You never answered my question: do you believe that negative integers are numbers?

Post 1562 :--)

 

Because one cannot measure the totality of information in a sine wave, and because qubits cannot be measured despite their working with information, I propose:

INFORMATION An immeasurable infinite continuum that may be partially discretized into quantities

Both "infinite" and "continuum" are far too conceptually loaded for me to accept in an axiom.

However, sine waves are an interesting informational case study. On one level, a sine has infinite values, but on another level, a sine has only three degrees of freedom: we can change its amplitude, its frequency, or its phase. Once these are fixed, everything about the sine is known. Consequently, a single sine wave conveys precisely zero information. Like a neverending sequence of a single symbol -- "11111111111..." -- a sine doesn't have enough structure to convey anything.

In order to convey information, a sine wave must change. We can modulate a sine's amplitude, frequency, or phase with an information source, and thereby convey the information to a receiver, but in doing so we destroy the sine. In other words, for non-constant functions f and Φ, neither of these are sines: \[ f(t) \sin(\omega_0 t) \qquad \sin( f(\omega) t) \qquad \sin(\omega_0 t + \phi(t)) \] There is an informational distinction between the infinite values of sine being "surprise values" and the infinite values of sine being completely known. In the former case, the recipient would presumably learn that the signal is not changing -- not conveying information -- after several periods of repetition, so the "surprise" aspect seems to be a temporary effect. Once a sine is understood (conceptualized), the sine becomes a zero information phenomenon.

 

Post 1562 :--)

Ah, I completely missed that post, thanks.

 

Both "infinite" and "continuum" are far too conceptually loaded for me to accept in an axiom.

However, sine waves are an interesting informational case study. On one level, a sine has infinite values, but on another level, a sine has only three degrees of freedom: we can change its amplitude, its frequency, or its phase. Once these are fixed, everything about the sine is known. Consequently, a single sine wave conveys precisely zero information. Like a neverending sequence of a single symbol -- "11111111111..." -- a sine doesn't have enough structure to convey anything.

In order to convey information, a sine wave must change. We can modulate a sine's amplitude, frequency, or phase with an information source, and thereby convey the information to a receiver, but in doing so we destroy the sine. In other words, for non-constant functions f and Φ, neither of these are sines: \[ f(t) \sin(\omega_0 t) \qquad \sin( f(\omega) t) \qquad \sin(\omega_0 t + \phi(t)) \] There is an informational distinction between the infinite values of sine being "surprise values" and the infinite values of sine being completely known. In the former case, the recipient would presumably learn that the signal is not changing -- not conveying information -- after several periods of repetition, so the "surprise" aspect seems to be a temporary effect. Once a sine is understood (conceptualized), the sine becomes a zero information phenomenon.

If we assume infinity is not a process, but as a stand-alone concept apart from quantity, for a moment...

To describe a sine wave spatially, we draw an S-shape element on a 2D material plane, or gesticulate an undulation in air to communicate the meaning of what it is.

Could it be posited, that if a sine wave has infinite values, it could exist as an infinite something that is not a function of something else and can be visualized internally as something?

 

A computer is NOT seeing the fraction 2/5 + 3/4 in the same way a human does, for the love of all that is holy.

How do humans see it? We use an algorithm to compute the sum of fractions. The computer also uses an algorithm. In both cases, the algorithm for "+" between fractions is different from the algorithm for "+" between natural numbers.

Someone once said "THERE ARE NO NUMBERS IN A COMPUTER." ;--)

That must have been a wise person. Wise, I tell you! ;--)

What precisely IS 2/5 + 3/4 to a computer? It is nothing but binary bits put through numerous (bill) gates.

Indeed, but we have to put them through different gates than we do for adding natural numbers. If there were only one addition, why would we have to do that?

First it is put through an insane number of steps to convert each term of those fractions into discrete binary logic states. Then it evaluates each term through human-induced sequential binary instructions, an algorithmic process that renders each term into a string of unique bits that represent the fraction by putting each string through special gates that do division (which is multiplication and therefore addition in disguise) through most likely NAND gates. THEN it takes both terms and puts it through an adder.

Depending on the algorithm and hardware involved, the computer might convert the fraction into scaled natural numbers, add those, and then deal with the denominator as a natural division, possibly with a remainder. Or it might use a floating-point unit to treat each fraction in the sum as a division operation, and then add the floating-point values. The floating-point rules of addition are different than the "+" of base-2 integer arithmetic.

Is there a proof of this? I'd say ℚ is a subset of processes upon ℕ, since ℚ is composed of integer algorithms.

A proof that ℕ ⊂ ℚ? Pick any element in ℕ, call it n. Then n/1 = n is in ℚ.

A different circuit which in the end is performing some kind of addition? As if the "i" is something special to the computer and is seen "conceptually" as different? As if (5 + 3i) + (4 + 2i) = (9 + 5i) is not just more of the same addition?

It's not the same addition, which is demonstrated by the fact that a computer must do "pre-production" on the numbers in order to treat them as if they were the same addition.

. . . (which, by the way, to answer your question — no, I don't believe in negative numbers, although I will use them colloquially). I believe in the subtraction process, which is shown in a computer to be a form of addition using 2's complement algorithm. 5 is a number. -5 says "subtract the number 5 wherever it is found." "5 + -5" is 0. "5 - -5" is 10. The negative sign is yet another operator assigned to the pure number of 5.

It's strange to me that you would want to go back to iron age mathematics, where the only acceptable numbers were counting numbers. We've learned a lot since then, but it's your choice.

 

How do humans see it? We use an algorithm to compute the sum of fractions. The computer also uses an algorithm. In both cases, the algorithm for "+" between fractions is different from the algorithm for "+" between natural numbers.

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

Indeed, but we have to put them through different gates than we do for adding natural numbers. If there were only one addition, why would we have to do that?

Again, it's different levels of addition upon various components of the numbers. Addition all the same.

Depending on the algorithm and hardware involved, the computer might convert the fraction into scaled natural numbers, add those, and then deal with the denominator as a natural division, possibly with a remainder. Or it might use a floating-point unit to treat each fraction in the sum as a division operation, and then add the floating-point values. The floating-point rules of addition are different than the "+" of base-2 integer arithmetic.

"Rules" simply mean "more steps to elementize." It's a matter of converting the numbers to the same 0 and 1 binary states to do it. More steps to do the same addition does not imply different addition. "Partiality" here just cannot exist in terms of saying "more steps" == "different addition." It's why I'm bent on calling the fractions as "composed of NUMBERS" (post 1574 above) so as to see the machine needs to break the fractions down first into numbers as how to add them to yield a result.

A proof that ℕ ⊂ ℚ? Pick any element in ℕ, call it n. Then n/1 = n is in ℚ.

This is the VERY reason why I'm bent on proving the elementary connection.

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.

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?

It's not the same addition, which is demonstrated by the fact that a computer must do "pre-production" on the numbers in order to treat them as if they were the same addition.

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).

It's strange to me that you would want to go back to iron age mathematics, where the only acceptable numbers were counting numbers. We've learned a lot since then, but it's your choice.

No, I want to see what "what we are calling numbers" are at their deepest essence, beyond "member of a set" definition, based on examining their "periodic table of elements" if you will. In everyday use I'm not going to NOT call -5 a number, or a fraction NOT a number, in the same way I have no problem calling my house a geometry. But your insistence on proper terminology I'm insistent applies to this level. We have not bridged the concept of spatiality, feeling, meaning, numbers, infinity, etc. To do so, we have to be very rigid, as you said.

The numbers are molecules — I want to study the atoms to figure out how the molecules behave. If the brain is JUST a Von Neumann "binary" machine, then we need to study how we are approaching the problem of manipulating numbers in a material device to do so as a reflection of how we truly reason with them. Check post 1574 for more of my take on why I call fractions as fractions and not numbers.

 

Both "infinite" and "continuum" are far too conceptually loaded for me to accept in an axiom.

However, sine waves are an interesting informational case study. On one level, a sine has infinite values, but on another level....

We need to zoom in on this significantly from a "machine's perspective."

A "sine" has infinite values. A "sine" is being called 1 "thing" here, despite "infinite values," known or unknown. The presence of the sine is still conveying one piece of information: its existence.

This is where the "1" element has an axiomatic basis in the ontology: A machine calling any n components as "1" needs to be hella accounted for. This is my whole issue with the word "concept." I cannot accept any definition of "concept" until we can justify calling "n-values" as "1 thing."

Remember, a "thing" gets its entire conceptual basis from physical space. We first learn to call things "things" from space. How is it we can divorce this basic element of semantic framework from the "undefined dog in the light" spatial referent? I can understand calling the dog's "fur color" a judgment or "the smell of its breath." Or even arguing its full dimensional scope. But can we argue that it's NOT there? This seems utterly insane to me, because we cannot divorce our ontology from our mathematical axioms to that degree. We must call even OURSELVES "1 thing" before we can justify using our brains to do any "THING!"

The concept of "thereness" I think is something that MUST be folded into the concept of truth. Why show partiality to the "theorems in the brain" and not to the light that allows us to use the same words to even talk about them, write them down, or learn the language to discuss them in the first place? At bare minimum, one has NO idea mathematical theorems exist until he can crack open a book, go to a lecture, read from a computer online. In EVERY case, there is one bare minimum truth one can rely on: The fact that a THING is THERE to do such a thing.

Thoughts?

 

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

So what are you trying to say here? That something can have infinite values and still be self contained?
In every case i know of except a truly nondescript object the "thing" has properties that are understood to exist within even though all that info is packed into one simple description.
Even a random sequence has properties that are understood even though i call it "a random sequence".
This is different than having a set of random sequences, but then that is different than having sets of sets of random sequences, and even that may be part of a larger set of sets.
So then we could have an infinite set of sets that each have an infinite number of elements, and even that set might be part of another larger all containing set.
The rationale i think is when we start to have symmetries that stand out enough to make sense of it. But that is really what we want to look for in order to understand larger sets.
What exactly is a "thing" if it does not have some recognizable property that exists relative to something else we know already, a crude form of symmetry. Except for the definition of "thing" there is no such thing as a thing without some property that is understood and that makes it whole, complete, and usable even as part of a construct in a larger 'thing'.

The usual discourse proceeds in a manner to try to find symmetries so that we can lump things together and build a knowledge set under the required constraints. Also, transformations abound in converting infinite sets into very simple finite sets that can be dealt with as a single object. So we can often go from the abstract to the less abstract mathematically.

Not sure if this is what you are looking for but thought i would add a few thoughts.

 

So what are you trying to say here? That something can have infinite values and still be self contained?
In every case i know of except a truly nondescript object the "thing" has properties that are understood to exist within even though all that info is packed into one simple description.
Even a random sequence has properties that are understood even though i call it "a random sequence".
This is different than having a set of random sequences, but then that is different than having sets of sets of random sequences, and even that may be part of a larger set of sets.
So then we could have an infinite set of sets that each have an infinite number of elements, and even that set might be part of another larger all containing set.
The rationale i think is when we start to have symmetries that stand out enough to make sense of it. But that is really what we want to look for in order to understand larger sets.
What exactly is a "thing" if it does not have some recognizable property that exists relative to something else we know already, a crude form of symmetry. Except for the definition of "thing" there is no such thing as a thing without some property that is understood and that makes it whole, complete, and usable even as part of a construct in a larger 'thing'.

The usual discourse proceeds in a manner to try to find symmetries so that we can lump things together and build a knowledge set under the required constraints. Also, transformations abound in converting infinite sets into very simple finite sets that can be dealt with as a single object. So we can often go from the abstract to the less abstract mathematically.

Not sure if this is what you are looking for but thought i would add a few thoughts.

Great thoughts... The basis of reason is about "things" relating to other "things", and this seems quite contrary to a contraptional or componental machine that is dealing with n discrete values.

To me, we always have to go back to the limitations of what we're dealing with to discuss the term.

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. In which case, that is "1" thing. Even if a function mapped it, to "see" it internally means to see it on some "2D" or "3D" screen to see the values visualized. There is no such thing. A thought form is "1" thing. You can't compromise its definition by dissociating values from it or it loses the capacity to have a word mapped to it as a workable definition.

This is why I've had an abiding problem using the word "concept" to a computer. I do not see any "concepts" from a computer's perspective. They do not exist. There are discrete voltages, switches, high or low, lighting up pixels, etc. There is an emulsifier within us that permits us to call any observable element a "thing" — so it's more about simple presence or absence as the baseline of whether or not something is there before introducing the abstract "qualia" elements to delineate it further.

 

Nonetheless, I stand by my assertion that numbers and logic states are different types of things. They use different languages, with different syntax, different rules of inference, and different axioms.

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.

;--)

 

Nonetheless, I stand by my assertion that numbers and logic states are different types of things. They use different languages, with different syntax, different rules of inference, and different axioms.

There are three people standing on a street corner waiting for the light to change so they could cross the street. One was John, another was Judy and the third was Debbie. They each wore different color coats. The colors were red, green, yellow.
John wore green, Debbie wore yellow. What color did Judy wear?

The players in the logic puzzle are not the logic itself, the logic is applied to them in some way.

State variables usually associate with numbers. The numbers describe mathematical state of the system. They take on the role of the colors in the above logic puzzle, while something else such as nodes take the role of the people.
There are three nodes, A, B, C, and three voltages, 3, 7, 1. A is 7v C is 1v, what voltage is node B?

So logic then is a connective that shows the relationship between objects or their state which simply means that there is something associated with each main thing (the main things are the peoples names and the state variables A, B, C), however the logic may not be reversible because we end up with a non sequitur. That is, we can tell what value each has from it's state variable but we can not tell which state variable it is from its value in general. Maybe this helps explain why some processes are not reversible.

"It takes money to make money"
By that logic then:
"If we dont make any money we dont need any money".
So we are able to go one way with the logic but not the other.
If we produce state Y from state X, we may not be able to go back to state X knowing only state Y.
There are probably a lot of reasons for this depending on the situation.

 

Why can’t physical space and things therein be used as part of an argument if the very brain making the argument claims, informationally, it‘s in this space as its first axiom?

 

Why can’t physical space and things therein be used as part of an argument if the very brain making the argument claims, informationally, it‘s in this space as its first axiom?

Example?

 

Example?

Curious your thoughts on post 1472 as the basis of a proof that numbers are logic states by bypassing framework knowledge.

 

To describe a sine wave spatially, we draw an S-shape element on a 2D material plane, or gesticulate an undulation in air to communicate the meaning of what it is.

Could it be posited, that if a sine wave has infinite values, it could exist as an infinite something that is not a function of something else and can be visualized internally as something?

Whatever shape we write on a piece of paper, whatever motion we gesticulate in the air, does not have infinite values. We know this because your pen does not have infinite ink -- whatever is drawn on the paper is necessarily composed of finite markings. Likewise, the motion of your finger has a finite trajectory -- we can easily see it start and stop. The motion itself requires some finite amount of energy. Whence infinity?

Remember, sine is a mathematical object, it is a purely abstract concept (refers only to internal states). There are no sines at the physical (ground-floor) levels. If you wish to show that "infinity" is some ground-floor thing, you'll need to describe it in physical terms. Sines won't do.

 

I don't know what you're trying to say here. By convention, the symbols in {0,1} refers to a set of numbers, and the symbols in {F,T} refers to a set of logic states. We can of course choose any symbols we want, but if we're specifically trying to compare numbers to logic states, we should use different symbols. Otherwise, how do we know if we're talking about the numbers or the logic states?

Anyway, I thought about my argument this morning and realized that all I have done is prove that a two-element set cannot be ordered. When I wrote it, I hadn't realized that a ring or field of characteristic 2 does not admit an order. Researching the matter, I learned that a necessary (though not sufficient) property of an ordered ring or field is that it have characteristic 0. In plain English, to have a well-defined ">" relation on a set of numbers, the set must be infinite.

So, my argument above is invalid. As an aside, this speaks to the incredible amount of attention to detail that must be paid to such endeavors. It shows how easy it is to make mistakes even when we're very careful about language -- and strongly suggests that, if we're not careful about language, the task is hopeless.

Nonetheless, I stand by my assertion that numbers and logic states are different types of things. They use different languages, with different syntax, different rules of inference, and different axioms. Consequently, the formulas (statements) of one cannot be interpreted as the formulas of the other.

Also, the drawing is a "best as we can reflection" of what's going on in the physical brain (not according to me, but using the brain as the basis of limitations).

You seem to think that the crucial part of the logic system is the true/false values, but that's just the semantics (interpretation) that gets applied to the formulas, which are purely syntactic. The logic doesn't change, regardless of our choice of semantics. For example, here's a valid formula of propositional logic: \[ A \wedge (A \vee B) \vee C \] Notice there are no truth/false values. Using the axioms and rules of inference, we can reduce it to the simpler formula \( A \vee C \): \[ \begin{align} A \wedge ( A \vee B) \vee C &\to (A \wedge A) \vee ( A \wedge B) \vee C \\ &\to [ A \vee (A \wedge B)] \vee C \\ &\to A \vee C \end{align} \] However we interpret \( A \), \( B \), and \( C \) -- i.e., whatever true/false values we give them -- the two formulas are equivalent. This is the entire point of any logic system: to show us the form of pure reason without any content. As soon as we apply an interpretation and give the symbols content, every valid formula disappears, becoming either a single "T" or a single "F". The meaningful part is not the Ts and Fs, it's the formulas!

Hopefully you can recognize that arithmetic formulas use an entirely different language built from different rules and axioms. Arithmetic formulas do not tell us the same kinds of things that logical formulas do -- they speak to different types of abstractions. So then, what's the connection between them that computers seem to exploit? It's none other than the trivial coincidence that in a two-element boolean ring, the truth table of logical AND is compatible with multiplication, and the truth table of logical XOR is compatible with addition.

It's crucial that you understand that a boolean ring is an arithmetic system, not a logical system. The formulas of boolean rings -- e.g., \( a^2 = a \) -- are not valid formulas in logic. By definition, boolean rings have two arithmetical operations, addition and multiplication. With only two elements (numbers), there are exactly sixteen different possible multiplication tables and sixteen possible addition tables. Notice that, whichever we choose, we are guaranteed to find a compatible logical operation (because there are sixteen of those, too).

Now, the axioms of a boolean ring dictate that every element of the ring is nilpotent under the addition operation: a + a = 0, and every element is idempotent under multiplication: a x a = a. With these constraints, there is only one possible multiplication table and one possible addition table. When we compare these with the truth tables of the sixteen logical operations, we find that logical AND and logical XOR are compatible, i.e., they have the same form. This simple coincidence means that we can interpret the result of the logical formula \( A \wedge B \) as if it had been the result of the arithmetic formula \( A \times B \) in a boolean ring. Likewise, we can interpret the logical formula\( A \oplus B \), where \( \oplus \) is the XOR operation, as if it had been the result of the arithmetic formula \( A + B \).

In other words, the connection happens in the human brain that is interpreting the result. Their "connection" is very simply that they both use two-element sets.

At the digital circuit level, we design binary gates to implement the logical operations. But, as we saw earlier, logical formulas are not arithmetic formulas. So how do computers do general arithmetic? We combine the results of individual gates and, using extra circuitry, treat the combined result as if they were base-2 digits. In other words, we group individual gates, whose results we interpret as performing a boolean ring arithmetic, to stand for numbers in ℕ. In general, these numbers are not defined at the boolean ring level, so we have to leave boolean rings to do any further arithmetic.

And I suspect this is the level where you get most confused. Let's review the situation at this level: we have a group of logic gates, each of whose operation is being treated as an arithmetic result in a boolean ring, At each gate, we interpret the voltage value as representing a 0 or a 1, which are valid in the boolean ring. We combine these gates -- which is a physical manifestation of the abstract notion of a bit string -- and treat them as numbers, like 5 and 42. However, to add 5 and 42, we cannot use the arithmetic of boolean rings (where such numbers have no definition), we have to use the arithmetic of integer rings. Integer arithmetic in base-2 requires the notion of carrying, so we introduce circuitry to implement "carry bits" and arrange our gates appropriately.

Et voila, we have results that we can interpret as base-2 integer arithmetic. Pretty awesome use of abstractions, but please see it for what it is. If numbers and logic states weren't distinct types of abstractions, we wouldn't need all of this extra complexity to allow us to treat them as if they were the same.

Whatever shape we write on a piece of paper, whatever motion we gesticulate in the air, does not have infinite values. We know this because your pen does not have infinite ink -- whatever is drawn on the paper is necessarily composed of finite markings. Likewise, the motion of your finger has a finite trajectory -- we can easily see it start and stop. The motion itself requires some finite amount of energy. Whence infinity?

Remember, sine is a mathematical object, it is a purely abstract concept (refers only to internal states). There are no sines at the physical (ground-floor) levels. If you wish to show that "infinity" is some ground-floor thing, you'll need to describe it in physical terms. Sines won't do.

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.

 

I don't know what you're trying to say here. By convention, the symbols in {0,1} refers to a set of numbers, and the symbols in {F,T} refers to a set of logic states. We can of course choose any symbols we want, but if we're specifically trying to compare numbers to logic states, we should use different symbols. Otherwise, how do we know if we're talking about the numbers or the logic states?

Anyway, I thought about my argument this morning and realized that all I have done is prove that a two-element set cannot be ordered. When I wrote it, I hadn't realized that a ring or field of characteristic 2 does not admit an order. Researching the matter, I learned that a necessary (though not sufficient) property of an ordered ring or field is that it have characteristic 0. In plain English, to have a well-defined ">" relation on a set of numbers, the set must be infinite.

So, my argument above is invalid. As an aside, this speaks to the incredible amount of attention to detail that must be paid to such endeavors. It shows how easy it is to make mistakes even when we're very careful about language -- and strongly suggests that, if we're not careful about language, the task is hopeless.

Nonetheless, I stand by my assertion that numbers and logic states are different types of things. They use different languages, with different syntax, different rules of inference, and different axioms. Consequently, the formulas (statements) of one cannot be interpreted as the formulas of the other.

You seem to think that the crucial part of the logic system is the true/false values, but that's just the semantics (interpretation) that gets applied to the formulas, which are purely syntactic. The logic doesn't change, regardless of our choice of semantics. For example, here's a valid formula of propositional logic: \[ A \wedge (A \vee B) \vee C \] Notice there are no truth/false values. Using the axioms and rules of inference, we can reduce it to the simpler formula \( A \vee C \): \[ \begin{align} A \wedge ( A \vee B) \vee C &\to (A \wedge A) \vee ( A \wedge B) \vee C \\ &\to [ A \vee (A \wedge B)] \vee C \\ &\to A \vee C \end{align} \] However we interpret \( A \), \( B \), and \( C \) -- i.e., whatever true/false values we give them -- the two formulas are equivalent. This is the entire point of any logic system: to show us the form of pure reason without any content. As soon as we apply an interpretation and give the symbols content, every valid formula disappears, becoming either a single "T" or a single "F". The meaningful part is not the Ts and Fs, it's the formulas!

Hopefully you can recognize that arithmetic formulas use an entirely different language built from different rules and axioms. Arithmetic formulas do not tell us the same kinds of things that logical formulas do -- they speak to different types of abstractions. So then, what's the connection between them that computers seem to exploit? It's none other than the trivial coincidence that in a two-element boolean ring, the truth table of logical AND is compatible with multiplication, and the truth table of logical XOR is compatible with addition.

It's crucial that you understand that a boolean ring is an arithmetic system, not a logical system. The formulas of boolean rings -- e.g., \( a^2 = a \) -- are not valid formulas in logic. By definition, boolean rings have two arithmetical operations, addition and multiplication. With only two elements (numbers), there are exactly sixteen different possible multiplication tables and sixteen possible addition tables. Notice that, whichever we choose, we are guaranteed to find a compatible logical operation (because there are sixteen of those, too).

Now, the axioms of a boolean ring dictate that every element of the ring is nilpotent under the addition operation: a + a = 0, and every element is idempotent under multiplication: a x a = a. With these constraints, there is only one possible multiplication table and one possible addition table. When we compare these with the truth tables of the sixteen logical operations, we find that logical AND and logical XOR are compatible, i.e., they have the same form. This simple coincidence means that we can interpret the result of the logical formula \( A \wedge B \) as if it had been the result of the arithmetic formula \( A \times B \) in a boolean ring. Likewise, we can interpret the logical formula\( A \oplus B \), where \( \oplus \) is the XOR operation, as if it had been the result of the arithmetic formula \( A + B \).

In other words, the connection happens in the human brain that is interpreting the result. Their "connection" is very simply that they both use two-element sets.

At the digital circuit level, we design binary gates to implement the logical operations. But, as we saw earlier, logical formulas are not arithmetic formulas. So how do computers do general arithmetic? We combine the results of individual gates and, using extra circuitry, treat the combined result as if they were base-2 digits. In other words, we group individual gates, whose results we interpret as performing a boolean ring arithmetic, to stand for numbers in ℕ. In general, these numbers are not defined at the boolean ring level, so we have to leave boolean rings to do any further arithmetic.

And I suspect this is the level where you get most confused. Let's review the situation at this level: we have a group of logic gates, each of whose operation is being treated as an arithmetic result in a boolean ring, At each gate, we interpret the voltage value as representing a 0 or a 1, which are valid in the boolean ring. We combine these gates -- which is a physical manifestation of the abstract notion of a bit string -- and treat them as numbers, like 5 and 42. However, to add 5 and 42, we cannot use the arithmetic of boolean rings (where such numbers have no definition), we have to use the arithmetic of integer rings. Integer arithmetic in base-2 requires the notion of carrying, so we introduce circuitry to implement "carry bits" and arrange our gates appropriately.

Et voila, we have results that we can interpret as base-2 integer arithmetic. Pretty awesome use of abstractions, but please see it for what it is. If numbers and logic states weren't distinct types of abstractions, we wouldn't need all of this extra complexity to allow us to treat them as if they were the same.

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.

So if we cannot prove they are the same thing in a physical machine where contrasted voltage states are the basis for every bit, AND numbers are not physical things per what you've insisted, then numbers exist in a non-physical or meta-physical state somewhere (which is my Pythagorean-esque point). Which to me is a QED.

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.

I think there are different ways of going about conceptual relationships that lead down certain roads of what is considered legitimate reasoning based in internal-referrent axioms set up, but if any of those axioms are partially blind, or not seeing "presence or absence" of physical existence in physical space as legal tender for the proof, the proof will in turn be partially blind.

 

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?

To me the sine wave IS conveying information... precisely that it exists. That the ultimate basis of knowledge is infinite in nature, because “what created the infinite sine?” A mathematical object that must itself exist as an infinite thing.

 

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.

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

Thoughts?

 

The first step is to clearly distinguish the symbols involved. Using "1" to stand for both a number and a logic state is sloppy, at best. We'll use {0,1,2,3,4,...} for numbers and {F,T} for logic states.

So that I don't have to spell them all out, let's assume that we already know the rules of propositional logic. In particular, we know that there are sixteen possible operations, four unary and twelve binary. We can test how numbers might respond to these operations.

For instance, we know how unary NOT (~) behaves with logic states: ~F → T and ~T → F.

How does it behave with numbers? Well, what does ~5 mean? It's nonsense. Let's try a different operator, say, binary OR.

We know that F OR F → F, F OR T → T, and so on.

What about numbers? What does 3 OR 4 mean? It's nonsense.

Perhaps we can try going the other way, using logic symbols within arithmetic. We know that 1 < 2, but is F < T?

What about addition. What does F + T mean? Well, we can try to derive its meaning from the "<" relation.

Let's suppose that F < T. An arithmetical property of the "<" relation is that, for any given a, b, c, if a < b then a + c < b + c.

Therefore, F < T implies two things (because c can be either F or T): F + F < T + F and F + T < T + T

Now, if F + F is less than T + F, it must be the case that they are distinct. Since we only have two symbols to work with, we'll say that F + F = F and T + F = T. Because both our arithmetic and our logic system are commutative, we know that F + T = T, too.

From before, we know that F + T < T + T. We have a problem, though: what are we supposed to label T + T? We can't label it T, because then F + T < T + T would become T < T, which is a contradiction. We can't label it F, because then we would have T < F, which contradicts our initial assumption that F < T. We can reverse our last labeling action and say that F + F = T and T + F = F, but that immediately leads to the contradiction T < F.

What have we learned? We cannot say that F < T, else we will get an inconsistent system. Clearly, then, we'll say that T < F. Repeating the a + c < b + c process, we now know that both of the following statements must be correct:

T + F < F + F and T + T < F + T

Again, we need to give these expressions labels, so we pick T + F = T and F + F = F (if we went the other way, we'd immediately get a contradiction with the T < F assumption).

From our assumption T < F, and our labels T+F = T and F+F = F, we can say that T + F < F + F is mathematically valid. That still leaves T + T < F + T. If we replace T + T with T, then we're saying that T < T, which is a contradiction. If instead we replace T + T with F, then we have F < T. But this contradicts our assumption that T < F!

What I have demonstrated is that we cannot make a consistent arithmetical system out of propositional logic. The two things are entirely different kinds of things! QED

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. It’s night and day. Simultaneously I want to prove that numbers and sets are derivative of 1 and 0 as the only numbers that really exist, and all others are higher concepts including any frameworks and sets to work with them. Brutally rasa here.

Can we see what it looks like when you compare the binary-base components of any number to truth states? (e.g., of course ~5 is nonsense from a higher base perspective.)