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# Snowgirl's Take on the Analogy of the Divided Line

#### snowgirl (978879) writes | more than 6 years ago

28

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

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.

### Interesting Stuff (1)

#### johndiii (229824) | more than 6 years ago | (#20987577)

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 into place. Math is fun. :-)

That understanding of the true nature of math is what makes it fun, I think. At least to me.

The divided line abstraction seems a useful one.

In the definition of the inc(x) function, I think that you meant the second line to be "x!=0 : powerset(inc(x-1)) }".

I'm not sure that this is actually a definition of mathematics out of nothing, though. It starts out with the concept of a "thing", and then goes to a set as a collection of things. Those ideas are abstractions based on reality, which would be a third-level notion. I'd accept the idea that an algebraic group, for example, is not a reflection of anything in reality. Of course, even something as simple as a function is that. So a fourth-level concept can have its origins in third-level reasoning.

Hmmm. This bears more thought. :-) Thanks!

### Re:Interesting Stuff (1)

#### snowgirl (978879) | more than 6 years ago | (#21048813)

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 stating that it's a combination and not a permutation, is to indicate that order is independent, and thus the concept of a set, is simply a collection of things.

### sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#20999503)

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 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.

### Odd notion (1)

#### johndiii (229824) | more than 6 years ago | (#21000069)

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 point that the abstraction of the divided line makes, in that there exist some things that do not have a reflection in the real world. I think that the point that snowgirl was making is that much the usefulness of reasoning with abstract concepts (fourth level reasoning) stems from this very malleability of expression of those concepts. By defining mathematics from some minimal set theory, the point is that math has a reality apart from its use in the real world, and that the key to understanding it is assimilation of that notion.

### Re:Odd notion (1)

#### Eivind (15695) | more than 6 years ago | (#21006629)

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.

### Re:Odd notion (1)

#### DaedalusHKX (660194) | more than 6 years ago | (#21007335)

"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 perceive. Much like currency, or even life itself, they do not truly have value until we are taught to value them in a certain manner. (This can obviously be used to relate to suicide bombers and the ease with which they throw away what we, in the west are taught is priceless.)

By the same, almost coincidental (on the surface) relationship, one can even argue the old "I think, therefore I am" idea using such logic. Frankly, I'm glad to see this being discussed on slashdot... even if it isn't on the main page. (Thankfully so because the trolls are kept away.)

### Re:Odd notion (1)

#### Eivind (15695) | more than 6 years ago | (#21009205)

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, exactly, for something to be "real".

### Re:Odd notion (1)

#### snowgirl (978879) | more than 6 years ago | (#21048873)

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.

### Re:Odd notion (1)

#### Eivind (15695) | more than 6 years ago | (#21070025)

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.

### Re:Odd notion (1)

#### snowgirl (978879) | more than 6 years ago | (#21074279)

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.)

### Re:Odd notion (1)

#### Eivind (15695) | more than 6 years ago | (#21082163)

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" ?

### Re:Odd notion (1)

#### snowgirl (978879) | more than 6 years ago | (#21090413)

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, pure math does say something about reality. If I push a ~1 kg weight with ~1 Newton for ~1 meter, then I know that I have exerted ~1 Joule. The uncertainty is certainly there, but the "pure" math saying that 1 kg*N/m is 1 J... well, it's useful insofar as I don't need to get indefinitely accurate.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21049053)

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, because as soon as there are two things, there exist inclusions and exclusions, and pairings, and groupings.

nothing
a set of nothing // since a set can include a single object
a set of ( (a set of nothing) and (nothing) )
a set of ( (a set of ( (a set of nothing) and (nothing) )) and (a set of nothing) and (nothing))

Thus, I have proven the existence of the natural numbers, upon which all other math derives its use.

I think the problem that you're having with this, is that you are approaching this from a third-level point of view.

Later you say "unless you want to get into a philosophical argument..."

This is a philosophical argument. Simply represented by nothing and groups which results in numbers, which results in math.

While in reality, a set of NAND gates arranged to make an XOR gate is not equivalent (and as such nothing but an XOR gate would be an XOR gate), in theory and philosophy they both operate in the same way, and thus are equivalent (and as such all logic equal to an XOR gate in operation are XOR gates, as an XOR gate is defined as simply anything that produces the results of an XOR gate.)

Once you have your theoretical definitions, you then turn to cold hard reality in order to project that shadow into reality, and what method you choose to represent that theoretical XOR gate is based on many different decisions, but can only be solidified into one aspect of the meaning of an XOR gate, because it is simply a shadow cast upon the wall of "reality" cast by the truthful existence of the idea of an XOR gate.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21070055)

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 together 4 wheels, an engine and various other parts doesn't make an object equivalent to a car. It makes an *actual* car.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21074111)

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 cars on blocks in a few yards may not even have engines.

The reason why a set of NAND-gates arranged to produce the same values from inputs as an XOR gate is not an XOR gate, is because ontologically, an XOR gate is a single item, while the NAND-gates are separate items.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21082145)

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 are at the same time one object, *AND* the set of many simpler objects. Claiming that the correctly assembled NAND-gates aren't really a XOR-gate because they're a collection of NAND-gates is a bit like claiming that the correctly assembled set of cells that make you out aren't really a human, because ontologically a human is a single item while the cells that together form your body are separate items.

Indeed, the natural numbers you talk of (and "prove") in the original post are *ALSO* constructed from a collection of simpler items. So if being constructed from smaller objects disqualifies, well, then you just disproved yourself in a nice twist of logic.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21089627)

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 something X.

Filling in variables for your instantiation of the above reasoned argument:
• The definition of "car" is defined as having "four wheels", "an engine", "various other parts"
• The given any object that has "four wheels", "an engine", and "various other parts"
• That object must be a car.

Provided in my example were the cases: a) a 4-wheeler, which has four wheels, an engine, and various other parts, yet is not a car. Thus providing a direct counter example to your proposition. b) an object known to be a car, but which has only 3-wheels.

Your definition of "is" is not the same as the philosphical "is".

No less, there can be no doubt that something is equivalent to itself (the proposition: X = X' iff X' = X), at least this statement is necessary for the entirety of equivalence to exist.

My assertion is that a set of NAND gates arranged to function as an XOR gate is equivalent to an XOR gate. Your criticism of this assertion is that a set of NAND gates arranged as an XOR gate *are* and XOR gate, which would necessarily imply the necessity of equivalence between the two. My statement is a weaker assertion of your own argument, because I say, "we cannot know for sure that a set of NAND gates arranged to function as an XOR gate is an XOR gate, we simply know that it functions equivalently."

a bit like claiming that the correctly assembled set of cells that make you out aren't really a human, because ontologically a human is a single item while the cells that together form your body are separate items.

No one would claim that a human is ontologically a single item. That's absurd. We have arms, we have heads, we have legs, we have organs, which are made of tissue, which are made of cells, which are made of proteins, which are made of molecules, which are made of atoms, which are made of electrons, neutrons, and protons, which are made of quarks. This is empirically known as true, so the ontological statement that "a human is a single item" is improper. Rather, it is a singular set of a number of smaller systems.

I would relate this NAND gates to XOR gate issue rather as this:
• a fuse connects a circuit, and will fail at a specific electric load
• a penny is electrically conductive, thus will connect a circuit
• a penny exposed to sufficient electric load will melt and fail to conduct two points
• a penny therefore a penny must be a fuse

The problem here is the limited definition of "fuse", (and in our case "XOR gate"). While it is true that a penny will act as a fuse, the amperage required to melt a penny is significantly higher than the copper wires ability to carry the current. The missing defintion is that a fuse is designed to fail at a load strictly less than the load at which the rest of the circuit would fail.

At this point, we see that the previous definition of "fuse" is bad. Similarly, I could construct a NAND gate flow that would impede the flow of the electrical current indefinitely:

inner_box(a,b) := ((a nand b) nand a) nand ((a nand b) nand b)
xor(a,b) := inner_box(a, b)
| nand(0, nand(0, xor(a,b)))
While yes, the first box is indeed equivalent to an XOR gate (not to say anything about equality of existence) then unfortunately, the set of NAND gates that this can describe would include an indefinitely large series of paired NAND gates negating the output, which would result in something that is impractical to use as an XOR gate. In order for a set of something X to be equal to something else scalar y, I would present the proposition:

There does not exist any condition C, implying there exists an element x in set X where x is not a y.

Given that I have already shown that there exists a specific instance of a set of NAND gates that is functionally equivalent to an XOR gate, yet it's impractical (intractable?) nature prevents it from use in Electronics:

Therefor since there exists such a condition, whereby an element of x (the indefinitely double negated box presented above) in set X (sets of NAND gates that are functionally equivalent to an XOR gate) where x (the indefinitely double negated box presented above) is not a y (an XOR gate).

Either the proposition is false, or the set of X is not necessarily each equivalent to y.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21110369)

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 correspond to what average people mean when they say "car". I already knew that. But that's merely a weakness in that particular definition.

our distinction to philosopphical stumbles because you confuse identifiers with grouping. X == X, true. But X may (or may not) be A Y, that is, X may or may not belong to the set Y.

"Cars" is a set. All objects that fulfill whatever definition of car you choose to use is a member of the set "Cars", in everyday speech we then say: "That is a car".

You say that we can't know that the set of NAND-gates is a xor-gate, but merely that the set of nand-gates *function* like a xor-gate. I guess the missing step that leads to this misunderstanding is that a XOR-gate is most commonly (and by me) defined ONLY by function. That is, "any object that *functions* as a XOR-gate *IS* a xor-gate", a two-input xor-gate is any object with two inputs and one output where input and output can have precisely 2 distinct states, and the output is in one state if-and-only-if the two inputs are in different states. That's it.

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.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21133727)

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 used as a fuse. It's being used as a fuse bypass. (The same as the penny.) While anything that could complete some arbitrary circuit with the intention that it would fail before any other part of the circuit can be called a "potential fuse", only those objects that are actually placed in such a circuit that it suits the appropriate and defined use of a fuse are actually fuses. All others are fuse-bypasses.

You'll notice that I specifically used a penny in this example, because the penny is an item which would not normally be considered a fuse, and would only be considered a fuse if it were in a circuit where it would be the first object of the circuit to fail. If it's placed in place of a fuse where it would not be the first to fail, it is not a fuse. It does not fufill the necessary functional condition of a fuse. Thus, it is not a fuse, but a fuse-proxy, or a fuse-bypass. It is something that completes the circuit in place of a fuse, which will not function as a fuse in that particular circuit. The definition of an objects "utility" (what it's designed to do) and an objection "utilization" (what it's actually being used for.)

You say that we can't know that the set of NAND-gates is a xor-gate, but merely that the set of nand-gates *function* like a xor-gate. I guess the missing step that leads to this misunderstanding is that a XOR-gate is most commonly (and by me) defined ONLY by function. That is, "any object that *functions* as a XOR-gate *IS* a xor-gate", a two-input xor-gate is any object with two inputs and one output where input and output can have precisely 2 distinct states, and the output is in one state if-and-only-if the two inputs are in different states. That's it.

You have a theoretical definition of an XOR-gate. Functionally an XOR-gate must operate in a circuit within a time indicated by the tolerances of that circuit. I can grab a CMOS XOR gate from Radio Shack, and use it as a decorative keychain. While that particular item were to have the potential to function as an XOR-gate, it is not being used as an XOR-gate and the mere potential that such a gate could be used as an XOR-gate does not make it an XOR-gate... it's in a superstate before hand, where it is unknown if the XOR-gate will even function properly.

I'm sure that you would agree that a broken XOR-gate would fail your functional definition, but still be in the total subset of "XOR-gate", because it's designed to be an XOR-gate. However, since it does not actually perform the functionality required for an XOR-gate to be a functional XOR-gate, it is a "broken" XOR-gate.

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.

The issue that I have with the definition, is that the definition you presented is not equivalent to my definition of a car. Thus, we end up with a situation where we are equivocating one set into two sets, which will lead to issues down the line.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21154671)

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 definitions at will to suit our goal. Now, most of the time it makes sense to strive for shared definitions, because otherwise you get pointless debates where both sides are missing the point.

You see this often in philosophy. I see it most commonly in rec.arts.sf.science. Two or more people argue for a long time over something or other, say if it's reasonable to expect that future computers will be intelligent or not. What typically happens is, both sides *assume* that the word "intelligent" is somehow agreed on, and that the other holds a definition similar to their own. Then they argue why, in their opinion, a computer fulfills, or doesn't fulfill their definition. They do however, neglect to mention what that definition actually is.

After a while, it turns out both people really agreed from the start about likely properties of computers. The thing they wheren't agreeing on was, what exactly does it mean for an object to be "intelligent". That question is a quite different one from "is X intelligent.

You see the same thing here: Is a penny a fuse ? Well, that depends on what you mean when you say "fuse", doesn't it ? You seem to be saying that you consider something a fuse only if it can a) complete a electrical circuit and b) will break the circuit at a lower current than the one needed to destroy other components in the circuit. Under that definition a penny ain't a fuse, if inserted in a low-current circuit. (but it'd be one if you inserted it in a high-current circuit)

Assuming that is indeed something close to your definition of "fuse", it's fine. Though it -does- have the weakness that one cannot in general say if a object is a fuse or not from the object itself, it's a feature of the usage, not a feature of the item. Even a conventional fuse, is, or isn't a fuse depending on what you use it for. Aslong as it's lying in your pocket it isn't a fuse. (It could -become- a fuse, by being inserted in the apropriate circuit though)

Put more simply, yes it'll lead to issues down the line. And yes, it's very common to start discussing "Is X Y?" without making sure that everyone involved agrees on the definition of "Y". There's a reason many scientific papers start with a section formally defining the words used, it is to try to avoid issues like these. Not that it works a 100%, but it's atleast worth being aware of.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21049121)

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 this proof is merely because we've spent a very long time looking at the nature of numbers in the third level.

Still, you are seeking to explain my proof as a descriptive definition of numbers, where as I am not defining numbers at all, but rather proving their existence, and thus making an assertion of existence, rather than a declarative description of a thing.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21070067)

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 is, I think, that you refer to them as natural numbers because they have the same properties as the things we invented millenia before we ever got across set-theory and used for counting, has. This too, is true.

My point is, there are *other* ways of describing natural numbers in terms of objects that are simpler than they are, for some choosen definition of "simple" anyway, and there's no real reason that set-theory is *THE* *SINGLE* correct one. I would argue that any such description that gives numbers with the same properties we know and love is equivalent, and none of them are fundamentally more correct than the others.

Put differently, I would propose that natural numbers are any collection of whatevers that follow the rules for natural numbers. As such, the set-theory description is one possible description. There are others, more or less fanciful ones.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21074067)

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 which we name "the natural numbers", meanwhile, you're still stuck looking at the shadow that it makes.

My point is, there are *other* ways of describing natural numbers in terms of objects that are simpler than they are, for some choosen definition of "simple" anyway, and there's no real reason that set-theory is *THE* *SINGLE* correct one. I would argue that any such description that gives numbers with the same properties we know and love is equivalent, and none of them are fundamentally more correct than the others.

To add emphasis where you're missing the point. "there are other ways of describing natural numbers". The point is that you're stuck thinking that I'm describing the natural numbers. Wrong, I am proving the natural numbers. Do you know of another proof of the natural numbers?

The reason why this proof works, is that both the "empty set" and "sets" and "power sets" are simply abstract ideas, and as they are abstract ideas they need not exist substantially in order to provide this proof. However, ontologically, we can prove that each one exists, because by the very nature of their definitions they must exist.

"nothing" must exist. Because even if the universe didn't contain a single thing, then it would still contain "nothing". "Nothing" being the absence of anything (and more specifically the empty set, as a set containing no items) must exist, because if you have something that is not nothing, then you can always remove something from that something, until at last there is no part of that something to remove. At which point, you have nothing.

"sets" must exist. Because if the universe exists, (Which it must, another ontological argument left for a later journal) then it must be composed of something. If that something is not "nothing", then we have what is defined as a "set" that the universe encompasses. Namely, meaning that the universe must be the set of all things that exist, and thus "sets" must exist.

The notion of a "power set" is easy to prove from the existence of "set", "nothing" and "universe". If we only have those three things to work with, then the universe cannot contain anything but "nothing". But if the universe contains nothing, then we could conceive of a greater universe that contains that previous universe, and nothing, and then we could conceive of a greater universe that contains all previous universes, and nothing.

Do you see that this proof not description of the natural numbers works? It's ok if you don't, they did throw the first person to think this up in a mental ward. Heck, they thought he was crazy. "Defining the numbers from nothing? This guy is nuts."

But I assure you, standing on the fourth level of knowledge about math where I am at, I am certain of the existence of the natural numbers by method of an dialectic proof, rather than a simple assumption that they must exist.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21082051)

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" ?

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21082113)

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 "everything", for example)
Can sets be nested, and does that make them distinct ? By most common definitions of "set", the answer is yes, but that's because they're defined that way, not because of any fundamental truth.

The "greater universe" you perceive of, for example, the one that contains "our universe and nothing", isn't that just plain our universe than ? In what sense is it greater ? Do there exist a "greater Norway" that contains "Norway and nothing", and is this distinct from the normal "Norway" ?

Again, please understand: I am both familiar with and understand your argument. As is everyone whos taken math-101. I find it vaguely annoying that you persist in claiming my arguments are simply a reflection of me not understanding the great truth. It's possible to -understand- and still -disagree-. Assuming that anyone who -disagrees- with you must not -understand- is sligthly silly, and rejects the idea that it's possible, in a debate, *all* participants could learn something from oneanother.

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21090327)

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 divisible, it cannot be the case that once divided into parts, removal of one part results in an indivisble object.

This line of reasoning as a consequence of the supposition that "nothing" doesn't exist, requires that everything that does exist be infinitely divisble. I would refute this assertion as simply absurd. Using "cogito ergo sum", I know that I must in fact exist even if I am the only thing that does exist, (I cannot prove this to you, however you can prove to yourself that you must exist in the same way) my existence is unrefutable, as I have empirical prove beyond all requirements of proof that I exist. (This proof cannot, by the veil of perception, be demonstrated to anyone else.)

Even if you refute my physical existence (solipsism), you would still have to believe that I am at least a representation within your mind, and thus while I may not exist in physicality, the fact that we are arguing, indicates to you that I must actually exist in some way, even if only by your perceptions.

It depends on the definitions of "exist".

Ontologically, I can't really define existence. I presume on the basis of my own known existence, that existence must exist, and there for accept that it is possible that something else must exist. There are many realms of existence however. There is actual existence, physical existence, mental existence, etc.

While physical existence can be demonstrated under the context of physicallity by its non-arbitrary presentation upon your senses and other's senses.

Mental existence can be demonstrated by my own internal mental interaction either pseudo-physically (hallucination), or simply conceptually.

Actual existence is naturaly significantly harder to define. However, it is safe to presume that "existence" encompasses all means of existence.

Thus regardless of if "nothing" actually exists, or simply conceptually exists, it still "exists", being that "existence" is a superset of all possible existences.

Does an empty set exist?

An empty set is a set which contains no elements. If sets exist, then an empty set must exist. (For the same reasons given above for the existence of nothing.)

Is a set that contains "nothing and a set of nothing" distinct from a set that contains nothing?

This is necessarily so. Recall that X is X' iff X' is X. Therefore a set containing an element which is a set which contains no elements itself, is necessarily distinct from a set that contains no elements itself.

This is because X ({}) subsets X' ({ {} }), but X' ({ {} }) does not subset X. ({})

This is positively known because by defintion the empty set contains no elements, thus nothing may subset it, except itself, and the set containing the empty set contains one element, which while equivalent to the empty set, it still does not subset the empty set.

Depending on your definitions, those sets may be the same (they are both the complementary set to "everything", for example)

The set containing the empty set cannot be the complementary set to "everything". In order to be complementary no element of either set can be in the other set. While in the case of the empty set, there are no elements, and any element in "everything" is thus not in the empty set, they are complementary. However "everything" also includes "nothing", which is contained in the set containing the empty set. Thus the two cannot be complementary.

Can sets be nested, and does that make them distinct? By most common definitions of "set", the answer is yes, but that's because they're defined that way, not because of any fundamental truth.

Not particularly accurate, ontologically, a set must be able to hold subsets. If a set is an object that is a collection of objects, then this allows for the recursive nature of sets.

Plainly, if a set cannot contain a subset then that set cannot be divisble (otherwise, one could build a set between what was divided on one side vs. the other)

The only set fitting this definition is the empty set, as it cannot be divided into two sets. There are no elements that can be taken from the empty set in order to make it distinct from itself. However, the set containing the empty set can. One can remove the empty set, and be left with the empty set. Being that a condition can be placed upon the set containing the empty set that the empty set is different than the empty set, then they must be distinct.

### Re:sort of.. (1)

#### Eivind (15695) | more than 6 years ago | (#21110315)

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 your opinion, exist ? To me it sounds rather as if in your concepts, everything exists, because as you say, you may not exist, but atleast I must acknowledge that my mental model of you exists.

But mental concepts of flying pigs, pink elephants and green jabberwockys living on Mars also exist, so this doesn't help us. If literally everything anyone ever thougth about or mentioned /exists/, then the statement "X exists" is a null statement, which makes it sorta pointless to provide a proof that "X exists".

### Re:sort of.. (1)

#### snowgirl (978879) | more than 6 years ago | (#21134201)

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 existence. Thus creating yet another set of "existence", this time by declaring something that cannot be conceived, nor could exist in reality, but never the less has been given a definition.

Lastly there are things that we cannot create definitions for, we cannot conceive, nor can we say exist in reality as we understand it. I would give an example, yet by very definition, I cannot know of any to exist. (This set could obviously contain zero items in it, but by definition, we'll never know.)

Now, to say something "exists" is fairly insignificant of a statement, considering that everything that we can consider, by necessity must "exist", at least conceptually, or by reason of having a definition (which I will agree, is a fairly fine distinction between conceputal existence.) So, what do I mean by "exists" when I say that nothing, and sets must exist. Simply, I mean that they exist in actuality. They exist in more ways than simply conceptually, and this is the most common meaning of the phrase.

What does it mean to exist in reality? (conceptually, we have a fairly good understanding of). By the meaning of existence in reality, or actuality, means that it exist such that it can be conceived of, although it cannot by necessity have any contradicting definitions, and its existence in "reality" must be guarenteed or else reality itself could not exist. Thus, this objects existence cannot contradict with what reality already is dialectically known to be.

I presented the position that if nothing did not exist in reality, then it would create a contradiction in the universe, and thus it must exist in reality. I proved (albeit to myself) that I must exist in reality, else I would not be able to sit here and ponder this material in the first place. Given two items that must exist in reality, lest it present a contradiction in reality, then there must be something that can be defined as collecting neither item, one item, the other item, or both of the items, and in fact, those collections must themselves exist in reality, else there could be no separable between me and nothing, we I know must be false.

Now, given this reality, I know that there must be a conceptual reality, as I am provided stimuli, which I then interpret into concepts and ideas, and which I generate on my own. Were conceptual reality not to exist within actual reality, then I could not conceive of anything. The matter that I am conceiving this topic, is proof sufficient to me that conception must exist.

However, much of my guarentees of the world then break down at this point. I cannot guarentee that most things truly exist outside of conceptual reality. Even though I participate in dialog with you, that dialog and interaction is held exclusively through my perceived stimuli. Thus, I perceive you to exist in actuality, however I do not know that within the actuality of reality, you do actually exist. Presuming you to exist in actuality does not lead to contradiction of known reality, although presuming you to not exist in actuality does not lead to contradiction of known reality. So, while I treat you as if you exist in actuality, and reality, I cannot say "you exist" within the context of actual reality.
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