Numbers are MADE of logic states, and this is axiomatically, positively correct at the ontological level for a digital system.

Answer this question: given that we agree that analog computers can do computations, what are numbers "made of" in analog systems?

You're pointing to digital computers and saying, "See! Numbers are logic states!" But if numbers are logic states, then how do analog computations work? <-- This fact needs to be addressed by your ontology.

In my ontology, there are no numbers inside of either analog or digital computers. There are no trues or falses, either. Those things -- numbers, logic values -- are abstract concepts. To perform calculations, we devise formal systems and then build machines that map physical phenomena to the symbols of our formal systems. In every computation, the laws of physics does the manual labor and we interpret the resulting physical state as a value.

In your ontology, logic values somehow have physical extent (otherwise, how do machines use them?). Please explain where they live, how much they weigh, etc.

The computer is working with voltage that is on (1 or T or "minimal presence").

Stop. Just stop with "voltage that is on", as that is not at all what's physically happening. From the 10,000 ft view, you see a computer as a bunch of discrete states, but this is an ILLUSION. There is no such thing as a discrete voltage. Voltage is the potential between two points in the electric field. No matter where you measure it, there will ALWAYS be a voltage. And when we measure it, it's value is not a discrete state.

So, on one hand you have this constantly varying quantity that can never be measured exactly, and on the other you have "true/false" logic states. These are EXTREMELY different things. It takes an enormous amount of effort to create a machine that pretends that they are the same. Yet, somehow you are claiming that ontologically they are the same. That's absurd.

At the end, the voltage is using "multiple instances of itself" to manifest a number

"The voltage is using multiple instances of itself to manifest a number." Seriously?

Category error, for you, again—a machine—is another way for you to get away with saying CONCEPT.

What's the category, and how did I make an error?

Let 1 (voltage) = a UNIVERSE. What's the universe? More voltage in different areas?

Let 1 = DOG in space. (you can use let J3gjLF = DOG in space. In the end, those glyphs need to boil down to tons of unary voltages, or voltages that are contrasting for greater space efficiency).

Do you see what I'm getting at here yet? I'm looking at the voltage states as what's going on, not any further abstractions.

No "further abstractions"? What do you think "1 = universe" is? Besides being a misunderstanding of what Boole was trying to say, it's a certified, grade-A abstraction!

 

Um. We can't pretend that any number, if it does not terminate, can have a terminating element, and thus, set it off from any other element. Under what intellectual justification are you saying those points, and intervals, are elements of this "set" without invoking Harry Potter? E.g., what "cells" in the C array are those again? You don't know where 42 is in there.. it's in between 41.9197917012910... and 42.4189027501827510... and infinite other numbers. ℝ is no more countable than it is addressable in "reality". Intervals are addressing actual, discretely addressable elements in a set that are delimited by commas (or some other delimiter) so you can address them. My point is, the very concept of R as having all its infinite numbers as being rationally "addressable as part of it" is madness. It's either 1 continuous line, or it doesn't exist. It is not a number "set," like "N," it is a stand-alone phenomenon.

I have no idea what you're trying to say. You claimed that we can't "access" or "delineate" an interval. I then gave you an example of an interval. What does any of this have to do with non-terminating representations?

You seem to believe that a non-terminating representation is somehow indicative of a non-terminating number. But that's grossly confused. Terminating and non-terminating are properties of representations, not the numbers themselves. The number represented by 1/3 has a non-terminating representation in its base-10 decimal expansion. The same number has a terminating representation in its base-3 expansion.

I can't help but think that most of your problems with ℝ stem from a few fundamental misconceptions about numbers and their representations.

Let me be clear: there are no non-terminating numbers in ℝ (or any other number set).

Because ℕ is a discrete, delimited number set, and ℝ is something else entirely. It is improper to call it a set in the same way.

This is simply not true of set theory. You're making up facts to suit your conclusion.

You also can’t say ℕ is a subset of ℝ, I’m realizing, because that again is incongruous for the same reason.

There is a precise definition of what "is a subset of" means for two sets. When that definition is applied to ℕ and ℝ, we find that the former is a subset of the latter.

Just because you say "NO!" does not make it invalid. You have to show that a logical contradiction follows from the assumption. Good luck with that.

You’re gonna love this one: Every discrete element of ℕ is actually a name for a unique packetized manifestation of ℝ, ordinalized into a comma delimited set. The other sets don’t exist as “sets” because they, too, are attempting to manifest ℝ, but fail because you can’t terminate elements with commas. The “all animals vs. ones with wings” comparison thing doesn’t work here, because of the above.

You're right, I hate it.

What exactly is a manifestation of ℝ? Do you mean a number in ℝ? If so, then it is provable that there are not enough elements in ℕ to cover ℝ.

What are they in the brain hardware, is what I mean. An infinite-value sine wave doesn't "exist" to you in the brain, but you seem to have a function to create one from an infinite continuum in your brain??? WTF all day long.

You still haven't explained what a "stand-alone continuous element" is, so I have no idea if or how it can exist in the brain hardware. The function sine is a concept in my head -- there is no physical wave that corresponds to the sine function.

Mapping what? How exactly is the notion of continuity existing in a discrete system???

Functions map elements in one set to elements in another set. When we write \( f: A \to B \), we are saying "f is a function that maps elements from the set A to elements in the set B".

In your second question, I take it you're asking how a brain -- which you're assuming is discrete -- can hold the notion of continuity. I don't assume the brain is discrete, though I'm happy to acknowledge that it may be. (My biggest problem with it is that "discrete" and "continuous", like "geometric 3D", are human concepts that may or may not apply to the universe.)

But let's assume the brain is a "discrete computer" (even though actual computers are fundamentally not discrete, lol). How can a set of discrete set of states hold the concept of a continuum? In the same way that a finite set of states can hold the concept of INFINITE. The concept of continuous is not itself continuous.

 

Let me be clear: there are no non-terminating numbers in ℝ (or any other number set).

What??? Do tell what this is about, right here.

 

010101011010101101010101010110101010101011101.... is the legs of a chicken

10101010101100101001001....is the treads of a tire

101101101101001010101010101010101... is a wine bottle

1101011001011010110100101010101... is infinity.

With 1’s and 0’s being discrete logic states created by letting electricity flow or not (open or closed) in any binary computer.

Concepts, right? 10-4. You get REALITY yet? Lol.

When I speak of discrete voltage, I know you can measure voltage around the whole damn thing. I’m talking about the fact that we are representing “things in physical space” or discrete concepts with high and low contrasts.

 

This is great right here:

ℝ is literally ALL the numbers and their interrelations in one long line.

No, ℝ does not contain "all the numbers". Counterexample: \(\sqrt{-1} \notin \mathbb{R} \).

Earlier you asked "What is a number?" I gave you a general answer, but I should have been more precise.

A number is an element of a number set. Full stop.

If the domain of discourse is ℕ, then you can happily and validly say that root-2 is not a number. If we want to talk about magnitudes, however, experience tells us that we need a bigger number set. We don't necessarily have to choose ℝ, but we can describe more magnitudes if we do.

But even ℝ is not big enough to solve all polynomial equations, so -- if we want to find the roots of polynomials -- we change the domain of discourse to ℂ.

On the other hand, if we're trying to describe rotations in three-dimensional space, we find that it can be inconvenient to work with ℝ or even ℂ. So we turn to \( \mathbb{H} \), the set of quaternions.

What we call a number is a choice. We find that some number sets are more useful in everyday life than others, but that doesn't make them anymore "REAL" than our other options.

This is not a delimit-able, elemental set in the mathematical sense, because we CANNOT terminate each element.

Hopefully you now recognize that non-terminating applies only to representations, not the elements of a number set.

 

Thought experiment based on the above:

Envision that REAL line, now extend it into a REAL plane. Now extend it up to create "walls". Now top it off with another REAL plane. What do you have? A CUBE made out of "The REAL stuff."

Incidentally, this cube was NOT seen in physical space. It was manufactured internally out of definitions that COULD NOT HAVE COME FROM PHYSICAL SPACE (So important)! You can now address elements of it!

"WHERE is that thing located in your physical discrete-state brain, if it's not a function, but an addressable spatial thing you can "walk around in the space of your mind?"

The thing so-described is a concept.

Here's the same thought experiment put more simply: think of the color red; where is the red located in your brain? What object are you "seeing" as red?

 

Hopefully you now recognize that non-terminating applies only to representations, not the elements of a number set.

Before number sets were invented, can you give me an answer concerning “what a number is” and what “non-terminating” are?
;—)

- Tabulo Rasini

 

And so the likes of pi is non-terminating to you?

 

Just a friendly neighborhood town reminder:

The inventor of calculus, Sir Isaac Newton, had no sets of any kind. Neither did Pythagoras, Euclid, or Des Cartes, Leibniz, or hundreds of others.

Somehow they could “get on” with the usage of these things called “numbers” without Cantor’s 19th century inventions?

“Most mathematicians ignore them [because they don't exist outside of a construct of convenience]” as a statement... Could apply to the grandfathers of mathematics themselves.

Computers tend to ignore them too, and all is fine. ;—)

 

What??? Do tell what this is about, right here.

Good, let's get to the heart of the confusion and stomp it out.

Numbers are abstractions, they have no physical representation. This is what makes them supremely useful -- n sheep or n cows, we're talking about the same quantity. But how do we physically talk about or use abstract concept? We give them a label.

Labels are arbitrary. Whether we say "three" or "3" or "III" or "tres" or whatever, the labels are describing the same thing, the abstract number.

Now, there are far more possible numbers than possible labels. Furthermore, it would be horrendously painful to have to a memorize unique label for each number. So, over the years, we've come up with systematic ways to label never-ending streams of numbers with a small, finite set of symbols. The Greeks used the same symbols as in their alphabet, but that was kind of confusing to read. The Romans improved this by using a small subset of symbols from their alphabet -- "I", "V", "X", and a few others -- and setting them off in texts that included both numbers and words.

A big improvement came with Indian (aka Arabic) numerals, which used its own small, dedicated set of symbols to represent the numbers zero through nine. These handful of digits allowed us to write any number using a positional notation system, by assigning different weights to the position a digit has. Adding the weighted digits gives us the number so represented.

So, to label "one-hundred and twenty-one", we write "121": \[ 1 \times 10^2 + 2 \times 10^1 + 1 \times 10^0 = 100 + 2 + 1 = 121\]
We can, of course, label bigger numbers by writing more digits to the left, each weighted by an increasing power of ten.

To label smaller numbers, we write a decimal point to the right of the "ones place". Any digits thereafter get weighted by negative powers of ten. So, to label the number represented by "five-sixteenths", we write ".3125": \[ 3 \times 10^{-1} + 1 \times 10^{-2} + 2 \times 10^{-3} + 5 \times 10^{-4} = 0.3125\]
Now, due to our choice of digits (0 through 9) and our weighting system (powers of ten), it turns out that some numbers have a non-terminating expression in this representation. For example, how do we write "one-third"? When we sit down to try and write it, taking into account all the properties of it (e.g., more than "three-tenths" but less than "four-tenths"), we end up with a non-terminating expression: \[ 3 \times 10^{-1} + 3 \times 10^{-2} + 3 \times 10^{-3} + \cdots = \frac{1}{3} \] Does this mean that the number represented by "one-third" is a non-terminating number? No! Only the base-10 representation of it is non-terminating.

In base-3 representations, we have three symbols (0, 1, 2) and each weight is a power of three. What's "one-third" in base-3? It's simply "0.1": \[ 1 \times 3^{-1} = 0.1_{\text{base-3}} \] Clearly, "one-third" is perfectly terminating in base-3 (and base-9 and so on).

It's crucial that we make the distinction between a number and its representation. In particular, there are no non-terminating numbers. This applies to any number, including the irrationals, the transcendentals, etc. That they have non-terminating representations does not make them "infinite" or any such thing. They are all pefectly finite, and we can label them with perfectly finite symbols, like Pi.

 

010101011010101101010101010110101010101011101.... is the legs of a chicken

10101010101100101001001....is the treads of a tire

101101101101001010101010101010101... is a wine bottle

1101011001011010110100101010101... is infinity.

With 1’s and 0’s being discrete logic states created by letting electricity flow or not (open or closed) in any binary computer.

Concepts, right? 10-4. You get REALITY yet? Lol.

When I speak of discrete voltage, I know you can measure voltage around the whole damn thing. I’m talking about the fact that we are representing “things in physical space” or discrete concepts with high and low contrasts.

No idea what you're trying to say with all of this.

I would never say "1010101" or whatever is the leg of a chicken. I might say that "1010101" is a representation of the image of a chicken leg, or a representation of a particular computer's concept of a chicken. But a bit string is a language from a formal system; bit strings are abstractions.

 

Two things:

1) "Numbers are abstractions, they have no physical representation" and "but how do we physically talk about or use abstract concepts? " There he goes again! LOL. And so, you claim you are just physical. Where are these abstractions actually stored if they're not physical? Please clarify here.

2) So you're saying that "pi" is non-terminating only because of base-10 representation, and not due to any other reason whatsoever? I.e., all of the non-terminating phenomenon is all due to its representation alone?

 

No idea what you're trying to say with all of this.

I would never say "1010101" or whatever is the leg of a chicken. I might say that "1010101" is a representation of the image of a chicken leg, or a representation of a particular computer's concept of a chicken. But a bit string is a language from a formal system; bit strings are abstractions.

Sure, so now abstractions/concepts are just "groups of bits" are they not? Is not a bit representing a physical state in the computer to hold the information that is the abstraction? So how are you using abstractions that are not physically represented again?

 

Before number sets were invented, can you give me an answer concerning “what a number is” and what “non-terminating” are?

Not a very good one! You're acting as if cavemen found a bushel of numbers and started using them. But numbers are abstact. The notion of what a number is took thousands of years for experts to get a firm hold of -- most people have no idea!

 

And so the likes of pi is non-terminating to you?

Absolutely not! It is a perfectly finite number with a perfectly terminating representation: \( \pi \).

 

No idea what you're trying to say with all of this.

I would never say "1010101" or whatever is the leg of a chicken. I might say that "1010101" is a representation of the image of a chicken leg, or a representation of a particular computer's concept of a chicken. But a bit string is a language from a formal system; bit strings are abstractions.

I'm saying, in the binary computer, instead of writing:

Voltage-high, voltage-low, voltage-high, voltage-low, etc. etc.

I'm saying:

1010001010 etc...

I'm only using 1's and 0's only has placeholders for the voltage. Perhaps that the problem. I'm using them to reference the actual hardware.

1010101010111101001000101010100101.... = dog in real life

1010101111101010101001... = dog on screen

1010110001010... = chicken leg

Tabula rasa to the max!!

 

The inventor of calculus, Sir Isaac Newton, had no sets of any kind. Neither did Pythagoras, Euclid, or Des Cartes, Leibniz, or hundreds of others.

Yawn. Newton and those other dudes were not exploring the nature of numbers. When you actually investigate numbers as their own entities, you end up in the post-Cantorian world we now inhabit.

 

Yawn. Newton and those other dudes were not exploring the nature of numbers. When you actually investigate numbers as their own entities, you end up in the post-Cantorian world we now inhabit.

And 99% of it is in the REAL set, if they're all subsets of it, correct? So what's the big deal?

REAL is not much different than my proposal of calling them "NUMBERS" as u-its.

Btw, you're back to talking non-tabula rasa almost exclusively (even beyond when I deviate)...apparently we can't ever stay there. I'm using tabula-rasa level definitions based on the hardware. Your "abstractions/concepts" to me don't exist, because you're elevating anthro-centric elements there in my estimation. In my estimation, we have dirt, voltages and grunts. That's TRUE tabula rasa.

 

Btw, I'm pretty sure I'm waiting for Candid Camera to appear on this thread.

https://techxplore.com/news/2019-03-pi-trillion-decimals-google-day.html

If you represent those digits as u-its, bits, or whatever else, you have a number that keeps going irrespective of base.

We call that number "pi", but "pi" stands for 3.14... into oblivion or "infinity."

It's part of the infinite points between 3 and 4 on the REAL number line.

But perhaps we need to call in some other people to get their opinion?

 

1) "Numbers are abstractions, they have no physical representation" and "but how do we physically talk about or use abstract concepts? " There he goes again! LOL. And so, you claim you are just physical. Where are these abstractions actually stored if they're not physical? Please clarify here.

Physically, concepts are just states of information, like anything else. An information processor forms concepts by associating different states with each other. It can form concepts from external states through physical interactions, or it can form concepts from internal states, or some mix of the two. We may call internal concepts "abstract", as they do not refer to external states.

Siri has a concept of the weather. Yes, she was programmed with the associations that comprise the concept, but it is a concept nonetheless. Siri's concept of weather is not abstract; her associations relate primarly to external states. My concept of ℝ is abstract -- its associations are primarily internal.

Abstractions themselves -- like numbers -- are concepts that may or may not be "abstract concepts", i.e., purely internal. "Voltage" is an abstraction but it is not an abstract concept. That is, "voltage" is an element of the electrical field model of electricity (an abstraction) that we use to characterize a particular form of physical phenomena. Because it associates states that are primarily external, we don't call voltage an "abstract concept". Number sets, however, are an "abstract concept".

2) So you're saying that "pi" is non-terminating only because of base-10 representation, and not due to any other reason whatsoever? I.e., all of the non-terminating phenomenon is all due to its representation alone?

What "non-terminating" phenomenon? Pi is not a phenomenon, nor does it "have" phenomena.