Precisely the point, though — you are rejecting it because it "feels" like a mess. If it didn't "feel" like a mess, you'd have a different take toward it potentially. How much is "feeling" some base line element of "this works" is part of the picture?

LOL! This is the first time that I've ever seen a purported "proof" of an argument based on its rejection. Props for creativity, lol.

 

You may not see them as numbers, but other humans do. And these other humans use computers to add them together. Or are you suggesting that a computer can't compute 2/5 + 3/4?

You mean like the other humans that use the term "geometry" improperly? (ducks)

A computer is NOT seeing the fraction 2/5 + 3/4 in the same way a human does, for the love of all that is holy. A computer has not one scintilla of one iota of one modicum of a clue about what "numbers" it is computing. Someone once said "THERE ARE NO NUMBERS IN A COMPUTER." ;--)

So what are there?

Logic states that are representing numbers. But in my estimation, heaven forbid there is a connection between these things at some deep level(!?) because it makes too much sense experientially from the age of 3 (but out of respect, we shall not show partiality(!) here to spatial physical-based truths (judgments) until somehow it can be formally proven).

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

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

The machine, the computer, is doing a full numeric computation on the logic-state binary incarnation of these numeric relationships. It must break these relationships down into simpler, more elemental states, per the human's binary instructions.

There's no sense in which the number 5 in N is different from the number 5 in Q. Indeed, N is a subset of Q, which means that every one of your beloved counting numbers has a p/q expression.

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

Uh, no. The reason an ALU will add natural numbers is because we designed them specifically to perform the "+" of boolean rings. Computers can also add two complex numbers, but they just need a different circuit. BECAUSE THE "+" IS DIFFERENT.

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

A complex "number" which is not a number at all, but a theoretical algorithmic element to deal with the problem of trying to create a square root of a negative number (which, by the way, to answer your question — no, I don't believe in negative numbers, although I will use them colloquially). I believe in the subtraction process, which is shown in a computer to be a form of addition using 2's complement algorithm. 5 is a number. -5 says "subtract the number 5 wherever it is found." "5 + -5" is 0. "5 - -5" is 10. The negative sign is yet another operator assigned to the pure number of 5.

 

LOL! This is the first time that I've ever seen a purported "proof" of an argument based on its rejection. Props for creativity, lol.

"But he didn't answer the question, you know"

 

What the computer does is, as you've said, up to its programmer. What "value" is computed is solely a function of human attribution to it, and human's direct intention to place values through gates to arrive at other values.

You're missing the point, as this is in reference to proofs. Computer proving programs aren't following a set of instructions to show that a proof is true. They prove statements by deriving them from given axioms. This is entirely mechanical. The human's only role is to give the computer the rules of the game, as it were. The computer simply follows the rules, and if the statement can be reduced to a set of axioms, then the statement is necessarily proven. There is no "feeling" involved.

"Information is a measurable quantity" someone once said. No quantity, no information.

No, wtf?! Please tell me you're joking, that you don't actually believe that "No quantity, no information" logically follows from "Information is a measurable quantity". Please.

A fourier transform is a multiplicity of values.

Well, in so far that you can characterize a function F(x) as an infinite set of values, I suppose. But in no way could you characterize it as a discrete value. A Fourier transform is not a quantity.

When you make a fourier transform in a computer, the computer has broken down the mathematically describable wave values into constituent-part wave values that are ALL represented as stored values.

Nope. A computer cannot perform a Fourier transform. The closest a computer can do is a DFT.

A protein folding sequence is certainly, again, multiplicity of values.

You are stretching the meaning of "value" beyond recognition.

Analog and digital computers are doing the same thing — yielding information.

They process information. They are information processors, as I've described them all along. Now, what does this have to do with "values"?

A computer doesn't "KNOW" it's doing any computations.

Says you. Somehow the computer gives me the right answer whether I ask for "2+2" or I ask for "John's telephone number". If it doesn't know the difference, then how can it do that?

Voltages are sent through gates to yield voltages. It's simply doing precisely what it's designed to do. There is no experience.

Your brain works in a similar way. The molecules in your brain were designed by the genes in your DNA.

The child sees the block and has a sense of "feeling of goodness attributed to the presence and potentiality of interacting with it." Playing with the block is an extension of that desire upon the object.

That is quite the speculation.

 

You're missing the point, as this is in reference to proofs. Computer proving programs aren't following a set of instructions to show that a proof is true. They prove statements by deriving them from given axioms. This is entirely mechanical. The human's only role is to give the computer the rules of the game, as it were. The computer simply follows the rules, and if the statement can be reduced to a set of axioms, then the statement is necessarily proven. There is no "feeling" involved.

...aaaannnd there is no "knowledge" anything was proven, though. Something that follows rules does not indicate knowledge of said function. By this definition, if I were to hook up a light switch to a motor and it turned on and off, does the light switch "know" it's turning on and off? No? Then what is the justification of saying any n-size group of them are? Because some other switches are reflecting their states? Ummm...

No, wtf?! Please tell me you're joking, that you don't actually believe that "No quantity, no information" logically follows from "Information is a measurable quantity". Please.

Haha... No. I'm saying that if "information is a measurable quantity," and "quantity is itself a form of information," then if there's no quantity relayed, no information is relayed about the computation, and there's no point to it!

Well, in so far that you can characterize a function F(x) as an infinite set of values, I suppose. But in no way could you characterize it as a discrete value. A Fourier transform is not a quantity.

Of course not. It's a load of values. Still values. A fourier transform is a quantity of quantities.

You are stretching the meaning of "value" beyond recognition.

If there is information there that is cognizable, it is value or values.

They process information. They are information processors, as I've described them all along. Now, what does this have to do with "values"?

Information = value?

Says you. Somehow the computer gives me the right answer whether I ask for "2+2" or I ask for "John's telephone number". If it doesn't know the difference, then how can it do that?

What is your specific definition system about "things in space" utilizing your own discrete states? You don't know what an object is in space by your own admission, so (important question!) why are you treating it as "one thing" as a discrete-state processor, if all you have to work with are non-dimensional bits from light to conceive of "1" something there?

That is quite the speculation.

I do not think so whatsoever. "Playing" is a natural desire of a child that carries well into adolescence. You can see a baby begin to incline and even cry to interact with things, and we give it toys to act on that impulse. It starts to map attributes to that toy, and we further that cognition along.

 

Let us say you were tasked to demonstrate that on some level there WAS a deep unified connection between addition of numbers 1 and 0 and the logic states 1 and 0. How would you go about doing this using a proof-based approach (which I am coming up to speed with as we speak)?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

C’mon!! You totally know I’m talking about comparing base 2 {0,1} to logic states {0, 1}! OMG.

You know I’m talking about base 2 (I have stressed that over and over, and you certainly didn’t get or ignored the gist of my 1472 post)! You are too funny! Of course it doesn’t work as base 2+n! Duh!! Lol. You have not gotten my point since the beginning. :--( I’m not sure to laugh or cry.

Now, can we please see it as a computer/machine does and try it again!! This is very important!!!!

 

I want to prove formally what a computer is doing: That there is foundational equivalency to binary logic states represented as 0 and 1 and integers 0 and 1 and their overlap in computational phenomena (and later, that all bases and number sets are constructed ultimately of this phenomenon). I would do it if I had experience writing proofs. I do not. I demonstrated addition equivalency in 1472 if you read it!! I know there is equivalency to the tune of $n million cash if I had it.

 

No, wtf?! Please tell me you're joking, that you don't actually believe that "No quantity, no information" logically follows from "Information is a measurable quantity". Please.

Jennifer makes this fallacious argument frequently. Along with others.

You use this argument all the time...

It is a fallacious argument, of affirming the consequent. That is the antecedent in an indicative conditional is claimed to be true because the consequent is true;

if A, then B; B, therefore A.

 

Jennifer makes this fallacious argument frequently. Along with others.

No, I do not... I clarified what I meant.

You should do yourself a favor and read more of the whole thread, because you accuse me out of ignorance more than once and piggyback on bogosort because you have almost no arguments of your own.

 

No, I do not... I clarified what I meant.

You should do yourself a favor and read more of the whole thread, because you accuse me out of ignorance more than once and piggyback on bogosort because you have no arguments.

Also, why don't you attempt to fully understand what it is I'm trying to prove, when you yourself said "I may be on the fringe of understanding" which is a parallelism of a most respected logician in history (L. Kronecker), whose quote concerning the integers is the very title of a Stephen Hawking book ("God Created the Integers") about the greatest contributions to logic and math... and that this is potentially "new ground" here, instead of "brushing me off" as "that nutbag chick on the Internet." If you don't fully grasp the core of what we're discussing, exactly what right do you have to accuse me of anything, much less in making judgments on probably less than 50% of the thread ("analog vs. digital processing?" I've said it several times I believe analog processing is fundamental). So feel free to stop randomly "thumb-upping" bogosort and responding out of ignorance when you jump into an established discussion and don't even give a baseline of respect of knowing the totality of what is being discussed and pursued?

Thank you.

 

No, I do not... I clarified what I meant.

You should do yourself a favor and read more of the whole thread, because you accuse me out of ignorance more than once and piggyback on bogosort because you have almost no arguments of your own.

I’ve read the entire thread up to this point. And I don’t think I’ve accused you of ignorance. And as to your accusation that I have no arguments of my own, I have noticed that you don’t respond to my arguments. Is it because that you don’t like the points that I make?

 

I’ve read the entire thread up to this point. And I don’t think I’ve accused you of ignorance. And as to your accusation that I have no arguments of my own, I have noticed that you don’t respond to my arguments. Is it because that you don’t like the points that I make?

I didn't say that. I said you accuse me OUT of ignorance. Again, proof you don’t read. I have responded to a few of your posts, including the one about Von Neumann's automata. You never responded back. Your entire argument about "base 10" being used "just the same" shows you are NOT understanding my point that numbers are COMPOSED of underlying logic state phenomena and higher bases are derivative abstractions. This is proved in hardware and software, and I demonstrated this clear as day in post 1472 when I broke base 10 numbers up into base-2 and showed that you can treat the T/F logic states as 0 and 1 integers.

Clearly you have not understood the totality of what we've discussed here, then... and again, above, you wrote without consideration of the fact that I clarified what I meant about information vs. quantity, and you didn't read that clarification.

It's clear you're looking to accuse me of making fallacious arguments and piggybacking on Javier/bogosort where possible. We don't need this bullsh*t here.

 

As I said, I don’t like doing this, but I need to address the nature of intelligence, because mythology abounds on what it means to know how to think clearly and incisively. I conceived, researched, wrote, skype-directed, and scored (played most instruments) the following 3D video ad for the Internet firm I worked for:

Having no formal training on how to do so...using desktop technology that is generally only seen in Hollywood movies, but I sleuthed out the apps and decided to push it to the max for a novel concept of doing marketing videos without hiring a camera crew. That is a to-scale 3D animated model of the building and property that I co-built over Skype video-conferencing with a guy overseas. It took hundreds of hours and months. It would be a $70K video I did for $4000. I hired another offshore motion graphics guy to overlay graphics and call-outs. It is a pioneering video concept.

My point is not at all to brag, but I took not one course in any of the skills required, and that video “could be used in university courses.“ Raw reasoning power and observation employed, shows that there is such thing as observational objective order, as not one person who has eyes would say that is not “proof of ability”, and I don’t care who did it... there is such thing as old-fashioned, Tesla-style empirical observation and invention as the basis of understanding feeling and truth as elements of existence. I do not believe it is relegated to information only.

 

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

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

 

I realize why you don’t like the word geometry... because to you geometries are derivative of vector-space processes, and not stand-alone phenomena indpendent of computation, am I correct?

 

C’mon!! You totally know I’m talking about comparing base 2 {0,1} to logic states {0, 1}! OMG.

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

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

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

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

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

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

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

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

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

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

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

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

 

Because expressions involving operators are not numbers. They are combinatorial numeric expressions. Fractions are NOT "numbers", they are combinatorial expressions of two or more integers using a manmade fraction-bar or decimal point to denote "divisional algorithm". We call them "numbers" for short because we "think" of them that way, but this isn't correct.

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

 

I conceived, researched, wrote, skype-directed, and scored (played most instruments) the following 3D video ad for the Internet firm I worked for:

I dig the piano solo in the middle, nice. ;--)

 

I dig the piano solo in the middle, nice. ;--)

Thanks