system (proof below). It's superpositional Shannon bits all day long. A qubit is still outputting ZEROES and ONES, or FALSE OR TRUES *in the end*, or you can't get any reasonable data from it. Quantum computers use binary, but each switch can have multiple binary states at the same time.

A QED *IS* an undeniable... TRUTH. QED. You must make the statement, "It is true I exist," before you make any mathematical inquiry.
In fact, this is so important, it must be the foundation of the ToE, lest no one believe it exists!

Lol

 

Qubits are described in terms of binary states, because TRUE or FALSE is the foundational basis of sane, mathematical reasoning!

 

A mathematical statement is a TRUE one or a FALSE one in the end.

FALSE, though this is far more nuanced than you might think. First, we have to be clear about what a "mathematical statement" means. By definition, a mathematical theorem is always true and a mathematical contradiction is always false. But, in general, mathematical expressions do not carry truth values. For instance, the polynomial \( x^2 -3x + 1\) is neither true nor false. We can put it into an equation and make it look like a statement: \[ x^2 -3x + 1 = 0 \] But, again, the equation itself is neither true nor false. It is not a proposition.

Truth values are the domain of logic systems. Thus, to make an arbitrary mathematical expression have a truth value, we need to frame it as a statement in some system of logic. A naive way of doing this is to simply couch a mathematical expression in a propositional statement:

It is true that "y = 2x"

But whether that statement is true, false, or undefined depends on the domain of discourse and what we mean by 'y', 'x', and the '=' symbol. We can choose any number of models in which the statement is true, false, or undefined. We can even choose a model in which the statement is neither true nor false nor undefined (e.g., as the definition of a function). Therefore, the truthness of this mathematical statement depends on the given model.

And this brings up several subtle and important issues. Assuming we fix a model, can every mathematical statement -- i.e., every logical statement about mathematics -- be said to have a TRUE or FALSE value? The answer, unfortunately, is no.

In basic boolean (propositional) logic, every statement is decidable. That is, for any propositional statement -- no matter how complex -- we can decide whether it is true or false within the model. This is a very nice property of boolean logic (especially suitable for designing computer circuits). Alas, the price of decidability is lack of expressive power. For example, basic boolean logic is not powerful enough to make general statements about the natural numbers ℕ. We can't, for instance, express the notion of an "even number" within propositional logic. So, if we wanted to make a truthy mathematical statement about even numbers -- say, "every even number has the form 2k for any integer k" -- we can't do it in propositional logic.

To get more power into propositional logic, we have to extend it with quantification over variables, giving us predicate logic. Instead of just "A" and "not A" and such, we can express ideas like "for all A". But with this power comes a dark side -- first-order logics (like predicate logic) necessarily include statements that are undecidable. That is, there are statements for which we cannot say whether they are true or false, regardless of the chosen model.

In short, we need (at least) first-order logic to make truth statements about "complicated" mathematical theories, and any such logic necessarily contains undecidable statements. For example, a famous open problem in mathematics is the Goldbach conjecture: every integer larger than 2 can be expressed as the sum of two primes. We can write this as a statement in first-order logic. Let \( \mathbb{P} \) be the set of primes, then \[ (\forall n \in \mathbb{N}_{> 2})(\exists p,q \in \mathbb{P})( n = p + q) \] We don't know if this statement is true or false. It may turn out to be undecidable.