Ok, but the essence of ourselves as "state machines" is that T/F is what matters to *us* as humans. I don't care how many transistors, wires, voltage levels, etc. In the end, whether classical or quantum, T/F is the basis of propositional logic, the basis of human reason, the basis of computers and their behavior as "computers." They "compute T/F" outputs. We amalgamate those output strings into higher abstractions.

Yes?

I don't agree. When you compute "174 +149", whether in your head or on paper, you're not using true/false values. Do you agree with that?

 

Sqrt. 2 is NOT a number. It is a computation. It just happens to relate in PART to other components of a geometric figure, that TOO is an obfuscation.

Let's be clear. You're claiming that magnitudes, such as the length of a stick, are not numbers?

In order to blaze new trails and find core connections, we have to question all existing presumptions, no matter how popular... and we cannot necessarily invoke consistency checkpoints from existing systems in the process!

I'm fine with questioning presumptions -- that's what I do -- but we have to accept the logical implications of anything we overthrow. For instance, we use numbers to solve equations. But if root-2 is not a number, then the equation "x^2 - 2 = 0" is somehow fundamentally different from the equation "x^2 - 4 = 0", as the latter has a solution, while the former does not. What, then, is the ontological difference between the two equations?

 

100% agreed! But that is a massive problem... because what if there are either errors or inadequacies in a given system? It's as if we need to build a semantic eco-system from scratch and THEN work it into existing formalities.

This is exactly why formal systems are so useful, because we can reason about the system directly, independent of any semantics or interpretations.

We know, for instance, that propositional logic is inadequate to express general statements about the natural numbers; it's simply not powerful enough. So, we augment the system and end up with predicate logic. This gives us the power to formalize arithmetic statements, but in return we have given up decidability -- we can no longer be certain that every statement is true or false in the system.

This uncertainty kind of sucks, but it's not crippling -- we do know that the system is sound (every provable statement will be provable in every model, under every possible interpretation). So, if we can manage to cast a notion as a provable statement in first-order logic, we know that it will hold in every model.

Where it gets especially confusing is when we try to reason about a formal system within a formal system! It's a perfectly valid thing to do, we just need to be especially careful about what we're saying and how we're saying it.

 

You have to somehow disengage the "category errors" element while we work on this, because I'm rendered one big blue screen of death with 404 emblems made out of old ANSI characters otherwise.

Category errors are a problem because they reveal a mismatch between what was said and what was meant. In terms of typed programming languages, if you declare a character string as having a type of integer, the compiler would instantly stop production. Strings and integers are different, incompatible types of things. The actions we do to strings don't make sense for integers, and vice versa.

When you say that a voltage is a T/F value, you are making an equivalent type error. Voltage has type "electricity", which is a sub-type of "physical phenomenon". T/F values have type "logic", which is a sub-type of "abstract formal system". They're completely different things. We measure voltage between two points, and can invert the measurement by simply swapping the order of the probes. A voltage magnitude can have any value. In contrast, we assign T/F values to boolean statements and, once assigned, they remain fixed. We cannot measure T/F values; they do not have a magnitude. Totally different kinds of things!

Keeping this in mind prevents us from making conceptual mistakes, like assuming that transistors are digital devices, or that computers compute with T/F values. I'm not asking you to become an expert on transistors, but I am asking you to recognize that it takes considerable effort to wrangle voltages into behaving as if they were T/F values. Even the very best digital circuits are imperfect models of boolean logic because, fundamentally, they are not boolean! Every digital circuit has noise, leakage current, finite transition times, fan-out restrictions, undefined voltage states, etc. We design them to be as impervious to these non-boolean errors as much as possible, but we cannot eliminate them because -- at least at the circuit level -- physics is not boolean.

Being mindful of category errors helps us conceptualize things more clearly.

 

I propose we start a new number set to begin, called I.

This number set is unary and infinite. Any interval created has infinite 1's in between them.

T. Rasa—

1) What is a number in this set?

By unary, you seem to mean base-1 representation. But numbers are independent of their representation. We can write any base-1 representation in, say, base-10. So, the "unary" part is unnecessary.

Let's tackle the "infinite 1s" part. Since we can do it in any base, we'll do it in base-1. Because we can make intervals, I'll assume I is ordered.

Let a and c be two numbers in I, with a < c. We're given that the amount of numbers between a and c is unbounded; in particular, there is some number b in I such that a < b < c. For concreteness, let a = 11 and c = 111. What is c?

Well, we can't write it as a simple string of 1s as we've exhausted the space of concatenated 1s. We can try taking half of their sum: \[ c = \frac{11 + 111}{11} = \frac{11111}{11} \] In base-10, we'd call this 5/2.

Now that we have b, what does d such that a < d < b look like? Well, \[ d = \frac{11 + \frac{11111}{11}}{11} = \frac{111111111}{1111} \] In base-10, we'd call this 9/4.

Since we can continue in this way for any number in I, I claim that I = ℚ.

Congrats, you've invented the rational numbers!

 

By unary, you seem to mean base-1 representation. But numbers are independent of their representation. We can write any base-1 representation in, say, base-10. So, the "unary" part is unnecessary.

Let's tackle the "infinite 1s" part. Since we can do it in any base, we'll do it in base-1. Because we can make intervals, I'll assume I is ordered.

Let a and c be two numbers in I, with a < c. We're given that the amount of numbers between a and c is unbounded; in particular, there is some number b in I such that a < b < c. For concreteness, let a = 11 and c = 111. What is c?

Well, we can't write it as a simple string of 1s as we've exhausted the space of concatenated 1s. We can try taking half of their sum: \[ c = \frac{11 + 111}{11} = \frac{11111}{11} \] In base-10, we'd call this 5/2.

Now that we have b, what does d such that a < d < b look like? Well, \[ d = \frac{11 + \frac{11111}{11}}{11} = \frac{111111111}{1111} \] In base-10, we'd call this 9/4.

Since we can continue in this way for any number in I, I claim that I = ℚ.

Congrats, you've invented the rational numbers!

By unary, you seem to mean base-1 representation. But numbers are independent of their representation. We can write any base-1 representation in, say, base-10. So, the "unary" part is unnecessary.

Let's tackle the "infinite 1s" part. Since we can do it in any base, we'll do it in base-1. Because we can make intervals, I'll assume I is ordered.

Let a and c be two numbers in I, with a < c. We're given that the amount of numbers between a and c is unbounded; in particular, there is some number b in I such that a < b < c. For concreteness, let a = 11 and c = 111. What is c?

Well, we can't write it as a simple string of 1s as we've exhausted the space of concatenated 1s. We can try taking half of their sum: \[ c = \frac{11 + 111}{11} = \frac{11111}{11} \] In base-10, we'd call this 5/2.

Now that we have b, what does d such that a < d < b look like? Well, \[ d = \frac{11 + \frac{11111}{11}}{11} = \frac{111111111}{1111} \] In base-10, we'd call this 9/4.

Since we can continue in this way for any number in I, I claim that I = ℚ.

Congrats, you've invented the rational numbers!

Darn, I edited it a few moments ago to “@“ for that reason!

Can you look again at it from that perspective?

 

Another pot on the stove:

Mathematics is, among other things, about the relationship between numbers. Why does your brain insist on any relationship at all?

Why does my brain insist on relationships between numbers? That's a weird way of putting it. Very likely, the first numeric relationships I knew about were taught to me by someone else. In any case, I didn't insist on the relationships. Indeed, I seem to be completely oblivious to a relationship until either it's pointed out to me, or I happen to discover it for myself. But, like a wallpaper pattern, once the relationship is recognized it becomes a concept.

I think patterns are a good way to get at what you're trying to get to. I believe mathematical relationships exist independently of my brain, just as the shapes in a wallpaper or rug exist independently from me. That the shapes are arranged periodically -- a pattern -- is extra information that my brain can latch on to. Similarly, there seem to be "patterns" in mathematics that we are able to recognize.

 

Darn, I edited it a few moments ago to “@“ for that reason!

Can you look again at it from that perspective?

It's precisely the same thing. Between @@ and @@@ is, among other numbers, "@@@@@/@@". Numbers are independent of their representation.

 

I don't agree. When you compute "174 +149", whether in your head or on paper, you're not using true/false values. Do you agree with that?

If the brain is a discrete state machine, and each number is a bank of states for representing those values, what are we “using” to make the computation? 174 and 149 are addresses for banks only, no?

 

It's precisely the same thing. Between @@ and @@@ is, among other numbers, "@@@@@/@@". Numbers are independent of their representation.

I’m trying to get at the basest of assumptions.

Why do we know 1 or @@ is “one unit” vs. &(&*$#*?

1 bit is 1 photon, one neuron, etc.

 

Why does my brain insist on relationships between numbers? That's a weird way of putting it. Very likely, the first numeric relationships I knew about were taught to me by someone else. In any case, I didn't insist on the relationships. Indeed, I seem to be completely oblivious to a relationship until either it's pointed out to me, or I happen to discover it for myself. But, like a wallpaper pattern, once the relationship is recognized it becomes a concept.

I think patterns are a good way to get at what you're trying to get to. I believe mathematical relationships exist independently of my brain, just as the shapes in a wallpaper or rug exist independently from me. That the shapes are arranged periodically -- a pattern -- is extra information that my brain can latch on to. Similarly, there seem to be "patterns" in mathematics that we are able to recognize.

What does “independent of your brain” mean? You are only your brain’s states. Does MathLab software know this relationship until it is arranged in hardware and gates to put the voltages through them and yield other voltages?

 

If the brain is a discrete state machine, and each number is a bank of states for representing those values, what are we “using” to make the computation? 174 and 149 are addresses for banks only, no?

I don't know if the brain is a discrete state machine. Perhaps it is. In any case, the numbers 174 and 149 are not addresses, they're concepts. The computation doesn't have much to do with their concepts, though. When I compute "174 + 149", I manipulate the symbols according to an algorithm. This uses a bunch of states, most of which have nothing to do with my concept of numbers 174 and 149.

My original question to you, which you didn't answer, was: do you personally use true/false values when computing "174 + 149"? If so, how? If not, then how can you say that computation is based on true/false?

 

Let's be clear. You're claiming that magnitudes, such as the length of a stick, are not numbers?

Correct. I addressed this in my grunt thesis. Magnitudes only exist when numbers are suffixed by something.

6 vs. 3 is not a comparison because numbers are independent of what they’re representing, such as size, and sequentiality until user-defined. 6 notches vs. 3 notches is reckoned when we biject the set or notches to the set of numbers. You can’t say the stick is “longer” otherwise without invoking some arbitrary non-numeric apparatus (feeling? Lol). The brain is a measurement device only.

 

I don't know if the brain is a discrete state machine. Perhaps it is. In any case, the numbers 174 and 149 are not addresses, they're concepts. The computation doesn't have much to do with their concepts, though. When I compute "174 + 149", I manipulate the symbols according to an algorithm. This uses a bunch of states, most of which have nothing to do with my concept of numbers 174 and 149.

My original question to you, which you didn't answer, was: do you personally use true/false values when computing "174 + 149"? If so, how? If not, then how can you say that computation is based on true/false?

“Concepts” again ... I don’t see them as anything other their discrete parts. I also don’t see any 2D symbols anywhere. Binary or unary is it at the hardware level, voltage is what we have to work with here.

I certainly can do the comp based on that, I can convert the numbers to binary true and false representation and do Boolean algebra on it to arrive at a string that I can call 323.

What if we invent a new non positional number system where I have a symbol for each number.

¥ is 149
£ is 147

How do I know when I add these things that they equate to symbol § which is 323, until I set up rules?

Abstract symbols is NOT where the computation is happening. It’s happening at the hardware level at the most elementary representation (grunts) thats why propositional logic and T/F, and {0,1} as both numbers and truth states fits so perfect to us innately.

 

I’m trying to get at the basest of assumptions.

Why do we know 1 or @@ is “one unit” vs. &(&*$#*?

Such a great question. What makes us think that "1" or "@@" or "&(*@&#" is a number? There is a single, simple answer: we do.

We define the symbols "@@" to represent something that is entirely conceptual, i.e., that has no intrinsic physical representation. As long as we agree on the notation, we can discuss as if the symbols were the numbers. But it's a category error to think that the symbols are the numbers; they're not. The symbols are entirely arbitrary; we can choose whatever we like and still be talking about the same thing, the concept of a number.

This is exactly what we do with the voltages in a computer. We agree that this range of voltages represents this concept, and another range of voltages represents another concept. Then we design the physical machine to manipulate the voltages as if it were manipulating the concepts themselves. The voltages are entirely arbitrary; we get to decide what they mean.

The concepts, however, are not arbitrary. This is why you ended up with ℚ from your definition of I. We can use whatever labels we want, but the thing described depends only on the conceptual relationships, not on the physical manifestation.

1 bit is 1 photon, one neuron, etc.

Again, weird way to say this. Bits are abstract; photons and neurons are physical. We say that a photon or neuron can encode 1 bit of information.

 

Such a great question. What makes us think that "1" or "@@" or "&(*@&#" is a number? There is a single, simple answer: we do.

We define the symbols "@@" to represent something that is entirely conceptual, i.e., that has no intrinsic physical representation. As long as we agree on the notation, we can discuss as if the symbols were the numbers. But it's a category error to think that the symbols are the numbers; they're not. The symbols are entirely arbitrary; we can choose whatever we like and still be talking about the same thing, the concept of a number.

This is exactly what we do with the voltages in a computer. We agree that this range of voltages represents this concept, and another range of voltages represents another concept. Then we design the physical machine to manipulate the voltages as if it were manipulating the concepts themselves. The voltages are entirely arbitrary; we get to decide what they mean.

The concepts, however, are not arbitrary. This is why you ended up with ℚ from your definition of I. We can use whatever labels we want, but the thing described depends only on the conceptual relationships, not on the physical manifestation.


Again, weird way to say this. Bits are abstract; photons and neurons are physical. We say that a photon or neuron can encode 1 bit of information.

This is the crux of the matter. You are assuming they are innate, when you haven’t defined yourself different from any other machine. They are not “anything” until something external PROGRAMS the machine to know the difference.

The so-called “concepts” in a computer, like the letter A on your keyboard, is nothing more than a bank of switches. To say “A” is special is to only say it’s wired to a bank, that when you press A, you are literally flicking a load of switches at once to initilaize them as high or low.

There are ZERO 2D symbols or concepts “to a 1D bit or uit machine.” You use this term, but from the ontological hardware perspective, there are just banks of states, the MEANING thereof is only programmed by something that gave it rules outside itself. There is only 0 or 1, standing for two contrasting states at the hardware level!

 

What does “independent of your brain” mean?

The SELF concept is based on the assumption that there are external states that are distinct from (independent of) the internal states. This is fundamentally an illusion -- we are all part of the universe -- but it's not entirely inaccurate. The information associated with the wallpaper seems to be independent of my internal states. There might be some trillionth-order correlation that I'm unaware of, but I can confidently behave as if the wallpaper's states are not my own.

Does MathLab software know this relationship until it is arranged in hardware and gates to put the voltages through them and yield other voltages?

The computer hardware only provides the capability, not the ability, to know mathematical relationships. The ability comes from the software that runs on the hardware. The software organizes the hardware such that the associations are made.

 

Take ANY geometric symbol or shape. WHERE is it in the brain *as-described* directly? It is ONLY represented in 1D discrete states. If you do an MRI or other scan, there are ZERO actual 2D geometric symbols or forms.

This was my entire starting question from page 3 of our discussion. WHERE is the 3D cube *as-described*? WHERE is the 3D dog in the light *as-described*? These “concepts” are NO WHERE outside of digital 1D representation!!

 

The SELF concept is based on the assumption that there are external states that are distinct from (independent of) the internal states. This is fundamentally an illusion -- we are all part of the universe -- but it's not entirely inaccurate. The information associated with the wallpaper seems to be independent of my internal states. There might be some trillionth-order correlation that I'm unaware of, but I can confidently behave as if the wallpaper's states are not my own.


The computer hardware only provides the capability, not the ability, to know mathematical relationships. The ability comes from the software that runs on the hardware. The software organizes the hardware such that the associations are made.

And the software is nothing more than a gateway to hardware that uses 1D states to initialize other 1D states as high and low! You are using abstractions to describe banks of switches. The banks are the meat and potatoes. The “concept” is nothing more than an address of banks. There is nary a 2D or 3D anything anywhere.

 

Magnitudes only exist when numbers are suffixed by something.

What is the ratio of 10 meters to 5 meters? Is it 2 meters? Nope, it's just 2. That "2" is a magnitude without any suffix.

6 vs. 3 is not a comparison because numbers are independent of what they’re representing, such as size, and sequentiality until user-defined. 6 notches vs. 3 notches iswhen we biject the set or notches to the set of numbers. You can’t say the stick is “longer” otherwise without invoking some arbitrary non-numeric apparatus (feeling? Lol). The brain is a measurement device only.

You're saying that I can't compare 6 vs 3? In other words, I can't meaningfully say "3 < 6"?