Check this out:

AXIOM: Numbers that involve a mantissa, operator, or fractions are not actual elementary numbers, they are numeric terms involving an arithmetical operator denoting a PROCESS on a number.

As a side note, the general idea behind an axiomatic system is to have a few, simple axioms and derive the rest of the theorems from them. Your axiom here includes a bunch of ideas, some of which aren't precisely defined.

In any case, a problem with this is that mantissas and fractions and such are elements of particular number representations, not the numbers themselves. Case in point: the number corresponding to "three-eighths" can be written without a fraction.

√2 is therefore not a number, it is a numeric PROCESS requiring additional computation to cognize than just a number. It is a question and answer involving the number 2: "What number when multiplied by itself yields 2?"

My problem with this is that we define what you call "elementary" numbers using a process, as well. The number two is an abstraction of counting/grunting more than once and less than thrice.

In the standard construction (Peano), the number two is the successor of the successor of the empty set. The process is "successor of".

 

"No, Javier. It's a throw yourself into the trash on trash night if you don't get the answer right kinda question."

Which is why I ask "What do you mean?" If she just needs some kind of affirmation, I'd be happy to give it. If the question comes from a place of intellectual curiosity, I'd probably start with "What isn't real?"

 

Wait. You are a state processor machine. Time to you is...? And how can you define it outside a state?

Time is the concept we give to the recognition that state transformations do not occur all at once. Time is not a thing; it is not even a particular configuration of states. I can form a concept of NOW, but I cannot point to NOW, and my conception of when NOW is must necessarily be different than any another state processor's conception.

 

Time is the concept we give to the recognition that state transformations do not occur all at once. Time is not a thing; it is not even a particular configuration of states. I can form a concept of NOW, but I cannot point to NOW, and my conception of when NOW is must necessarily be different than any another state processor's conception.

Ugh. The same issue you have with me is the same one I have with you with the nebulosity.

Concepts ARE groups of states. You only KNOW states! Why are you insisting on a difference between a state and a process, for example, when you ARE the states alone that reflect their ultimate computation?

 

Which is why I ask "What do you mean?" If she just needs some kind of affirmation, I'd be happy to give it. If the question comes from a place of intellectual curiosity, I'd probably start with "What isn't real?"

So you'd be saying that to your wife not because you believe she's real. Or that you can use this built-in token "real" to describe her. I don't understand why you show partiality to CONCEPT and STATES and PROCESSES as built-in tokens, but not REAL?

 

Ugh. The same issue you have with me is the same one I have with you with the nebulosity.

Concepts ARE groups of states. You only KNOW states! Why are you insisting on a difference between a state and a process, for example, when you ARE the states alone that reflect their ultimate computation?

Concepts are groups of associated states. A process is a transformation that changes one or more states. Following me so far?

The relationship between states and processes is easier to see with a computer (where there are no confusing meta-levels of "I" or "me"). A computer is made up of a bunch of "physical states" (which we call plastic, silicon, etc.). Some of these physical states have been arranged in such a way that they respond to certain external states by transforming their internal states. These internal transformations are computation or information processing.

I intentionally made that description very simple to highlight how it's all just states and transformations. Awareness comes into the picture when some of the internal states become dependent on other internal states, creating a feedback loop. As a very simple example, a computer can access information about its CPU's temperature because it has an appropriate sensor. However, the computer has awareness of its CPU's temperature when the computer reduces the CPU's speed or voltage when the temperature is too hot.

In other words, the computer has some set of internal states that are specifically associated with CPU temperature, and these states are used (process) to control the CPU's speed.

It's all states and processes. Things get interesting when the states have many associations.

 

So you'd be saying that to your wife not because you believe she's real. Or that you can use this built-in token "real" to describe her. I don't understand why you show partiality to CONCEPT and STATES and PROCESSES as built-in tokens, but not real?

I feel like I've said it 42 times. The word "real" is empty, devoid of meaning to me. It's a throw-away word that we use in casual conversation. "I'm the real MC."

In contrast, "concept" and 'state" carry precise meaning. They're useful in formal discourse.

 

As a side note, the general idea behind an axiomatic system is to have a few, simple axioms and derive the rest of the theorems from them. Your axiom here includes a bunch of ideas, some of which aren't precisely defined.

In any case, a problem with this is that mantissas and fractions and such are elements of particular number representations, not the numbers themselves. Case in point: the number corresponding to "three-eighths" can be written without a fraction.


My problem with this is that we define what you call "elementary" numbers using a process, as well. The number two is an abstraction of counting/grunting more than once and less than thrice.

In the standard construction (Peano), the number two is the successor of the successor of the empty set. The process is "successor of".

Concepts are groups of associated states. A process is a transformation that changes one or more states. Following me so far?

The relationship between states and processes is easier to see with a computer (where there are no confusing meta-levels of "I" or "me"). A computer is made up of a bunch of "physical states" (which we call plastic, silicon, etc.). Some of these physical states have been arranged in such a way that they respond to certain external states by transforming their internal states. These internal transformations are computation or information processing.

I intentionally made that description very simple to highlight how it's all just states and transformations. Awareness comes into the picture when some of the internal states become dependent on other internal states, creating a feedback loop. As a very simple example, a computer can access information about its CPU's temperature because it has an appropriate sensor. However, the computer has awareness of its CPU's temperature when the computer reduces the CPU's speed or voltage when the temperature is too hot.

In other words, the computer has some set of internal states that are specifically associated with CPU temperature, and these states are used (process) to control the CPU's speed.

It's all states and processes. Things get interesting when the states have many associations.

What do you mean by "associated" — I need this VERY specific. Do you mean in "proximity?" And why do you know any states are associated, if each is discrete?

 

What do you mean by "associated" — I need this VERY specific. Do you mean in "proximity?" And why do you know any states are associated, if each is discrete?

Not proximity. States are associated when there is a dependence between them. The way you describe "each" state as being discrete, I think we also need to make clear that state does not imply a single binary switch. It is correct to say that a binary switch has two possible states, but that does not in any way mean that every state is a binary switch.

For a simple example, consider a RAM chip as an m-row by n-column matrix of independent binary cells. Each cell has its own state. Likewise, the entire matrix has its own state. Likewise, every possible sub-matrix of cells have their own state. This property gives a computer m*n degrees of freedom to organize information.

Let's say that a computer uses a 10x10 sub-matrix of RAM to store information about the temperature of its CPU. We'll also say that it uses a 50x50 sub-matrix of cells, different from the 10x10 matrix, to control the CPU's fan setting. Finally, we'll say that the computer uses a distinct 1024x1024 sub-matrix of RAM to store code that controls the CPU's fan based on the CPU's temperature.

In other words, the state of the 1024x1024 matrix associates the state of the 10x10 matrix with the state of 50x50 matrix. Does this make it clearer what I mean?

Note that the cells do not have to be in proximity to each other. The only restriction on their physical locations is the laws of physics. Also, note that I used binary cells only out of familiarity. I could have equally used 42-level cells, or analog cells, or whatever. I could have even used non-discrete media, like waves instead of cells. RAM is just one possible example, not a foundational example.

 

I feel like I've said it 42 times. The word "real" is empty, devoid of meaning to me. It's a throw-away word that we use in casual conversation. "I'm the real MC."

In contrast, "concept" and 'state" carry precise meaning. They're useful in formal discourse.

It's not a throw-away word, though. It's literally "the word we all universally use to denote existence."

I would go so far to say as "words" are a default phenomenon in the set REAL.

"I'm the real MC" is short-hand for, "I can define myself as an MC worth calling an MC by n parameters in time and space that we all share."

I know you've said it a lot of times. But bear in mind, most human beings (including myself) would not believe the world "real" is devoid of default meaning (no offense). So it's difficult to believe this. I'm talking about even in a sense of perhaps quasi "formal" use. If you went into a bio lab and you insisted the petri dish wasn't "REAL", they'd say, "Get outta town — what are you talking about (in the space we all share as "REALITY")? It's right here."

If you went into a math class and pointed at the chalkboard and said, "THE CHALKBOARD ISN'T REAL!" They'd say, "What are you talking about? It's right on the wall." "The ticket isn't real, officer". "What do you mean, son— I just gave it to you." If you insisted on calling nothing real, they may want to commit you, because words are "by default" reflecting this thing called "real things in space."

Your wife certainly would not need additional definition for "real." Ask your wife. I bet she'd do something like, "of couse I'm real?" and then maybe pinch her cheek and say "I'm real, Javier!" and then say "I'm in the room with you right now, this room is real! You're real!" It's a synonym for "there-ness" in O-Space or "Reality", the "set of all real things."

"Real" is not useful in formal discourse perhaps because it's not yet scientifically defined, and this is something we have to do here in my estimation.

 

Not proximity. States are associated when there is a dependence between them. The way you describe "each" state as being discrete, I think we also need to make clear that state does not imply a single binary switch. It is correct to say that a binary switch has two possible states, but that does not in any way mean that every state is a binary switch.

For a simple example, consider a RAM chip as an m-row by n-column matrix of independent binary cells. Each cell has its own state. Likewise, the entire matrix has its own state. Likewise, every possible sub-matrix of cells have their own state. This property gives a computer m*n degrees of freedom to organize information.

Let's say that a computer uses a 10x10 sub-matrix of RAM to store information about the temperature of its CPU. We'll also say that it uses a 50x50 sub-matrix of cells, different from the 10x10 matrix, to control the CPU's fan setting. Finally, we'll say that the computer uses a distinct 1024x1024 sub-matrix of RAM to store code that controls the CPU's fan based on the CPU's temperature.

In other words, the state of the 1024x1024 matrix associates the state of the 10x10 matrix with the state of 50x50 matrix. Does this make it clearer what I mean?

Note that the cells do not have to be in proximity to each other. The only restriction on their physical locations is the laws of physics. Also, note that I used binary cells only out of familiarity. I could have equally used 42-level cells, or analog cells, or whatever. I could have even used non-discrete media, like waves instead of cells. RAM is just one possible example, not a foundational example.

As a side note, the general idea behind an axiomatic system is to have a few, simple axioms and derive the rest of the theorems from them. Your axiom here includes a bunch of ideas, some of which aren't precisely defined.

In any case, a problem with this is that mantissas and fractions and such are elements of particular number representations, not the numbers themselves. Case in point: the number corresponding to "three-eighths" can be written without a fraction.


My problem with this is that we define what you call "elementary" numbers using a process, as well. The number two is an abstraction of counting/grunting more than once and less than thrice.

In the standard construction (Peano), the number two is the successor of the successor of the empty set. The process is "successor of".

Ha!— I knew you'd come back with just that. I will further hone and repost either tomorrow or in 3 months. ;--)

 

What is your opinion of this "micro-proof":

Axiom: The number 1 ("one") is the foundational number from which all other numbers are conceptualized.

Proof: All modern computers are able to perform any computation of any kind using the number 1 and its absence (0) as represented by classical bits, and yield results in any base. QED.

 

Not proximity. States are associated when there is a dependence between them. The way you describe "each" state as being discrete, I think we also need to make clear that state does not imply a single binary switch. It is correct to say that a binary switch has two possible states, but that does not in any way mean that every state is a binary switch.

For a simple example, consider a RAM chip as an m-row by n-column matrix of independent binary cells. Each cell has its own state. Likewise, the entire matrix has its own state. Likewise, every possible sub-matrix of cells have their own state. This property gives a computer m*n degrees of freedom to organize information.

Let's say that a computer uses a 10x10 sub-matrix of RAM to store information about the temperature of its CPU. We'll also say that it uses a 50x50 sub-matrix of cells, different from the 10x10 matrix, to control the CPU's fan setting. Finally, we'll say that the computer uses a distinct 1024x1024 sub-matrix of RAM to store code that controls the CPU's fan based on the CPU's temperature.

In other words, the state of the 1024x1024 matrix associates the state of the 10x10 matrix with the state of 50x50 matrix. Does this make it clearer what I mean?

Note that the cells do not have to be in proximity to each other. The only restriction on their physical locations is the laws of physics. Also, note that I used binary cells only out of familiarity. I could have equally used 42-level cells, or analog cells, or whatever. I could have even used non-discrete media, like waves instead of cells. RAM is just one possible example, not a foundational example.

I understand exactly what you’re saying, but the word CONCEPT needs even more specificity to work for me in that context. The way you describe it, the only difference between one concept or another is essentially the number of states, and whose “qualitative” characteristic is merely “more on-off valves permitting the same water to flow (or not) to another area.” So your multi-D wife vs. your car are only really different due to the number of 1D states. It also does not make sense to me that infinite D Hilbert-space vectors are what they are to you, when you have essentially 0% of those dimensions represented.

 

No! GR started with the recognition that acceleration and gravity are physically indistinguishable. Einstein was visualizing window-less elevators, not "bent space". Using this equivalence, and what he learned in SR, Einstein followed the physical implications. The result was a set of ten coupled differential equations that characterize the dynamics of motion in a gravitational field. If we model this geometrically, we find that the metric tensor (the thing that tells us the spatial distance between objects) is not a constant, it's a function of spacetime. Ergo, the geometry of spacetime is not flat (constant metric), it's curved.

This was not an assumption that Einstein started out with, nor is it an ontological declaration.

You’re 100% positive the “light around Mercury” question had absolutely nothing to do with it, when he even wrote a paper on the applicability of the theory to it?

 

"I'm the real MC" is short-hand for, "I can define myself as an MC worth calling an MC by n parameters in time and space that we all share."

LOL no. "I'm the real MC" is a boast of someone's oratorical skill, not their spatial parameters. A rival might counter with "You're a sucka MC." (Though I'd definitely laugh if someone countered with "You're so insignificant, you're not even in ℝ.")

But bare in mind, most human beings (including myself) would not believe the world "real" is devoid of default meaning (no offense). So it's difficult to believe this. I'm talking about even in a sense of perhaps quasi "formal" use. If you went into a bio lab and you insisted the petri dish wasn't "REAL", they'd say, "Get outta town — what are you talking about (in the space we all share as "REALITY")? It's right here."

Again, in casual conversation the opposite of "real" is "fake". Suppose one of the biolab researchers was known as a practical joker, one who might, for instance, swap your experiment's petri dishes with tap water. Then, "Hey Jim, are these the real petri dishes?" would be a perfectly reasonable question.

But, outside of the mathematical connection with ℝ, what does "real" mean in formal discourse? Can you give an example of something that is not "real"? If not, you're using an empty word.

 

This potentially needs academic polish, but see how grokable it is as starting point:

Axiom 1: A number is a concept with a name.

Axiom 2: A number can be grouped together in order to “number” or "count."

In the most elementary of ways possible (unary), this means to add successive copies of the number 1 to create different numbers that can be given a unique name. They can also be labeled as a shortcut symbol. For example, “1111” in unary is “4” in base-10.

Axiom 3: Beyond counting, one can create special computational numeric terms that are the result of appending a separate component of arithmetical operation(s).

These are no longer numbers, but higher abstractive expressions that involve more numbers with analysis to cognize beyond simple counting.

For example, the expression 4.5 is using the number 4 concatenated by a decimal point to denote the addition of the result of another operation on 2 numbers: the number 5 divided by the number 10. It can be read as (4 + 5/10) or (4 + 1/2). This mantissa can be further arithmetically reduced to 1 divided by 2.

A number cannot be divided without creating a separate system to denote an abstract rule system for fractionation beyond counting.

Another example is what is commonly labeled as “Root 2” or “ √2.” This again is not a number, but an operation upon a number. Here we have 2 (or 11 in unary) as a number, with a “radical operation symbol” attached to it. The symbol is calling for a non-existent number that when multiplied by itself yields the number 2. This number is non-existent, so we call it “irrational,” which can also mean “unknowable.” In practical terms, this also means non-computable, uncountable as-is and therefore should not be treated as a number, or thought of as comparable to a number. Irrational numbers are the result of dividing two numbers that don’t rationally divide, and so an infinitesimal suffix is created that approaches another countable, rational number but doesn’t ever do so. It should therefore be called a “numeric expression” only.

“Transcendental numbers” are also not numbers, but expressions of multiple numbers written as single symbols. Pi/π is the ratio of two numbers. e is the base of the natural log.

Theorem: Based on the axioms above, |ℕ| < |ℝ| is an irrational comparison akin to saying "cat" is less than "dog." The first set is a set of numbers that only have 1 level of addition to create them (counting). The second set is a set of numeric computational expressions or computational results of variable complexity combined with countable numbers to create a set that shouldn’t exist as a true “numeric set” but “an abstract collection of numeric expressions.”

One cannot logically compare either set and say the cardinality of one is greater or less than the other’s due to this fundamental definition disparity.

 

LOL no. "I'm the real MC" is a boast of someone's oratorical skill, not their spatial parameters. A rival might counter with "You're a sucka MC." (Though I'd definitely laugh if someone countered with "You're so insignificant, you're not even in ℝ.")

There’s an implied definition there, though... “real” can mean here, “I exist as the very incarnation of MC, bruh, and you or anyone else don’t cut it.” He would then proceed to rattle off elements of the real set MC vs. poser set MC. “Check my rhymes, my ride, my flow, my dope, my girls, my backups, my Lex, my hoes; then tell me I ain’t the REAL DEAL, fulfilling the definition of MC Hilbert in physical space, in light of the definitions in your mind.”

Again, in casual conversation the opposite of "real" is "fake". Suppose one of the biolab researchers was known as a practical joker, one who might, for instance, swap your experiment's petri dishes with tap water. Then, "Hey Jim, are these the real petri dishes?" would be a perfectly reasonable question.”

But, outside of the mathematical connection with ℝ, what does "real" mean in formal discourse? Can you give an example of something that is not "real"? If not, you're using an empty word.

The concept dog is “real” to you, as is the dog in physical space. Which one is “more real?” Are we going to ignore the “gradation element?”

 

What is your opinion of this "micro-proof":

Axiom: The number 1 ("one") is the foundational number from which all other numbers are conceptualized.

Proof: All modern computers are able to perform any computation of any kind using the number 1 and its absence (0) as represented by classical bits, and yield results in any base. QED.

It has a few problems. The biggest are 1) the proof uses two numbers to suggest that one of them is "foundational", and 2) the statement "All computers can perform any computation using two numbers" does not imply that computers must use two numbers.

Friendly technical note: axioms aren't proved, they're assumed. Using a set of given axioms, we prove theorems. Once we have a few theorems, we can use them to make other theorems. The set of all theorems is a theory.

Humor me to lay out a simple example, so you can get a better feel for what these things look like. We'll use group theory, which is one of the simplest algebraic structures (few axioms) that nevertheless is very rich (in theorems).

A group is just a set of elements \( S \) and a single binary operator \( \star \). The elements can be anything -- numbers, geometric shapes, colors, types of monkeys -- and the operator can be anything that obeys the group axioms, which are the same for every group.

Suppose a, b, and c are elements in \(S\). Then, the group axioms are:

• \( S \) is closed under the operation: \( a \star b \in S \)
• There is an identity element \( e \), such that \( a \star e = a \)
• The operator is associative: \( a \star (b \star c) = (a \star b) \star c \)
• Every element has an inverse under the operation: \( a \star a^{-1} = e \)

That's it. We don't have to prove any of the axioms, we just assume them. Using these axioms, we can prove theorems.

For instance, we know that the following is a theorem of any group:

If \( a \star a = a \), then \( a \) is the identity element of the group.

Proof:

1. \( a \star a = a \)
2. \( a \star a = a \star e \) (by identity axiom)
3. \( a \star a = a \star (a \star a^{-1}) \) (by inverse axiom)
4. \( a \star a = (a \star a) \star a^{-1} \) (by associativity axiom)
5. \( a \star a = a \star a^{-1} \) (by step 1)
6. \( a \star a = e \) (by inverse axiom)

QED. We've proven the theorem. This means that, for any group we can think of, we know that \( a \star a = a\) implies that \( a \) is the identity element. For example, the integers under addition form a group, \((\mathbb{Z}, +)\). From elementary school, we know that 0 is the identity element of this group. Indeed, 0 is the only element of \( \mathbb{Z} \) for which \(a + a = a\).

What about the integers under multiplication? Is \((\mathbb{Z}, \times)\) a group? It is not, because axiom 4 (inverses) is not obeyed for every element. For example, the number 2 has no multiplicative inverse in the integers.

Group theory, being so simple, is quite general. We can add more structure -- and, hence, more power -- by adding stuff to it. We can introduce new axioms (e.g., adding the commutative property leads to the theory of Abelian groups), or, even better, introduce more operators, which leads to richer structures, such as the rings and fields we learn about in elementary school arithmetic. Using these structures, we can create even larger palaces, such as vector spaces, Banach spaces, Hilbert spaces, etc.

This is generally how theories build on one another.

 

I understand exactly what you’re saying, but the word CONCEPT needs even more specificity to work for me in that context. The way you describe it, the only difference between one concept or another is essentially the number of states, and whose “qualitative” characteristic is merely “more on-off valves permitting the same water to flow (or not) to another area.” So your multi-D wife vs. your car are only really different due to the number of 1D states. It also does not make sense to me that infinite D Hilbert-space vectors are what they are to you, when you have essentially 0% of those dimensions represented.

Using states of RAM, the computer I described has a (very simple) concept of "heat affects my performance". We can imagine this network of state associations becoming more complex by adding more types of sensors, more code, and more state, giving the computer a concept of its own general health.

Contrast this with another type of concept. Again, using states of RAM affected by different types of sensors and code, the computer can hold a concept of the current weather. To the computer, these two concepts are indeed distinct, which we can deduce by noticing that it responds differently to increases in internal vs external temperature. (In the former case, it might increase its fan speed; in the latter, it might suggest a day at the beach.)

 

You’re 100% positive the “light around Mercury” question had absolutely nothing to do with it, when he even wrote a paper on the applicability of the theory to it?

The only things I'm 100% certain of is logical and mathematical theorems. From what I understand of the history of physics, Einstein was enormously dissatisfied that his "special" theory of relativity was not general enough to account for gravity. While thinking about ways to include gravity, Einstein had what he literally called his "happiest thought", which was the phyiscal equivalence of gravity and acceleration. This lead to GR, which later lead to [i[demonstrations[/i] of GR, which included the inexplicable wobble of Mercury's orbit.

Googling for Einstein's "happiest thought" brought up this page, which -- reading the GR section -- seems to confirm my account:
https://en.wikipedia.org/wiki/Einstein's_thought_experiments