we really haven't agreed on any specific type of order yet. — Metaphysician Undercover

Yes, when 'we' includes you. But with math, we do specify specific kinds of order.

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

The notion of infinite sets is used calculus, which is mathematics for the sciences, which is mathematics for the technology you enjoy.

So we really haven't agreed on any specific type of order yet. — Metaphysician Undercover

That’s a cop out. You claimed that the diagram has an inherent order. Specify that order. Which dots are the start and end points of that order? This needn’t imply a temporal start and end. For example, winning poker hands have a rank from lowest to highest; from a pair to a royal flush. This is not a spatial or temporal order of rank.

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

A newborn doesn't have a hat but it can acquire one.

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.

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

You're confusing presentation with the set itself. 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?

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

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.

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

Whatever. I can't add anything. The points you're making are too silly to require response.

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

Yes.

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

They're different as ordered sets, but the same as sets.

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

If I stopped responding would that be ok? I've been done with this for a while. You're not making any points worthy of response.

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

That might be the dumbest thing you've ever said. If you said that no two physical spheres could be identical, that would be true. But why can't I have two conceptual, abstract spheres? There can't be a center point to a perfect circle? Look @Meta if you deny the unit circle in the Euclidean plane, with its center at the origin, we're done here. We're done here anyway, you are not making any points that seem reasonable to me. Pi is a computable real number anyway, so even if the universe is a simulation, the great computer in the sky would know about pi.

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 would you fix math?

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

Examples like that is how fishfry convinced me otherwise.

You claimed that the diagram has an inherent order. Specify that order. — Luke

I believe I already did. 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, requiring a different diagram. So that order is inherent to that diagram.

Specify that order. Which dots are the start and end points of that order? — Luke

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. "Order" is defined as "the condition in which every part, unit, etc., is in its right place". That's why the diagram has an inherent order. If any of the dots were in a different place it would not be the diagram which it is, because some part would be in the wrong place for it to be that particular diagram.

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

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.

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

Now, do you see that Sonny and Cher, Meta and fishfry, as individual people, have spatial temporal positioning, therefore an inherent order? 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.

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? There is nothing which could fulfill the condition of having no order.

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. 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. Without any order, they cannot be logical, and are simply nothings, not even abstract objects. It appears like you want to abstract the order out of the thing, but that's completely incoherent. 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"?

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

Yes. Now do you see that these three things have order, regardless of the order in which you name them? 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. 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. 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. It's false because a complete lack of order would be absolute nothing, therefore nothing to be ordered, and absolutely zero possibility for order. 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.

But why can't I have two conceptual, abstract spheres? — fishfry

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.

It is not I who is making the dumb propositions.

It's a spatial order, each dot has its own specific position on the plane. — Metaphysician Undercover

That is not specification of an order, let alone of "THE inherent order".

Of course, we could move left to right and also up and down, zigzagging to specify an ordering of the dots. But we could also do it right to left, or from spiraling from the center outward, or from one particular dot anywhere. So there is not one single order that is "THE inherent order".

It's amazing that you don't get this. Or that you simply evade, as you are Metaphysician Undercover from Evasionsville.

"Order" is defined as "the condition in which every part, unit, etc., is in its right place" — Metaphysician Undercover

That's not the sense of the word 'order' we're talking about! You have yourself even being using 'order' in a sense not expressed as "in its right place". Wow, you really do have a cognitive problem.

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

There are two issues here: "order" and "inherent".

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.

You also presume that the elements of the diagram have an inherent order. 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.

"Order" is defined as "the condition in which every part, unit, etc., is in its right place". — Metaphysician Undercover

A more suitable definition in the context of this discussion might be “the arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern, or method.”

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

Those "start" and "end" points would be non-temporal. That was the point of the counterexamples.

Nearly all of those counterexamples have elements which can be ordered first to last, lowest to highest, worst to best, etc. Those orderings (e.g. poker hands, military ranks, letters of the alphabet) therefore have "start" and "end" points in terms of the arrangement of their elements.

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

I specified the order. It is a spatial order, the one demonstrated by the diagram. Why is this difficult for you to understand? 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. 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. 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.

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

Yes I did explain that. 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. That is why it is not true to call it a random arrangement, unless you are using "random" to signify something other than no order.

This is an important constituent of the distinction between actual order, and possible order. A distinction which fishfry rejected as not with principle. But there is such a principle, which fishfry simply denied, that things must have an actual order, to have existence. I believe it's called the principle of sufficient reason. And this principle renders "the set", as being a unity composed of parts, without any inherent order, as an incoherent notion. We could say that there are many possible orders which the parts could have, but if they do not have an actual order, the supposition of "unity" and therefore "set" is unjustified.

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

You continue to assert that the elements have an order without specifying what that order is. "What is that order? An arrangement. And what is that arrangement? The order it has."

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

What algorithm might someone use to re-create the same pattern?

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

"It's not random because it has an order and it has an order so it isn't random." Empty words.

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

"They have that order and not some other order because there was a cause of that order."

Seriously? That does not explain why the elements must have this particular order instead of some other order. Why wasn't some other order or arrangement caused instead?

an actual order — Metaphysician Undercover

That's just another undefined term by you as is 'inherent order'. It adds nothing to your incorrect argument.

There are many orders. You have not defined what it means to say that one of the orders is 'AN actual order' or 'THE inherent order' while none of the other orders are an actual order or the inherent order. It is a remarkable feat of stubbornness for you to persist in not recognizing that clear and simple fact.

And we still have this classic:

"Order" is defined as "the condition in which every part, unit, etc., is in its right place". — Metaphysician Undercover

Clearly, we have not been talking about that sense of 'order'. Credit to you though for your sophistical resourcefulness in looking at a dictionary to find a different sense of a word to distract from the sense that has been used (even by you) throughout the discussion.

That's just another undefined term by you as is 'inherent order'. — TonesInDeepFreeze

From earlier in the discussion, “inherent” is being used to mean “non-arbitrary”.

what is the mathematical definition of 'abritrary'?

Anyway, I don't think we need 'inherent', 'actual' or 'non-arbitrary'.

It is enough to observe that sets of cardinality greater than 1 have more than 1 ordering, and that any other privileged designation is only by stipulation (such as 'the standard ordering' for specific sets is fine, since it is stipulated and defined), and there is no overall rubric of individuation for the orderings of sets in general.

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

No, it's your own private concept of a set that's incoherent. But what I do find noteworthy is that you genuinely believe (unless everything you post on this site is an elaborate troll, which I do suspect) that you are "explaining" anything to me. On the contrary, you're demonstrating your mathematical ignorance, which I labor mightily, and without hope of succeeding, to correct. Like Sisyphus rolling his boulder uphill, only to watch it roll down again; in vain do I endlessly explain to you that mathematical sets have no inherent order, only to suffer yet more sophistry from you.

Now, do you see that Sonny and Cher, Meta and fishfry, as individual people, have spatial temporal positioning, therefore an inherent order? — Metaphysician Undercover

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.

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

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?

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

No. Consider for example the vertices of an equilateral triangle. We may call them v1, v2, and v3, realizing that this labeling is completely arbitrary and that labels could be assigned in many different ways. Six different ways in fact. Now we have a SET of vertices which we can denote {v1, v2, v3} or {v2, v1, v3} or {v3, v2, v1}. In each case the set of vertices doesn't change. There are three vertices and they are the same set of vertices regardless of how we list them.

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.

There is nothing which could fulfill the condition of having no order. — Metaphysician Undercover

I just gave you a nice example, but I'm sure you'll argue. I push the boulder up the hill, it rolls down again.

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

Well, I take heart in your at least acknowledging my position. Just as the three vertices of a triangle have no inherent order. And that if you do assign them an order based on "leftmost" or some such, that order is contingently based on the orientation of the triangle. But a triangle's orientation is not an inherent part of its trianglitude. It's the same triangle no matter how we spin it.

Or would you say that the earth right now isn't the same as the earth five minutes from now, because it's spun on its axis? I think you either have to admit that a triangle is the same triangle no matter how it's oriented; OR you have to claim that the earth isn't the same earth from moment to moment because it's spinning. As if you could rearrange your living room by moving your couch, and it somehow becomes a different couch.

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

Vertices of a triangle. Inherently without order. Any spatial order is a function of the triangle's contingent orientation. I think this is a good example.

Without any order, they cannot be logical, and are simply nothings, not even abstract objects. — Metaphysician Undercover

The vertices of a triangle are not nothing, they're the vertices of a triangle.

It appears like you want to abstract the order out of the thing, but that's completely incoherent. — Metaphysician Undercover

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.

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

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.

Yes. Now do you see that these three things have order, regardless of the order in which you name them? — Metaphysician Undercover

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.

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

Vertices of an equilateral triangle.

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

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.

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

You're wrong. Or a troll. Lately you're starting to convince me of the latter.

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

Repeating the same ignorant falsehood doesn't work in mathematics. Only in politics.

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

Repetitive and wrong. And boring. At least say something interesting once in a while.

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

Take two identical sphere, of radius 1, say, in Euclidean three-space. You might say that one is to the "left" or "above" the other, as the case may be; but that is only a function of the coordinate system. And changing the coordinate system doesn't change the essential nature of an object. So if you had a universe consisting of two identical unit spheres and nothing else, how would you tell them apart? For ease of visualization, take them as two unit circles in the plane. How do you tell them apart without reference to a coordinate system?

(Edit) -- Ah, I see your point. Let me rephrase that. I'll stipulate that if they are identical, they are the same sphere. You have corrected me and I stand corrected. Consider two congruent spheres of radius one. The rest of my argument stands as stated.

It is not I who is making the dumb propositions. — Metaphysician Undercover

What I referred to as a dumb proposition is your claim that you can't have two identical spheres [Edit -- congruent] because pi is irrational. That's just such a bad argument that you should be embarrassed. The unit circle in Euclidean space has a circumference of 2 pi. I am sorry to have to be the one to break that news to you. But why do you care? Pi is a computable real number. We have many finite-length algorithms that exactly and uniquely characterize it.

Repeating the same ignorant falsehood doesn't work in mathematics. Only in politics. — fishfry

That should become a classic. Perfectly said.

The set of all primes between one and twenty-one has no order dependent upon its definition. However, when I begin to write down the elements of the set I establish an order in which they appear. :roll:

What more needs to be said? (but I am a victim of naive set theory)

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

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?

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

Well, it might be the case, that this "is simply how mathematical sets are conceived", but the question is whether this is a misconception.

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

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. 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.

I just gave you a nice example, but I'm sure you'll argue. — fishfry

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.

The issue is whether or not there can be a group of things without any such inherent order. 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.

In your example of "equilateral triangle" you have granted the points an inherent order with that designation. 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. 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.

Vertices of a triangle. Inherently without order. — fishfry

You think it's a good example, I see it as a contradiction. "Vertices of a triangle" specifies an inherent order.

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

The point though, is that to remove all order from a group of things is physically impossible. 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.

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

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.

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

The "inherent order" is the order that the things have independently of the order that we assign to them. 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. 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.

Now, you want to assume "a set" of points or some such thing without any inherent order at all. Of course we can all see that such points cannot have any real spatial temporal existence, they are simply abstract tools. To deny them of all inherent order is to deny them of all spatial-temporal existence. 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. 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.

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

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. By what means do you say that there is a possibility for ordering them? 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.

You're wrong. — fishfry

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?

You have corrected me and I stand corrected. — fishfry

Accepted, and I think that course of two identical spheres is a dead end route not to be pursued.

The set of all primes between one and twenty-one has no order dependent upon its definition. — jgill

You have stipulated an order, "primes" indicates a relation to each other.

What is the set then? — Metaphysician Undercover

It is the unique object whose members are all and only those specified by the set's definition.

{0 1} is the unique set whose members are all and only 0 and 1.

the law of identity is an important law to uphold — Metaphysician Undercover

Set theory adheres to the laws of identity.

The sun, earth, and moon, as three unique points, have an order inherent to them — Metaphysician Undercover

What is that "inherent order"? On what basis would you say one of these is the "inherent" order but not the others?:

<Bob Sue Tom>

<Bob Tom Sue>

<Sue Bob Tom>

<Sue Tom Bob>

<Tom Bob Sue>

<Tom Sue Bob>

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"?

For a finite set S of cardinality n, there are n! strict linear ordering of S. What is the general definition of "THE inherent ordering" among n! orderings?

it makes no sense whatsoever to assume something without any order, — Metaphysician Undercover

We don't assume that sets don't have orderings. Indeed, for a set S with cardinality n, there are n! strict linear orderings of S. But when you say that one of them is "THE inherent ordering" then that requires saying what "THE inherent ordering" means.

"primes" indicates a relation to each other. — Metaphysician Undercover

'is prime' is a predicate, not an ordering.

[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

Ordering was not "removed". There are n! orderings of the set. Remarking that you have not defined "THE inherent ordering" is not "removing" anything.

By what means do you say that there is a possibility for ordering them? — Metaphysician Undercover

There is not just a possibility. There exist n! orderings of the set. We prove that from axioms.

They have no spatial-temporal separation, therefore no means for distinguishing one from the other — Metaphysician Undercover

Members of mathematical sets are distinguished by properties other than spatial-temporal. And members of sets in non-mathematical contexts may be distinguished by means other than spatial-temporal. The queen of hearts is distinguished from the ace of spades, without having to refer to their positions in time and space.

abstraction has removed any possibility of order, so to speak of possible orders now is contradiction. — Metaphysician Undercover

Not just "possibility" but existence.

S = {queen-hearts ace-spades} = {ace-spades queen-hearts}

Abstraction has not "removed" orderings. S has two orderings:

{<queen-hearts ace-spades>}

and

{<ace-spades queen-hearts>}

Neither is "THE inherent ordering", unless you first give a definition of "THE inherent ordering".

"primes" indicates a relation to each other. — Metaphysician Undercover

'is prime' is a predicate, not an ordering.

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

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? You can't say leftmost or rightmost or top or bottom-most, because that only depends on the position of the observer. In modern physics you can't even line things up by temporal order since even that depends on one's frame of reference, and there is no frame of reference.

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.

Well, it might be the case, that this "is simply how mathematical sets are conceived", — Metaphysician Undercover

Ok!! Well we have made progress. You agree finally that mathematical sets have no inherent order, until we impose one. This point is made more strongly by mathematical objects that may not be familiar to you, such as topological spaces. A set may have many different topologies. A topological space is first a SET with no inherent topological structure. Then we impose a topological structure on it by associating the set with a SECOND set called the "topology," which is a particular collection of subsets of the first set. Given a set there are many different topologies that can be put on it. No one topology has any primacy over any other.

This pattern is so pervasive in math that it soon becomes second nature. You have a bare set with no structure. You impose on it an order to make it an ordered set. Or you impose a topology to make it a topological space. Or you impose a binary operation or two to make it a group or a ring or a field. That's the power of mathematical abstraction. You start with a bare set and toss in the ingredients you want. Like making a salad. You start with a bowl. The bowl is not initially any kind of salad. It's not even inherently a salad, it might turn out to be a bowl of oatmeal. You start with the bowl and add in the ingredients you want to get a particular object that you're interested in.

but the question is whether this is a misconception. — Metaphysician Undercover

It's been a long time, but we've made progress. 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. I don't want to start that now, I want to make sure we're in agreement. 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.

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

What is the natural, inherent order of the vertices of a triangle? This I really want to hear.

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

What is the inherent order of the vertices of a triangle? Which one is first, which second, which third? How do you know? I want you to answer this.

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

I get that. So what is the inherent order, the "order which the things have independently of such an imposed order," of the vertices of an equilateral triangle? I am standing by for your response.

The issue is whether or not there can be a group of things without any such inherent order. — Metaphysician Undercover

I'd prefer the word "collection," since a group is a specific mathematical object that's not at issue here. But I would say the vertices of an equilateral triangle are a pretty good example of a collection of three things that have no inherent order. If you disagree, tell me which one is first.

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

Triangle triangle triangle. Please answer.

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

This is sophistry. Clearly there is more than one point in math. I daresay there's a physical analogy here, because other than position, all electrons in the universe are the same. All points on the real line or in Euclidean space are the same. There's a point here and a point there. You can't deny and wish to retain any intellectual credibility.

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

Come on, man. The point at (0,0) and the point at (1,1) are two distinct points. Or two distinct locations in the plane, if you like to think of it this way. You can't pretend to throw out analytic geometry by denying there are points.

However I will give you this. We can use the word congruent instead of identical. Two geometric objects are congruent if they have the exact same shape, even if they are in different locations or have different orientations. I trust that handles your objections to saying they are identical.

You think it's a good example, I see it as a contradiction. "Vertices of a triangle" specifies an inherent order. — Metaphysician Undercover

Tell me what the order is so that I may know.

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

Mathematical order is inspired by physical order, but goes far beyond it. Graph theory for example is all about partially ordered sets. Big deal in computer science, social media, and machine learning.

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

Which are the first, second, and third vertices of an equilateral triangle?

The "inherent order" is the order that the things have independently of the order that we assign to them. — Metaphysician Undercover

Which is what? What is the inherent order of the earth, the sun, and a bowl of spaghetti? What is the inherent order of the vertices of a triangle? Would this order be the same for any observer in the universe? Make your case. You don't seem to be able to grapple with any specific examples.

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

And what is that order? You keep saying they have an inherent order but you won't say what that order is.

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

Then what is their inherent order, one that would be recognized by any intelligent observer anywhere in the universe? When we meet Martian mathematicians we expect they will know pi (or one of its multiples such as 2pi or pi/2 etc.) I would not expect them to agree on the order of the vertices of a triangle as you seem to claim they would.

Now, you want to assume "a set" of points or some such thing without any inherent order at all. — Metaphysician Undercover

For purposes of founding all the diverse set-based mathematical structures such as totally ordered sets, partially ordered sets, well-ordered sets, topological spaces, measure spaces, groups, rings, and field, vector spaces, yes. Exactly. That's the formalism. You can't argue with a formalism any more than you can argue with how the knight moves in chess.

Of course we can all see that such points cannot have any real spatial temporal existence, they are simply abstract tools. — Metaphysician Undercover

Yes, has it really taken you this long to understand that?

To deny them of all inherent order is to deny them of all spatial-temporal existence. — Metaphysician Undercover

Mathematical abstractions don't have spacio-temporal existence. This is news to you?

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

Of course we can. We have a bare set. We order it this way. We order it that way. We put on a partial order, a linear order, a well-order. We make it into a topological space in several different ways. We make it a group or a ring or a field. I'm sorry you haven't seen any modern math but you must recognize your own limitations in this regard.

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

More repetitive falsehoods.

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

Ok. Good.

By what means do you say that there is a possibility for ordering them? — Metaphysician Undercover

Define a binary relation on the set that is antisymmetric, reflexive, and transitive. As explained in painful detail in the Wiki article on order theory.

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

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.

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

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.

Accepted, and I think that course of two identical spheres is a dead end route not to be pursued. — Metaphysician Undercover

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.

It is the unique object whose members are all and only those specified by the set's definition. — TonesInDeepFreeze

As I explained, the objects, as existing objects, have an inherent order, so it is wrong to deny that the objects have an inherent order.

What is that "inherent order"? On what basis would you say one of these is the "inherent" order but not the others?: — TonesInDeepFreeze

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.

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

I assume that there are three individual human beings indicated by those names. The inherent order, if we were to attempt to describe it, would contain all the truthful relations between those beings. Order is the condition under which every part is in its right place. Therefore everything said about the relations between these people would be true if we were describing the inherent order.

'is prime' is a predicate, not an ordering. — TonesInDeepFreeze

It is a predicate which refers to relations with others, therefore an order.

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

Yes, according to fishfry ordering was removed, abstracted away, to leave the members of the set without any inherent order. If you are having difficulty with "inherent order", it is fishfry's term as well, but I suggest that it means order which inheres within the mentioned object (set). A set has been described by fishfry as a type of unity, but it was also said that this unity has no inherent order to its parts.

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

In a collection of things in the physical world, everything has the place that it has. This is the order of that collection. The fact that we cannot adequately define that order only indicates that we do not adequately understand the positioning of things in the world.

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're missing the point. If the triangle has any existence, then each of its vertices has a position relative to the others, and therefore an order. You can say "equilateral triangle" and insist that this refers to an abstract object with no inherent order to its vertices, but you'd be speaking falsely. There is clearly an inherent order to the vertices signified by "equilateral triangle", or else you could have something other than an equilateral triangle.

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

I never denied that this was how you conceived "set". I just argued that it is incoherent. Which I still do. If that's progress, then great.

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

No, rather than say "currently understood", I would class it as a misunderstanding. The reason, as I explained, I think it is impossible to have a set without order, regardless of what mathematicians believe. You continue to assert that this is possible, but have not addressed my arguments against it, nor have you demonstrated how a group of things could exist as a unity (a set), without any order to that group. so I still believe that your assertions, and those of other mathematicians, if they assert similar things, are reflections of a deep misunderstanding.

Clearly there is more than one point in math. — fishfry

That is untrue, just like your claim of multiple identical spheres is untrue. If the point is truly non-dimensional then there is nothing to distinguish one point from another, point therefore only one point in math. We make representations of points, in speaking and drawing diagrams, but these are representations, they are not a true point which must be purely abstract. Once you abstract a pure point, how would you make another?

Tell me what the order is so that I may know. — fishfry

As I said above in my reply to Tones, 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. This is the issue similar to the issue with the law of identity. The identity of a thing is within the thing itself, that's what the law of identity states, a thing is the same as itself, its identity is itself. If you asked, me, tell me, what is the thing's identity, well very clearly I cannot tell you, because I'd be assigning an identity and this is not the true identity which inheres within the thing. Likewise, I cannot tell you the order which inheres within the group of things, because iIwould just be giving you an order which I impose on that group from an external perspective.

Which are the first, second, and third vertices of an equilateral triangle? — fishfry

Each vertex is distinct from the others, and necessarily unique, separated from the others and having a specific relation with the others, or else it would not be the mentioned object. It is not a matter of "first, second, and third", that is simply how you might order them. However, there must be three distinct vertices each with its own unique identity, and a spatial order between these three, indicated by "equilateral triangle".

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

That is not a set, it's a category mistake. Sun and moon refer to particular objects existing with an inherent order, but "tuna sandwich" refers not to a particular, it signifies a universal, an abstraction.

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

You do not think that there is an inherent order between your ass and your elbow? You're just being ridiculous now. Do you even know what "order" means? Look it up in the dictionary please. Maybe then we might proceed. However, it appears like we are far apart as to what that word actually means. I'm starting to see now the misunderstanding in mathematics. You give "order" some special meaning, as a magical thing which you can take away from unities, and give to unities without affecting the unity of that unity.

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

There's contradiction in your description. You say that they have distinguishable locations, yet you can't distinguish their locations. In any case to have an object here, and an object over there, at the same time, is sufficient to say that they are distinct objects and therefore not identical.

You know, you can simply reject the identity of indiscernibles if you don't like it. You can just say that you do not believe it, and you think that two distinct objects might be exactly the same. It's an ontological principle based in inductive reasoning, so it's not necessarily true. It's just that it's a strong inductive principle.

As I explained, the objects, as existing objects, have an inherent order — Metaphysician Undercover

As you dogmatically claim. You keep skipping the central challenge to your claim. That challenge will be repeated in this post.

The inherent order is the true order, which inheres in the arrangement of objects. — Metaphysician Undercover

Petitio principii!

You just shift the answer among various still undefined terms: "actual", "inherent", and now "true".

What is "THE true order" such that the other orders are not "true" orders?

The inherent order, if we were to attempt to describe it, would contain all the truthful relations between those beings — Metaphysician Undercover

Before, it was temporal/spatial. But now, for just three people it's an even more complicated big deal with you just to attempt to describe "THE inherent order". And "all the truthful relations". I guess you mean the various comparisons, connections, associations, differences, shared properties, contrary properties, contexts, etc. among those three people. That could be a vast number of things. And still you haven't defined how "THE inherent order" is determined by a vast number of associated properties among the three people.

Here is a set of three living people you know about:

{Angela Merkel, Lance Armstrong, Justin Bieber}

Now, please tell me "THE inherent order" of them. Please tell me how you used "all the truthful relations" to determine "THE inherent order".

The inherent order is the true order, which inheres in the arrangement of objects. — Metaphysician Undercover

"The true order that inheres", but you can't say what it is. Sounds pretty "abstract", nay mystical, to me.

Order is the condition under which every part is in its right place. — Metaphysician Undercover

From "inherent" to "actual" to "true" to "in its right place". None of them defined by you.

'is prime' is a predicate, not an ordering.
— TonesInDeepFreeze

It is a predicate which refers to relations with others, therefore an order. — Metaphysician Undercover

So any predicate that involves "relations with others" is an order?

'the queen of hearts is a red card' and 'the four of clubs is a black card' are statements about predicates of the two cards. So what is "THE inherent order" - <QH 4C> or <4C QH>? Do we also have to consider all the "truthful relations between them"? What are some of those "truthful relations"? How did you determine "The inherent order" based on those "truthful relations"?

By the way, what do you mean by 'truthful' that wouldn't be said by just 'true'? And why even say 'truthful relations' when you could just say 'relations'?

according to fishfry ordering was removed, abstracted away, — Metaphysician Undercover

I don't speak for him, but I would imagine he's using such locutions as figures of speech.

I'll say it for you again, without recourse to figures of speech such as "removing" as I did before without recourse to figures of speech such as "removing":

{Bob Sue Tom} is the set whose members are all and only Bob, Sue and Tom.

There are 6 strict linear orderings of the set {Bob Sue Tom}.

If you want to say there is one of those orderings that is "THE inherent/actual/true/truthful/tutti-fruiti" ordering while the others are not "THE inherent/actual/true/truthful/tutti-fruiti" ordering, then you need to DEFINE the terminology "THE inherent/actual/true/truthful/tutti-fruiti" ordering".

If you are having difficulty with "inherent order", it is fishfry's term as well — Metaphysician Undercover

Again, I don't speak for him, but I understood him to be making a correct point that does not depend on whether he used a figure of speech such as "inherent". His point was that the set is determined by its members and not by the orderings of the set. I don't speak for him, but I have little doubt that he would agree (without using "inherent"):

(1) A set is determined by its members. Axiomatically, S =T if and only if every member of S is a member of T and every member of T is a member of S.

(2) A finite set of cardinality n has n! strict linear orderings. So for sets with cardinality greater than 1, there are more than 1 strict linear orderings of the set.

See, no need to use the word 'inherent' and especially not "THE inherent ordering".

Saying "the set has no inherent ordering" boils down as a locution to saying (1) and (2).

But then YOU jumped to say there IS "THE inherent ordering", and you are floundering to meet the challenge of defining it.

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

And you chided ME for my interest in mathematics that you deem not empirically justified. 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!

Just by saying which you think is "THE inherent order" you would necessarily not be choosing "THE inherent order"? So just by saying which ordering you think is "THE inherent order" you would necessarily be wrong!?

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

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?

In other words, how could the inherent order be known? If it cannot be known then how do you know there is one?

Before, it was temporal/spatial. — TonesInDeepFreeze

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.

Now, please tell me "THE inherent order" of them. — TonesInDeepFreeze

I explained very clearly in the last post why i cannot tell you the inherent order. It's not something that can be spoken,. Just like the identity of a thing, as stated by the law of identity, is not something that can be spoken. It is what is proper to the thing itself, not what is said about the thing. The order which inheres within the thing is proper to the thing itself, and not what is said about the thing. Do you understand this principle of identity?

So any predicate that involves "relations with others" is an order? — TonesInDeepFreeze

Of course, any relation with another is an order.

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 you wish to view the law of identity as a "mystical" principle, I have no problem with that. I would consider that most good ontology is based in mysticism.

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

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. Kant seems to describe the noumena as fundamentally unknowable. Others argue that it is fundamentally knowable, but only to a divine intellect, and not a lower intellect like the human being.

In other words, how could the inherent order be known? If it cannot be known then how do you know there is one? — Luke

We assume that there is a way that the world is (inherent order), because this is what makes sense intuitively. If we assume that there is no such thing, then we assume that the world is fundamentally unintelligible. To the philosophically inclined mind, which has the desire to know, the assumption that the world is fundamentally unintelligible is self-defeating. Therefore the rational choice is to assume that there is a way that the world is (inherent order). So we don't know that there is an inherent order, we assume that there is, because that is the rational choice.

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

If order is not restricted to "temporal/spatial", then order is not restricted to unknowable noumena.

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

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.

But now you enlist the view of other posters, with whom you so strongly disagreed, to wiggle out of your own untenable view!

What has happened is that it has finally gotten through to you that ordering is NOT only temporal-spatial, so you shifted to saying that "THE inherent order" is based on "all the relations". A complete reversal of your position, except you still cling to your notion of "THE inherent order".

And you still can't say what "THE inherent order" is with regard to your new bases of "all relations".

truth of a determined order is dependent ONLY [bold and all-caps added] on our concepts of space and time. — Metaphysician Undercover

Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".

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

Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".

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

Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".

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

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"!

Metaphysician Undercover fails to distinguish between two facts:

(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.)

(2) For a finite set of cardinality n>1, there are n! strict linear orderings of the set. And there has not been given a definition of "THE inherent ordering" from among those n! orderings (expect Metaphysician Undercover's mumbo jumbo about an ordering based (based in what way?) on all relations among the members and such that no one can correctly identify it but that it exists by virtue of identity).

If you wish to view the law of identity as a "mystical" principle — Metaphysician Undercover

I don't.

If we assume that there is no [the inherent order], then we assume that the world is fundamentally unintelligible. — Metaphysician Undercover

Non sequitur.

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 am sympathetic to the idea of assuming frameworks for making sense our experience. And, indeed, the mathematical notion of ordering can be part of that. It doesn't require an undefined notion of "THE inherent order".

I would consider that most good ontology is based in mysticism. — Metaphysician Undercover

To each his own.

See, not dogmatic. You expressed a philosophical preference, which is not itself an ignorant, confused and incorrect claim about mathematical logic and set theory, so I don't begrudge you having that preference.

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

I changed my mind on that days ago, when fishfry offered an ordering based on best. Then we moved along to "inherent order".

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

Right, I cannot say what the inherent order is, for the reasons explained. Do you have a problem with those reasons? Or do you just not understand what I've already repeated? You understand what "inherent" means don't you?

(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

The question is whether or not it is possible for a set to be free from inherent order, i.e. having no inherent order, as fishfry claimed. You still don't seem to be grasping the issue.