we really haven't agreed on any specific type of order yet. — Metaphysician Undercover
I apprehend, that at the base of the idea of infinity in natural numbers, is the desire, or intention to allow that numbers can be used to count anything. — Metaphysician Undercover
So we really haven't agreed on any specific type of order yet. — Metaphysician Undercover
The problem is that your demonstration was unacceptable because you claimed to start with a set that had no order. A newborn is not a thing without order, so the newborn analogy doesn't help you. — Metaphysician Undercover
So we start with the unordered set {a,b,c}.
— fishfry
You are showing me an order, "a" is to the right of "b" which is to the right of "c". And even if you state that there is a set which consists of these three letters without any order, that would be unacceptable because it's impossible that three letters could exist without any order. — Metaphysician Undercover
If you insist that it's not the letters you are talking about, but what the letters stand for, or symbolize, then I ask you what kind of things do these letters stand for, which allows them to be free from any order? To me, "a", "b", and "c" signify sounds. How can you have sounds without an order? Maybe have them all at the same time like a musical chord? No, that constitutes an order. Maybe suppose they are non-existent sounds? But then they are not sounds. So the result is contradiction. — Metaphysician Undercover
I really do not understand, and need an explanation, if you think you understand how these things in the set can exist without any order. What do "a", "b", and "c" signify, if it's something which can exist without any order? Do you know of some type of magical "element" which has the quality of existing in a multitude without any order? I don't think so. I think it's just a ploy to avoid the fundamental laws of logic, just like your supposed "two spheres" which cannot be distinguished one from the other, because they are really just one sphere. — Metaphysician Undercover
Huh, all my research into the axiom of extensionality indicates that it is concerned with equality. I really don't see it mentioned anywhere that the axiom states that a set has no inherent order. Are you sure you interpret the axiom in the conventional way? — Metaphysician Undercover
I have no problem admitting that two equal things might consist of the same elements in different orders. We might say that they are equal on the basis of having the same elements, but then we cannot say that the two are the same set, because they have different orders to those elements, making them different sets, by that fact. — Metaphysician Undercover
Are you serious? If I can imagine them as distinct things, I know that they cannot be identical. That's the law of identity, the uniqueness of an individual,. A fundamental law of logic which you clearly have no respect for. — Metaphysician Undercover
The argument of Max Black fails because pi is irrational. There is no such thing as a perfectly symmetrical sphere. The irrationality of pi indicates that there cannot be a center point to a perfect circle. Therefore we cannot even imagine an ideal sphere, let alone two of them. — Metaphysician Undercover
Yes, it seems like mathematics has really taken a turn for the worse. If you really believe that mathematicians are aware of this problem, why do you think they keep heading deeper and deeper in this direction of worse? I really don't think they are aware of the depth of the problem. — Metaphysician Undercover
How does this account for the order of rank of military officers, or of suits in a game of bridge? Or the order of values of playing cards in Blackjack? Or the order of the letters of the alphabet? Or monetary value? — Luke
You claimed that the diagram has an inherent order. Specify that order. — Luke
Specify that order. Which dots are the start and end points of that order? — Luke
But the deeper point is that YOUR CONCEPT of a set has inherent order, and that's fine. But the mathematical concept of a set has no inherent order. So you have your own private math. I certainly can't argue with you about it. — fishfry
The sets {a,b,c}, {c,b,a}, and {a,c,b} are exactly the same set. Just as Sonny and Cher are the same singing group as Cher and Sonny. They're the same two people. You and I are the same two people whether we're described as Meta and fishfry or fishfry and Meta. If you can't see that, what the heck could I ever say and why would I bother? — fishfry
Whatever man. You're talking nonsense. The Sun, the Moon, and the stars are the same collection of astronomical objects as the Moon, the stars, and the Sun. — fishfry
But why can't I have two conceptual, abstract spheres? — fishfry
It's a spatial order, each dot has its own specific position on the plane. — Metaphysician Undercover
"Order" is defined as "the condition in which every part, unit, etc., is in its right place" — Metaphysician Undercover
It's a spatial order, each dot has its own specific position on the plane. To change the position of one would change the order — Metaphysician Undercover
"Order" is defined as "the condition in which every part, unit, etc., is in its right place". — Metaphysician Undercover
There is no need to specify a start and end. After giving me examples of order which is not a temporal order, you cannot now turn around and insist that "order" implies a known start and end. — Metaphysician Undercover
You claim that the elements of the diagram have a "spatial order". Effectively, this is to say that the elements "have the order they have" without specifying what that order is. — Luke
You have provided no explanation as to why the diagram has the spatial order it has instead of any other possible spatial ordering of the the same elements. — Luke
When a diagram shows us an arrangement of dots, it shows us the spatial order of those dots, where the dots must be on a spatial plane to fulfill the order being demonstrated. What is the diagram? An arrangement of dots. What does it demonstrate to us? An ordering of those dots. — Metaphysician Undercover
Someone could proceed with that diagram to lay out the same pattern with other objects, with the ground, or some other surface as the plane. — Metaphysician Undercover
Just because fishfry called it a "random" arrangement doesn't mean that it does not demonstrate an order. Fishfry used "random" deceptively, as I explained already. — Metaphysician Undercover
There was a process which put those dots where they are, a cause, therefore a reason for them being as they are and not in any other possible ordering. — Metaphysician Undercover
an actual order — Metaphysician Undercover
"Order" is defined as "the condition in which every part, unit, etc., is in its right place". — Metaphysician Undercover
That's just another undefined term by you as is 'inherent order'. — TonesInDeepFreeze
Right, and do you also see now, that the mathematical concept of a set is incoherent? I hope so, after all the time I've spent explaining that to you. — Metaphysician Undercover
Now, do you see that Sonny and Cher, Meta and fishfry, as individual people, have spatial temporal positioning, therefore an inherent order? — Metaphysician Undercover
I am here, you are there, etc.. We can change the order, and switch places, or move to other places, but at no time is there not an order. — Metaphysician Undercover
So, you propose a set [a,b.c], [c,b,a], or phrase it however you like. You have these three elements. Do you agree that the three things referred to by "a", "b", and "c", must have an order, just like three people must have an order, or else the set is really not a set of anything? — Metaphysician Undercover
There is nothing which could fulfill the condition of having no order. — Metaphysician Undercover
I know, you'll probably say it's abstract objects, mathematical objects, referred to by the letters as members of the set, therefore there is no spatial-temporal order. — Metaphysician Undercover
But even this type of "thing" must have an order as defined by, or as being part of a logical system, or else they can't even qualify as conceptions or abstract objects. — Metaphysician Undercover
Without any order, they cannot be logical, and are simply nothings, not even abstract objects. — Metaphysician Undercover
It appears like you want to abstract the order out of the thing, but that's completely incoherent. — Metaphysician Undercover
Order is what is intelligible to us, so to remove the order is to render the concept unintelligible. What's the point to an unintelligible concept of "set"? — Metaphysician Undercover
Yes. Now do you see that these three things have order, regardless of the order in which you name them? — Metaphysician Undercover
And all things have some sort of order regardless of whether you recognize the order, or not. If there was something without any order it would not be sensible, cognizable or recognizable at all. — Metaphysician Undercover
In fact it makes no sense whatsoever to assume something without any order, or even to claim that such a thing is a real possibility. — Metaphysician Undercover
So to propose that there could be a complete lack of order, and start with this as a premise, whereby you might claim infinite possibility for order, you'd be making a false proposition. — Metaphysician Undercover
It's false because a complete lack of order would be absolute nothing, therefore nothing to be ordered, and absolutely zero possibility for order. — Metaphysician Undercover
But you want to say that there is "something" which has no order, and this something provides the possibility of order. By insisting that there is no order to this "something" you presume it to be unintelligible. — Metaphysician Undercover
If you think about what it means to be a conceptual abstract sphere, the answer ought to become apparent to you. What makes one sphere different from another is their physical presence. If you have two distinct concepts of a sphere, then they are only both the exact same concept of sphere through the fallacy of equivocation. If you have one concept of an abstract sphere then it is false to say that this is two concepts. It is simply impossible to have two distinct abstract concepts which are exactly the same, because you could not tell them apart. It's just one concept. — Metaphysician Undercover
It is not I who is making the dumb propositions. — Metaphysician Undercover
Repeating the same ignorant falsehood doesn't work in mathematics. Only in politics. — fishfry
As sets, they have no order. If you ADD IN their spatio-temporal position, that gives them order. The positioning is something added in on top of their basic setness. For some reason this is lost on you. — fishfry
Yes. That is true. But the SETNESS of these elements has no order. Not for any deep metaphysical reason, but rather because that is simply how mathematical sets are conceived. It is no different, in principle, than the way the knight moves in chess. Do you similarly argue with that? Why not? — fishfry
Now consider. You claim that their position in space defines an inherent order. But what if I rotate the triangle so that the formerly leftmost vertex is now on the bottom, and the uppermost vertex is now on the left? The set of vertices hasn't changed but YOUR order has. So therefore order was not an inherent part of the set, but rather depends on the spatial orientation of the triangle. — fishfry
I just gave you a nice example, but I'm sure you'll argue. — fishfry
Vertices of a triangle. Inherently without order. — fishfry
Well no, not really. I do abstract out order, for the purpose of formalizing our notions of order. I'm not making metaphysical claims. I'm showing you how mathematicians conceive of abstract order, which they do so that they can study order, in the abstract. But you utterly reject abstract thinking, for purposes of trolling or contrariness or for some other motive that I cannot discern. — fishfry
It clarifies our thinking, by showing us how to separate the collection-ness of some objects from any of the many different ways to order it. — fishfry
Vertices of an equilateral triangle. Let's drill down on that. It's a good example.
But take the sun, the earth, and the moon. Today we might say they have an inherent order because the sun is the center of the solar system, the earth is a planet, and the moon is a satellite of the earth.
But the ancients thought it was more like the earth, sun, and moon. The earth is the center, the sun is really bright, and the moon only comes out at night.
Is "inherent order" historically contingent? What exactly do YOU think is the "inherent order" of the sun, the earth, and the moon? You can't make a case. — fishfry
A mathematical set is such a thing. And even if you claim that your own private concept of a set has inherent order, you still have to admit that the mathematical concept of a set doesn't. — fishfry
You're wrong. — fishfry
You have corrected me and I stand corrected. — fishfry
The set of all primes between one and twenty-one has no order dependent upon its definition. — jgill
What is the set then? — Metaphysician Undercover
the law of identity is an important law to uphold — Metaphysician Undercover
The sun, earth, and moon, as three unique points, have an order inherent to them — Metaphysician Undercover
it makes no sense whatsoever to assume something without any order, — Metaphysician Undercover
"primes" indicates a relation to each other. — Metaphysician Undercover
[a mathematical set] consists of points, or some similar type of thing which have had all principles for ordering removed from them, therefore these points (or whatever) have no inherent order. — Metaphysician Undercover
By what means do you say that there is a possibility for ordering them? — Metaphysician Undercover
They have no spatial-temporal separation, therefore no means for distinguishing one from the other — Metaphysician Undercover
abstraction has removed any possibility of order, so to speak of possible orders now is contradiction. — Metaphysician Undercover
"primes" indicates a relation to each other. — Metaphysician Undercover
What is the set then? You already said it's not the names. If it's the individual people named, then they necessarily have spatial temporal positioning. You cannot remove the necessity of spatial-temporal positioning for those individuals, and still claim that the name refers to the individuals. That would be a falsity. So I ask you again, what constitutes "the set"? It's not the symbols, and it's not the individuals named by the symbol (which necessarily have order). What is it? — Metaphysician Undercover
Well, it might be the case, that this "is simply how mathematical sets are conceived", — Metaphysician Undercover
but the question is whether this is a misconception. — Metaphysician Undercover
This is not true, the order has not changed . The vertices still have the same spatial-temporal relations with each other, and this is what constitutes their order. By rotating the triangle you simply change the relations of the points in relation to something else, something external. — Metaphysician Undercover
So it is nothing but a change in perspective, similar to looking at the triangle from the opposite side of the plane. It appears like there is a different ordering, but this is only a perspective dependent ordering, not the ordering that the object truly has. — Metaphysician Undercover
Clearly your example fails to give what you desire. We are talking about "inherent order". This is the order which inheres within the group of things. It is not the perspective dependent order, which we assign to the things in an arbitrary manner, which is an extrinsically imposed order, it is the order which the things have independently of such an imposed order. — Metaphysician Undercover
The issue is whether or not there can be a group of things without any such inherent order. — Metaphysician Undercover
It is only by denying all inherent order that one can claim that an arbitrarily assigned order has any truth, thereby claiming to be able to attribute any possible order to the individuals. — Metaphysician Undercover
In your example of "equilateral triangle" you have granted the points an inherent order with that designation. [/quote[
How so?
— Metaphysician Undercover
You can only remove the order with the assumption that each point is "the same". However, it is necessary that each point is different, because if they were the same you would just have a single point, not a triangle. — Metaphysician Undercover
Therefore, it is necessary to assume that each point is different, with its own unique identity, and cannot be exchanged one for the other as equal things are said to be exchangeable. So when the triangle is rotated, each point maintains its unique identity, and its order in relation to the other points, and there is no change of order. A change of order would destroy the defined triangle. — Metaphysician Undercover
You think it's a good example, I see it as a contradiction. "Vertices of a triangle" specifies an inherent order. — Metaphysician Undercover
The point though, is that to remove all order from a group of things is physically impossible.[/quote[
I disagree with that even physically, since time and space are not absolute in modern physics. But in math, a collection of things has no order. The vertices of an equilateral triangle are a crystal clear example. If you disagree, tell me which one is first in such a way that a Martian mathematician would make the same determination.
— Metaphysician Undercover
And, "order" is a physically based concept. So the effort to remove all order from a group of things in an attempt to "conceive of abstract order" will produce nothing but misconception. If this is the mathematician's mode of studying order, then the mathematician is lost in misunderstanding. — Metaphysician Undercover
There is a fundamental principle which must be respected when considering "the many different ways to order" a group of things. That is the fact that such possibility is restricted by the present order. This is a physical principle. Existing physical conditions restrict the possibility for ordering. Therefore whenever we consider "the many different ways to order" a group of things, we must necessarily consider their present order, if we want a true outlook. To claim "no order" and deny the fact that there is a present order, is a simple falsity. — Metaphysician Undercover
The "inherent order" is the order that the things have independently of the order that we assign to them. — Metaphysician Undercover
This is the reason why the law of identity is an important law to uphold, and why it was introduced in the first place. It assigns identity to the thing itself, rather than what we say about it. We can apply this to the "order" of related things like the ones you mentioned. The sun, earth, and moon, as three unique points, have an order inherent to them, which is distinct from any order which we might assign to them. — Metaphysician Undercover
The order we assign to them is perspective dependent. The order which inheres within them is the assumed true order. In our actions of assigning to them a perspective dependent order, we must pay attention to the fact that they do have an independent, inherent order, and the goal of representing that order truthfully will restrict the possibility for orders which we can assign to them. — Metaphysician Undercover
Now, you want to assume "a set" of points or some such thing without any inherent order at all. — Metaphysician Undercover
Of course we can all see that such points cannot have any real spatial temporal existence, they are simply abstract tools. — Metaphysician Undercover
To deny them of all inherent order is to deny them of all spatial-temporal existence. — Metaphysician Undercover
The point which you do not seem to grasp, is that once you have abstracted all order away from these points, to grant to them "no inherent order" by denying them all spatial-temporal relations, you cannot now turn around and talk about their possible orders. — Metaphysician Undercover
Order is a spatial-temporal concept, and you have removed this from those points, in your abstraction. That abstraction has removed any possibility of order, so to speak of possible orders now is contradiction. — Metaphysician Undercover
Assuming you have understood the paragraphs I wrote above, let's say that "a mathematical set is such a thing". It consists of points, or some similar type of thing which have had all principles for ordering removed from them, therefore these points (or whatever) have no inherent order. — Metaphysician Undercover
By what means do you say that there is a possibility for ordering them? — Metaphysician Undercover
They have no spatial-temporal separation, therefore no means for distinguishing one from the other, they are simply assumed to exist as a set. How do you think it is possible to order them when they have been conceived by denying all principles of order.? To introduce a principle of order would contradict the essential nature of these things. — Metaphysician Undercover
I'm waiting for a demonstration to support this repeated assertion. How would you distinguish one from another if you remove all principles which produce inherent order? — Metaphysician Undercover
Accepted, and I think that course of two identical spheres is a dead end route not to be pursued. — Metaphysician Undercover
It is the unique object whose members are all and only those specified by the set's definition. — TonesInDeepFreeze
What is that "inherent order"? On what basis would you say one of these is the "inherent" order but not the others?: — TonesInDeepFreeze
There are no spatial-temporal parameters for you to reference, just 3 people, with 3!=6 strict linear orderings. Order them temporally by birth date? Temporally by the day you first met them? Spatially east to west? Spatially west to east? Which is "THE inherent order" such that the other orders are not "THE inherent order"? — TonesInDeepFreeze
'is prime' is a predicate, not an ordering. — TonesInDeepFreeze
Ordering was not "removed". There are n! orderings of the set. Remarking that you have not defined "THE inherent ordering" is not "removing" anything. — TonesInDeepFreeze
In the case of a collection of things in the physical world, they have spatio-temporal positoin, but there is no inherent order. How would you define it? — fishfry
The point is even stronger when considering abstract objects such as the vertices of a triangle, which can not have any inherent order before you arbitrarily impose one. If the triangle is in the plane you might again say leftmost or topmost or whatever, but that depends on the coordinate system; and someone else could just as easily choose a different coordinate system. — fishfry
You agree finally that a mathematical set has no inherent order, and you ask whether that's the right conceptual model of a set. THIS at last is a conversation we can have going forward. — fishfry
You agree that mathematical sets as currently understood have no order, and you question whether Cantor and Zermelo may have gotten it wrong back in the day and led everyone else astray for a century. This is a conversation we can have. — fishfry
Clearly there is more than one point in math. — fishfry
Tell me what the order is so that I may know. — fishfry
Which are the first, second, and third vertices of an equilateral triangle? — fishfry
Of course the elements can be distinguished from each other, just as the elements of the set {sun, moon, tuna sandwich} can. There's no inherent order on the elements of that set. — fishfry
How about {ass, elbow}. Can you distinguish your ass from your elbow? That's how. And what is the 'inherent order" of the two? A proctologist would put the asshole first, an orthopedist would put the elbow first. — fishfry
If I use the word congruent, the example stands. And what it's an example of, is a universe with two congruent -- that is, identical except for location -- objects, which can not be distinguished by any quality that you can name. You can't even distinguish their location, as in "this one's to the left of that one," because they are the only two objects in the universe. This is proposed as a counterexample to identity of indiscernibles. I take no position on the subject. but I propose this thought experiment as a set consisting of two objects that can not possibly have any inherent order. — fishfry
As I explained, the objects, as existing objects, have an inherent order — Metaphysician Undercover
The inherent order is the true order, which inheres in the arrangement of objects. — Metaphysician Undercover
The inherent order, if we were to attempt to describe it, would contain all the truthful relations between those beings — Metaphysician Undercover
The inherent order is the true order, which inheres in the arrangement of objects. — Metaphysician Undercover
Order is the condition under which every part is in its right place. — Metaphysician Undercover
'is prime' is a predicate, not an ordering.
— TonesInDeepFreeze
It is a predicate which refers to relations with others, therefore an order. — Metaphysician Undercover
according to fishfry ordering was removed, abstracted away, — Metaphysician Undercover
If you are having difficulty with "inherent order", it is fishfry's term as well — Metaphysician Undercover
if I stated an order, it would be a representation, imposed from my perspective, and therefore not the order which inheres within the object, the inherent order. — Metaphysician Undercover
The inherent order is the true order, which inheres in the arrangement of objects. If I stated an order, this would be an order which I assign to those objects, from an external perspective, and therefore not the inherent order. — Metaphysician Undercover
Before, it was temporal/spatial. — TonesInDeepFreeze
Now, please tell me "THE inherent order" of them. — TonesInDeepFreeze
So any predicate that involves "relations with others" is an order? — TonesInDeepFreeze
Now you're resorting to a mystical "true order that inheres in an object" but such that if we even attempted to state what that order would be, then we would be wrong! — TonesInDeepFreeze
If the true order cannot be assigned from an external perspective, then what is the "internal perspective" of an arrangement of objects? Will I know the "true order" of its vertices if I stand in the middle of a triangle? — Luke
In other words, how could the inherent order be known? If it cannot be known then how do you know there is one? — Luke
Temporal/spatial was just one type of order, fishfry and Lluke gave examples of many other types. — Metaphysician Undercover
Are you aware of Kant;s distinction between phenomena and noumena? As human beings, we do not know the thing itself, we only know how it appears to us. — Metaphysician Undercover
Temporal/spatial was just one type of order, fishfry and Lluke gave examples of many other types. So we're not restricted to temporal/spatial order in our attempts at understanding the nature of inherent order. — Metaphysician Undercover
truth of a determined order is dependent ONLY [bold and all-caps added] on our concepts of space and time. — Metaphysician Undercover
in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time [...] I think it can be demonstrated that each and every order imaginable is dependent on a spatial or temporal relation. To the right, left, or any such pattern, is spatial, and ANY INTELLIGIBLE SENSE OF "PRIOR" IS REDUCIBLE TO A TEMPORAL RELATION. I REALLY DO NOT THINK THERE IS ANY TYPE OF ORDER WHICH IS NOT BASED IN A SPATIAL OR TEMPORAL RELATION [bold and all-caps added]. — Metaphysician Undercover
If you truly think that there is some type of order which is intelligible without any spatial or temporal reference, you need to do a better job demonstrating and explaining it. — Metaphysician Undercover
to grant to them "no inherent order" by denying them all spatial-temporal relations, you cannot now turn around and talk about their possible orders. ORDER IS A SPATIAL-TEMPORAL CONCEPT [bold and all-caps added] — Metaphysician Undercover
Surely, "first" does not mean "highest quality", or "best", in mathematics, so if it's not a temporal reference, what is it? — Metaphysician Undercover
You cannot remove the necessity of spatial-temporal positioning for those individuals, and still claim that the name refers to the individuals. — Metaphysician Undercover
It was my suggestion that "order" is fundamentally temporal — Metaphysician Undercover
If it can't be understood without spatial or temporal reference, then there clearly is a need for space and time in math, or else all mathematics would be simply unintelligible. — Metaphysician Undercover
And understanding them is what requires spatial and temporal reference. The number 5 has no meaning, and cannot be understood without such reference. — Metaphysician Undercover
the ordering of numbers requires a spatial or temporal reference. — Metaphysician Undercover
If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. I've seen some people argue for a "logical order" which is neither temporal nor spatial, but this so-called logical order, which is usually expressed in terms of first and second, is always reducible to a temporal order. — Metaphysician Undercover
What is this act which you call "the collecting of the objects into a set"? Wouldn't such an act necessarily create an order, if only just a temporal order according to which ones are collected first? — Metaphysician Undercover
I have opted for a sort of compromise to this problem of justifying the pure a priori, by concluding that time itself is non-empirical, thus justifying the temporal order of first, second, third, etc., as purely a priori. — Metaphysician Undercover
If you want to define numbers by order, then you assign temporality as the difference between 1 ,2,3 and 4. — Metaphysician Undercover
i cannot tell you the inherent order. It's not something that can be spoken,. — Metaphysician Undercover
If you wish to view the law of identity as a "mystical" principle — Metaphysician Undercover
If we assume that there is no [the inherent order], then we assume that the world is fundamentally unintelligible. — Metaphysician Undercover
So we don't know that there is an inherent order, we assume that there is, because that is the rational choice. — Metaphysician Undercover
I would consider that most good ontology is based in mysticism. — Metaphysician Undercover
It was pointed out to you that there are orderings that are not temporal-spatial. But you insisted, over and over, that temporal-spatial position is required for ordering. — TonesInDeepFreeze
Not only cannot you tell us what "THE inherent order" is for ANY set, but you can't even define the rubric.
You can't even say what is "THE inherent order" of {your-mouse your-keyboard} even though they are right in front of you. And you said a really crazy thing: If you identified "THE inherent order" of a set then we will identified it incorrectly. You have a 50% chance of correctly identifying "THE inherent order" of {your-mouse your-keyboard} but you say that you would be wrong no matter which order you identified as "THE inherent order"! — TonesInDeepFreeze
(1) For every set, there is a determined set of strict linear orderings of the set. (That is a simple and intelligible alternative to his confused notion of "THE inherent order" ensuing from identity.) — TonesInDeepFreeze
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