I must also state this openly, for my own sanity, with plenty of bold and even color this time :--D:

The elementary concept of numbers have existed for thousands of years, and computations performed with them by the greatest of mathematicians of all time, such as Pythagorus, Gauss, Euler, Euclid, Fermat, Des Cartes, Galois, Boole, Fibonacci, Newton, long before any "sets" of them were created. Even today, computers perform computations with zero awareness of any "sets" these quantitative elements belong to.

Therefore, numbers and their utility are positively NOT in any way defined as a function of their belonging to any specific man-made modern set. Period. QED.

This is an utterly fallacious notion of post-Cantoral thinking that the greatest of mathematical minds such as Pythagorus and Newton would openly and vehemently repudiate.

Man-manufactured sets exist ONLY for the convenience of mathematical measurement and to see relationships between types of numeric expressions. They do NOT define whether or not the number is a number.

This is not debatable in any way, shape or form, and is self-evidentiary on multiple levels.

 

You’re making it sound like the “decimal expansion” is not part and parcel of pi!

It is not. There is no limit to the number of different ways that we can express Pi that are not decimal expansions. Furthermore, there are as many numbers in ℕ as there are numbers with a non-terminating decimal expansion. In other words, having a non-terminating decimal expansion is hardly unique to Pi, therefore, it cannot be a defining characteristic of Pi.

What makes Pi unique has nothing to do with the fact that it has a non-terminating decimal expansion.

Numbers are hardware states to machines/you!

You're confusing abstraction levels. Perhaps this will make it more clear. Let's call the physical states -- the stuff of space -- first-order states. Using first-order states, an information processor can create second-order states. Using second-order states, a processor can create nth-order states. There's a hierarchy of organization, and each level conveys its own information -- each level has its own domain of discourse.

Concepts, as states of states, are minimally second-order states. We can of course form concepts from other concepts, forming nth-order states. Numbers are an nth-order state.

As physical machines, computers work with first-order states. Computations are changes in first-order state -- computations are not conceptual.

The states that we associate with bits, aka {F,T} or {0, 1}, are nth-order states. Memories, qualia, and the phenomena of conscious experience are nth-order states. The notion of Pi is an nth-order state. The representations of Pi are necessarily first-order, they're physical manifestations. But the first-order thing is not the nth-order thing, it's a representation. In exactly the same way, the nth-order concept of dog is not the same thing as the first-order stuff that we associate with dogs.

1) Precisely how much “more“ than 3 (@@@) is required to represent the quantity of pi?

Precisely? \( \pi - 3 \). If you're not satisfied with that, then \[ 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \cdots \right) - 3\]

2) Did they, or did not they not, computer the 2 quadillionth digit of pi’s mantissa? If not, what did they compute?

They computed the quadrillionth or whatever digit of the base-10 decimal expansion of Pi. That quadrillionth digit would be different in the base-2 decimal expansion, or the base-16 decimal expansion, and so on.

I do not understand how you don't get this. A label is not the thing it labels.

From this point on, I seriously want to reference NOTHING without seeing it as hardware at the same time...

Fine, but we need to be explicit about the level of abstraction. If you're thinking in terms of bits, you're not thinking at the hardware level.

 

That means “numbers” ARE represented as some thing (voltage).

Correct, we can represent numbers using voltages. Never in dispute.

They are NOT “independent of their representation” at the hardware level.

Hopefully, after you read my post just above, you'll agree that this statement needs qualification. Numbers are not defined at the hardware level, i.e., first-order states. At the nth-order level where they are defined, numbers are indeed independent of their representations.

If you disagree, then how can the same number be represented in two different ways? We can represent numbers using voltages, or water levels, or gas pressures, or symbols on paper, or words, or whatever. If a number were dependent on its representation, then all of these different representations would be representing a different number!

How can you not see this?

So now, please define a number based on voltage states alone. Define it as {#,!}. Please define pi that way alone.

It doesn't make sense to me to define a number based on voltage states.

If we can’t at all know “the way anything is,” then we should close up shop here, because why even bother with knowing “any” “zoom level” of the information/machinery QED’s you made earlier? They might not be the way things are in the end?

This doesn't make sense to me either. It's like saying, Why bother living if we're going to die in the end? We'll almost certainly never know how things "really are", but we can still learn a lot about how they seem to be.

What if GR makes sense mathematically, but is not ontologically by empirical measurement?

We know full well that GR is not a complete theory; it's missing a bunch of stuff. But we also know that any better theory will only augment GR, not contradict it. GR didn't "overturn" Newtonian theory, it only augmented it.

 

The result of it is continuous (fundamental), and the “worth” of that continuity computation-wise is identical to a quantum continuity: either a discrete number, expression, or “whether or not it’s TRUE” it’s the same or different to something else.

Is this sentence English?

The result of the Lissajouz phase calculation is not discrete and cannot be labeled as true or false.

 

Unary is a logic value. @ = True that something is quantifiable

TTTTT + TTT = TTTTTTTT
Or
@@@@@ + @@@ = @@@@@@

T is “it’s true something is there”. There are 5 values of truth in 5 volts and 3 levels of truth in 3 volts = 8 “presence” truth levels

So you've finally come around to the validity of unary computation?

 

Of course the theorem is correct. But what is “talking” about the Shannon theorem? More discrete states that don’t the difference.

You seem to think the term “know” can apply to discrete elements that don’t know they’re connected to any other.

Does a single flip-flop in memory "know" about the rest of the flip-flops? No, I would say not. Does the CPU know about all of the flip-flops, and how to treat them as separate groups? Absolutely, otherwise we wouldn't be having this conversation.

What is so magical about "knowing"?

 

It is indirectly discerned by an algorithm created by a human that directly knows spatiality, or that the dog exists in physical space.

"Indirectly discerned" is still discerned, though, yes? What if humans indirectly discern spatiality, as well -- would we be able to tell the difference between directly and indirectly discerned spatiality?

A machine is an information processor. It has no “questions” about what it’s processing about.

And what makes you think machines could never question what they're processing? What is magical about questioning?

Life, as the dog, as “feeling and knowing” are qualia beyond the physical mechanics! Why do you think, with all our studying of biology, we still have no definition for life?? Because life is not physical machinery.

Why do I think we "have no definition for life"? I think we have several. I would say we have no perfect definitions for two primary reasons: 1) biological systems are the most complex systems we've ever come across, so it's very difficult to capture all of its nuances in a single definition; and 2) we didn't invent life -- we tend to have better definitions for things we invent. Our definitions of "rock" or "ocean" are not perfect either, but we can still fruitfully study geology and oceanography.

But all of that is irrelevant. You haven't answered the question: what property does a blood cell have that a computer cannot?

Qualia is NOT just states of voltages. A strawberry is different from a shower faucet in myriad ways. The living, conscious senses are not the “sensors” of a non-living machine.

What specifically prevents a computer from knowing the difference between a strawberry and a shower faucet? We can make sensors that measure any of the things our physical senses measure. So, what's the difference?

 

Does a single flip-flop in memory "know" about the rest of the flip-flops? No, I would say not. Does the CPU know about all of the flip-flops, and how to treat them as separate groups? Absolutely, otherwise we wouldn't be having this conversation.

What is so magical about "knowing"?

Ah, but what is your very specific definition of knowing? You won't say any other flip flop "knows" what any other flip flop is doing, but yet you will define the CPU and the wires that go to the flip flops as "knowing?" The switch that turns on the ceiling light: Does it "know" it's turning on the ceiling light? Does a faucet handle that's letting water come through the faucet "know" that the water is now going through as a result of its state?

 

It is not. There is no limit to the number of different ways that we can express Pi that are not decimal expansions. Furthermore, there are as many numbers in ℕ as there are numbers with a non-terminating decimal expansion. In other words, having a non-terminating decimal expansion is hardly unique to Pi, therefore, it cannot be a defining characteristic of Pi.

What makes Pi unique has nothing to do with the fact that it has a non-terminating decimal expansion.


You're confusing abstraction levels. Perhaps this will make it more clear. Let's call the physical states -- the stuff of space -- first-order states. Using first-order states, an information processor can create second-order states. Using second-order states, a processor can create nth-order states. There's a hierarchy of organization, and each level conveys its own information -- each level has its own domain of discourse.

Concepts, as states of states, are minimally second-order states. We can of course form concepts from other concepts, forming nth-order states. Numbers are an nth-order state.

As physical machines, computers work with first-order states. Computations are changes in first-order state -- computations are not conceptual.

The states that we associate with bits, aka {F,T} or {0, 1}, are nth-order states. Memories, qualia, and the phenomena of conscious experience are nth-order states. The notion of Pi is an nth-order state. The representations of Pi are necessarily first-order, they're physical manifestations. But the first-order thing is not the nth-order thing, it's a representation. In exactly the same way, the nth-order concept of dog is not the same thing as the first-order stuff that we associate with dogs.


Precisely? \( \pi - 3 \). If you're not satisfied with that, then \[ 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \cdots \right) - 3\]


They computed the quadrillionth or whatever digit of the base-10 decimal expansion of Pi. That quadrillionth digit would be different in the base-2 decimal expansion, or the base-16 decimal expansion, and so on.

I do not understand how you don't get this. A label is not the thing it labels.

OMG! What you are not seeing is your Harry Potter Abstractor Wand (HPAW) again from the hardware perspective!

\[ 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \cdots \right) - 3\]

Please re-write this as unary as the only level of abstraction we are seeing. The unary would then stand for the voltage presence.

Why do you insist the above is finite? You suffixed a "..." to it to denote non-termination even in the representation above! Those are numeric digits that are ADDING to 3, correct? Pi is NOT just 3, it is 3 + an additional quantity. However you want to "label it" is fine, but what is that quantity?

 

It’s not asking questions about the nature of what the information is representing, or questioning the difference between information and its spatial representation.

The airplane computer system most certainly questions its data. Remember, these are 500,000 pound machines carrying hundreds of people at 300 mph. Airplanes are incredibly complex machines, and a sensor malfunction can be deadly. One of the most important capabilities of an airplane's computer is to issue an alert (question!) when the data doesn't make sense.

Granted, an airplane's computer doesn't ask philosophical questions, but neither does my meathead friend.

It does not “feel” or “mean.”

What is the nature of "feel"? Does a blood cell "feel"? What prevents a computer from having a similar experience of "feel"?

Something knows that the image (information is connected to the dog, which means that something in the being has a discerning mechanism that differentiates! "Knows that it knows the difference!" Does that make sense? If it does, then why and how??

Differentiation -- organizing information -- is key, I agree. But a computer can organize information; computers have discerning mechanisms. What's the difference between a blood cell and a computer?

 

If there are no numbers in a machine/computer, there are no numbers in you, a machine/computer, therefore you should not refer to things coming from your brain as numbers.

yes?

Little dramatic with the bold there, eh?

I 100% agree that there are no numbers in my brain. My brain holds concepts of numbers, but not numbers themselves. Similarly, my brain holds the concept of unicorn, but there are no unicorns in my brain.

 

So you've finally come around to the validity of unary computation?

Lol, I've said this before more than once. I believe fundamentally everything is unary "or" binary — binary is a more efficient and compact version of unary essentially, by using a contrasting "absence" version of itself for more efficient space handling. Are you groking the overlap between T and @ though?

 

Little dramatic with the bold there, eh?

I 100% agree that there are no numbers in my brain. My brain holds concepts of numbers, but not numbers themselves. Similarly, my brain holds the concept of unicorn, but there are no unicorns in my brain.

I told you, you can ignore all my boldings and italics — I need living emotion and this text-only is suffocating at times. ;--D Therein lies the difference between a computer and a person. A living person "feels" desire, intention, etc. Not just computes. A non-living person does not, nor does a computer, nor n red and white blood cells. Because life is not the machine. Life is IN the machine.

"Concepts" of numbers?? I thought a number was a concept!

 

The elementary concept of numbers have existed for thousands of years, and computations performed with them by the greatest of mathematicians of all time . . .

And it is just as true that, for thousands of years, medicine has been performed in various way, all without the modern notions of physiology and molecular biology. Therefore . . . what?

Just because people have used numbers does not make them experts on numbers. Newton was literally ridiculed for his idea of fluxions (infinitesimals) -- Berkeley called them the "ghosts of departed quantities". And Berkeley was right to criticize, because without precise, rigorous justification, the very notion is absurd. But Newton was focused on the mechanics, not the mathematics -- calculus was a tool for him, not an end in itself. Much later, mathematicians would clean up the conceptual mess that Newton and Leibniz had made.

Number theorists study the consequences of ℕ. Algebraic geometers study the consequences of ℤ. Analysts study the consequences of ℝ and ℂ. Go tell them that they're not studying numbers.

 

Ah, but what is your very specific definition of knowing? You won't say any other flip flop "knows" what any other flip flop is doing, but yet you will define the CPU and the wires that go to the flip flops as "knowing?"

Let's make it perfectly binary for you. True or false: The CPU knows what the states of its flip-flops are.

If the answer is "true", then what's your problem with computers knowing things? If the answer is "false", then how does the CPU differentiate between, say, opening a JPEG and playing a WAV file?

The switch that turns on the ceiling light: Does it "know" it's turning on the ceiling light? Does a faucet handle that's letting water come through the faucet "know" that the water is now going through as a result of its state?

Seriously? We are talking about machines of sufficient complexity here. Light switches and faucets are extremely simple machines.

 

This conversation has been very fun, stimulating, entertaining, etc. But it's going to go on forever in as many areas as pi has decimal-10 digits, and I'm sure you agree, we gotta get more ordered if it's going to actually be productive.

And formalizing is 100% your gift and shtick.

So I'm now going to look to you to direct the flow of questions in a formal manner so we stay ultra disciplined in one area of focus to arrive at *actual agreed conclusions*.

Cool? We need to get down to parity of abstraction with hardware, unary, voltage, etc...

 

And it is just as true that, for thousands of years, medicine has been performed in various way, all without the modern notions of physiology and molecular biology. Therefore . . . what?

Just because people have used numbers does not make them experts on numbers. Newton was literally ridiculed for his idea of fluxions (infinitesimals) -- Berkeley called them the "ghosts of departed quantities". And Berkeley was right to criticize, because without precise, rigorous justification, the very notion is absurd. But Newton was focused on the mechanics, not the mathematics -- calculus was a tool for him, not an end in itself. Much later, mathematicians would clean up the conceptual mess that Newton and Leibniz had made.

Number theorists study the consequences of ℕ. Algebraic geometers study the consequences of ℤ. Analysts study the consequences of ℝ and ℂ. Go tell them that they're not studying numbers.

Nah, they study "sets of numeric expressions." "There are only grunts, and if necessary, one grunt set. All else is the work of man."

 

Let's make it perfectly binary for you. True or false: The CPU knows what the states of its flip-flops are.

Ah, good. No, not at all.

"Know" for me here is like "real" for you.

First, calling it a "CPU" from you as a machine yourself, is an "nth" order abstraction that doesn't really exist outside of Harry Potter. The "CPU" is a spatial catch-all term for a thing in reality you don't really know what is made of.

Second, it's all 1st order for me until you can prove other abstractions earn the right to be called nth order.
The CPU has flip flops that register state value changes. Those flip-flops are the only thing that would qualify as knowing anything, if you were to use the term "know." A registration of a change of state is no different than saying the paper knows it has a check in a checkbox. "Awareness" implies some kind of innateness to it, I will not ascribe to papers or switches that have no idea why their state changed. Besides, if consciousness it something outside the switches that is not "a result of n switches", it has something to do with it. I propose feeling is independent of information, and is the very mystery element responsible for "knowing" the dog is something independent of information.

The CPU is a componental device. Just because there are wires between each portion of the salad's carrots, onions, tomatoes, and lettuce, and some power source sent enough juice between them to alter their states into quasi-melted elements, means nothing. In short, level of functionality is not "knowing."

 

Please re-write this as unary as the only level of abstraction we are seeing. The unary would then stand for the voltage presence.

Why would I bother to do that? You can do it yourself: replace every "5" with "@@@@@" and so on. Who cares? Representation is not significant.

Why do you insist the above is finite? You suffixed a "..." to it to denote non-termination even in the representation above!

Why do I insist it is finite? Because it fit on your screen, lol. You're missing the point and getting lost in the "...".

All representations are necessarily finite (otherwise they wouldn't fit in the universe). A non-terminating expansion has no "last digit", so to speak. If a non-terminating expansion has a representation (i.e., is computable), then that representation is finite.

What does this mean in plain English? If we can write a program to spit out the nth digit of a non-terminating expansion, for any n we choose, then we have a finite representation of that number. Namely, the program is the representation. Pi is such a number. Therefore, Pi has a finite representation.

Note that "3.1415..." is not a representation of Pi, because it is the representation of countless numbers, namely, all those that start with "3.1415". But the program that spits out Pi uniquely specifies Pi, which makes it a representation of Pi.

What's an example of a number that does not have a representation? Chaitin's constant, which I believe you've mentioned before. Chaitin's constant is not computable -- there is no program that can spit out its nth digit -- and so it has no representation. It is a finite number that we cannot represent. Pi, on the other hand, is a finite number that we can represent.

Those are numeric digits that are ADDING to 3, correct? Pi is NOT just 3, it is 3 + an additional quantity. However you want to "label it" is fine, but what is that quantity?

No, if you multiply every term in the series by 4 and then subtract its value from 4, you get Pi. So, Pi minus 3 tells us how much more Pi is than 3. Note that Pi - 3 is an irrational number with a non-terminating decimal expansion. But we can finitely represent it. It is computable.

 

Lol, I've said this before more than once. I believe fundamentally everything is unary "or" binary — binary is a more efficient and compact version of unary essentially, by using a contrasting "absence" version of itself for more efficient space handling. Are you groking the overlap between T and @ though?

Am I grokking? I've said it since the beginning. In a two-valued logical system, the domain of discourse is {T, F}. In a single-valued system, the domain of discourse is {@}.