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Journal snowgirl's Journal: Snowgirl's Take on the Analogy of the Divided Line 28

I started reading about The Analogy of the Divided Line whereby the reasoning of an individual is divided into four parts.

The first part is the blanket assumption that what is seen is reality.
The second part is the higher order of understanding the particulars of reality.
The third part is an understanding that the objects around us are merely reflections, and that there is something more meaningful that gives a object its reality.
And the last part is inexistential, the understanding that there are things that exist but have no image in reality.

As I was reading this, it occured to me that Arithmetic itself is a wonderful example of this sort of reasoning.

The first level is physically tied arithmetic. Two apples and two apples are four apples, and four rows of five apples makes 20 apples, this is known to be true as you can actually count them.

The second level is route memorization. One knows that the concept of math is more than simply counting numbers, and that many operations have simple systematic answers, that do not require you to count the entirety of the result. Thus,2+2=4 and 4*5=20 because that's what the results of the arithmetic operation are.

The third level is an understanding of the principles behind math. That there is an abstract level behind math, that guides all the principles. You might realize that scalar math is simply 1x1 matrix math. You know that 2+2=4 not because you have been told so, but that the definition of the elements hold that the operation must produce that, but that "+" means much more than simply addition, but can include a number of ideas, such as the idea of logical or, or the construction of a set which strictly supersets both sets used as operands (Union operation). You know that 4*5=20 not because you have been told so, but because you know and understand how multiplication works, and that it is an agumentation of addition, thus n*m = sum(1-n, m), and inserting the values four and five, you retreive 5+5+5+5, which is defined in the context of the vector field we are using is 20.

The final level, is the cause of many people being considered to be raving lunatics. It's the understanding that numbers themselves do not having meaning, but rather that everything is derived from nothing. 0 is the cardinality of the empty set, while 1 is the cardinality of the powerset of the empty set, while 2 is defined as the cardinaltiy of the powerset of one, and that every number N is the cardinality of the powerset of N-1. Maths working in this field are like lambda calculus is to computation, because the lambda operation is the most basic of all operations, and breaking down math into the most basic of all elements, the empty set, you end up able to prove not only that 4*5=20 because it equals sum(1-4, 5) = 5+5+5+5 = 20, but because you know that n+m = incr(1-n, m) (where inc is like the Sigma Summation operator and the Pi Product operator, showing a sequential series of incrementations). Thus you know:
4*5 = sum(1-4, 5
        = 5+5+5+5
        = incr(1-5, 5) + 5 + 5
        = incr(1-(incr(1-5, 5)), 5) + 5
        = incr(1-(incr(1-(incr(1-5, 5), 5), 5)

And knowing that a single incrementation step is defined as
inc(x) = { x=0 : emptyset
                      x!=0 : powerset(f(x-1)) }

Thus that "numbers" are inexistential sets of literally nothing (various combinations (not permutations) of empty sets) and that finally collapsing them to a definition results in us taking the cardinality of the set in order to understand the value. Thus continuing above:
4*5 = incr(1-(incr(1-(incr(1-5, 5), 5), 5)
        = incr(1-(incr(1-inc(inc(inc(inc(inc(5))))), 5), 5)
        = incr(1-(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(5))))))))))), 5)
        = inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(5)))))))))))))))

Inserting the definition of five:
        = inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(inc(emptyset))))))))))))))))))))

By applying the inc function we end up with:
        = powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(powerset(emptyset))))))))))))))))))))

Whose cardinality is 20.

Having defined and proven addition, and the very definition of numbers themselves, you have broken free of any physical manifestation of mathematics. You are truly looking at what mathematics really is a priori, you are not bound to "it is defined this way" or "it was taught to me to be this value" or "I counted it myself", but rather you understand that all of those are merely images and reflections on a cave wall with regard to what really really is math.

You no longer "perform" math based on axioms, instruction, or the crude rudimentary counting, but rather based off of a priori proofs. You know of the existance and nature of math, because you understand what it really is.

At this point, do you become truly enlightened about math, and you understand why such variations as there are exist, because you can extract all of them from nothing.

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Snowgirl's Take on the Analogy of the Divided Line

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  • I'm not sure if the capability to reason on the third and fourth levels in innate, or can be taught. Certainly, the number of people who can reason on the level that you illustrate (the fourth) is a fairly small subset of the general population. And that is fairly elementary for math, if more esoteric considered solely as logic. I found group theory hard at first, because I had not wrapped my mind around the concepts, so to speak. As with many things, there was a "click" moment when it all just fell int
    • Yes, I did mean inc() not f(), that was an earlier draft...

      The concept here is that what is required in order to being mathematics, is simply nothing. Nothing at all. Nothing must exist as a concept, because it's defined as the absense of anything. Thus, it must exist, at the very least as a concept.

      Sets are essentially collections of things, which is naturally known to exist. By having one thing and another thing you can conceive of a relationship which would contain both things. The necessity of stat
  • Sort of.

    You -CAN- define numbers in terms of set-theory. Infact you can define numbers in any number of ways, aslong at that still leads to them having the same properties, they're all the same numbers.

    Analyzing something complex by splitting it up in a larger count of simpler operations is useful. But you shouldn't confuse that with the complex thing really -BEING- the set of simpler operations.

    In binary logic you -can- express everything in terms of NAND-gates. But this does not apply that, for example th
    • How do you know what something "really" is? You allude to this when you talk about the XOR operation, saying that expressing that operation with a combination of NAND gates is just one of an infinite number of ways of expressing the nature of the operation. Which of these is "really" XOR? Looked at one way, the answer is "all of them", and another way the answer is "none". I would conclude from that that this notion, of what an abstract concept "really" is, is not a useful one.

      This speaks to exactly the
      • by Eivind ( 15695 )
        That's my point. None of them "really" are XOR. Any definition that leads to a function which output true if one of the inputs is true, and false otherwise is XOR. Arguing that one of them is "really" xor is invalid. The choice of definition is arbitrary, aslong as the function defined is the correct one.

        "really" xor doesn't exist, other than as the abstract concept.
        • "really" xor doesn't exist, other than as the abstract concept.

          The same can be said for everything in mathematics, as even the number 2, or the number 4, or any other number, or operation, are all merely symbols. They represent a symbollic relationship, which is itself derived merely from our own perceptions of things.

          In fact, these symbols can be used to refer to love, hate, identity, etcetera. These are all symbols which represent relationships between objects which we perceive or have been taught to pe
          • by Eivind ( 15695 )
            Yeah. Certainly. Maths doesn't concern itself with reality. It concerns itself only with definitions, and the consequences of those definitions.

            But math is -rooted- in reality. The concept 1, 2 and 3 -originally- came into being as an abstraction of something human beings do regularily, and have great utility from, counting objects/persons/whatevers.

            So, while "3" is only an abstract concept, "3 apples" is real.

            Well, assuming we don't go wandering off into philosophy lala-land and start arguing what it means
            • But math is -rooted- in reality.

              This is third level reasoning. That math came from reality.

              While it is true that we came to the truth of math by passing through a stage where it was presumed that "math" was a reflection of reality, it is in fact actually reality that is a reflection of math.
              • by Eivind ( 15695 )
                Actually, neither. As far as math is pure -- it says nothing about reality. And as far as it says something about reality, it is no longer pure.
                • How is pure math less pure for describing reality?

                  We are simply using a subset of math, (which is not pure math of its own, but rather simply an inextricable part of pure math), which is useful only in describing reality, leaving various other parts of math that do not have relation to reality.

                  You're letting your definition of math change, which is a Fallacy of Equivocation. (Which is not to say that your statement is false, merely by being a fallacy, as is the reason for my assertion above.)
                  • by Eivind ( 15695 )
                    Think about it. It's not as if this is my idea. Einstein said it famously, though many other thinkers had arrived at similar conclusions prior to him. "Sidelights of relativity" was published in 1983 and contains; As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Yeah ok, perhaps "certain" describes it better than my choice of word, "pure" ?
                    • Yeah ok, perhaps "certain" describes it better than my choice of word, "pure"

                      No, it doesn't just describe it better, but rather presents a whole different argument.

                      My argument was that there are two forms of math, those that relate to reality, and those that do not relate to reality. The proposition that those that relate to reality are not certain, and that those that do not relate to realty are certain is a completely different argument from "pure math" does not "say something about reality".

                      In fact, pur

    • But you shouldn't confuse that with the complex thing really -BEING- the set of simpler operations.

      In binary logic you -can- express everything in terms of NAND-gates. But this does not apply that, for example the operation XOR is "really" a combination of NAND-gates. That's merely one of an infinite number of ways of defining that function in terms of another function.

      I am not working in binary logic. I'm working from literally nothing. Ontologically "nothing" must exist. Also "sets" must exist, becau

      • by Eivind ( 15695 )
        You argue as if I don't *understand* what you say. I do. Just don't *agree* with all of it.

        For example, you claim that this proves that natural numbers exist. And I am curious, what, exactly, do you mean when you say "exist" ? What distinguishes things that exist from things that do not ?

        In which sense is a set of NAND-gates arranged to form a XOR-gate not equivalent to a XOR-gate ? Infact I'm arguing that this arrangement isn't merely /equivalent/ to a XOR-gate; it really *IS* a XOR-gate. Putting togethe
        • Putting together 4 wheels, an engine and various other parts doesn't make an object equivalent to a car. It makes an *actual* car.

          Ah... you're falling into a pit of definitions. Something with 4 wheels, an engine and various other parts, may be for example a four wheeler, which is certainly not a car. You're making assumptions as to the definition of car that those criteria that you specify are necessary and sufficient to describe a car. I have seen a few cars actually on the road with 3 wheels. Some ca

          • by Eivind ( 15695 )
            Nope. I'm not. I'm saying, something which fulfills the definition of "car" (whatever that definition is) is not "equivalent" to a car, it *IS* a car. Yes, I agree cars may or may not have 4 wheels, an engine and any number of other features. That was however not my point.

            If you -DO- set a definition for X, and then, thereafter, you find an object Y that infact fulfills the definition for X, then Y isn't an object "equivalent" to an X, it *IS* an X.

            Your other objection also is somewhat strange. Most objects
            • If you -DO- set a definition for X, and then, thereafter, you find an object Y that infact fulfills the definition for X, then Y isn't an object "equivalent" to an X, it *IS* an X.

              I will restate your statement here in the form of a proposition:

              • The definition of X is defined as having properties P
              • The object Y has properties Q
              • Given that Q subsets P
              • Y must be an X

              The failure here is in the counter example, which I provided. Philosophically something X is only something Y if and only if something Y is also so

              • by Eivind ( 15695 )
                I agree with your first proposition. If I *define* "car" as "any object having the following properties .....", then it follows that by my definition, any object which can be demonstrated to have all properties, is indeed, by my definition (not nessecarily everyone elses) a CAR.

                I already stated that the point wasn't to provide a real definition of car. Your so-called "counter-examples" merely demonstrate that the sketch definition that I provided is probably not a good one to use if I want my "cars" to corr
                • I'm actually perfectly happy to allow a penny as a fuse. Indeed, given your choosen definition it's inevitable. A 50A fuse in a 10A circuit will ALSO cause the circuit to fail before the fuse, that doesn't mean the 50A fuse is not a fuse. Sure it is. It's not one apropriate for that particular circuit, but that's a completely different matter.

                  Merely because there is a condition in which an object may function appropriate does not make it a fuse. A 50A fuse in a 10A circuit, is most definitely not being use

                  • by Eivind ( 15695 )
                    Now you're missing -my- point. My point is that words, as such, doesn't have any meaning whatsoever. They mean -only- what we agree to make them mean. "fuse" is just 4 letters, and even that statement is only true if we already agree on what a 'letter' is, or at the least our definitions for 'letter' overlap to the point that both you and me consider f,u,s and e to be examples of letters.

                    I'm not arguing what a fuse *really* is. Because it *really* isn't anything, or it is anything, we are free to choose def
    • You -CAN- define numbers in terms of set-theory. Infact you can define numbers in any number of ways, aslong at that still leads to them having the same properties, they're all the same numbers.

      I just recalled something else regarding this statement. No, I am not defining numbers such that they have the same properties as natural numbers. I'm defining numbers from ontologically true statements, and as such proving the existence of numbers. The mere fact that our understanding of numbers conforms to thi

      • by Eivind ( 15695 )
        Well, you are talking about -SOMETHING-. You're saying that given that "nothing" exist and "sets" that are capable of holding, or not holding, any given thing exist, then you can make an infinite number of distinct such sets by following a simple recursive procedure. This is true. Given your assumptions, the conclusion holds.

        But why do you choose to refer to these sets as "natural numbers" ? You could just aswell refer to them as "pink giraffes" and claim that this proves that pink giraffes exist.

        The answer
        • But why do you choose to refer to these sets as "natural numbers" ? You could just aswell refer to them as "pink giraffes" and claim that this proves that pink giraffes exist.

          You're still dealing with the lower levels, three or two specifically. An in fact, what you're doing here is attempting to make a third-level attack against my fourth-level statement. You are saying that regardless of what actually exists, we can name the shadow that it casts anything. What I am looking at is the actual extant thing

          • by Eivind ( 15695 )
            You seem to have missed one essential question in this debate:

            You keep repeating that this line of reasoning prove that natural numbers exist.

            What, exactly, do you mean when you say they "exist" ? What separates things that "exist" from things that don't "exist" ?
          • by Eivind ( 15695 )
            As to your other question, yes there are other similar sequences that can be claimed to be *nessecarily* true. And it can be claimed that your argument isn't nessecarily true. Any number of ways. Does "nothing" exist ? It depends on the definitions of "exist". Does an empty set exist ? It depends on definitions. Is a set that contains "nothing and a set of nothing" distinct from a set that contains nothing ? Depending on your definitions, those sets may be the same (they are both the complementary set to "e
            • Critic #1: Does "nothing" exist?

              I answered this already, ontologically nothing must exist. If we presume that "nothing" does not exist, then given anything that exists:

              1st)it cannot be nothing itself, as we have already presumed that "nothing" doesn't exist, and as we are already given something that exists, this would create a contradiction.
              2nd) it cannot be the case that the object is indivisible, as one could simply remove the indivisible object and be left with "nothing".
              3rd) since the object must be d
              • by Eivind ( 15695 )
                Yes you answered that, but there's still the elephant in the living-room:

                What do you -MEAN- when you say that something (or nothing) "exists", what distinguishes things that "exist" from things that don't ?

                You give some examples in this post. But you still don't come close to explaining what it means, for something to "exist".

                So, in the absence of an actual definition, we'll try to narrow the concept in your mind down by example.

                Could you give me a few examples of different types of objects that do not, in
                • The assertion of "exists" as a wider concept means that it either exists as a concept, or it exists in actuality. Now, it must be known that while something may "exist" conceptually, it does not necessarily exist in reality, and we are actually able to name things that cannot possibly exist, nor can be conceived as existing, yet we can name.

                  For example, a sphere of size > 0 in a two universe. We can create a set of constraints that exist as a defintion, yet cannot have any conceptual or realistic exist

Faith may be defined briefly as an illogical belief in the occurence of the improbable. - H. L. Mencken