Where does Kant say this? — Luke

I don't know if Kant ever said, but it's pretty obvious how intention must fit in.

Also, do you have any intention of accounting for your latest blatant contradiction: — Luke

No, sorry I must have made a mistake, or perhaps you just misunderstood. More likely, you intentionally misinterpreted, as usual. I'm very well acquainted with your strawman interpretations designed at creating the appearance of contradiction.

Before your claim was that the inherent order is what's shown. Now you claim that the inherent order is what's hidden. It can't be both. — Luke

There is no contradiction in saying that I am showing you an order which I cannot describe. Try again. But I suggest you try to understand rather than trying to misunderstand.

How is intention phenomenal (in the relevant Kantian sense)? — Luke

Intention is an integral part of the phenomenal system, the world as it appears to us, as the fulfillment of our wants and needs have shaped the way that we perceive the world evolutionarily, and have much influence over our perceptions on a day to day basis..

You are trying to draw an analogy between order/inherent order and phenomena/noumena. However, phenomena and noumena are both temporal-spatial, which makes order and inherent order also temporal-spatial by analogy. — Luke

Right, inherent order, which I classed as noumenal, appears to be spatial-temporal. But the type of ordering which fishfry demonstrated to me, ordering by best, or better, cannot be inherent order because it is relative to intention, therefore phenomenal.

I don't see the problem.

So there you are, still demanding that order must be temporal-spatial. — TonesInDeepFreeze

Yes, I agree that "order", when reduced in the extreme, seems to require necessarily, spatial-temporal conceptions. .I agree that some types of ordering such as those presented, which I called order by best, or better, appear to be free from spatial-temporal conceptions. But ultimately there must be something which is being ordered, individual objects, and this requires spatial separations. Perhaps however, we can order intentions themselves, as better or worse, as objects in the sense of goals, and we might give ends an ordering which is completely void of spatial-temporal conceptions. But this requires that we determine what type of existence an intention has.

And after so many days on end of you claiming that orderings are necessarily temporal-spatial, now you recognize that orderings do not have to be temporal-spatial, so what took you so long? It's piercingly clear that there are orderings that are not not temporal-spatial, but you could not see that because you are stubborn and obtuse. — TonesInDeepFreeze

Earlier in the thread, it was flatly denied as nonsensical, that the type of objects which existed in sets, are intentions. As explained above, that is the only way I can apprehend an "object" which has no inherent order, if it were an intention. So, if the things in sets are said to have no inherent order, the issue remains. How do you conceive of individual things with no spatial-temporal ordering, such that they can exist in a set without inherent ordering if these things are not intentions?

I have rebutted great amounts of your confusions. You either skip the most crucial parts of those rebuttals or get them all mixed up in your mind.

Anyway, to say that there is "THE inherent ordering" of a set, but not be able to identify it for a set as simple as two members is, at the least, problematic. But more importantly, you cannot even define the "THE inherent ordering" as a general notion. That is, you cannot provide a definition like: — TonesInDeepFreeze

I have defined "inherent order", in relation to the law of identity. It is you who is skipping the most crucial parts of what I write.

In set theory and abstract mathematics. EVERY property of an object is inherent to the object. (Mathematical) objects don't change properties. They have the exact properties they have - always - and no other properties - always. — TonesInDeepFreeze

OK then, two distinct ordering of the same elements constitutes two distinct sets. Do you agree? If order is inherent to the object, as you claim, then two distinct orderings of the same elements constitutes two distinct objects, therefore two distinct sets. Do you see this?

But the point you keep missing is that you have not defined what it means to say that one of the orderings in particular is "THE inherent ordering". They are all orderings of the set, and they are all inherent to the set. I have put 'THE' in all caps about a hundred times now. The reason I do that is obvious, but you still don't get it. — TonesInDeepFreeze

It's you who keeps missing the point. The "inherent order" is as "inherent" implies, the one which inheres within the object, as its identity, stipulated by the law of identity. It is categorically different from any order which we might assign to the object. Therefore it is not "one of the orderings" which we lay out, it is distinct from these. And the question you keep asking me, which of these orders is the inherent order is nonsensical because i keep telling you it's none of those orders.

This started with discussion of the axiom of extensionality. With that axiom, sets are equal if they have the same members. — TonesInDeepFreeze

Do you agree with me, that "equal" does not mean "the same"? Therefore equal sets are not the same set. Two sets with the same members in different orders can be said to be equal, but they cannot be said to be the same set. This is the part that fishfry doesn't get. Fishfry believes that if the sets are equal, they are necessarily the same set.

And it seems the reason you don't get that is because you started out needing to deny the sense of the axiom of extensionality itself, even though you are ignorant of what it does in set theory and you are ignorant of virtually the entire context of logic, set theory and mathematics. — TonesInDeepFreeze

I am not denying the axiom of extensionality, I am denying a particular interpretation of it, which says that equal sets are the same set. I look at this as a misunderstanding.

I made no argument for a philosophy regarding truth. — TonesInDeepFreeze

Then what would you call the following?

We are concerned not just with the object-language and a meta-language, but the object-theory and a meta-theory.

With a meta-theory, there a models of the object-theory. Per those models, sentences of the object-language have truth values. So the Godel-sentence is not provable in the object-language but it in a meta-theory, we prove that the Godel-sentence is true in the standard model for the language of arithmetic. Also, as you touched on, in the meta-theory, we prove the embedding of the Godel-sentence into the language of the meta-theory (which is tantamount to proving that the sentence is true in the standard model). That is a formal account of the matter. And in a more modern context than Godel's own context, if we want to be formal, then that is the account we most likely would adopt.

Godel himself did not refer to models. Godel's account is that the Godel-sentence is true per arithmetic, without having to specify a formal notion of 'truth'. And we should find this instructive. It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold. — TonesInDeepFreeze

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

Of course, that's why we have to acknowledge the difference between the order we say that a group of things has, and the inherent order of that group of things. They are both called "order". That they are different accounts for the fact that we make mistakes in understanding the order of things..

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.

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.

I haven't argued a philosophy. — TonesInDeepFreeze

Take a look at my quote above, and the context from where it's taken. You are arguing a philosophy of truth.

empirical validation isn't relevant. — Wayfarer

That's exactly what makes arguing for mathematics as the purveyor of truth, dogmatism.

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.

Let me explain it clearly then, since you seem to be having trouble understanding. When someone accepts, believes in, and adheres to principles which have not been empirically proven and argues a philosophy which gives the highest esteem to such principles, that is dogmatism. Many principles employed in modern mathematics, axioms, have not been empirically proven. So to believe strongly in, and argue from such principles, even labeling those who doubt these unproven principles as cranks, is dogmatism.

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.

Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. — TonesInDeepFreeze

When you reject such, and insist on the other, it's dogmaticism.

What dogmatism do you think you have witnessed? — TonesInDeepFreeze

Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.

(1) Sentences are not true in a language. They are true or false in a model for a language.

(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.

(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.

(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system. — TonesInDeepFreeze

Thanks. I have learned from this thread to avoid discussion of this topic in future. — Wayfarer

Wise decision, the dogmatic don't provide reasonable discourse.

Sextus Empiricus against the dogmatist's criteria of truth:

At the end of Sextus’ discussion in PH II, he clearly signals, as one would expect, that he suspends judgment on whether there are criteria of truth:

You must realize that it is not our intention to assert that standards of truth are unreal (that would be dogmatic); rather, since the Dogmatists seem plausibly to have established that there is a standard of truth, we have set up plausible-seeming arguments in opposition to them, affirming neither that they are true nor that they are more plausible than those on the contrary side, but concluding to suspension of judgement because of the apparently equal plausibility of these arguments and those produced by the Dogmatists. (PH II 79; cf. M VII 444) — Stanford Encyclopedia of Philosophy

According to Chisholm, there are only three responses to the Problem of the Criterion: particularism, methodism, and skepticism. The particularist assumes an answer to (1) and then uses that to answer (2), whereas the methodist assumes an answer to (2) and then uses that to answer (1). The skeptic claims that you cannot answer (1) without first having an answer to (2) and you cannot answer (2) without first having an answer to (1), and so you cannot answer either. Chisholm claims that, unfortunately, regardless of which of these responses to the Problem of the Criterion we adopt we are forced to beg the question. It will be worth examining each of the responses to the Problem of the Criterion that Chisholm considers and how each begs the question against the others. — Internet Encyclopedia of Philosophy

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.

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.

erhaps “numerical” wasn’t the right word. The context of the post and the preceding discussion indicates that a “before and after” ordinality was implied, of the sort Tones recently mentioned: — Luke

Before and after, are temporal terms. Fishfry had rejected the notion that "order" is based in spatial-temporal relations, and wanted an order based in quantity. But a quantity based order is what produces the problem I first referred to. If "2" refers to a quantity of objects in the context of "order", then it does not refer to a single object, the number 2. In any case, what distinguishes one thing from another, allowing for individuals, and quantity itself is spatial relations. So we're back to spatial relations as the bases of order. It's very clear that the subject was order of any sort, when "no inherent order" was mentioned.

It was my suggestion that "order" is fundamentally temporal, but fishfry produced examples of an order based on a judgement of better and worse, to discount that theory. So we really haven't agreed on any specific type of order yet. This is probably because we haven't agreed on what type of existence the things which are supposed to have no inherent order, but are capable of being ordered, have.

Yes, it certainly looks like that faulty brick in the foundations of math will surely cause the giant structure to collapse. I wish I had known this before becoming a mathematician. :cry: — jgill

I don't foresee any imminent collapse, but the structure ought to be dismantled because it supports the faulty worldview which is prevalent today, which is a sort of scientism. Fishfry perceives that physics has reached a sort of dead end in its endeavours, but refuses to acknowledge that the dead end is brought about by the principles employed (mathematics included) rather than the unintelligibility of the world itself.

The use of "infinity" which is the topic of this thread (believe it or not) is a very good example. 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. There will never be something which cannot be counted because the numbering system has been designed to allow that the numbers can always go higher. This gives the appearance that everything is measurable.

The only drawback is that this renders infinity itself, (the principle which provides this capacity, that everything is measurable) as immeasurable. So in reality everything is measurable except the principles we use to measure with. What this means is that to understand the nature of measurement and infinity, we must place these into a different category from the category of things which are measurable, and understand it on those terms, as immeasurable. If we attempt to bring infinity into the category of things which are measurable, as is the trend in modern mathematics, because we want mathematics to enable science to be applicable everything, even the thing which by its own design is immeasurable, then we introduce contradiction (the immeasurable is measurable) and therefore unintelligibility into our principles. That is where we are today, we have allowed unintelligibility to inhere within our principles of measurement. As fishfry said "math has experienced a loss of certainty".

Fishfry and I really agree that the big picture is very hazy (uncertain). But when it comes down to determining the specific points where the haziness arises from, fishfry refuses to follow. It's like seeing smoke on the horizon and wondering why it's there. But when I point to the fire, fishfry refuses to acknowledge a relationship between the fire and the smoke on the horizon. Maybe if fishfry would accept the possibility of a relationship, a closer look would reveal smoke rising from the fire.

If not specified, then at least strongly implied in the same post: — Luke

I really don't see how the qualification "numerical" is relevant , or even meaningful in the context of dots on a plane. So I don't see why you think it was implied. Fishfry is not sloppy and would not have forgotten to mention a special type of order was meant when "no inherent order" was said numerous times.

You wrote 'Russel' twice. It's 'Russell'. — TonesInDeepFreeze

Sorry "Bertie", as fishfry says.

Tracking recent points Metaphysician Undercover has either evaded or failed to recognize that he was mistaken. — TonesInDeepFreeze

I believe all the relevant points were addressed. You don't seem to know how to read very well.

I wasn't born wearing a hat but I can go buy one. The idea that a thing can't gain stuff it didn't have before is false. But I've already shown you how we impose order on a set mathematically. We start with the bare (newborn if you like) set. Then we pair it with another, entirely different set that consists of all the ordered pairs of elements of the first set that define the desired order. — fishfry

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.

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.

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.

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.

Also: "How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction?" Like ordering schoolkids by height, or alphabetically by name. Two different ways to order the same set. I can't understand why this isn't obvious to you. — fishfry

Those are different ways, but not contrary ways.

A set has no inherent order. That's the axiom of extensionality. — fishfry

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? 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.

You can't visualize two identical spheres floating in an otherwise empty universe? How can you not be able to visualize that? I don't believe you. — fishfry

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.

I did not make this up. I'm not clever enough to have made it up. I read it somewhere. Now you've read it. — fishfry

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.

No, on the contrary. Mathematicians love festering falsities. And of course even this point of view is historically contingent and relatively recent. In the time of Newton and Kant, Euclidean geometry was the true geometry of the world, and Euclidean math was true in an absolute, physical sense. Then Riemann and others showed that non-Euclidean geometry was at least logically consistent, so that math itself could not distinguish truth from falsity. Then Einstein (or more properly Minkowski and Einstein's buddy Marcel Grossman) realized that Riemann's crazy and difficult ideas were exactly what was needed to give general relativity a beautiful mathematical formalism. The resulting "loss of certainty" was documented in Morris Kline's book, Mathematics: The Loss of Certainty. Surely the title alone must tell you that mathematicians are not unaware of the issues you raise. — fishfry

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.

Hopefully others will correct me if I'm wrong but, as I understand it, the point iof the diagram of "dots" is that the elements of the set have no inherent numerical order or sequence. Otherwise, you should be able to number the elements from 1 to n and explain why that is their inherent order. — Luke

Actually "numerical order" (whatever that is supposed to mean in reference to a diagram of dots) was not specified. It was simply asserted that the elements have no inherent order.

I don't speak for fishfry, but the second 'you' above appears to include both of us. But I have never said, implied, or remotely suggested, that truth and falsity are irrelevant to pure mathematics. So you are lying to suggest that I did. — TonesInDeepFreeze

Are you taking lessons from Luke on how to make strawman interpretations? Read the quoted passage. Fishfry and Tones are in denial of the logical fallacies, and "you" (directed at fishfry only) talk about truth and falsity not being relevant to pure mathematics. How is that sentence so difficult for you to read properly?

You ignore what I said. That is your favorite argument tactic: — TonesInDeepFreeze

Yes, I am finding that to be the best tactic in dealing with the type of nonsense you throw at me.

What is "THE INHERENT" order you claim that the dots have? — TonesInDeepFreeze

The one in the diagram. Take a look at it yourself, and see it.

Start with what people say in everyday language. — TonesInDeepFreeze

I have no problem with what people say in everyday language, about the number of students in the class, the number of chairs in the room, the number of trees in the forest, etc., where I have the problem is with what mathematicians say about numbers alone, without referring to "the number of ..."

Sets of cardinality greater than 1 have more than one ordering. — TonesInDeepFreeze

Look at it this way Tones. As you describe sets, order is an attribute, or property of the set. How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction? Fishfry resolves this by saying that a set has no order, so order is not a property of a set. But then it appears like fishfry wants to smuggle order in, with some notion of possible orders. However the set is already defined as not having the property of order, therefore order is impossible.

Between you and fishfry, the two of you do not even seem to be in agreement as to whether a set has order or not. Fishfry says that a set has no inherent order. You say that not only does a set have order, but it has a multitude of different orders at the same time. See what happens when you employ contradictory axioms? Total confusion.

Nobody is claiming math is absolute truth but you. — fishfry

Wow, that's an even worse interpretation of what I'm saying than TIDF's terrible interpretation. I argue to demonstrate untruths in math, and you say I'm claiming math is absolute truth. This thread has gone too far. I think you're cracking up.

Don't you think he was recognizing and responding to exactly the point you are making? — fishfry

No, I have read a fair bit of Russel and he was in no way responding to the same points I'm making. More precisely he was helping to establish the situation which I am so critical of. Remarks like that only inspire mathematicians to produce more nonsense.

Try understanding the axiom of extensionality. — fishfry

We've been through the axiom of extensionality you and I, in case you've forgotten. It's where you get the faulty idea that equal to, means the same as.

One example that comes to mind is the famous counterexample to the identity of indiscernibles, in which we posit a universe consisting of two identical spheres. You can't distinguish one from the other by any property, even though the spheres are distinct. This example also serves as a pair of objects without any inherent order. — fishfry

Contradiction again. If you cannot distinguish one from the other, you cannot say that there are two. To count two, you need to apprehend two distinct things. But to say that you cannot distinguish one from the other means that you cannot apprehend two distinct things. Therefore it is false to say that there are two. So you are just proposing a contradictory scenario, that there are two distinct spheres which cannot be distinguished as two distinct spheres (therefore they are not two distinct spheres), hoping that someone will fall for your contradiction. Obviously, if one cannot be distinguished from the other, they are simply two instances of the same sphere, and you cannot say that there are two. And to count one and the same sphere as two spheres is a false count.

Reality? I make no claims of reality of math. I've said that a hundred times. YOU are the one who claims math is real then complains that this can't possibly true. — fishfry

Coming from the Platonic realist who claims the reality of "mathematical objects".

But math makes no claims as to the truth of "this." — fishfry

Right, just like the statement from Russel. That's why there is a real need for metaphysicians to rid mathematics of falsity. The mathematicians obviously do not care about festering falsities.

.

A contradiction is a statement and its negation. — TonesInDeepFreeze

Contradiction may be implied. Here's Wikipedia's opening statement:
'In traditional logic, a contradiction consists of a logical incompatibility or incongruity between two or more propositions."

The problem is that you refuse to recognize that an arrangement of points on a plane, logically implies order, therefore "an arrangement of points on a plane without order" is contradictory.

This was in reference to my question, Why don't you treat math like chess, and accept it on its own terms? — fishfry

Don't you see that I said math is not like chess. Therefore I do not treat math like chess. I answered your question.

Then your complaint is with the physicists, engineers, and others; and not the mathematicians, who frankly are harmless. — fishfry

Obviously not, as you've already noticed,

Again, your complaint is with those mis-applying math or applying math to bad ends. — fishfry

No, my complaint is with the fundamental principles of mathematicians, As explained already to you, violation of the law of identity, contradiction, and falsity. You, and Tones alike (please excuse me Tones, but I love to mention you, and see your response. Still counting?), are simply in denial of these logical fallacies existing in the fundamental principles of mathematics, and you say truth and falsity is irrelevant to the pure mathematicians.

Particles? Dots? What are those? In math, the elements of sets are other sets. There are no particles or dots. Again, you confuse math with physics. — fishfry

In case you forgot, you posted a diagram with dots, intended to represent a plane with an arrangement of points without any order. This is what I argued is contradictory, "an arrangement... without order". And this was representative of our disagreement about the ordering of sets. You insisted that it is possible to have a set in which the elements have no order. You implied that there was some special, magical act of "collection" by which the elements could be collected together, and exist without any order. What you are in denial of, is that if the elements exist, in any way, shape, or form, then they necessarily have order, because that's what existence is, to be endowed with some type of order.

You tell me, just imagine a plane, with points on the plane, without any order, and I tell you I can't imagine such a thing because it's clearly contradictory. If the points are on the plane, then they have order. And you just want to pretend that it has been imagined and proceed into your smoke and mirrors tricks of the mathemajicians. I'm sorry, but I refuse to follow such sophistry.

I can't argue with the fantasies in your head. Set theory is what it is. — fishfry

Why not give it a try? I can argue with the fantasies in your head, demonstrating that they are contradictory. So please explain to me how you think you can have a collection of elements, points, or anything, and that collection has no order. Take this fantasy out of your head and demonstrate the reality of it.

There are no dots. I don't know what dots are. I tried to give you a visual example but perhaps that was yet another rhetorical error. I should just refer you to the axiom of extensionality and be done with it, because in truth that is all there is to the matter. — fishfry

The dots. I believe, were supposed to be a representation of points on a plane. The points on a plane, I believe, were supposed to be a representation of elements in a set. And you were using these representations in an attempt to show me that there is no inherent order within a set. So, are you ready to give it another try? Demonstrate to me how there could be a set with elements, and no order to these elements.

I've explained to you the problem. You describe the set as a sort of unity. And you want to say that the parts which compose this unity have no inherent order. Do you recognize that to be a unity, the parts must be ordered? There is no unity in disordered parts. Or are you going to continue with your denial and refusal to recognize the fundamental flaws of set theory?

First thing I really want to know what are the bad things that you think mathematicians and scientists are going to cause to happen? — TonesInDeepFreeze

Come on TIDF, don't you see that as a ridiculous question? If one could predict the bad things that were going to happen, before they happened, then we could take the necessary measures to ensure that they don't happen. It's like asking me what accident are you going to have today. It's a matter of risk management. If the mathematics employed in any given situation is faulty, the risk is increased. The biggest problem, I think, is the complete denial of the faults, from people like you. This creates a false sense of certainty. That's why it's like religion, you completely submit to the power of the mathematics, with your faith, believing that your omnibenevolent "God", the mathematics would never mislead you.

Suppose the number 2 is not distinct from the numeral '2'. Suppose also that the number 2 is not distinct from the Hebrew numeral for 2. Then both the numeral '2' and the Hebrew numeral for 2 are the same. But they are not. — TonesInDeepFreeze

The symbols are not the same, nor ought they be said to be the same, or to say the same thing. They ought not be said to say the same thing, because different cultures have different ways of looking at the world. Where's the problem with that? If someone translates a passage of philosophy from ancient Greece, we ought not say that the translation says the same thing as the original. Something is always lost in translation. Likewise, we ought not say that the numeral 2 says the same thing as the Hebrew symbol. This would be very clear to you if you would consider all the different numbering systems discussed on this forum, natural, rational, real, cardinals, ordinals, etc.. The same symbol has a different meaning depending on the system. If we do not keep these distinguished, and adhere to the rules of the specific system, we have equivocation.

That's a picture of dots in a disk. It's not an ordering. — TonesInDeepFreeze

If you refuse to acknowledge that there's an order to those dots, then I don't see any point in proceeding with this discussion.

. That means for you to state which dots come before other dots, for each dot. — TonesInDeepFreeze

Order is not necessarily temporal. And, modern physics looks at time as the fourth dimension of space. So if you cannot see order in an arrangement on a two dimensional plane, I don't see any point in discussing "order" with you.

The symbols do represent how many individuals there are. What do you mean by “directly”? — Luke

If you follow what is taught in math, the symbol "2" represents a mathematical object which is called a number. The number represents how many individuals there are.

Do you recognize that the word 'tree' is not a tree? — TonesInDeepFreeze

Of course, the word "tree" might be used as a symbol, to represent a tree.

But you fail to recognize that the word 'two' or the symbol '2' are not the number 2. — TonesInDeepFreeze

You misunderstand. What I am asking is why can't the symbol "2" be used to represent a quantity of two individuals, just like the word "tree" is used to represent a tree? Why must the symbol "2" represent a mathematical object, the number two, and the number two represents a quantity of two individuals? We don't say that the word "tree" represents a conceptual object, tree, and this concept represents the individual tree.

In reality we simply use the word "tree" to represent a tree, and we use the symbol "2" to represent a quantity of two individuals. There is no conceptual object, or mathematical object in between. So if someone states as a premise, that "2" represents a mathematical object, the number two, this would be a false premise.

The number does not represent how many individuals there are.

The number is how many individuals there are. — Luke

Well no, this is not true. The number is how many individuals it is said that there are. The number is supposed to be what the numeral stands for. It is conceptual, and a representation of a particular quantity of individuals. Being universal, we cannot say that it is actually a feature of the individuals involved, but a feature of our description, therefore a representation. That's why the OED defines "number" as "an arithmetical value representing a particular quantity and used in counting and making calculations." If the number is not a representation of how many individuals there are, but actually "how many individuals there are", there would be no possibility of error, or falsity. If I said "there are 2 chairs", and the supposed mathematical object, the number 2 which is said to be signified by the numeral "2" was "how many individuals there are", rather than how many there are said to be, how could I possibly lie?

You said your teacher insisted that "the numeral is not the number" and that you couldn't understand it. But you also said that you had no problem with basic arithmetic. My point was that you must have understood that "the numeral is not the number" in order to do basic arithmetic. — Luke

No, as I explained. The numeral 2 represents how many objects there are. We could also call that symbol the number 2, which represents how many objects there are. There is no need to assume that the number 2 is distinct from the symbol, to do basic arithmetic..

But "1" or "2" are the number of individuals, not the individuals. — Luke

If that were the case, I'd be fine with it, but it's not what I was told. I was told that "1" and "2" are numerals, symbols, and there is also something else, called the numbers 1 and 2. The numbers are distinct from the numerals, as what is represented by the numerals. So, I was told that "1" and "2" are symbols, which represent the numbers 1 and 2, and the number represent how many individuals there are. Why not just say that the symbols "'1" and "2" represent how many individuals there are, directly?

That is one of the best, most risible, evasions of a challenge I've ever read. What is "the order they actually have" as opposed to all the others? Saying that they have the order they "actually" have is not telling us what you contend to be the order nor how other orderings are not the "actual" ordering. You are so transparently evading and obfuscating here. — TonesInDeepFreeze

Fishfry posted the order, it's right here:

What more do you want?

When confronted with the challenge of points in a plane, a reasonable response by you would be "Let me think about that." But instead you reflexively resort to the first specious and evasive reply that comes to you and post it twice with supposed serious intent. That indicates once again your lack of intellectual curiosity, honesty or credibility. — TonesInDeepFreeze

Look, if the dots exist on a plane, they have positions on that plane, and therefore an exact order which is specific to that particular positioning. They do not have any other order, or else they would not be those same dots on that plane. Take a look at that posting of fishfry's and see the order which the dots have, on that plane, and tell me how they could have a different order, or no order at all, and still be those same dots on that same plane.

If you cannot apprehend this simple fact, then tell me what is so difficult for you.

You were presented with points in a plane, without being given a stated particular ordering. — TonesInDeepFreeze

How can you not see that 'points in a plane without a particular ordering' is a blatant contradiction? If the points exist on a plane, then they each have a particular position on that plane, as demonstrated in fishfry's post, and it is impossible that they have no particular order, because the particular order has been posted. Can you grasp this fact?

One could just as well say 'unstated'. — TonesInDeepFreeze

The problem is, that it is stated. It is stated that they exist on a plane. Therefore each point has a position on that plane unique to itself. Not one of these points makes a line, nor occupies a section of the plane, they each have a specific position. Therefore there is necessarily an order to these points, their positions on that plane, according to what is stated. To give them no order you'd have to remove them from their positions on the plane.

Suppose we just assume a multitude of points, without any spatial reference, no dimensions or anything, just points. Then we have the question of what distinguishes one point from another. It is stipulated that there is a multitude of points. If there is no spatial reference, therefore no space separating one point from another, what makes them distinct from one another? How can we assume a multitude of points when we posit no principle whereby one point is distinguished from another point? And if we posit a principle of separation other than space, (suppose one is later in time than another, or something like that), then isn't this a principle of order. it is impossible to posit a multitude of points without implying order.

It isn't I who is evading the issue. All those people who simply assume that it is possible for a multitude of points to exist without any order, are the one's evading the issue, because such a scenario is logically impossible.

It is not the case that there are not orderings. The point though is that there is not a single ordering that is "THE actual ordering". There are many orderings and they are actual even though 'actual' is gratuitious. — TonesInDeepFreeze

Again, look at fishfry's post:
Do you not see that there is an actual order to those dots on the plane? How could there be "many orderings" if to give them a different order would be to change their positions? Then it would no longer be those dots on that plane. And if your intent is to abstract them, remove them from that plane, then they are no longer those dots on that plane. Why is something so simple so difficult for you to understand?

First, of course, is that we may take a collection of dots as given, without stipulating that a particular person placed the dots herself. — TonesInDeepFreeze

OK, but do you agree that something must have caused those dots to be where they are, i.e. given them that order?

Second, let's even suppose that "actual order" is a function of a person placing the dots. Say that Joe places the dots in temporal succession and Val places the dots in a different temporal succession. But that both collection of dots look exactly the same to us. So there's "Joes actual (temporal) order" and "Val's actual (temporal) order", but no one can say which is THE actual order of the collection of dots we are looking at without Joe and Val there to tell us (if they even remember) the different order of placement they used. — TonesInDeepFreeze

I am talking about their spatial ordering, their positioning on the plane, like what is described by a Cartesian system. Do you not apprehend spatial arrangements as order?

Meanwhile, you're not even familiar with the distinction between semantics and syntax and the notion of model theoretic truth. — TonesInDeepFreeze

That's not true. I simply don't accept it as a realistic notion of "truth", and don't want to waste my time discussing it.

In any event, can you please respond to my point about chess? Surely if you learned to play chess, or any other artificial game -- monopoly, bridge, checkers, baseball -- you were willing to simply accept the rules as given, without objecting that they don't have proper referents in the real world or that they make unwarranted philosophical assumptions. If you could see math that way, even temporarily, for sake of discussion, you might learn a little about it. And then your criticisms would have more punch, because they'd be based on knowledge. I wonder if you can respond to this point. Why can't you just treat math like chess? Take it on its own terms and shelve your philosophical objections in favor of the pleasure of the game. — fishfry

I didn't answer, because it's not relevant. Philosophy is not a game in which you either accept the rules of play or you don't,, neither is theoretical physics such a game, nor is what you call "pure mathematics" (or as close to "pure" as is possible). In these fields we determine, and create rules which are deemed applicable. So your analogy is not relevant, because the issue here is not a matter of "will you follow the rules or not", it's a matter of making up the rules. And there's no point to arguing that people must follow rules in the act of making up rules because this is circular, and does not account for how rules come into existence in the first place.

It makes no sense to anyone else either. This is well known. Especially in terms of quantum fields being "probability waves." That makes no sense to me. Physics has perhaps lost its way. Many argue so. You and I might well be in agreement on this. — fishfry

Ok, we've found a point of agreement, physics has lost it's way. Do you ever think that there must be a reason for this? And, since physics is firmly based in mathematics, don't you see the implication, that perhaps the root of the problem is actually that mathematics has lost its way.

Ok. I get that. And I've asked you this many times. You don't want to play the game of math. So then why the energetic objection to it? After all if someone invites me to play Parcheesi and I prefer not to, I don't then go on an anti-Parcheesi crusade to convince the enthusiasts of the game that they are mis-allocating their time on a philosophically wrong pursuit. So there must be more to it than that. With respect to a perfectly harmless pastime like Parcheesi or modern math, one can be for, against, or indifferent. You have explained why you are indifferent; but NOT why you are so vehemently against. — fishfry

Physicists, engineers, and others, applying mathematics in the world have a huge impact on the world in which I live, unlike Parcheesi players. Despite arguments that mathematical objects exist in some realm of eternal truth where they are ineffectual, non-causal, I think it is undeniable, that the mathematical principles which are applied, have an impact on our world. I believe it is inevitable that bad mathematics will have a bad effect.

That people vehemently support and defend fundamental axioms which may or may not be true, refusing to analyze and understand the meaning of these axioms, simply accepting them on faith, and applying them in the conventional way, in new situations, with little or no understanding of the situation, or the axioms, to me is a clear indication that bad results are inevitable.

Makes no sense. It's perfectly clear that you can order a random assemblage of disordered points any way you like, and that no one order is to be preferred over any other. — fishfry

You do not seem to be making any effort to understand this fundamental principle, which is the key to understanding what I am arguing. A group of particles, or dots (we cannot really use "points" here because they are imaginary) existing in a spatial layout, have an order by that very fact that they are existing in a spatial arrangement. Yes, they can be "ordered any way you like", but not without changing the order that they already have. The order which they have is their actual order, whereas all those others are possible orders.

Do you understand and accept this? Or do you dispute it, and know some way to demonstrate how a spatial arrangement of dots or particles could exist without any order? It's one thing to move to imaginary points, and claim to have a specific number of imaginary points, in your mind, which have no spatial arrangement, but once you give them a spatial arrangement you give them order. Even if we just claim "a specific number of points", we need to validate that imaginary number of points without ordering them. This is what Tones and I discussed earlier. How can we count a specific number of points without assigning some sort of order to them? To count them we need to distinguish one from the other by some means or else we do not know which ones have been counted and which have not been counted. So even to have "a specific number of points", imaginary, in your mind, requires that they have an order, or else that specific number cannot be validated.

Well yes, the random number generator I used was actually determined at the moment of the big bang, if one believes in determinism. But you're making a point about randomness, not about the order of the points. You are not persuading me with your claim that a completely random collection of points has an inherent order. — fishfry

Yes, I'm making a point about "randomness" because you are using the term "random" to justify your claim that a bunch of dots in a spatial arrangement could have no order. You simply say, the points are "randomly distributed" and you think that just because you say "randomly", this means that there actually could be existing dots in a spatial assemblage, without any order. But your use of the term does not support your claim. There was a process which placed the dots where they are, therefore they were ordered by that process, regardless of whether you call that process "random" or not.

You don't want to read the Wiki piece on order theory. — fishfry

I looked at the Wikipedia entry, and it does not appear to cover the issue of whether existing things necessarily have an order or not. So it seems to provide nothing which bears on the point which I am trying to get you to understand.

Actually it doesn't make initial sense. Moving from one letter to the next is always a whole step, except from B to C and from E to F. And then double flats move you down a letter except from C to Cbb and from F to Fbb, and double sharps move you up a step except from B to B## and from E to E##. — TonesInDeepFreeze

All this, what you say, comes later, it's not "initial". What is "initial" is that you learn a specific fingering, and it sounds good, therefore it makes sense. The part that doesn't make sense is when you move deeper into the theory. This is just like mathematics. The initial practice makes sense, learning addition, multiplication, pi, Pythagorean theorem, etc.. All these simple procedures make perfect sense, you learn a procedure, apply it, and it works. However, then there is layers of theory piled on after the fact, and this is where the sense gets lost, because the theory doesn't necessarily follow what is actually the case.

You could see the quantity of objects but not the number of objects? — Luke

Right, I don't look at two chairs and see the number 2 there.

You must have already understood that the number is not the numeral in order to do simple arithmetic. Otherwise, the addition of any two numbers (i.e. numerals) would always equal 2 (numerals). — Luke

No, the numeral represents a quantity, and a quantity must consist of particulars, or individual things. So "2"" represents a quantity, or number of individuals, two, and "1" represents a quantity of one individual. What is added or multiplied is the quantity or number of individuals. The number is of the individual, a predication, and what is added or subtracted is the individuals, not the number. Have two individuals, add two more individuals, and you have four individuals. See, the operation is a manipulation of individuals, not a manipulation of some imaginary "numbers". And, the fact that we can make the individuals imaginary, such that the manipulation of individuals involves imaginary individuals, does not change the reality that the individuals are what is manipulated, not the numbers.

I didn't need a teacher to make me aware that numerals are not numbers. '2' and 'two' refer to the same thing. But '2' is not 'two'. So whatever they refer to is something else, which is a number, which is an abstraction. Rather than be a benighted bloviating ignoramus (such as you), I could see that thought uses concepts and abstraction and our explanations, reasoning and knowledge are not limited to always merely pointing at physical objects. — TonesInDeepFreeze

When I was seven years old I had no idea what an abstraction is, or what a concept is. I didn't understand this until much later when I studied philosophy. This is why mathematics really is like religion. We are required just to accept the rules, on faith, follow and obey, without any real understanding. 'There really is a number two there, accept and obey'. 'There really is a God there, accept and obey'.

That about sums it up. Math is like religion, a whole bunch of bullshit which we are told to accept on faith.

Sorry Tones, but we're so far apart on these principles of truth and falsity, that I see no place to start, or any point to it. I look at truth as corresponding with reality.

It's like saying that learning to play a musical instrument is tremendously difficult at first so people should just give up. — fishfry

I don't think the analogy is good. I learned to play a musical instrument, and it always made sense to me, right from the start. I learned philosophy and it always made sense to me. The point is that I proceeded because it made sense to. If mathematics requires self-deception, then this does not make sense to me, and so I will not proceed. Many people would not become athletes because there is much physical pain involved in the practice. This might be similar to my refusal to learn math because physical pain, and self-deception may both be viewed as harmful. Some people though will put up with the physical pain because they see a greater good in being an athlete. Maybe you put up with the self-deception of mathematics because you apprehend a greater good.

It's true of virtually EVERYTHING that at first, the subject makes no sense. You just do as you're told, do the exercises, do the homework, do the problem sets without comprehension, till one day you wake up and realize you've learned something. It must be that you've learned nothing at all in your life, having given up the moment something doesn't make immediate sense to you. — fishfry

Again I don't agree with this. Many things I've learned made sense to me right from the start. Even learning the numerals, how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense.

I had a similar experience later with physics. We learned basic physics, then we learned about waves, and got to experiment in wave tanks. We learned that waves were an activity within a medium and we were shown through diagrams how the particles of the medium moved to formulate such an activity. All of this made very much sense to me. Then we were shown empirical proof that light existed as waves, and we were told that light waves had no medium. Of course this made no sense to me.

When you learned to play chess, or any game -- bridge, poker, whist -- do you say, "Oh this is nonsense, no knight REALLY moves this way," and quit? Why can't you learn a formal game on its own terms? If for no other reason than to be able to criticize it from a base of knowledge rather than ignorance? If you've never seen a baseball game, it makes no sense. As you watch, especially if you are lucky enough to have a companion who is willing to teach you the fine points of the game, you develop appreciation. Is that not the human activity called LEARNING? Why are you morally opposed to it? — fishfry

When I learn a game, I must learn the rules before I play. If the rules are such that I have no desire to play the game, I do not play. It's not a question of whether the game makes sense or not, so the analogy is not a good one.

Finally, even your basic objection to unordered sets is wrong. Imagine a bunch (infinitely many, even) of points randomly distributed on the plane or in 3-space. Can't you see that there is no inherent order? Then you come by and say, "Order them left to right, top to bottom." Or, "Order them by distance from the origin, and break ties by flipping a coin." Or, "Call this one 1, call this one 2, etc." — fishfry

Finally, you decided to address the issue. If there are points distributed on a plane, or 3d space, the positioning of those points relative to each other is describable, therefore there is an inherent order to them. If there was no order their positioning relative to each other could not be described..

You say that they are "randomly distributed", to create the illusion that there is no order. But the fact is that they must have been distributed by some activity, and their positioning posterior to that activity is a reflection of that activity, therefore their positioning is necessarily ordered, by that activity.

So just adopt the formalist perspective. There are only numerals and the rules for manipulating them. It's a game. What on earth is your objection? Were you like this when you learned to play chess? "There is no knight!" "The Queen has her hands full with Harry and that witch Meghan!" etc. Surely you're not like this all the time, are you? — fishfry

If you think you can interpret the rules as we go, then I'd advise you not to play any games with me.

What is the inherent order of the points in this set? Can you see that the points are inherently disordered or unordered, and that we may impose order on them arbitrarily in many different ways? Pick one and call it the first. Pick another and call it the second. Etc. What's wrong with that? — fishfry

The inherent order is the exact spatial positioning shown in the diagram. If any point changes location, then the order is broken. Is that so difficult to understand? A spatial ordering is not a matter of first and second, that is a temporal ordering.

Is it possible for an axiom to be false? Please explain. Don't refer to inconsistency. :roll: — jgill

For sure it's possible, the difficulty would be to demonstrate falsity, and this would require reference to some sort of inconsistency. What else could demonstrate falsity other than a reference to some form of inconsistency?.

An axiom is expressed as a bunch of symbols, so it must be interpreted. Interpretation requires that it be related to something else, and here we can have inconsistency and contradiction. So the author of an axiom will intentionally avoid internal inconsistency, or contradiction, but to understand, or employ the axiom it must be related to something external to it. If in interpretation, there is a contradiction with another principle then one or both must be false. If the other is a principle one holds to be true, then the axiom must be viewed as false.

Take the axiom of extensionality for example. Here's how Wikipedia states it:
".Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.
(It is not really essential that X here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)"

Further, Wikipedia says it is interpreted like this:
" To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members."

Notice there is an exchange of "equal" and "same". As I've argued in other threads, if we adhere to the law of identity, this is a false use of "same". To resolve this issue, one might deny the law of identity, or insist on a faulty interpretation of that law. I think that approach is futile, so we must look directly at the axiom of extensionality and see what "equal" means in that context. If we can interpret in a way which does not employ "same" we might avoid the falsity.

Which axioms of finite set theory do you think are false? — TonesInDeepFreeze

Read above.

I think you are just not cut out for mathematical abstraction and should pick another major. — fishfry

I found that out at about tenth grade, despite living in an extremely mathematically inclined family. Prior to that though, I had difficulty even in grade school, when the teachers insisted on distinguishing numbers from numerals. Where are these "numbers" that the teacher kept trying to tell us about, I thought. All I could see is the numerals, and the quantity of objects referred to by the numeral. But the teacher insisted no, the numeral is not the number. So it took me very long to figure out that the numeral was not the "number" which the teacher was talking about, and that the number was just some fictitious thing existing in the teacher's mind, so I shouldn't even bother looking for it because I have to make up that fiction in my own mind, for there to be a number for me to "see". Of course, I chose philosophy as a major instead, because philosophy provides a solid, grounded understanding of abstraction, rather than simply insisting on the existence of fictitious "numbers" existing in people's minds, and trying to convince people to create those fictitious things in their minds.

Enough. You win. You wore me out. — fishfry

Simply put, I'm right and you're wrong. Nah, nah, let's go back to grade school. You should have chosen philosophy instead of math, if you wanted to learn the truth about abstraction.

Pick another major. — fishfry

I have picked another major, philosophy. That's why I'm discussing this in a philosophy forum. Care to join me? Or will you simply assert that mathematics is far superior to philosophy, then run off and hide under some numbers somewhere when the unintelligibility of your principles is demonstrated to you?

You may well have a philosophical point to make, but you are preventing yourself from learning the subject. And it's learning the subject that would allow you to make more substantive rather than naive and obfuscatory objections. — fishfry

You seem to have a very naive outlook. How do you propose that one proceed toward "learning the subject", when the most basic principles in that subject do not make any sense to the person? To me, the distinction between a numeral and a number is fundamentally unintelligible, as a falsity, because it requires producing a fictitious thing in my mind, and then talking about that fictitious thing as if it is a truth. Therefore proceeding into "learning the subject" requires an initial step of dishonesty, self-deception, then deceiving others in talking about this issue I have deceived myself about. I am not prepared to make that step of dishonesty. Making that initial step of self-deception is the first step toward misunderstanding, not toward understanding.

We prove from axioms. — TonesInDeepFreeze

If an axiom is false then the proof is unsound.

You also told us that you assume numbers are objects. — Luke

Notice, the quoted passage says numbers are assumed when "you" count. And, it's your count that I argue is false. .

You are back to your pathetic strawman misinterpretation, for the sake of ridicule.

It's not that sets don't have orderings. It's that sets have many orderings (though in some cases we need a choice axiom or an axiom weaker than choice but still implying linear ordering). So the point is that there is no single ordering that is "the ordering". — TonesInDeepFreeze

If we assume that a set necessarily has an ordering, but it could be one of many possible orderings, by what principle can we say that each of these many possible orderings constitutes the same set? What type of entity is an "element", such that the identity of a unity of numerous elements is based solely in the identity of its parts with complete disregard for the relations between those parts? Isn't this a sort of fallacy of composition?

You assume that numbers are objects but argue that numbers are not objects? Sounds about right given your confusion. — Luke

Do you understand the meaning of the word "if"? I don't think it's me who's the confused one.

By the axiom of extensionality, a set is entirely characterized by its elements, without regard to order. So the set {a,b,c} is the exact same set as {b,c,a} or {c,b,a}. — fishfry

That "elements" may exist without an order is the falsity I've explained to you already. And if we say that "element" indicates an abstraction, then it is a universal, not a particular, and to assume that an abstraction is a particular is a category mistake.

Let's assume a special type of "element", created, or imagined specifically for set theory. This type of element can exist in a multitude without that multitude having any order. Each of these elements would have no spatial or temporal relation to any other element, or else there would be an order, according to that relation. We could say that they are like points, but without a spatial reference, so that we cannot draw lines between them etc., because there is no order to them. But if they were like points, without spatial relations constituting order, there would be no way to distinguish one from another

Unlike points though, there is something which distinguishes one element from another, so that in the set of (a,b,c,), "a" does not represent the same thing as "b" does. Can I conclude, that the distinct elements are separated from one another, and distinguished one from another, by something other than space? To make them distinct and individual, they must have separation, but the separation cannot be spatial or else they would have an order, by that spatial relation.

Do you see, that from this premise alone, we cannot give any order to any set? To give a set an order would be a violation of the fundamental meaning of "element" which allows that elements can exist as particulars without any spatial temporal; relations. To be able to talk about an order within a set, would require that we transform the elements into something other than "elements", something which could have spatial or temporal relations and therefore an order. Remember, even quantity requires spatial-temporal separation between one and the other, to distinguish separate individuals.

Now we want to layer on the concept of order. To do that, we define a binary relation, which I'll call <, and we list or designate all the true pairs x < y in our set. So for example to designate the order relation a,b,c, we would take the base set {a,b,c}, and pair it with the set of ordered pairs {a < b, a < c, b < c}. Then the ordered set is designated as the PAIR ({a,b,c,}, {a < b, a < c, b < c}). I hope this is clear. — fishfry

No, sorry, it's not clear at all. You have imagined distinct "elements" which exist without any spatial or temporal relations, thereby having no order, though they are somehow distinct individuals. Now you want to add order. You have already defined order out of the set, to add it in, is blatant contradiction.

The basic takeaway is that a set has no inherent order. We impose an order on a set by PAIRING the set with an order relation. — fishfry

What I need, is a clear explanation of what an "order relation" is. What type of relation are you attempting to give to these elements, which gives them an order, when you've already stipulated the premise that they have no order?

The point is, that to give them existence without order requires a special conceptualization which I described above. Now if we want to proceed with that conceptualization, and now bring in principles of order, we must do so in a consistent way. So, we need to describe what separates one element from another, since it's clearly not space, and what makes it distinct as an element, in terms which do not give it a relationship to the others, to allow that the multitude of them do not already have any order, Then we need a principle by which order can be initiated within this non-ordered type of separation.

A set is a collection of elements, regarded as an individual thing, a set. — fishfry

Clearly, for a group of things to be regarded as an individual thing, "unity" is implied. And, it is quite clear that for a group of parts to form a unity, it is necessary that the parts exist with some sort of order. So this statement directly contradicts you assertion that a set has no inherent order.

Perhaps it's the distinction between a bunch of athletes and a team, or a collection of birds and a flock. I'm sure some philosophers have found ways to describe this. A set is a collection of elements, along with the concept of their set-hood. That's the best I can do! — fishfry

Do you see the contradiction? You describe a "set" as a thing, a unity, like a team, but then you say that there is no inherent order to this unified thing. Do you see how ridiculous this is, to say that there exists a unified thing, composed of parts, but there is no order to the parts? How in the world are we supposed to conceive of a unity of parts which have no order? To say that they are a unity is to say that they have order.

A set is inherently without order and without any kind of structure. — fishfry

Take a look again. You are proposing a type of unity, a "set", without any structure. By what principle do you say that it is a collection?

A set is entirely characterized by its elements; but a set is more than just its elements. It's the elements along with the collecting of the objects into a set. — fishfry

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? I'm trying to figure out how you get around the need for the elements to have an order. I mean, it's one thing to assert, "I've got this collection of elements and they have no inherent order" (to which I'd say you're lying), and another thing to demonstrate how you've collected a group of elements into a unified whole, without them having any order.

I welcome you to provide a non-circular reason for why "determining a quantity" is (true) counting and why "reciting the natural numbers in ascending order" is not (true) counting. — Luke

I explained this already. Your "ascending order" is based on quantity, therefore your supposed "count" of ascending order means nothing unless it is determining a quantity. This is why "numbers" as objects are assumed, so that when you count up to ten you have counted ten objects, (numbers).

Without this assumption that the symbol represents a quantity, you have a meaningless order of symbols which cannot be said to be "ascending". But if you allow that the symbol represents a quantity, then you have an ascending order. However, if the symbol does represent a quantity, then there must be objects which are counted, to validate the use of the symbols. Therefore, it is proposed that numbers are the objects which are counted, to validate the fact that the symbol must represent a quantity.

Therefore your "reciting the natural numbers in ascending order" is nothing but an act of determining a quantity of numbers. And, if numbers are not true objects, as I argue is the case, then this is not a true act of counting at all.

To determine a quantity is equally to make reference to an ascending order. — Luke

This is not true, as I argued with TIDF earlier in the thread. There are many ways to determine a quantity without referencing an ascending order.

In a logical proceeding, it is imperative that the symbol employed maintains the same meaning, to avoid the fallacy of equivocation. If "beating" means something different when used to describe beating eggs, from what it means when used to describe beating drums, and we proceed with a logic process, there could be a fallacious conclusion. For example, after the eggs are beaten, the internal parts are all mixed up into a new order, therefore if I beat the drums the internal parts will become all mixed up into a new order.

You might argue that “counting” in the sense of reciting the natural numbers in ascending order is not the proper meaning of the word, but why is it not? Why is “counting” in the sense of determining a quantity the only proper meaning of the word? These are both counting. — Luke

I explained this already, I think more than once. It is my opinion that there is no such thing as numbers which serve as a medium between the numeral (symbol) and its meaning, or what it represents. So this sense of "counting", which you describe, or define, as "reciting the natural numbers in ascending order", has a false description, or definition. This false definition would act as an unsound premise.

When we "count" in that sense, we are making an expression of symbols. As fishfry has explained, there is no inherent order to those symbols. To say that a particular order is "ascending order", is simply to make a reference to quantity. Therefore the meaning of that sense off "counting" is derived from, or based in that other sense, which is determining a quantity. So determining a quantity is the primary, and proper sense. If we remove that reference to quantity then there is no basis for any specific order, and you cannot say that "counting" involves an "ascending order", because "ascending" is not justified.

What I've been trying to tell fishfry, is that there is better sense of "order" which is not based in quantity, but it is temporal. If we refer to a temporal order, then we need some reason other than quantity to support any proposed order, showing why one symbol ought to be prior in time to another, when we "count" in the sense of expressing an order. This reason for ordering in this manner would provided the alternative name.

I think you need to reread my post. I have no desire to respond to your misinterpretation.


Thanks for the clarification fishfry, but here's a couple more things still to clear up.

To me, the following statements contradict each other.

But the set of natural number may nonetheless be ordered in many alternative ways. — fishfry

But no set has order. That's the axiom of extensionality. Will you kindly engage with this point? — fishfry

Which is the case, no set has order, or a set may be ordered in many different ways. Do you apprehend the contradiction? Which is it, ordered in different ways, or not ordered?

Let me go back to my question from the last post. What exactly constitutes "the set"? Is it the description, or is it the elements which are the members of the set. If it is the description, or definition, then order is excluded by the definition. But if the set is the actual participants, then as I explained already they cannot exist without having an order. If the supposed participants have no existence then they cannot constitute the set.

That's why I ask, which is it? Can a set be ordered, or is it inherently without order? Surely it cannot be both.

So I can use the phrase mathematical, but not mathematical objects? But mathematical is an adjective and mathematical object is a noun. You've still not answered the question.

But are you saying that if I call 5 a "mathematical concept" you're ok with that, but NOT with my calling it a mathematical object? Ok, I can almost live with that. Although to me, it's a mathematical object. — fishfry

Let's look at "concept" as a noun, as if a concept is a thing. Do you agree that a concept is the product, or result of conception, which is a mental activity? There's different mental activity involved, understanding, judgement, conclusion, and effort to remember. Would you agree that the effort to remember is what maintains the concept as a static thing, So if a "concept" is used as a noun, and is said to be a thing, it is in the same sense that a memory is said to be a thing. Would you agree that if a mathematical concept is "a thing", it is a thing in the same sense that a memory is a thing?

I propose instead that we reserve the term "counting" for counting the natural numbers and counting imaginary things, and that we should use the term "measuring" (instead of "counting") for "determining a quantity". — Luke

Now you just have a vicious circle. What does the numeral "2" refer to? The imaginary object which is the number 2. What is the number 2? The imaginary object which the numeral "2" refers to. See, vicious circle.

If you give the number 2 meaning, a definition, to validate its existence as a conceptual object, you might say that it means a quantity of two, but then you justify my argument, that counting is counting a quantity of objects, and "2" refers to two objects, not one object, the number 2. If you give the number 2 meaning by saying that it is the number which comes after 1, then you justify my argument that what you are doing is expressing an order, rather than counting. Either way, you'd be validating what I called justification, and you refused to acknowledge as justification. Or, would you like to give the number 2 some other type of definition, to validate its existence as a conceptual object? Prefer just remain within your vicious circle?

Furthermore, you ought to see that there is no need to assume "an object", or "number", as the intermediary between the sign "2", and its definition. When we say "square" there is no need to assume a conceptual object which is a square, as an intermediary between the word "square", and its definition, "equilateral rectangle".

But now you are saying that space and time have "conceptual" meaning; at the same time you deny that 5 or other numbers can have conceptual meaning. — fishfry

I surely have not denied that "5" has conceptual meaning. To say that the numeral "5", when it is properly used, must refer to five distinct particular things, is to give it conceptual meaning. It is a universal statement, therefore conceptual. I am not saying that it must refer to one specific group of five, as a name of that group, I am saying that it could refer to any group of five, therefore it is a universal, and this indicates that the "5" in my usage refers to a concept, what you've called an abstraction, rather than any particular group of five.

For example, if I said that to properly use "square", it must refer to an equilateral rectangle, or "circle" must refer to a plane round figure with a circumference which has each point equidistant from its center point, I give these terms conceptual meaning, because I do not say that the words must refer to a particular figure, I allow them to refer to a class or category of figures.

Even if I said that "5" must refer only to one particular group of five, or that "square" must refer only to one particular figure, it could still be argued that this is "conceptual meaning", because to understand this phrase "must refer only to one particular", is to understand something conceptual. In reality any meaning assigned to word usage is conceptual, so this position you've thrust at me, that I deny the conceptual meaning of 5, is nonsense. What I say is that the conceptual meaning given to "5", in some situations, namely that it refers to a type of object called a number (as described by platonic realism), ought to be considered as wrong. Do you accept the fact that concepts can be wrong? For instance, your example of "justice". A group of people could have a wrong idea about what "justice" means. Likewise, a group of people could have a wrong idea about what "5" means.

How about "inspired by" rather than grounded? As in Moby Dick being a work of fiction nevertheless inspired by a real historical event. Of course we get our concept of number from real, physical things. Nobody's denying that. — fishfry

Why would you want to make this change to "inspired" rather than "grounded"? Logic is grounded in true premises, and this is an important aspect of soundness. If your desire is to remove that requirement, and insist that the axioms of mathematics need not be true, they need only to be "inspired", like a work of fiction, the result would be unsound mathematics. Sure this unsound mathematics might be fun to play with for these people whom you call "pure mathematicians", and I call "mathemagicians", but unsound mathematics can't be said to provide acceptable principles for a discipline like physics.

Well the "first" element of a total order is an element that is less than any other element. Some orders have a first element, such as 1 in the positive integers. Some orders don't. There's no first positive rational number.

That's what first means. — fishfry

OK, I assume that "less than" refers to quantity. So we're right back to my original argument then. Numerals like "1", "2", "3", "4", refer to a quantity of objects, "3' indicating a quantity which is less than that indicated by "4", and "first" indicates a lower quantity. How do you propose to remove the quantitative reference to produce a pure order, not grounded in a physical quantity?

Now that's funny, as we got off onto this conversation by pointing out to you that numbers can indicate order as well as quantity. But of course ordinals are different than cardinals. Two distinct ordinals can have the same cardinal. — fishfry

If you are grounding your definition of "order" in "less than", as you have, then numbers simply indicate quantity, and your "order" is just implied. It is not the case that "2" indicates "first" in relation to "3" and "4", it is the case that "2" indicates a quantity which is less than the quantity indicated by "3" and by "4". And by your premise, that the "first "is the one which is less than the others, you conclude that "2" is first.

Therefore "order" as you have presented it is not indicated by the numbers, only quantity is indicated by the numbers. Order is indicated by something other than the numbers, it's indicated by your premise that the numeral signifying a quantity less than the others, is first.

red, blue, green. Three words ordered by length. There is no time involved. You are stuck on this point through stubborness, not rational discourse. The player who finishes first in a golf tournament is the one with the lowest score, NOT the one who races around the course first. — fishfry

You clearly haven't followed what I've been saying, and I realize that I did not make myself clear at all. The point is that if we remove the reference to a quantity of individual objects, from numerals, then the ordering of numbers requires a spatial or temporal reference. You seemed to believe that we could remove the quantitative reference, and have numbers with their meanings understood in reference to order only. Clearly, "less than" does not provide this for us. And your example of the length of the word here, is a spatial reference.

Your other example, of the best score being first is only made relevant through a quantitative interpretation. How is 3 better than 4? Because it's less than. So you have not removed the reference to quantity as the necessary aspect of numerals, to provide a purely ordinal definition. Therefore I am still waiting for you to prove your claims.

I have already given many counterexamples such as rationals, reals, complex numbers, p-adics, hyperreals, and various other exotic classes of numbers studied by mathematicians. What quantity or order does 3+5i3+5i represent?

There is no general definition of number in math. That's kind of a curiosity, and it's kind of an interesting philosophical point, and it's also factually true. — fishfry

You have shown me absolutely nothing in the sense of a number not dependent on quantity for its meaning.

I've made my point and all you have is mathematical ignorance. — fishfry

If your point is that "order" is defined by " less than", and this is supposed to be an order which is independent from quantity, then you've failed miserably at making your point.

You haven't seen them in the playground at recess. Of course that's only when I was a kid. These days I gather they don't let the kids run around randomly at recess. — fishfry

Obviously, "in the playground" is not "in the classroom", and you're clutching at straws in defense of a lost cause.

If you don't know that sets have no inherent order, there is no point in my arguing with your willful mathematical ignorance. — fishfry

Instead of addressing my argument you portray me as mathematically ignorant. It's not a matter of ignorance on my part, it's a refusal to accept a mathematical axiom which is clearly false. So I'd correct this to say that this is an instance of your denial, and willful ignorance of the truth, for the sake of supporting a false mathematical axiom.

No that is not true. It's entirely contrary to the concept of set. A set has no inherent order. An order is a binary relation that's imposed on a given set. If I have a set and don't bother to supply an order relation, then the set has no order. Sets inherently have no order. That's what a set is. You can sit here all day long and make up your own definitions, but that's of no use or interest to anyone. — fishfry

Show me that set which has no order then. And remember, there is a difference between a thing itself, and the description of a thing. Therefore to describe a set which has no order is not to show me a set which has no order.

I think you need to make clear what "set" means. Does it refer to a group of things, or does it refer to the category which those things are classed into? The two are completely different. Take your example of "schoolkids" for instance. Does "set" refer to the actual kids, in which case there is necessarily an order which they are in, even if they are running around and changing their order? Or, does "set" refer to the concept, the category "schoolkids", in which case there are no particular individuals being referred to, and no necessary order? Which is it that "set" refers to, the particulars or the universal? Or is "set" just a clusterfuck, a massive category mistake?

I'm asking you, if you don't accept the phrase mathematical object, what phrase do you use to name or label conceptual entities that are mathematical, as opposed to conceptual entities like justice that are not mathematical? — fishfry

You just named it for me. "Mathematical" is the word I use to refer to mathematical concepts. In ethics there are ethical concepts like justice, in biology there are biological concepts like evolution, and in physics there are physical concepts like mass. Why do you think mathematics ought to be afforded the luxury of treating their concepts as if they are objects?

You have attempted to argue that counting natural numbers, or counting imaginary things, is not true counting, and that to call this "counting" is a misnomer. — Luke

Right, and the reason why I argued this is that we ought not have two distinct activities going by the same name in a rigorous logical system, because equivocation is inevitable. So, one ought to be called "counting" and the other something else. I propose the obvious, for the other, expressing an order.

And, I provided all the required justification. You just do not accept it. So in your mind it has not been justified. That's the nature of justification, regardless of how sound the argument is, if it is not accepted the proposal does not qualify as "justified".

I however, expect nothing less from you. This is consistent with your previous behaviour. No matter what explanation I provide, as to why specific words ought to be restricted in certain ways, to enhance the epistemic capacity of a logical system, you'll reject it. It's quite clear to me that you reject these proposals because they would incapacitate your principal means of argumentation, which is equivocation.

But what is the justification for your stipulation that counting natural numbers is not real counting or that real counting must involve "determining a quantity"? — Luke

The point is to avoid equivocation which is a logical fallacy. Since one sense of "counting" involves counting real things, then why not call this "real counting"?

If a flame be a dumpster fire. — TonesInDeepFreeze

That's me, the dumpster arsonist. Easiest way to dispose of garbage is to burn it. Not so good for the environment though. But neither is garbage.

Everyone else considers "counting up to ten" to be counting (you also called it "counting", by the way). — Luke

Yes, i call it "counting", but the point is that there's two very distinct senses of "counting" and to avoid ambiguity and equivocation we ought to have two distinct names for the activity, like fishfry explained with the distinct names for the numerals used, cardinals and ordinals.

Why should we care about your unjustified stipulation that counting the natural numbers is not real counting or that real counting must involve "determining a quantity"? — Luke

Don't mathematicians and other logicians harbour a goal of of maintaining validity, and avoiding fallacies such as equivocation? If it is the case, that when a person expresses the order of numerals, one to ten, and the person calls this "counting", it is interpreted that the person has counted a quantity of objects, a bunch of numbers, rather than having expressed an ordering of numerals, then the interpretation is fallacious due to equivocation between the distinct meanings of "counting".

The issue which fishfry and I have now approached is the idea of a set without any order. I have argued that this is a contradictory idea because if the set exists as a set, its members must have existence in the order which they have in the existing set. It is only by removing existence from the set that we can say the members of the set have any possible order. But then the set itself is not an actual set, it's just the possibility of a set. This would be like a definition without the necessity of anything fulfilling that definition. We could say it's an imaginary set, whereas a real set has real existing members and therefore a real existing order.

In the case of mathematics the question becomes what is supposed to be in the set, the symbols (numerals) or what the symbols represent (numbers). If it is the latter, then the set can be defined with the symbols, and the members within the set, being imaginary, have no existence, and therefore can be said to have no order, or any possible order. But such a set is necessarily non-existent and imaginary, and it cannot be used to represent any real things in the physical world, because real things have an order.

So we have a distinction to be made between two different uses of "set". We can refer to a group of existing objects which necessarily have an order, as a "set". And this type of set is "countable" in the sense that we can determine the quantity of objects within the set. And we can also can use "set" to refer to an imaginary group of objects, having no order because they have no existence. But this type of "set" is not "countable" in the sense that we cannot determine the quantity of objects within such a set. In other words, any set which is stated as having no order, but only possible orders, ought to be considered as imaginary and therefore of indeterminate quantity.

What unit of measurement is required for counting the natural numbers? Metres? Litres? Hours? Bananas? Obviously, no unit of measurement is required. You can count to ten without having to determine any unit of measurement. Therefore, counting is independent of measuring. Counting is not a "form of" measuring. — Luke

"Counting the natural numbers", as described here, is a matter of established an order. It is not an instance of counting in the sense of determining a quantity. There are no objects (numbers) being counted.

You are equivocating between these two senses of "counting". To count, in the sense of determining a quantity, is an act of measuring. To "count" in the sense of counting up to ten, is a case of expressing an order, two comes after one, three comes after two, etc.. To call this "counting the natural numbers" is a misnomer because this is nothing being counted, no quantity being determined. That is why we can theoretically "count the natural numbers" infinitely, without end, because we are just stating an order, not determining a quantity.

Perhaps you're right that meaning isn't the correct word. If I said we remove a concept from its worldly or physical referent, would that be better? We care about first, second, third, and not first base, second base, third base. So how would you describe that? I'm focusing on ordinality itself and not the things ordered. So you're right, meaning was an imprecise word. — fishfry

Let's get this straight. I am not talking physical referents here. I am talking space and time, which are conceptual. The issue is that when we remove the physical referents (required for "counting" in the sense of determining a quantity, as the things counted), for the sake of what you might call purely abstract numbers, the meaning of the numbers is grounded in the abstract concepts of space and time. Numbers no longer refer to physical objects being counted, they refer to these abstract concepts of space and time.

Now, we have only deferred the need to refer to physical existence, because if our conceptions of space and time are inaccurate, and the ordering of our numbers is based in these conception of space and time, then our ordering of the numbers will be faulty as well. You seem to think that in pure mathematics, a logician is free to establish whatever one wants as "an order", but this is not true, because the logician is bound by the precepts of "logic" in order that the order be logical. For example, a self-contradicting premise is not allowed. So there are fundamental rules as to the criteria for "order" which cannot be broken. And even if you argue that the order could be a completely random ordering of numbers, the rule here is that each thing in the order must be a number. And every time a logician tries to escape the rule, by establishing a principle allowing oneself to go outside that rule, there must be a new rule created, or else the logician goes outside the field of logic. And the point, is that if the rule is not grounded in empirical fact (physical existence) the logic produced is faulty, and the proposed rule ought to be rejected as a false premise.

There is no temporal reference. — fishfry

Surely, "first" does not mean "highest quality", or "best", in mathematics, so if it's not a temporal reference, what is it?

Ok. I agree that I'm having trouble precisely defining abstraction and I sort of see your point. But ordinal numbers are purely about order, but they're not about any particular things being ordered. How would you describe that? It's not meaningless, yet it refers to nothing in the world at all other than the pure concept of order. Which you don't seem to believe in. — fishfry

Yes that is my point as to how counting order is different from counting a quantity. To count a quantity requires particular things, but to count an order requires only time. However, time is something in the world, and that's why I don't believe in what you call "the pure concept of order".

But order is not essential to numbers, it's imposed afterward. — fishfry

If order is not essential to numbers, then something else must be, because to be a concept is to be definable according to essential properties. I propose, then that quantity is essential to numbers. Do you agree? If for example you make an order, or a category, of odd numbers, or even numbers, or prime numbers, it is something about the quantity represented by the number which makes it belong in one or more of these categories. If it's not quantity which is essential to numbers, as the defining feature of "number" then what do you think is? You've already ruled out order.

. I get that you are drawing a distinction between the mathematical formalism, in which order is secondary to the existence of numbers; and philosophy, in which order is an essential aspect of numbers. — fishfry

No, I am saying that if order is secondary to the existence of numbers, then quantity must be primary.

A schoolkid must have a height, but it could be any height. — fishfry

That's not true at all, it's the fallacy I referred to. The schoolkid must have height, and that height must be the height that the schoolkid has. Therefore it is impossible that the schoolkid has a height other than the height that the schoolkid has, and very obviously impossible that "it could be any height". To make such a claim is clearly fallacious, in violation of the law of identity, because you are implying that a thing could have properties other than those that it has, saying it could have any property. Obviously this is not true because a thing can only have the properties that it has, otherwise it is not the thing that it is.

You see it that way. I see it as providing beautifully logical clarity. We have the set of natural numbers, and we have the standard order and we have a lot of other orders, and we can even consider the entire collection of all possible orders, which itself turns out to be a very interesting mathematical object. It's quite a lovely intellectual structure. I'm sorry it gives you such distress. — fishfry

It gives me distress to see you describe something so obviously fallacious as "providing beautiful logical clarity". If you consider circumventing the law of identity as beautiful logical clarity, I have pity.

But I have not asserted that a set must have any order at all. The set NN has no inherent order at all. Just like a classroom full of kids has no inherent order till the teacher tells them to line up by height or by alpha firstname or reverse alpha lastname or age or test score or age. Why can't you see that? — fishfry

Again, you're continuing with your fallacy. A classroom full of kids must have an order, or else the kids have no spatial positions in the classroom. Clearly though, they are within the classroom, and whatever position they are in is the order which they have. To deny that they have an order is to deny that they have spatial existence within the room, but that contradicts your premise "a classroom full of kids".

A contradiction is a proposition P such that both P and not-P may be proven from the axioms. Perhaps you would CLEALY state some proposition whose assertion and negation are provable from the concept of order as I've presented it. I don't think you can. — fishfry

Above, is your CLEAR example of contradiction "a classroom full of kids has no inherent order". By saying "there is a classroom full of kids", you are saying that there is an order to these kids, they exist with determinate positions, in a defined space. You contradict this by saying they have no inherent order.

Absolutely agreed. Yes. The essence of a set of numbers is NOT in their order, since we can easily impose many different orders on the same underlying set. Just as the ordering by height is not essential to the classroom of kids, since we can impose a different order; or by letting them loose in the playground at recess, we can remove all semblance of order! Surely you must take this point. — fishfry

So, if "a set" is like the kids in the classroom, then it must have an order to exist as a set. We can say that the order is accidental, it is not an essential feature, so that the same set could change from one order to another, just like the kids in the class, and still maintain its status as the same set. However, we cannot say that a set could have any order by reason of the fallacy described above, because this is to say that it has no actual order which implies that it does not exist.

Ok. But that's not good enough. I asked how do you call mathematical objects like topological spaces. But justice and property are concepts and abstractions, yet they are not mathematical objects.

If you don't like the phrase, "mathematical object," what do you call them? Sure they're an abstraction, but that's way too general. You see that I'm sure. — fishfry

No, I don't see that at all. They are all concepts, ideas. By what principle do you say that mathematical concepts are "objects", but concepts like "justice" are not objects. I mean where is your criteria as to what constitutes a conceptual "object". I know it's not the law of identity.

An object is not a goal. An (American) football is an object, and the goal is to get it across the goal line. You would not say the football is a goal. I think you're way off the mark with your claim that an object is a goal or objective. 5 has no object or purpose. It's just the number 5. A mathematical object. An abstract object, as all mathematical objects are. — fishfry

You've never heard "the object of the game"?

No, not in the least. How can you say that? That's not even the meaning of the words in everyday speech in the real world. The winner takes first place and the runner up takes second place sometimes (as in a foot race) but not always (as in a weight lifting contest) by being temporally first. You must know this, why are you using such a weak argument? First place in golf goes to the player with the lowest score, not to the player who finishes the course first. This is a TERRIBLE argument you're making here. — fishfry

So in this context, "first" means best. Clearly this is not how "first" is used in mathematics. In mathematics, "first" has a temporal reference of prior to, as I said, not a qualitative reference as "best". Your attempt at equivocation is not very good, I'm happy to say, for your sake. Ask Luke who is the master of equivocation for guidance, if you want to learn. I think you ought to stay away from that though.

Math just has the number 5. — fishfry

.The problem obviously, is that you, and mathematicians in general, according to what you said above, haven't got a clue as to what a number is. It's just an imaginary thing which you claim is an object. It appears like you can't even tell me how to distinguish the number 4 from the number 5, because you refuse to recognize the importance of quantity. And if you would recognize that it is by means of quantity that we distinguish 4 from 5, then you would see that "4", and "5" cannot each represent an object, because one represents four objects, and the other five objects. Why do you take numbers for granted?

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