Still the same old addition at every step, though. Just more of it.

But it's not as if it's the same steps, just more of them. You're ignoring the fact that different steps have to be taken to perform the addition.

Let me be very clear about this. Addition is defined in terms of the set of things that can be added, and by the axioms that addition obeys. Changing either the set or any of the axioms changes the operation. This is indisputable. It's the reason why 1 + 1 = 0 in a boolean ring, but 1 + 1 = 2 in an integer ring. They can't both be the same "+" operation, otherwise we'd have a contradiction in math. All of math would break down!

The way we add things depends on what we're adding. For example, if we're adding the velocities of two objects, we're performing vector addition. This is a different type of addition than adding, say, apples. Indeed, relativistic velocity addition isn't even commutative! You say that "addition is addition", but if \( u \) and \( v \) are the velocities of two objects, then relativity tells us that \( u + v \ne v + u \). This can't be the case for adding apples!

I tell you, while that might be a "true statement" on the "surface," it is an illussory fallacious implication at its core and is in violation of the core laws of reason, and clouding the true nature of numbers. This makes it sounds like ℚ is more foundational than ℕ, which has Herr Kronecker rolling in his grave.

Illusory and fallacious implication? Then it should be easy to contradict. Go ahead and try. ;--)

A MUCH better question is: How would you prove every element of ℚ is a derivative of compounded combinatorial elements of, and processes upon ℕ, the counting numbers that are the very building blocks of every fractional expression? And if you could, would you not then conclude ℚ is derivative of ℕ, rather than ℕ a subset of ℚ?

What you don't seem to get is that all numbers are "derivative" abstractions. We derived the numbers in ℕ from abstracting the counting process. We derived arithmetic over ℕ from counting subsets: I have two rocks in my left pocket, three rocks in my right pocket, so I have a total of five rocks: 2 + 3 = 5. Take three rocks away from my five and I'm left with two: 5 - 3 = 2.

Arithmetic over ℕ gave us the capability to generalize problems, and so find general solutions to problems. Once we understood this abstraction, we realized that the numbers in ℕ were insufficient to answer all of the different types of problems we could pose. The history of the number sets is the history of humanity learning how to answer increasingly general mathematical questions:

\[ \begin{align} x + 1 = a \qquad &\to \qquad \mathbb{N} \\ x + a = b \qquad &\to \qquad \mathbb{Z} \\ ax + b = c \qquad &\to \qquad \mathbb{Q} \\ ax^2 + bx + c = d \qquad &\to \qquad \mathbb{R}, \mathbb{C} \end{align}\] It's not that the solutions to \( x + 1 = a\) are more fundamental than the solutions to \( x^2 + 1 = a\), it's just that we need more numbers to be able to solve the latter than the former.

What you're calling pre-production I call "breaking down into their TRUE elemental states" to perform the multiple level of addition (same addition, just multiple levels).

What you're describing doesn't actually work. For example, we can think of a number in \( \mathbb{C} \) as a pair of two numbers in \( \mathbb{R} \). Taken strictly as sets, with no additional structure, then \( \mathbb{C} \) is indeed isomorphic to \( \mathbb{R} \times \mathbb{R} \). We can even include more structure to the sets by defining addition between elements of the sets and scalar multiplication, in which case \( \mathbb{C} \) as a vector space is isomorphic to \( \mathbb{R}^2 \) as a vector space. We know this because both vector spaces have two basis vectors; they are two-dimensional vector spaces, and any n-dimensional vector space is isomorphic to any other n-dimensional vector space.

However, to do general arithmetic on these sets, we need to introduce even more structure and treat \( \mathbb{C} \) and \( \mathbb{R}^2 \) as rings or fields. And as soon as we do this, we find that \( \mathbb{C} \) and \( \mathbb{R}^2 \) are incompatible. Indeed, \( \mathbb{C} \) is a full-fledged field, while \( \mathbb{R}^2 \) is not even an integral domain (because it has zero divisors). In particular, multiplication over \( \mathbb{C} \) is incompatible with multiplication over \( \mathbb{R}^2 \).

So, there is no sense in which arithmetic over \( \mathbb{C} \) can be "broken down" to arithmetic over \( \mathbb{R} \). We can repeat this story with arithmetic over the rings of nxn matrices, or arithmetic over boolean rings, or whatever. Fundamentally, these objects are all different, and so their arithmetic is necessarily different.

No, I want to see what "what we are calling numbers" are at their deepest essence . . .

At the deepest essence, numbers are abstractions of abstractions. We find that there are logical consequences to these abstractions, and we categorize them into number sets based on these consequences (their properties).

The numbers are molecules — I want to study the atoms to figure out how the molecules behave.

If numbers are molecules, then their atoms are particular state configurations.

 

A "sine" has infinite values. A "sine" is being called 1 "thing" here, despite "infinite values," known or unknown.

No, not at all. "Sine" is an equivalence class of innumerable things. This fact is explicit in its description: \( \sin(\omega_0 t + \phi \) describes not a single thing, but an uncountable number of mathematical objects, each of which is comprised of an uncountable number of different values.

The presence of the sine is still conveying one piece of information: its existence.

Again, no. Mr. Al had some good points here -- noting the "presence" of a sine includes a whole bunch of other information, not the least of which is knowing what a sine is in the first place. There is no sense in which "presence" can adequately characterize the thing being present. If things are different, then "presence" alone is insufficient to distinguish between them.

Remember, a "thing" gets its entire conceptual basis from physical space. We first learn to call things "things" from space.

I disagree for two reasons. One, we don't in fact know how we conceptualize things -- it's possible that we conceptualize physical space itself independently from physical space (i.e., there need not be a physical space as we perceive it). Two, purely abstract concepts, such as mathematics, do not require a physical referent.

The concept of "thereness" I think is something that MUST be folded into the concept of truth.

We can't even be sure a thing is "there" to begin with.

 

And this is why I posed the original problem in the first few pages: Where is the 3D cube with infinite points as visually described in a discrete state-processing machine? Not in some vector space. I'm talking about in physical space, in the brain. Nowhere that it is aware of. The values are discrete. Is a Euclidean line with infinite values existing "somewhere" as a stand-alone geometric entity within the mind? Sure appears that way to me.

You're assuming that a cube with infinite points is a physical thing, external to the brain. But no such thing exists! Euclidean cubes only exist as abstract concepts.

 

Axiom 1: A physical machine is defined as such by using a definition that involves making a distinction between information and a physical space referent as its basis.

Axiom 2: Numbers are non-physical concepts.

Axiom 3: There are no non-physical things in a machine — all is physical, that's why numbers don't exist.

Axiom 4: You are a machine.

Theorem: If there are numbers in you, you are not a machine. QED.

;--)

Never said there were numbers in me. ;--) Numbers don't have physical existence as numbers. The concept of "government" is not a physical thing -- we can't, unfortunately, pick up a government, turn it upside down, and shake it. Do governments exist? Yes, as abstract concepts. Do humans use these abstract concepts when they interact with the physical world? Yup. "Where" is the government? In our brains.

 

But there's another one of those "inverta-proofs". You are a physical device — every constituent element of every concept in the physical device must be represented by physical states. There is no self-referencing "information" without each discrete thing having a physical representation in physical space.

So what? I don't know what you're arguing for/against. In my brain there are physical states that correspond to my conception of "government". And?

 

Axiom: Existence is an uncomputed state, and is ontologically the foundation of all logic.

Any human being, before constructing or identifying any mathematical frameworks must first declare the truth that he or she exists, and by extension, they exist to do so. Prior to one’s birth, one cannot be proven to physically exist in physical space. After this point in time, per axiom 1, one’s existence is the basis of propositional truth. QED

yes?

 

Never said there were numbers in me. ;--) Numbers don't have physical existence as numbers. The concept of "government" is not a physical thing -- we can't, unfortunately, pick up a government, turn it upside down, and shake it. Do governments exist? Yes, as abstract concepts. Do humans use these abstract concepts when they interact with the physical world? Yup. "Where" is the government? In our brains.

No, no, no!! :$-) NOT without a “spatializer” or “conscious knower” unifying those states ”as one”! Every physical element must be addressed to define the concept! You want to say numbers exist apart from their discrete constituent physical elemental things!

You are drawing imaginary lines between states and calling them “supra-states!” No can do! THE most important element in this thread thus far.

 

To a machine they are the same thing. To say that voltages and switches have a "partiality to frameworks" is not elemental thinking when it comes to understanding this ontologically.

But clearly the machine treats the various groups of switches differently. You keep saying that the machine doesn't know the difference, but clearly it does!

"Oh, but a human programmer told it the difference." So what? The machine is treating the groups of switches differently -- that's all that matters at the elemental level. How it got there -- whether it was programmed by humans, or by genetic code, or by parents and teachers -- is just details.

In the same way I don't think it's proper to see "N” as a subset of Q, but that Q is an algorithmic derivative of N.

You're free to think it's not "proper", but it is indisputable that N is a subset of Q. By the definition of subsets, a set A is a subset of B if and only if every element of A is an element of B. So all you need to do is pick a number from N that is not in Q and you will have proved your argument! Let me know when you find one. ;--)

(In case you it helps, Kronecker believed in Q as much as he did in N.)

 

So what? I don't know what you're arguing for/against. In my brain there are physical states that correspond to my conception of "government". And?

We must grind this out before moving further. This is the entire core issue we’re discussing since day 1 here.

But clearly the machine treats the various groups of switches differently. You keep saying that the machine doesn't know the difference, but clearly it does!

"Oh, but a human programmer told it the difference." So what? The machine is treating the groups of switches differently -- that's all that matters at the elemental level. How it got there -- whether it was programmed by humans, or by genetic code, or by parents and teachers -- is just details.

The computer nor brain does not exist as “1” thing to you if you are the machine doing the observing, in a discrete bit machine. We seriously need to grind this out, because it is the entire issue here.

You're free to think it's not "proper", but it is indisputable that N is a subset of Q. By the definition of subsets, a set A is a subset of B if and only if every element of A is an element of B. So all you need to do is pick a number from N that is not in Q and you will have proved your argument! Let me know when you find one. ;--)

(In case you it helps, Kronecker believed in Q as much as he did in N.)

Yes, it is an illusory magic trick of illusory amalgamations based on a machine treating groups of states with more weight than others.

If an apple is a number (a grunt!), and a bushel of apples is made up of several apples, an apple is a set of a bushel? Or is the bushel a combinatorial derivative of the definition of an apple? The latter!

Q can “exist!“ But it does NOT exist without more foundational processes. If a number is defined by a unary GRUNT, please tell me how N is derivative of Q, if Q requires MORE grunts to define each element and contains more algorithms using the same grunts?

 

Ultra-important element here in my mind:

Information is separate from your brain.

You are your brain.

Your brain is insisting information exists. But with information you can't insist your brain exists.

Do tell!?

 

You insist on using frameworks to reason, so let us create our own. No prior knowledge other than basic logic, arithmetic and inference.

AXIOM: A number is a grunt

Yes?

 

But clearly the machine treats the various groups of switches differently. You keep saying that the machine doesn't know the difference, but clearly it does!

"Oh, but a human programmer told it the difference." So what? The machine is treating the groups of switches differently -- that's all that matters at the elemental level. How it got there -- whether it was programmed by humans, or by genetic code, or by parents and teachers -- is just details.

It operates with a programmatic difference. It does not “know” it operates, because the brain is an object in physical space that you have not defined (a la “dog”), nor know how it relates to information, but insist it knows what information is to define itself, when information is separate from itself!! ??

 

But clearly the machine treats the various groups of switches differently. You keep saying that the machine doesn't know the difference, but clearly it does!

"Oh, but a human programmer told it the difference." So what? The machine is treating the groups of switches differently -- that's all that matters at the elemental level. How it got there -- whether it was programmed by humans, or by genetic code, or by parents and teachers -- is just details.


You're free to think it's not "proper", but it is indisputable that N is a subset of Q. By the definition of subsets, a set A is a subset of B if and only if every element of A is an element of B. So all you need to do is pick a number from N that is not in Q and you will have proved your argument! Let me know when you find one. ;--)

(In case you it helps, Kronecker believed in Q as much as he did in N.)

I’m going to take this time to address the audience. Lol.

THIS is at heart of the philsophical debates created with the post-Cantoral world of sets and transfinites.

The difference between defining a number as a stand-alone (a “1”) element vs. only seeing it as a member of a set is to make the concept of God, which is “1“ and “infinity“ combined, to be dependent upon external classification, or an effect of some other concept other than God defining what God is. Cantor was clear to state he still believed in the former, but believed this other relationship was also true. I would call this the basis of “fantasy truth,” where a transfinite number is a bizarre-concept that lies in between numbers and infinity.

We have therefore two simultaneous truth systems. But I aim to prove the Cantoral system is a derivative of the original absolute one.

If we start from the premise that a number is a stand-alone label and not part of a set, how does this affect your reasoning?

 

If you were tasked to prove infinity is a non-numeric stand-alone thing composed of endless elements and not a process, how might you go about that?

That task doesn't make logical sense to me. You might as well ask me to prove that ham is an ocean of elephant ears.

To me the sine wave IS conveying information... precisely that it exists.

Think about that statement carefully. In order to know that a particular sine wave exists, you need to already know what a sine wave is, which means you already need to know what a real-valued periodic function is, which means you already need to know what ℝ is. That's an enormous amount of prerequisite information before "it exists" can be conveyed.

Think about the original conception of sine. Before we knew of sine waves, we knew of triangles. While exploring the mathematical consequences of right triangles, we found that the ratios of the lengths of a triangle of some given angles always stay the same, regardless of how big or small the triangle is. In other words, we noticed that these ratios are a property of triangles, which is a useful thing to know. As the the values of the ratios change depending on the given angle, the values of the ratios are a function of the angle, and so were able to abstract these trigonometric properties from the triangle and treat them more generally as functions.

Any way you look at it, sine is an abstraction of an abstraction.

 

Axiom 1: Information is a measurable quantity.

It is the foundation of what it means for a human to know anything about anything.

Axiom 2: Infinity is not a measurable quantity.

Lemma: Infinity is not information.

Theorem: INFINITY is not a measurable quantity, and therefore not information per axiom 1. But if information can be used to describe the very existence of infinity, information must be derivative of it in some way.

Here's your argument with "government" replacing "infinity":

Information is a measurable quantity.
Government is not a measurable quantity; therefore, government is not information.

Thereom: GOVERNMENT is not a measurable quantity, and therefore not information. But if information can be used to describe the very existence of government, information must be derivative of government in some way,

Are you willing to accept that information must be derived from government? If not, then your logic for infinity doesn't work.

In this respect, Euclid’s definitions can axiomatically stand-alone before a vector space computation is used to describe them.

If your goal is to show that geometry -- i.e., the mathematical abstraction of spatiality -- is somehow more fundamental than a vector space -- i.e., the mathematical abstraction of linearity -- then you're doomed to fail.

Note that the linearity of vector spaces has nothing to do with the lines of geometry.

 

The thing is, this proof is not taking into consideration that numbers can all be represented as base 1. Why is this important here? Because of the parity of positional representation ontologically with respect to strings of logic states.

You're not getting this. You say "string of logic states" as if that has some kind of independent physical existence. In order to make strings, we need a language to define the symbols and syntax we use for constructing the strings. In the language of propositional logic, a string such as "FFTTFT" is not well-formed -- it has no meaning in the language.

Can we create a language in which "FFTTFT" is well-formed? Sure. The elevator control system of a six-story building might use a protocol that has such strings. It might associate each floor with a position in the string, and interpret the corresponding symbol as determining whether or not it should stop on that floor.

We might create a language in which each symbol in "FFTTFT" is taken to represent a 0 or 1, each position (starting from the right) is weighted by increasing non-negative powers of 2, and the resulting value treated as an integer. Using such a language, we could define how strings are added, subtracted, multiplied, etc. Notice, however, that our language is not the language of logic and so -- crucially -- it is not using "logic states". Just because some of the symbols in our language look like some of the symbols in logic languages does not make them the same types of things. The language itself defines what they are. This is why the same string in elevator language can mean something entirely different in "binary computation" language.

Can you grok this uber fundamental distinction?

 

Axiom: Existence is an uncomputed state, and is ontologically the foundation of all logic.

What does "uncomputed state" mean? What does "foundation of all logic" mean? It sounds way too important to be stuffed into an axiom. Remember, axioms are the very basic building blocks for a theory.

Any human being, before constructing or identifying any mathematical frameworks must first declare the truth that he or she exists, and by extension, they exist to do so. Prior to one’s birth, one cannot be proven to physically exist in physical space. After this point in time, per axiom 1, one’s existence is the basis of propositional truth.

This seems like an awkwardly worded version of cogito ergo sum. Regardless, your conclusion -- existence is the basis of propositional truth -- whatever that means, does not follow from the premise.

 

Every physical element must be addressed to define the concept!

What does this mean? Precision of language, please. When you say "physical element", do you mean the gross, high-level physical elements -- like gates and neurons -- or the ground-level physical elements, the raw physical stuff? When you say "addressed", do you mean as in "we need to address this in our discussion", or do you mean like a CPU addresses a location in RAM?

You want to say numbers exist apart from their discrete constituent physical elemental things!

Again, I don't understand what you mean. I want to say what now?

You are drawing imaginary lines between states and calling them “supra-states!” No can do! THE most important element in this thread thus far.

Please elaborate because I don't know what you're trying to say. States-within-states seems fundamental to me, and I don't think it's deniable. The way heterogenous molecules mix in a liquid or gas depends on this state-of-states phenomenon, and it has nothing to do with consciousness or some "spatializer". So, unless you're arguing that liquids and gases don't exist without a human to conceive of them, you'll need to accept the notion of "supra-states".

 

The computer nor brain does not exist as “1” thing to you if you are the machine doing the observing, in a discrete bit machine.

Huh? I don't have a clue what "exists as '1' thing" is supposed to mean. A computer knows the difference between groups of switches. <-- If you do not believe so, then speak directly to that. Forget that I exist, it's just you and the computer. Does it know the difference between switches?

Yes, it is an illusory magic trick of illusory amalgamations based on a machine treating groups of states with more weight than others.

Well, in this sense it's all illusory -- there is no ℕ, either.

If a number is defined by a unary GRUNT, please tell me how N is derivative of Q, if Q requires MORE grunts to define each element and contains more algorithms using the same grunts?

There are precisely as many grunts in N as there are in Q.

Both N and Q (and R and all the rest) come from processes. There is no fundamental bedrock of numbers that existed before the abstraction of processes. We make number sets to solve equations. Different types of equations need different types of numbers. That's it.

 

Information is separate from your brain.

"Separate" implies a spatiality that doesn't make sense at this level. My brain is a set of states that store (and convey) information.

Your brain is insisting information exists. But with information you can't insist your brain exists.

The second sentence is incoherent. "With information I can't insist that my brain exists." I have no idea what you're trying to say.