The substance of our disagreement seems to be the following. I think that if physical objects are real (not simply ideas), then the relations between them must also be real (not just ideas). You seem to believe that physical objects are real, but the relations between them are not real (the relations are simply ideas).

I've tried to explain how I understand your belief to be inconsistent. If objects like the earth and moon are real, then their movements must be real as well, and their relations reals too.

There is another possible resolution which we have not explored, and that is that neither objects nor their relations are real, they are all ideas. That is known as idealism, and I think it is well supported by modern physics. If we follow the principles of quantum physics we will find that all objects are composed of particles, which are features of fields. The fields are constructed from ideas about relations.

Would you consider this as a possible way of understanding "reality"? My point has been, that we cannot make a separation between a physical object and its relations with other objects, to say that one is real and the other ideal, without separating the object from its activities, one real, the other ideal, and this necessarily produces an inconsistency in our representation of "reality". Since you appear to be dead set against allowing the proposition that both, objects and their relations are real, would you be more inclined toward the other option, the one which is well supported by the science of physics, that both, objects and their relations, are ideal?

That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. — Ludwig V

In the thought experiment, the allotted amount of time cannot pass. The switch must complete an infinite (endless) cycle of on/off before the allotted time can pass. The endless cycle cannot ever be finished, so there will always be more switching to do, and the allotted amount of time will never pass. This is just like Zeno's thought experiment, Achilles will never pass the tortoise, because there will always be more distance to cover first..

Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. That is reality So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider.

We'd think that the rational human being ought to choose "reality" over the ideas of the thought experiment, then we'd reject the nonsense. But this "reality" is concerned with "what will happen", and Hume's problem of induction applies. How do we know that there will be a tomorrow? Because there always has been in the past. How do we know there will be a next hour? Because there always has been. How do we know that there will be a next moment? Because there always has been. However, "because there always has been" does not provide proof that there will continue to be into the future.

Therefore "reality" concerning "what will happen" is lacking in certainty, due to the problem of induction. And, theoretically, a system which prevents the allotted amount of time from passing, through a mechanism similar to the one of the thought experiment could possibly be arranged. Imagine that what is really represented is a continuous slowing down of "our time". Imagine that the mechanism is in a different time frame, so that in the different time frame, the switching on/off is at a constant speed. From our perspective, the switching appears to get faster and faster, but what is really happening is that our time is passing slower and slower in comparison to the other time frame. As it slows more and more and more, it approaches a complete stop, without ever reaching that complete stop, so that every tiny fraction of a second which goes by in our frame, is extremely long in the other. Then it is actually going so slow in comparison to the other time frame, that a very large number of switching can occur in a very short time, and so on as it approaches an infinite amount.

The issue is about how to perform a thought experiment - how much of reality you can import into the story. — Ludwig V

That's exactly right. And as I told Michael, way back, in the beginning, if the rules allow for zero importation from the "real world", (which is distinct from the possible world of the supertask), then the allotted amount of time known as the limit, can never pass. And the supertask dilemma never gets off the ground. When Michael insists that the duration must pass, "reality" is needlessly being imported into the thought experiment. We have two incompatible possible worlds.

That the allotted amount of time must pass, if true, is enough evidence to reject the supertask as impossible. However, if we attempt to prove that the amount of time must pass, we run into problems, like those exposed by Hume, namely a lack of necessity in the continuity of time.

Something central to my understanding is something I learned from Klein - "the myth of anamnesis or recollection." A myth Klein points out that Socrates tells from hearsay. The significance of this should not be missed. It is something he has heard, a story, not something he knows from the experience of recollection. He shows how and why the story is problematic. — Fooloso4

The theory of recollection is a very small aspect of Plato's writing. And, where it is displayed in the Meno, it is a matter of interpretation as to whether Plato is supporting this idea, or is simply describing it. The theories of recollection, and participation, are the two principal supports of Pythagorean idealism. Socrates questions Pythagorean idealism endlessly, in the manner of skepticism. Plato exposes the weaknesses of Pythagorean idealism as the theories of participation, and recollection.

To be "Platonist", in the sense of supporting Plato as a very formative philosopher, it is is not necessary to follow Pythagorean idealism, or Neoplatonism. Aristotle rejected Pythagorean idealism, and is often claimed to have effectively refuted that form of idealism, to proceed with another form of idealism which he learned from Plato.

I think Fooloso's approach discourages people from reading Plato, and that's unfortunate. — Leontiskos

I tend to agree with this. Fooloso4 has a very hardened way of looking at Plato. It appears like an opinion of Plato as useless. But to make that argument, there is a tendency to portray Plato as misleading. There's a very big difference between these two. Useless is simply non-productive, having no effect, and that is basically to say that there is no substance there at all. To say that Plato is misleading, is to acknowledge philosophical substance, and claim that it is wrong, pointing us in the wrong direction. Fooloso4 tends to argue both about Plato, without distinguishing one from the other, and without revealing what is truly believed.

The point of the story is that this problem hasn't been solved. And it comes down to the problem in the story. — ssu

That's one problem. I'm sure there's many more. Find another, and start another thread.

You win. — tim wood

Hooray! I'm going to go celebrate. Care to join me for a glass of champagne? Fuck the queen, or the biscuit, or whatever you're talking about, let's just celebrate!

I hope you meant that actions taken outside the system are neither consistent nor inconsistent with the rules. Could we not express this by saying that the rules don't apply, or that it is not clear how to apply the rules, in the new context? — Ludwig V

No, I mean they are inconsistent. To be consistent with the rules is to act according to the rules. Actions which are outside of the rules are not according to the rules, therefore they are inconsistent with the rules.

Physical collections have inherent order. Sets don't. That's all I'm saying. You seem to agree. What are you concerned with then? — fishfry

The reason why physical collections are different from sets, in this way, is that physical objects are different from intelligible (including mathematical) objects. What I am concerned about is that the law of identity, as formulated from Aristotle, is specifically designed from a recognition of this difference, and intentionally designed to protect, and maintain the understanding and acceptance of that difference. To put it simply, an abstraction, intelligible object, is a universal, and a physical object is a particular. The law of identity refers to the identity of a particular. And, because intelligible objects are different from physical objects, as you recognize and acknowledge, they cannot be held to this law. So mathematical ideas, if they are called "objects", are objects which naturally violate the law of identity. In short, that's how we distinguish physical objects from ideas, with the law of identity.

In classical sophistry physical objects are confused, mixed up, and conflated with intelligible objects. The difference between the particular and the universal, as "objects" is ignored. This allows sophists to logically prove things which are absurd. The law of identity is intended to enforce that difference, and expose the faults of the sophist. The head sophist at TPF, TIDF, continues to defend sophistry by arguing that intelligible objects are consistent with the law of identity.

Is this related to the intensional and extensional meaning of symbols as has been discussed previously? — fishfry

I'd have to say, no, not really. Internal/external properties is a distinction we make concerning the properties of particular physical objects, the object's internal relations, and the object's external relations. Intensional/extensional meaning is a distinction concerning the meaning of a word, how the word relates to ideas, and possibly physical objects. This is a matter of semiotics, and Charles Peirce provides some very good insight into the use of symbols. But that is a completely different matter from what I was discussing, as the internal/external properties of a physical object.

I have no idea what topic you are discussing at this point. I've agreed with you about the playground and I thought I'd explained to you about sets. What is left to discuss? — fishfry

The problem, is that you continually cross the boundary of separation between physical objects, and intelligible objects, in your manner of speaking, in the sophistic way, without even noticing it. That's what happened with your example of schoolkids. In order for the example to work, "schoolkids" must refer to a multitude of particular physical objects. Yet "set" must refer to an intelligible object. So in speaking the example you cross the category separation, back and forth in the way of sophistry, without even realizing it.

Imagine if we were to maintain the boundary. Instead of having schoolkids in a playground, we would be talking about the idea of "schoolkid", or an imaginary schoolkid. This appears to deny the possibility of any extensional meaning. Further, if we want a number of schoolkids, then we need a principle of separation to distinguish one from the other. But that principle of separation would either create an order amongst the imaginary schoolkids, or else produce a complete separation of type, making distinct types of schoolkids.

I understand what you're saying, I just don't have much interest in the subject. — fishfry

OK, you have no interest in the difference between a subject to be studied and an object to be studied. That's fine by me, but until you learn this difference you are likely to continue to speak in a way which mixes these two up, and makes your examples and arguments appear like nothing more than sophistry, and arguing by equivocation, just like Tones. This is what happens when a subject is called an object (mathematical) and the difference between the physical object and the mathematical object, (as defended by the law of identity) is ignored.

I am not your philosophy professor and this is not going to get you a good grade in my class. Why are you telling me all this? Honestly. I don't get it. I'm sorry. — fishfry

That's right, you are not my philosophy professor, that would reverse credentials. I am your philosophy professor, and your lack of interest deserves a failing grade.

I'm not referring to internal properties of anything. 2 and 3 are MODELED within set theory as sets. — fishfry

Right, this is why a set is not an object, objects have internal properties and external properties, sets have meaning.

There's only one instance of each set. You seem to disagree. Don't know what to say. — fishfry

There is no "instance" of any set. You recognize that there is a difference between physical objects an sets, why do you not see that there is no such thing as an instance of a set? Sets are not the type of thing which have an instantiation. "Instance" refers to a particular, a set is a universal. That sort of misleading statement is where the sophistry kicks in, even though I know you are not intending to be misleading..

How can you disbelieve in a rule in a game? — fishfry

That's a simple question with a simple answer. When a rule in a game contradicts another rule in a game, this is cause for disbelief in the whole game. That was the point of the example I gave you of waves in physics.

Nobody but you is making ontological or metaphysical claims about sets. — fishfry

That has become obvious to me. But in a philosophy forum, things ought to be the other way around. We ought to be discussing the ontology of sets and working through the problems which arise.

If you could just clearly summarize your concerns, it would help. The internal and external stuff, I'm sure it's interesting, but I was not able to relate it to anything we've ever talked about. So just toss me a clue if you would. — fishfry

There's too many concerns to summarize. But let's look at a most fundamental problem of set theory as an example. You recognize the difference between physical objects, and sets, so let's start there. Now, consider the elements of a set, these might be sets as well. The elements of a set are not physical objects, just like sets are not physical objects. The elements are ideas, universals, they are not particulars or individuals. Since they are not particulars the set cannot be measured as particulars. A set cannot have a cardinality. That's a basic problem.

If you flip a coin it might be heads or tails. That doesn't mean it can't be both at different times. — fishfry

This doesn't make sense. Each flip of the coin is an individual act, and it has a single outcome. Once the outcome is achieved, that outcome stands until there is another flip. The outcome "can't be both at different times", because a different outcome requires a different flip. However, there can be different outcomes from different flips.

The claim was that multiple possible outcomes of a process is inconsistent. Not so. Each outcome is consistent with the rules of the problem. There's nothing inconsistent about a lamp being on sometimes and off other times. — fishfry

The claim was directed at your example of choosing a direction at a fork in the road. The only way that you could have multiple possible outcomes is by assuming a principle that overrules the rules, i.e. transcends the rules. Freedom of choice, allows you to choose rather than follow a rule. If your example is analogous, then multiple possible outcomes being consistent with the rules, implies that choice is allowed, i.e. the rules allow one to transcend the rules. Strictly speaking the actions taken when the rules are transcended are not consistent with the rules, because these actions transcend the rules. The rules may allow for such acts, acts outside the system of rules, but the particular acts taken cannot be said to be consistent with the rules because they are outside the system.

The question is the nature of the existence of that relationship. — tim wood

See, you actually do believe in the reality of such relations, you just do not understand the nature of the existence of that relationship. As I said, it is one of the many things which are unknown. Here is the mother of all such relationships, the relation between space and time. You seem to like describing relations in terms of spacetime, and within those descriptions is assumed a relation between space and time. But in reality the nature of the existence of that specific relationship itself, is the biggest of all the unknowns.

They wouldn't fit in the universe - on the assumption they take up space, however small! — tim wood

Like ideas, they are immaterial. Why would they take up space? Then again, maybe what we call "matter" is simply the manifestation of a particular type of relations, which do take up space. Since the nature of the existence of relations is unknown, we really cannot exclude anything, can we?

The expression the "One" has a different life in different texts as do so many other ideas and perspectives. — Paine

Have you read how these differences are explained in the Metaphysics, especially Bk3, ch4 and BK13, ch6? The issue I think, is that all of the different ways which "the One" is held to be first principle, are unacceptable.

That's why the task was for the philosophers "to tell a way to feed all the dogs on the beach without any dog being left out hungry and Themis would make this instantly to happen". — ssu

Then why isn't Plato's way the proper way? There's no need to determine the dog which eats the most or the dog which eats the least, just keep feeding in the way Plato described.

What is it, then, that the relation refers to that might be real. — tim wood

"Relation" refers to what the one has to do with the other, what I described as orbiting and you described as corkscrewing. If the things (E&M) are real, don't you think that their motions are also real? And since their motions involve each other, isn't this a real relation?

And the E and the M apparently alter spacetime. The alterations apparently effecting the exact path of both through spacetime. — tim wood

That looks like a relation (what one has to do with the other) to me. Why deny that it's real?

All of this a description, and no mention of relation or the existence or causative efficacy of any relation. — tim wood

Of course we can describe a relation without mention of the word "relation". But don't you recognize that when you say the E and M alter spacetime, and this effects the path of both, that you are speaking of "causative efficacy"?

Anyway, that's an unnecessary point because "relation" does not necessarily imply causative efficacy. When two things are doing something together (as in your description of the E and M) there is a relation, regardless of causation.

Why are you so hellbent on denial, that you describe the relation between the earth and moon in such an extraordinarily strange way, trying to be careful about what you say, and what you mean, intent on hiding the fact that you actually believe there is a real relation between the earth and moon?

Great, what is relation? — tim wood

I told you numerous times, "relation" is what one thing has to do with another. Please quit accusing me of not answering.

Great, what is relation? I keep asking and you keep not answering. I ask what they do, and you answer that they do things and their activities are related. All you're telling me is that your not very good - or no good at all - at reflectively questioning your own thinking. This the state of a naive thinker who has taken certain things for granted and having done so, is incapable of further testing them or thinking about them. If relations are real in your sense please provide an example, which of course cannot be an idea. — tim wood

I don't see any reason for this assessment. Each relation between individual things is distinct and unique. You asked me for the general, "what is relation" and I answered. I also gave you an example of a distinct and particular relation, that between the earth and the moon, and you criticized me for not describing this relation properly. That does not mean that I did not give you a particular example, it only means that I did not describe the relation, which served as my example, to your satisfaction. And, as much as the earth and moon are not ideas, neither is the relation between them an idea.

We're corkscrewing around each other in circles, because you have a mental block which prevents you from understanding the meaning of "relation". You force an unwarranted restriction on the meaning of this word, 'a relation is necessarily an idea'. This is the mental disability of a closed mind, a disorder which you willfully inflict upon yourself.

We have differed in the past on what the consequences of De Caelo are on the divinity of the celestial sphere and I remember you do not accept the account of divinity in Metaphysics book Lamda. So, I will leave all that be. — Paine

Do you not agree, that in De Caelo Aristotle begins by agreeing with those who promote it, explaining that eternal circular motion is a valid concept, and a real logical possibility. However, he then proceeds to assert that anything moving in a circular motion must be a material body, and as a material body it must have been generated and will be destroyed. If you agree with these two aspects of De Caelo, you ought to also agree that what Aristotle has done is that even though he has accepted the logic of eternal circular motion as a valid logical possibility, he has dismissed eternal circular motion as 'physically impossible'.

I am glad we could find common ground on the role of forms in the dialectic. — Paine

Yes, I agree that is an accomplishment. But the significant issue is the direction which Plato points us. And since Plato appears to be pointing in a number of different directions, people can take up a number of different ontological, or metaphysical positions, and claim the position to be Platonic. This we see in the difference between Aristotle and Plotinus for example. Aristotle argues that the first principle, i.e. anything that is eternal, must be something actual. Plotinus assumes the first principle "the One" to be pure potential. Notice above, that the eternal circular motions are for Aristotle, possibilities whose actuality is denied.

To me, the direction taken by the Neoplatonists which gives priority to mathematical objects, in the manner of maintaining Pythagorean idealism, is a dead end. The inquiring in this direction culminates with Plotinus, who meets the brick wall of assuming the first principle "the One" as a pure potential, because then he has no principle of causation to account for the emanation or procession of Forms and beings from the One, in a hierarchical order. The hierarchical order is very good, and well thought out and constructed, but the problem is with the first principle, pure potential provides no source of causation.

Aristotle, on the other hand effectively refutes Pythagorean idealism, and those Platonists who follow that path, by demonstrating that anything eternal must be actual, and showing that mathematical, and geometrical forms, like the one, and the circle, exist only as potential, prior to being discovered by the human mind. Therefore such forms cannot be eternal. Notice that Christian theology, following Aquinas, represents God as pure act.

And I thought in my ignorance, that there's at least this obvious limit in Physics! Of course, what is Physics else than the study of change and movement? So there's big problems to get funding for a research on the effects of temperatures of negative millions of Celsius. Fortunately there's an actual reality to seek something else. — ssu

I've read some speculations showing that the hottest temperature will actually end up being the same as the coldest temperature. Strange.

The task was to feed all the dogs. — ssu

If that's the case then both Plato's dog and Zeno's dogs are irrelevant, all one needs to do is point the dogs to the food and tell them to go to it. When they all get fed the task is complete. Since it is stipulated that the quantities are unlimited, the task will never be completed, some dogs will not finish eating before Plato and Zeno pass on.

However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile: — javi2541997

Not under the assumption that quantities are unlimited.

Err, isn't there actually an absolute lowest temperature, - 273,15 Celsius? We cannot talk then about a temperature of - 2 000 000 Celsius or lower temperatures to my knowledge. So this isn't similar to the problematics of the Zeno's dogs in the story (or at least the other one). — ssu

That is the lowest temperature realizable from our methods of measurement. In other words it is a restriction created by our choice of dog to use for comparison, the movement of atoms. It does not mean that a lower temperature will not be discovered, if we devise a different measurement technique. Notice there is no such limit to the hottest possible temperature, because we move to different measuring principles.

Are you suggesting that it is irrelevant to Plato whether there is a dog who eats the most and another who eats the least? Well, maybe. — javi2541997

That is exactly what I am suggesting. Plato was given the task of measurement, and he took that task and proceeded. That the task will never be completed because the quantities are unlimited, is irrelevant. Therefore whether or not there is a dog that eats the least or the most, is also irrelevant.

But the rules stated by Athena say: ‘All the dogs eat the same food, which is divisible, and there is enough of it for every dog’ but Zeno argues (and I agree with that) that by randomly picking up a dog and then starting to count from it the various quantities other dogs, was missing at least these two dogs, one that ate the least and one that ate the most. — javi2541997

The "other two dogs" referred to by Zeno is a sophistic ruse, just like Plato says. Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved. Instead, Zeno said you are "forgetting" these two dogs. But Plato is not "forgetting" them, he has not yet found them, so there is no need for them to have ever entered his mind.

Ok. For things in the real world, they are already in some order, even if it's a complete state of disorder. Even a completely disordered collection of gas molecules in a container, at every instant each molecule is wherever it is. And that set of coordinates, locating every molecule in space, is the order.

I get that. But by the same token, there is no preferred order. Suppose for example that I got my schoolkids from the playground to line up single-file in order of height. And now YOU come along and say, "Ah, that is the inherent order, and all other orders are disorders of that."

But of course your observation was a complete accident. I could have lined them up alphabetically by last name.

So even among physical objects, if we allow that they are always in some order, even if it's disorderly; but nevertheless, there is no preferred or inherent order.

I believe you are saying there's an inherent order, have I got that right? — fishfry

I think we have to look at context here. What is our subject of discussion, what are we talking about here? Are we talking about things (individuals), of which there is a multitude, or are we talking about a group (set) of individuals, of which there is one? Your description above, seems to imply the former. You are talking about separate things, many schoolkids, and there is many possibilities as to the order they could have. On the other hand, if you were talking about the group as a whole, as your subject, then the parts of that group, the individuals, must have the order that they have at that time, even though it could be different in past or future times. If you were talking about the same individuals in a different order, this would require a change to that specific group, so you would be talking about that group, at a different time, because you'd be talking about the individuals, changing places.

You might understand this better through what is known as internal and external properties. To each individual, as a subject, its relations to other individuals are external properties. To the group, as a unit, and the subject, the relations between the individuals is an internal property.

You talk about the schoolkids as distinct individuals, where the various relations between them are the external properties of each and everyone of them. There are no internal relations here. Each schoolkid is a subject to predication, age, height, etc.. and you might produce an order according to those predications. The order is external to each schoolkid, people say it transcends, and changing the transcendent order does not change any of the schoolkids in anyway.

Now, let's take the group as a whole, as an object, and produce a corresponding subject, the set, and make that our subject. Since the whole group is our object of study, any change to the order of the individuals is an internal change to that object, therefore a change to that object itself. The order of the individuals (as the parts of the whole) is an internal property of that object, and a change to that order constitutes a change to the object, which we must respect as predicable to the corresponding subject. Therefore we can say that the order of the individuals, as the parts of the whole, is an intrinsic property of the whole, which is represented as the set.

Notice however, the switch from "subject" to "object", and this I believe is the key to understanding these principles. There is an implicit gap, a separation, between the meaning of "logical subject" and "physical object". When we make a predication, "the sky is blue" for example, "the sky" is the subject, and if there is an object which corresponds with that subject, the predication may be judged for truth. However, we can manufacture subjects and predications with complete disregard for any physical objects, and so long as we have consistency, we have a valid "subject", with no corresponding object.

Consider the following proposition, "There is a group of schoolkids". We have a propositional subject, without a corresponding object, what some people would call "a possible world". Since there is no assumed corresponding object which would cause a need for conformity, we can predicate any possible order we want, so long as it is not contradictory. The hidden problem of formalism which I referred to lies in the naming of the group, "schoolkids". That name needs to be clearly defined and the definition will place restrictions on what can be predicated without contradiction.

Perhaps, we can remove these restrictions, by making the things within the group, the elements of the set that is, completely nondescript. "There is a group of nondescript things". We still have the name "things", with implied meaning, so this name has to be defined, and this would put restrictions on what we can predicate. So we go to a simple symbol, "x" for example, and assume that the symbol on its own, has absolutely no meaning, and this would allow any individual predication whatsoever without any risk of self-contradiction. X is a subject which has absolutely no inherent properties.

It might appear like we have resolved the problem in this way, we have a subject "x" which can hold absolutely any predication, so long as the predications don't contradict.. However, when we assume that the subject has no inherent properties, we disallow any predication because the predication would be a property and this would contradict the initial assumption. So this starting point allows no procedure without contradiction.

Now look what happens when we say "there is a group of x's". There is actually something implied about x, which is implied simply by saying that there is a group of them. It is implied that x has a boundary, separation, etc.. We may start with the assumption that there is no intrinsic properties of X, but as soon as we start to predicate, we negate that assumption. And the symbol, x, without any predications is absolutely useless.

Well now that you mention it, no. 1, 2, 3, ... is NOT the inherent order of the set N

, believe it or not. On the other hand it sort of is, in a sneaky way. Von Neumann defined the symbols 1, 2, 3, ... in such a way that n∈m

∈

if it happens to be the case that we want n < m to be true. — fishfry

I agree, what I meant is that this appears to be the inherent order, but it's not necessarily, that's why I went on to say that we can deny that order.

I know this is hard for normal humans to accept, since it's pretty obvious that 1 < 2 < 3 and so on. But mathematicians insist on being picky about how numbers and other things are defined. In the set-theoretic view of modern math, the numbers 1, 2, 3, ... are defined as particular sets, with no inherent order; and then we impose their order by leveraging the ∈
∈
operator. — fishfry

I think I see the need for this, and so I understand it.

Have I got any of that right? — fishfry

I think so, but I also think, that sort of inherent order has minimal effect, and the real issue comes up with the restrictions, or limitations to order which are constructed. What I am arguing is that how the inherent order manifests, is as a limitation to the order which one can select. If there is absolutely no inherent order, then we can select any order, but if there is limitations to what can be selected, we cannot choose any order. The examples you give are, I believe, selected, therefore they're probably no true inherent order. The example I gave, is that we cannot give 2 and 3 the same place in the order, they cannot be equal, so we need to proceed toward understanding how this limitation exists.

Anyway, back to the question. How do we know that 2 and 3 are not the same set?

Well 2∈3
2
∈
3
, but 2∉2
2
∉
2
.

Therefore by extensionality, 2≠3
2
≠
3
, because they don't have exactly the same elements.

Perhaps you can begin to see the virtues of working a the set level separately from its order properties. We can see the mechanics of how to use the axiom of extensionality. No order properties are needed to determine that 2 and 3 are different sets. It's just a matter of ignoring hypotheses that you don't need for a particular argument.

Nobody is saying that a given set doesn't have an order, as well as a lot of other stuff. A topology, some algebraic operations, a manifold structure perhaps. But we can learn a lot just from restricting our attention to the membership relation and seeing what we can learn just about that. — fishfry

So this is where the real problem lies, in defining a symbol, such as 2 or 3, as a set. Check back to what I said about the difference between internal and external properties. The subject now is a set, say 2, and a set necessarily has internal properties. We have the elements which compose the set, 0,1, which are also sets. As the set is also related to other sets, it has external properties, represented by the ∈
operator. The external properties are not necessary, and are a matter of choice, but whatever choice is made, that choice dictates the nature of the internal properties.

Now here's where I think the illusion lies. A set necessarily has internal properties, even though there may be infinite possibility as to the nature of the internal properties, making the specific nature of the internal properties dependent on choice, in this case von Neumann's choice. The illusion is that since the specific nature of the internal properties is dependent on a choice from infinite possibilities, it would therefore be possible to have a set with no internal properties. Clarification of the illusion implies that a set cannot exist prior to the choice of external properties, which dictate the internal properties. Internal properties are essential to "a set", and so a set has no existence prior to the choice of external properties, which determine the internal properties. This makes the empty set, as a set with no internal properties, impossible. The problem now, is what is zero? It can't be a number, because numbers are sets, and an empty set is impossible.

Then you have been proven wrong. I don't need to mention or consider or use any of the order properties of 2 and 3 to determine that they're different numbers. — fishfry

I think you misunderstand. As I explain above, you refer exactly to the internal (intrinsic) properties of 2 and 3, as sets, to show that they are different numbers. What the set theory has done is denied order as an external property of those things, 2 and 3, as numbers with order relative to other numbers, and made it into an internal property of those things, as sets. An internal property is an intrinsic order. The fact that the intrinsic order is ultimately dependent on choice is irrelevant, because some order must be chosen for, or else the system would be meaningless.

Entirely without rational basis. This para is a wild generalization of your complaint about 2 and 3, but I already showed how we can distinguish 2 and 3 using only their membership properties and not their order properties. — fishfry

No, you've simply shown how external order has been switched for internal order. And now I've shown the problem which arises from this switch, the contradictory, therefore impossible "empty set", which makes the inclusion of zero an inconsistency.

You are thrashing away at a strawman you've created out of your imagination, and under the mistaken belief that we can't tell 2 from 3 without their order properties. But we can. — fishfry

As I say, the idea that you've gotten rid of the order properties is just an illusion. The order inheres within each individual number, as the definition of that specific set. Rather than simply being an external property of a number, as an object, and how it relates to other numbers, order is now an internal property of the number itself, as a set..

No, you are consistently wrong about this. If A and B are sets and I can prove that A = B, then A and B are the same set. They are in fact the identical set, of which there is only one instance in the entire universe. They are NOT "two copies" or two distinct entities that we are calling the same by changing the meaning of the word "same." — fishfry

I argue the exact opposite, that you are consistently wrong about this. It is exactly "two copies", just like the word "same" here, and the word "same" here, are two distinct copies, even though we say it's the same word. Look, we are talking the meaning of symbols here. "A=B" means that that symbol A has the same meaning as B, it does not mean that A signifies the same entity as B, without additional information. However, the additional information in this case indicates that what is signified by A and B is a set, "the same set". But a set is not a thing, it is a group of things, grouped by a categorization such as type. Therefore this is an instance of "the same meaning", signified by A and B (indicated by "type"), not an instance of the same entity signified by A and B. This is just like when we use the same word twice when the word has meaning, rather than referencing a particular object. We say that the word has the same meaning, just like we might say A and B have the same meaning, in your example.

DUH that is what it MEANS to be the same set. That is the ONLY thing it means to be the same set. — fishfry

Exactly, and this is a different meaning of "same" from the meaning of "same" in the law of identity. That is the point. In the law of identity "same" means a lot more than simply having the same members (what I called a qualified "same"), it means to be the same in every possible way ("same" in an absolute, unqualified way),

Yes that is what it MEANS for two sets to be the same. That they have the same members. That's ALL it means and EVERYTHING it means.

You simply can't accept that and I don't know why. — fishfry

I totally agree with that, that's what "same" means in this context. The problem is that it does not mean what you stated above: "They are in fact the identical set, of which there is only one instance in the entire universe". The set is an imaginary thing, indicated by meaning, it is not something in the universe. So it's not even coherent to say that there is one instance of that set, it's not even a thing which has an instance of existence, it's just the meaning of a symbol. So you speak of "the same set", and claim there is only one instance of that set, but this would be taking a different meaning of "same", which refers to instantiated things, and applying it to "same set", which really means having the same meaning, and not referring to one instantiated thing. Do you see the difference between referring to one and the same thing with a name, "MU", and using a word which has meaning, like "person", without any particular thing referred to? Person refers to a type, so it has meaning, just like "set" refers to a type, so it has meaning. These do not refer to instantiated things of which we could say there is one instance of, they refer to ideas.

Why? I drive down the road and come to a fork. One day I turn left. Then next day I drive down the same road and turn right.? What logical inconsistency do you see to there being multiple possible outcomes to a process that are inconsistent with each other, but each consistent with the rules of the game? — fishfry

You have a hidden element here, known as freedom of choice. The "multiple possible outcomes" are only the result of this hidden premise, you have freedom to choose. That premise overrules "the rules of the game", such that the two are inconsistent. In other words, by allowing freedom of choice, you allow for something which is not "consistent with the rules of the game", this is something outside the rules, the capacity to choose without rules.

Consider this example, suppose we want to set a scale to measure all possible degrees of heat in the vast variety of things we encounter, a temperature scale. We could start by determining the highest possible temperature, and the lowest possible temperature, (analogous to Zeno's dogs) and then scale every temperature of every circumstance we encounter, as somewhere in between. Alternatively, we could start with one temperature, the freezing point of water for example (analogous to Plato's dog), and scale the temperature of all other things we encounter relative to this. Whether there is a hottest or coldest possible temperature is irrelevant to this alternative way of scaling.

Incidentally, I think this issue is relevant to the way that we judge goodness and badness in moral actions, and create codes of ethics. Some would argue that we need a best, the omnibenevolent God, and a worst, the evil devil, and all moral acts are judged in relation to these two. Others however, argue that we take any random act, and judge whether other acts are better or worse than it. I would argue that the latter is the common way that people make decisions. If a person is inclined toward a particular act (this represents "Plato's dog"), they will look at other possibilities, and judge these possibilities, each one, as to whether it is a better or worse course of action in relation to the one that the person is inclined toward. The person will choose accordingly. I believe that it is not often, that in making a decision, the person judges the possible act as to whether it is closer to what God would choose, than what the devil would choose.

I mentioned the "corkscrewing through spacetime" only as against your idea/relation/model of the moon "orbiting" the earth. Now, take a moment and try to think through exactly what the earth is doing and what the moon is doing. I think you will see that any "relation" between them is an idea that comes from you. — tim wood

No, I recognize that the earth and the moon are doing things, and that their activities are related. You apparently recognize this to, by describing it as a 'corkscrewing" activity. You, however refuse to separate the description "corkscrewing", which is an idea, from the reality of what the relation actually is, which is not an idea. So you keep insisting that the map, (model, or description, which is an idea) is the territory, (the relation itself). Then you contradict yourself by claiming that the earth and the moon which are engaged in this interrelated activity exist independently of human ideas, yet the interrelated activity which locks them together as an essential aspect of the existence which they have, is just a human idea.

To show one way how an at least 2400 year old (but likely older) difficulty in mathematics emerges, which hasn't gone away. You should read the answer that I gave to L'éléphant and @javi2541997 here. It gives also a question for further thinking. — ssu

Yes, I already read that, and I didn't see much to disagree with, except your question at the end.

Do you have a source that touches on how Aristotle's text was produced? — Paine

I did some quick Google research for you, and dug up the following. References are at the bottom.

What is called "Metaphysics" is a collection of works, which were taught in a school in Rhodes. This was a separate school from Aristotle's school the Peripatetic school in the Lyceum at Athens. These papers were put together into a single collection, "Metaphysics" some time (a couple centuries I believe) after Aristotle's death by Andronicus of Rhodes. It is not known how much of the material was produced by Aristotle himself, because Andronicus provided no indication of how he authenticated the individual writings he collected together under that title.

Apparently, the writings now known as "Metaphysics" were in the possession of one of Aristotle's students, Eudemus of Rhodes, after Aristotle's death. The writings were unpublished and Eudemus supposedly had the only copy. Eudemus and Theophrastus were two of Aristotle's top students. Aristotle appointed Theophrastus to head his Peripatetic School, and Eudemus went back to Rhodes (supposedly with the only copy of what is now known as the Metaphysics) to open his own school.

You can see why there would be much debate concerning the authenticity of the Metaphysics, as truly Aristotelian writings. Eudemus of Rhodes apparently provided no solid evidence to support his claims that this material he taught was actually authored by Aristotle. It is commonly believed that this material was notes taken by Eudemus, from Aristotle's teaching.


https://plato.stanford.edu/entries/aristotle-commentators/supplement.html
https://www.britannica.com/biography/Andronicus-of-Rhodes
https://www.philosophy.ox.ac.uk/event/workshop-in-ancient-philosophy-thursday-week-4-tt21
https://www.philosophie.hu-berlin.de/de/lehrbereiche/antike/mitarbeiter/menn/editors.pdf
https://en.wikipedia.org/wiki/Eudemus_of_Rhodes
https://mathshistory.st-andrews.ac.uk/Biographies/Eudemus/

Are you suggesting that when "Platonists" are mentioned in Aristotle that others are speaking in his name? — Paine

You'll notice that much of the material which Eudemus of Rhodes taught was derived from notes taken from Aristotle's teachings, and this is probably the case with the Metaphysics. I believe that at the time when Aristotle was teaching, the divisions between different schools of Platonism had not yet been established. Eudemus is well known for his work on mathematics and principles of geometry, and Plato's academy was a school of skepticism. In the metaphysics, where there is a pronounced separation from the other Platonists it has to do with the nature of mathematical and geometrical ideas, forms. Iam not familiar with Eudemus' famous works on mathematics, so I cannot confirm this speculation.

While we can guess the first Academicians would have taken issue with Aristotle challenging the separate land of the forms, it is unlikely they would have disagreed with Parmenides who sharply protects the boundary between the divine and the world of becoming that we muck about in: — Paine

This boundary is exactly what is attacked in Aristotle's De Caelo. The eternal circular motions of the heavenly bodies are supposed to support the eternal existence of the divine. But Aristotle demonstrates how anything which moves in a circular motion must be a body composed of matter, and is therefore generated and will be destroyed. This effectively breaks the boundary between the divine (eternal) and the earthly world of becoming.

This is a far cry from the mono-logos of Plotinus where the divine is a continuity from the highest reality to the lowest. The dialectic descends into the silence of contemplation. — Paine

Very true, and this indicates the separation between Neoplatonism and Aristotelianism very well. This passage forms the foundation for Aristotle's hylomorphism. Each individual material thing has a form specific to itself (the law of identity), as well as its matter:

“Indeed, Socrates,” said Parmenides, “the forms inevitably possess these difficulties and many others 135A besides these, if there are these characteristics of things that are, and someone marks off each form as something by itself. And the person who hears about them gets perplexed and contends that these forms do not exist, and even if they do it is highly necessary that they be unknowable to human nature. And in saying all this he seems to be making sense, and as we said before, it is extraordinarily difficult to persuade him otherwise. Indeed, this will require a highly gifted man who will have the ability to understand that there is, for each, some kind, a being just by itself, 135B and someone even more extraordinary who will make this discovery and be capable of teaching someone else who has scrutinized all these issues thoroughly enough for himself.”

“I agree with you, Parmenides,” said Socrates. “What you are saying is very much to my mind.”

“Yet on the other hand Socrates,” said Parmenides, “if someone, in the light of our present considerations and others like them, will not allow that there are forms of things that are, and won’t mark off a form for each one, he will not even have anywhere to turn his thought, since he does not allow that a characteristic 135C of each of the things that are is always the same. And in this way he will utterly destroy the power of dialectic. However, I think you are well aware of such an issue.” — Plato, Parmenides, 133e, translated by Horan

You appear to have a third category; the essence(?) of which you have yet to make clear. — tim wood

What do you mean i haven't made clear the essence of this third category. I've stated it over and over, it is the relations between things, what one thing has to do with another. Like ideas, relations are immaterial, unlike ideas, relations are independent of human minds.

So its up to you to make clear how they exist, as non-material, non-mind-based what-evers. — tim wood

There's a lot of unknowns in the world, and this is one of them.

To keep it simple, you say they're related, I say they are not, on two causes, 1) that relations are ideas and things don't have ideas, and 2) the "relationship" of earth and moon is a convenient fiction and artifact of ideas, and that the two have actually nothing to do with each other. — tim wood

Ok, that's what you think. I think you are in denial. You've already told me how you think that the moon and earth have something to do with each other, here:

What the moon and earth actually do in terms of these descriptions is that both revolve around a common moving center as they cork-screw their way along curved geodesics in space-time - or at least I think that's the most recent and accurate description. — tim wood

So now you are just contradicting yourself, to uphold your denial. Why are you afraid to admit that the reality of the immaterial extends far beyond the reality of human ideas.

Well, think in the story about how much all dogs eat, then remember the rules. — ssu

There is no general statement about how much all dogs eat. It is explicitly stated "every dog eats a different quantity of food", and " There are no limitations on the quantities". Therefore we cannot make any inductive conclusion about "how much all dogs eat", because each eats a different amount, nor can we make a conclusion as to how much all the dogs would eat, because this is stated to be unlimited.

What are you trying to get at?

So what are they? And also what are ideas? — tim wood

The definition of "relation" I already told you, "what one thing has to do with another". The definition of "Idea" is more difficult, my OED says "a conception or plan formed by mental effort". Does that suffice for you?

I believe that if material objects exist independently of a conception formed by mental effort, then it is also the case that what one thing has to do with another (such as the interrelated activities of the earth and moon which we discussed already) also exist independently of conceptions formed by human minds.

Why is this perspective so hard for you to apprehend? Imagine the earth and moon existing independently of human conception. Would these two things not be doing anything at all? And if they are doing anything, wouldn't they be doing something with one another, as being related to one another?

So what is it that you believe? To me, it seems like you believe that the earth and moon exist independently of mental effort, but you do not believe that they are doing anything. You seem to believe that doing something requires mental effort.

Apparently for you ideas are independent of mind, existing without mind... — tim wood

Tim, are you having trouble reading? I just got though explicitly telling you the opposite of this, twice in one short post. I told you:

you misrepresent my axiom as 'ideas exist independently'

...

therefore I do not conclude, as you claim, that ideas exist independently. — Metaphysician Undercover

In between those two quotes is the explanation of why I do not believe that ideas exist independently. Please, stop the straw man representation. How many more times will I have to make this request?

Obviously they can and do- called theories - and on the basis of applied criteria - experiments - work or are disproved. The word for this is "science." — tim wood

You seem to be having difficulty understanding the difference between a relation, and the description of a relation (map and territory confusion). The theory is not the relation. The relation between the theory and the experiment is not an idea. The relation between the theory and the experiment is represented with ideas.

Which implies you understood well-enough to judge it inaccurate — tim wood

I apprehended it as needing clarification, not as inaccurate. Therefore I did not judge it as inaccurate, I judged it as requiring clarification. Your use of "right" and "exists" in that context didn't make sense to me. The typo, which you edited in the reproduction also contributed to the not making sense to me.

Our contention is the existence of ideas. — tim wood

No, our contention is not the existence of ideas. It is the existence of relations. Your assumption (which I take to be wrong) is that all relations are ideas. From this faulty premise, that all relations are ideas, you conclude that our contention is the existence of ideas. That is not our contention, our contention is whether there are relations which are not ideas.

I infer that for you truth is comportment with some set of criteria. I call that truth-according-to. And with that standard, you can, for example, represent the movement of the earth and moon on a very Euclidean piece of paper and say that the moon orbits, goes around, the earth. And that would be true-on-paper, but not really true. Thus "truth" itself a possible source of great confusion. — tim wood

Yes, "truth" is a source of great confusion, especially if you represent it like this, that something could be "true-on-paper, but not really true". This confuses me immensely, such that I cannot understand what you are trying to say at all, in this paragraph.

An axiom for me is that material things, in terms of their existence, truly exist. For you to hold - in my view - that an idea exists independently is to hold that in some sense the idea is true apart from any notion of "true-according-to." But ideas can obviously be wrong, even impossible. And that would suggest that as ideas, they cannot so exist. - Unless you separate idea from its content. Do you claim not that ideas exist, but instead idea as an "empty vessel" exists without content? A very odd thing to claim if you do. — tim wood

You are completely avoiding the issue with this straw man representation. You have an axiom which says 'material things truly exist'. I have an axiom which says 'the relations between material things truly exist just as much as the material things themselves truly exist. You apparently cannot understand how a relation can be anything other than an idea, so you misrepresent my axiom as 'ideas exist independently' just like material things do.

The issue is, that we cannot get to your misrepresentation (straw man) unless we accept your principle that all relations are ideas. I do not accept that principle. Therefore your claim that my belief is of independent ideas is false. My belief is of independent relations. And, I do not accept 'all relations are ideas', therefore I do not conclude, as you claim, that ideas exist independently.

I offer this quick distinction for convenience without claiming rigour: that you discover them and I invent them. — tim wood

Try this instead. I say we represent relations, with models and such. That's why I use the map/territory analogy. You say we invent relations. I say your position makes no sense, because if we actually were free to invent the relations, then these ideas (what I call models or representations, and you call inventions), could consist of absolutely anything, and one would not be more true (in the sense of corresponding with reality) than another.

I invite you to try even to think about what that relation might be without yourself putting into it exactly what you're trying to find in it. — tim wood

This is exactly the sort of question, or invitation which totally confuses me and therefore I ask for clarification. I cannot at all understand "without yourself putting into it exactly what you're trying to find in it". When we model a relation, i.e. try to describe it, we are neither trying to put something into the relation (that is an act of art, construction, manufacturing, or production). nor are we trying to find something in the relation. When we represent a relation, by describing it or modeling it, we are attempting to understand it. That is neither finding something in it, nor putting something in it.

On the other hand, with Plato's dog, we can do something as important as count and measure. The first thing that mathematics evolved from, and something that smart animals can also in their way do. — ssu

Yes, Plato's dog is the point of comparison, the paradigm we might say, and this is the basis of measurement.

And in my view this measurement creates the confusion. Here with dogs that simply cannot be measured as their definition relies on this (if you could measure it, they wouldn't eat the least or most as Plato is totally right in this way). And I think this is the problem when we want to view mathematics as a logical system, but start from the natural numbers and assume something like addition is a meaningful operation with everything. Yet Mathematics, as a logical system, holds true mathematical objects that aren't countable or directly provable. I think we are still missing something very essential here. — ssu

Doesn't mathematics start with the unit, one, as the point of comparison, just like Plato\s dog, and from here we allow an unlimited number of units and also unlimited divisions of the unit. The actual problem is when we try to measure the system of measurement. The system of measurement is designed to allow for the measurement of any possibility, hence the unlimited, or infinite, numbers, and this makes it inherently unmeasurable. Then we need to go to another system of comparison, another paradigm other than measurement. This produces a problem because anything unlimited is fundamentally unintelligible because the way that the intellect apprehends things is through their limits. So the goal of measuring the system of measurement is self-defeating.

This is what I tried to refer to, when I said " it's a limitation, when you start from Plato's dog." Perhaps better wording would be simply a rejection. — ssu

I can see how starting (in this case at a dog), is a limitation, because the start produces a particular perspective. However, this is not a limitation on any quantity, physical or otherwise, as dictated by the premises: "There are no limitations on the quantities (physical or other), and hence on the dogs." The starting point, "Plato's dog" is a limitation on the act of measuring, imposed by choice, it is not a limitation on any dogs.

That is a constant question when reading Plato that does not come up in theories presented directly by others. — Paine

I think this is something which really needs to be respected when talking about "Platonism". For the most part, Plato, following the Socratic method, worked to expose problems. He offered very little in the way of theories presented as a means of resolution to the exposed problems. So what we have is Plato analyzing the theories of others, and poking holes in them. The reader has a very challenging task of understanding his logic, and why the holes are holes, in order to understand why the theories analyzed are deficient.

This leads to a big difference in interpretation, and a corresponding difference in proposed resolutions which follow, such as the difference between Neoplatonism and Aristotelianism. I believe the true test, for determining the most accurate, or "best" interpretation, is to look for the places where Plato actually presents some direction for resolution of the problems in knowledge, exposed by Socrates.

There is a number of such places, such as in The Republic, and in The Timaeus. The offerings are generally metaphorical, as the pathway to resolution is very unclear to Plato, and metaphor provides a broad route due to an even wider range of possible interpretations. Some examples are the comparison of "the good" to the light of the sun, and the related cave allegory, in The Republic, along with his discussion of "matter" as the recipients of the Forms, in The Timaeus.

I believe That Aristotle demonstrates a better understanding of these principles than the classical Neoplatonists such as Plotinus. He provides a proper representation of "the good", as "that for the sake of which", final cause, and does much work on this subject in The Nicomachean Ethics, and Metaphysics. This principle allows for activity, actuality, within the realm of Ideas and Forms, breaking the interaction problem of idealism. Participation theory renders the Forms as inactive, and cannot explain how Form is an active cause in the generation of individual material beings.

You'll notice in Aristotle's Metaphysics, (much of this being material produced from his school, after his death), how the Aristotelians distance themselves from those other Platonists, whom we call Neoplatonists. The other Platonists adhere more strongly to Pythagorean idealism, not understanding the problems of participation theory, revealed by Plato. They understand "the One" as the first principle, and try to make this consistent with Plato's "the good". Aristotle's Metaphysics shows how "the One" is really just a first Form, and cannot be "the good" itself; "the good" being something beyond the realm of Forms, as cause of the intelligibility of intelligible objects. So we commonly see modern day Neoplatonists representing "the good" of Plato as "the idea of good", or "Form of good", when Plato talks of "the good" itself. And the ancient Neoplatonists, such as Plotinus, claim that "the One" is something which transcends the realm of Forms, as an attempt to make it consistent with "the good". But the Aristotelians show that "the One" cannot be anything other than a first Form, and Aristotle continues onward to address "the good" in its causal relation with the Forms, as distinct from the Forms, like it is described in Plato's Republic.

I made an edit, a correction, from "what what," to "what with what." — tim wood

Good start tim. Maybe a few more acts of clarification, and I might be able to understand what you are asking.

Here's the problem. You take a couple snippets from my paragraph, present them out of context, and then draw a very strange conclusion containing three words "right" "true" and "exist". All I was talking about in that passage was "truth".

You should have simply asked, if it agrees with my criteria of truth, do I conclude that it is true, and I would have answered "yes". But "right" and "exist" are different words with different criteria, making your question appear absurd.

If you read the OP, I'm not asking how we distinguish objects. I'm asking how such distinctions are physical, not just ideals.
I give many examples illustrating what I'm after. — noAxioms

All distinctions are ideal, and not physical, aren't they?

That's a difficult question, concerning a difficult subject. Space and time were not well understood back then, and still aren't. Notice even today, the common convention is space-time, which considers time to be a dimension of space. Under such conceptions time is not separable from space. It could even be the case the examples were presented by Plato as a way of demonstrating a difference between space and time. Aristotle gave principles to understand space and time separately.

it is interesting that Parmenides does not introduce the sail metaphor as a rebuttal of Socrates' statement but as a description of it. — Paine

I believe that what Parmenides does with the sail metaphor is convert Socrates' temporal description to a spatial description. By the temporal description, we can say that it is the very same time in many different places, therefore one and the same time, in many different places. The sail, produces a spatial description, where each place has a different part of the sail. So I think that Parmenides really alters the image by switching from a temporal representation to a spatial one.

"This is what we observe.., and if we deny the validity..., then how can there be any truth ...?So then it would appear that for you, what you "observe" and that comports what what you think is the case, so that it agrees with your criteria for truth, must be right and true and exist.

Is that accurate? — tim wood

Sorry, I don't understand what you're asking.

But it's a limitation, when you start from Plato's dog.

Yet doesn't the dog that eats more than any other dog define it different from all other dogs? No matter if there's an Apeiron (endless amount) of dogs that eat less. — ssu

I don't understand what you're saying here. Can you explain?

There are an infinite number of quantities between 1 bowl of food and 2 bowls, just as there are between 1/infinity and infinity. — LuckyR

How is that relevant?

Couldn’t classical philosophy ascribe the unintelligibility of the world to the treachery of the senses? It wouldn’t have regarded ‘the world’ as possessing intrinsic intelligibility in the first place, would it? — Wayfarer

This is exactly the case. Plato, in his mind/body distinction claims that the body, along with its sensations misleads us, away from the good. The good is the source of intelligibility. So unintelligibility is proper to the world which the senses provides for us. This is Heraclitus' world of becoming, change and movement, where nothing "is". Aristotle goes further, to name a specific principle at the base of unintelligibility, "matter". Matter is placed in the category of "potential" what may or may not be, and this violates the law of excluded middle.

I think the counting here is for the sake of discussing how participation (μετέχειν) in forms is supposed to work now that the Stranger has brought the sharp separation between being and becoming into question. This leads to the discussion of "blending" forms which wraps up as: — Paine

I find that Socrates' best attempt at describing "participation" is when he explains how a form is like the day. I believe its in the Parmenides. It does not matter how many different places do or do not participate in "the day", the day is still the day, and participation by all these different places, no matter where they are, or how many there are, does not alter the day in any way.

Ultimately though, I think the theory of participation does not hold up to analysis.

So this got me thinking, and I could only conclude that what constitutes an 'object' is entirely a matter of language/convention. There's no physical basis for it. I can talk about the blue gutter and that, by convention, identifies an object distinct from the red gutter despite them both being parts of a greater (not separated) pipe. — noAxioms

I believe that the principal way which we distinguish objects is with the sense of sight. We see boundaries which mark the edges of things, and when a boundary encompasses an area this is seen as an individual object. The sense of touch confirms what sight shows us. I consider this to be the "empirical basis" of what constitutes "an object".

Notice though, that "physical" (meaning of the body) is derived from the assumption of objects, as a body is an object. So it is actually impossible to separate "physical" from "object" in the way you seem to suggest, as this is an essential relation, physical is necessarily, by definition, of the object. What is actually the case then, is that there is an objective (of the object) basis of "the physical", and the inverse cannot be the case. "Physical" is based in the object, and "object" is not based in the physical.