You didn't get the molecule analogy so I went back to the language one, I believe I used it earlier. Now you claim not to get the language one, but that appeared to me like an intentional misinterpretation. You pretended as if you didn't understand that apprehending language is understanding meaning.

Each is an example of sensing something without apprehending what is being shown by the thing being sensed.

You think it must be "hidden", if we sense something without understanding it, but I think that idea is what's misleading you. It's not at all hidden, the mind is just lacking in the capacity to understand what is being sensed. Thinking it is "hidden" is a feature of your accusative nature. When you can't understand a person you blame the other, instead of introspecting your own capacity. And if you cannot see what is right in front of you, you think it must be "hidden" from you, instead of considering the possibility that your eyes are actually sensing it, but your mind is just not apprehending it.,

You said that we sense a foreign language without apprehending it. — Luke

To apprehend the language being spoken, is to understand the meaning. You work very hard to make understanding difficult for yourself.

Metaphysics is much more valuable than that. It's priceless.

Now you say that we neither sense nor perceive the meaning of a foreign language: — Luke

I don't see the problem. Do you not grasp a difference between hearing people talking, and apprehending the meaning? Meaning as analogous with order, was the example.

Do you have an attention deficit or memory disorder? You seem incapable of maintaining your own position. A moment ago you were talking about "the reality of the situation" that we see the inherent order without apprehending it, and now you've switched back to saying that we can neither see nor apprehend the inherent order. If you think there are two different senses of the word "see" involved here, then define them both. Because one of them seems to refer to what we cannot see, which is not a familiar definition of the word "see". — Luke

You seem to be going through great effort to create problems where there are none. Oh well, it's what I've come to expect from you.

The truth of the assumption that a photon is massless is questionable:
https://www.sciencedirect.com/science/article/pii/S2211379719330943

Can you not hear foreign languages? This is a terrible analogy. This is something which we can perceive but cannot apprehend. Your analogy with molecules is equally bad, since it is something we can apprehend but cannot perceive. It is (or very recently was) your position that we can neither perceive nor apprehend the inherent order. Remember? You said that order is "not visible". — Luke

I don't think the analogies are bad. That there is order inherent within the thing seen is something inferred, just like that there is meaning in the foreign language which is heard, is something inferred, and that there are molecules in the object seen is something inferred. We neither perceive nor apprehend the inherent order but we infer that it is there, just like we infer that there is meaning in the foreign language, and that there are molecules within the thing seen. But we neither perceive nor apprehend the meaning in the foreign language, nor do we perceive or apprehend the actual molecules in the object seen. We apprehend a representation of the molecules, just like we apprehend a representation of the inherent order. And, when we come to understand the language we apprehend a representation of the meaning intended (what is meant) by the author.

There's no contradiction here, I take it? — Luke

I assume there are many different senses to the word "see". The word is used sometimes to refer strictly to what is sensed, and other times to what is apprehended by the mind.

So how can it be seen? — Luke

I really don't know how, it's just the reality of the situation. We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. There is a matter of distinguishing the individual elements, one from another, which the sense organ does not necessarily do, despite sensing the elements together as a composite.

From your side it must seem like you're being tag-teamed by Luke and myself, but I'm not reading his posts. I'm not aware of that half of the conversation. — fishfry

Don't worry about that, the conversations are completely different. Luke is on a completely different plane.

It's an abstraction intended to formalize an aspect of nature. If you think it's a generalization of something, you might be missing the point. Hard to say. — fishfry

I don't see the distinction you're trying to make here, between an inductive conclusion, and "an abstraction intended to formalize an aspect of nature". What do you mean by "formalize" other than to state an inductive conclusion.

I see the majority of definitions as inductive conclusions. Either they are like the dictionary, giving us a formalization (inductive conclusion) of how the word is commonly used, or they are intended to say something inductive (state a formalization) about some aspect of nature.

I think you are missing the point. If I drop a hundred bowling balls and I say, "Bowling balls fall down. That's a law of nature," then THAT is an inductive conclusion.

But if you see 100 bowling balls fall down and you go, F = ma, that is an abstraction and a mathematical formalization. You don't seem to have a firm grasp on this. Do you follow my point here? — fishfry

I think it's you who is missing the point. I do not have a firm grasp on the distinction you are trying to make, because there are no principles, or evidence to back up your claim of a difference between these two.

F=ma says something about a much broader array of things than just bowling balls. So one could not produce that generalization just from watching bowling balls, you'd have to have some information telling you that other things behave in a similar way to bowling balls. Mass is a property assigned to all things, and the statement "f=ma" indicates that a force is required to move mass. How can you not see this as an inductive conclusion? It's not just a principle dreamed up with no empirical evidence. In all cases where an object starts to move, a force is required to cause that motion. It might have been the case that "force" was a word created, thought up, or taken from some other context and handed that position, as being what is required to produce motion (acceleration), but this does not change the inductive nature of the statement.

Your notion of induction is wrong. "All bowling balls fall down," is an inductive conclusion. F = ma is a formalization. — fishfry

As I said, I really do not understand how a "formalization" as used here, is anything other than an inductive conclusion. So I do not understand how you think my notion of induction is wrong. Perhaps you should look into what inductive reasoning is, and explain to me how you think a "formalization" is something different. I think induction is usually defined as the reasoning process whereby general principles are derived from our experiences of circumstances which are particular.

But there are non-physical parts of the world that we are interested in, such as quantity, order, shape, symmetry, and so forth. Those are the non-physical parts of the world that are formalized by math. — fishfry

That such things are non-physical is what I dispute. How could there be a quantity which is not physical? "Quantity" implies an amount of something, and if that something were not physical it would be nothing. "Order" implies something which is ordered, and if there was no physical things which are ordered, there would be no order. And so on, for your other terms. It makes no sense to say that properties which only exist as properties of physical things are themselves non-physical.

Things in the world have order, and we have a mathematical theory of order that seeks to formalize the idea. — fishfry

When you say "formalize" here, do you mean to express in a formal manner, to state in formal terms? If it is physical things in the world which have order, and mathematics seeks to express this order in a formal way, then how is this not making a generalization about the order which exists in the phyiscal world, i.e. making an inductive conclusion?

On the other hand, of course there are non-physical, non-part-of-the-world abstractions too. Chess, for instance. Chess is a formal game, it's its own little world, it has a self-consistent set of rules that correspond to nothing at all in the real world. Knights don't "really" move that way. Right? Say you agree. How can anyone possibly disagree? — fishfry

How can I agree with this? Chess is a game of physical pieces, and a physical board, with rules as to how one may move those physical pieces, and the results of the movements. The physical board and pieces are not "nothing at all in the real world", they are all part of the world.

What's with your motive here? Why do you insist on taking rules like those of mathematics, which clearly refer to parts of the real world, and remove them from that context, insisting that they do not refer to any part of the real world? Your analogy clearly does not work for you. The chess game is obviously a part of the world and so its rules refer to a part of the real world, just like quantity, order, shape, and symmetry are all parts of the real world, and so the rules (or formalities) of these also refer to parts of the real world.

But you are the one that insists that physical collections of things have an inherent order. And that's what the mathematical concept of order is intended to formalize. Things in the world have order, and we have a mathematical theory of order that seeks to formalize the idea.

Right? When mathematicians formalize numbers, they're abstracting and formalizing familiar counting and ordering. When they create abstract sets, they are formalizing the commonplace idea of collections. A bag of groceries becomes, in the formalization, a set of groceries. Surely you can see that. Why would you claim math is not based on everyday, common-sense notions of the world? — fishfry

Yes, I agree with this here. Now the issue is how can you say that there is a collection of things which has no inherent order. If things in the world have order, and mathematicians seek to formalize that order, then where does the idea of "no inherent order" come from? That notion of "no inherent order" is obviously not derived from any instance of order, and if mathematicians are seeking to formalize the idea of order, the idea of "no order" has no place here. It is in no way a part of the order which things have, and therefore ought not enter into the formalized idea of "order".

Like what? Can you name some of these? Sets correspond to collections. — fishfry

Have you lost track of our conversation? The idea of "no inherent order" is what we are talking about, and this is what I say does not correspond with our observations of the world. We observe order everywhere in the world. Sets do not correspond to collections, because any collection has an inherent order, existing as the group of particular things which it is, in that particular way, therefore having that order, yet as a "set" you claim to remove that order.

But again I ask you, exactly WHICH mathematical ideas are not based on or inspired by the natural world? You must have something in mind, but I am not sure what. — fishfry

I'll repeat. It's what we've been discussing, your idea of "a set", as a collection of things with no inherent order. Something having no inherent order is not based in, nor inspired by the real world, we don't see this anywhere in the world. We can also look at the idea of the infinite. It is not inspired by anything in the natural world. It is derived completely from the imagination.

That's a useful mindset to have, so that we don't allow our everyday intuitions interfere with our understanding of the formalism. But of course historically, math is inspired by the real world. Even though the formalisms can indeed get way out there. — fishfry

Let's try this. We'll say that a "formalism" relates to the real world in one way or another, and then we can avoid the issue of whether it is an inductive conclusion. We'll just say that it relates to the world. Now, can we make a category of ideas which do not relate to the real world? Then can we place things like "infinity", and "no order" into this category of ideas? But rules about quantifying things, and rules about chess games do relate to the real world, as formalisms.

Can you see that these ideas are not formalisms, nor formalizations in any way? Because they are purely imaginary, and not grounded in any real aspects of the natural world, there is no real principles whereby we can say that they are true or false, correct or incorrect. They cannot be classed as formalizations because they do not formalize anything, they are just whimsical imaginary principles. To use your game analogy, they are rules for a game which does not exist. People can just make up rules, and claim these are the rules to X game, but there is no such thing as X game, just a hodgepodge of rules which some people might choose to follow sometimes, and not follow other times, because they are not ever really playing game X, just choosing from a vast array of rules which people have put out there. Therefore there is nothing formal, so we cannot call these ideas formalisms or formalizations.

The truth is in the thing. — fishfry

I disagree with your notion of truth. I think truth is correspondence, therefore not in the thing itself, but attributable to the accuracy of the representation of the thing. Identity is in the thing, as per the law of identity, but "true" and "false" refer to what we say about the thing.

If I want to study the planets I put little circles on paper and draw arrows representing their motion. The truth is in the planets, not the circles and arrows. I hope you can see this and I don't know why you act like you can't. — fishfry

I think this is a completely unreasonable representation of "truth", one which in no way represents how the term is commonly used. We say that a proposition is true or false, and that is a judgement we pass on the interpreted meaning of the proposition. We never say that truth is within the thing we are talking about, we say that it is a property of the talk. or a relation between the talk and the thing.

First, sets are intended to model our everyday notion of a collection. And in order to do a nice formalization, we like to separate ideas. So we have orderless sets, then we add in order, then we add in other stuff. If I want to put up a building, you can't complain that a brick doesn't include a staircase. First we use the bricks to build the house, then we put in the staircase. It's a process of layering. — fishfry

Take a look at your example. The bricks are never "orderless". They come from the factory on skids, very well ordered. Your idea of "orderless sets" in no way models our everyday notion of a collection.

Our formalization begins with pure sets. It's just how this particular formalization works. — fishfry

The point is that orderlessness is in no way a formalization. A formalization is fundamentally, and essentially, a structure of order. Therefore you cannot start with a formalization of "no order". This is self-contradictory. As I proposed above, the idea of orderlessness, just like the idea of infinite, must be removed from the category of formalizations because it can in no way be something formal. To make it something formal is to introduce contradiction into your formalism.

If I represent a planet as a circle, you don't complain that my circle doesn't have rocks and and atmosphere and little green men. I'll add those in later. — fishfry

What I'm complaining about is your attempt to represent nothing, and say that it is something. You have an idea, "no inherent order", which represents nothing real, It's not a planet, a star, or any part of the universe, it's fundamentally not real. Then you say that this nothing exists as something, a set. So this nothing idea "no inherent order" as a set. Now you have represented nothing (no inherent order), as if it is the property of something, a set.

You act like all this is new to you. Why? — fishfry

The idea of contradictory formalisms is not at all new to me. I am very well acquainted with an abundance of them. That's why I work hard to point them out, and argue against them.

I refer you to Galileo's sketch of Jupiter's moons. With this picture he started a scientific and philosophical revolution. Yet anyone can see that these little circles are not planets! There are no rocks, no craters, no gaseous Jovian atmosphere. Why do you pretend to be mystified by this obvious point? — fishfry

I don't see how this is analogous. Galileo represented something real, existing in the world, the motions of Jupiter's moons. What I object to is representing something which is not real, i.e. having no existence in the world, things like "no inherent order". This is not a representation, it is a fundamental assumption which does not represent anything. If a formalism is a representation, then the fundamental assumption, "no inherent order" cannot be a part of the formalism.

Do you feel the same way about maps? — fishfry

Consider this analogy. The idea of "no inherent order" describes nothing real, anywhere. So why is it part of the map? Obviously it's a misleading part of the map because there is nowhere out there where there is no inherent order, therefore I would not want it as part of my map.

Tell me this, Meta. When you see a map, do you raise all these issues? — fishfry

Yes, I get very frustrated when the map shows something which is not there. I look for that thing as a marker or indicator of where I am, and when i can't find it I start to feel lost. Then I realize that it was really the maker of the map who was lost.

An order that is shown can be seen: — Luke

But we cannot see the inherent order: — Luke

I already explained in what sense we see the inherent order, and do not see it, just like when we look at an object and we see the molecules of that object. The order is there, just like the molecules are there, and what our eyes are seeing, yet we do not distinguish nor apprehend the molecules nor the order, so we cannot say that we see it. We are always seeing things without actually seeing them, because it is a different sense of the word "see".

There was no change to my position, just a need to go deeper in explanation, to clarify the use of common terms, to assist you in understanding.

The idea, as with anything else mathematical, is that we have some aspect of the real world, in this case "order"; and we create a mathematical formalism that can be used to study it. And like many mathematical formalisms, it often seems funny or strange compared to our everyday understanding of the aspect of the world we're trying to formally model. — fishfry

I follow this, it seems to be exactly what I've been trying to explain to Luke, so we're on the same page here.

After all bowling balls fall down, and the moon orbits the earth. To help us understand why, Newton said things like F=maF=ma, and F=m1m2r2F=m1m2r2. And E=12mv2E=12mv2, and things like that. And you could just as easily say, "Well this doesn't seem to be about bowling balls. These are highly artificial definitions that Newton just made up." And you'd essentially be right, while at the same time totally missing the point of how we use formalized mathematical models in order to clarify our understanding of various aspects of the real world. — fishfry

These are what I would call universals, generalities produced from inductive reasoning, sometimes people call them laws, because they are meant to have a very wide application. As inductive conclusions they are derived from empirical observations of the physical world

So if you can see the difference between a real world thing like order, on the one hand; and how mathematicians formalize it, on the other; and if you are interested in the latter, if for no other reason than to be better able to throw rocks at it, I'm at your service. — fishfry

The issue is with what you call the purely abstract. It appears to me, that you believe there are some sort of "abstractions" which are completely unrelated to the physical world. They are not generalizations, not produced from inductive reasoning, therefore not laws, or "artificial definitions", in the sense described above. You seem to think axioms of "pure mathematics" are like this, completely unrelated to, and not derived from, the physical world.

That's the mindset for understanding how math works. You seem to object to math because it's a formalized model and not the thing itself, but that's how formal models and formal systems like chess work. They are not supposed to be reality and it's no knock agains them that they are not reality. They're formal systems. If you can see your way to taking math on its own terms, you'd be in a better position to understand it. And like I say, for no other reason than to have better arguments when you want to throw rocks at it. — fishfry

I object to the parts of these formalizations which do not correspond with our observations of the world. These would be faulty inductive conclusions, falsities. You claim that they do not need to correspond, that they a completely unrelated to the physical world. Yet when you go to describe what they are, you describe them as inductive conclusions, above, which are meant to correspond, in order that they might accurately "clarify our understanding of various aspects of the real world.".

So I see a disconnect here, an inconsistency. You describe "pure abstractions" as being related to the world in the sense of being tools, or formalizations intended to help us understand the world. Yet you insist that those who create these formalizations need not pay any attention to truth or falsity, how they correspond with the physical world, in the process of creating them. And you claim that when mathematicians dream up axioms, they do not pay any attention to how these axioms correspond with the world, because they are working within some sort of realm of pure abstraction.

As an example consider what we've discussed in this thread concerning " a set". It appears to me, that mathematicians have dreamed up some sort of imaginary object, a set, which has no inherent order. This supposed object is inconsistent with inductive conclusions which show all existing objects as having an inherent order. You seem to think, that's fine so long as this formalized mathematical system helps us to understand the world. I would agree that falsities, such as the use of counterfactuals, may help us to understand the world in some instances. But if we do not keep a clear demarcation between premises which are factual, and premises which are counterfactual, then the use of such falsities will produce a blurred or vague boundary between understanding and misunderstanding, where we have no principles to distinguish one from the other. If axioms, as the premises for logical formalizations are allowed to be false, then how do we maintain sound conclusions?

I don't care about your latest position. In case you missed it, my entire point for the last three or four pages is that you changed your position three or four pages ago. This is obvious from the quote that you somehow managed to overlook: — Luke

As I explained, I haven't changed my position. You have not yet understood it.

Here you say that "the exact spatial positioning" is not what is being demonstrated (i.e. shown) in the diagram. — Luke

No that's not a good interpretation. You need to respect the fact that what is being shown to the observer, as inhering within the physical thing being used in the demonstration, is not the same order as that which exists in the mind of the person performing the demonstration. I said there is a demonstration of "an order". I also said "the order which the mind apprehends, based on the demonstration is not the same order as that which inheres within the things shown." Then I said "the order apprehended in the mind is not the same as the order in the object. Therefore 'the exact spatial positioning' is not what is being demonstrated." The exact spatial positioning is what inheres within the object, and though it is what is being shown by the one doing the demonstration, it is not the same order as what the person is trying to demonstrate. This is why I said fishfry's claim that the order was random is false. That's what the person doing the demonstration was trying to demonstrate, but it was not what the demonstration actually showed.

What is "being demonstrated" is an order which exists in the mind of the person making the demonstration. This is the first line, the "demonstration of an order". What appears to the person making the interpretation, as what is "shown", is the physical object with an inherent order. This is a representation of the order which exists in the mind of the person making the demonstration. It is not the same order, but a representation of it. So the order being demonstrated is not the same as the order which inheres within the representation, (as a representation is different from the thing it represents), and the order in the mind of the person interpreting what is shown, is not the same as the order which inheres in the object. And, because of this medium, which exists between the one demonstrating and the one interpreting, the physical object as symbols, the order on the minds of the two individuals is not the same. That as I said is why we misunderstand each other.

Which is it? — Luke

As I said, numerous times, the mind creates an order to account for the order assumed to be in the thing Therefore the order in the mind it is not the order shown by the thing. No change of position, just a difficult ontological principle to describe to someone with a different worldview.

But you didn't say any of those things. Instead you just accepted the mathematical definition I repeatedly gave you, and then kept arguing from your own private definition. When I finally figured out what you were doing, I was literally shocked by your bad faith and disingenuousness. I'm willing to have you explain yourself, or put your deliberate confusion-inducing equivocation into context, but failing that I no longer believe you are arguing in good faith at all. You have no interest in communication, but rather prefer to waste people's time by deliberately inducing confusion. — fishfry

Let me remind you. You started in the discussion with the repeated assertion "sets have no inherent order". Check this post, I think you'll see that claim stated a number of times.

My God, you wield your ignorance like a cudgel. I could have just as easily notated the two ordered sets as:

* ({1,2,3,4,…},<)({1,2,3,4,…},<) and

* ({1,2,3,4,…},≺)({1,2,3,4,…},≺)

which shows that these two ordered sets consist of the exact same underlying set of elements but different linear orders. Remember that sets have no inherent order. So {1,2,3,4,...} has no inherent order. The order is given by << or ≺≺. — fishfry

When I said order is spatial and temporal, you claimed a completely "abstract order", which I didn't understand, and still don't understand because you haven't yet explained this in a coherent way. So I continued to insist order was spatial or temporal until you gave me examples of first and second place in competition, what I called order relative to best. I accepted this as non-spatial or temporal ordering, but I still don't see it as completely abstract because it still is based in concrete criteria for judgement.

You proceeded to define order in terms of "less than", as if you thought that this is purely abstract. However, I had already explained how "less that" is dependent on, defined in relation to, quantity. So you only contradicted your earlier claim that order is logically prior to quantity, by defining order in relation to quantity. And, since quantity is dependent on spatial separation between individuals you have not really escaped the spatial aspect of order, to get to a purely abstract order.

So this is where we stand. You have claimed a purely abstract order, but given me an order based in "less than" which is based in quantity. And quantity relies on spatial conception, so you have really given me a concept of order based in spatial conceptions..

By "shown" you do not mean "displayed". After all, "it is not visible". Therefore, you must not be making the argument - as you did earlier - that we see the order but do not apprehend it. — Luke

Luke. I very consistently said, over and over again, that we do not see the order.

You are saying that the "exact spatial positioning" is logically demonstrated by
("shown" in) the diagram, but it is not apprehended? If the exact spatial positioning is not (or cannot be) apprehended, then how has it been logically demonstrated by the diagram? — Luke

I went through that already, more than once. There is a logical demonstration of an order. The order which the mind apprehends, based on the demonstration is not the same order as that which inheres within the things shown. You keep neglected the principal point of the argument, that the order apprehended in the mind is not the same as the order in the object. Therefore "the exact spatial positioning" is not what is being demonstrated. So, do not ask again, this same strawman question. Check out these quotes:

Of course, location is intelligible, conceptual, so it's not actually sensed it's something determined by the mind, just like meaning. When you tell someone something, they do not sense the meaning. This is "showing" in the sense of a logical demonstration. And the point is that upon seeing, or hearing what is shown, the mind may or may not produce the required conceptualization, which would qualify for what we could call apprehending the principle. More specifically, the conceptualization produced in the mind being shown the demonstration is distinct and uniquely different from the conceptualization in the mind which is showing the demonstration, therefore they are not the same. That is why people misunderstand each other.. — Metaphysician Undercover

I answered the question, it's just like a logical demonstration. Like when one person shows another something, and the other makes a logical inference. All instances of being shown something intelligible, i.e. conceptual, bear this same principle. We perceive something with the senses and conclude something with the mind. The things that I sense with my senses are showing me something intelligible, so I produce what serves me as an adequate representation of that intelligible thing, with my mind. — Metaphysician Undercover

Inherent order is a wider concept, applying in particular to biological systems and natural phenomena. Ordering is more specific having to do with listing. I think you are discussing the latter. — jgill

Fishfry claimed that a set has no inherent order, and I questioned whether it is possible that there could be a thing with no inherent order.

i really do not think that "listing" is the subject here, because listing Is an ordering of symbols, not the things represented by the symbols. The list may represent an order, but the reason for the order is something other than the spatial order of the symbols. And fishfry insisted on the reality of a purely abstract order, which could not be a spatial relation of symbols, as listing is. We would need to find a principle of order which is purely abstract.

Sorry, I haven't kept up. Are you speaking of inherent order or inherent ordering? — jgill

I think I said "inherent order", but I don't quite understand the point to making the difference.

But you now concede that sense perception is involved in showing. — Luke

Of course sense perception is involved! Where have you been? We've been talking about seeing things and inferring an order. My point was that we do not sense the order which inheres within the thing, we produce an order in the mind. I never said anything ridiculous like we do not use the senses to see the thing, when we produce a representation of order for the thing.

I would say that the word "shown" here means what is visible in, or displayed by, the diagram; not what is demonstrated or proved by the diagram. — Luke

But we went through this already. I explained that this is not what I meant by "shown"., and the reason why, being that order is inferred by the mind, it is not visible. I went through a large number of posts explaining this to you. It seemed to be a very difficult thing for you to grasp. And now, when you finally seemed to grasp the meaning associated with the way I used the term, you've gone right back to assuming that this is not the way I used it, despite all those explanations. Why? We just go around in circles, it's stupid. You pretend to have understood my explanation as to what I meant by "shown", then all of a sudden you say but obviously that's not what you meant. It's ridiculous. It' like you're saying 'I would not have used "shown" that way, therefore you did not'. And when i go through extreme lengths to explain that this is actually how I used the term, to the point where you seem to understand, you turn right back to the starting point, claiming but I would not have used it that way, therefore obviously you didn't. What's the point?

All that remains for you to explain is your contradictory pair of claims that (i) the inherent order is the exact spatial positioning that we do apprehend in the diagram; and that (ii) we are unable to apprehend the inherent order. — Luke

I stated repeatedly that we do not apprehend the exact spatial positioning, so you have a strawman here.. You don't seem to be capable of understanding any of what I am saying, we're just going around in circles of misunderstanding. it's pointless.

You avoid the question instead of answering it. How can location be shown to someone without it being sensed? — Luke

I answered the question, it's just like a logical demonstration. Like when one person shows another something, and the other makes a logical inference. All instances of being shown something intelligible, i.e. conceptual, bear this same principle. We perceive something with the senses and conclude something with the mind. The things that I sense with my senses are showing me something intelligible, so I produce what serves me as an adequate representation of that intelligible thing, with my mind.

"Shown" in the sense of a logical demonstration is different to "shown" in the sense of your statement: "the exact spatial positioning shown in the diagram". That's obvious. — Luke

Sorry, you've lost me. You appear to be making up a difference in the meaning of "shown", for the sake of saying that I contradict myself. I take back my apology, I'm going back to thinking that you do it intentionally.

There's an odd sort of reality of life which is almost a sort of paradox. The individual living being in the extreme complexity of its existence, as a delicately balanced organism, is very susceptible to death. We all run the risk of dying on a daily basis. Death may be waiting for you around any corner, or curve. But life in general, as a beautiful vast array of all sorts of different organisms, is extremely robust, and resilient in consequence to the occurrence of any possible extermination events.

The beauty of life is found in its diversity, and this provides its strength. Take a look at all the different colours of flowers there are, and think about how difficult it would be to produce such an extensive array. And that's just one simple property, colour.

The fragility of the individual, although it results in the death of each and every one of us, is not a weakness however, because this is the means by which life tests all the different boundaries of the the environment which it inhabits, thereby producing all the diverse individuals which provide its overall strength. We ought not seek to limit diversity, because that would be a self-imposed weakness, making the vulnerability of the individual, universal.

Do you think that location can be shown to someone without it being sensed? — Luke

Of course, location is intelligible, conceptual, so it's not actually sensed it's something determined by the mind, just like meaning. When you tell someone something, they do not sense the meaning. This is "showing" in the sense of a logical demonstration. And the point is that upon seeing, or hearing what is shown, the mind may or may not produce the required conceptualization, which would qualify for what we could call apprehending the principle. More specifically, the conceptualization produced in the mind being shown the demonstration is distinct and uniquely different from the conceptualization in the mind which is showing the demonstration, therefore they are not the same. That is why people misunderstand each other..

I now see why we have such a hard time understanding each other, we seem to be very far apart on some fundamental ideas, which form the basis for our conceptualizing what is shown by the other. I thought you were intentionally misreading me, in order to say that I contradict myself. But now I see that this is the way you actually understand those words. My apologies for the accusation.

As a result you can find yourself living with people who are simple in thought who don't give the extra effort to think from a philosophers perspective. — Tiberiusmoon

Each person has one's own place within a society, and many of these places do not require complex philosophical thought, so there is not need to compel these people toward it. What is a problem though, I believe, is bad habits of thought. Bad habits may enter into any field or discipline involved in complex thought, and may in some cases be associated with a form of laziness. For instance, in some cases we are encouraged to accept the principles presented by others whom we apprehend as authorities, without asking for justification. This form of laziness seems to pervade modern academia.

Sorry Luke, now it's me who can't understand what you're saying. Do you think that we sense location?

Given that (1) we do not perceive order via the senses and that (2) we cannot apprehend inherent order, then how can the inherent order be the exact spatial positioning shown in the diagram? — Luke

I think I've answered this about three times, so either I don't understand your question, or you don't understand my answer, or both.

Even if I grant you that there is an inherent order to the universe, how can you say that the inherent order of the diagram is the same as the order that we perceive via the senses, or "the exact spatial positioning shown"? — Luke

I told you, we don't perceive order with the senses, we create orders with the mind. Judging by this statement, I'm thinking it's you who is the one not understanding.

I note your change in position. You are no longer arguing that the inherent order cannot be apprehended. You have now adopted the weaker claim that the inherent order cannot be completely apprehended. — Luke

Again, you're looking for hidden meaning to make unjustified inferences. The inherent order cannot be apprehended by us. I can't even imagine what it would mean to partially understand an order. If my use of "completely" misled you, I retract it as a mistake, and apologize.

More importantly, as far as I can tell, inherent order is not the kind of order that mathematicians are concerned with. — Luke

Maybe, but fishfry claimed that a set has no inherent order. So if mathematicians are making such assumptions in their axioms, then they are concerned with it; concerned enough to exclude it from the conceptions of set theory. The issue I'm concerned with is the question of whether a thing without inherent order is a logically valid conception.

You're factually wrong.

Is the set {0,1,2,3,4} identical to the set {0,1,2,3,4}? I have to assume you'd say yes.

But 2 + 3 and 5 are both representations of the set {0,1,2,3,4}. So they're identical. — fishfry

The way you described sets in this thread, a set is something which cannot have an identity because it has no inherent order. Therefore I cannot agree that the set {0,1,2,3,4} is identical to the set {0,1,2,3,4}. It seems like a set is an abstraction, a universal, rather than a particular, and therefore does not have an identity as a "thing". It is particulars, individual things, which have identity according to the law of identity. Notice that the law of identity says something about things, a thing is the same as itself.

The law of identity is intended to make that category separation between particular things, and abstractions which are universals, so that we can avoid the category mistake of thinking that abstractions are things. "The set {0,1,2,3,4}" refers to something with no inherent order, so it does not have an identity and is therefore not a thing, by the law of identity, To say that it is a thing with an identity is to violate the law of identity.

Well, math does not violate this principle. 2 + 3 and 5 are identical. They are both representations of the set represented by {0,1,2,3,4}, which of course is not actually "the" set, but is rather yet another representation of that abstract concept of 5. — fishfry

This is the whole point of the law of identity, to distinguish an abstract concept from a thing, so that we have a solid principle whereby we can avoid the category mistake of thinking of concepts as if they are things. A thing has an identity which means that it has a form proper to itself as a particular. To have a form is to have an order, because every part of the thing must be in the required order for the thing to have the form that it has. So to talk about something with no inherent order, is to talk about something without a form, and this is to talk about something without an identity, and this is therefore not a thing.

Only to the extent that you don't seem to think a thing is identical to itself. Because when mathematicians use equality, they mean identity, and this is provable from first principles. They either mean identity as sets; which is easy to show; or, they often mean identical structurally. This is a more subtle philosophical point. — fishfry

The problem is not that I don't think a thing is the same as itself. That is the law of identity, which I adhere to. The problem is that you make the category mistake of believing that abstract conceptions are things. Because you will not admit that a concept is not a thing, you make great effort to show that two distinct concepts, like what "2+3" means, and what "5" means, which have equal quantitative value, refer to the same "thing". Obviously though, "2+3" refers to a completely different concept from "5".

If you would just recognize the very simple, easy to understand, fact, that "2+3" does not mean the same thing as "5" does, you would understand that the two expressions do not refer to the same concept. So even if concepts were things, we could not say that "2+3" refers to the same thing as "5", because they each have different associated concepts. And it's futile to argue as you do, that the law of identity is upheld in your practice of saying that they refer to the same "mathematical object", because all you are doing is assuming something else, something beyond the concepts of "2+3", and "5", as your "mathematical object". This supposed "object" is not a particular, nor a universal concept, but something conjured up for the sake of saying that there is a thing referred to. But there is no basis for this object. It is not the concept of "2+3" nor is it the concept of "5", it is just a fiction, a false premise you produce for the sake of begging the question in your claim that the law of identity is not violated.

You said that "1) We do not perceive order with the senses" and that "2) We cannot apprehend the inherent order". Therefore, how do you know that what's shown in the diagram is the exact positioning of the parts (i.e. the inherent order)? — Luke

It is a fundamental ontological assumption based in the law of identity, that a thing has an identity. In Kant we see it as the assumption that there is noumena, which is intelligible, just not intelligible to us. In Descartes we see skepticism as to whether there even is external objects.

So we assume that there is something, the sensible world, and we assume it to be intelligible, it has an inherent order. To answer your question of how do we "know" this, it is inductive. We sense things, and we conclude that there is reality there. Also, we have some capacity to understand and manipulate what is there, so we conclude that there is intelligibility there, intelligibility being dependent on ordering. We have some degree of reliability in our understanding of the ordering therefore there must be some ordering.

Isn't the positioning of the dots that I perceive the "perspective dependent order" which you earlier stated was not the inherent order? So how can the diagram show the inherent order to anybody?

The inherent order cannot be perceived by the senses and we can't apprehend it, anyway. — Luke

Yes, you perception provides for you, the basis for a perspective dependent order, which your mind produces. What the object is showing you, its inherent order, and the order which you are producing towards understanding the inherent order, are two distinct things. As I described, there is some degree of inconsistency, constituting a difference, between what is shown to you, and what you apprehend from that showing. The claim of difference is justified by our failures. The inherent order is shown. It is not perceived by the senses. If you try to understand the inherent order, your mind will produce an order which you think best represents that which inheres in the object.

Consider, that in seeing objects we do not see the molecules, atoms or other fundamental particles, we have to figure those things out as a representation of the order which inheres within. But we cannot completely apprehend that order because our minds are deficient. This doesn't mean that sentient beings will never be able to apprehend it, or that there isn't an omniscient being which already can apprehend it. And even if it is impossible that human beings or any sentient beings will ever be able to understand it in perfection, like an omniscient being is supposed to be able to, we can still improve our understanding, i.e. get a better understanding, and decrease our failures.

If "We cannot apprehend the inherent order", then how do you know that our representations are inaccurate? — Luke

I think we judge the accuracy of our understanding mostly by the reliability of our predictions. But reliability is perspective dependent and subjective. So where some people see reliability, I see unreliability. It all depends on what type of predictions you are looking for the fulfillment of.

If the inherent order is unintelligible to the human mind by definition, then what makes inherent order preferable to (or distinguishable from) randomness? — Luke

Order is fundamentally intelligible. So assuming order is to assume the possibility of being understood, which is to inspire the philosophical mind which has the desire to understand. To assume randomness is to assume unintelligibility which is repugnant to philosophical mind which has the desire to understand.

So, as I explained. If the object appears (seems) to be unintelligible (without inherent order), we need to determine why. Is it our approach (are we applying the wrong principles in our attempt to understand), or is it the reality, that the object truly has no inherent order? The latter is repugnant to the philosophical mind, and even if it were true, it cannot be confirmed until the possibility of the former is excluded. Therefore, when the object appears to be unintelligible (without inherent order), we must assume that our approach is faulty (we are applying the wrong principles in our attempt to understand), and we must subject all principles to extreme skepticism, before we can conclude that this object is truly unintelligible (without inherent order). The rational approach is to assume that we are applying the wrong principles, and to assume that the object has no inherent order is irrational.

On the contrary. 2 + 3 and 5 are mathematically identical. There is not the slightest question, controversy, or doubt about that. — fishfry

As per the quotes above, from Wikipedia, the mathematical notion of identical , as equal, is not consistent with the philosophical notion of identity, described by the law of identity. In other words, mathematicians violate the law of identity to apply a different concept of identity, making two things of equal value mathematically identical. You might accept this, and we could move on to visit the possible consequences of what I believe is an ontological failure of mathematics, or you could continue to deny that mathematicians violate this principle. The latter is rather pointless.

You spoke of an "external perspective", which implies an internal perspective. You might recall I asked you about it and you responded: — Luke

If all perspectives are external to the object, then "perspective" is necessarily external, and saying "external perspective" just emphasizes the fact that perspective is external. Internal perspective is not implied, just like saying "cold ice" doesn't imply that there is warm ice. That's why I referred to Kant, to show how our perspective of the thing in itself is external to the thing.

Your current argument is that we do not perceive order with the senses, and that we cannot apprehend inherent order at all. Therefore, how is it possible that the inherent order is the exact spatial positioning shown in the diagram? — Luke

Why not? I don't understand your inability to understand. Let me go through each part of your question. 1) We do not perceive order with the senses. No problem so far, as we understand order with the mind, not the senses. 2) We cannot apprehend the inherent order. Correct, because the order which we understand is created by human minds, as principles of mathematics and physics, and we assign this artificially created order to the object, as a representation of the order which inheres within the object, in an attempt to understand the inherent order. But that representation, the created order is inaccurate due to the deficiencies of the human mind. 3)The inherent order is the exact positioning of the parts, which is what we do not understand due to the deficiencies of the human mind.

Does that help? You have emboldened the word "shown". Why? Do you understand that something can be shown to you which you do not have the capacity to understand? The physical world shows us many things which we do not have the capacity to understand. For example, the theologians used to argue that the physical world shows us the existence of God. Most people would claim that the physical world is not evidence of God, and in no way does the physical world show us God. The theologian would say that you just do not understand what is being shown to you. the exact order is being shown to us but we do not understand it.

Fishfry will argue fervently that "pure mathematics" is not influenced by physics. Perhaps some mathematicians actually have no respect for physical principles, and that's why infinities have become the norm, rather than the abnormal.

Apparent order is not perceived? Do you know what "apparent" means? — Luke

In that context, "apparent" must mean "seems". If you used "apparent" to mean "perceived by the senses", I would say that you had stated an oxymoron. We apprehend order with the mind, we do not perceive it with the senses.

If apparent order is not perceived, then your earlier distinction between "internal" and "external" perspective is irrelevant; it's not a matter of perspective at all. So why did you introduce the distinction between "internal" and "external" perspective? — Luke

I don't believe I said anything about an internal perspective. I distinguished between the order which inheres within the thing itself (which we assume must be real to account for the consistency we note from observations), and the order which we assign to things, from our external perspective of them. This is the reason why our knowledge of the order of things is fallible, the order which we say something has is not the same as the order which it actually has. The best we can say is that we have created a representation of the order that things have. We create the representations through the means of analysis of empirical observations, which do not provide us with the inherent order, in conjunction with theorizing, hypotheses. So there is a separation, a medium consisting of observation and theory, which lies between the apparent order (the order which things seem to have), and the true order which inheres within the things themselves.

The reason why I introduced this distinction is because fishfry claimed that there is a sort of thing, "a set", which has no inherent order at all. I said that such a thing does not exist, because to exist is to have some sort of inherent order. Fishfry scoffed at this. So I proceeded to ask fishfry to explain this type of unity of parts, within which the parts have no order. How could there be such a unity? The point being, that this is simply an imaginary thing stated, 'parts without order', which doesn't correspond to any reality, which is really a logical inconsistency representing falsity, because it is impossible to have parts without order. To be a part of something implies an order in relation to a whole, without that order it cannot be said to be a part. We would have to call it something other than a "part". The point being, that the whole, which the so-called part is said to be a part of, "the set", is not a true whole because it provides no order relations to the so-called parts. Things existing with absolutely no relations of order cannot be said to form a whole, or unity of any kind.

Do we perceive both the apparent order and the inherent order? Is there a difference between the apparent order and the inherent order? If so, what is the difference between them? — Luke

The apparent order is made up, a created order, assigned to the group of things, so it is not perceived, it is produced by the mind.

If there is a difference between the apparent order and the inherent order, then why did you state: — Luke

Why not? The order which we assign to things is clearly not the same as the "exact" order which inheres within things or else we'd have an absolutely perfect understanding of the order of the universe.

Not a word of this is even on topic relative to whether 2 + 3 and 5 are identical. Since mathematically they are, and as a mathematical expression it must necessarily be interpreted in terms of mathematics, nothing you say can make the slightest difference. Excluded middle? Did Aristotle anticipate intuitionism? That's interesting. — fishfry

As I've explained to you already, the idea that 2+3 is mathematically the same as 5, is simply a misunderstanding of the difference between equality and identity. They are equal, but equal is distinct from identity. I've told you this numerous times before, but you do not listen. Nor do you seem to pay any attention to my references, only repeating your misunderstanding in ignorance.

However, I'll reproduce for you the opening lines from the Wikipedia entries on both "equality" and "identity" below, just to remind you of how bad your interpretation really is.

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

In philosophy, identity, from Latin: identitas ("sameness"), is the relation each thing bears only to itself.

Are these both the inherent order (bolded)? If so, then why do you say "along with the order"? — Luke

Yes, you see the object along with the order which inheres within, meaning you see the order, you just do not apprehend it. Consider the dots, we see them, we must see the order because it's there, yet fishfry claimed that the dots were randomly arranged, indicating the order was not apprehended

So, a professional philosopher. At one point in the article he says: "We are indeed rationally justified in thinking 2 plus 3 will always be 5, because 2 plus 3 is not distinct from but rather identical with 5." My emphasis. So at least one professional philosopher would object to your claim that they are not identical. — fishfry

There are numerous philosophers who argue against the law of identity as stated by Aristotle, Hegel opposed it, as is evident here: https://thephilosophyforum.com/discussion/9078/hegel-versus-aristotle-and-the-law-of-identity/p1

What I see as an issue which arises from rejecting the idea that each particular object has its own unique identity (law of identity), is a failure of the other two interrelated laws, non-contradiction, and excluded middle. Some philosophers in the Hegelian tradition, like dialectical materialists, and dialetheists, openly reject the the law of non-contradiction. When the law of identity is dismissed, and a thing does not have an identity inherent to itself, the law of non-contradiction loses its applicability because things, or "objects" are imaginary, and physical reality has no bearing on how we conceive of objects.

There are specific issues with the nature of the physical world that we observe with our senses, which make aspects of it appear to be unintelligible. There must be a reason why aspects of it appear as unintelligible. We can assume that unintelligibility inheres within the object itself, it violates those fundamental laws of intelligibility, or we can assume that our approach to understanding it is making it appear.as unintelligible. I argue that the latter is the only rational choice, and I look for faults in mathematical axioms, and theories of physics, to account for the reason why aspects appear as unintelligible. I believe this is the only rational choice, because if we take the other option, and assume that there is nothing which distinguishes a thing as itself, making it distinct from everything else (aspects of reality violate the law of identity), or that the same thing has contradictory properties at the same time (aspects of reality violate the law of non-contradiction), we actually assume that it is impossible to understand these aspects of reality. So I say it is the irrational choice, because if we start from the assumption that it is impossible to understand certain aspects of reality, we will not attempt to understand them, even though it may be the case that the appearance of unintelligibility is actually caused by the application of faulty principles. Therefore it is our duty subject all fundamental principles to skeptical practices, to first rule out that possibility before we can conclude that unintelligibility inheres within the object.

Aristotle devised principles whereby the third fundamental law, excluded middle would be suspended under certain circumstances, to account for the appearance of unintelligibility. Ontologically, there is a very big difference between violating the law of excluded middle, and violating the law of non-contradiction. When we allow that excluded middle is violated we admit that the object has not been adequately identified by us. When we allow that non-contradiction is violated we assume that the object has been adequately identified, and it simply is unintelligible.

If you were talking about the inherent order the entire time, and if the inherent order is not perceived or apprehended, then why did you say: — Luke

I don't see any problem with those quotes. As I said, the order is right there, in the object, as shown by the object, and seen by you, as you actually see the object, along with the order which inheres within the object, yet it's not apprehended by your mind.

Sorry L:uke, but I find it extremely ridiculous that you are trying to tell me what I was talking about. As I said, you need to go back and reread the entire section, with the understanding, and commitment, that it's all about the inherent order, therefore the order which inheres within the object. And quit trying to force your nonsense interpretation, insisting that you know better than I do, what I was trying to say, simply so that you can say that I was trying to contradict myself. It's foolish of you.

If you see now, that the entire time, I was talking about the order which inheres within the thing itself, as "inherent order", rather than some perceived, apprehended, or creatively imagined order, you can go back and reread the entire section and clear up your misunderstanding. I recommend that anytime you feel the inclination to interpret in the latter sense, and thereby apprehend contradiction, you suppress this inclination, and remain true to the intentions of the author. Afterwards you can ask me for clarification if any points appear to be unclear.

But now you say that Kant's phenomena-noumena distinction is not the basis for your argument. — Luke

Right, this principle has a long tradition, it goes back at leas to Aristotle, with the law of identity, so it is definitely not based in Kant. Kant has simply presented the similar principle in his own way. I present it in my way. The principle is not "the same" it is a similar principle, needing to be refined and understood in the unique way of each particular individual mind who desires to understand..

In Aquinas, we see that independent Forms are fundamentally "intelligible" but not intelligible to the human intellect, because that intellect is united with a material body. This position, of being dependent on a body and the sense organs makes the intellect deficient. We find this same principle in Kant. The human intellect produces knowledge from phenomena which is dependent on sensation, and sense appearances. Notice that Kant refers to the noumenon as "intelligible", though it is not intelligible to us human beings, due to this predicament, which is not a contradiction.

Yes, but in the posts before you introduced Kant, you were clearly saying that the appearances were the reality (i.e. direct realism), as demonstrated by the quotes. — Luke

No, I don't believe I mentioned "appearances". And "inherent" clearly means within the object, as what inheres within. So if you interpreted me as saying the "inherent order" is part of the appearance of the object within a mind, rather than within the thing itself, I think this was a matter of misinterpretation. You did demonstrate some confusion as to what "inherent" means, as if you were somewhat unfamiliar with the word, so perhaps you thought I was talking about an order abstracted from an appearance, rather than an inherent order at that time.. But if you understood what "inherent" means, and what "appearance" means, you would not have interpreted in this contradictory way.

I think perhaps the issue was confused because we were talking about a diagram, which is intended to show something. Therefore there is a number of levels of representation which adds ambiguity. The diagram is an actual thing itself, with an inherent order. But it is also made to represent an order (an apparent order), which a human mind apprehends. This produced the problem with fishfry claiming it was "random", lacking order, because that is the intended (apparent order) which it was made to represent, However, I argued that there is necessarily an inherent order within the thing shown, and fishfry's claim that it did not show an order, that it was "random", is a false claim. If you had understood this argument from me, you would have recognized that I was making the same distinction at that time.

You asked us here (prior to your introduction of Kant) to take a look at the diagram and see the order the dots have, and that they could not have any other order. Yet now (after your introduction of Kant) you are trying to convince us of the opposite: that there must be another order - the inherent order - which is different to the order we can see in the diagram. Moreover, you have claimed that the appearance of order and the inherent order could not be the same just by chance, despite your admission that you don't know whether or not they could be the same. — Luke

That's a misinterpretation. I was asking the same thing both times, to look at the thing, and see that there is an order within the thing itself. What seems to be causing you confusion is the fact that we can look at a thing, and conclude that there is order inherent within (that's what makes a thing intelligible) without actually understanding the order., i.e. we see order without understanding it.

To return to my recent point, you have conceded that there are "many other types" of order which are not "temporal-spatial", therefore your references to phenomena-noumena (or indirect realism or whatever) do not apply to these many other types of order. Therefore, you cannot claim that there is some hidden order to these other types. While that might be irrelevant to your claims, it is not irrelevant to the criticisms of your claims made by the other posters here. You are the only one arguing that order must involve spatio-temporal phenomena (and/or noumena). — Luke

I don't understand your point. Your reference to "hidden order" doesn't make sense. I'm not talking about a hidden order, and this idea seems to be the source of your misunderstanding. The order is right there in plain view, as things are, but it is just not understood, because we do not have the capacity to understand it.

So you don't know whether intention has anything to do with Kant's phenomena-noumena distinction?
And yet you still use this distinction as the basis of your argument regarding inherent order? — Luke

I don't use that distinction as the basis for my argument, I gave that distinction as an example which i thought you might be able to understand.

You tried to draw an analogy between your supposed inherent order and Kant's noumena. When I pointed out that you had already conceded that "many other types" of order are not spatio-temporal and therefore not noumenal, you said that one other type (best to worst) "is relevant to intention, therefore phenomenal". If you don't know whether intention has anything to do with Kant's phenomena-noumena distinction, as you now admit, then you cannot claim that best-to-worst order is "relevant to intention, therefore phenomenal". — Luke

Come on Luke, use some intelligence. Kant did not have to name every instance of what contributes to phenomena for us to place things in that category. If you think I am wrong, and intention ought not be placed in that category, then just tell me. But please give reasons. Simply saying Kant didn't explicitly say it therefore, you're wrong in your analogy, is pointless.

What strawman interpretation? Instead of empty accusations, go ahead and explain how or what I have misinterpreted. — Luke

I told you how you misinterpreted., You claimed a contradiction when I said I couldn't describe something which was shown. That is just an indication of the limits to human intelligence, and word use, not a contradiction.

Pure contradiction. — Luke

Thanks for all the quotes removed from context. To be shown, or demonstrated does not mean to be stated, I went through that in the last post, and again above. And, "the positioning of those points relative to each other is describable" does not mean that I have the capacity to describe them. I do believe I mentioned that it would require an intelligence superior to a human intelligence, like a divine intellect. I was arguing the deficiencies of the human intellect, in being incapable of describing what is inherently describable.

This is exactly the problem which quantum physics actually has. The physicists are incapable of adequately describing the positioning, therefore "order" of the particles. We can either conclude that the particles have no inherent order, because the order cannot be determined by the human techniques, or we can conclude that they have an order, but other principles, and a higher intelligence, are required to figure out the order. As I explained to you already, (which you've left out of your inflammatory interpretation), is that the latter choice is the rational choice.

Therefore I reaffirm my accusation, that you are intentionally misinterpreting what I write for the sake of making it appear as contradictory, instead of putting any effort into trying to understand it. This is very consistent with my observations of your mode of operation at this forum.

In that last sentence, I merely say that it seems to me that when we make correct computations in arithmetic, we can take the results as true. — TonesInDeepFreeze

Strictly speaking, when we make correct computations in arithmetic, the results are logically valid. Following correct procedure results in a valid conclusion. Do you recognize the commonly held distinction between true and valid? A valid conclusion is not necessarily true, because it requires also that the premises are true. If we hold that axioms (as premises) are neither true nor false, or that truth and falsity is not relevant to axioms, then we cannot claim that correct computations provide us with truth.

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.